General System of Monotone Nonlinear Variational Inclusions Problems with Applications

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Volume 2009, Article ID 364615,13pages doi:10.1155/2009/364615

Research Article

General System of

A

-Monotone Nonlinear

Variational Inclusions Problems with Applications

Jian-Wen Peng

1

_{and Lai-Jun Zhao}

2

1_{College of Mathematics and Computer Science, Chongqing Normal University, Chongqing 400047, China} 2_{Management School, Shanghai University, Shanghai 200444, China}

Correspondence should be addressed to Lai-Jun Zhao,zhao laijun@163.com

Received 25 September 2009; Accepted 2 November 2009

Recommended by Vy Khoi Le

We introduce and study a new system of nonlinear variational inclusions involving a combination ofA-Monotone operators and relaxed cocoercive mappings. By using the resolvent technique of theA-monotone operators, we prove the existence and uniqueness of solution and the convergence of a new multistep iterative algorithm for this system of variational inclusions. The results in this paper unify, extend, and improve some known results in literature.

Copyrightq2009 J.-W. Peng and L.-J. Zhao. 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

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A-monotonicity is defined in terms of relaxed monotone mappings—a more general notion than the monotonicity or strong monotonocity—which gives a significant edge over the H-monotonocity. Very recently, Verma 6 studied the solvability of a system of variational inclusions involving a combination ofA-monotone and relaxed cocoercive mappings using resolvent operator techniques ofA-monotone mappings. Since relaxed cocoercive mapping is a generalization of strong monotone mappings, the main result in6is more general than the corresponding results in1,2.

Inspired and motivated by recent works in 1, 2, 6, the purpose of this paper is to introduce a new mathematical model, which is called a general system of A-monotone nonlinear variational inclusion problems, that is, a family of A-monotone nonlinear variational inclusion problems defined on a product set. This new mathematical model contains the system of inclusions in1,2,6, the variational inclusions in7,8, and some variational inequalities in literature as special cases. By using the resolvent technique for the A-monotone operators, we prove the existence and uniqueness of solution for this system of variational inclusions. We also prove the convergence of a multistep iterative algorithm approximating the solution for this system of variational inclusions. The result in this paper unifies, extends, and improves some results in1,2,6–8and the references therein.

2. Preliminaries

We suppose thatHis a real Hilbert space with norm and inner product denoted by · and ·,·, respectively, 2H _{denotes the family of all the nonempty subsets of}_{H. If}_M _:_{H →} ₂H

be a set-valued operator, then we denote the eﬀective domainDMofMas follows:

DM {x∈ H:Mx_/∅}. 2.1

Now we recall some definitions needed later.

Definition 2.1see2,6,7. LetA:H → Hbe a single-valued operator and letM:H → 2H be a set-valued operator.Mis said to be

im-relaxed monotone, if there exists a constantm >0 such that

x−y, u−v≥ −mu−v2_, _{∀u, v}_∈_D_M_{, x}_∈_{Mu, y}_∈_Mv, _2.2

iiA-monotone with a constantmif

aMism-relaxed monotone,

bAλMis maximal monotone forλ >0 i.e.,AλMH H, for allλ >0.

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It follow from3, Lemma 7.11we know that ifXis a reflexive Banach space withX∗ its dual, andA:X → X∗bem-strongly monotone andf :X → Ris a locally Lipschitz such that∂fisα-relaxed monotone, then∂fisA-monotone with a constantm−α.

Definition 2.3see1,7,8. LetA, T :H → H, be two single-valued operators.T is said to be

imonotone if

Tu−Tv, u−v ≥0, ∀u, v∈ H; 2.3

iistrictly monotone ifTis monotone and

Tu−Tv, u−v0, iﬀuv; 2.4

iiiγ-strongly monotone if there exists a constantγ >0 such that

Tu−Tv, u−v ≥γu−v2_, _{∀u, v}_{∈ H;} _2.5

ivs-Lipschitz continuous if there exists a constants >0 such that

Tu−Tv ≤su−v, ∀u, v∈ H; 2.6

vr-strongly monotone with respect toAif there exists a constantγ >0 such that

Tu−Tv, Au−Av ≥ru−v2_{, u, v}_{∈ H.} _2.7

Definition 2.4see2. LetA:H → Hbe aγ-strongly monotone operator and letM:H → 2Hbe anA-monotone operator. Then the resolvent operatorRA_M,λ:H → His defined by

RA

M,λx AλM−1x, ∀x∈ H. 2.8

We also need the following result obtained by Verma2.

