stationary in distributions of numerical solutions for stochastic partial differential ... ·...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 752953 12 pageshttpdxdoiorg1011552013752953

Research ArticleStationary in Distributions of Numerical Solutions for StochasticPartial Differential Equations with Markovian Switching

Yi Shen1 and Yan Li12

1 Department of Control Science and Engineering Huazhong University of Science and Technology Wuhan 430074 China2 College of Science Huazhong Agriculture University Wuhan 430079 China

Correspondence should be addressed to Yi Shen yeeshen0912gmailcom

Received 30 December 2012 Accepted 24 February 2013

Academic Editor Qi Luo

Copyright copy 2013 Y Shen and Y Li This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate a class of stochastic partial differential equations with Markovian switching By using the Euler-Maruyama schemeboth in time and in space of mild solutions we derive sufficient conditions for the existence and uniqueness of the stationarydistributions of numerical solutions Finally one example is given to illustrate the theory

1 Introduction

The theory of numerical solutions of stochastic partialdifferential equations (SPDEs) has been well developed bymany authors [1ndash5] In [2] Debussche considered the errorof the Euler scheme for the nonlinear stochastic partialdifferential equations by using Malliavin calculus Gyongyand Millet [3] discussed the convergence rate of space timeapproximations for stochastic evolution equations Shardlow[5] investigated the numerical methods of the mild solutionsfor stochastic parabolic PDEs derived by space-time whitenoise by applying finite difference approach

On the other hand the parameters of SPDEs mayexperience abrupt changes caused by phenomena such ascomponent failures or repairs changing subsystem intercon-nections and abrupt environmental disturbances [6ndash9] andthe continuous-timeMarkov chains have been used to modelthese parameter jumps An important equation is a class ofSPDEs with Markovian switching

119889119883 (119905) = [119860119883 (119905) + 119891 (119883 (119905) 119903 (119905))] 119889119905

+ 119892 (119883 (119905) 119903 (119905)) 119889119882 (119905) 119905 ge 0

(1)

Here the state vector has two components 119883(119905) and 119903(119905) thefirst one is normally referred to as the state while the secondone is regarded as the mode In its operation the system willswitch from one mode to another one in a random way and

the switching among the modes is governed by the Markovchain 119903(119905)

Since only a few SPDEs with Markovian switchinghave explicit formulae numerical (approximate) schemes ofSPDEs with Markovian switching are becoming more andmore popular In this paper we will study the stationarydistribution of numerical solutions of SPDEs withMarkovianswitching Bao et al [10] investigated the stability in distribu-tion of mild solutions to SPDEs Bao and Yuan [11] discussedthe numerical approximation of stationary distribution forSPDEs For the stationary distribution of numerical solu-tions of stochastic differential equations in finite-dimensionalspace Mao et al [12] utilized the Euler-Maruyama schemewith variable step size to obtain the stationary distributionand they also proved that the probability measures inducedby the numerical solutions converge weakly to the stationarydistribution of the true solution But since the mild solutionsof SPDEs with Markovian switching do not have stochasticdifferential a significant consequence of this fact is that theIto formula cannot be used for mild solutions of SPDEs withMarkovian switching directly Consequently we generalizethe stationary distribution of numerical solutions of the finitedimensional stochastic differential equationswithMarkovianswitching to that of infinite dimensional cases

Motived by [11ndash13] we will show in this paper that themild solutions of SPDE with Markovian switching (1) have aunique stationary distribution for sufficiently small step size

2 Abstract and Applied Analysis

So this paper is organised as follows in Section 2 we givenecessary notations and define Euler-Maruyama scheme ofmild solutions In Section 3 we give some lemmas and themain result in this paper Finally we will give an example toillustrate the theory in Section 4

2 Statements of Problem

Throughout this paper unless otherwise specified we let(ΩF F

119905119905ge0P) be complete probability space with a

filtration F119905119905ge0

satisfying the usual conditions (ie it isincreasing and right continuous while F

0contains all P-

null sets) Let (119867 ⟨sdot sdot⟩119867 sdot

119867) be a real separable Hilbert

space and 119882(119905) an 119867-valued cylindrical Brownian motion(Wiener process) defined on the probability space Let 119868

119866be

the indicator function of a set 119866 Denote by (L(119867) sdot )and (LHS(119867) sdot HS) the family of bounded linear operatorsand Hilbert-Schmidt operator from 119867 into 119867 respectivelyLet 119903(119905) 119905 ge 0 be a right-continuous Markov chain on theprobability space taking values in a finite state space S =

1 2 119873 with the generator Γ = (120574119894119895)119873times119873

given by

P 119903 (119905 + 120575) = 119895 | 119903 (119905) = 119894 = 120574119894119895120575 + 119900 (120575) if 119894 = 1198951 + 120574119894119895120575 + 119900 (120575) if 119894 = 119895

(2)

where 120575 gt 0 Here 120574119894119895gt 0 is the transition rate from 119894 to 119895 if

119894 = 119895 while

120574119894119894= minussum

119895 = 119894

120574119894119895 (3)

We assume that the Markov chain 119903(sdot) is independent of theBrownian motion 119882(sdot) It is well known that almost everysample path of 119903(sdot) is a right-continuous step function withfinite number of simple jumps in any finite subinterval ofR+= [0 +infin)Consider SPDEs with Markovian switching on119867

119889119883 (119905) = [119860119883 (119905) + 119891 (119883 (119905) 119903 (119905))] 119889119905

+ 119892 (119883 (119905) 119903 (119905)) 119889119882 (119905) 119905 ge 0

(4)

with initial value 119883(0) = 119909 isin 119867 and 119903(0) = 119894 isin S Here119891 119867 times S rarr 119867 119892 119867 times S rarr LHS(119867) Throughout thepaper we impose the following assumptions

(A1) (119860D(119860)) is a self-adjoint operator on 119867 generatinga 1198620-semigroup 119890119860119905

119905ge0 such that 119890119860119905 le 119890minus120572119905 for

some120572 gt 0 In this caseminus119860 has discrete spectrum 0 lt1205881le 1205882le sdot sdot sdot le lim

119894rarrinfin120588119894= infin with corresponding

eigenbasis 119890119894119894ge1

of119867(A2) Both 119891 and 119892 are globally Lipschitz continuous That

is there exists a constant 119871 gt 0 such that

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

119867or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le 1198711003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 forall119909 119910 isin 119867 119895 isin S

(5)

(A3) There exist 120583 gt 0 and 120582119895gt 0 (119895 = 1 2 119873) such

that

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS

+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172


(6)

It is well known (see [1 8]) that under (A1)ndash(A3) (4) hasa unique mild solution 119883(119905) on 119905 ge 0 That is for any 119883(0) =119909 isin 119867 and 119903(0) = 119894 isin S there exists a unique 119867-valuedadapted process119883(119905) such that

119883(119905) = 119890119905119860119909 + int

119905

0

119890(119905minus119904)119860119891 (119883 (119904) 119903 (119904)) 119889119904

+ int

119905

0

119890(119905minus119904)119860119892 (119883 (119904) 119903 (119904)) 119889119882 (119904)

(7)

Moreover the pair119885(119905) = (119883(119905) 119903(119905)) is a time-homogeneousMarkov process

Remark 1 We observe that (A2) implies the following lineargrowth conditions

1003816100381610038161003816119891 (119909 119895)10038161003816100381610038162

119867or1003817100381710038171003817119892 (119909 119895)

10038171003817100381710038172

HS le 119871 (1 + 1199092

119867) forall119909 isin 119867 119895 isin S

(8)

where 119871 = 2max119895isinS(119871 or |119891(0 119895)|

2

119867or 119892(0 119895)

2

HS)

Remark 2 We also establish another property from (A3)

2120582119895⟨119909 119891 (119909 119895)⟩

119867+ 120582119895

1003817100381710038171003817119892 (119909 119895)10038171003817100381710038172

HS +119873

sum

119897=1

1205741198951198971205821198971199092

119867

le 2120582119895⟨119909 119891 (119909 119895) minus 119891 (0 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (0 119895)10038171003817100381710038172

HS +119873

sum

119897=1

1205741198951198971205821198971199092

119867

+ 2120582119895⟨119909 119891 (0 119895)⟩

119867

+ 2120582119895⟨119892 (119909 119895) minus 119892 (0 119895) 119892 (0 119895)⟩HS + 120582119895

1003817100381710038171003817119892 (0 119895)10038171003817100381710038172

HS

le minus1205831199092

119867+120583

41199092

119867+41205822

119895

1003817100381710038171003817119891 (0 119895)1003817100381710038171003817119867

120583

+120583

4119871

1003817100381710038171003817119892 (119909 119895) minus 119892 (0 119895)10038171003817100381710038172

HS

+41198711205822

119895

120583

1003817100381710038171003817119892 (0 119895)10038171003817100381710038172

HS + 1205821198951003817100381710038171003817119892 (0 119895)

10038171003817100381710038172

HS

le minus1205831199092

119867+120583

41199092

119867+120583

41199092

119867+41205822

119895

1003817100381710038171003817119891 (0 119895)1003817100381710038171003817119867

120583

Abstract and Applied Analysis 3

+41198711205822

119895

120583

1003817100381710038171003817119892 (0 119895)10038171003817100381710038172

HS + 1205821198951003817100381710038171003817119892 (0 119895)

10038171003817100381710038172

HS

le minus120583

21199092

119867+ 1205721 forall119909 isin 119867 119895 isin S

(9)

where1205721= max

119895isinS[(41205822

119895 119891(0 119895)

2

119867120583) +(4119871120582

2

119895120583) 119892(0 119895)

2

HS + 120582119895 119892(0 119895)2

HS] and ⟨119879 119878⟩HS = suminfin

119894=1⟨119879119890119894 119878119890119895⟩119867for

119878 119879 isinLHS(119867)

Denote by119885119909119894(119905) = (119883119909119894(119905) 119903119894(119905)) themild solution of (4)starting from (119909 119894) isin 119867 times S For any subset 119860 isin B(119867) 119861 subS let P

119905((119909 119894) 119860 times 119861) be the probability measure induced by

119885119909119894(119905) 119905 ge 0 Namely

P119905((119909 119894) 119860 times 119861) = P (119885

119909119894isin 119860 times 119861) (10)

whereB(119867) is the family of the Borel subset of119867Denote byP(119867timesS) the family by all probabilitymeasures

on 119867 times S For 1198751 1198752isin P(119867 times S) define the metric 119889L as

follows

119889L (1198751 1198752)

= sup120593isinL

10038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119895=1

int119867

120593 (119906 119895) 1198751(119889119906 119895) minus

119873

sum

119895=1

int120593 (119906 119895) 1198752(119889119906 119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11)

where L = 120593 119867 times S rarr R |120593(119906 119895) minus 120593(V 119897)| le 119906 minus V119867+

|119895 minus 119897| and |120593(119906 119895)| le 1 for 119906 V isin 119870 119895 119897 isin S

Remark 3 It is known that the weak convergence of probabil-ity measures is a metric concept with respect to classes of testfunction In other words a sequence of probability measures119875119896119896ge1

of P(119867 times S) converges weakly to a probabilitymeasure 119875

0isin P(119867timesS) if and only if lim

119896rarrinfin119889L(119875119896 1198750) = 0

Definition 4 The mild solution 119885(119905) = (119883(119905) 119903(119905)) of (4) issaid to have a stationary distribution 120587(sdot times sdot) isin P(119867 times S) ifthe probability measure P

119905((119909 119894) (sdot times sdot)) converges weakly to

120587(sdottimessdot) as 119905 rarr infin for every 119894 isin S and every 119909 isin 119880 a boundedsubset of119867 that is

lim119905rarrinfin

119889L (P119905 (119909 119894) 120587 (sdot times sdot))

= lim119905rarrinfin

(sup120593isinL

10038161003816100381610038161003816100381610038161003816100381610038161003816

E120593 (119885119909119894(119905))

minus

119873

sum

119895=1

int119867

120593 (119906 119895) 120587 (119889119906 119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

) = 0

(12)

By Theorem 31 in [10] and Theorem 31 in [14] we havethe following

Theorem 5 Under (A1)ndash(A3) the Markov process 119885(119905) has aunique stationary distribution 120587(sdot times sdot) isin P(119867 times S)

For any 119899 ge 1 let 120587119899 119867 rarr 119867

119899= Span119890

1 1198902 119890

119899

be the orthogonal projection Consider SPDEs with Marko-vian switching on119867

119899

119889119883119899(119905) = [119860

119899119883119899(119905) + 119891

119899(119883119899(119905) 119903 (119905))] 119889119905

+ 119892119899(119883119899(119905) 119903 (119905)) 119889119882 (119905)

(13)

with initial data 119883119899(0) = 120587119899119909 = sum

119899

119894=1⟨119909 119890119894⟩119867119890119894 119909 isin 119867 Here

119860119899= 120587119899119860 119891119899= 120587119899119891 119892119899= 120587119899119892

Therefore we can observe that

119860119899119909 = 119860119909 119890

119905119860119899119909= 119890119905119860119909 ⟨119909 119891

119899⟩119867= ⟨119909 119891⟩

119867

⟨119909 119892119899⟩119867= ⟨119909 119892⟩

119867 forall119909 isin 119867

119899

(14)

By the property of the projection operator and (A2) we have

1003817100381710038171003817119860119899 (119909 minus 119910)10038171003817100381710038172

119867or1003817100381710038171003817119891119899 (119909 119895) minus 119891119899 (119910 119895)

10038171003817100381710038172

119867

or1003817100381710038171003817119892119899 (119909 119895) minus 119892119899 (119910 119895)

10038171003817100381710038172

HS

le 1205822

119899

1003817100381710038171003817(119909 minus 119910)10038171003817100381710038172

119867or1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)

10038171003817100381710038172

119867

or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS le (1205822

119899or 119871)1003817100381710038171003817(119909 minus 119910)

10038171003817100381710038172

119867

forall119909 119910 isin 119867119899 119895 isin S

(15)

Hence (13) admits a unique strong solution 119883119899(119905)119905ge0

on119867119899

(see [8])We now introduce an Euler-Maruyama based computa-

tionalmethodThemethodmakes use of the following lemma(see [15])

Lemma 6 Given Δ gt 0 then 119903(119896Δ) 119896 = 0 1 2 is adiscrete Markov chain with the one-step transition probabilitymatrix

119875 (Δ) = (119875119894119895(Δ))119873times119873= 119890ΔΓ (16)

Given a fixed step size Δ gt 0 and the one-step transitionprobability matrix 119875(Δ) in (16) the discrete Markov chain119903(119896Δ) 119896 = 0 1 2 can be simulated as follows let119903(0) = 119894

0 and compute a pseudorandom number 120585

1from the

uniform (0 1) distributionDefine

119903 (Δ)

=

119894 119894 isin S minus 119873

such that119894minus1

sum119895=1

119875119903(0)119895(Δ)

le 1205851

lt

119894

sum119895=1

119875119903(0)119895(Δ)

119873

119873minus1

sum119895=1

119875119903(0)119895(Δ) le 120585

1

(17)


where we set sum0119895=1119875119903(0)119895(Δ) = 0 as usual Having computed

119903(0) 119903(Δ) 119903(119896Δ) we can compute 119903((119896 + 1)Δ) by drawinga uniform (0 1) pseudorandom number 120585

119896+1and setting

119903 ((119896 + 1) Δ)

=

119894 119894 isin S minus 119873


sum119895=1

119875119903(119896Δ)119895

(Δ)

le 120585119896+1lt

119894

sum119895=1

119875119903(119896Δ)119895

(Δ)

119873

119873minus1

sum119895=1

119875119903(119896Δ)119895

(Δ) le 120585119896+1

(18)

The procedure can be carried out repeatedly to obtain moretrajectories

We now define the Euler-Maruyama approximation for(13) For a stepsize Δ isin (0 1) the discrete approximation119884119899

(119896Δ) asymp 119883119899(119896Δ) is formed by simulating from 119884119899(0) =

120587119899119909 119903(0) = 119903

0 and

119884119899

((119896 + 1) Δ)

