The Solvability of a New System of Nonlinear Variational-Like Inclusions

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Volume 2009, Article ID 609353,9pages doi:10.1155/2009/609353

Research Article

The Solvability of a New System of Nonlinear

Variational-Like Inclusions

Zeqing Liu,

1

_{Min Liu,}

1

_{Jeong Sheok Ume,}

2

_{and Shin Min Kang}

3

1_{Department of Mathematics, Liaoning Normal University, P.O. Box 200, Dalian Liaoning 116029, China} 2_{Department of Applied Mathematics, Changwon National University, Changwon 641-773, South Korea} 3_{Department of Mathematics and Research Institute of Natural Science, Gyeongsang National University,}

Jinju 660-701, South Korea

Correspondence should be addressed to Jeong Sheok Ume,[email protected]

Received 23 November 2008; Accepted 1 April 2009

Recommended by Marlene Frigon

We introduce and study a new system of nonlinear variational-like inclusions involvings-G, η -maximal monotone operators, strongly monotone operators, η-strongly monotone operators, relaxed monotone operators, cocoercive operators, λ, ξ-relaxed cocoercive operators,ζ, ϕ,

-g-relaxed cocoercive operators and relaxed Lipschitz operators in Hilbert spaces. By using the resolvent operator technique associated withs-G, η-maximal monotone operators and Banach contraction principle, we demonstrate the existence and uniqueness of solution for the system of nonlinear variational-like inclusions. The results presented in the paper improve and extend some known results in the literature.

Copyrightq2009 Zeqing Liu et al. 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

It is well known that the resolvent operator technique is an important method for solving various variational inequalities and inclusions1–20. In particular, the generalized resolvent operator technique has been applied more and more and has also been improved intensively. For instance, Fang and Huang5introduced the class ofH-monotone operators and defined the associated resolvent operators, which extended the resolvent operators associated with η-subdiﬀerential operators of Ding and Luo3and maximalη-monotone operators of Huang and Fang 6, respectively. Later, Liu et al. 17 researched a class of general nonlinear implicit variational inequalities including theH-monotone operators. Fang and Huang4

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enriched and improved the class of generalized resolvent operators. Lan 10 studied a system of general mixed quasivariational inclusions involvingA, η-accretive mappings in q-uniformly smooth Banach spaces. Lan et al. 14 constructed some iterative algorithms for solving a class of nonlinear A, η-monotone operator inclusion systems involving nonmonotone set-valued mappings in Hilbert spaces. Lan9investigated the existence of solutions for a class ofA, η-accretive variational inclusion problems with nonaccretive set-valued mappings. Lan 11 analyzed and established an existence theorem for nonlinear parametric multivalued variational inclusion systems involving A, η-accretive mappings in Banach spaces. By using the random resolvent operator technique associated withA, η -accretive mappings, Lan 13 established an existence result for nonlinear random multi-valued variational inclusion systems involvingA, η-accretive mappings in Banach spaces. Lan and Verma15studied a class of nonlinear Fuzzy variational inclusion systems with A, η-accretive mappings in Banach spaces. On the other hand, some interesting and classical techniques such as the Banach contraction principle and Nalder’s fixed point theorems have been considered by many researchers in studying variational inclusions.

Inspired and motivated by the above achievements, we introduce a new system of nonlinear variational-like inclusions involving s-G, η-maximal monotone operators in Hilbert spaces and a class ofζ, ϕ, -g-relaxed cocoercive operators. By virtue of the Banach’s fixed point theorem and the resolvent operator technique, we prove the existence and uniqueness of solution for the system of nonlinear variational-like inclusions. The results presented in the paper generalize some known results in the field.

2. Preliminaries

In what follows, unless otherwise specified, we assume that Hi is a real Hilbert space

endowed with norm · _iand inner product·,·_i, and 2Hi_{denotes the family of all nonempty} subsets ofHifori∈ {1,2}.Now let’s recall some concepts.

