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Verma A new system of nonlinear variational inclusions involving A, η-monotone mappings in the framework of Hilbert space is introduced and then based on the generalized resolvent operat

Volume 2008, Article ID 681734, 6 pages

doi:10.1155/2008/681734

Research Article

A System of Nonlinear Variational Inclusions with

A, η-Monotone Mappings

Zhiguo Wang 1 and Changqun Wu 2

1 School of Mathematics and Information Sciences, Henan University, Kaifeng 475001, China

2 School of Business and Administration, Henan University, Kaifeng 475001, China

Correspondence should be addressed to Zhiguo Wang, henumath@yahoo.cn

Received 17 December 2007; Accepted 2 January 2008

Recommended by Ram U Verma

A new system of nonlinear variational inclusions involving A, η-monotone mappings in the

framework of Hilbert space is introduced and then based on the generalized resolvent operator technique associated withA, η-monotonicity, the approximation solvability of solutions using an

iterative algorithm is investigated Since A, η-monotonicity generalizes A-monotonicity and

H-monotonicity, our results improve and extend the recent ones announced by many others.

Copyright q 2008 Z Wang and C Wu This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

Variational inclusions problems are among the most interesting and intensively studied classes

of mathematical problems and have wide applications in the fields of optimization and control, economics and transportation equilibrium, and engineering sciences Variational inclusions problems have been generalized and extended in different directions using the novel and in-novative techniques Various kinds of iterative algorithms to solve the variational inequalities and variational inclusions have been developed by many authors For details, we can refer to

1 9 In most of the resolvent operator methods, the maximal monotonicity has played a key role, but more recently introduced notions of A-monotonicity 7 and H-monotonicity 2, 3 have not only generalized the maximal monotonicity, but gave a new edge to resolvent oper-ator methods Recently, Verma9 generalized the recently introduced and studied notion of

A-monotonicity to the case of A, η-monotonicity, while examining the sensitivity analysis for

a class of nonlinear variational inclusion problems based on the generalized resolvent operator technique Resolvent operator techniques have been in use for a while in literature, especially with the general framework involving set-valued maximal monotone mappings, but it got a new empowerment by the recent developments ofA-monotonicity and H-monotonicity

In-spired and motivated by the recent research going on in this area, we introduce and analyze

a new class of variational inclusions problems involving A, η-monotone mappings 9 in the framework of Hilbert space SinceA, η-monotonicity generalizes A-monotonicity 7 and

H-monotonicity 2, 3, our results improve and extend the recent ones announced by many others

2 Preliminaries

In this section, we explore some basic properties derived from the notion of A,

η-monotonicity Let H denote a real Hilbert space with the norm · and inner product ·, ·,

respectively Letη : H × H : →H be a single-valued mapping The map η is called τ-Lipschitz

continuous if there is a constantτ > 0 such that ηu, v ≤ τy − v for all u, v ∈ H.

LetM : H→2 H be a multivalued mapping from a Hilbert spaceH to 2 H, the power set

ofH We recall the following.

i The set DM, defined by DM  {u ∈ H : Mu/∅}, is called the effective domain of M.

ii The set RM, defined by RM u∈H Mu, is called the range of M.

iii The set GM, defined by GM  {u, v ∈ H × H : u ∈ DM, v ∈ Mu}, is the graph

ofM.

Definition 2.1 Let η : H × H→H be a single-valued mapping and let M : H→2 H be a multival-ued mapping onH The map M is said to be

i r, η-strongly monotone if



u∗− v∗, ηu, v≥ ru − v ∀u, u∗, v, v∗ ∈ GM, 2.1

ii m, η-relaxed monotone if there exists a positive constant m such that



u∗− v∗, ηu, v≥ − mu − v2 ∀u, u∗, v, v∗ ∈ GM. 2.2

Definition 2.2see 9 Let A : H→H and η : H × H→H be two single-valued mappings The

mapM : H→2 H is said to beA, η-monotone if

i M is m, η-relaxed monotone,

ii RA  ρM  H for ρ > 0.

Note that, alternatively, the mapM : H→2 H is said to beA, η-monotone if

i M is m, η-relaxed monotone,

ii A  ρM is η-pseudomonotone for ρ > 0.

Definition 2.3 Let A : H→H be an r, η-strong monotone mapping and let M : H→H be an

A, η-monotone mapping Then the generalized resolvent operator J M,ρ A,η :H→H is defined by

J M,ρ A,η u  A  ρM−1u for all u ∈ H, where ρ > 0 is a constant.