Lemma 2.5. LetA : H → Hbe aγ-strongly monotone operator and letM : H → 2H be an A-monotone operator. Then, the resolvent operatorRA_M,λ : H → His Lipschitz continuous with constant1/γ−mλfor0< λ < γ/m, that is,

RA

M,λx−RAM,λ

y≤ _γ₋1_mλx−y, ∀x, y∈H. 2.9

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Definition 2.6. LetH1,H2, . . . ,Hpbe Hilbert spaces and · 1denote the norm ofH1, also let

A1:H1 → H1andN1:p_j₁Hj → H1be two single-valued mappings:

iN1is said to beξ-Lipschitz continuous in the first argument if there exists a constant

ξ >0 such that

_N₁

x1, x2, . . . , xp−N1

y1, x2, . . . , xp₁ ≤ξx1−y1₁,

∀x1, y1∈ H1, xj∈ Hj

j2,3, . . . , p; 2.10

iiN1is said to be monotone with respect toA1in the first argument if

N1

x1, x2, . . . , xp−N1

y1, x2, . . . , xp, A1x1−A1

y1

≥0, ∀x1, y1∈ H1, xj∈ Hj j2,3, . . . , p;

2.11

iiiN1is said to beβ-strongly monotone with respect toA1in the first argument if there

exists a constantβ >0 such that

N1

x1, x2, . . . , xp−N1

y1, x2, . . . , xp, A1x1−A1

y1

≥βx1−y12₁,

∀x1, y1∈ H1, xj∈ Hjj2,3, . . . , p;

2.12

ivN1is said to beγ-cocoercive with respect toA1in the first argument if there exists

a constantγ >0 such that

N1

x1, x2, . . . , xp−N1

y1, x2, . . . , xp, A1x1−A1

y1

≥γN1

x1, x2, . . . , xp−N1

y1, x2, . . . , xp2₁, ∀x1, y1∈ H1, xj∈ Hj j2,3, . . . , p;

2.13

vN1is said to beγ-relaxed cocoercive with respect toA1in the first argument if there

exists a constantγ >0 such that

N1

x1, x2, . . . , xp−N1

y1, x2, . . . , xp, A1x1−A1

y1

≥ −γN1

x1, x2, . . . , xp−N1

y1, x2, . . . , xp2₁, ∀x1, y1∈ H1, xj∈ Hj j 2,3, . . . , p;

2.14

viN1is said to beγ, r-relaxed cocoercive with respect toA1in the first argument if

there exists a constantγ >0 such that

N1

x1, x2, . . . , xp−N1

y1, x2, . . . , xp, A1x1−A1

y1

≥ −γN1

x1, x2, . . . , xp−N1

y1, x2, . . . , xp2₁rx1−y12₁,

∀x1, y1∈ H1, xj∈ Hj j2,3, . . . , p.

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In a similar way, we can define the Lipschitz continuity and the strong monotonicity

monotonicity, relaxed cocoercivity cocoercivity ofN_i : p_j₁H_j → H_i with respect to Ai:Hi → Hiin theith argumenti2,3, . . . , p.

3. A System of Set-Valued Variational Inclusions

In this section, we will introduce a new system of nonlinear variational inclusions in Hilbert spaces. In what follows, unless other specified, for eachi 1,2, . . . , p, we always suppose thatH_iis a Hilbert space with norm denoted by · _i,A_i : H_i → H_i,F_i : p_j₁H_j → H_i are single-valued mappings, andMi : Hi → 2Hi is a nonlinear mapping. We consider the following problem of findingx1, x2, . . . , xp∈pi1Hisuch that for eachi1,2, . . . , p,

0∈Fix1, x2, . . . , xp

Mixi. 3.1

Below are some special cases of3.1.

