research article on the positive operator solutions to an...
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Hindawi Publishing CorporationISRNMathematical AnalysisVolume 2013 Article ID 145606 4 pageshttpdxdoiorg1011552013145606
Research ArticleOn the Positive Operator Solutions to an Operator Equation
Kai-Fan Yang1 Fang-An Deng1 and Hong-Ke Du2
1 School of Mathematics and Computer Science Shaanxi University of Technology Shaanxi 723001 China2 College of Mathematics and Information Science Shaanxi Normal University Xirsquoan 710062 China
Correspondence should be addressed to Kai-Fan Yang ykf201126com
Received 3 May 2013 Accepted 9 June 2013
Academic Editors M Lindstrom and K A Lurie
Copyright copy 2013 Kai-Fan Yang et al 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
The necessary conditions and the sufficient condition for the existence of positive operator solutions to the operator equation119883119904
+ 119860lowast
119883minus119905
119860 = 119876 are established An iterative method for obtaining the positive operator solutions is proposed
1 Introduction
Let B(H) be the set of all bounded linear operators on theHilbert space H In this paper we consider the nonlinearoperator equation
119883119904
+ 119860lowast
119883minus119905
119860 = 119876 (1)
where 119860 119876 isin B(H) 119876 gt 0 and 119883 is an unknown operatorinB(H) Both 119904 and 119905 are positive integers
This type of equation often arises from many areas suchas dynamic programming [1] control theory [2 3] stochasticfiltering and statistics and so forth [4 5] In the recentyears for matrices (1) has been considered by many authors(see [6ndash13]) and different iterative methods for computingthe positive definite solutions to (1) are proposed in finite-dimensional space The case 119904 = 119905 = 1 has been extensivelystudied by several authors In this paper we extend the studyof the operator equation (1) from a finite-dimensional spaceto an infinite-dimensional Hilbert space We derive somenecessary conditions for the existence of positive solutions tothe operator equation (1) Moreover conditions under whichthe operator equation (1) has positive operator solutions areobtained Based on Banachrsquos fixed-point principle we obtainthe positive operator solution to the operator equation (1)First we introduce some notations and terminologies whichare useful later For 119860 isin B(H) if (119860119909 119909) ge 0 for all 119909 isin Hthen119860 is said to be a positive operator and is denoted by119860 ge
0 If119860 is a positive operator and invertible then denote119860 gt 0
For 119860 and 119861 inB(H) 119860 ge 119861 means that 119860 minus 119861 is a positiveoperator For 119860 isin B(H) 119860lowast 120596(119860) 120590(119860) and 119903(119860) denotethe adjoint the radius of numerical range the spectrum andthe spectral radius of 119860 respectively
For positive operators in B(H) the following facts arewell known
(1) If 119875 ge 119876 gt 0 then 119875minus1 le 119876minus1(2) Denote 120582max(119875) = max120582 120582 isin 120590(119875) 119875 gt 0
120582min(119875) = min120582 120582 isin 120590(119875) 119875 gt 0 Then120582min(119875)119868 le 119875 le 120582max(119875)119868
(3) If the sequence 119883119899+infin
119899=1of positive operator is mono-
tonically increasing and has upper bound that is119883119896le 119883119896+1
le 1198621 or is monotonically decreasing and
has lower bound that is 1198622le 119883119896+1
le 119883119896 then this
sequence is convergent to a positive limit operatorwhere 119862
1 1198622isinB(H) are given operators
2 Main Results and Proofs
In order to prove ourmain result we beginwith some lemmasas follows
Lemma 1 Let 119860 119861 isinB(H) If 119860 ge 119861 ge 0 then 119860 ge 119861
Lemma 2 Let 119860 and 119861 be self-adjoint operators in B(H) If119860 le 119861 then for every 119879 isinB(H) one has 119879lowast119860119879 le 119879lowast119861119879
In this section we give our main results and proofs
2 ISRNMathematical Analysis
Theorem 3 If the operator equation (1) has a positive operatorsolution 119883 then (1) 119905radic119860119876minus1119860lowast le 119883 le
119904radic119876 and (2) 119903(119860) le
(119887119905(119904 + 119905))1199052119904
(119887119904(119904 + 119905))12 where 119887 = 119876
Proof (1) If the operator equation (1) has a positive operatorsolution119883 then 0 le 119860lowast119883minus119905119860 le 119876 Hence we obtain119883119904 le 119876that is119883 le
119904radic119876 From 119860
lowast
119883minus119905
119860 le 119876 it follows that
119876minus12
119860lowast
119883minus1199052
119883minus1199052
119860119876minus12
le 119868
119883minus1199052
119860119876minus12
119876minus12
119860lowast
119883minus1199052
le 119868
119860119876minus1
119860lowast
le 119883119905
(2)
That is119883 ge119905radic119860119876minus1119860lowast
(2) From (1) we have
119883119905+119904
+ 1198831199052
119860lowast
119883minus119905
1198601198831199052
= 1198831199052
1198761198831199052
le 120582max (119876)119883119905
(3)
Then
119903(119860)2
= 1199032
(1198831199052
119860119883minus1199052
)
le100381710038171003817100381710038171198831199052
119860119883minus1199052
10038171003817100381710038171003817
2
le10038171003817100381710038171003817120582max (119876)119883
119905
minus 119883119905+11990410038171003817100381710038171003817
(4)
According to the spectral decomposition of119883 we have
120582max (119876)119883119905
minus 119883119905+119904
= int120590(119883)
(120582max (119876) 120582119905
minus 120582119905+119904
) 119889119864120582 (5)
Denote 120582max(119876) = 119876 = 119887 Then10038171003817100381710038171003817120582max (119876)119883
119905
minus 119883119905+11990410038171003817100381710038171003817le max120582isin(0
119904radic119887]
10038161003816100381610038161003816119887120582119905
minus 120582119905+11990410038161003816100381610038161003816
= (119887119905
119904 + 119905)
119905119904
(119887119904
119904 + 119905)
(6)
Therefore 119903(119860) le (119887119905(119904 + 119905))1199052119904(119887119904(119904 + 119905))12
Theorem4 If119860 is invertible and119860lowast119883minus119905119860 le 119876minus(119860119876minus1
119860lowast
)119904119905
for all 119883 isin [119905radic119860119876minus1119860lowast
119904radic119876] then (1) has a positive operator
solution
Proof Let Φ(119883) = (119876 minus 119860lowast
119883minus119905
119860)1119904 then Φ is continuous
for 119883 isin [119905radic119860119876minus1119860lowast
119904radic119876] Φ(119883) le
119904radic119876 and Φ(119883) ge
[119876 minus (119876 minus (119860119876minus1
119860lowast
)119904119905
)]1119904
= (119860119876minus1
119860lowast
)1119905 So Φ maps
[119905radic119860119876minus1119860lowast
119904radic119876] into itself FurthermoreΦ is continuous on
[119905radic119860119876minus1119860lowast
119904radic119876] because 119883 gt 0 hence Φ has a fixed point
in [119905radic119860119876minus1119860lowast