Definition 2.1. LetA:H1 → H2, f, g:H1 → H1, η:H1×H1 → H1be mappings.

1Ais said to beLipschitz continuous, if there exists a constantα >0 such that

Ax−Ay₂≤αx−y₁, ∀x, y∈H1; 2.1

2Ais said to ber-expanding,if there exists a constantr >0 such that

Ax−Ay₂≥rx−y₁, ∀x, y∈H1; 2.2

3fis said to beδ-strongly monotone,if there exists a constantδ >0 such that

fx−fy, x−y₁≥δx−y2₁, ∀x, y∈H1; 2.3

4fis said to beδ-η-strongly monotone,if there exists a constantδ >0 such that

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5f is said to beζ, ϕ, -g-relaxed cocoercive,if there exist nonnegtive constantsζ, ϕ andsuch that

fx−fy, gx−gy₁≥ −ζfx−fy2₁−ϕgx−gy2₁x−y2₁, ∀x, y∈H1; 2.5

6gis said to beζ-relaxed Lipschitz,if there exists a constantζ >0 such that

gx−gy, x−y₁≤ −ζx−y2₁, ∀x, y∈H1. 2.6

Definition 2.2. LetN:H2×H1×H2 → H1, A, C:H1 → H2, B:H2 → H1be mappings.N is called

1 λ, ξ-relaxed cocoercive with respect to A in the first argument, if there exist nonnegative constantsλ, ξsuch that

NAu, x, y−NAv, x, y, u−v₁

≥ −λAu−Av2

2ξu−v21 , ∀u, v, x∈H1, y∈H2;

2.7

2θ-cocoercivewith respect toBin the second argument, if there exists a constantθ >0 such that

Nx, Bu, y−Nx, Bv, y, u−v₁≥θBu−Bv2₁, ∀u, v, x, y∈H2; 2.8

3τ-relaxed Lipschitzwith respect toCin the third argument, if there exists a constant τ >0 such that

Nx, y, Cu−Nx, y, Cv, u−v₁≤ −τu−v2₁, ∀u, v, y∈H1, x∈H2; 2.9

4τ-relaxed monotonewith respect toCin the third argument, if there exists a constant τ >0 such that

Nx, y, Cu−Nx, y, Cv, u−v₁≥ −τu−v2₁, ∀u, v, y∈H1, x∈H2; 2.10

5Lipschitz continuousin the first argument, if there exists a constantμ >0 such that

Nu, x, y−Nv, x, y₁≤μu−v₁, ∀u, v, y∈H2, x∈H1. 2.11

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Definition 2.3. Fori∈ {1,2}, j ∈ {1,2} \ {i}, letMi :Hj×Hi → 2Hi, ηi :Hi×Hi → Hibe

mappings. For each givenx2, x1∈H1×H2andi∈ {1,2}, Mixi,·:Hi → 2Hiis said to be

si-ηi-relaxed monotone,if there exists a constantsi>0 such that

x∗−y∗, ηi

x, y_i≥ −six−y2_i, ∀x, x∗,

y, y∗∈graphMixi,·. 2.12

Definition 2.4. Fori∈ {1,2}, j ∈ {1,2}\{i}, letMi:Hj×Hi → 2Hi, Gi:Hi → Hibe mappings.

For any givenx2, x1∈H1×H2andi∈ {1,2}, Mixi,· :Hi → 2Hi is said to besi-Gi, ηi

-maximal monotone,ifB1Mixi,·issi-ηi-relaxed monotone;B2 GiρiMixi,·Hi Hi

forρi>0.

Lemma 2.5see8. LetHbe a real Hilbert space,η:H×H → Hbe a mapping,G:H → Hbe a

d-η-strongly monotone mapping andM:H → 2H_{be a}_s_-_{G, η}_{-maximal monotone mapping. Then}

the generalized resolvent operatorRG,η_M,ρ GρM−1:H → His singled-valued ford > ρs >0.