Ifp2, then3.1becomes the following problem of findingx1, x2∈ H1× H2such

that

0∈F1x1, x2 M1x1,

0∈F2x1, x2 M2x1.

3.2

However, 3.2 is called a system of set-valued variational inclusions introduced and researched by Fang and Huang1,9and Verma2,6.

Ifp 1, then3.1becomes the following variational inclusion with anA-monotone operator, which is to findx1∈ H1such that

0∈F1x1 M1x1, 3.3

problem3.3is introduced and studied by Fang and Huang8. It is easy to see that the mathematical model2studied by Verma7is a variant of3.3.

4. Existence of Solutions and Convergence of an Iterative Algorithm

In this section, we will prove existence and uniqueness of solution for 3.1. For our main results, we give a characterization of the solution of3.1as follows.

Lemma 4.1. Fori 1,2, . . . , p, letAi : Hi → Hi be a strictly monotone operator and letMi : Hi → 2Hi be anAi-monotone operator. Thenx1, x2, . . . , xp∈p_i1Hiis a solution of 3.1if and

only if for eachi1,2, . . . , p,

xiRA_Mi_i_,λ_iAixi−λiFix1, x2, . . . , xp, 4.1

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Proof. It holds thatx1, x2, . . . , xp∈p_i1Hiis a solution of3.1

⇐⇒θi∈Fix1, x2, . . . , xpMixi, i1,2, . . . , p,

⇐⇒Aixi−λiFix1, x2, . . . , xp∈AiλiMixi, i1,2, . . . , p,

⇐⇒xiRA_Mi_i_,λ_iAixi−λiFix1, x2, . . . , xp

, i1,2, . . . , p.

4.2

LetΓ {1,2, . . . , p}.

Theorem 4.2. For i 1,2, . . . , p, let Ai : Hi → Hi be γi-strongly monotone and let τi -Lipschitz continuous, M_i : H_i → 2Hi _{be an} A_i_{-monotone operator with a constant} m_i_{, let}

Fi :p_j1Hj → Hibe a single-valued mapping such thatFi isθi, ri-relaxed cocoercive monotone

with respect toA_iands_i-Lipschitz continuous in theith argument,F_iisl_ij-Lipschitz continuous in thejth arguments for eachj∈Γ, j /i. Suppose that there exist constantsλ_i>0i1,2, . . . , psuch that

1 γ1−m1λ1

τ2

1θ21−2λ1r12λ1θ1s21λ1 2_s2

1

p

k2

lk1λk γk−mkλk <1,

1 γ2−m2λ2

τ2

2θ22−2λ2r22λ2θ2s22λ2 2_s2

2

k∈Γ, k /2

lk2λk γk−mkλk <1,

. . . ,

1 γp−mpλp

τ2

pθp2−2λprp2λpθps2pλp2s2p p−1

k1

lk,pλk γk−mkλk <1.

4.3

Then,3.1admits a unique solution.

Proof. For i 1,2, . . . , p and for any given λ_i > 0, define a single-valued mapping T_i,λ_i :

p

j1Hj → Hiby

Ti,λi

x1, x2, . . . , xpR_MAi_i_,λ_iAixi−λiFix1, x2, . . . , xp, 4.4

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For anyx1, x2, . . . , xp,y1, y2, . . . , yp∈p_i1Hi, it follows from4.4andLemma 2.5

that fori1,2, . . . , p,

_T_i,λ

i

x1, x2, . . . , xp−Ti,λi

y1, y2, . . . , yp_i

RAi

Mi,λi

Aixi−λiFix1, x2, . . . , xp−RA_Mi_i_,λ_iAiyi−λiFiy1, y2, . . . , yp_i

≤ _γ 1 i−miλi

_A_i_x_i₋_A_i

yi−λiFix1, x2, . . . , xp−Fiy1, y2, . . . , yp_i

_A_i_x_i₋_A_i

yi

−λiFix1, x2, . . . , xi−1, xi, xi1, . . . , xp−Fix1, x2, . . . , xi−1, yi, xi1, . . . , xp_i

λi

γi−miλi

⎛ ⎝

j∈Γ,j /i

_F_i

x1, x2, . . . , xj−1, xj, xj1, . . . , xp

−Fix1, x2, . . . , xj−1, yj, xj1, . . . , xp_i

⎞ ⎠_.

