Mathematical and Computer Modelling 51 (2010) 63–71

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Mathematical and Computer Modelling

journal homepage: www.elsevier.com/locate/mcm

Perturbed Mann iterative method with errors for a new system ofgeneralized nonlinear variational-like inclusionsZeqing Liu a, Min Liu a, Shin Min Kang b,∗, Sunhong Lee ba Department of Mathematics, Liaoning Normal University, P. O. Box 200, Dalian, Liaoning 116029, People’s Republic of Chinab Department of Mathematics and the Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Korea

a r t i c l e i n f o

Article history:Received 8 April 2009Accepted 10 August 2009

Keywords:System of generalized nonlinearvariational-like inclusionsPerturbed Mann iterative method witherrorsConvergenceStability

a b s t r a c t

In this paper, a new system of generalized nonlinear variational-like inclusions isintroduced and investigated in Hilbert spaces. By means of the resolvent operatortechnique, the existence and uniqueness of solution for the systemof generalized nonlinearvariational-like inclusions is demonstrated. Moreover, a perturbed Mann iterative methodwith errors for approximating the solution of the system of generalized nonlinearvariational-like inclusions is constructed and the convergence and stability of the iterativesequence generated by the algorithm is discussed. The results presented in this papergeneralize and unify many known results in the literature.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Variational inclusions are important generalizations and extensions of the classical variational inequalities and havebeen studied intensively in recent years. For more details, we refer to [1–17] and the references therein. Among themethods for solving variational inclusion problems, it is worthwhile to mention that the resolvent operator techniqueshave become more and more popular. In particular, the applications of the generalized resolvent operator technique havebeen explored and improved recently. For instance, Fang and Huang [3] introduced a class of H-monotone operators anddefined the associated class of resolvent operators, which extends the classes of the resolvent operators associated withη-subdifferential operators of Ding and Luo [2], maximal monotone operators of Liu, Agarwal and Kang [13] and maximalη-monotone operators of Huang and Fang [6], respectively. Fang andHuang [4] created a class of (H, η)-monotone operatorsand solved a system of variational inclusions regarding with the (H, η)-monotone operators. In a recent paper [12], Lanfurther introduced a class of (A, η)-accretive operators, which offers a unifying framework for the classes of maximalmonotone operators, maximal η-monotone operators,H-monotone operators and (H, η)-monotone operators. On the otherhand, some classical techniques have been considered by many researchers in studying variational inequalities. Ansariand Yao [1] studied a system of variational inequalities by using fixed-point theorem. Huang and Fang [7] introduced asystem of order complementary problems and established some results by utilizing fixed-point theorem. Liu, Chen, Shimand Kang [15] investigated a generalized nonlinear quasi-variational-like inclusion dealing with (h, η)-proximal mappingby applying fixed-point theorem. Kassay and Kolumban [9] introduced a system of variational inequalities and proved anexistence theorem in view of Fan’s lemma.Inspired and motivated by the above achievements, in this paper, we study a new system of generalized nonlinear

variational-like inclusions in Hilbert spaces, in which the s-(G, η)-maximal monotone operators are involved. In Section 2,we give some necessary concepts and lemmas. In Section 3, we supply a lemma which instructs the characterization of

∗ Corresponding author. Fax: +82 55 755 1917.E-mail addresses: zeqingliu@dl.cn (Z. Liu), min_liu@yeah.net (M. Liu), smkang@gnu.ac.kr (S.M. Kang), sunhong@gnu.ac.kr (S. Lee).

0895-7177/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2009.08.041

64 Z. Liu et al. / Mathematical and Computer Modelling 51 (2010) 63–71

solution of the system of generalized nonlinear variational-like inclusions and prove the existence and uniqueness of itssolution. In Section 4, we construct a perturbed Mann iterative method with errors for approximating the solution of thesystem of generalized nonlinear variational-like inclusions and discuss the convergence and stability of iterative sequencegenerated by the algorithm. The results presented in this paper improve and extend many known results in the literature.