Lemma 2.6see8. LetHbe a real Hilbert space,η:H×H → Hbe aσ-Lipschitz continuous

mapping,G : H → Hbe a d-η-strongly monotone mapping, andM : H → 2H _{be a}_s_-_{G, η}

-maximal monotone mapping. Then the generalized resolvent operatorRG,η_M,ρ:H → Hisσ/d−ρs -Lipschitz continuous ford > ρs >0.

Fori ∈ {1,2}andj ∈ {1,2} \ {i}, assume thatAi, Ci : Hi → Hj, Bi : Hj → Hi, ηi :

Hi×Hi → Hi, Ni :Hj×Hi×Hj → Hi, fi, gi :Hi → Hiare single-valued mappings,Mi :

Hj×Hi → 2Hisatisfies that for each givenxi∈Hj, Mixi,·issi-Gi, ηi-maximal monotone,

whereGi : Hi → Hi isdi-ηi-strongly monotone and Rangefi−gi

domMixi,·/∅.We

consider the following problem of findingx, y∈H1×H2such that

x∈N1

A1x, B1y, C1x

M1

y,f1−g1

x,

y∈N2A2y, B2x, C2yM2x,f2−g2y, 2.13

wherefi−gix fix−gixforx ∈ Hi andi ∈ {1,2}. The problem2.13is called the

system of nonlinear variational-like inclusions problem. Special cases of the problem2.13are as follows.

IfA1 B1B2 C2 f1−g1 f2−g2 I,N1x, y, z N1x, y x,N2u, v, w N2v, w w,M1x, y M1y,M2u, v M2vfor eachx, z, v∈H2, y, u, w∈H1, then the problem2.13collapses to findingx, y∈H1×H2such that

0∈N1

x, yM1x,

0∈N2x, yM2y, 2.14

which was studied by Fang and Huang4with the assumption thatMiisGi, ηi-monotone

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IfHi H, AiA, BiB, CiC, MiM, fi f, gig,andNiu, v, w Nu, v, for

allu, v, w∈Hfori∈ {1,2}, then the problem2.13reduces to findingx∈Hsuch that

0∈NAx, Bx Mx,f−gx, 2.15

which was studied in Shim et al.19.

It is easy to see that the problem 2.13 includes a number of variational and variational-like inclusions as special cases for appropriate and suitable choice of the mappingsNi, Ai, Bi, Ci, Mi, fi, gifori∈ {1,2}.

3. Existence and Uniqueness Theorems

In this section, we will prove the existence and uniqueness of solution of the problem2.13.

Lemma 3.1. Letρ1 and ρ2 be two positive constants. Thenx, y ∈ H1 ×H2 is a solution of the

problem2.13if and only ifx, y∈H1×H2satisfies that

f1x g1x RG1,η1 M1y,·,ρ1

xG1f1−g1x−ρ1N1A1x, B1y, C1x ,

f2yg2yRG2,η2 M2x,·,ρ2

yG2f2−g2y−ρ2N2A2y, B2x, C2y ,

3.1

where RG1,η1

M1y,·,ρ1u G1ρ1M1y,·

−1_u_{, R}G2,η2

M2x,·,ρ2v G2ρ2M2x,·

−1_v_{, for all} u, v∈H1×H2.

Theorem 3.2. Fori ∈ {1,2}, j ∈ {1,2} \ {i},let ηi : Hi×Hi → Hi be Lipschitz continuous

with constantσi,Ai, Ci : Hi → Hj, Bi : Hj → Hi, fi, gi : Hi → Hi be Lipschitz continuous

with constantsαi, γi, βi, ϑfi, ϑgi respectively,Ni :Hj×Hi×Hj → Hibe Lipschitz continuous in

the first, second and third arguments with constantsμi, νi, ωi respectively, letNi beλi, ξi-relaxed

cocoercive with respect toAiin the first argument, andτi-relaxed Lipschitz with respect toCi in the

third argument,fibeζi, ϕi, i-gi-relaxed cocoercive,fi−gibeδfi,gi-strongly monotone,Gi:Hi → Hibeti-Lipschitz continuous anddi-ηi-strongly monotone, andGifi−gibeζi-relaxed Lipschitz,