= 119890Δ119860119899 119884119899

(119896Δ) + 119891119899(119884119899

(119896Δ) 119903 (119896Δ)) Δ

+119892119899(119884119899

(119896Δ) 119903 (119896Δ)) Δ119882119896

(19)

where Δ119882119896= 119882((119896 + 1)Δ) minus 119882(119896Δ))

To carry out our analysis conveniently we give thecontinuous Euler-Maruyama approximation solution whichis defined by

119884119899(119905) = 119890

119905119860119899120587119899119909 + int

119905

0

119890(119905minuslfloor119904rfloor)119860

119899119891119899(119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor)) 119889119904

+ int

119905

0


119899119892119899(119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor)) 119889119882 (119904)

= 119890119905119860119899120587119899119909 + int

119905

0



+ int

119905

0



(20)

where lfloor119905rfloor = [119905Δ]Δ and [119905Δ] denotes the integer part of 119905Δand 119884119899(0) = 119884119899(0) = 120587

119899119909 and 119884119899(119896Δ) = 119884119899(119896Δ)

It is obvious that119884119899(119905) coincides with the discrete approx-imation solution at the gridpoints For any Borel set 119860 isinB(119867119899) 119909 isin 119867

119899 119894 119895 isin S let 119885

119899

(119896Δ) = (119884119899

(119896Δ) 119903(119896Δ))

P119899Δ((119909 119894) 119860 times 119895)

= P (119885119899

(Δ) isin 119860 times 119895 | 119885119899

(0) = (119909 119894))

P119899Δ

119896((119909 119894) 119860 times 119895)

= P (119885119899

(119896Δ) isin 119860 times 119895 | 119885119899

(0) = (119909 119894))

(21)

Following the argument of Theorem 5 in [13] we have thefollowing

Lemma 7 119885 119899(119896Δ)119896ge0

is a homogeneous Markov processwith the transition probability kernel P119899Δ((119909 119894) 119860 times 119895)

To highlight the initial value we will use notation119885119899(119909119894)

(119896Δ)

Definition 8 For a given stepsize Δ gt 0 119885119899(119909119894)

(119896Δ)119896ge0

issaid to have a stationary distribution 120587119899Δ(sdot times sdot) isin P(119867

119899times

S) if the 119896-step transition probability kernel P119899Δ119896((119909 119894) sdot times sdot)

converges weakly to 120587119899Δ(sdot times sdot) as 119896 rarr infin for every (119909 119894) isin119867119899times S that is

lim119896rarrinfin

119889L (119875119899Δ

119896((119909 119894) sdot times sdot) 120587

119899Δ(sdot times sdot)) = 0 (22)

We will establish our result of this paper in Section 3

Theorem 9 Under (A1)ndash(A3) for a given stepsize Δ gt 0and arbitrary 119909 isin 119867

119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0

has a uniquestationary distribution 120587119899Δ(sdot times sdot) isin P(119867

119899times S)

3 Stationary in Distribution ofNumerical Solutions

In this section we shall present some useful lemmas andproveTheorem 9 In what follows119862 gt 0 is a generic constantwhose values may change from line to line

For any initial value (119909 119894) let 119884119899119909119894(119905) be the continuousEuler-Maruyama solution of (20) and starting from (119909 119894) isin119867 times S Let 119883119909119894(119905) be the mild solution of (4) and startingfrom (119909 119894) isin 119867 times S

Lemma 10 Under (A1)ndash(A3) then

E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119909119894(lfloor119905rfloor)10038171003817100381710038171003817

2

119867

le 3 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

(23)

Proof Write 119884119899119909119894(119905) = 119884119899(119905) 119884119899119909119894(lfloor119905rfloor) = 119884119899(lfloor119905rfloor) From(20) we have

119884119899(lfloor119905rfloor) = 119890

lfloor119905rfloor119860120587119899119909 + int

lfloor119905rfloor

0

119890(lfloor119905rfloorminuslfloor119904rfloor)119860


+ int

lfloor119905rfloor

0



(24)


Thus

119884119899(119905) minus 119884

119899(lfloor119905rfloor)

= 119890(119905minuslfloor119905rfloor)119860

times (119890lfloor119905rfloor119860120587119899119909

+ int

lfloor119905rfloor

0


times 119891119899(119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor)) 119889119904

+ int

lfloor119905rfloor

0


times 119892119899(119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor)) 119889119882 (119904) )

minus 119884119899(lfloor119905rfloor)

+ int

119905

lfloor119905rfloor



+ int

119905

lfloor119905rfloor



= (119890(119905minuslfloor119905rfloor)119860

minus 1) 119884119899 (lfloor119905rfloor)

+ int

119905

lfloor119905rfloor



+ int

119905

lfloor119905rfloor



(25)

Then by theHolder inequality and the Ito isometrywe obtain

E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 3E10038171003817100381710038171003817(119890(119905minuslfloor119905rfloor)119860

minus 1) 119884119899 (lfloor119905rfloor)100381710038171003817100381710038172

119867

+ Eint119905

lfloor119905rfloor

1003817100381710038171003817119891119899 (119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor))

10038171003817100381710038172

119867119889119904

+Eint119905

lfloor119905rfloor


10038171003817100381710038172

HS119889119904

(26)

From (A1) we have

E10038171003817100381710038171003817(119890(119905minuslfloor119905rfloor)119860


119867

= E

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119894=1

(119890minus120588119894(119905minuslfloor119905rfloor)

minus 1) ⟨119884119899(lfloor119905rfloor) 119890

119894⟩119867119890119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867

le (1 minus 119890minus120588119899(119905minuslfloor119905rfloor)

)2

E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 1205882

119899Δ2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

(27)

here we use the fundamental inequality 1 minus 119890minus119886 le 119886 119886 gt 0And by (8) it follows that

Eint119905

lfloor119905rfloor


10038171003817100381710038172

119867119889119904

+ Eint119905

lfloor119905rfloor


10038171003817100381710038172

HS119889119904

le 2119871Δ (1 + E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

(28)

Substituting (27) and (28) into (26) the desired assertion (23)follows

Lemma 11 Under (A1)ndash(A3) if Δ lt min1 13(1205882119899+ 2119871)

((4120572119901 + 120583)(8119902 + 41199021205882

119899119871+4119902119871 + 24119902 + 6119902119871(120588

2

119899+2119871)))

2

thenthere is a constant119862 gt 0 that depends on the initial value 119909 butis independent ofΔ such that the continuous Euler-Maruyamasolution of (20) has

sup119905ge0

E10038171003817100381710038171003817119884119899119909119894(119905)10038171003817100381710038171003817119867le 119862 (29)

where 119902 = max1le119894le119873

120582119894 119901 = min

1le119894le119873120582119894

Proof Write 119884119899119909119894(119905) = 119884119899(119905) 119903119894(119896Δ) = 119903(119896Δ) From (20) wehave the following differential form

119889119884119899(119905)

= 119860119884119899(119905) + 119890

(119905minuslfloor119905rfloor)119860119891119899(119884119899(lfloor119905rfloor) 119903 (lfloor119905rfloor)) 119889119905

+ 119890(119905minuslfloor119905rfloor)119860


(30)

with 119884119899(0) = 120587119899119909

Let119881(119909 119894) = 120582119894 1199092

119867 By the generalised Ito formula for

any 120579 gt 0 we derive from (30) that

119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1205821198941199092

119867+ Eint

119905

0

119890120579119904120582119903(119904)

times 1205791003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ 2⟨119884

119899(119904) 119860119884

119899(119904)⟩119867

+ 2⟨119884119899(119904) 119890(119904minuslfloor119904rfloor)119860

119891119899(119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor))⟩

119867

+1003817100381710038171003817119892119899 (119884

119899(lfloor119904rfloor) 119903 (lfloor119904rfloor))

10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904


le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) 119890(119904minuslfloor119904rfloor)119860

119891 (119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor))⟩

119867

+1003817100381710038171003817119892 (119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor))

10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

(31)

By the fundamental transformation we obtain that

⟨119884119899(119905) 119890(119905minuslfloor119905rfloor)119860


119867

= ⟨119884119899(119905) 119891 (119884

119899(119905) 119903 (119905))⟩

119867

+ ⟨119884119899(119905) (119890

(119905minuslfloor119905rfloor)119860minus 1) 119891 (119884119899 (119905) 119903 (119905))⟩

119867

+ ⟨119884119899(119905) 119890(119905minuslfloor119905rfloor)119860

(119891 (119884119899(lfloor119905rfloor) 119903 (lfloor119905rfloor))

minus119891 (119884119899(119905) 119903 (119905))) ⟩

119867

(32)

By Hold inequality we have


10038171003817100381710038172

HS

=1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

minus (119892 (119884119899(119905) 119903 (119905)) minus 119892 (119884

119899(lfloor119905rfloor) 119903 (lfloor119905rfloor)))

10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

10038171003817100381710038172

HS + (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

(33)

Then from (31) we have

119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2⟨119884119899(119904) 119891 (119884

119899(119904) 119903 (119904))⟩

119867

+1003817100381710038171003817119892 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867

+ 2 ⟨119884119899(119904) 119890(119904minuslfloor119904rfloor)119860

times (119891 (119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor))

minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))

minus119892 (119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor)))

10038171003817100381710038172

HS 119889119904

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus120583

2Eint119905

0

119890120579119904119884 (119904)

2

119867119889119904 + 120572

1int

119905

0

119890120579119904119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

= 1198691(119905) + 119869

2(119905) + 119869

3(119905) + 119869

4(119905)

(34)

By the elemental inequality 2119886119887 le (1198862120581)+1205811198872 119886 119887 isin R 120581 gt0 and (8) (27) we obtain that for Δ lt 1

1198692(119905) le Eint

119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817

(119890(119904minuslfloor119904rfloor)119860

minus 1)

times 119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904


le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δminus121205882

119899Δ2119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ121205882

119899119871)

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

(35)

By (A2) and (8) we have

1003817100381710038171003817(119891 (119884119899(lfloor119905rfloor) 119903 (lfloor119905rfloor)) minus 119891 (119884

119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 21003817100381710038171003817(119891 (119884

119899(lfloor119905rfloor) 119903 (lfloor119905rfloor)) minus 119891 (119884

119899(lfloor119905rfloor) 119903 (119905)))

10038171003817100381710038172

119867

+ 21003817100381710038171003817(119891 (119884

119899(lfloor119905rfloor) 119903 (119905)) minus 119891 (119884

119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(36)

Similarly we have

1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))) minus 119892 (119884


10038171003817100381710038172

HS

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(37)

Thus we obtain from (36) that

1198693(119905)

le 2Eint119905

0

119890120579119904120582119903(119904)⟨119884119899(119904) 119890(119904minuslfloor119904rfloor)119860


minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867119889119904

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + Δ

minus1210038171003817100381710038171003817119890(119904minuslfloor119904rfloor)11986010038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884


minus119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δminus128119871

times (1 +1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119868119903(119904) = 119903(lfloor119904rfloor)

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(38)

By Markov property we compute

E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

]

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

| 119903 (lfloor119905rfloor)])

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) | 119903 (lfloor119905rfloor)])

times E [119868119903(119905) = 119903(lfloor119905rfloor)

| 119903 (lfloor119905rfloor)]

= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

P (119903 (119905) = 119894 | 119903 (lfloor119905rfloor) = 119894)

= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

times sum

119895 = 119894

(120574119894119895(119905 minus lfloor119905rfloor) + 119900 (119905 minus lfloor119905rfloor))

= E (1 + 119884 (lfloor119905rfloor)2

119867) (max119894isinS(minus120574119894119894) Δ + 119900 (Δ))sum

119894isinS

119868119903(lfloor119905rfloor)=119894

le ΔE (1 + 119884 (lfloor119905rfloor)2

119867)

(39)

where = 119873[1 +max1le119894le119873

(minus120574119894119894)] Substituting (39) into (38)

gives

1198693(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

+ 119902int

119905

0

8119890120579119904Δ12119871E (1 + 119884 (lfloor119904rfloor)

2

119867) 119889119904

(40)

Furthermore due to (37) and (39) we have

1198694(119905)

= Eint119905

0

119890120579119904120582119903(119904)

times Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)1003817100381710038171003817119892 (119884

119899(119904) 119903 (119904)))

minus119892 (119884119899(lfloor119904rfloor) 119903 (lfloor119904rfloor))

10038171003817100381710038172

HS 119889119904


le 119902Eint119905

0

119890120579119904Δ12119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904 + 119902 (1 + Δ

minus12)

times int

119905

0

1198901205791199048119871 (1 +

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119868119903(119904) = 119903(lfloor119904rfloor)

119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

le 119902Δ12119871Eint119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 16119902Δ12119871int

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(41)

On the other hand by Lemma 10 when 3(1205882119899+ 2119871)Δ le 1 we

have

E1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867

le 2E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 6 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 4E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867+ 2

(42)

Putting (35) (40) and (41) into (34) we have

119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 1199021205882

119899Δ12119871

+24119902Δ12 119871 + 119902Δ

12119871] 119889119904

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus

120583

2+ 119902Δ12(2 + 120588

2

119899119871) + 119902Δ

12119871]

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + 24119902Δ

12 119871E

times int

119905

0

1198901205791199041003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ (4119902Δminus12119871 + 2119902119871)Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(43)

By Lemma 10 and the inequality (42) we obtain that

119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583

+ 31199021205882

119899Δ12119871 + 24119902Δ

12 119871

+3119902Δ12119871 + 4119902Δ

12] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+4119902Δ12119871 + 24119902Δ

12 119871]1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ 6119902119871Δ12(1205882

119899+ 2119871)Eint

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119889119904

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583 + 3119902120588

2

119899Δ12119871

+ 24119902Δ12 119871 + 3119902Δ

12119871 + 4119902Δ

12

+6119902119871Δ12(1205882

119899+ 2119871)] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+ 4119902Δ12119871 + 24119902Δ

12 119871

+6119902119871Δ12(1205882

119899+ 2119871)]

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(44)

Let 120579 = (4120572119901+120583)4119902 for Δ lt ((4120572119901+120583)(8119902+41199021205882119899119871+4119902119871+

24119902 + 6119902119871(1205882

119899+ 2119871)))

2 then

119901119890120579119905E (119884 (119905)

2

119867) le 119902119909

2

119867+ int

119905

0

119890120579119904[1205721+4120572119901 + 120583

2] 119889119904

(45)

That is

sup119905ge0

E (119884 (119905)2

119867) le 119862 (46)

Lemma 12 Let (A1)ndash(A3) hold If Δ lt min1 118(1205882119899+

2119871) ((2120572119901 + 120583)(4119902 + 2119902119871 + 21199021205882

119899119871 + 12119902119871))

2 then

lim119905rarrinfin

E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119910119894(119905)10038171003817100381710038171003817

2

119867= 0 119906119899119894119891119900119903119898119897119910 119891119900119903 119909 119910 isin 119880

(47)

where 119880 is a bounded subset of119867119899


Proof Write 119884119899119909119894(119905) = 119884119909(119905) 119884119899119910119894(119905) = 119884119910(119905) 119903119894(119896Δ) =119903(119896Δ) From (20) it is easy to show that

(119884119909(119905) minus 119884

119910(119905)) minus (119884

119909(lfloor119905rfloor) minus 119884

119910(lfloor119905rfloor))

= (119884119909(119905) minus 119884

119909(lfloor119905rfloor)) minus (119884

119910(119905) minus 119884



minus 1) (119884119909 (lfloor119905rfloor) minus 119884119910 (lfloor119905rfloor))

+ int

119905

lfloor119905rfloor


(119891119899(119884119909(lfloor119904rfloor) 119903 (lfloor119904rfloor))

minus119891119899(119884119910(lfloor119904rfloor) 119903 (lfloor119904rfloor))) 119889119904

+ int

119905

lfloor119905rfloor



minus119892119899(119884119910(lfloor119904rfloor) 119903 (lfloor119904rfloor))) 119889119882 (119904)

(48)