2. Preliminaries

In what follows, unless otherwise specified, we assume that Hi is a real Hilbert space endowedwith norm ‖ · ‖i and innerproduct 〈·, ·〉i, and 2Hi denotes the family of all nonempty subsets of Hi for i ∈ {1, 2}. Now let us recall some concepts.

Definition 2.1. Let A : H1 → H2, f : H1 → H1, η : H1 × H1 → H1 be mappings.(1) A is said to be Lipschitz continuous if there exists a constant α > 0 such that

‖Ax− Ay‖2 ≤ α‖x− y‖1, ∀x, y ∈ H1;(2) A is said to be λ-expanding if there exists a constant λ > 0 such that

‖Ax− Ay‖2 ≥ λ‖x− y‖1, ∀x, y ∈ H1;(3) f is said to be δ-strongly monotone if there exists a constant δ > 0 such that

〈fx− fy, x− y〉1 ≥ δ‖x− y‖21, ∀x, y ∈ H1;(4) f is said to be δ-η-strongly monotone if there exists a constant δ > 0 such that

〈fx− fy, η(x, y)〉1 ≥ δ‖x− y‖21, ∀x, y ∈ H1;(5) f is said to be ζ -relaxed Lipschitz if there exists a constant ζ > 0 such that

〈fx− fy, x− y〉1 ≤ −ζ‖x− y‖21, ∀x, y ∈ H1.

Definition 2.2. Let N : H2 × H1 × H1 × H2 → H1, A : H1 → H2, h, B : H2 → H1, C : H1 → H1,D : H2 → H2 be mappings.The mapping N is called(1) δ-strongly monotonewith respect to A in the first argument if there exists a constant δ > 0 such that

〈N(Au, x, y, z)− N(Av, x, y, z), u− v〉1 ≥ δ‖u− v‖21, ∀u, v, x, y ∈ H1, z ∈ H2;(2) ξ -h-cocoercive in the second argument if there exists a constant ξ > 0 such that

〈N(x, u, y, z)− N(x, v, y, z), h(u)− h(v)〉1 ≥ ξ‖N(x, u, y, z)− N(x, v, y, z)‖21, ∀u, v, y ∈ H1, x, z ∈ H2;(3) ι-expanding in the second argument, if there exists a constant ι > 0 such that

‖N(x, u, y, z)− N(x, v, y, z)‖1 ≥ ι‖u− v‖1, ∀u, v, y ∈ H1, x, z ∈ H2;(4) τ -relaxed Lipschitz with respect to C in the third argument if there exists a constant τ > 0 such that

〈N(x, y, Cu, z)− N(x, y, Cv, z), u− v〉1 ≤ −τ‖u− v‖21, ∀u, v, y ∈ H1, x, z ∈ H2;(5) θ-h-relaxed monotonewith respect to D in the fourth argument if there exists a constant θ > 0 such that

〈N(x, y, z,Du)− N(x, y, z,Dv), h(u)− h(v)〉1 ≥ −θ‖u− v‖21, ∀u, v, x ∈ H2, y, z ∈ H1;(6) (θ, ϕ, %)-h-relaxed cocoercive in the fourth argument if there exist nonnegative constants θ, ϕ and % such that

〈N(x, y, z, u)− N(x, y, z, v), h(u)− h(v)〉1 ≥ −θ‖N(x, y, z, u)− N(x, y, z, v)‖21 − ϕ‖h(u)− h(v)‖21

+ %‖u− v‖21, ∀u, v, x ∈ H2, y, z ∈ H1;(7) Lipschitz continuous in the first argument if there exists a constant µ > 0 such that

‖N(u, x, y, z)− N(v, x, y, z)‖1 ≤ µ‖u− v‖1, ∀u, v, z ∈ H2, x, y ∈ H1.

Similarly, we can define the Lipschitz continuity of N in the second, third and fourth arguments, respectively.For i ∈ {1, 2}, j ∈ {1, 2} \ {i}, letMi : Hj × Hi → 2Hi be a multi-valued mapping. For each (x2, x1) ∈ H1 × H2, define the

graph ofMi(xi, ·) : Hi → 2Hi bygraph(Mi(xi, ·)) = {(y, z) : z ∈ Mi(xi, y)},

where the domain ofMi(xi, ·) is defined bydomMi(xi, ·) = {y ∈ Hi : there exists z ∈ Hi such that (y, z) ∈ graph(Mi(xi, ·))}.