Definition 2.4 The map N : H ×H is said to be relaxed β, γ-cocoercive with respect to A in the

first argument if there exist two positive constantsα, β such that Tx, u − Ty, u, Ax − Ay ≥

−βTx, u − Ty, u2 γx − y2for allx, y, u ∈ H × H × H.

Proposition 2.5 see 9 Let η : H × H→H be a single-valued mapping, let A : H→Hr, η-strongly monotone mapping, and let M : H→2 H be an A, η-monotone mapping Then the mapping

A  ρM−1is single valued.

3 Results on algorithmic convergence analysis

Let N : H × H→H and η : H × H→H be two nonlinear mappings Let M : H→2 H be an

A, η-monotone mapping Then we have the nonlinear system of variational inclusion NSVI

problem: determine an elementu, v ∈ H × H such that

0∈ Au − Av  ρ1Nv, u  Mu,

0∈ Av − Au  ρ2Nu, v  Mv. 3.1

Ifu  v and ρ1  ρ2in NSVI3.1, we have the following NVI problem: find elements

u, v ∈ H such that

0∈ Nu, u  Mu, 3.2 which was considered by Verma8

In order to prove our main results, we need the following lemmas

Lemma 3.1 see 10 Assume that {a n } is a sequence of nonnegative real numbers such that a n1≤

1 − λ n a n  b n for all n ≥ n0, where n0is some nonnegative integer, {λ n } is a sequence in 0, 1 with

∞

n1 λ n  ∞, b n  ◦λ n , then lim n→∞ a n  0.

Lemma 3.2 see 9 Let H be a real Hilbert space and let η : H ×H→H be a τ-Lipschitz continuous nonlinear mapping Let A : H→H be an r, η-strongly monotone and let M : H→2 H be A, η-monotone Then the generalized resolvent operator J M,ρ A,η :H→H is τ/r − ρm, that is,

J A,η M,ρ x − J M,ρ A,η y ≤ τ r − ρm x − y ∀x, y ∈ H. 3.3

Lemma 3.3 Let H be a real Hilbert space, let A : H→H be r, η-strongly monotone, and let M :

H→2 H be A, η-monotone Let η : H × H→H be a τ-Lipschitz continuous nonlinear mapping Then

u, v is the solution of NSVI 3.1 if and only if it satisfies

u  J M,ρ A,η

1



Av − ρ1Nv, u,

v  J M,ρ A,η

2



Au − ρ2Nu, v. 3.4 Proof The fact directly follows fromDefinition 2.4

Algorithm 3.4 For any u0, v0∈ H, compute the sequences {u n } and {v n} by the iterative process

u n1  1 − α n u n  α n J M,ρ A,η

1



Av n − ρ1Nv n , u n,

v n  J M,ρ A,η

2



Au n − ρ2Nu n , v n. 3.5

Ifu n  v nfor alln and ρ1 ρ2inAlgorithm 3.4, then we have the following algorithm

Algorithm 3.5 For any u0 ∈ H, compute the sequence {u n} by the iterative processes:

u n1  1 − α n u n  α n J M,ρ A,ηAu n − ρNu n , u n. 3.6

Theorem 3.6 Let H be a real Hilbert space, let A : H × H→H be r, η-strongly monotone and

s-Lipschitz continuous, and let M : H→2 H be A, η-monotone Let η : H × H→H be a τ-Lipschitz continuous nonlinear mapping, let N : H × H→H be relaxed β, γ-cocoercive (with respect to A) and μ-Lipschitz coninuous in the first variable, and let N i be ν-Lipschitz continuous in the second variable Let u∗, v∗ be the solution of NSVI problem 3.1, and {u n } and {v n } sequences generated by

Algorithm 3.4 Suppose the following conditions are satisfied:

i α n ⊂ 0, 1,∞n0 α n ∞;

ii τθ1θ2θ3  ρ1ν < r − ρ1m, where θ1  s2− 2ρ1γ  2ρ1βμ2 ρ2

1μ2, θ2 

s2− 2ρ2γ  2ρ2βμ2 ρ2

2μ2, and θ3  τ/r − ρ2m − τρ2ν;

iii ρ2< r/m  τν.

Then the sequences {u n } and {v n } converge strongly to u∗and v∗, respectively.