119904radic119876] that is there exists 119883 ge 0 such that
(119876 minus 119860lowast
119883minus119905
119860)1119904
= 119883 this implies that (1) has a positiveoperator solution
Theorem 5 If the operator equation (1) has a positive operatorsolution119883 then 120582min(119879
lowast
119879) le (119905(119905 + 119904))119905119904
(119904(119905 + 119904)) and119883 le
119904radic119876 when 120582min(119879
lowast
119879) = 1 where 119879 = 119876minus1199052119904119860119876minus12 and is asolution to the equation120572119905119904(1minus120572) = 120582min(119879
lowast
119879) in [119905(119905+119904) 1]
Proof We consider the sequence
1205720= 1 120572
119896+1
= 1
minus
120582min ((119876minus1199052119904
119860119876minus12
)lowast
119876minus1199052119904
119860119876minus12
)
120572119905119904
119896
119896 = 0 1 2
(7)
Let 119883 be a positive solution to (1) Then 119883119904
= 119876 minus
119860lowast
119883minus119905
119860 le 119876 = 1205720119876 that is 119883 le
119904radic1205720119876 Assuming that
119883 le119904radic120572119896119876 then
119883119904
= 119876 minus 119860lowast
119883minus119905
119860 le 119876 minus119860lowast
119876minus119905119904
119860
120572119905119904
119896
= 11987612
[119868 minus119876minus12
119860lowast
119876minus1199052119904
119876minus1199052119904
119860119876minus12
120572119905119904
119896
]11987612
le 11987612
times [
[
1 minus
120582min ((119876minus1199052119904
119860119876minus12
)lowast
119876minus1199052119904
119860119876minus12
)
120572119905119904
119896
]
]
times 11987612
= 120572119896+1119876
(8)
Hence 119883 le119904radic120572119899119876 for all 119899 = 0 1 2 It is straightforward
to check that the sequence 120572119899 is monotonically decreasing
also 120572119899 is bounded below hence 120572
119899 is convergent Denote
119879 = 119876minus1199052119904
119860119876minus12 and let lim
119899rarrinfin120572119899
= then =
1 minus 120582min(119879lowast
119879)119905119904 that is is a solution to the equation
120572119905119904
(1 minus 120572) = 120582min(119879lowast
119879) Let 119891(119909) = 119909119905119904(1 minus 119909) Then
max119909isin[01]
119891 (119909) = 119891(119905
119905 + 119904) = (
119905
119905 + 119904)
119905119904
(119904
119905 + 119904) (9)
Thus it follows that 120582min(119879lowast
119879) le (119905(119905 + 119904))119905119904
(119904(119905 + 119904))Since
120582min (119879lowast
119879) le (119905
119905 + 119904)
119905119904
(119904
119905 + 119904) (10)
Then the equation 120572119905
(1 minus 120572) = 120582min(119879lowast
119879) may have twosolutions one of these solutions is in the interval [119905(119905+119904) 1]
In order to prove that the limit of the sequence 120572119899 is
in [119905(119905 + 119904) 1] we assume that 120572119894ge 119905(119905 + 119904) (obviously 120572
0=
1 gt 119905(119905 + 119904)) then
120572119894+1
= 1 minus120582min (119879
lowast
119879)
120572119905119904
119894
ge 1 minus 120582min (119879lowast
119879) (119905 + 119904
119905)
119905119904
ge 1 minus (119905
119905 + 119904)
119905119904
(119904
119905 + 119904) (
119905 + 119904
119905)
119905119904
=119905
119905 + 119904
(11)
ISRNMathematical Analysis 3
Therefore 120572119894ge 119905(119905+119904) for each 119894 = 1 2 Since119883 le
119904radic120572119899119876
then119883 le119904radic119876 and isin [119905(119905 + 119904) 1] The proof is completed
Consider the following iterative sequence
119883119904
0= 120574119876 119883
119904
119896+1= 119876 minus 119860
lowast
119883minus119905
119896119860
119896 = 0 1 120574 gt 0
(12)
We will prove the following theorem
Theorem6 If the operator equation (1) has a positive operatorsolution 119883 then it has a maximal one 119883
119871 Moreover the
sequence 119883119894 in (12) for 120574 isin [ 1] is monotonically decreasing
and converges to119883119871 where is defined in Theorem 5
Proof Consider the iterative sequence (12) with 120574 isin [ 1]According to Theorem 4 we have 119883119904 le 119876 le 120574119876 = 119883
119904
0for
any positive operator solution119883 to (1)Suppose that119883119904
119894ge 119883119904 Then
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 ge 119876 minus 119860
lowast
119883minus119905
119860 = 119883119904
(13)
Hence119883119894ge 119883 hold for each 119894 According to the definition
of and the monotonicity of the function 119891(119909) = 119909119905119904(1 minus 119909)on [119905(119905 + 119904) 1] we have
120574119905119904
(1 minus 120574) 119868 le 119905119904
(1 minus ) 119868 = 120582min (119879lowast
119879) 119868 le 119879lowast
119879 (14)
for all 120574 isin [ 1] where 119879 = 119876minus1199052119904119860119876minus12 We compute
119883119904
1= 119876 minus 119860
lowast
(120574119876)minus119905119904
119860
= 11987612
[119868 minus 119876minus12
119860lowast
(120574119876)minus119905119904
119860119876minus12
]11987612
= 11987612
(119868 minus 120574minus119905119904
119879lowast
119879)11987612
(15)
Using inequality (14) and equality (15) we obtain1198831199041le 120574119876 =
119883119904
0Assuming that119883119904
119894le 119883119904
119894minus1 then
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 le 119876 minus 119860
lowast
119883minus119905
119894minus1119860 = 119883
119904
119894 (16)
Therefore 119883119904119894+1
le 119883119904
119894for all 119894 = 0 1 that is the sequence
119883119904
119899 is monotonically decreasing Hence 119883119904
119899 converges to
the positive operator solution119883119871to (1) Since119883119904
119871ge 119883119904 for any
positive operator solution119883 it follows that119883119871is themaximal
solution
Theorem 7 Let 120582min(119879lowast
119879) lt 120582max(119879lowast
119879) le
(119905(119905 + 119904))119905119904
(119904(119905 + 119904)) and 1205731 1205732are solutions to the equation
120573119905119904
(1 minus 120573) = 120582max(119879lowast
119879) in [0 119905(119905 + 119904)] and [119905(119905 + 119904) 1]respectively Then there are the following conclusions
(i) If 120574 isin [1205731 1205732] then the sequence 119883
119899 in (12) is mono-
tonically increasing and converges to a positive operatorsolution119883
120574isin [119904radic120574119876
119904radic119876] to (1)
(ii) If 120574 isin [ 1] then the sequence 119883119899 in (12) is monoton-
ically decreasing and converges to the maximal positiveoperator solution119883
119871isin [119904radic1205732119876119904radic119876] to (1)
(iii) If 120574 isin (1205732 ) and (119905119904)119860
2
le (1205732120582min(119876))
(119904+119905)119904then the sequence 119883
119899 in (12) converges to the unique
solution 119883119871
isin [119904radic1205732119876119904radic119876] where and 119879 are
defined in Theorem 5
Proof Consider the function119891(119909) = 119909119905119904(1minus119909) 119909 isin [0 1] Itismonotonically increasing on [0 119905(119905+119904)] andmonotonicallydecreasing on [119905(119905 + 119904) 1] and