Mi:Hj×Hi → 2Hisatisfy that for each fixedxi∈Hj, Mixi,·:Hi → 2Hiissi-Gi, ηi-maximal

monotone, Rangefi−gi∩domMixi,·/∅and

RGi,ηi

Miyi,·,ρix−R

Gi,ηi

Mizi,·,ρix i

≤ryi−zi_j, ∀x∈Hi, yi, zi∈Hj, i∈ {1,2}, j ∈ {1,2} \ {i}.

3.2

If there exist positive constantsρ1, ρ2, andksuch that

di> ρisi, i∈ {1,2}, 3.3

kmax

m1 σ1 d1−ρ1s1

c1ρ1l1 σ2 d2−ρ2s2

χ2, m2 σ2 d2−ρ2s2

c2ρ2l2 σ1 d1−ρ1s1

χ1

r <1,

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where

mi

1−2δfi,gi

ϑ2

fi2

ζiϑfiϕiϑgi−i ϑ2 gi , ci

1−2ζit2_i

ϑfiϑgi 2

,

li

μ2_iα2_i 2λiαi−ξi 1

ω2_iγ_i2−2τi1,

χiρiνiβi, i∈ {1,2},

3.5

then the problem2.13possesses a unique solution inH1×H2.

Proof. For anyx, y∈H1×H2, define

Fρ1

x, yx−f1−g1

xRG1,η1 M1y,·,ρ1

xG1

f1−g1

x−ρ1N1

A1x, B1y, C1x ,

Fρ2

x, yy−f2−g2yRG2,η2 M2x,·,ρ2

yG2f2−g2y−ρ2N2A2y, B2x, C2y . 3.6

For eachu1, v1,u2, v2∈H1×H2,it follows fromLemma 2.6that

Fρ1u1, v1−Fρ1u2, v21

≤u1−u2−f1−g1u1−f1−g1u2 ₁ σ1 d1−ρ1s1

×u1−u2G1f1−g1u1−G1f1−g1u2₁ ρ1N1A1u1, B1v1, C1u1−N1A1u2, B1v2, C1u21

rv1−v22.

3.7

Becausef1−g1isδf1,g1-strongly monotone,f1, g1andG1are Lipschitz continuous, andG1f1−

g1isζ1-relaxed Lipschitz, we deduce that

u1−u2−

f1−g1

u1−

f1−g1

u2 2₁

≤1−2δf1,g1

ϑ_f2

12

ζ1ϑf1ϕ1ϑg1−1

ϑ2_g₁u1−u22₁,

3.8

_u1₋_u2_G1

f1−g1u1−G1f1−g1u22₁

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SinceA1, B1, C1are all Lipschitz continuous,N1isλ1, ξ1-relaxed cocoercive with respect to A1,τ1-relaxed Lipschitz with respect toC1, and is Lipschitz continuous in the first, second and third arguments, respectively, we infer that

N1A1u1, B1v1, C1u1−N1A1u2, B1v1, C1u1−u1−u221

≤μ2₁α2₁2λ1α1−ξ1 1

u1−u221,

3.10

N1A1u2, B1v2, C1u1−N1A1u2, B1v2, C1u2 u1−u22₁

≤ω2₁γ₁2−2τ11

u1−u221,

3.11

N1A1u2, B1v1, C1u1−N1A1u2, B1v2, C1u1 ≤ν1β1v1−v22.

