on nonlinear discrete weakly singular inequalities and applications to volterra-type difference...

Zheng et al. Advances in Difference Equations 2013, 2013:239http://www.advancesindifferenceequations.com/content/2013/1/239

RESEARCH Open Access

On nonlinear discrete weakly singularinequalities and applications to Volterra-typedifference equationsKelong Zheng1, Hong Wang1 and Chunxiang Guo2*

*Correspondence:[email protected] of Business, SichuanUniversity, Chengdu, Sichuan610064, P.R. ChinaFull list of author information isavailable at the end of the article

AbstractSome nonlinear discrete weakly singular inequalities, which generalize some knownresults are discussed. Under suitable parameters, prior bounds on solutions tononlinear Volterra-type difference equations are obtained. Two examples arepresented to show the applications of our results in boundedness and uniqueness ofsolutions of difference equations, respectively.MSC: 34A34; 45J05

Keywords: discrete integral inequalities; weakly singular; Volterra differenceequation; multi-step

1 IntroductionAs an important branch of the Gronwall-Bellman inequality, various weakly singular inte-gral inequalities and their discrete analogues have attracted more and more attention andplay a fundamental role in the study of the theory of singular differential equations andintegral equations (for example, see [–] and []). When many problems such as the be-havior, the perturbation and the numerical treatment of the solution for the Volterra typeweakly singular integral equation are studied, they often involve some certain integral in-equalities and discrete inequalities. Dixon andMcKee [] investigated the convergence ofdiscretization methods for the Volterra integral and integro-differential equations, usingthe following inequalities

x(t)≤ ψ(t) +M∫ t

x(s)(t – s)α

ds, m > , < α <

and

xi ≤ ψi +Mh–α

i–∑j=

xj(i – j)α

, i = , , . . . ,N ,n > ,Nh = T . (.)

As for the second kind Abel-Volterra singular integral equation, Beesack [] also dis-cussed the inequalities

x(t)≤ ψ(t) +M∫ t

sσx(s)(tβ – sβ )α

ds

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and

xi ≤ ψi +Mh+σ–αβ

i–∑j=

jσxj(iβ – jβ )α

, (.)

where < α < , β ≥ and σ ≥ β – . It should be noted that the above mentioned resultsare based on the assumption that themesh size is uniformly bounded. Unfortunately, suchtechnique leads to weak results, which do not reflect the true order of consistency of thescheme and may not even yield a convergence result at all. To avoid the shortcoming ofthese results, Norbuy and Stuart [, ] presented some new inequalities to describe thenumerical method for weakly singular Volterra integral equations, which is based on avariable mesh.Another purpose of studying weakly singular integral inequalities and their discrete ver-

sions is related to the theory of the parabolic equation (for example, see [–] and thereferences therein). Consider the weakly singular integral inequality

x(t)≤ a(t) +∫ t

(t – s)β–b(s)ω

(x(s)

)ds, < β < ,

and the corresponding discrete inequality of multi-step,

xn ≤ an +n–∑k=

(tn – tk)β–τkbkω(xk), (.)

where t = , τk = tk+ – tk , supk∈N τk = τ and limt→∞ tk = ∞. Henry [] and Slodicka []discussed the linear case ω(ξ ) = ξ of the two inequalities above and obtained the estimateof the solution. Furthermore, Medveď [, ] studied the general nonlinear case. How-ever, his results are based on the ‘(q) condition’: () ω satisfies e–qt[ω(u)]q ≤ R(t)ω(e–qt)uq;() there exists c > such that ane–τ tn ≤ c. Later, Yang and Ma [] generalized the resultsto a new case

xαn ≤ an +

n–∑k=

(tn – tk)β–τkbkxλk . (.)