max119909isin[01]
119891 (119909) = 119891(119905
119905 + 119904) = (
119905
119905 + 119904)
119905119904
(119904
119905 + 119904) (17)
Since 120582max(119879lowast
119879) le (119905(119905+119904))119905119904
(119904(119905+119904)) then for each 1205741 1205742
such that 0 lt 1205731le 1205741le 1205732lt le 120574
2le 1 the inequalities
120574119905119904
2(1 minus 120574
2) 119868 le 120582min (119879
lowast
119879) le 119879lowast
119879
le 120582max (119879lowast
119879) le 120574119905119904
1(1 minus 120574
1) 119868
(18)
are satisfied(i) Let 120574 isin [120573
1 1205732] We will prove that the operator
sequence 119883119899 in (12) is monotonically increasing and is
bounded aboveAccording to the fourth inequality in (18) we compute
119883119904
1= 119876 minus 119860
lowast
(120574119876)minus119905119904
119860
= 11987612
(119868 minus 120574minus119905119904
119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
= 11987612
(119868 minus 120574minus119905119904
119879lowast
119879)11987612
ge 120574119876 = 119883119904
0
(19)
Assuming that119883119904119894ge 119883119904
119894minus1 then
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 ge 119876 minus 119860
lowast
119883minus119905
119894minus1119860 = 119883
119904
119894 (20)
Therefore 119883119904119894+1
ge 119883119904
119894for all 119894 = 0 1 that is the sequence
119883119904
119899 is monotonically increasing Obviously 119883119904
0= 120574119876 le
1205732119876 le 119876 We suppose that 119883119904
119894le 119876 from the first
inequality in (18) we obtain
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 le 119876 minus 119860
lowast
(119876)minus119905119904
119860 le 119876 (21)
Hence 119883119904119899 converges to a positive operator solution119883119904
120574
to (1) Since 119883119894isin [
119904radic120574119876
119904radic119876] for all 119894 = 0 1 we have
119883120574isin [119904radic120574119876
119904radic119876]
(ii) Let 120574 isin [ 1] From the proving procession ofTheorem 6 the sequence (12) is monotonically decreasingSince 119883
0=119904radic120574119876 ge
119904radic1205732119876 and supposing that 119883119904
119894ge 1205732119876
it follows that
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 ge 119876 minus 119860
lowast
(1205732119876)minus119905119904
119860
= 11987612
(119868 minus 120573minus119905119904
2119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
ge 11987612
(119868 minus 120573minus119905119904
2120582max (119879
lowast
119879) 119868)11987612
= 1205732119876
(22)
By induction principle119883119904119894ge 1205732119876 hold for each 119894 = 0 1 2
Hence 119883119904119899 is convergent FromTheorem 6 119883119904
119899 converges
to119883119904119871such that119883
119871isin [119904radic1205732119876119904radic120574119876]
4 ISRNMathematical Analysis
(iii) Considering the sequence 119883119904119899 in (12) for 120574 isin (120573
2 )
that is 1198831199040isin (1205732119876 119876) and supposing that 119883119904
119894isin (1205732119876 119876)
then for119883119904119894+1
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860
lt 11987612
(119868 minus 120573minus119905119904
2119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
le 119876
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860
gt 11987612
(119868 minus minus119905119904
119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
ge 1205732119876
(23)
Hence119883119904119894isin (1205732119876 120574119876) for all 119894 = 0 1 2 Now we consid-
ering 119883119904119894+1
minus 119883119904
119894
1003817100381710038171003817119883119904
119894+1minus 119883119904
119894
1003817100381710038171003817 =10038171003817100381710038171003817(119876 minus 119860
lowast
119883minus119905
119894119860) minus (119876 minus 119860
lowast
119883minus119905
119894minus1119860)10038171003817100381710038171003817
le1
119904
1
(119904radic1205732120582min (119876))
119904minus1
times1199051198602
(119904radic1205732120582min (119876))
119905+1
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817
=119905
1199041198602
1
(119904radic1205732120582min (119876))
119905+119904
times1003817100381710038171003817119883119894 minus 119883119894minus1
1003817100381710038171003817
(24)
and (119905119904)1198602 le (1205732120582min(119876))
(119904+119905)119904 it follows that 119883119899 in
(12) is a Cauchy sequence in the Banach space [ 119904radic1205732119876119904radic119876]
Hence it has a limit 119883120574in [ 119904radic120573
2119876119904radic119876] and 119883
120574is a unique
solution to (1) in [ 119904radic1205732119876119904radic119876] According to Theorem 6 it
is the maximal solution that is 119883120574= 119883119871 The proof is
completed
Acknowledgments
This paper is supported by the NSF of China Grant no11171197 and by the Natural Science Foundation of ShaanxiEducational Committee Grant no 12JK0884
References
[1] W Pusz and S L Woronowicz ldquoFunctional calculus forsesquilinear forms and the purification maprdquo Reports on Math-ematical Physics vol 8 no 2 pp 159ndash170 1975
[2] J C Engwerda ldquoOn the existence of a positive definite solutionof the matrix equation119883+119860lowast119883minus120572119860 = 119868rdquo Linear Algebra and itsApplications vol 194 pp 91ndash108 1993
[3] W L Green and E W Kamen ldquoStabilizability of linear sys-tems over a commutative normed algebra with applications tospatially-distributed and parameter-dependent systemsrdquo SIAMJournal onControl andOptimization vol 23 no 1 pp 1ndash18 1985
[4] R S Bucy ldquoA priori bounds for the Riccati equationrdquo inProceedings of the 6th Berkeley Symposium on Mathematical
Statistics and Probability vol 111 of ProbabilityTheory pp 645ndash656 University of California Press Berkeley Calif USA 1972
[5] V I Hasanov Solutions and perturbation theory of nonlinearmatrix equations [PhD thesis] Sofia Bulgaria 2003
[6] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation119883+119860lowast119865(119883)119860 = 119876 solutions and perturbation theoryrdquoLinear Algebra and its Applications vol 346 pp 15ndash26 2002
[7] V I Hasanov ldquoPositive definite solutions of the matrix equa-tions 119883 plusmn 119860
lowast
119883minus119902
119860 = 119876rdquo Linear Algebra and its Applicationsvol 404 pp 166ndash182 2005
[8] A C M Ran and M C B Reurings ldquoThe symmetric linearmatrix equationrdquo Electronic Journal of Linear Algebra vol 9 pp93ndash107 2002
[9] K F Yang and H K Du ldquoStudies on the positive operatorsolutions to the operator equations 119883 + 119860
lowast
119883minus119905
119860 = 119876rdquo ActaMathematica Scientia A vol 26 no 2 pp 359ndash364 2009(Chinese)
[10] Y Yang F Duan and X Zhao ldquoOn solutions for the matrixequation 119883
119904
+ 119860lowast
119883minus119905
119860 = 119876rdquo in Proceedings of the 7thInternational Conference on Matrix Theory and Its Applicationsin China pp 21ndash24 2006