4.5

Fori 1,2, . . . , p, sinceAi isτi-Lipschitz continuous, Fi isθi, ri-relaxed cocoercive with respected toAiandsi-Lipschitz continuous in theith argument, we have

_A_i_x_i₋_A_i

yi

−λiFix1, x2, . . . , xi−1, xi, xi1, . . . , xp−Fix1, x2, . . . , xi−1, yi, xi1, . . . , xp2_i

≤Aixi−Aiyi2_i

−2λiFix1, x2, . . . , xi−1, xi, xi1, . . . , xp

−Fix1, x2, . . . , xi−1, yi, xi1, . . . , xp, Aixi−Aiyi

λi2Fix1, x2, . . . , xi−1, xi, xi1, . . . , xp−Fix1, x2, . . . , xi−1, yi, xi1, . . . , xp2_i

≤τ2

ixi−yi2_i −2λirixi−yi2_i 2λiθis2ixi−yi2_i λi2s2ixi−yi2_i ≤τ2

i −2λiri2λiθis2i λi2s2i

xi−yi2_i.

4.6

Fori1,2, . . . , p, sinceFiislij-Lipschitz continuous in thejth argumentsj∈Γ, j /i, we have

_F_i

x1, x2, . . . , xj−1, xj, xj1, . . . , xp−Fix1, x2, . . . , xj−1, yj, xj1, . . . , xp_i ≤lijxj−yj_j,

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It follows from4.5–4.7that for eachi1,2, . . . , p,

_T_i,λ_i

x1, x2, . . . , xp

−Ti,λi

y1, y2, . . . , yp_i

τ2

i −2λiri2λiθis2i λi2s2ixi−yii λi γi−miλi

⎛ ⎝

j∈Γ,j /i

lijxj−yj_j

⎞ ⎠_.

4.8

Hence,

p

i1

_T_i,λ_i

x1, x2, . . . , xp−Ti,λi

y1, y2, . . . , yp_i

≤ p i1

⎡ ⎣ 1

γi−miλi

τ2

i −2λiri2λiθis2i λi2s2ixi−yi_i_γ λi i−miλi

⎛ ⎝

j∈Γ,j /i

lijxj−yj_j

⎞ ⎠ ⎤ ⎦ 1 γ1−m1λ1

τ2

1θ21−2λ1r12λ1θ1s21λ1 2_s2

1

p

k2

lk1λk γk−mkλk

_x₁₋_y₁

1

⎛

⎝ 1

γ2−m2λ2

τ2

2θ22−2λ2r22λ2θ2s22λ2 2_s2

2

k∈Γ,k /2

lk2λk γk−mkλk

⎞

⎠_x₂₋_y₂

2

· · ·

1 γp−mpλp

τ2

pθ2p−2λprp2λpθps2pλp2s2p p−1

k1

lk,pλk γk−mkλk

_x_p₋_y_p

p

≤ξ

_p

k1

_x_k₋_y_k

k

,

4.9

where

ξmax

1 γ1−m1λ1

τ2

1θ21−2λ1r12λ1θ1s21λ1 2_s2

1

p

k2

lk1λk

γk−mkλk,

1 γ2−m2λ2

τ2

2θ22−2λ2r22λ2θ2s22λ2 2_s2

2

k∈Γ,k /2

lk2λk γk−mkλk,

. . . ,

1 γp−mpλp

τ2

pθ2p−2λprp2λpθps2pλp2s2p p−1

k1

.