In this paper, we are concerned with the following weakly singular inequality on a vari-able mesh

xμn ≤ an +

n–∑k=

(tαn – tαk

)β–tγ–k τkbkxλk , (.)

where μ > , λ > , and < β ,γ ≤ . Our proposed method can avoid the so-called‘q-condition,’ and under a new assumption, the more concise results are derived. More-over, to show the application of the more general inequality to a Volterra-type differenceequation, some examples are presented.

2 PreliminariesIn what follows, we denote R by the set of real numbers. Let R+ = (,∞) and N ={, , , . . .}. C(X,Y ) denotes the collection of continuous functions from the set X to theset Y . As usual, the empty sum is taken to be .



Lemma . (Discrete Jensen inequality, see []) Let A,A, . . . ,An be nonnegative realnumbers, and let r > be a real number. Then

(A +A + · · · +An)r ≤ nr–(Ar +Ar

+ · · · +Arn).

Lemma . (Discrete Hölder inequality, see []) Let ai, bi (i = , , . . . ,n) be nonnegativereal numbers, and p, q be positive numbers such that

p +q = (or p = , q = ∞). Then

n∑i=

aibi ≤( n∑

i=api

)/p( n∑i=

bqi

)/q

.

Definition . (See []) Let [x, y, z] be an ordered parameter group of nonnegative realnumbers. The group is said to belong to the first-class distribution and is denoted by[x, y, z] ∈ I if conditions x ∈ (, ], y ∈ (/, ), z ≥ / – y and z > y are satisfied; it issaid to belong to the second-class distribution and is denoted by [x, y, z] ∈ II if conditionsx ∈ (, ], y ∈ (, /] and z > –y

–y are satisfied.

Lemma . (See []) Let α, β , γ and p be positive constants. Then

∫ t

(tα – sα

)p(β–)sp(γ–) ds = tθ

αB[p(γ – ) +

α,p(β – ) +

], t ∈ R+,

where B[ξ ,η] =∫ s

ξ–( – s)η– ds (Re ξ > , Reη > ) is the well-known B-function andθ = p[α(β – ) + γ – ] + .

Lemma . Suppose that ω(u) ∈ C(R+,R+) is nondecreasing with ω(u) > for u > . Letan, cn be nonnegative and nondecreasing in n. If yn is nonnegative such that

yn ≤ an + cnn–∑k=

bkω(yk), n ∈N.

Then

yn ≤ –

[ (an) + cn

n–∑k=

bk

], ≤ n≤ M,

where (v) =∫ vv

ω(s) ds, v ≥ v, – is the inverse function of , and M is defined by

M = sup

{i : (ai) + ci

i–∑k=

bk ∈ Dom( –)}.

Remark . Martyniuk et al. [] studied the inequality yn ≤ c +∑n–

k= bkω(yk), n ∈ N.Obviously, our result is a more general case of the nonlinear difference inequality.



Lemma . If [α,β ,γ ] ∈ I , then p = β; if [α,β ,γ ] ∈ II , then p = +β

+β . Furthermore, forsufficiently small τk , we have

n–∑k=

(tαn – tαk

)pi(β–)tpi(γ–)k τk

≤∫ tn

(tαn – sα

)pi(β–)spi(γ–) ds=tθinαB[pi(γ – ) +

α,pi(β – ) +

](.)

for i = , , where θi = pi[α(β – ) + γ – ] + .

Proof By the definition of θi, θi ≥ . For its proof, see []. On one hand, when [α,β ,γ ] ∈ I ,it follows from Definition . that γ > β . On the other hand, when [α,β ,γ ] ∈ II , that is,α ∈ (, ], β ∈ (, ], we have that

γ > – β

– β >

≥ β (.)

holds, since

– β

– β >

⇔ – β > – β ⇔ – β > . (.)

According to the condition β ∈ (, ], – β > holds, which yields (.) holds directly.Thus, when [α,β ,γ ] ∈ I or [α,β ,γ ] ∈ II , we have

γ > β .

Next, we consider the integrated function in the B-function in (.).