[11] Z-y Peng S M El-Sayed and X-l Zhang ldquoIterative methodsfor the extremal positive definite solution of thematrix equation119883 + 119860
lowast
119883minus120572
119860 = 119876rdquo Journal of Computational and AppliedMathematics vol 200 no 2 pp 520ndash527 2007
[12] Y Yueting ldquoThe iterative method for solving nonlinear matrixequation119883119904 +119860lowast119883minus119905119860 = 119876rdquo Applied Mathematics and Compu-tation vol 188 no 1 pp 46ndash53 2007
[13] V I Hasanov and S M El-Sayed ldquoOn the positive definitesolutions of nonlinear matrix equation 119883 + 119860
lowast
119883minus120575
119860 = 119876rdquoLinear Algebra and its Applications vol 412 no 2-3 pp 154ndash160 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Stochastic AnalysisInternational Journal of
2 ISRNMathematical Analysis
Theorem 3 If the operator equation (1) has a positive operatorsolution 119883 then (1) 119905radic119860119876minus1119860lowast le 119883 le
119904radic119876 and (2) 119903(119860) le
(119887119905(119904 + 119905))1199052119904
(119887119904(119904 + 119905))12 where 119887 = 119876
Proof (1) If the operator equation (1) has a positive operatorsolution119883 then 0 le 119860lowast119883minus119905119860 le 119876 Hence we obtain119883119904 le 119876that is119883 le
119904radic119876 From 119860
lowast
119883minus119905
119860 le 119876 it follows that
119876minus12
119860lowast
119883minus1199052
119883minus1199052
119860119876minus12
le 119868
119883minus1199052
119860119876minus12
119876minus12
119860lowast
119883minus1199052
le 119868
119860119876minus1
119860lowast
le 119883119905
(2)
That is119883 ge119905radic119860119876minus1119860lowast
(2) From (1) we have
119883119905+119904
+ 1198831199052
119860lowast
119883minus119905
1198601198831199052
= 1198831199052
1198761198831199052
le 120582max (119876)119883119905
(3)
Then
119903(119860)2
= 1199032
(1198831199052
119860119883minus1199052
)
le100381710038171003817100381710038171198831199052
119860119883minus1199052
10038171003817100381710038171003817
2
le10038171003817100381710038171003817120582max (119876)119883
119905
minus 119883119905+11990410038171003817100381710038171003817
(4)
According to the spectral decomposition of119883 we have
120582max (119876)119883119905
minus 119883119905+119904
= int120590(119883)
(120582max (119876) 120582119905
minus 120582119905+119904
) 119889119864120582 (5)
Denote 120582max(119876) = 119876 = 119887 Then10038171003817100381710038171003817120582max (119876)119883
119905
minus 119883119905+11990410038171003817100381710038171003817le max120582isin(0
119904radic119887]
10038161003816100381610038161003816119887120582119905
minus 120582119905+11990410038161003816100381610038161003816
= (119887119905
119904 + 119905)
119905119904
(119887119904
119904 + 119905)
(6)
Therefore 119903(119860) le (119887119905(119904 + 119905))1199052119904(119887119904(119904 + 119905))12
Theorem4 If119860 is invertible and119860lowast119883minus119905119860 le 119876minus(119860119876minus1
119860lowast
)119904119905
for all 119883 isin [119905radic119860119876minus1119860lowast
119904radic119876] then (1) has a positive operator
solution
Proof Let Φ(119883) = (119876 minus 119860lowast
119883minus119905
119860)1119904 then Φ is continuous
for 119883 isin [119905radic119860119876minus1119860lowast
119904radic119876] Φ(119883) le
119904radic119876 and Φ(119883) ge
[119876 minus (119876 minus (119860119876minus1
119860lowast
)119904119905
)]1119904
= (119860119876minus1
119860lowast
)1119905 So Φ maps
[119905radic119860119876minus1119860lowast
119904radic119876] into itself FurthermoreΦ is continuous on
[119905radic119860119876minus1119860lowast
119904radic119876] because 119883 gt 0 hence Φ has a fixed point
in [119905radic119860119876minus1119860lowast
119904radic119876] that is there exists 119883 ge 0 such that
(119876 minus 119860lowast
119883minus119905
119860)1119904
= 119883 this implies that (1) has a positiveoperator solution
Theorem 5 If the operator equation (1) has a positive operatorsolution119883 then 120582min(119879
lowast
119879) le (119905(119905 + 119904))119905119904
(119904(119905 + 119904)) and119883 le
119904radic119876 when 120582min(119879
lowast
119879) = 1 where 119879 = 119876minus1199052119904119860119876minus12 and is asolution to the equation120572119905119904(1minus120572) = 120582min(119879
lowast
119879) in [119905(119905+119904) 1]
Proof We consider the sequence
1205720= 1 120572
119896+1
= 1
minus
120582min ((119876minus1199052119904
119860119876minus12
)lowast
119876minus1199052119904
119860119876minus12
)
120572119905119904
119896
119896 = 0 1 2
(7)
Let 119883 be a positive solution to (1) Then 119883119904
= 119876 minus
119860lowast
119883minus119905
119860 le 119876 = 1205720119876 that is 119883 le
119904radic1205720119876 Assuming that
119883 le119904radic120572119896119876 then
119883119904
= 119876 minus 119860lowast
119883minus119905
119860 le 119876 minus119860lowast
119876minus119905119904
119860
120572119905119904
119896
= 11987612
[119868 minus119876minus12
119860lowast
119876minus1199052119904
119876minus1199052119904
119860119876minus12
120572119905119904
119896
]11987612
le 11987612
times [
[
1 minus
120582min ((119876minus1199052119904
119860119876minus12
)lowast
119876minus1199052119904
119860119876minus12
)
120572119905119904
119896
]
]
times 11987612
= 120572119896+1119876
(8)
Hence 119883 le119904radic120572119899119876 for all 119899 = 0 1 2 It is straightforward
to check that the sequence 120572119899 is monotonically decreasing
also 120572119899 is bounded below hence 120572
119899 is convergent Denote
119879 = 119876minus1199052119904
119860119876minus12 and let lim
119899rarrinfin120572119899
= then =
1 minus 120582min(119879lowast
119879)119905119904 that is is a solution to the equation
120572119905119904
(1 minus 120572) = 120582min(119879lowast
119879) Let 119891(119909) = 119909119905119904(1 minus 119909) Then
max119909isin[01]
119891 (119909) = 119891(119905
119905 + 119904) = (
119905
119905 + 119904)
119905119904
(119904
119905 + 119904) (9)
Thus it follows that 120582min(119879lowast
119879) le (119905(119905 + 119904))119905119904
(119904(119905 + 119904))Since
120582min (119879lowast
119879) le (119905
119905 + 119904)
119905119904
(119904
119905 + 119904) (10)
Then the equation 120572119905
(1 minus 120572) = 120582min(119879lowast
119879) may have twosolutions one of these solutions is in the interval [119905(119905+119904) 1]
In order to prove that the limit of the sequence 120572119899 is
in [119905(119905 + 119904) 1] we assume that 120572119894ge 119905(119905 + 119904) (obviously 120572
0=
1 gt 119905(119905 + 119904)) then
120572119894+1
= 1 minus120582min (119879
lowast
119879)
120572119905119904
119894
ge 1 minus 120582min (119879lowast
119879) (119905 + 119904
119905)
119905119904
ge 1 minus (119905
119905 + 119904)
119905119904
(119904
119905 + 119904) (
119905 + 119904
119905)
119905119904
=119905
119905 + 119904
(11)