By using the argument of Lemma 10 we derive that if Δ lt 1

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884



10038171003817100381710038172

119867

le 3 (1205882

119899+ 2119871)ΔE


119910(lfloor119905rfloor)10038171003817100381710038172

119867

(49)

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172

119867

= E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884



+ (119884119909(lfloor119905rfloor) minus 119884


10038171003817100381710038172

119867

le (1 + 2)E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))

minus (119884119909(lfloor119905rfloor) minus 119884


10038171003817100381710038172

119867

+ (1 +1

2)E1003817100381710038171003817119884119909(lfloor119905rfloor) minus 119884

119910(lfloor119905rfloor)10038171003817100381710038172

119867

le 9 (1205882

119899+ 2119871)ΔE

1003817100381710038171003817(119884119909(lfloor119905rfloor) minus 119884


10038171003817100381710038172

119867

+ 15E1003817100381710038171003817(119884119909(lfloor119905rfloor) minus 119884


10038171003817100381710038172

119867

(50)

If Δ lt 118(1205882119899+ 2119871) then

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172

119867le 2E1003817100381710038171003817(119884119909(lfloor119905rfloor) minus 119884


10038171003817100381710038172

119867 (51)

Using (30) and the generalised Ito formula for any 120579 gt 0 wehave

119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 120582119894

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

+ Eint119905

0

119890120579119904120582119903(119904)1205791003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ 2 ⟨119884119909(119904) minus 119884

119910(119904)

119860119884119909(119904) minus 119884

119910(119904)⟩119867

+ 2 ⟨119884119909(119904) minus 119884

119910(119904) 119890(119904minuslfloor119904rfloor)119860

times (119891119899(119884119909(lfloor119904rfloor) 119903 (lfloor119904rfloor))

minus119891119899(119884119910(lfloor119904rfloor) 119903 (lfloor119904rfloor)))⟩

119867

+1003817100381710038171003817119892119899 (119884


minus119892119899(119884119910(lfloor119904rfloor) 119903 (lfloor119904rfloor))

10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ 119902120579Eint

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2 ⟨119884

119909(119904) minus 119884



minus119891 (119884119910(lfloor119904rfloor) 119903 (lfloor119904rfloor))) ⟩

119867



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(52)


⟨119884119909(119905) minus 119884


times (119891 (119884119909(lfloor119905rfloor) 119903 (lfloor119905rfloor)) minus 119891 (119884

119910(lfloor119905rfloor) 119903 (lfloor119905rfloor))) ⟩

119867

= ⟨119884119909(119905) minus 119884

119910(119905) 119891 (119884

119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))⟩

119867

+ ⟨119884119909(119905) minus 119884

119910(119905) (119890

(119905minuslfloor119905rfloor)119860minus 1)

times (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

+ ⟨119884119909(119905) minus 119884




minus (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

(53)

By the Hold inequality we have

1003817100381710038171003817119892 (119884119909(lfloor119905rfloor) 119903 (lfloor119905rfloor)) minus 119892 (119884


10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905))

10038171003817100381710038172

HS


+ (1 + Δminus12)1003817100381710038171003817(119892 (119884

119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905)))

minus (119892 (119884119909(lfloor119905rfloor) 119903 (lfloor119905rfloor))


10038171003817100381710038172

HS

(54)

Then from (52) and (A3) we have

119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ (119902120579 minus 2120572119901 minus 120583)E

times int

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884

119910(119904) (119890


times (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904))) ⟩

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884




minus (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904)))⟩

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δ12 1003817100381710038171003817119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904)))


minus 119892 (119884119910(lfloor119904rfloor)

119903 (lfloor119904rfloor) ) )10038171003817100381710038172

HS 119889119904

= 1198661(119905) + 119866

2(119905) + 119866

3(119905) + 119866

4(119905)

(55)

By (A2) and (27) we have for Δ lt 1

1198662(119905)

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ321205882

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(56)

It is easy to show that

1198663(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus1210038171003817100381710038171003817(119890(119904minuslfloor119904rfloor)119860

)10038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884


minus 119891 (119884119910(lfloor119904rfloor) 119903 (lfloor119904rfloor))

minus (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904 + 119866

3(119905)

(57)

By (39) we have

1198663(119905)

le 2119902Δminus12

Eint119905

0

119890120579119904[1003817100381710038171003817119891 (119884



10038171003817100381710038172

119867

+1003817100381710038171003817(119891 (119884

119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867]

times 119868119903(119904) = 119903(lfloor119904rfloor)

119889119904

le 4119902Δminus12119871Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867


119889119904

le 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(58)

Therefore we obtain that

1198663(119905) le 119902Δ

12Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(59)


On the other hand using the similar argument of (58) wehave

1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ

times1003817100381710038171003817119884119909(lfloor119904rfloor) minus 119884

119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)

By (50) we obtain that

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+

12119902119871))2 then the desired assertion (47) follows

We can now easily prove our main result

Proof of Theorem 9 Since 119867119899

is finite-dimensional byLemma 31 in [12] we have

lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)

uniformly in 119909 119910 isin 119867119899 119894 119895 isin S

By Lemma 7 there exists 120587119899Δ(sdot times sdot) isin P(119867119899times S) such

that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


By the triangle inequality (63) and (64) we have

lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587

119899Δ(sdot times sdot))

le lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((0 1) sdot times sdot))

+ lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587

119899Δ(sdot times sdot)) = 0

(65)

4 Corollary and Example

In this section we give a criterion based119872-matrices whichcan be verified easily in applications(A4) For each 119895 isin S there exists a pair of constants 120573

119895and

120575119895such that for 119909 119910 isin 119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)

MoreoverA = minus diag(21205731+1205751 2120573

119873+120575119873) minus Γ is

a nonsingular119872-matrix [8]

Corollary 13 Under (A1) (A2) and (A4) for a given stepsizeΔ gt 0 and arbitrary 119909 isin 119867

119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0

has aunique stationary distribution 120587119899Δ(sdot times sdot) isin P(119867

119899times S)

Proof In fact we only need to prove that (A3) holdsBy (A4) there exists (120582

1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)

In the following we give an example to illustrate theCorollary 13

Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)


We take 119867 = 1198712(0 120587) and 119860 = 12059721205971205852 with domainD(119860) = 1198672(0 120587) cap1198671

0(0 120587) then 119860 is a self-adjoint negative

operator For the eigenbasis 119890119896(120585) = (2120587)

12 sin(119896120585) 120585 isin[0 120587] 119860119890

119896= minus1198962119890119896 119896 isin N It is easy to show that

1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)

This further gives that1003817100381710038171003817100381711989011990511986010038171003817100381710038171003817le 119890minus119905 (70)

where 120572 = 1 thus (A1) holdsLet 119882(119905) be a scalar Brownian motion let 119903(119905) be a

continuous-time Markov chain values in S = 1 2 with thegenerator

Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286

Moreover 119892 satisfies

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)

It is easy to compute

1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)

So the matrixA becomes

A = diag (00838 00015) minus Γ = (20838 minus2

minus1 10015) (75)

It is easy to see that A is a nonsingular119872-matrix Thus(A4) holds By Corollary 13 we can conclude that (68) has aunique stationary distribution 120587119899Δ(sdot times sdot)

Acknowledgment

This work is partially supported by the National NaturalScience Foundation of China under Grants 61134012 and11271146

References

[1] G Da Prato and J Zabczyk Stochastic Equations in Infinite-Dimensions CambridgeUniversity Press CambridgeUK 1992

[2] A Debussche ldquoWeak approximation of stochastic partial differ-ential equations the nonlinear caserdquoMathematics of Computa-tion vol 80 no 273 pp 89ndash117 2011

[3] I Gyongy and A Millet ldquoRate of convergence of space timeapproximations for stochastic evolution equationsrdquo PotentialAnalysis vol 30 no 1 pp 29ndash64 2009

[4] A Jentzen and P E Kloeden Taylor Approximations forStochastic Partial Differential Equations CBMS-NSF RegionalConference Series in Applied Mathematics 2011

[5] T Shardlow ldquoNumerical methods for stochastic parabolicPDEsrdquoNumerical Functional Analysis andOptimization vol 20no 1-2 pp 121ndash145 1999

[6] M Athans ldquoCommand and control (C2) theory a challenge tocontrol sciencerdquo IEEE Transactions on Automatic Control vol32 no 4 pp 286ndash293 1987

[7] D J Higham X Mao and C Yuan ldquoPreserving exponentialmean-square stability in the simulation of hybrid stochasticdifferential equationsrdquo Numerische Mathematik vol 108 no 2pp 295ndash325 2007

[8] X Mao and C Yuan Stochastic Differential Equations withMarkovian Switching Imperial College London UK 2006

[9] M Mariton Jump Linear Systems in Automatic Control MarcelDekker 1990

[10] J Bao Z Hou and C Yuan ldquoStability in distribution of mildsolutions to stochastic partial differential equationsrdquo Proceed-ings of the American Mathematical Society vol 138 no 6 pp2169ndash2180 2010

[11] J Bao and C Yuan ldquoNumerical approximation of stationarydistribution for SPDEsrdquo httparxivorgabs13031600

[12] X Mao C Yuan and G Yin ldquoNumerical method for stationarydistribution of stochastic differential equations withMarkovianswitchingrdquo Journal of Computational and Applied Mathematicsvol 174 no 1 pp 1ndash27 2005

[13] C Yuan and X Mao ldquoStationary distributions of Euler-Maruyama-type stochastic difference equations with Marko-vian switching and their convergencerdquo Journal of DifferenceEquations and Applications vol 11 no 1 pp 29ndash48 2005

[14] C Yuan and X Mao ldquoAsymptotic stability in distribution ofstochastic differential equations with Markovian switchingrdquoStochastic Processes and their Applications vol 103 no 2 pp277ndash291 2003

[15] W J Anderson Continuous-Time Markov Chains SpringerNew York NY USA 1991

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So this paper is organised as follows in Section 2 we givenecessary notations and define Euler-Maruyama scheme ofmild solutions In Section 3 we give some lemmas and themain result in this paper Finally we will give an example toillustrate the theory in Section 4

2 Statements of Problem

Throughout this paper unless otherwise specified we let(ΩF F

119905119905ge0P) be complete probability space with a

filtration F119905119905ge0

satisfying the usual conditions (ie it isincreasing and right continuous while F

0contains all P-

null sets) Let (119867 ⟨sdot sdot⟩119867 sdot

119867) be a real separable Hilbert

space and 119882(119905) an 119867-valued cylindrical Brownian motion(Wiener process) defined on the probability space Let 119868

119866be

the indicator function of a set 119866 Denote by (L(119867) sdot )and (LHS(119867) sdot HS) the family of bounded linear operatorsand Hilbert-Schmidt operator from 119867 into 119867 respectivelyLet 119903(119905) 119905 ge 0 be a right-continuous Markov chain on theprobability space taking values in a finite state space S =

1 2 119873 with the generator Γ = (120574119894119895)119873times119873

given by

P 119903 (119905 + 120575) = 119895 | 119903 (119905) = 119894 = 120574119894119895120575 + 119900 (120575) if 119894 = 1198951 + 120574119894119895120575 + 119900 (120575) if 119894 = 119895

(2)

where 120575 gt 0 Here 120574119894119895gt 0 is the transition rate from 119894 to 119895 if

119894 = 119895 while

120574119894119894= minussum

119895 = 119894

120574119894119895 (3)

We assume that the Markov chain 119903(sdot) is independent of theBrownian motion 119882(sdot) It is well known that almost everysample path of 119903(sdot) is a right-continuous step function withfinite number of simple jumps in any finite subinterval ofR+= [0 +infin)Consider SPDEs with Markovian switching on119867

119889119883 (119905) = [119860119883 (119905) + 119891 (119883 (119905) 119903 (119905))] 119889119905

+ 119892 (119883 (119905) 119903 (119905)) 119889119882 (119905) 119905 ge 0

(4)

with initial value 119883(0) = 119909 isin 119867 and 119903(0) = 119894 isin S Here119891 119867 times S rarr 119867 119892 119867 times S rarr LHS(119867) Throughout thepaper we impose the following assumptions

(A1) (119860D(119860)) is a self-adjoint operator on 119867 generatinga 1198620-semigroup 119890119860119905

119905ge0 such that 119890119860119905 le 119890minus120572119905 for

some120572 gt 0 In this caseminus119860 has discrete spectrum 0 lt1205881le 1205882le sdot sdot sdot le lim

119894rarrinfin120588119894= infin with corresponding

eigenbasis 119890119894119894ge1

of119867(A2) Both 119891 and 119892 are globally Lipschitz continuous That

is there exists a constant 119871 gt 0 such that

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

119867or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le 1198711003817100381710038171003817119909 minus 119910

10038171003817100381710038172


(5)

(A3) There exist 120583 gt 0 and 120582119895gt 0 (119895 = 1 2 119873) such

that

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS

+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172


(6)

It is well known (see [1 8]) that under (A1)ndash(A3) (4) hasa unique mild solution 119883(119905) on 119905 ge 0 That is for any 119883(0) =119909 isin 119867 and 119903(0) = 119894 isin S there exists a unique 119867-valuedadapted process119883(119905) such that

119883(119905) = 119890119905119860119909 + int

119905

0

119890(119905minus119904)119860119891 (119883 (119904) 119903 (119904)) 119889119904

+ int

119905

0

119890(119905minus119904)119860119892 (119883 (119904) 119903 (119904)) 119889119882 (119904)

(7)

Moreover the pair119885(119905) = (119883(119905) 119903(119905)) is a time-homogeneousMarkov process

Remark 1 We observe that (A2) implies the following lineargrowth conditions

1003816100381610038161003816119891 (119909 119895)10038161003816100381610038162

119867or1003817100381710038171003817119892 (119909 119895)

10038171003817100381710038172

HS le 119871 (1 + 1199092

119867) forall119909 isin 119867 119895 isin S

(8)

where 119871 = 2max119895isinS(119871 or |119891(0 119895)|

2

119867or 119892(0 119895)

2

HS)

Remark 2 We also establish another property from (A3)

2120582119895⟨119909 119891 (119909 119895)⟩

119867+ 120582119895

1003817100381710038171003817119892 (119909 119895)10038171003817100381710038172

HS +119873

sum

119897=1

1205741198951198971205821198971199092

119867

le 2120582119895⟨119909 119891 (119909 119895) minus 119891 (0 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (0 119895)10038171003817100381710038172

HS +119873

sum

119897=1

1205741198951198971205821198971199092

119867

+ 2120582119895⟨119909 119891 (0 119895)⟩

119867

+ 2120582119895⟨119892 (119909 119895) minus 119892 (0 119895) 119892 (0 119895)⟩HS + 120582119895

1003817100381710038171003817119892 (0 119895)10038171003817100381710038172

HS

le minus1205831199092

119867+120583

41199092

119867+41205822

119895

1003817100381710038171003817119891 (0 119895)1003817100381710038171003817119867

120583

+120583

4119871

1003817100381710038171003817119892 (119909 119895) minus 119892 (0 119895)10038171003817100381710038172

HS

+41198711205822

119895

120583

1003817100381710038171003817119892 (0 119895)10038171003817100381710038172

HS + 1205821198951003817100381710038171003817119892 (0 119895)

10038171003817100381710038172

HS

le minus1205831199092

119867+120583

41199092

119867+120583

41199092

119867+41205822

119895

1003817100381710038171003817119891 (0 119895)1003817100381710038171003817119867

120583


+41198711205822

119895

120583

1003817100381710038171003817119892 (0 119895)10038171003817100381710038172

HS + 1205821198951003817100381710038171003817119892 (0 119895)

10038171003817100381710038172

HS

le minus120583

21199092


(9)

where1205721= max

119895isinS[(41205822

119895 119891(0 119895)

2

119867120583) +(4119871120582

2

119895120583) 119892(0 119895)

2

HS + 120582119895 119892(0 119895)2


119894=1⟨119879119890119894 119878119890119895⟩119867for

119878 119879 isinLHS(119867)



119885119909119894(119905) 119905 ge 0 Namely

P119905((119909 119894) 119860 times 119861) = P (119885

119909119894isin 119860 times 119861) (10)



follows

119889L (1198751 1198752)