B[pi(γ – ) +

α,pi(β – ) +

]=

∫

( – s)pi(β–)+–s

pi(γ–)+α – ds

:=∫

( – s)n–sn– ds. (.)

Denote f (s) := ( – s)n–sn– for s ∈ (, ), where n = pi(γ–)+α

and n = pi(β – ) + . Ifn = n, then f (s) is symmetric about s =

. In fact, because of α ∈ (, ], we get

> n =pi(γ – ) +

α> pi(γ – ) + > pi(β – ) + = n > ,

i.e.,

> n – > n – > –. (.)

Moreover, we can obtain the zero-point of f ′(s) as follows

s =n –

n + n – <. (.)



Therefore, the function f (s) is decreasing on the interval (, s] while increasing sharplyon the interval [s, ). So, for some given sufficiently small τk , by the properties of the left-rectangle integral formula, we have

n–∑k=

(α – tαk

)n–tn–k τk ≤∫

( – sα

)n–sn– ds= B

[pi(γ – ) +

α,pi(β – ) +

], (.)

where = t < t < · · · < tn = .As for the general interval (, tn], we can easily obtain the corresponding result (.),

which is similar to (.). We omit the details here. �

3 Main resultTo state our result conveniently, we fist introduce the following function

(u) =∫ u

u

sλμ

ds, u≥ u > ,μ > ,λ > .

Thus, we have

(u) =

⎧⎨⎩ln u

u , μ = λ,u > ,μ

μ–λu

μ–λμ , μ = λ,u =

and

–(ξ ) =

⎧⎨⎩u exp ξ ,(μ–λ

μξ )

μμ–λ .

Denote an = max≤k≤n,k∈N ak , where τ = max≤k≤n–,k∈N τk .

Theorem . Suppose that an, bn are nonnegative functions for n ∈ N. Let μ > , λ > ,μ = λ. If xn is nonnegative function such that (.), then for some sufficiently small τk :() [α,β ,γ ] ∈ I , letting p =

β, q =

–β, we have

xn ≤[(

β

–β a

–β

n)μ–λ

μ +μ – λ

μ

β–β τ

(tθnα

) β–β

Bβ

–β

n–∑k=

b

–β

k

] –βμ–λ

(.)

for n ∈N (μ > λ) or ≤ n≤ N (μ < λ), where

θ = p[α(β – ) + γ –

]+ =

α(β – ) + γ – β

+ ,

B = B[

γ – + β

αβ,β –

β

],



and N is the largest integer number such that,

μ

μ – λ

(

β–β a

–β

n)μ–λ

μ + β

–β τ

(tθnα

) β–β

Bβ

–β

n–∑k=

b

–β

k > ; (.)

() [α,β ,γ ] ∈ II , letting p = +β

+β , q =+β

β, we have

xn ≤[(

+β

β a+β

β

n)μ–λ

μ +μ – λ

μ

+ββ τ

(tθnα

) +ββ

B+β

β

n–∑k=

b+β

β

k

] +ββ(μ–λ)

(.)

for n ∈N (μ > λ) or ≤ n≤ N (μ < λ), where

θ = + β

+ β[α(β – ) + γ –

]+ ,

B = B[γ ( + β) – β

α( + β),β

+ β

],

and N is the largest integer number such that

μ

μ – λ

(

+ββ a

+ββ

n)μ–λ

μ + +β

β τ

(tθnα

) +ββ

(B)+β

β

n–∑k=

b+β

β

k > . (.)

Proof By the definition of an, obviously, an is nonnegative and nondecreasing, that is,an ≥ an. It follows from (.) that

xμn ≤ an +

n–∑k=

(tαn – tαk

)β–tγ–k τkbkxλk , (.)

where τk is the variable time step.Next, for convenience, we take the indices pi, qi. Denote that if [α,β ,γ ] ∈ I , let p =

β

and q = –β

; if [α,β ,γ ] ∈ II , let p = +β

+β , and let q = +β

β. Then

pi+

qi= holds for

i = , .Using Lemma . with indices pi, qi in (.), we have

xμn ≤ an +

n–∑k=

(tαn – tαk

)β–tγ–k τpik τ

qik bkxλ

k

≤ an + τqi

n–∑k=

(tαn – tαk

)β–tγ–k τpik bkxλ

k

≤ an + τqi

[ n–∑k=

(tαn – tαk


] pi

( n–∑k=

bqik xqiλk

) qi

. (.)