ISRNMathematical Analysis 3
Therefore 120572119894ge 119905(119905+119904) for each 119894 = 1 2 Since119883 le
119904radic120572119899119876
then119883 le119904radic119876 and isin [119905(119905 + 119904) 1] The proof is completed
Consider the following iterative sequence
119883119904
0= 120574119876 119883
119904
119896+1= 119876 minus 119860
lowast
119883minus119905
119896119860
119896 = 0 1 120574 gt 0
(12)
We will prove the following theorem
Theorem6 If the operator equation (1) has a positive operatorsolution 119883 then it has a maximal one 119883
119871 Moreover the
sequence 119883119894 in (12) for 120574 isin [ 1] is monotonically decreasing
and converges to119883119871 where is defined in Theorem 5
Proof Consider the iterative sequence (12) with 120574 isin [ 1]According to Theorem 4 we have 119883119904 le 119876 le 120574119876 = 119883
119904
0for
any positive operator solution119883 to (1)Suppose that119883119904
119894ge 119883119904 Then
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 ge 119876 minus 119860
lowast
119883minus119905
119860 = 119883119904
(13)
Hence119883119894ge 119883 hold for each 119894 According to the definition
of and the monotonicity of the function 119891(119909) = 119909119905119904(1 minus 119909)on [119905(119905 + 119904) 1] we have
120574119905119904
(1 minus 120574) 119868 le 119905119904
(1 minus ) 119868 = 120582min (119879lowast
119879) 119868 le 119879lowast
119879 (14)
for all 120574 isin [ 1] where 119879 = 119876minus1199052119904119860119876minus12 We compute
119883119904
1= 119876 minus 119860
lowast
(120574119876)minus119905119904
119860
= 11987612
[119868 minus 119876minus12
119860lowast
(120574119876)minus119905119904
119860119876minus12
]11987612
= 11987612
(119868 minus 120574minus119905119904
119879lowast
119879)11987612
(15)
Using inequality (14) and equality (15) we obtain1198831199041le 120574119876 =
119883119904
0Assuming that119883119904
119894le 119883119904
119894minus1 then
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 le 119876 minus 119860
lowast
119883minus119905
119894minus1119860 = 119883
119904
119894 (16)
Therefore 119883119904119894+1
le 119883119904
119894for all 119894 = 0 1 that is the sequence
119883119904
119899 is monotonically decreasing Hence 119883119904
119899 converges to
the positive operator solution119883119871to (1) Since119883119904
119871ge 119883119904 for any
positive operator solution119883 it follows that119883119871is themaximal
solution
Theorem 7 Let 120582min(119879lowast
119879) lt 120582max(119879lowast
119879) le
(119905(119905 + 119904))119905119904
(119904(119905 + 119904)) and 1205731 1205732are solutions to the equation
120573119905119904
(1 minus 120573) = 120582max(119879lowast
119879) in [0 119905(119905 + 119904)] and [119905(119905 + 119904) 1]respectively Then there are the following conclusions
(i) If 120574 isin [1205731 1205732] then the sequence 119883
119899 in (12) is mono-
tonically increasing and converges to a positive operatorsolution119883
120574isin [119904radic120574119876
119904radic119876] to (1)
(ii) If 120574 isin [ 1] then the sequence 119883119899 in (12) is monoton-
ically decreasing and converges to the maximal positiveoperator solution119883
119871isin [119904radic1205732119876119904radic119876] to (1)
(iii) If 120574 isin (1205732 ) and (119905119904)119860
2
le (1205732120582min(119876))
(119904+119905)119904then the sequence 119883
119899 in (12) converges to the unique
solution 119883119871
isin [119904radic1205732119876119904radic119876] where and 119879 are
defined in Theorem 5
Proof Consider the function119891(119909) = 119909119905119904(1minus119909) 119909 isin [0 1] Itismonotonically increasing on [0 119905(119905+119904)] andmonotonicallydecreasing on [119905(119905 + 119904) 1] and
max119909isin[01]
119891 (119909) = 119891(119905
119905 + 119904) = (
119905
119905 + 119904)
119905119904
(119904
119905 + 119904) (17)
Since 120582max(119879lowast
119879) le (119905(119905+119904))119905119904
(119904(119905+119904)) then for each 1205741 1205742
such that 0 lt 1205731le 1205741le 1205732lt le 120574
2le 1 the inequalities
120574119905119904
2(1 minus 120574
2) 119868 le 120582min (119879
lowast
119879) le 119879lowast
119879
le 120582max (119879lowast
119879) le 120574119905119904
1(1 minus 120574
1) 119868
(18)
are satisfied(i) Let 120574 isin [120573
1 1205732] We will prove that the operator
sequence 119883119899 in (12) is monotonically increasing and is
bounded aboveAccording to the fourth inequality in (18) we compute
119883119904
1= 119876 minus 119860
lowast
(120574119876)minus119905119904
119860
= 11987612
(119868 minus 120574minus119905119904
119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
= 11987612
(119868 minus 120574minus119905119904
119879lowast
119879)11987612
ge 120574119876 = 119883119904
0
(19)
Assuming that119883119904119894ge 119883119904
119894minus1 then
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 ge 119876 minus 119860
lowast
119883minus119905
119894minus1119860 = 119883
119904
119894 (20)
Therefore 119883119904119894+1
ge 119883119904
119894for all 119894 = 0 1 that is the sequence
119883119904
119899 is monotonically increasing Obviously 119883119904
0= 120574119876 le
1205732119876 le 119876 We suppose that 119883119904
119894le 119876 from the first
inequality in (18) we obtain
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 le 119876 minus 119860
lowast
(119876)minus119905119904
119860 le 119876 (21)
Hence 119883119904119899 converges to a positive operator solution119883119904
120574
to (1) Since 119883119894isin [
119904radic120574119876
119904radic119876] for all 119894 = 0 1 we have
119883120574isin [119904radic120574119876
119904radic119876]
(ii) Let 120574 isin [ 1] From the proving procession ofTheorem 6 the sequence (12) is monotonically decreasingSince 119883
0=119904radic120574119876 ge
119904radic1205732119876 and supposing that 119883119904
119894ge 1205732119876
it follows that
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 ge 119876 minus 119860
lowast
(1205732119876)minus119905119904
119860
= 11987612
(119868 minus 120573minus119905119904
2119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
ge 11987612
(119868 minus 120573minus119905119904
2120582max (119879
lowast
119879) 119868)11987612
= 1205732119876
(22)
By induction principle119883119904119894ge 1205732119876 hold for each 119894 = 0 1 2
Hence 119883119904119899 is convergent FromTheorem 6 119883119904
119899 converges
to119883119904119871such that119883
119871isin [119904radic1205732119876119904radic120574119876]
4 ISRNMathematical Analysis
(iii) Considering the sequence 119883119904119899 in (12) for 120574 isin (120573
2 )
that is 1198831199040isin (1205732119876 119876) and supposing that 119883119904
119894isin (1205732119876 119876)
then for119883119904119894+1
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860
lt 11987612
(119868 minus 120573minus119905119904