= sup120593isinL

10038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119895=1

int119867

120593 (119906 119895) 1198751(119889119906 119895) minus

119873

sum

119895=1

int120593 (119906 119895) 1198752(119889119906 119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11)






119896rarrinfin119889L(119875119896 1198750) = 0




lim119905rarrinfin



(sup120593isinL

10038161003816100381610038161003816100381610038161003816100381610038161003816

E120593 (119885119909119894(119905))

minus

119873

sum

119895=1

int119867

120593 (119906 119895) 120587 (119889119906 119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

) = 0

(12)




119899= Span119890

1 1198902 119890

119899


119899

119889119883119899(119905) = [119860

119899119883119899(119905) + 119891

119899(119883119899(119905) 119903 (119905))] 119889119905

+ 119892119899(119883119899(119905) 119903 (119905)) 119889119882 (119905)

(13)


119899

119894=1⟨119909 119890119894⟩119867119890119894 119909 isin 119867 Here

119860119899= 120587119899119860 119891119899= 120587119899119891 119892119899= 120587119899119892


119860119899119909 = 119860119909 119890

119905119860119899119909= 119890119905119860119909 ⟨119909 119891

119899⟩119867= ⟨119909 119891⟩

119867

⟨119909 119892119899⟩119867= ⟨119909 119892⟩

119867 forall119909 isin 119867

119899

(14)


1003817100381710038171003817119860119899 (119909 minus 119910)10038171003817100381710038172

119867or1003817100381710038171003817119891119899 (119909 119895) minus 119891119899 (119910 119895)

10038171003817100381710038172

119867

or1003817100381710038171003817119892119899 (119909 119895) minus 119892119899 (119910 119895)

10038171003817100381710038172

HS

le 1205822

119899

1003817100381710038171003817(119909 minus 119910)10038171003817100381710038172

119867or1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)

10038171003817100381710038172

119867

or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS le (1205822

119899or 119871)1003817100381710038171003817(119909 minus 119910)

10038171003817100381710038172

119867


(15)


on119867119899




119875 (Δ) = (119875119894119895(Δ))119873times119873= 119890ΔΓ (16)



1from the


119903 (Δ)

=

119894 119894 isin S minus 119873


sum119895=1

119875119903(0)119895(Δ)

le 1205851

lt

119894

sum119895=1

119875119903(0)119895(Δ)

119873

119873minus1

sum119895=1

119875119903(0)119895(Δ) le 120585

1

(17)




119896+1and setting

119903 ((119896 + 1) Δ)

=

119894 119894 isin S minus 119873


sum119895=1

119875119903(119896Δ)119895

(Δ)

le 120585119896+1lt

119894

sum119895=1

119875119903(119896Δ)119895

(Δ)

119873

119873minus1

sum119895=1

119875119903(119896Δ)119895

(Δ) le 120585119896+1

(18)




120587119899119909 119903(0) = 119903

0 and

119884119899

((119896 + 1) Δ)

= 119890Δ119860119899 119884119899

(119896Δ) + 119891119899(119884119899

(119896Δ) 119903 (119896Δ)) Δ

+119892119899(119884119899

(119896Δ) 119903 (119896Δ)) Δ119882119896

(19)

where Δ119882119896= 119882((119896 + 1)Δ) minus 119882(119896Δ))


119884119899(119905) = 119890

119905119860119899120587119899119909 + int

119905

0



+ int

119905

0



= 119890119905119860119899120587119899119909 + int

119905

0



+ int

119905

0



(20)


119899119909 and 119884119899(119896Δ) = 119884119899(119896Δ)


119899 119894 119895 isin S let 119885

119899

(119896Δ) = (119884119899

(119896Δ) 119903(119896Δ))

P119899Δ((119909 119894) 119860 times 119895)

= P (119885119899

(Δ) isin 119860 times 119895 | 119885119899

(0) = (119909 119894))

P119899Δ

119896((119909 119894) 119860 times 119895)

= P (119885119899

(119896Δ) isin 119860 times 119895 | 119885119899

(0) = (119909 119894))

(21)


Lemma 7 119885 119899(119896Δ)119896ge0



(119896Δ)


(119896Δ)119896ge0


119899times



lim119896rarrinfin

119889L (119875119899Δ

119896((119909 119894) sdot times sdot) 120587




119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)





E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119909119894(lfloor119905rfloor)10038171003817100381710038171003817

2

119867

le 3 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

(23)


119884119899(lfloor119905rfloor) = 119890


lfloor119905rfloor

0



+ int

lfloor119905rfloor

0



(24)


Thus

119884119899(119905) minus 119884




+ int

lfloor119905rfloor

0



+ int

lfloor119905rfloor

0




+ int

119905

lfloor119905rfloor



+ int

119905

lfloor119905rfloor





+ int

119905

lfloor119905rfloor



+ int

119905

lfloor119905rfloor



(25)


E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867



119867

+ Eint119905

lfloor119905rfloor


10038171003817100381710038172

119867119889119904

+Eint119905

lfloor119905rfloor


10038171003817100381710038172

HS119889119904

(26)

From (A1) we have



119867

= E

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119894=1



119894⟩119867119890119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867


)2

E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 1205882

119899Δ2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

(27)


Eint119905

lfloor119905rfloor


10038171003817100381710038172

119867119889119904

+ Eint119905

lfloor119905rfloor


10038171003817100381710038172

HS119889119904


119867)

(28)



((4120572119901 + 120583)(8119902 + 41199021205882

119899119871+4119902119871 + 24119902 + 6119902119871(120588

2

119899+2119871)))

2


sup119905ge0

E10038171003817100381710038171003817119884119899119909119894(119905)10038171003817100381710038171003817119867le 119862 (29)

where 119902 = max1le119894le119873

120582119894 119901 = min

1le119894le119873120582119894


119889119884119899(119905)

= 119860119884119899(119905) + 119890




(30)

with 119884119899(0) = 120587119899119909

Let119881(119909 119894) = 120582119894 1199092



119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1205821198941199092

119867+ Eint

119905

0

119890120579119904120582119903(119904)

times 1205791003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ 2⟨119884

119899(119904) 119860119884

119899(119904)⟩119867



119867

+1003817100381710038171003817119892119899 (119884


10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904


le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)



119867


10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

(31)




119867

= ⟨119884119899(119905) 119891 (119884

119899(119905) 119903 (119905))⟩

119867

+ ⟨119884119899(119905) (119890


119867



minus119891 (119884119899(119905) 119903 (119905))) ⟩

119867

(32)



10038171003817100381710038172

HS

=1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

minus (119892 (119884119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

(33)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2⟨119884119899(119904) 119891 (119884

119899(119904) 119903 (119904))⟩

119867

+1003817100381710038171003817119892 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus120583

2Eint119905

0

119890120579119904119884 (119904)

2

119867119889119904 + 120572

1int

119905

0

119890120579119904119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

= 1198691(119905) + 119869

2(119905) + 119869

3(119905) + 119869

4(119905)

(34)


1198692(119905) le Eint

119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times 119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904


le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δminus121205882

119899Δ2119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ121205882

119899119871)

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

(35)



119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 21003817100381710038171003817(119891 (119884


119899(lfloor119905rfloor) 119903 (119905)))

10038171003817100381710038172

119867

+ 21003817100381710038171003817(119891 (119884


119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(36)

Similarly we have

1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))) minus 119892 (119884


10038171003817100381710038172

HS

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(37)


1198693(119905)

le 2Eint119905

0



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867119889119904

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + Δ


2

times1003817100381710038171003817119891 (119884


minus119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δminus128119871


119867) 119868119903(119904) = 119903(lfloor119904rfloor)

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(38)


E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

]

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

| 119903 (lfloor119905rfloor)])

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) | 119903 (lfloor119905rfloor)])



= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

P (119903 (119905) = 119894 | 119903 (lfloor119905rfloor) = 119894)

= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

times sum

119895 = 119894


= E (1 + 119884 (lfloor119905rfloor)2


119894isinS

119868119903(lfloor119905rfloor)=119894


119867)

(39)

where = 119873[1 +max1le119894le119873


gives

1198693(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

+ 119902int

119905

0

8119890120579119904Δ12119871E (1 + 119884 (lfloor119904rfloor)

2

119867) 119889119904

(40)


1198694(119905)

= Eint119905

0

119890120579119904120582119903(119904)

times Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)1003817100381710038171003817119892 (119884

119899(119904) 119903 (119904)))


10038171003817100381710038172

HS 119889119904


le 119902Eint119905

0

119890120579119904Δ12119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904 + 119902 (1 + Δ

minus12)

times int

119905

0

1198901205791199048119871 (1 +

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119868119903(119904) = 119903(lfloor119904rfloor)

119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

le 119902Δ12119871Eint119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 16119902Δ12119871int

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(41)


have

E1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867

le 2E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 6 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867


119867+ 2

(42)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 1199021205882

119899Δ12119871

+24119902Δ12 119871 + 119902Δ

12119871] 119889119904

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus

120583

2+ 119902Δ12(2 + 120588

2

119899119871) + 119902Δ

12119871]

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + 24119902Δ

12 119871E

times int

119905

0

1198901205791199041003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ (4119902Δminus12119871 + 2119902119871)Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(43)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583

+ 31199021205882

119899Δ12119871 + 24119902Δ

12 119871

+3119902Δ12119871 + 4119902Δ

12] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+4119902Δ12119871 + 24119902Δ

12 119871]1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ 6119902119871Δ12(1205882

119899+ 2119871)Eint

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119889119904

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583 + 3119902120588

2

119899Δ12119871

+ 24119902Δ12 119871 + 3119902Δ

12119871 + 4119902Δ

12

+6119902119871Δ12(1205882

119899+ 2119871)] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+ 4119902Δ12119871 + 24119902Δ

12 119871

+6119902119871Δ12(1205882

119899+ 2119871)]

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(44)

Let 120579 = (4120572119901+120583)4119902 for Δ lt ((4120572119901+120583)(8119902+41199021205882119899119871+4119902119871+

24119902 + 6119902119871(1205882

119899+ 2119871)))

2 then

119901119890120579119905E (119884 (119905)

2

119867) le 119902119909

2

119867+ int

119905

0

119890120579119904[1205721+4120572119901 + 120583

2] 119889119904

(45)

That is

sup119905ge0

E (119884 (119905)2

119867) le 119862 (46)


2119871) ((2120572119901 + 120583)(4119902 + 2119902119871 + 21199021205882

119899119871 + 12119902119871))

2 then

lim119905rarrinfin

E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119910119894(119905)10038171003817100381710038171003817

2

119867= 0 119906119899119894119891119900119903119898119897119910 119891119900119903 119909 119910 isin 119880

(47)




(119884119909(119905) minus 119884

119910(119905)) minus (119884



= (119884119909(119905) minus 119884


119910(119905) minus 119884




+ int

119905

lfloor119905rfloor




+ int

119905

lfloor119905rfloor




(48)


E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884



10038171003817100381710038172

119867

le 3 (1205882

119899+ 2119871)ΔE


119910(lfloor119905rfloor)10038171003817100381710038172

119867

(49)

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172

119867

= E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884





10038171003817100381710038172

119867

le (1 + 2)E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))



10038171003817100381710038172

119867

+ (1 +1


119910(lfloor119905rfloor)10038171003817100381710038172

119867

le 9 (1205882

119899+ 2119871)ΔE



10038171003817100381710038172

119867



10038171003817100381710038172

119867

(50)

If Δ lt 118(1205882119899+ 2119871) then

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172



10038171003817100381710038172

119867 (51)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 120582119894

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

+ Eint119905

0

119890120579119904120582119903(119904)1205791003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ 2 ⟨119884119909(119904) minus 119884

119910(119904)

119860119884119909(119904) minus 119884

119910(119904)⟩119867

+ 2 ⟨119884119909(119904) minus 119884




119867

+1003817100381710038171003817119892119899 (119884



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ 119902120579Eint

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2 ⟨119884

119909(119904) minus 119884




119867



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(52)


⟨119884119909(119905) minus 119884




119867

= ⟨119884119909(119905) minus 119884

119910(119905) 119891 (119884

119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))⟩

119867

+ ⟨119884119909(119905) minus 119884

119910(119905) (119890


times (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

+ ⟨119884119909(119905) minus 119884




minus (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

(53)




10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905))

10038171003817100381710038172

HS


+ (1 + Δminus12)1003817100381710038171003817(119892 (119884

119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905)))



10038171003817100381710038172

HS

(54)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ (119902120579 minus 2120572119901 minus 120583)E

times int

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884

119910(119904) (119890


times (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904))) ⟩

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884




minus (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904)))⟩

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δ12 1003817100381710038171003817119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904)))



119903 (lfloor119904rfloor) ) )10038171003817100381710038172

HS 119889119904

= 1198661(119905) + 119866

2(119905) + 119866

3(119905) + 119866

4(119905)

(55)


1198662(119905)

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ321205882

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(56)


1198663(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867


)10038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884



minus (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904 + 119866

3(119905)

(57)

By (39) we have

1198663(119905)

le 2119902Δminus12

Eint119905

0

119890120579119904[1003817100381710038171003817119891 (119884



10038171003817100381710038172

119867

+1003817100381710038171003817(119891 (119884

119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867]


119889119904

le 4119902Δminus12119871Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867


119889119904

le 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(58)


1198663(119905) le 119902Δ

12Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(59)



1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ


119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)


119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+





lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)



that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587



lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587



119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ



119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


(65)



119895and


⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)


119873+120575119873) minus Γ is



119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)


1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)


Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)





12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




+41198711205822

119895

120583

1003817100381710038171003817119892 (0 119895)10038171003817100381710038172

HS + 1205821198951003817100381710038171003817119892 (0 119895)

10038171003817100381710038172

HS

le minus120583

21199092


(9)

where1205721= max

119895isinS[(41205822

119895 119891(0 119895)

2

119867120583) +(4119871120582

2

119895120583) 119892(0 119895)

2

HS + 120582119895 119892(0 119895)2


119894=1⟨119879119890119894 119878119890119895⟩119867for

119878 119879 isinLHS(119867)



119885119909119894(119905) 119905 ge 0 Namely

P119905((119909 119894) 119860 times 119861) = P (119885

119909119894isin 119860 times 119861) (10)



follows

119889L (1198751 1198752)

= sup120593isinL

10038161003816100381610038161003816100381610038161003816100381610038161003816

119873

sum

119895=1

int119867

120593 (119906 119895) 1198751(119889119906 119895) minus

119873

sum

119895=1

int120593 (119906 119895) 1198752(119889119906 119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

(11)






119896rarrinfin119889L(119875119896 1198750) = 0




lim119905rarrinfin



(sup120593isinL

10038161003816100381610038161003816100381610038161003816100381610038161003816

E120593 (119885119909119894(119905))

minus

119873

sum

119895=1

int119867

120593 (119906 119895) 120587 (119889119906 119895)

10038161003816100381610038161003816100381610038161003816100381610038161003816

) = 0

(12)




119899= Span119890

1 1198902 119890

119899


119899

119889119883119899(119905) = [119860

119899119883119899(119905) + 119891

119899(119883119899(119905) 119903 (119905))] 119889119905

+ 119892119899(119883119899(119905) 119903 (119905)) 119889119882 (119905)

(13)


119899

119894=1⟨119909 119890119894⟩119867119890119894 119909 isin 119867 Here

119860119899= 120587119899119860 119891119899= 120587119899119891 119892119899= 120587119899119892


119860119899119909 = 119860119909 119890

119905119860119899119909= 119890119905119860119909 ⟨119909 119891

119899⟩119867= ⟨119909 119891⟩

119867

⟨119909 119892119899⟩119867= ⟨119909 119892⟩

119867 forall119909 isin 119867

119899

(14)


1003817100381710038171003817119860119899 (119909 minus 119910)10038171003817100381710038172

119867or1003817100381710038171003817119891119899 (119909 119895) minus 119891119899 (119910 119895)