Definition 2.3. For i ∈ {1, 2}, j ∈ {1, 2} \ {i}, let Mi : Hj × Hi → 2Hi , ηi : Hi × Hi → Hi be mappings, for each xi ∈ Hj,Mi(xi, ·) : Hi → 2Hi is said to be si-ηi-relaxed monotone if there exists a constant si > 0 such that

〈x∗ − y∗, ηi(x, y)〉i ≥ −si‖x− y‖2i , ∀(x, x∗), (y, y∗) ∈ graph(Mi(xi, ·)).

Z. Liu et al. / Mathematical and Computer Modelling 51 (2010) 63–71 65

Definition 2.4. For i ∈ {1, 2}, j ∈ {1, 2} \ {i}, let Mi : Hj × Hi → 2Hi ,Gi : Hi → Hi be mappings, for each xi ∈ Hj,Mi(xi, ·) : Hi → 2Hi is said to be si-(Gi, ηi)-maximal monotone if(B1) Mi(xi, ·) is si-ηi-relaxed monotone;(B2) (Gi + ρiMi(xi, ·))Hi = Hi for ρi > 0.

Lemma 2.1 ([12]). Let H be a real Hilbert space, η : H × H → H be a mapping, G : H → H be a d-η-stronglymonotone mapping and M : H → 2H be a s-(G, η)-maximal monotone mapping. Then the generalized resolvent operatorRG,ηM,ρ = (G+ ρM)

−1: H → H is singled-valued for d > ρs > 0.

Lemma 2.2 ([12]). Let H be a real Hilbert space, η : H × H → H be a σ -Lipschitz continuous mapping, G : H → H be a d-η-strongly monotonemapping, and M : H → 2H be a s-(G, η)-maximal monotone mapping. Then the generalized resolvent operator RG,ηM,ρ : H → His σd−ρs -Lipschitz continuous for d > ρs > 0.

Let i ∈ {1, 2} and j ∈ {1, 2} \ {i}. Assume that A1,D2, h2 : H1 → H2, B1, C2, h1 : H2 → H1, C1, B2 : H1 → H1,D1, A2 :H2 → H2, ηi : Hi × Hi → Hi,Ni : H2 × H1 × H1 × H2 → Hi, fi, gi : Hi → Hi are mappings, Mi : Hj × Hi → 2Hisatisfies that for each xi ∈ Hj,Mi(xi, ·) is si-(Gi, ηi)- maximal monotone, where Gi : Hi → Hi is di-ηi-strongly monotone andRange(fi − gi)

⋂domMi(xi, ·) 6= ∅. We consider the following problem of finding (x, y) ∈ H1 × H2 such that

x ∈ N1(A1x, B1y, C1x,D1y)− h1(y)+M1(y, (f1 − g1)x),y ∈ N2(A2y, B2x, C2y,D2x)− h2(x)+M2(x, (f2 − g2)y),

(2.1)

where (fi − gi)x = fi(x) − gi(x) for x ∈ Hi and i ∈ {1, 2}. The problem (2.1) is called a system of generalized nonlinearvariational-like inclusions.Special cases:If H1 = H2 = H , x = y, A1 = A2 = A, B1 = B2 = B, C1 = C2 = C , D1 = D2 = D, M1(u, v) = M2(u, v) = M(v, u),

f1 = f2 = f , g1 = g2 = g , h1(z) = h2(z) = x0 − z, N1(u, v, w, ν) = N2(u, v, w, ν) = N(u, v) − N ′(w, ν) for eachz, u, v, w, ν ∈ H , then the problem (2.1) reduces to finding x ∈ H such that

x0 ∈ N(Ax, Bx)− N ′(Cx,Dx)+M((f − g)x, x), (2.2)which was studied in Liu, Agarwal and Kang [13]. Furthermore, if x0 = 0, N(Ax, Bx) − N ′(Cx,Dx) = N(x) − N ′(x) for anyx ∈ H , then the problem (2.2) changes into finding x ∈ H such that