2119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
le 119876
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860
gt 11987612
(119868 minus minus119905119904
119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
ge 1205732119876
(23)
Hence119883119904119894isin (1205732119876 120574119876) for all 119894 = 0 1 2 Now we consid-
ering 119883119904119894+1
minus 119883119904
119894
1003817100381710038171003817119883119904
119894+1minus 119883119904
119894
1003817100381710038171003817 =10038171003817100381710038171003817(119876 minus 119860
lowast
119883minus119905
119894119860) minus (119876 minus 119860
lowast
119883minus119905
119894minus1119860)10038171003817100381710038171003817
le1
119904
1
(119904radic1205732120582min (119876))
119904minus1
times1199051198602
(119904radic1205732120582min (119876))
119905+1
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817
=119905
1199041198602
1
(119904radic1205732120582min (119876))
119905+119904
times1003817100381710038171003817119883119894 minus 119883119894minus1
1003817100381710038171003817
(24)
and (119905119904)1198602 le (1205732120582min(119876))
(119904+119905)119904 it follows that 119883119899 in
(12) is a Cauchy sequence in the Banach space [ 119904radic1205732119876119904radic119876]
Hence it has a limit 119883120574in [ 119904radic120573
2119876119904radic119876] and 119883
120574is a unique
solution to (1) in [ 119904radic1205732119876119904radic119876] According to Theorem 6 it
is the maximal solution that is 119883120574= 119883119871 The proof is
completed
Acknowledgments
This paper is supported by the NSF of China Grant no11171197 and by the Natural Science Foundation of ShaanxiEducational Committee Grant no 12JK0884
References
[1] W Pusz and S L Woronowicz ldquoFunctional calculus forsesquilinear forms and the purification maprdquo Reports on Math-ematical Physics vol 8 no 2 pp 159ndash170 1975
[2] J C Engwerda ldquoOn the existence of a positive definite solutionof the matrix equation119883+119860lowast119883minus120572119860 = 119868rdquo Linear Algebra and itsApplications vol 194 pp 91ndash108 1993
[3] W L Green and E W Kamen ldquoStabilizability of linear sys-tems over a commutative normed algebra with applications tospatially-distributed and parameter-dependent systemsrdquo SIAMJournal onControl andOptimization vol 23 no 1 pp 1ndash18 1985
[4] R S Bucy ldquoA priori bounds for the Riccati equationrdquo inProceedings of the 6th Berkeley Symposium on Mathematical
Statistics and Probability vol 111 of ProbabilityTheory pp 645ndash656 University of California Press Berkeley Calif USA 1972
[5] V I Hasanov Solutions and perturbation theory of nonlinearmatrix equations [PhD thesis] Sofia Bulgaria 2003
[6] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation119883+119860lowast119865(119883)119860 = 119876 solutions and perturbation theoryrdquoLinear Algebra and its Applications vol 346 pp 15ndash26 2002
[7] V I Hasanov ldquoPositive definite solutions of the matrix equa-tions 119883 plusmn 119860
lowast
119883minus119902
119860 = 119876rdquo Linear Algebra and its Applicationsvol 404 pp 166ndash182 2005
[8] A C M Ran and M C B Reurings ldquoThe symmetric linearmatrix equationrdquo Electronic Journal of Linear Algebra vol 9 pp93ndash107 2002
[9] K F Yang and H K Du ldquoStudies on the positive operatorsolutions to the operator equations 119883 + 119860
lowast
119883minus119905
119860 = 119876rdquo ActaMathematica Scientia A vol 26 no 2 pp 359ndash364 2009(Chinese)
[10] Y Yang F Duan and X Zhao ldquoOn solutions for the matrixequation 119883
119904
+ 119860lowast
119883minus119905
119860 = 119876rdquo in Proceedings of the 7thInternational Conference on Matrix Theory and Its Applicationsin China pp 21ndash24 2006
[11] Z-y Peng S M El-Sayed and X-l Zhang ldquoIterative methodsfor the extremal positive definite solution of thematrix equation119883 + 119860
lowast
119883minus120572
119860 = 119876rdquo Journal of Computational and AppliedMathematics vol 200 no 2 pp 520ndash527 2007
[12] Y Yueting ldquoThe iterative method for solving nonlinear matrixequation119883119904 +119860lowast119883minus119905119860 = 119876rdquo Applied Mathematics and Compu-tation vol 188 no 1 pp 46ndash53 2007
[13] V I Hasanov and S M El-Sayed ldquoOn the positive definitesolutions of nonlinear matrix equation 119883 + 119860
lowast
119883minus120575
119860 = 119876rdquoLinear Algebra and its Applications vol 412 no 2-3 pp 154ndash160 2006
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
ISRNMathematical Analysis 3
Therefore 120572119894ge 119905(119905+119904) for each 119894 = 1 2 Since119883 le
119904radic120572119899119876
then119883 le119904radic119876 and isin [119905(119905 + 119904) 1] The proof is completed
Consider the following iterative sequence
119883119904
0= 120574119876 119883
119904
119896+1= 119876 minus 119860
lowast
119883minus119905
119896119860
119896 = 0 1 120574 gt 0
(12)
We will prove the following theorem
Theorem6 If the operator equation (1) has a positive operatorsolution 119883 then it has a maximal one 119883
119871 Moreover the
sequence 119883119894 in (12) for 120574 isin [ 1] is monotonically decreasing
and converges to119883119871 where is defined in Theorem 5
Proof Consider the iterative sequence (12) with 120574 isin [ 1]According to Theorem 4 we have 119883119904 le 119876 le 120574119876 = 119883
119904
0for
any positive operator solution119883 to (1)Suppose that119883119904
119894ge 119883119904 Then
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 ge 119876 minus 119860
lowast
119883minus119905
119860 = 119883119904
(13)
Hence119883119894ge 119883 hold for each 119894 According to the definition
of and the monotonicity of the function 119891(119909) = 119909119905119904(1 minus 119909)on [119905(119905 + 119904) 1] we have
120574119905119904
(1 minus 120574) 119868 le 119905119904
(1 minus ) 119868 = 120582min (119879lowast
119879) 119868 le 119879lowast
119879 (14)
for all 120574 isin [ 1] where 119879 = 119876minus1199052119904119860119876minus12 We compute
119883119904
1= 119876 minus 119860
lowast
(120574119876)minus119905119904
119860
= 11987612
[119868 minus 119876minus12
119860lowast
(120574119876)minus119905119904
119860119876minus12
]11987612
= 11987612
(119868 minus 120574minus119905119904
119879lowast
119879)11987612
(15)
Using inequality (14) and equality (15) we obtain1198831199041le 120574119876 =
119883119904
0Assuming that119883119904
119894le 119883119904
119894minus1 then
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 le 119876 minus 119860
lowast
119883minus119905
119894minus1119860 = 119883
119904
119894 (16)
Therefore 119883119904119894+1
le 119883119904
119894for all 119894 = 0 1 that is the sequence
119883119904
119899 is monotonically decreasing Hence 119883119904
119899 converges to
the positive operator solution119883119871to (1) Since119883119904
119871ge 119883119904 for any
positive operator solution119883 it follows that119883119871is themaximal
solution