Proof Let u∗, v∗ ∈ H be the solution of NSVI problem 3.1, we have

u∗ 1 − α n u∗ α n J M,ρ A,η

1



Av∗− ρ1Nv∗, u∗,

v∗ J M,ρ A,η

2



Au∗− ρ2Nu∗, v∗. 3.7

It follows that

u n1 − u∗

1 − α

n u n  α n J M,ρ A,η

1



Av n − ρ1Nv n , u n− u∗

≤1−α nun −u∗ τα n

r−ρ1m Av n −Av∗−ρ1



Nv n , u n −Nv∗, u nρ

1Nv∗, u n −Nv∗, u∗

.

3.8

It follows from relaxedβ, γ-cocoercive monotonicity and μ-Lipschitz continuity of N in the

first variable that



Av n − Av∗− ρ1Nv n , u n − Nv∗, u n 2

Av n − Av∗2− 2ρ1Nv n , u n − Nv∗, u n , Av n − Au∗

 ρ2

1Nv n , u n − Nv∗, u n 2

≤ θ2

1v n − v∗2,

3.9 where θ1  s2− 2ρ1γ  2ρ1βμ2 ρ2

1μ2 Observe that the ν-Lipschitz continuity of N in the

second argument yields that

Nv∗, u n − Nv∗, u∗

n − u∗. 3.10

On the other hand, we have

v n − v∗ J A,η

M,ρ2



Au n − ρ2Nu n , v n − v∗

≤ r − ρ τ

2m Au n − Au∗− ρ2Nu n , v n − Nu∗, v∗

≤ τ

r − ρ2m Au n − Au∗− ρ2Nu n , v n − Nu∗, v n 2Nu∗, v n − Nu∗, v∗

3.11

It follows from relaxedβ, γ-cocoercive monotonicity and μ-Lipschitz continuity of N in the

first variable that

Au n − Au∗− ρ2Nu n , v n − Nu∗, v n 2

Au n − Au∗2− 2ρ2Nu n , v n − Nu∗, v n , Au n − Au∗

 ρ2

2Nu n , v n − Nu∗, v n 2

≤ θ2

2u n − u∗2,

3.12 whereθ2  s2− 2ρ2γ  2ρ2βμ2 ρ2

2μ2 Again, observe that theν-Lipschitz continuity of N2in the second argument yields that

Nu∗, v n − Nu∗, v∗ n − v∗. 3.13 Substituting3.12 and 3.13 into 3.11 yields that

v n − v∗ ≤ τ r − ρ

2m



θ2u n − u∗  ρ2νv n − v∗. 3.14

It follows from conditioniii that

v n − v∗ ≤ θ3θ2u n − u∗, 3.15 whereθ3 τ/r − ρ2m − τρ2ν Substituting 3.15 into 3.9, we obtain

Av n − Av∗− ρ1Nv n , u n − Nv∗, u n 1θ2θ3u n − u∗. 3.16 Again, substituting3.10 and 3.16 into 3.8, we arrive at

u n1 − u∗

≤ 1 − α nun − u∗  τα n

r − ρ1m



θ1θ2θ3u n − u∗  ρ1νu n − u∗





1− α n



1− r − ρ τ

1m

θ1θ2θ3 ρ1ν n − u∗. 3.17

Using conditionsi–iii and applyingLemma 3.1to3.17, we can obtain the desired conclu-sion This completes the proof

FromTheorem 3.6, we have the following result immediately.

Corollary 3.7 Let H be a real Hilbert space, let A : H×H be r, η-strongly monotone and s-Lipschitz

continuous, and let M : H→2 H be A, η-monotone Let η : H × H→H be a τ-Lipschitz continuous nonlinear mapping, let N : H × H→H be relaxed β, γ-cocoercive (with respect to A) and μ-Lipschitz coninuous in the first variable, and let N be ν-Lipschitz continuous in the second variable Let u∗be the solution of NVI problem3.2, and {u n } a sequence generated by Algorithm 3.5 Suppose the following conditions are satisfied:

i α n ⊂ 0, 1, ∞n0 α n ∞;

ii τθ  ρν < r − ρm, where θ  s2− 2ργ  2ρβμ2 ρ2μ2.

Then the sequence {u n } converges strongly to u∗.

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On the other hand, we have

v n − v∗ J A, η... and applyingLemma 3.1to3.17, we can obtain the desired conclu-sion This completes the proof

FromTheorem