0 ∈ N(x)− N ′(x)+M((f − g)x, x),which was considered by Huang [5].If A1 = B1 = B2 = C2 = f1 − g1 = f2 − g2 = I , h1(x) = h2(x) = x0 − x, N1(u, v, w, ν) = N1(u, v), N2(u, v,

w, ν) = N2(v,w),M1(y, x) = M1(x),M2(x, y) = M2(y) for each v,w, x ∈ H1, u, ν, y ∈ H2, then the problem (2.1) collapsesto finding (x, y) ∈ H1 × H2 such that

0 ∈ N1(x, y)+M1(x),0 ∈ N2(x, y)+M2(y),

(2.3)

which was studied by Fang and Huang [4] with the assumptions thatMi is (Gi, ηi)-monotone for i ∈ {1, 2}.For appropriate and suitable choices of themappingsNi, Ai, Bi, Ci,Di, hi,Mi, fi, gi for i ∈ {1, 2}, the problem (2.1) includes

a number of variational and variational-like inclusions as special cases.

3. Existence and uniqueness theorem

In this section, we prove the existence and uniqueness of solution of the system of generalized nonlinear variational-likeinclusions.

Lemma 3.1. (x, y) ∈ H1 × H2 is a solution of the system of generalized nonlinear variational-like inclusions if and only if(x, y) ∈ H1 × H2 satisfies

f1(x) = g1(x)+ RG1,η1M1(y,·),ρ1

[x+ G1((f1 − g1)x)+ ρ1h1(y)− ρ1N1(A1x, B1y, C1x,D1y)],

f2(y) = g2(y)+ RG2,η2M2(x,·),ρ2

[y+ G2((f2 − g2)y)+ ρ2h2(x)− ρ2N2(A2y, B2x, C2y,D2x)],

where ρ1 and ρ2 are two positive constants,

RG1,η1M1(x1,·),ρ1(u) = (G1 + ρ1M1(x1, ·))−1(u), RG2,η2M2(x2,·),ρ2

(v) = (G2 + ρ2M2(x2, ·))−1(v), ∀(u, v) ∈ H1 × H2.

Theorem 3.1. For i ∈ {1, 2}, let ηi : Hi × Hi → Hi be Lipschitz continuous with constant σi, A1,D2, h2 : H1 → H2, B1, C2,h1 : H2 → H1, C1, B2 : H1 → H1, D1, A2 : H2 → H2, fi, gi : Hi → Hi be Lipschitz continuous with constants α1, κ2, ϑh2 ,

66 Z. Liu et al. / Mathematical and Computer Modelling 51 (2010) 63–71

β1, γ2, ϑh1 , γ1, β2, κ1, α2, ϑfi , ϑgi , respectively, fi−gi be δfi,gi-strongly monotone, Bi be λi-expanding, Ni : H2×H1×H1×H2 →Hi be Lipschitz continuous in the first, second, third and fourth arguments with constants µi, νi, ωi, υi, respectively, δi-stronglymonotone with respect to Ai in the first argument, be ξi-hi-cocoercive and ιi-expanding in the second argument, be τi-relaxedLipschitz with respect to Ci in the third argument, be (θi, ϕi, %i)-hi-relaxed cocoercive in the fourth argument, Gi : Hi → Hibe ti-Lipschitz continuous and di − ηi-strongly monotone, and Gi(fi − gi) be ζi-relaxed Lipschitz. Let M1 : H2 × H1 →2H1 , M2 : H1 × H2 → 2H2 satisfy that for each (x2, x1) ∈ H1 × H2,Mi(xi, ·) : Hi → 2Hi is si − (Gi, ηi)-maximal monotone,Range(fi − gi)

⋂domMi(xi, ·) 6= ∅ and

(C1) ‖RGi,ηiMi(yi,·),ρi(x)− RGi,ηiMi(zi,·),ρi

(x)‖i ≤ c‖yi − zi‖j, ∀x ∈ Hi, yi, zi ∈ Hj, i ∈ {1, 2}, j ∈ {1, 2} \ {i}.