Theorem 7 Let 120582min(119879lowast
119879) lt 120582max(119879lowast
119879) le
(119905(119905 + 119904))119905119904
(119904(119905 + 119904)) and 1205731 1205732are solutions to the equation
120573119905119904
(1 minus 120573) = 120582max(119879lowast
119879) in [0 119905(119905 + 119904)] and [119905(119905 + 119904) 1]respectively Then there are the following conclusions
(i) If 120574 isin [1205731 1205732] then the sequence 119883
119899 in (12) is mono-
tonically increasing and converges to a positive operatorsolution119883
120574isin [119904radic120574119876
119904radic119876] to (1)
(ii) If 120574 isin [ 1] then the sequence 119883119899 in (12) is monoton-
ically decreasing and converges to the maximal positiveoperator solution119883
119871isin [119904radic1205732119876119904radic119876] to (1)
(iii) If 120574 isin (1205732 ) and (119905119904)119860
2
le (1205732120582min(119876))
(119904+119905)119904then the sequence 119883
119899 in (12) converges to the unique
solution 119883119871
isin [119904radic1205732119876119904radic119876] where and 119879 are
defined in Theorem 5
Proof Consider the function119891(119909) = 119909119905119904(1minus119909) 119909 isin [0 1] Itismonotonically increasing on [0 119905(119905+119904)] andmonotonicallydecreasing on [119905(119905 + 119904) 1] and
max119909isin[01]
119891 (119909) = 119891(119905
119905 + 119904) = (
119905
119905 + 119904)
119905119904
(119904
119905 + 119904) (17)
Since 120582max(119879lowast
119879) le (119905(119905+119904))119905119904
(119904(119905+119904)) then for each 1205741 1205742
such that 0 lt 1205731le 1205741le 1205732lt le 120574
2le 1 the inequalities
120574119905119904
2(1 minus 120574
2) 119868 le 120582min (119879
lowast
119879) le 119879lowast
119879
le 120582max (119879lowast
119879) le 120574119905119904
1(1 minus 120574
1) 119868
(18)
are satisfied(i) Let 120574 isin [120573
1 1205732] We will prove that the operator
sequence 119883119899 in (12) is monotonically increasing and is
bounded aboveAccording to the fourth inequality in (18) we compute
119883119904
1= 119876 minus 119860
lowast
(120574119876)minus119905119904
119860
= 11987612
(119868 minus 120574minus119905119904
119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
= 11987612
(119868 minus 120574minus119905119904
119879lowast
119879)11987612
ge 120574119876 = 119883119904
0
(19)
Assuming that119883119904119894ge 119883119904
119894minus1 then
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 ge 119876 minus 119860
lowast
119883minus119905
119894minus1119860 = 119883
119904
119894 (20)
Therefore 119883119904119894+1
ge 119883119904
119894for all 119894 = 0 1 that is the sequence
119883119904
119899 is monotonically increasing Obviously 119883119904
0= 120574119876 le
1205732119876 le 119876 We suppose that 119883119904
119894le 119876 from the first
inequality in (18) we obtain
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 le 119876 minus 119860
lowast
(119876)minus119905119904
119860 le 119876 (21)
Hence 119883119904119899 converges to a positive operator solution119883119904
120574
to (1) Since 119883119894isin [
119904radic120574119876
119904radic119876] for all 119894 = 0 1 we have
119883120574isin [119904radic120574119876
119904radic119876]
(ii) Let 120574 isin [ 1] From the proving procession ofTheorem 6 the sequence (12) is monotonically decreasingSince 119883
0=119904radic120574119876 ge
119904radic1205732119876 and supposing that 119883119904
119894ge 1205732119876
it follows that
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860 ge 119876 minus 119860
lowast
(1205732119876)minus119905119904
119860
= 11987612
(119868 minus 120573minus119905119904
2119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
ge 11987612
(119868 minus 120573minus119905119904
2120582max (119879
lowast
119879) 119868)11987612
= 1205732119876
(22)
By induction principle119883119904119894ge 1205732119876 hold for each 119894 = 0 1 2
Hence 119883119904119899 is convergent FromTheorem 6 119883119904
119899 converges
to119883119904119871such that119883
119871isin [119904radic1205732119876119904radic120574119876]
4 ISRNMathematical Analysis
(iii) Considering the sequence 119883119904119899 in (12) for 120574 isin (120573
2 )
that is 1198831199040isin (1205732119876 119876) and supposing that 119883119904
119894isin (1205732119876 119876)
then for119883119904119894+1
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860
lt 11987612
(119868 minus 120573minus119905119904
2119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
le 119876
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860
gt 11987612
(119868 minus minus119905119904
119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
ge 1205732119876
(23)
Hence119883119904119894isin (1205732119876 120574119876) for all 119894 = 0 1 2 Now we consid-
ering 119883119904119894+1
minus 119883119904
119894
1003817100381710038171003817119883119904
119894+1minus 119883119904
119894
1003817100381710038171003817 =10038171003817100381710038171003817(119876 minus 119860
lowast
119883minus119905
119894119860) minus (119876 minus 119860
lowast
119883minus119905
119894minus1119860)10038171003817100381710038171003817
le1
119904
1
(119904radic1205732120582min (119876))
119904minus1
times1199051198602
(119904radic1205732120582min (119876))
119905+1
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817
=119905
1199041198602
1
(119904radic1205732120582min (119876))
119905+119904
times1003817100381710038171003817119883119894 minus 119883119894minus1
1003817100381710038171003817
(24)
and (119905119904)1198602 le (1205732120582min(119876))
(119904+119905)119904 it follows that 119883119899 in
(12) is a Cauchy sequence in the Banach space [ 119904radic1205732119876119904radic119876]
Hence it has a limit 119883120574in [ 119904radic120573
2119876119904radic119876] and 119883
120574is a unique
solution to (1) in [ 119904radic1205732119876119904radic119876] According to Theorem 6 it
is the maximal solution that is 119883120574= 119883119871 The proof is
completed
Acknowledgments
This paper is supported by the NSF of China Grant no11171197 and by the Natural Science Foundation of ShaanxiEducational Committee Grant no 12JK0884
References
[1] W Pusz and S L Woronowicz ldquoFunctional calculus forsesquilinear forms and the purification maprdquo Reports on Math-ematical Physics vol 8 no 2 pp 159ndash170 1975
[2] J C Engwerda ldquoOn the existence of a positive definite solutionof the matrix equation119883+119860lowast119883minus120572119860 = 119868rdquo Linear Algebra and itsApplications vol 194 pp 91ndash108 1993
[3] W L Green and E W Kamen ldquoStabilizability of linear sys-tems over a commutative normed algebra with applications tospatially-distributed and parameter-dependent systemsrdquo SIAMJournal onControl andOptimization vol 23 no 1 pp 1ndash18 1985
[4] R S Bucy ldquoA priori bounds for the Riccati equationrdquo inProceedings of the 6th Berkeley Symposium on Mathematical