By Lemma ., the inequality above can be rewritten as

xqiμn ≤ qi–aqin + qi–τ[ n–∑

k=

(tαn – tαk


] qipi

( n–∑k=

bqik xqiλk

). (.)



By Lemma .,

n–∑k=

(tαn – tαk

)pi(β–)tpi(γ–)k τk ≤∫ tn

(tαn – sα

)pi(β–)spi(γ–) ds=tθinαB[pi(γ – ) +

α,pi(β – ) +

]=tθinαBi, (.)

where Bi = B[ pi(γ–)+α

,pi(β – ) + ], we get

xqiμn ≤ qi–aqin + qi–τ(tθinα

) qipi(Bi)

qipi

( n–∑k=

bqik xqiλk

), (.)

and θi is given in Lemma . for i = , .

Let yn = xqiμn . Then xqiλk = xqiμ λ

μ

k = yλμ

k . It follows from (.) that

yn ≤ qi–aqin + qi–τ(tθinα

) qipi(Bi)

qipi

( n–∑k=

bqik yλμ

k

). (.)

According to Lemma ., we have

yn ≤ –

[

(qi–aqin

)+ qi–τ

(tθinα

) qipi(Bi)

qipi

n–∑k=

bqik

](.)

for ≤ n≤ M, whereM is the largest integer number such that

(qi–aqin

)+ qi–τ

(tθinα

) qipi(Bi)

qipi

n–∑k=

bqik ∈ Dom( –).

() For [α,β ,γ ] ∈ I , p = β, q =

–β. By the definitions of and –, we can compute

that

(q–aqn

)=

μ

μ – λ

(

β–β a

–β

n)μ–λ

μ ,

θ = p[α(β – ) + γ –

]+ =

α(β – ) + γ – β

+ , (.)

B = B[

γ – + β

αβ,β –

β

].

Then

yn ≤ –

[

(q–aqn

)+ q–τ

(tθnα

) qp B

qp

n–∑k=

bqk

]

≤ –

[μ

μ – λ

(

β–β a

–βn

)μ–λμ +

β–β τ

(tθnα

) β–β

Bβ

–β

n–∑k=

b

–β

k

]

≤[(

β

–β a

–βn

)μ–λμ +

μ – λ

μ

β–β τ

(tθnα

) β–β

Bβ

–β

n–∑k=

b

–β

k

] μμ–λ

. (.)



Observe the second formula in (.). To ensure that

μ

μ – λ

(

β–β a

–βn

)μ–λμ +

β–β τ

(tθnα

) β–β

Bβ

–β

n–∑k=

b

–β

k ∈ Dom( –),

we may take M = ∞ for μ > λ and M = N for μ < λ, respectively, where N is the largestinteger number such that

μ

μ – λ

(

β–β a

–βn

)μ–λμ +

β–β τ

(tθnα

) β–β

Bβ

–β

n–∑k=

b

–β

k > .

Since xn ≤ y

qμn = y

–βμ

n , substituting it into (.), we can get (.).() For [α,β ,γ ] ∈ II , p = +β

+β , q =+β

β. Similarly to the computation above, we have

(q–aqn

)=

μ

μ – λ

(

+ββ a

+ββ

n)μ–λ

μ ,

θ = + β

+ β[α(β – ) + γ –

]+ , (.)

B = B[γ ( + β) – β

α( + β),β

+ β

].