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Define · _Γ on p_i₁Hi by x1, x2, . . . , xp_Γ x11 x22 · · · xp_p, for all x1, x2, . . . , xp ∈ p_i1Hi. It is easy to see that

p

i1Hi is a Banach space. For any given

λi>0i∈Γ, defineWΓ,λ1,λ2,...,λp :

p i1Hi →

p i1Hiby

WΓ,λ1,λ2,...,λp

x1, x2, . . . , xp

T1,λ1x1, x2, . . . , xp, T2,λ2x1, x2, . . . , xp, . . . , Tp,λp

x1, x2, . . . , xp,

4.11

for allx1, x2, . . . , xp∈p_i1Hi.

By4.3, we know that 0< ξ <1, it follows from4.9that

WΓ,λ1,λ2,...,λp

x1, x2, . . . , xp−WΓ,λ1,λ2,...,λp

x1, x2, . . . , xp_Γ

≤ξx1, x2, . . . , xp−y1, y2, . . . , yp_Γ.

4.12

This shows that WΓ,λ1,λ2,...,λp is a contraction operator. Hence, there exists a unique

x1, x2, . . . , xp∈p_i1Hi, such that

WΓ,λ1,λ2,...,λp

x1, x2, . . . , xp

, 4.13

that is, fori1,2, . . . , p,

xiRA_Mi_i_,λ_iAixi−λiFix1, x2, . . . , xp. 4.14

ByLemma 4.1,x1, x2, . . . , xpis the unique solution of3.1. This completes this proof.

Corollary 4.3. For i 1,2, . . . , p, let Hi : Hi → Hi beγi-strongly monotone andτi-Lipschitz continuous, letM_i :Hi → 2Hi be anHi-monotone operator, letFi :p_j1Hj → Hi be a

single-valued mapping such thatFiisri-strongly monotone with respect toHiandsi-Lipschitz continuous in theith argument,Fiislij-Lipschitz continuous in thejth arguments for eachj ∈Γ, j /i. Suppose that there exist constantsλ_i>0 i1,2, . . . , psuch that

1 γ1

τ2

1−2λ1r1λ1 2_s2

1

p

k2

lk1λk γk <1,

1 γ2

τ2

2 −2λ2r2λ22s22

k∈Γ,k /2

lk2λk

γk <1, ..

.

1 γp

τ2

p−2λprpλp2s2p p−1

k1

lk,pλk γk <1.

4.15

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Remark 4.4. Theorem 4.2andCorollary 4.3unify, extend, and generalize the main results in

1,2,6–8.

5. Iterative Algorithm and Convergence

In this section, we will construct some multistep iterative algorithm for approximating the unique solution of3.1and discuss the convergence analysis of these Algorithms.

Lemma 5.1see8,9. Let{cn}and{kn}be two real sequences of nonnegative numbers that satisfy the following conditions:

10≤k_n<1, n0,1,2, . . .andlim sup n kn<1,

2cn1≤kncn, n0,1,2, . . . ,

thencnconverges to 0 asn → ∞.

Algorithm 5.2. Fori 1,2, . . . , p, letA_i, M_i, F_ibe the same as inTheorem 4.2. For any given

x0

1, x02, . . . , x0p∈

p

j1Hj, define a multistep iterative sequence{x1n, xn2, . . . , xnp}by

xn1

i αnxni 1−αn

RAi Mi,λi

Aixni

−λiFi

xn

1, xn2, . . . , xnp

, 5.1

where

0≤αn <1, lim sup

n αn<1. 5.2

Theorem 5.3. For i 1,2, . . . , p, let A_i, M_i, F_i be the same as in Theorem 4.2. Assume that all the conditions of theorem 4.1 hold. Then {x₁n, x₂n, . . . , x_pn} generated byAlgorithm 5.2 converges strongly to the unique solutionx1, x2, . . . , xpof 3.1.