Statistics and Probability vol 111 of ProbabilityTheory pp 645ndash656 University of California Press Berkeley Calif USA 1972
[5] V I Hasanov Solutions and perturbation theory of nonlinearmatrix equations [PhD thesis] Sofia Bulgaria 2003
[6] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation119883+119860lowast119865(119883)119860 = 119876 solutions and perturbation theoryrdquoLinear Algebra and its Applications vol 346 pp 15ndash26 2002
[7] V I Hasanov ldquoPositive definite solutions of the matrix equa-tions 119883 plusmn 119860
lowast
119883minus119902
119860 = 119876rdquo Linear Algebra and its Applicationsvol 404 pp 166ndash182 2005
[8] A C M Ran and M C B Reurings ldquoThe symmetric linearmatrix equationrdquo Electronic Journal of Linear Algebra vol 9 pp93ndash107 2002
[9] K F Yang and H K Du ldquoStudies on the positive operatorsolutions to the operator equations 119883 + 119860
lowast
119883minus119905
119860 = 119876rdquo ActaMathematica Scientia A vol 26 no 2 pp 359ndash364 2009(Chinese)
[10] Y Yang F Duan and X Zhao ldquoOn solutions for the matrixequation 119883
119904
+ 119860lowast
119883minus119905
119860 = 119876rdquo in Proceedings of the 7thInternational Conference on Matrix Theory and Its Applicationsin China pp 21ndash24 2006
[11] Z-y Peng S M El-Sayed and X-l Zhang ldquoIterative methodsfor the extremal positive definite solution of thematrix equation119883 + 119860
lowast
119883minus120572
119860 = 119876rdquo Journal of Computational and AppliedMathematics vol 200 no 2 pp 520ndash527 2007
[12] Y Yueting ldquoThe iterative method for solving nonlinear matrixequation119883119904 +119860lowast119883minus119905119860 = 119876rdquo Applied Mathematics and Compu-tation vol 188 no 1 pp 46ndash53 2007
[13] V I Hasanov and S M El-Sayed ldquoOn the positive definitesolutions of nonlinear matrix equation 119883 + 119860
lowast
119883minus120575
119860 = 119876rdquoLinear Algebra and its Applications vol 412 no 2-3 pp 154ndash160 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 ISRNMathematical Analysis
(iii) Considering the sequence 119883119904119899 in (12) for 120574 isin (120573
2 )
that is 1198831199040isin (1205732119876 119876) and supposing that 119883119904
119894isin (1205732119876 119876)
then for119883119904119894+1
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860
lt 11987612
(119868 minus 120573minus119905119904
2119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
le 119876
119883119904
119894+1= 119876 minus 119860
lowast
119883minus119905
119894119860
gt 11987612
(119868 minus minus119905119904
119876minus12
119860lowast
119876minus119905119904
119860119876minus12
)11987612
ge 1205732119876
(23)
Hence119883119904119894isin (1205732119876 120574119876) for all 119894 = 0 1 2 Now we consid-
ering 119883119904119894+1
minus 119883119904
119894
1003817100381710038171003817119883119904
119894+1minus 119883119904
119894
1003817100381710038171003817 =10038171003817100381710038171003817(119876 minus 119860
lowast
119883minus119905
119894119860) minus (119876 minus 119860
lowast
119883minus119905
119894minus1119860)10038171003817100381710038171003817
le1
119904
1
(119904radic1205732120582min (119876))
119904minus1
times1199051198602
(119904radic1205732120582min (119876))
119905+1
1003817100381710038171003817119883119894 minus 119883119894minus11003817100381710038171003817
=119905
1199041198602
1
(119904radic1205732120582min (119876))
119905+119904
times1003817100381710038171003817119883119894 minus 119883119894minus1
1003817100381710038171003817
(24)
and (119905119904)1198602 le (1205732120582min(119876))
(119904+119905)119904 it follows that 119883119899 in
(12) is a Cauchy sequence in the Banach space [ 119904radic1205732119876119904radic119876]
Hence it has a limit 119883120574in [ 119904radic120573
2119876119904radic119876] and 119883
120574is a unique
solution to (1) in [ 119904radic1205732119876119904radic119876] According to Theorem 6 it
is the maximal solution that is 119883120574= 119883119871 The proof is
completed
Acknowledgments
This paper is supported by the NSF of China Grant no11171197 and by the Natural Science Foundation of ShaanxiEducational Committee Grant no 12JK0884
References
[1] W Pusz and S L Woronowicz ldquoFunctional calculus forsesquilinear forms and the purification maprdquo Reports on Math-ematical Physics vol 8 no 2 pp 159ndash170 1975
[2] J C Engwerda ldquoOn the existence of a positive definite solutionof the matrix equation119883+119860lowast119883minus120572119860 = 119868rdquo Linear Algebra and itsApplications vol 194 pp 91ndash108 1993
[3] W L Green and E W Kamen ldquoStabilizability of linear sys-tems over a commutative normed algebra with applications tospatially-distributed and parameter-dependent systemsrdquo SIAMJournal onControl andOptimization vol 23 no 1 pp 1ndash18 1985
[4] R S Bucy ldquoA priori bounds for the Riccati equationrdquo inProceedings of the 6th Berkeley Symposium on Mathematical
Statistics and Probability vol 111 of ProbabilityTheory pp 645ndash656 University of California Press Berkeley Calif USA 1972
[5] V I Hasanov Solutions and perturbation theory of nonlinearmatrix equations [PhD thesis] Sofia Bulgaria 2003
[6] A C M Ran and M C B Reurings ldquoOn the nonlinear matrixequation119883+119860lowast119865(119883)119860 = 119876 solutions and perturbation theoryrdquoLinear Algebra and its Applications vol 346 pp 15ndash26 2002
[7] V I Hasanov ldquoPositive definite solutions of the matrix equa-tions 119883 plusmn 119860
lowast
119883minus119902
119860 = 119876rdquo Linear Algebra and its Applicationsvol 404 pp 166ndash182 2005
[8] A C M Ran and M C B Reurings ldquoThe symmetric linearmatrix equationrdquo Electronic Journal of Linear Algebra vol 9 pp93ndash107 2002
[9] K F Yang and H K Du ldquoStudies on the positive operatorsolutions to the operator equations 119883 + 119860
lowast
119883minus119905
119860 = 119876rdquo ActaMathematica Scientia A vol 26 no 2 pp 359ndash364 2009(Chinese)
[10] Y Yang F Duan and X Zhao ldquoOn solutions for the matrixequation 119883
119904
+ 119860lowast
119883minus119905
119860 = 119876rdquo in Proceedings of the 7thInternational Conference on Matrix Theory and Its Applicationsin China pp 21ndash24 2006
[11] Z-y Peng S M El-Sayed and X-l Zhang ldquoIterative methodsfor the extremal positive definite solution of thematrix equation119883 + 119860
lowast
119883minus120572
119860 = 119876rdquo Journal of Computational and AppliedMathematics vol 200 no 2 pp 520ndash527 2007
[12] Y Yueting ldquoThe iterative method for solving nonlinear matrixequation119883119904 +119860lowast119883minus119905119860 = 119876rdquo Applied Mathematics and Compu-tation vol 188 no 1 pp 46ndash53 2007
[13] V I Hasanov and S M El-Sayed ldquoOn the positive definitesolutions of nonlinear matrix equation 119883 + 119860
lowast
119883minus120575
119860 = 119876rdquoLinear Algebra and its Applications vol 412 no 2-3 pp 154ndash160 2006
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of