3.12

In terms of3.7–3.12, we obtain that

_F_ρ

1u1−v1−Fρ1u2, v2

≤m1u1−u21 σ1 d1−ρ1s1

c1ρ1l1

u1−u21χ1v1−v22 rv1−v22. 3.13

Similarly, we deduce that

_F_ρ

2u1, v1−Fρ2u2, v2

≤m2v1−v22 σ2 d2−ρ2s2

c2ρ2l2v1−v2₂χ2u1−u2₁ ru1−u2₁.

3.14

Define · _∗onH1×H2 byu, v_∗ u₁v₁ for anyu, v∈H1×H2.It is easy to see thatH1×H2, · _∗is a Banach space. DefineLρ1,ρ2:H1×H2 → H1×H2by

Lρ1,ρ2u, v

Fρ1u, v, Fρ2u, v

, ∀u, v∈H1×H2. 3.15

By virtue of3.3,3.4,3.13and3.14, we achieve that 0< k <1 and

_L_ρ

1,ρ2u1, v1−Lρ1,ρ2u2, v2_∗≤ku1, v1−u2, v2∗, 3.16

which means thatLρ1,ρ2 :H1×H2 → H1×H2is a contractive mapping. Hence, there exists a

uniquex, y∈H1×H2such thatLρ1,ρ2x, y x, y.That is,

f1x g1x R_MG1,η₁_y,1_·_,ρ₁

xG1

f1−g1

x−ρ1N1

A1x, B1y, C1x ,

f2

yg2

yRG2,η2 M2x,·,ρ2

yG2

f2−g2

y−ρ2N2

A2y, B2x, C2y .

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ByLemma 3.1, we derive thatx, yis a unique solution of the problem2.13. This completes the proof.

Theorem 3.3. Fori∈ {1,2}, j ∈ {1,2} \ {i},letηi, Ai, Ci, Mi, fi, gi, fi−gi, Gibe all the same as in

Theorem 3.2,Bi :Hj → Hiberi-expanding,Ni :Hj×Hi×Hj → Hibe Lipschitz continuous in

the first, second and third arguments with constantsμi, νi, ωirespectively, andNibeλi, ξi-relaxed

cocoercive with respect toAiin the first argument, beθi-cocoercive with respect toBiin the second

argument, beτi-relaxed Lipschtz with respect to Ci in the third argument. If there exist constants

ρ1, ρ2andksuch that3.3and3.4, but

citi

ϑ2_f i2

ζiϑfiϕiϑgi−i

ϑ2gi, χi

ρ2_iν2_iβ2_i −2ρiθiri1, i∈ {1,2}, 3.18

then the problem2.13possesses a unique solution inH1×H2.

Theorem 3.4. Fori∈ {1,2}, j∈ {1,2} \ {i},letηi, Ai, Bi, Ci, Mi, fi, gi, fi−gi, Gi, Gifi−gibe all

the same as inTheorem 3.2,Ni : Hj×Hi×Hj → Hibe Lipschitz continuous in the first, second

and third arguments with constantsμi, νi, ωirespectively, andNi beλi, ξi-relaxed cocoercive with

respect toAiin the first argument, beθi-relaxed Lipschitz with respect toBiin the second argument,

beτi-relaxed monotone with respect toCiin the third argument. If there exist constantsρ1, ρ2andk

such that3.3and3.4, but

li

μiαiωiγi

2₂_λ

iαi−ξiτi 1, χiρi

ν2

iβ2i −2θi1, i∈ {1,2}, 3.19

then the problem2.13possesses a unique solution inH1×H2.

Remark 3.5. In this paper, there are three aspects which are worth of being mentioned as follows:

1Theorem 3.2extends and improves in4, Theorem 3.1and in19, Theorem 4.1;

2the class of ζ, ϕ, -g-relaxed cocoercive operators includes the class of α, ξ -relaxed cocoercive operators in8as a special case;

3the class ofs-G, η-maximal monotone operators is a generalization of the classes of η-subdiﬀerential operators in 3, maximal η-monotone operators in 6, H-monotone operators in5andH, η-monotone operators in4.

Acknowledgments

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