Then

yn ≤ –

[

(q–aqn

)+ q–τ

(tθnα

) qp B

qp

n–∑k=

bqk

]

≤ –

[μ

μ – λ

(

+ββ a

+ββ

n)μ–λ

μ + +β

β τ

(tθnα

) +ββ

B+β

β

n–∑k=

b+β

β

k

]

≤[(

+β

β a+β

βn

)μ–λμ +

μ – λ

μ

+ββ τ

(tθnα

) +ββ

B+β

β

n–∑k=

b+β

β

k

] μμ–λ

. (.)

Observe the second formula in (.). To ensure that

μ

μ – λ

(

+ββ a

+ββ

n)μ–λ

μ + +β

β τ

(tθnα

) +ββ

B+β

β

n–∑k=

b+β

β

k ∈ Dom( –),

we takeM = ∞ forμ > λ andM =N forμ < λ, respectively, whereN is the largest integernumber such that

μ

μ – λ

(

+ββ a

+ββ

n)μ–λ

μ + +β

β τ

(tθnα

) +ββ

B+β

β

n–∑k=

b+β

β

k > .

Because xn ≤ y

qμ

n = y–βμ

n , substituting it into (.), we can get (.). This completes theproof. �



For the case that μ = λ, let yn = xμn , we obtain from (.)

yn ≤ an +n–∑k=

(tαn – tαk

)β–tγ–k τkbkyk (.)

and derive the estimation of the upper bound as follows.

Theorem . Let an, bn be defined as in Theorem .. If yn is nonnegative function suchthat (.), then for some sufficiently small τk :() [α,β ,γ ] ∈ I , p =

β, q =

–β, we have

yn ≤[

β–β a

–βn exp

{

β–β τ

(tθnα

) β–β

Bβ

–β

n–∑k=

b

–β

k

}]–β

(.)

for n ∈N, where θ, B are defined in Theorem .;() [α,β ,γ ] ∈ II , p = +β

+β , q =+β

β, we have

yn ≤[

+ββ a

+ββ

n exp

{

+ββ τ

(tθnα

) +ββ

B+β

β

n–∑k=

b+β

β

k

}] β+β

(.)

for n ∈N, where θ, B are defined in Theorem ..

Proof In the proof of Theorem ., before we apply Lemma ., it is independent of thecomparison of μ and λ. Hence, taking yn = xμ

n = xλn in (.), we have

yqin ≤ qi–aqin + qi–τ(tθinα

) qipi B

qipii

( n–∑k=

bqik yqik

). (.)

We denote zn = yqin and get

zn ≤ qi–aqin + qi–τ(tθinα

) qipi B

qipii

( n–∑k=

bqik zk

).

By Lemma . and the definitions of and – for μ = λ, we have the following result

zn ≤ –

[

(qi–aqin

)+ qi–τ

(tθinα

) qipi B

qipii

n–∑k=

bqik

]

= –

[ln

(qi–aqin )u

+ qi–τ(tθinα

) qipi B

qipii

n–∑k=

bqik

]

= exp

{[ln

(qi–aqin

)+ qi–τ

(tθinα

) qipi B

qipii

n–∑k=

bqik

]}

= qi–aqin exp

{qi–τ

(tθinα

) qipi B

qipii

n–∑k=

bqik

}, (.)



which yields

yn ≤[qi–aqin exp

{qi–τ

(tθinα

) qipi B

qipii

n–∑k=

bqik

}] qi

. (.)

Finally, considering two situations for i = , and using paremeters α, β , γ to denote pi, qi,Bi and θi in (.), we can obtain the estimations, respectively. We omit the details here.

�

Remark . Henry [] and Slodicka [] discussed the special case of Theorem ., thatis, α = and γ = . Moreover, our result is simpler and has a wider range of applications.

Remark . Although Medveď [, ] investigated the more general nonlinear case, hisresult is under the assumption that ‘w(u) satisfies the condition (q).’ In our result, the ‘(q)condition’ is eliminated.

Letting μ = and λ = in Theorem ., we have the following corollary.