10038171003817100381710038172

119867

or1003817100381710038171003817119892119899 (119909 119895) minus 119892119899 (119910 119895)

10038171003817100381710038172

HS

le 1205822

119899

1003817100381710038171003817(119909 minus 119910)10038171003817100381710038172

119867or1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)

10038171003817100381710038172

119867

or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS le (1205822

119899or 119871)1003817100381710038171003817(119909 minus 119910)

10038171003817100381710038172

119867


(15)


on119867119899




119875 (Δ) = (119875119894119895(Δ))119873times119873= 119890ΔΓ (16)



1from the


119903 (Δ)

=

119894 119894 isin S minus 119873


sum119895=1

119875119903(0)119895(Δ)

le 1205851

lt

119894

sum119895=1

119875119903(0)119895(Δ)

119873

119873minus1

sum119895=1

119875119903(0)119895(Δ) le 120585

1

(17)




119896+1and setting

119903 ((119896 + 1) Δ)

=

119894 119894 isin S minus 119873


sum119895=1

119875119903(119896Δ)119895

(Δ)

le 120585119896+1lt

119894

sum119895=1

119875119903(119896Δ)119895

(Δ)

119873

119873minus1

sum119895=1

119875119903(119896Δ)119895

(Δ) le 120585119896+1

(18)




120587119899119909 119903(0) = 119903

0 and

119884119899

((119896 + 1) Δ)

= 119890Δ119860119899 119884119899

(119896Δ) + 119891119899(119884119899

(119896Δ) 119903 (119896Δ)) Δ

+119892119899(119884119899

(119896Δ) 119903 (119896Δ)) Δ119882119896

(19)

where Δ119882119896= 119882((119896 + 1)Δ) minus 119882(119896Δ))


119884119899(119905) = 119890

119905119860119899120587119899119909 + int

119905

0



+ int

119905

0



= 119890119905119860119899120587119899119909 + int

119905

0



+ int

119905

0



(20)


119899119909 and 119884119899(119896Δ) = 119884119899(119896Δ)


119899 119894 119895 isin S let 119885

119899

(119896Δ) = (119884119899

(119896Δ) 119903(119896Δ))

P119899Δ((119909 119894) 119860 times 119895)

= P (119885119899

(Δ) isin 119860 times 119895 | 119885119899

(0) = (119909 119894))

P119899Δ

119896((119909 119894) 119860 times 119895)

= P (119885119899

(119896Δ) isin 119860 times 119895 | 119885119899

(0) = (119909 119894))

(21)


Lemma 7 119885 119899(119896Δ)119896ge0



(119896Δ)


(119896Δ)119896ge0


119899times



lim119896rarrinfin

119889L (119875119899Δ

119896((119909 119894) sdot times sdot) 120587




119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)





E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119909119894(lfloor119905rfloor)10038171003817100381710038171003817

2

119867

le 3 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

(23)


119884119899(lfloor119905rfloor) = 119890


lfloor119905rfloor

0



+ int

lfloor119905rfloor

0



(24)


Thus

119884119899(119905) minus 119884




+ int

lfloor119905rfloor

0



+ int

lfloor119905rfloor

0




+ int

119905

lfloor119905rfloor



+ int

119905

lfloor119905rfloor





+ int

119905

lfloor119905rfloor



+ int

119905

lfloor119905rfloor



(25)


E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867



119867

+ Eint119905

lfloor119905rfloor


10038171003817100381710038172

119867119889119904

+Eint119905

lfloor119905rfloor


10038171003817100381710038172

HS119889119904

(26)

From (A1) we have



119867

= E

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119894=1



119894⟩119867119890119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867


)2

E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 1205882

119899Δ2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

(27)


Eint119905

lfloor119905rfloor


10038171003817100381710038172

119867119889119904

+ Eint119905

lfloor119905rfloor


10038171003817100381710038172

HS119889119904


119867)

(28)



((4120572119901 + 120583)(8119902 + 41199021205882

119899119871+4119902119871 + 24119902 + 6119902119871(120588

2

119899+2119871)))

2


sup119905ge0

E10038171003817100381710038171003817119884119899119909119894(119905)10038171003817100381710038171003817119867le 119862 (29)

where 119902 = max1le119894le119873

120582119894 119901 = min

1le119894le119873120582119894


119889119884119899(119905)

= 119860119884119899(119905) + 119890




(30)

with 119884119899(0) = 120587119899119909

Let119881(119909 119894) = 120582119894 1199092



119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1205821198941199092

119867+ Eint

119905

0

119890120579119904120582119903(119904)

times 1205791003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ 2⟨119884

119899(119904) 119860119884

119899(119904)⟩119867



119867

+1003817100381710038171003817119892119899 (119884


10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904


le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)



119867


10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

(31)




119867

= ⟨119884119899(119905) 119891 (119884

119899(119905) 119903 (119905))⟩

119867

+ ⟨119884119899(119905) (119890


119867



minus119891 (119884119899(119905) 119903 (119905))) ⟩

119867

(32)



10038171003817100381710038172

HS

=1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

minus (119892 (119884119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

(33)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2⟨119884119899(119904) 119891 (119884

119899(119904) 119903 (119904))⟩

119867

+1003817100381710038171003817119892 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus120583

2Eint119905

0

119890120579119904119884 (119904)

2

119867119889119904 + 120572

1int

119905

0

119890120579119904119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

= 1198691(119905) + 119869

2(119905) + 119869

3(119905) + 119869

4(119905)

(34)


1198692(119905) le Eint

119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times 119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904


le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δminus121205882

119899Δ2119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ121205882

119899119871)

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

(35)



119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 21003817100381710038171003817(119891 (119884


119899(lfloor119905rfloor) 119903 (119905)))

10038171003817100381710038172

119867

+ 21003817100381710038171003817(119891 (119884


119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(36)

Similarly we have

1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))) minus 119892 (119884


10038171003817100381710038172

HS

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(37)


1198693(119905)

le 2Eint119905

0



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867119889119904

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + Δ


2

times1003817100381710038171003817119891 (119884


minus119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δminus128119871


119867) 119868119903(119904) = 119903(lfloor119904rfloor)

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(38)


E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

]

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

| 119903 (lfloor119905rfloor)])

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) | 119903 (lfloor119905rfloor)])



= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

P (119903 (119905) = 119894 | 119903 (lfloor119905rfloor) = 119894)

= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

times sum

119895 = 119894


= E (1 + 119884 (lfloor119905rfloor)2


119894isinS

119868119903(lfloor119905rfloor)=119894


119867)

(39)

where = 119873[1 +max1le119894le119873


gives

1198693(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

+ 119902int

119905

0

8119890120579119904Δ12119871E (1 + 119884 (lfloor119904rfloor)

2

119867) 119889119904

(40)


1198694(119905)

= Eint119905

0

119890120579119904120582119903(119904)

times Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)1003817100381710038171003817119892 (119884

119899(119904) 119903 (119904)))


10038171003817100381710038172

HS 119889119904


le 119902Eint119905

0

119890120579119904Δ12119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904 + 119902 (1 + Δ

minus12)

times int

119905

0

1198901205791199048119871 (1 +

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119868119903(119904) = 119903(lfloor119904rfloor)

119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

le 119902Δ12119871Eint119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 16119902Δ12119871int

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(41)


have

E1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867

le 2E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 6 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867


119867+ 2

(42)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 1199021205882

119899Δ12119871

+24119902Δ12 119871 + 119902Δ

12119871] 119889119904

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus

120583

2+ 119902Δ12(2 + 120588

2

119899119871) + 119902Δ

12119871]

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + 24119902Δ

12 119871E

times int

119905

0

1198901205791199041003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ (4119902Δminus12119871 + 2119902119871)Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(43)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583

+ 31199021205882

119899Δ12119871 + 24119902Δ

12 119871

+3119902Δ12119871 + 4119902Δ

12] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+4119902Δ12119871 + 24119902Δ

12 119871]1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ 6119902119871Δ12(1205882

119899+ 2119871)Eint

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119889119904

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583 + 3119902120588

2

119899Δ12119871

+ 24119902Δ12 119871 + 3119902Δ

12119871 + 4119902Δ

12

+6119902119871Δ12(1205882

119899+ 2119871)] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+ 4119902Δ12119871 + 24119902Δ

12 119871

+6119902119871Δ12(1205882

119899+ 2119871)]

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(44)

Let 120579 = (4120572119901+120583)4119902 for Δ lt ((4120572119901+120583)(8119902+41199021205882119899119871+4119902119871+

24119902 + 6119902119871(1205882

119899+ 2119871)))

2 then

119901119890120579119905E (119884 (119905)

2

119867) le 119902119909

2

119867+ int

119905

0

119890120579119904[1205721+4120572119901 + 120583

2] 119889119904

(45)

That is

sup119905ge0

E (119884 (119905)2

119867) le 119862 (46)


2119871) ((2120572119901 + 120583)(4119902 + 2119902119871 + 21199021205882

119899119871 + 12119902119871))

2 then

lim119905rarrinfin

E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119910119894(119905)10038171003817100381710038171003817

2

119867= 0 119906119899119894119891119900119903119898119897119910 119891119900119903 119909 119910 isin 119880

(47)




(119884119909(119905) minus 119884

119910(119905)) minus (119884



= (119884119909(119905) minus 119884


119910(119905) minus 119884




+ int

119905

lfloor119905rfloor




+ int

119905

lfloor119905rfloor




(48)


E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884



10038171003817100381710038172

119867

le 3 (1205882

119899+ 2119871)ΔE


119910(lfloor119905rfloor)10038171003817100381710038172

119867

(49)

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172

119867

= E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884





10038171003817100381710038172

119867

le (1 + 2)E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))



10038171003817100381710038172

119867

+ (1 +1


119910(lfloor119905rfloor)10038171003817100381710038172

119867

le 9 (1205882

119899+ 2119871)ΔE



10038171003817100381710038172

119867



10038171003817100381710038172

119867

(50)

If Δ lt 118(1205882119899+ 2119871) then

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172



10038171003817100381710038172

119867 (51)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 120582119894

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

+ Eint119905

0

119890120579119904120582119903(119904)1205791003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ 2 ⟨119884119909(119904) minus 119884

119910(119904)

119860119884119909(119904) minus 119884

119910(119904)⟩119867

+ 2 ⟨119884119909(119904) minus 119884




119867

+1003817100381710038171003817119892119899 (119884



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ 119902120579Eint

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2 ⟨119884

119909(119904) minus 119884




119867



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(52)


⟨119884119909(119905) minus 119884




119867

= ⟨119884119909(119905) minus 119884

119910(119905) 119891 (119884

119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))⟩

119867

+ ⟨119884119909(119905) minus 119884

119910(119905) (119890


times (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

+ ⟨119884119909(119905) minus 119884




minus (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

(53)




10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905))

10038171003817100381710038172

HS


+ (1 + Δminus12)1003817100381710038171003817(119892 (119884

119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905)))



10038171003817100381710038172

HS

(54)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ (119902120579 minus 2120572119901 minus 120583)E

times int

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884

119910(119904) (119890


times (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904))) ⟩

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884




minus (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904)))⟩

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δ12 1003817100381710038171003817119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904)))



119903 (lfloor119904rfloor) ) )10038171003817100381710038172

HS 119889119904

= 1198661(119905) + 119866

2(119905) + 119866

3(119905) + 119866

4(119905)

(55)


1198662(119905)

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ321205882

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(56)


1198663(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867


)10038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884



minus (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904 + 119866

3(119905)

(57)

By (39) we have

1198663(119905)

le 2119902Δminus12

Eint119905

0

119890120579119904[1003817100381710038171003817119891 (119884



10038171003817100381710038172

119867

+1003817100381710038171003817(119891 (119884

119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867]


119889119904

le 4119902Δminus12119871Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867


119889119904

le 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(58)


1198663(119905) le 119902Δ

12Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(59)



1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ


119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)


119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+





lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)



that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587



lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587



119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ



119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


(65)



119895and


⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)


119873+120575119873) minus Γ is



119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)


1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)


Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)





12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014






119896+1and setting

119903 ((119896 + 1) Δ)

=

119894 119894 isin S minus 119873


sum119895=1

119875119903(119896Δ)119895

(Δ)

le 120585119896+1lt

119894

sum119895=1

119875119903(119896Δ)119895

(Δ)

119873

119873minus1

sum119895=1

119875119903(119896Δ)119895

(Δ) le 120585119896+1

(18)




120587119899119909 119903(0) = 119903

0 and

119884119899

((119896 + 1) Δ)

= 119890Δ119860119899 119884119899

(119896Δ) + 119891119899(119884119899

(119896Δ) 119903 (119896Δ)) Δ

+119892119899(119884119899

(119896Δ) 119903 (119896Δ)) Δ119882119896

(19)

where Δ119882119896= 119882((119896 + 1)Δ) minus 119882(119896Δ))


119884119899(119905) = 119890

119905119860119899120587119899119909 + int

119905

0



+ int

119905

0



= 119890119905119860119899120587119899119909 + int

119905

0



+ int

119905

0



(20)


119899119909 and 119884119899(119896Δ) = 119884119899(119896Δ)


119899 119894 119895 isin S let 119885

119899

(119896Δ) = (119884119899

(119896Δ) 119903(119896Δ))

P119899Δ((119909 119894) 119860 times 119895)

= P (119885119899

(Δ) isin 119860 times 119895 | 119885119899

(0) = (119909 119894))

P119899Δ

119896((119909 119894) 119860 times 119895)

= P (119885119899

(119896Δ) isin 119860 times 119895 | 119885119899

(0) = (119909 119894))

(21)


Lemma 7 119885 119899(119896Δ)119896ge0



(119896Δ)


(119896Δ)119896ge0


119899times



lim119896rarrinfin

119889L (119875119899Δ

119896((119909 119894) sdot times sdot) 120587




119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)





E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119909119894(lfloor119905rfloor)10038171003817100381710038171003817

2

119867

le 3 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

(23)


119884119899(lfloor119905rfloor) = 119890


lfloor119905rfloor

0



+ int

lfloor119905rfloor

0



(24)


Thus

119884119899(119905) minus 119884




+ int

lfloor119905rfloor

0



+ int

lfloor119905rfloor

0




+ int

119905

lfloor119905rfloor



+ int

119905

lfloor119905rfloor





+ int

119905

lfloor119905rfloor



+ int

119905

lfloor119905rfloor



(25)


E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867



119867

+ Eint119905

lfloor119905rfloor


10038171003817100381710038172

119867119889119904

+Eint119905

lfloor119905rfloor


10038171003817100381710038172

HS119889119904

(26)

From (A1) we have



119867

= E

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119894=1



119894⟩119867119890119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867


)2

E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 1205882

119899Δ2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

(27)


Eint119905

lfloor119905rfloor


10038171003817100381710038172

119867119889119904

+ Eint119905

lfloor119905rfloor


10038171003817100381710038172

HS119889119904


119867)

(28)



((4120572119901 + 120583)(8119902 + 41199021205882

119899119871+4119902119871 + 24119902 + 6119902119871(120588

2

119899+2119871)))

2


sup119905ge0

E10038171003817100381710038171003817119884119899119909119894(119905)10038171003817100381710038171003817119867le 119862 (29)

where 119902 = max1le119894le119873

120582119894 119901 = min

1le119894le119873120582119894


119889119884119899(119905)

= 119860119884119899(119905) + 119890




(30)

with 119884119899(0) = 120587119899119909

Let119881(119909 119894) = 120582119894 1199092



119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1205821198941199092

119867+ Eint

119905

0

119890120579119904120582119903(119904)

times 1205791003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ 2⟨119884

119899(119904) 119860119884

119899(119904)⟩119867



119867

+1003817100381710038171003817119892119899 (119884


10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904


le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)



119867


10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

(31)




119867

= ⟨119884119899(119905) 119891 (119884

119899(119905) 119903 (119905))⟩

119867

+ ⟨119884119899(119905) (119890


119867



minus119891 (119884119899(119905) 119903 (119905))) ⟩

119867

(32)



10038171003817100381710038172

HS

=1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

minus (119892 (119884119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

(33)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2⟨119884119899(119904) 119891 (119884