Proof. ByTheorem 4.2, problem3.1admits a unique solutionx1, x2, . . . , xp, it follows from

Lemma 4.1that for eachi1,2, . . . , p,

xiRA_Mi_i_,λ_iAixi−λiFix1, x2, . . . , xp

. 5.3

It follows from4.3,5.1and5.3that for eachi1,2, . . . , p,

xn1

i −xi_i αn

xn i −xi

1−αn

RAi Mi,λi

Aixn_i−λiFi

xn

1, xn2, . . . , xnp

−RAi Mi,λi

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≤αnxni −xi_i 1−αn_γ 1 i−miλi

×Aixn_i−Aixi−λi

Fi

xn

1, xn2, . . . , xni−1, xni, xni1, . . . , xnp

−Fi

xn

1, xn2, . . . , xni−1, xi, xin1, . . . , xnp_i

_γ λi

i−miλi

⎛ ⎝

j∈Γ,j /i

Fi

xn

1, xn2, . . . , xnj−1, xjn, xjn1, . . . , xnp

−Fi

xn

1, xn2, . . . , xnj−1, xj, xnj1, . . . , xnp_i

⎞ ⎠_.

5.4

Fori 1,2, . . . , p, sinceA_i isτ_i-Lipschitz continuous, F_i isθ_i, r_i-relaxed cocoercive with respected toAi, andsi-Lipschitz is continuous in theith argument, we have

Aixni

−Aixi

−λi

Fi

xn

1, xn2, . . . , xni−1, xni, xin1, . . . , xnp

−Fi

xn

1, xn2, . . . , xni−1, xi, xin1, . . . , xnp 2

i

≤τ2

i −2λiri2λiθis2i λi2s2i

xn i −xi2.

5.5

Fori1,2, . . . , p, sinceF_iisl_ij-Lipschitz continuous in thejth argumentsj∈Γ, j /i, we have

Fi

xn

1, xn2, . . . , xnj−1, xnj, xnj1, . . . , xnp

−Fi

xn

1, x2n, . . . , xjn−1, xj, xnj1, . . . , xpn_i ≤lijxnj −xj_j.

5.6

It follows from5.4–5.6that fori1,2, . . . , p,

xn1

i −xi_i≤αnxni −xii 1−αn 1 γi−miλi

τ2

i −2λiri2λiθis2i λi2s2ixni −xii

1−α_n λi γi−miλi

⎛ ⎝

j∈Γ,j /i

lijxnj −xj_j

⎞ ⎠_.

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Hence,

p

i1

xn1

i −xi_i≤ p

i1

⎡ ⎣_α_n_xn

i −xi_i 1−αn_γ 1 i−miλi

τ2

i −2λiri2λiθis2i λi2s2ixni −xi_i

1−α_n λi

γi−miλi

⎛ ⎝

j∈Γ,j /i

lijxnj −xj_j

⎞ ⎠ ⎤ ⎦

≤αn

_p

i1

_xn i −xi_i

1−αnξ

_p

i1

_xn i −xi_i

ξ 1−ξαn

_p

i1

_xn i −xii

,

5.8

where

ξmax

1 γ1−m1λ1

τ2

1θ21−2λ1r12λ1θ1s21λ1 2_s2

1

p

k2

lk1λk γk−mkλk,

1 γ2−m2λ2

τ2

2θ22−2λ2r22λ2θ2s22λ22s22

k∈Γ,k /2

lk2λk

γk−mkλk,

. . . ,

1 γp−mpλp

τ2

pθp2−2λprp2λpθps2pλp2s2p p−1

k1

.

5.9

It follows from hypothesis4.3that 0< ξ <1.

Letanp_i1xni −xii, ξnξ1−ξαn. Then,5.8can be rewritten asan1≤ξnan, n 0,1,2, . . . .By5.2, we know that lim sup_nξn<1, it follows fromLemma 5.1that

an p

i1

_xn

i −xiiconverges to 0 as n−→ ∞. 5.10

Therefore,{xn₁, xn₂, . . . , xn_p}converges to the unique solutionx1, x2, . . . , xpof3.1. This completes the proof.

Acknowledgment

This study was supported by grants from National Natural Science Foundation of China

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