Corollary . Suppose that an, bn are nonnegative functions for n ∈ N, and un is nonde-creasing such that


(tαn – tαk

)β–tγ–k τkbkxk , (.)

then for some sufficiently small τk :() when [α,β ,γ ] ∈ I , p =

β, q =

–β, we have

xn ≤[(

β

–β a

–β

n) +

β––β τ

(tθnα

) β–β

Bβ

–β

n–∑k=

b

–β

k

]–β

(.)

for n ∈N, where θ, B are defined in Theorem .;() when [α,β ,γ ] ∈ II , p = +β

+β , q =+β

β, we have

xn ≤[(

+β

β a+β

β

n) +

+ββ τ

(tθnα

) +ββ

B+β

β

n–∑k=

b+β

β

k

] +ββ

(.)

for n ∈N, where θ, B are defined in Theorem ..

Remark . Inequality (.) is the extension of the well-knownOu-Iang-type inequality.Clearly, our inequality enriches the results for such an inequality.

4 ApplicationsIn this section, we apply our results to discuss the boundedness and uniqueness of solu-tions of a Volterra-type difference equation with a weakly singular kernel.



Example Suppose that xn satisfies the equation

xn =+

n–∑k=

(tn – tk)– t–

k τkbkxk (.)

for n ∈N. Then we get

|xn| ≤ +

n–∑k=

(tn – tk)– t–

k τkbk|xk|. (.)

Letting |xn| = yn changes (.) into

yn ≤ +

n–∑k=

(tn – tk)– t–

k τkbkyk . (.)

From (.), we can see that

an =, α = , β =

, γ =

, γ > β .

Obviously, [α,β ,γ ] ∈ I . Letting p = , q = , we have

an =, bk = ,

θ =

[(–

)+–

]+ =

, B = B

[,

].

Using Corollary ., we get

|xn| ≤[√

+ nτ tn B

[,

]] , n ∈N, (.)

which implies that xn in (.) is upper bounded.

Example Consider the linear weakly singular difference equation


(tαn – tαk

)β–tγ–k τkbkxk (.)

and

yn ≤ cn +n–∑k=

(tαn – tαk

)β–tγ–k τkbkyk , (.)

where |an – cn| < ε, ε is an arbitrary positive number, and [α,β ,γ ] ∈ I or [α,β ,γ ] ∈ II .From (.) and (.), we get

|xn – yn| ≤ |an – cn| +n–∑k=

(tαn – tαk

)β–tγ–k τkbk|xk – yk|, (.)



which is the form of inequality (.). Applying Theorem ., we have

|xn – yn| ≤[

β–β ε

–β exp

{

β–β τ

(tθnα

) β–β

Bβ

–β

n–∑k=

b

–β

k

}]–β

or

|xn – yn| ≤[

+ββ ε

+ββ exp

{

+ββ τ

(tθnα

) +ββ

B+β

β

n–∑k=

b+β

β

k

}] β+β

for n ∈ N. If an = cn, let ε → , and we obtain the uniqueness of the solution of equa-tion (.).

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsAll authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in thesequence alignment, and read and approved the final manuscript.

Author details1School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, P.R. China. 2School ofBusiness, Sichuan University, Chengdu, Sichuan 610064, P.R. China.

AcknowledgementsThe authors are very grateful to both reviewers for carefully reading this paper and their comments. The authors alsothank Professor Shengfu Deng for his valuable discussion. This work is supported by the Doctoral Program ResearchFunds of Southwest University of Science and Technology (No. 11zx7129) and the Fundamental Research Funds for theCentral Universities (No. skqy201324).

Received: 22 May 2013 Accepted: 23 July 2013 Published: 8 August 2013

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doi:10.1186/1687-1847-2013-239Cite this article as: Zheng et al.: On nonlinear discrete weakly singular inequalities and applications to Volterra-typedifference equations. Advances in Difference Equations 2013 2013:239.


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