119899(119904) 119903 (119904))⟩

119867

+1003817100381710038171003817119892 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus120583

2Eint119905

0

119890120579119904119884 (119904)

2

119867119889119904 + 120572

1int

119905

0

119890120579119904119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

= 1198691(119905) + 119869

2(119905) + 119869

3(119905) + 119869

4(119905)

(34)


1198692(119905) le Eint

119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times 119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904


le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δminus121205882

119899Δ2119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ121205882

119899119871)

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

(35)



119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 21003817100381710038171003817(119891 (119884


119899(lfloor119905rfloor) 119903 (119905)))

10038171003817100381710038172

119867

+ 21003817100381710038171003817(119891 (119884


119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(36)

Similarly we have

1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))) minus 119892 (119884


10038171003817100381710038172

HS

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(37)


1198693(119905)

le 2Eint119905

0



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867119889119904

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + Δ


2

times1003817100381710038171003817119891 (119884


minus119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δminus128119871


119867) 119868119903(119904) = 119903(lfloor119904rfloor)

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(38)


E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

]

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

| 119903 (lfloor119905rfloor)])

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) | 119903 (lfloor119905rfloor)])



= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

P (119903 (119905) = 119894 | 119903 (lfloor119905rfloor) = 119894)

= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

times sum

119895 = 119894


= E (1 + 119884 (lfloor119905rfloor)2


119894isinS

119868119903(lfloor119905rfloor)=119894


119867)

(39)

where = 119873[1 +max1le119894le119873


gives

1198693(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

+ 119902int

119905

0

8119890120579119904Δ12119871E (1 + 119884 (lfloor119904rfloor)

2

119867) 119889119904

(40)


1198694(119905)

= Eint119905

0

119890120579119904120582119903(119904)

times Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)1003817100381710038171003817119892 (119884

119899(119904) 119903 (119904)))


10038171003817100381710038172

HS 119889119904


le 119902Eint119905

0

119890120579119904Δ12119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904 + 119902 (1 + Δ

minus12)

times int

119905

0

1198901205791199048119871 (1 +

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119868119903(119904) = 119903(lfloor119904rfloor)

119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

le 119902Δ12119871Eint119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 16119902Δ12119871int

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(41)


have

E1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867

le 2E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 6 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867


119867+ 2

(42)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 1199021205882

119899Δ12119871

+24119902Δ12 119871 + 119902Δ

12119871] 119889119904

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus

120583

2+ 119902Δ12(2 + 120588

2

119899119871) + 119902Δ

12119871]

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + 24119902Δ

12 119871E

times int

119905

0

1198901205791199041003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ (4119902Δminus12119871 + 2119902119871)Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(43)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583

+ 31199021205882

119899Δ12119871 + 24119902Δ

12 119871

+3119902Δ12119871 + 4119902Δ

12] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+4119902Δ12119871 + 24119902Δ

12 119871]1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ 6119902119871Δ12(1205882

119899+ 2119871)Eint

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119889119904

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583 + 3119902120588

2

119899Δ12119871

+ 24119902Δ12 119871 + 3119902Δ

12119871 + 4119902Δ

12

+6119902119871Δ12(1205882

119899+ 2119871)] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+ 4119902Δ12119871 + 24119902Δ

12 119871

+6119902119871Δ12(1205882

119899+ 2119871)]

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(44)

Let 120579 = (4120572119901+120583)4119902 for Δ lt ((4120572119901+120583)(8119902+41199021205882119899119871+4119902119871+

24119902 + 6119902119871(1205882

119899+ 2119871)))

2 then

119901119890120579119905E (119884 (119905)

2

119867) le 119902119909

2

119867+ int

119905

0

119890120579119904[1205721+4120572119901 + 120583

2] 119889119904

(45)

That is

sup119905ge0

E (119884 (119905)2

119867) le 119862 (46)


2119871) ((2120572119901 + 120583)(4119902 + 2119902119871 + 21199021205882

119899119871 + 12119902119871))

2 then

lim119905rarrinfin

E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119910119894(119905)10038171003817100381710038171003817

2

119867= 0 119906119899119894119891119900119903119898119897119910 119891119900119903 119909 119910 isin 119880

(47)




(119884119909(119905) minus 119884

119910(119905)) minus (119884



= (119884119909(119905) minus 119884


119910(119905) minus 119884




+ int

119905

lfloor119905rfloor




+ int

119905

lfloor119905rfloor




(48)


E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884



10038171003817100381710038172

119867

le 3 (1205882

119899+ 2119871)ΔE


119910(lfloor119905rfloor)10038171003817100381710038172

119867

(49)

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172

119867

= E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884





10038171003817100381710038172

119867

le (1 + 2)E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))



10038171003817100381710038172

119867

+ (1 +1


119910(lfloor119905rfloor)10038171003817100381710038172

119867

le 9 (1205882

119899+ 2119871)ΔE



10038171003817100381710038172

119867



10038171003817100381710038172

119867

(50)

If Δ lt 118(1205882119899+ 2119871) then

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172



10038171003817100381710038172

119867 (51)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 120582119894

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

+ Eint119905

0

119890120579119904120582119903(119904)1205791003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ 2 ⟨119884119909(119904) minus 119884

119910(119904)

119860119884119909(119904) minus 119884

119910(119904)⟩119867

+ 2 ⟨119884119909(119904) minus 119884




119867

+1003817100381710038171003817119892119899 (119884



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ 119902120579Eint

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2 ⟨119884

119909(119904) minus 119884




119867



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(52)


⟨119884119909(119905) minus 119884




119867

= ⟨119884119909(119905) minus 119884

119910(119905) 119891 (119884

119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))⟩

119867

+ ⟨119884119909(119905) minus 119884

119910(119905) (119890


times (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

+ ⟨119884119909(119905) minus 119884




minus (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

(53)




10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905))

10038171003817100381710038172

HS


+ (1 + Δminus12)1003817100381710038171003817(119892 (119884

119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905)))



10038171003817100381710038172

HS

(54)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ (119902120579 minus 2120572119901 minus 120583)E

times int

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884

119910(119904) (119890


times (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904))) ⟩

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884




minus (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904)))⟩

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δ12 1003817100381710038171003817119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904)))



119903 (lfloor119904rfloor) ) )10038171003817100381710038172

HS 119889119904

= 1198661(119905) + 119866

2(119905) + 119866

3(119905) + 119866

4(119905)

(55)


1198662(119905)

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ321205882

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(56)


1198663(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867


)10038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884



minus (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904 + 119866

3(119905)

(57)

By (39) we have

1198663(119905)

le 2119902Δminus12

Eint119905

0

119890120579119904[1003817100381710038171003817119891 (119884



10038171003817100381710038172

119867

+1003817100381710038171003817(119891 (119884

119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867]


119889119904

le 4119902Δminus12119871Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867


119889119904

le 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(58)


1198663(119905) le 119902Δ

12Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(59)



1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ


119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)


119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+





lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)



that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587



lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587



119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ



119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


(65)



119895and


⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)


119873+120575119873) minus Γ is



119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)


1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)


Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)





12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




Thus

119884119899(119905) minus 119884




+ int

lfloor119905rfloor

0



+ int

lfloor119905rfloor

0




+ int

119905

lfloor119905rfloor



+ int

119905

lfloor119905rfloor





+ int

119905

lfloor119905rfloor



+ int

119905

lfloor119905rfloor



(25)


E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867



119867

+ Eint119905

lfloor119905rfloor


10038171003817100381710038172

119867119889119904

+Eint119905

lfloor119905rfloor


10038171003817100381710038172

HS119889119904

(26)

From (A1) we have



119867

= E

1003817100381710038171003817100381710038171003817100381710038171003817

119899

sum

119894=1



119894⟩119867119890119894

1003817100381710038171003817100381710038171003817100381710038171003817

2

119867


)2

E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 1205882

119899Δ2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

(27)


Eint119905

lfloor119905rfloor


10038171003817100381710038172

119867119889119904

+ Eint119905

lfloor119905rfloor


10038171003817100381710038172

HS119889119904


119867)

(28)



((4120572119901 + 120583)(8119902 + 41199021205882

119899119871+4119902119871 + 24119902 + 6119902119871(120588

2

119899+2119871)))

2


sup119905ge0

E10038171003817100381710038171003817119884119899119909119894(119905)10038171003817100381710038171003817119867le 119862 (29)

where 119902 = max1le119894le119873

120582119894 119901 = min

1le119894le119873120582119894


119889119884119899(119905)

= 119860119884119899(119905) + 119890




(30)

with 119884119899(0) = 120587119899119909

Let119881(119909 119894) = 120582119894 1199092



119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1205821198941199092

119867+ Eint

119905

0

119890120579119904120582119903(119904)

times 1205791003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ 2⟨119884

119899(119904) 119860119884

119899(119904)⟩119867



119867

+1003817100381710038171003817119892119899 (119884


10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904


le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)



119867


10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

(31)




119867

= ⟨119884119899(119905) 119891 (119884

119899(119905) 119903 (119905))⟩

119867

+ ⟨119884119899(119905) (119890


119867



minus119891 (119884119899(119905) 119903 (119905))) ⟩

119867

(32)



10038171003817100381710038172

HS

=1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

minus (119892 (119884119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

(33)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2⟨119884119899(119904) 119891 (119884

119899(119904) 119903 (119904))⟩

119867

+1003817100381710038171003817119892 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus120583

2Eint119905

0

119890120579119904119884 (119904)

2

119867119889119904 + 120572

1int

119905

0

119890120579119904119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

= 1198691(119905) + 119869

2(119905) + 119869

3(119905) + 119869

4(119905)

(34)


1198692(119905) le Eint

119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times 119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904


le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δminus121205882

119899Δ2119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ121205882

119899119871)

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

(35)



119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 21003817100381710038171003817(119891 (119884


119899(lfloor119905rfloor) 119903 (119905)))

10038171003817100381710038172

119867

+ 21003817100381710038171003817(119891 (119884


119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(36)

Similarly we have

1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))) minus 119892 (119884


10038171003817100381710038172

HS

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(37)


1198693(119905)

le 2Eint119905

0



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867119889119904

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + Δ


2

times1003817100381710038171003817119891 (119884


minus119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δminus128119871


119867) 119868119903(119904) = 119903(lfloor119904rfloor)

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(38)


E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

]

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

| 119903 (lfloor119905rfloor)])

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) | 119903 (lfloor119905rfloor)])



= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

P (119903 (119905) = 119894 | 119903 (lfloor119905rfloor) = 119894)

= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

times sum

119895 = 119894


= E (1 + 119884 (lfloor119905rfloor)2


119894isinS

119868119903(lfloor119905rfloor)=119894


119867)

(39)

where = 119873[1 +max1le119894le119873


gives

1198693(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

+ 119902int

119905

0

8119890120579119904Δ12119871E (1 + 119884 (lfloor119904rfloor)

2

119867) 119889119904

(40)


1198694(119905)

= Eint119905

0

119890120579119904120582119903(119904)

times Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)1003817100381710038171003817119892 (119884

119899(119904) 119903 (119904)))


10038171003817100381710038172

HS 119889119904


le 119902Eint119905

0

119890120579119904Δ12119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904 + 119902 (1 + Δ

minus12)

times int

119905

0

1198901205791199048119871 (1 +

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119868119903(119904) = 119903(lfloor119904rfloor)

119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

le 119902Δ12119871Eint119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 16119902Δ12119871int

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(41)


have

E1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867

le 2E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 6 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867


119867+ 2

(42)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 1199021205882

119899Δ12119871

+24119902Δ12 119871 + 119902Δ

12119871] 119889119904

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus

120583

2+ 119902Δ12(2 + 120588

2

119899119871) + 119902Δ

12119871]

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + 24119902Δ

12 119871E

times int

119905

0

1198901205791199041003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ (4119902Δminus12119871 + 2119902119871)Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(43)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583

+ 31199021205882

119899Δ12119871 + 24119902Δ

12 119871

+3119902Δ12119871 + 4119902Δ

12] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+4119902Δ12119871 + 24119902Δ

12 119871]1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ 6119902119871Δ12(1205882

119899+ 2119871)Eint

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119889119904

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583 + 3119902120588

2

119899Δ12119871

+ 24119902Δ12 119871 + 3119902Δ

12119871 + 4119902Δ

12

+6119902119871Δ12(1205882

119899+ 2119871)] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+ 4119902Δ12119871 + 24119902Δ

12 119871

+6119902119871Δ12(1205882

119899+ 2119871)]

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(44)

Let 120579 = (4120572119901+120583)4119902 for Δ lt ((4120572119901+120583)(8119902+41199021205882119899119871+4119902119871+

24119902 + 6119902119871(1205882

119899+ 2119871)))

2 then

119901119890120579119905E (119884 (119905)

2

119867) le 119902119909

2

119867+ int

119905

0

119890120579119904[1205721+4120572119901 + 120583

2] 119889119904

(45)

That is

sup119905ge0

E (119884 (119905)2

119867) le 119862 (46)


2119871) ((2120572119901 + 120583)(4119902 + 2119902119871 + 21199021205882

119899119871 + 12119902119871))

2 then

lim119905rarrinfin

E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119910119894(119905)10038171003817100381710038171003817

2

119867= 0 119906119899119894119891119900119903119898119897119910 119891119900119903 119909 119910 isin 119880

(47)




(119884119909(119905) minus 119884

119910(119905)) minus (119884



= (119884119909(119905) minus 119884


119910(119905) minus 119884




+ int

119905

lfloor119905rfloor




+ int

119905

lfloor119905rfloor




(48)


E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884



10038171003817100381710038172

119867

le 3 (1205882

119899+ 2119871)ΔE


119910(lfloor119905rfloor)10038171003817100381710038172

119867

(49)

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172

119867

= E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884





10038171003817100381710038172

119867

le (1 + 2)E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))



10038171003817100381710038172

119867

+ (1 +1


119910(lfloor119905rfloor)10038171003817100381710038172

119867

le 9 (1205882

119899+ 2119871)ΔE



10038171003817100381710038172

119867



10038171003817100381710038172

119867

(50)

If Δ lt 118(1205882119899+ 2119871) then

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172



10038171003817100381710038172

119867 (51)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 120582119894

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

+ Eint119905

0

119890120579119904120582119903(119904)1205791003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ 2 ⟨119884119909(119904) minus 119884

119910(119904)

119860119884119909(119904) minus 119884

119910(119904)⟩119867

+ 2 ⟨119884119909(119904) minus 119884




119867

+1003817100381710038171003817119892119899 (119884



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ 119902120579Eint

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2 ⟨119884

119909(119904) minus 119884




119867



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(52)


⟨119884119909(119905) minus 119884




119867

= ⟨119884119909(119905) minus 119884

119910(119905) 119891 (119884

119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))⟩

119867

+ ⟨119884119909(119905) minus 119884

119910(119905) (119890


times (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

+ ⟨119884119909(119905) minus 119884




minus (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

(53)




10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905))

10038171003817100381710038172

HS


+ (1 + Δminus12)1003817100381710038171003817(119892 (119884

119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905)))



10038171003817100381710038172

HS

(54)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ (119902120579 minus 2120572119901 minus 120583)E

times int

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884

119910(119904) (119890


times (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904))) ⟩

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884




minus (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904)))⟩

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δ12 1003817100381710038171003817119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904)))



119903 (lfloor119904rfloor) ) )10038171003817100381710038172

HS 119889119904

= 1198661(119905) + 119866

2(119905) + 119866

3(119905) + 119866

4(119905)

(55)


1198662(119905)

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ321205882

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(56)


1198663(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867


)10038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884



minus (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904 + 119866

3(119905)

(57)

By (39) we have

1198663(119905)

le 2119902Δminus12

Eint119905

0

119890120579119904[1003817100381710038171003817119891 (119884



10038171003817100381710038172

119867

+1003817100381710038171003817(119891 (119884

119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867]


119889119904

le 4119902Δminus12119871Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867


119889119904

le 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(58)


1198663(119905) le 119902Δ

12Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(59)



1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ


119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)