If there exist positive constants ρ1 and ρ2 such that

(C2) d1 > ρ1s1, d2 > ρ2s2;

(C3) k := max{m1 +

σ1

d1 − ρ1s1(r1 + ρ1l1)+

σ2

d2 − ρ2s2ρ2χ2,m2 +

σ2

d2 − ρ2s2(r2 + ρ2l2)

+σ1

d1 − ρ1s1ρ1χ1

}+ c < 1,

where

mi =√1− 2δfi,gi + (ϑfi + ϑgi)2,

li =√µ2i α

2i − 2δi + 1+

√ω2i γ

2i − 2τi + 1,

ri =√1− 2ζi + t2i (ϑfi + ϑgi)2,

χi =

√(νiβi + υiκi)2 − 2(ξiιiλi − θiυiκi − ϕiϑhi + %i)+ ϑ

2hi

for i ∈ {1, 2}, then the system of generalized nonlinear variational-like inclusions possesses a unique solution in H1 × H2.

Proof. For each (x, y) ∈ H1 × H2, define

Fρ1(x, y) = x− (f1 − g1)x+ RG1,η1M1(y,·),ρ1

[x+ G1((f1 − g1)x)+ ρ1h1(y)− ρ1N1(A1x, B1y, C1x,D1y)],

Fρ2(x, y) = y− (f2 − g2)y+ RG2,η2M2(x,·),ρ2

[y+ G2((f2 − g2)y)+ ρ2h2(x)− ρ2N2(A2y, B2x, C2y,D2x)].

For each (u1, v1), (u2, v2) ∈ H1 × H2, it follows from (C1) and Lemma 2.2 that

‖Fρ1(u1, v1)− Fρ1(u2, v2)‖1 ≤ ‖u1 − u2 − [(f1 − g1)u1 − (f1 − g1)u2]‖1 +σ1

d1 − ρ1s1{‖u1 − u2

+G1((f1 − g1)u1)− G1((f1 − g1)u2)‖1 + ρ1‖h1(v1)− h1(v2)− [N1(A1u1, B1v1, C1u1,D1v1)−N1(A1u2, B1v2, C1u2,D1v2)]‖1} + c‖v1 − v2‖2. (3.1)

Because f1 − g1 is δf1,g1-strongly monotone, f1, g1,G1 are Lipschitz continuous, and G1(f1 − g1) is ζ1-relaxed Lipschitz, wededuce that

‖u1 − u2 − [(f1 − g1)u1 − (f1 − g1)u2]‖21 ≤ (1− 2δf1,g1 + (ϑf1 + ϑg1)2)‖u1 − u2‖21, (3.2)

‖u1 − u2 + G1((f1 − g1)u1)− G1((f1 − g1)u2)‖21 ≤ (1− 2ζ1 + t21 (ϑf1 + ϑg1)

2)‖u1 − u2‖21. (3.3)

Since A1, B1, C1,D1 are all Lipschitz continuous, B1 is λ1-expanding, N1 is δ1-strongly monotone with respect to A1,ξ1-h1-cocoercive in the second argument, τ1-relaxed Lipschitz with respect to C1, (θ1, ϕ1, %1) − h1-relaxed cocoercive inthe fourth argument, and is Lipschitz continuous in the first, second, third and fourth arguments, respectively, we get that

‖N1(A1u1, B1v1, C1u1,D1v1)− N1(A1u2, B1v1, C1u1,D1v1)− (u1 − u2)‖21

≤ (µ21α21 − 2δ1 + 1)‖u1 − u2‖

21, (3.4)

‖N1(A1u2, B1v2, C1u1,D1v1)− N1(A1u2, B1v2, C1u2,D1v1)+ u1 − u2‖21

≤ (ω21γ21 − 2τ1 + 1)‖u1 − u2‖

21, (3.5)

‖N1(A1u2, B1v1, C1u1,D1v1)− N1(A1u2, B1v2, C1u1,D1v1)+N1(A1u2, B1v2, C1u2,D1v1)− N1(A1u2, B1v2, C1u2,D1v2)− (h1(v1)− h1(v2))‖21