119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+





lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)



that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587



lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587



119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ



119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


(65)



119895and


⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)


119873+120575119873) minus Γ is



119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)


1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)


Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)





12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)



119867


10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

(31)




119867

= ⟨119884119899(119905) 119891 (119884

119899(119905) 119903 (119905))⟩

119867

+ ⟨119884119899(119905) (119890


119867



minus119891 (119884119899(119905) 119903 (119905))) ⟩

119867

(32)



10038171003817100381710038172

HS

=1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

minus (119892 (119884119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119905) 119903 (119905)) minus 119892 (119884


10038171003817100381710038172

HS

(33)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2⟨119884119899(119904) 119891 (119884

119899(119904) 119903 (119904))⟩

119867

+1003817100381710038171003817119892 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

le 1199021199092

119867+ 120579119902Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

minus120583

2Eint119905

0

119890120579119904119884 (119904)

2

119867119889119904 + 120572

1int

119905

0

119890120579119904119889119904

+ Eint119905

0

119890120579119904120582119903(119904)

times 2⟨119884119899(119904) (119890


119867



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867

+ Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172


times1003817100381710038171003817(119892 (119884

119899(119904) 119903 (119904))


10038171003817100381710038172

HS 119889119904

= 1198691(119905) + 119869

2(119905) + 119869

3(119905) + 119869

4(119905)

(34)


1198692(119905) le Eint

119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times 119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904


le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δminus121205882

119899Δ2119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ121205882

119899119871)

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

(35)



119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 21003817100381710038171003817(119891 (119884


119899(lfloor119905rfloor) 119903 (119905)))

10038171003817100381710038172

119867

+ 21003817100381710038171003817(119891 (119884


119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(36)

Similarly we have

1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))) minus 119892 (119884


10038171003817100381710038172

HS

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(37)


1198693(119905)

le 2Eint119905

0



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867119889119904

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + Δ


2

times1003817100381710038171003817119891 (119884


minus119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δminus128119871


119867) 119868119903(119904) = 119903(lfloor119904rfloor)

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(38)


E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

]

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

| 119903 (lfloor119905rfloor)])

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) | 119903 (lfloor119905rfloor)])



= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

P (119903 (119905) = 119894 | 119903 (lfloor119905rfloor) = 119894)

= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

times sum

119895 = 119894


= E (1 + 119884 (lfloor119905rfloor)2


119894isinS

119868119903(lfloor119905rfloor)=119894


119867)

(39)

where = 119873[1 +max1le119894le119873


gives

1198693(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

+ 119902int

119905

0

8119890120579119904Δ12119871E (1 + 119884 (lfloor119904rfloor)

2

119867) 119889119904

(40)


1198694(119905)

= Eint119905

0

119890120579119904120582119903(119904)

times Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)1003817100381710038171003817119892 (119884

119899(119904) 119903 (119904)))


10038171003817100381710038172

HS 119889119904


le 119902Eint119905

0

119890120579119904Δ12119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904 + 119902 (1 + Δ

minus12)

times int

119905

0

1198901205791199048119871 (1 +

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119868119903(119904) = 119903(lfloor119904rfloor)

119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

le 119902Δ12119871Eint119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 16119902Δ12119871int

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(41)


have

E1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867

le 2E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 6 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867


119867+ 2

(42)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 1199021205882

119899Δ12119871

+24119902Δ12 119871 + 119902Δ

12119871] 119889119904

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus

120583

2+ 119902Δ12(2 + 120588

2

119899119871) + 119902Δ

12119871]

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + 24119902Δ

12 119871E

times int

119905

0

1198901205791199041003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ (4119902Δminus12119871 + 2119902119871)Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(43)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583

+ 31199021205882

119899Δ12119871 + 24119902Δ

12 119871

+3119902Δ12119871 + 4119902Δ

12] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+4119902Δ12119871 + 24119902Δ

12 119871]1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ 6119902119871Δ12(1205882

119899+ 2119871)Eint

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119889119904

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583 + 3119902120588

2

119899Δ12119871

+ 24119902Δ12 119871 + 3119902Δ

12119871 + 4119902Δ

12

+6119902119871Δ12(1205882

119899+ 2119871)] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+ 4119902Δ12119871 + 24119902Δ

12 119871

+6119902119871Δ12(1205882

119899+ 2119871)]

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(44)

Let 120579 = (4120572119901+120583)4119902 for Δ lt ((4120572119901+120583)(8119902+41199021205882119899119871+4119902119871+

24119902 + 6119902119871(1205882

119899+ 2119871)))

2 then

119901119890120579119905E (119884 (119905)

2

119867) le 119902119909

2

119867+ int

119905

0

119890120579119904[1205721+4120572119901 + 120583

2] 119889119904

(45)

That is

sup119905ge0

E (119884 (119905)2

119867) le 119862 (46)


2119871) ((2120572119901 + 120583)(4119902 + 2119902119871 + 21199021205882

119899119871 + 12119902119871))

2 then

lim119905rarrinfin

E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119910119894(119905)10038171003817100381710038171003817

2

119867= 0 119906119899119894119891119900119903119898119897119910 119891119900119903 119909 119910 isin 119880

(47)




(119884119909(119905) minus 119884

119910(119905)) minus (119884



= (119884119909(119905) minus 119884


119910(119905) minus 119884




+ int

119905

lfloor119905rfloor




+ int

119905

lfloor119905rfloor




(48)


E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884



10038171003817100381710038172

119867

le 3 (1205882

119899+ 2119871)ΔE


119910(lfloor119905rfloor)10038171003817100381710038172

119867

(49)

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172

119867

= E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884





10038171003817100381710038172

119867

le (1 + 2)E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))



10038171003817100381710038172

119867

+ (1 +1


119910(lfloor119905rfloor)10038171003817100381710038172

119867

le 9 (1205882

119899+ 2119871)ΔE



10038171003817100381710038172

119867



10038171003817100381710038172

119867

(50)

If Δ lt 118(1205882119899+ 2119871) then

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172



10038171003817100381710038172

119867 (51)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 120582119894

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

+ Eint119905

0

119890120579119904120582119903(119904)1205791003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ 2 ⟨119884119909(119904) minus 119884

119910(119904)

119860119884119909(119904) minus 119884

119910(119904)⟩119867

+ 2 ⟨119884119909(119904) minus 119884




119867

+1003817100381710038171003817119892119899 (119884



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ 119902120579Eint

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2 ⟨119884

119909(119904) minus 119884




119867



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(52)


⟨119884119909(119905) minus 119884




119867

= ⟨119884119909(119905) minus 119884

119910(119905) 119891 (119884

119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))⟩

119867

+ ⟨119884119909(119905) minus 119884

119910(119905) (119890


times (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

+ ⟨119884119909(119905) minus 119884




minus (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

(53)




10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905))

10038171003817100381710038172

HS


+ (1 + Δminus12)1003817100381710038171003817(119892 (119884

119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905)))



10038171003817100381710038172

HS

(54)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ (119902120579 minus 2120572119901 minus 120583)E

times int

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884

119910(119904) (119890


times (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904))) ⟩

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884




minus (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904)))⟩

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δ12 1003817100381710038171003817119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904)))



119903 (lfloor119904rfloor) ) )10038171003817100381710038172

HS 119889119904

= 1198661(119905) + 119866

2(119905) + 119866

3(119905) + 119866

4(119905)

(55)


1198662(119905)

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ321205882

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(56)


1198663(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867


)10038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884



minus (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904 + 119866

3(119905)

(57)

By (39) we have

1198663(119905)

le 2119902Δminus12

Eint119905

0

119890120579119904[1003817100381710038171003817119891 (119884



10038171003817100381710038172

119867

+1003817100381710038171003817(119891 (119884

119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867]


119889119904

le 4119902Δminus12119871Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867


119889119904

le 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(58)


1198663(119905) le 119902Δ

12Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(59)



1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ


119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)


119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+





lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)



that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587



lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587



119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ



119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


(65)



119895and


⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)


119873+120575119873) minus Γ is



119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)


1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)


Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)





12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δminus121205882

119899Δ2119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ121205882

119899119871)

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δ121205882

119899119871 119889119904

(35)



119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 21003817100381710038171003817(119891 (119884


119899(lfloor119905rfloor) 119903 (119905)))

10038171003817100381710038172

119867

+ 21003817100381710038171003817(119891 (119884


119899(119905) 119903 (119905)))

10038171003817100381710038172

119867

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(36)

Similarly we have

1003817100381710038171003817119892 (119884119899(119905) 119903 (119905))) minus 119892 (119884


10038171003817100381710038172

HS

le 8119871 (1 +1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

+ 21198711003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867

(37)


1198693(119905)

le 2Eint119905

0



minus119891 (119884119899(119904) 119903 (119904))) ⟩

119867119889119904

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + Δ


2

times1003817100381710038171003817119891 (119884


minus119891 (119884119899(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867+ Δminus128119871


119867) 119868119903(119904) = 119903(lfloor119904rfloor)

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(38)


E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

]

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) 119868119903(119905) = 119903(lfloor119905rfloor)

| 119903 (lfloor119905rfloor)])

= E (E [(1 + 119884 (lfloor119905rfloor)2

119867) | 119903 (lfloor119905rfloor)])



= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

P (119903 (119905) = 119894 | 119903 (lfloor119905rfloor) = 119894)

= E (1 + 119884 (lfloor119905rfloor)2

119867)sum

119894isinS

119868119903(lfloor119905rfloor)=119894

times sum

119895 = 119894


= E (1 + 119884 (lfloor119905rfloor)2


119894isinS

119868119903(lfloor119905rfloor)=119894


119867)

(39)

where = 119873[1 +max1le119894le119873


gives

1198693(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867

+2Δminus121198711003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

+ 119902int

119905

0

8119890120579119904Δ12119871E (1 + 119884 (lfloor119904rfloor)

2

119867) 119889119904

(40)


1198694(119905)

= Eint119905

0

119890120579119904120582119903(119904)

times Δ121003817100381710038171003817119892 (119884

119899(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)1003817100381710038171003817119892 (119884

119899(119904) 119903 (119904)))


10038171003817100381710038172

HS 119889119904


le 119902Eint119905

0

119890120579119904Δ12119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904 + 119902 (1 + Δ

minus12)

times int

119905

0

1198901205791199048119871 (1 +

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119868119903(119904) = 119903(lfloor119904rfloor)

119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

le 119902Δ12119871Eint119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 16119902Δ12119871int

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(41)


have

E1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867

le 2E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 6 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867


119867+ 2

(42)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 1199021205882

119899Δ12119871

+24119902Δ12 119871 + 119902Δ

12119871] 119889119904

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus

120583

2+ 119902Δ12(2 + 120588

2

119899119871) + 119902Δ

12119871]

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + 24119902Δ

12 119871E

times int

119905

0

1198901205791199041003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ (4119902Δminus12119871 + 2119902119871)Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(43)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583

+ 31199021205882

119899Δ12119871 + 24119902Δ

12 119871

+3119902Δ12119871 + 4119902Δ

12] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+4119902Δ12119871 + 24119902Δ

12 119871]1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ 6119902119871Δ12(1205882

119899+ 2119871)Eint

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119889119904

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583 + 3119902120588

2

119899Δ12119871

+ 24119902Δ12 119871 + 3119902Δ

12119871 + 4119902Δ

12

+6119902119871Δ12(1205882

119899+ 2119871)] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+ 4119902Δ12119871 + 24119902Δ

12 119871

+6119902119871Δ12(1205882

119899+ 2119871)]

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(44)

Let 120579 = (4120572119901+120583)4119902 for Δ lt ((4120572119901+120583)(8119902+41199021205882119899119871+4119902119871+

24119902 + 6119902119871(1205882

119899+ 2119871)))

2 then

119901119890120579119905E (119884 (119905)

2

119867) le 119902119909

2

119867+ int

119905

0

119890120579119904[1205721+4120572119901 + 120583

2] 119889119904

(45)

That is

sup119905ge0

E (119884 (119905)2

119867) le 119862 (46)


2119871) ((2120572119901 + 120583)(4119902 + 2119902119871 + 21199021205882

119899119871 + 12119902119871))

2 then

lim119905rarrinfin

E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119910119894(119905)10038171003817100381710038171003817

2

119867= 0 119906119899119894119891119900119903119898119897119910 119891119900119903 119909 119910 isin 119880

(47)




(119884119909(119905) minus 119884

119910(119905)) minus (119884



= (119884119909(119905) minus 119884


119910(119905) minus 119884




+ int

119905

lfloor119905rfloor




+ int

119905

lfloor119905rfloor




(48)


E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884



10038171003817100381710038172

119867

le 3 (1205882

119899+ 2119871)ΔE


119910(lfloor119905rfloor)10038171003817100381710038172

119867

(49)

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172

119867

= E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884





10038171003817100381710038172

119867

le (1 + 2)E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))



10038171003817100381710038172

119867

+ (1 +1


119910(lfloor119905rfloor)10038171003817100381710038172

119867

le 9 (1205882

119899+ 2119871)ΔE



10038171003817100381710038172

119867



10038171003817100381710038172

119867

(50)

If Δ lt 118(1205882119899+ 2119871) then

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172



10038171003817100381710038172

119867 (51)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 120582119894

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

+ Eint119905

0

119890120579119904120582119903(119904)1205791003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ 2 ⟨119884119909(119904) minus 119884

119910(119904)

119860119884119909(119904) minus 119884

119910(119904)⟩119867

+ 2 ⟨119884119909(119904) minus 119884




119867

+1003817100381710038171003817119892119899 (119884



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ 119902120579Eint

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2 ⟨119884

119909(119904) minus 119884




119867



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(52)


⟨119884119909(119905) minus 119884




119867

= ⟨119884119909(119905) minus 119884

119910(119905) 119891 (119884

119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))⟩

119867

+ ⟨119884119909(119905) minus 119884

119910(119905) (119890


times (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

+ ⟨119884119909(119905) minus 119884




minus (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

(53)




10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905))

10038171003817100381710038172

HS


+ (1 + Δminus12)1003817100381710038171003817(119892 (119884

119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905)))



10038171003817100381710038172

HS

(54)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ (119902120579 minus 2120572119901 minus 120583)E

times int

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884

119910(119904) (119890


times (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904))) ⟩

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884




minus (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904)))⟩

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δ12 1003817100381710038171003817119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904)))



119903 (lfloor119904rfloor) ) )10038171003817100381710038172

HS 119889119904

= 1198661(119905) + 119866

2(119905) + 119866

3(119905) + 119866

4(119905)

(55)


1198662(119905)

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ321205882

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(56)


1198663(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867


)10038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884



minus (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904 + 119866

3(119905)

(57)

By (39) we have

1198663(119905)

le 2119902Δminus12

Eint119905

0

119890120579119904[1003817100381710038171003817119891 (119884



10038171003817100381710038172

119867

+1003817100381710038171003817(119891 (119884

119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867]


119889119904

le 4119902Δminus12119871Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867


119889119904

le 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(58)


1198663(119905) le 119902Δ

12Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(59)



1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ


119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)


119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+





lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)



that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587



lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587



119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ



119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


(65)



119895and


⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)


119873+120575119873) minus Γ is



119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)


1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)


Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)





12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




le 119902Eint119905

0

119890120579119904Δ12119871 (1 +

1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904 + 119902 (1 + Δ

minus12)

times int

119905

0

1198901205791199048119871 (1 +

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119868119903(119904) = 119903(lfloor119904rfloor)

119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

le 119902Δ12119871Eint119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 16119902Δ12119871int

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867) 119889119904

+ 2119871119902 (1 + Δminus12)int

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(41)


have

E1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867

le 2E1003817100381710038171003817119884119899(119905) minus 119884

119899(lfloor119905rfloor)10038171003817100381710038172

119867+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867

le 6 (1205882

119899+ 2119871)Δ (1 + E

1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867)

+ 2E1003817100381710038171003817119884119899(lfloor119905rfloor)10038171003817100381710038172

119867


119867+ 2

(42)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 1199021205882

119899Δ12119871

+24119902Δ12 119871 + 119902Δ

12119871] 119889119904

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus

120583

2+ 119902Δ12(2 + 120588

2

119899119871) + 119902Δ

12119871]

times1003817100381710038171003817119884119899(119904)10038171003817100381710038172

119867119889119904 + 24119902Δ

12 119871E

times int

119905

0

1198901205791199041003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ (4119902Δminus12119871 + 2119902119871)Eint

119905

0

1198901205791199041003817100381710038171003817119884119899(119904) minus 119884

119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(43)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119899(119905)10038171003817100381710038172

119867)

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583

+ 31199021205882

119899Δ12119871 + 24119902Δ

12 119871

+3119902Δ12119871 + 4119902Δ

12] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+4119902Δ12119871 + 24119902Δ

12 119871]1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

+ 6119902119871Δ12(1205882

119899+ 2119871)Eint

119905

0

119890120579119904(1 +1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867) 119889119904

le 1199021199092

119867+ int

119905

0

119890120579119904[1205721+ 2119902120579 minus 4120572119901 minus 120583 + 3119902120588

2

119899Δ12119871

+ 24119902Δ12 119871 + 3119902Δ

12119871 + 4119902Δ

12

+6119902119871Δ12(1205882

119899+ 2119871)] 119889119904

+ int

119905

0

119890120579119904[4119902120579 minus 8120572119901 minus 2120583 + 4119902Δ

12(2 + 120588

2

119899119871)

+ 4119902Δ12119871 + 24119902Δ

12 119871

+6119902119871Δ12(1205882

119899+ 2119871)]

1003817100381710038171003817119884119899(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(44)

Let 120579 = (4120572119901+120583)4119902 for Δ lt ((4120572119901+120583)(8119902+41199021205882119899119871+4119902119871+

24119902 + 6119902119871(1205882

119899+ 2119871)))

2 then

119901119890120579119905E (119884 (119905)

2

119867) le 119902119909

2

119867+ int

119905

0

119890120579119904[1205721+4120572119901 + 120583

2] 119889119904

(45)

That is

sup119905ge0

E (119884 (119905)2

119867) le 119862 (46)


2119871) ((2120572119901 + 120583)(4119902 + 2119902119871 + 21199021205882

119899119871 + 12119902119871))

2 then

lim119905rarrinfin

E10038171003817100381710038171003817119884119899119909119894(119905) minus 119884

119899119910119894(119905)10038171003817100381710038171003817

2

119867= 0 119906119899119894119891119900119903119898119897119910 119891119900119903 119909 119910 isin 119880

(47)




(119884119909(119905) minus 119884

119910(119905)) minus (119884



= (119884119909(119905) minus 119884


119910(119905) minus 119884




+ int

119905

lfloor119905rfloor




+ int

119905

lfloor119905rfloor




(48)


E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884



10038171003817100381710038172

119867

le 3 (1205882

119899+ 2119871)ΔE


119910(lfloor119905rfloor)10038171003817100381710038172

119867

(49)

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172

119867

= E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884





10038171003817100381710038172

119867

le (1 + 2)E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))



10038171003817100381710038172

119867

+ (1 +1


119910(lfloor119905rfloor)10038171003817100381710038172

119867

le 9 (1205882

119899+ 2119871)ΔE



10038171003817100381710038172

119867



10038171003817100381710038172

119867

(50)

If Δ lt 118(1205882119899+ 2119871) then

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172



10038171003817100381710038172

119867 (51)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 120582119894

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

+ Eint119905

0

119890120579119904120582119903(119904)1205791003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ 2 ⟨119884119909(119904) minus 119884

119910(119904)

119860119884119909(119904) minus 119884

119910(119904)⟩119867

+ 2 ⟨119884119909(119904) minus 119884




119867

+1003817100381710038171003817119892119899 (119884



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ 119902120579Eint

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2 ⟨119884

119909(119904) minus 119884




119867



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(52)


⟨119884119909(119905) minus 119884




119867

= ⟨119884119909(119905) minus 119884

119910(119905) 119891 (119884

119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))⟩

119867

+ ⟨119884119909(119905) minus 119884

119910(119905) (119890


times (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

+ ⟨119884119909(119905) minus 119884




minus (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

(53)




10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905))

10038171003817100381710038172

HS


+ (1 + Δminus12)1003817100381710038171003817(119892 (119884

119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905)))



10038171003817100381710038172

HS

(54)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ (119902120579 minus 2120572119901 minus 120583)E

times int

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884

119910(119904) (119890


times (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904))) ⟩

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884




minus (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904)))⟩

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δ12 1003817100381710038171003817119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904)))



119903 (lfloor119904rfloor) ) )10038171003817100381710038172

HS 119889119904

= 1198661(119905) + 119866

2(119905) + 119866

3(119905) + 119866

4(119905)

(55)


1198662(119905)

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ321205882

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(56)


1198663(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867


)10038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884



minus (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904 + 119866

3(119905)

(57)

By (39) we have

1198663(119905)

le 2119902Δminus12

Eint119905

0

119890120579119904[1003817100381710038171003817119891 (119884



10038171003817100381710038172

119867

+1003817100381710038171003817(119891 (119884

119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867]


119889119904

le 4119902Δminus12119871Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867


119889119904

le 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(58)


1198663(119905) le 119902Δ

12Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(59)



1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ


119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)


119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+





lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)



that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587



lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587



119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ



119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


(65)



119895and


⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)


119873+120575119873) minus Γ is



119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)


1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)


Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)





12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





(119884119909(119905) minus 119884

119910(119905)) minus (119884



= (119884119909(119905) minus 119884


119910(119905) minus 119884




+ int

119905

lfloor119905rfloor




+ int

119905

lfloor119905rfloor




(48)


E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884



10038171003817100381710038172

119867

le 3 (1205882

119899+ 2119871)ΔE


119910(lfloor119905rfloor)10038171003817100381710038172

119867

(49)

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172

119867

= E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905)) minus (119884





10038171003817100381710038172

119867

le (1 + 2)E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))



10038171003817100381710038172

119867

+ (1 +1


119910(lfloor119905rfloor)10038171003817100381710038172

119867

le 9 (1205882

119899+ 2119871)ΔE



10038171003817100381710038172

119867



10038171003817100381710038172

119867

(50)

If Δ lt 118(1205882119899+ 2119871) then

E1003817100381710038171003817(119884119909(119905) minus 119884

119910(119905))10038171003817100381710038172



10038171003817100381710038172

119867 (51)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 120582119894

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

+ Eint119905

0

119890120579119904120582119903(119904)1205791003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ 2 ⟨119884119909(119904) minus 119884

119910(119904)

119860119884119909(119904) minus 119884

119910(119904)⟩119867

+ 2 ⟨119884119909(119904) minus 119884




119867

+1003817100381710038171003817119892119899 (119884



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ 119902120579Eint

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

minus 2120572119901Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)2 ⟨119884

119909(119904) minus 119884




119867



10038171003817100381710038172

HS 119889119904

+ Eint119905

0

119890120579119904

119873

sum

119897=1

120574119903(119904)119897120582119897

1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(52)


⟨119884119909(119905) minus 119884




119867

= ⟨119884119909(119905) minus 119884

119910(119905) 119891 (119884

119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))⟩

119867

+ ⟨119884119909(119905) minus 119884

119910(119905) (119890


times (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

+ ⟨119884119909(119905) minus 119884




minus (119891 (119884119909(119905) 119903 (119905)) minus 119891 (119884

119910(119905) 119903 (119905))) ⟩

119867

(53)




10038171003817100381710038172

HS

le (1 + Δ12)1003817100381710038171003817119892 (119884119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905))

10038171003817100381710038172

HS


+ (1 + Δminus12)1003817100381710038171003817(119892 (119884

119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905)))



10038171003817100381710038172

HS

(54)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ (119902120579 minus 2120572119901 minus 120583)E

times int

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884

119910(119904) (119890


times (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904))) ⟩

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884




minus (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904)))⟩

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δ12 1003817100381710038171003817119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904)))



119903 (lfloor119904rfloor) ) )10038171003817100381710038172

HS 119889119904

= 1198661(119905) + 119866

2(119905) + 119866

3(119905) + 119866

4(119905)

(55)


1198662(119905)

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ321205882

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(56)


1198663(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867


)10038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884



minus (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904 + 119866

3(119905)

(57)

By (39) we have

1198663(119905)

le 2119902Δminus12

Eint119905

0

119890120579119904[1003817100381710038171003817119891 (119884



10038171003817100381710038172

119867

+1003817100381710038171003817(119891 (119884

119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867]


119889119904

le 4119902Δminus12119871Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867


119889119904

le 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(58)


1198663(119905) le 119902Δ

12Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(59)



1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ


119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)


119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+





lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)



that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587



lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587



119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ



119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


(65)



119895and


⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)


119873+120575119873) minus Γ is



119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)


1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)


Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)





12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014




+ (1 + Δminus12)1003817100381710038171003817(119892 (119884

119909(119905) 119903 (119905)) minus 119892 (119884

119910(119905) 119903 (119905)))



10038171003817100381710038172

HS

(54)


119890120579119905E (120582119903(119905)

1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ (119902120579 minus 2120572119901 minus 120583)E

times int

119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884

119910(119904) (119890


times (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904))) ⟩

119867119889119904

+ 2Eint119905

0

119890120579119904120582119903(119904)

times ⟨119884119909(119904) minus 119884




minus (119891 (119884119909(119904) 119903 (119904))

minus119891 (119884119910(119904) 119903 (119904)))⟩

119867119889119904

+ Eint119905

0

119890120579119904120582119903(119904)Δ12 1003817100381710038171003817119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904))

10038171003817100381710038172

HS

+ (1 + Δminus12)

times1003817100381710038171003817(119892 (119884

119909(119904) 119903 (119904))

minus119892 (119884119910(119904) 119903 (119904)))



119903 (lfloor119904rfloor) ) )10038171003817100381710038172

HS 119889119904

= 1198661(119905) + 119866

2(119905) + 119866

3(119905) + 119866

4(119905)

(55)


1198662(119905)

le Eint119905

0

119890120579119904120582119903(119904)Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ Δminus12 10038171003817100381710038171003817


minus 1)

times (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le Eint119905

0

119890120579119904120582119903(119904)(Δ12+ Δ321205882

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

le 119902Eint119905

0

119890120579119904Δ12(1 + 120588

2

119899119871)1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

(56)


1198663(119905)

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867


)10038171003817100381710038171003817

2

times1003817100381710038171003817119891 (119884



minus (119891 (119884119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867 119889119904

le 119902Eint119905

0

119890120579119904Δ121003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904 + 119866

3(119905)

(57)

By (39) we have

1198663(119905)

le 2119902Δminus12

Eint119905

0

119890120579119904[1003817100381710038171003817119891 (119884



10038171003817100381710038172

119867

+1003817100381710038171003817(119891 (119884

119909(119904) 119903 (119904))

minus119884119910(119904) 119903 (119904))

10038171003817100381710038172

119867]


119889119904

le 4119902Δminus12119871Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867


119889119904

le 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(58)


1198663(119905) le 119902Δ

12Eint119905

0

1198901205791199041003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ 4119902119871Δ12

Eint119905

0


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(59)



1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ


119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)


119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+





lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)



that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587



lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587



119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ



119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


(65)



119895and


⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)


119873+120575119873) minus Γ is



119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)


1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)


Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)





12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014





1198664(119905) le 119902Eint

119905

0

119890120579119904Δ121198711003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867

+ (1 + Δminus12) 4119871Δ


119910(lfloor119904rfloor)10038171003817100381710038172

119867 119889119904

(60)

Hence we have

119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

+ Eint119905

0

119890120579119904[119902120579 minus 2120572119901 minus 120583 + 119902 (1 + 120588

2

119899119871)Δ12

+119902Δ12+ 119902119871Δ

12]1003817100381710038171003817119884119909(119904) minus 119884

119910(119904)10038171003817100381710038172

119867119889119904

+ Eint119905

0

119890120579119904[4119902119871Δ

12+ (Δ12+ Δ) 4119902119871]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(61)


119901119890120579119905E (1003817100381710038171003817119884119909(119905) minus 119884

119910(119905)10038171003817100381710038172

119867)

le 1199021003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867+ Eint

119905

0

119890120579119904[2119902120579 minus 4120572119901 minus 2120583

+ 4119902Δ12+ 2119902119871Δ

12

+21199021205882

119899119871Δ12+ 12119902119871Δ

12]


119910(lfloor119904rfloor)10038171003817100381710038172

119867119889119904

(62)

Let 120579 = (2120572119901+120583)2119902 for Δ lt ((2120572119901+120583)(4119902+ 2119902119871+21199021205882119899119871+





lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ

119896((119910 119894) sdot times sdot)) = 0 (63)



that

lim119896rarrinfin

119889L (P119899Δ

119896((0 1) sdot times sdot) 120587



lim119896rarrinfin

119889L (P119899Δ

119896((119909 119894) sdot times sdot) 120587



119889L (P119899Δ

119896((119909 119894) sdot times sdot) P

119899Δ



119889L (P119899Δ

119896((0 1) sdot times sdot) 120587


(65)



119895and


⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867le 120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(66)


119873+120575119873) minus Γ is



119899 119894 isin S 119885

119899(119909119894)

(119896Δ)119896ge0


119899times S)


1 1205822 120582

119873)119879gt 0 such that

(1199021 1199022 119902

119873)119879= A(120582

1 1205822 120582

119873)119879gt 0

Set 120583 = min1le119895le119873

119902119895 by (66) we have

2120582119895⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩

119867

+ 120582119895

1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

le 2120582119895120573119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+ 120575119895120582119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867+

119873

sum

119897=1

120574119895119897120582119897

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

= (2120582119895120573119895+ 120575119895120582119895+

119873

sum

119897=1

120574119895119897120582119897)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

= minus119902119895

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867le minus1205831003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(67)


Example 14 Consider

119889119883 (119905 120585) = [1205972

1205971205852119883(119905 120585) + 119861 (119903 (119905))119883 (119905 120585)] 119889119905

+ 119892 (119883 (119905 120585) 119903 (119905)) 119889119882 (119905) 0 lt 120585 lt 120587 119905 ge 0

(68)





12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014







12 sin(119896120585) 120585 isin[0 120587] 119860119890


1003817100381710038171003817100381711989011990511986011990910038171003817100381710038171003817

2

119867=

infin

sum

119894=1

119890minus21198962

119905⟨119909 119890119894⟩2

119867le 119890minus2119905

infin

sum

119894=1

⟨119909 119890119894⟩2

119867 (69)




Γ = (minus2 2

1 minus1)

119861 (1) = 1198611= (minus03 minus01

minus02 minus02)

119861 (2) = 1198612= (minus04 minus02

minus03 minus02)

(71)

Then 120582max(119861119879

11198611) = 01706 120582max(119861

119879

21198612) = 03286


1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)10038171003817100381710038172

HS le 1205751198951003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867 (72)

where 1205751= 01 120575

2= 006

Defining 119891(119909 119895) = 119861(119895)119909 then

1003817100381710038171003817119891 (119909 119895) minus 119891 (119910 119895)10038171003817100381710038172

HS or1003817100381710038171003817119892 (119909 119895) minus 119892 (119910 119895)

10038171003817100381710038172

HS

le (120582max (119861119879

119895119861119895) or 120575119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867lt 033

1003817100381710038171003817119909 minus 11991010038171003817100381710038172

119867

⟨119909 minus 119910 119891 (119909 119895) minus 119891 (119910 119895)⟩119867

le1

2⟨119909 minus 119910 (119861

119879

119895+ 119861119895) (119909 minus 119910)⟩

119867

le1

2120582max (119861

119879

119895+ 119861119895)1003817100381710038171003817119909 minus 119910

10038171003817100381710038172

119867

(73)


1205731=1

2120582max (119861

119879

1+ 1198611) = minus00919

1205732=1

2120582max (119861

119879

2+ 1198612) = minus003075

(74)



minus1 10015) (75)


Acknowledgment


References























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014










Volume 2014




Journal of











Journal of


Function Spaces






Algebra







Volume 2014



stationary in distributions of numerical solutions for stochastic partial differential ... ·...

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