≤ [(ν1β1 + υ1κ1)2− 2(ξ1ι1λ1 − θ1υ1κ1 − ϕ1ϑh1 + %1)+ ϑ

2h1 ]‖v1 − v2‖

22. (3.6)

Z. Liu et al. / Mathematical and Computer Modelling 51 (2010) 63–71 67

In terms of (3.1)–(3.6), we obtain that

‖Fρ1(u1, v1)− Fρ1(u2, v2)‖ ≤ m1‖u1 − u2‖1 +σ1

d1 − ρ1s1[(r1 + ρ1l1)‖u1 − u2‖1

+ ρ1χ1‖v1 − v2‖2] + c‖v1 − v2‖2. (3.7)

Similarly, we infer that

‖Fρ2(u1, v1)− Fρ2(u2, v2)‖ ≤ m2‖v1 − v2‖2 +σ2

d2 − ρ2s2[(r2 + ρ2l2)‖v1 − v2‖2

+ ρ2χ2‖u1 − u2‖1] + c‖u1 − u2‖1. (3.8)

Linking (3.7) and (3.8) implies that

‖Fρ1(u1, v1)− Fρ1(u2, v2)‖1 + ‖Fρ2(u1, v1)− Fρ2(u2, v2)‖2 ≤ k(‖u1 − u2‖1 + ‖v1 − v2‖2). (3.9)

Define ‖ · ‖∗ on H1 × H2 by ‖(u, v)‖∗ = ‖u‖1 + ‖v‖2 for any (u, v) ∈ H1 × H2. It is easy to see that (H1 × H2, ‖ · ‖∗) is aBanach space. Define Lρ1,ρ2 : H1 × H2 → H1 × H2 by

Lρ1,ρ2(u, v) = (Fρ1(u, v), Fρ2(u, v)), ∀(u, v) ∈ H1 × H2.

By virtue of (C2), (C3) and (3.9), we get that 0 < k < 1 and

‖Lρ1,ρ2(u1, v1)− Lρ1,ρ2(u2, v2)‖∗ ≤ k‖(u1, v1)− (u2, v2)‖∗,

which means that Lρ1,ρ2 : H1 × H2 → H1 × H2 is a contraction. According to Banach fixed-point theorem, there exists aunique (x, y) ∈ H1 × H2 such that Lρ1,ρ2(x, y) = (x, y). That is,

f1(x) = g1(x)+ RG1,η1M1(y,·),ρ1

[x+ G1((f1 − g1)x)+ ρ1h1(y)− ρ1N1(A1x, B1y, C1x,D1y)],

f2(y) = g2(y)+ RG2,η2M2(x,·),ρ2

[y+ G2((f2 − g2)y)+ ρ2h2(x)− ρ2N2(A2y, B2x, C2y,D2x)].

By Lemma 3.1, we derive that (x, y) is a unique solution of the system of generalized nonlinear variational-like inclusions.This completes the proof. �

4. Algorithm and convergence

In this section, we construct an perturbed Mann iterative method with errors for approximating the unique solutionof the system of generalized nonlinear variational-like inclusions based on Lemma 3.1 and Theorem 3.1, and discuss theconvergence and stability of the iterative sequences generated by the algorithm.

Algorithm 4.1. For i ∈ {1, 2}, let ηi, Ai, Bi, Ci,Di, hi, fi, gi,Gi,Mi,Ni be the same as those in Theorem 3.1. For any (x0, y0) ∈H1 × H2, compute the iterative sequence {(xn, yn)}n≥0 by

xn+1 = (1− an)xn + an{xn − (f1 − g1)xn + RG1n,η1nM1n(yn,·),ρ1

[xn + G1((f1 − g1)xn)+ ρ1h1(yn)

− ρ1N1(A1xn, B1yn, C1xn,D1yn)]} + anjn,

yn+1 = (1− an)yn + an{yn − (f2 − g2)yn + RG2n,η2nM2n(xn,·),ρ2

[yn + G2((f2 − g2)yn)+ ρ2h2(xn)

− ρ2N2(A2yn, B2xn, C2yn,D2xn)]} + anj′n, ∀n ≥ 0,

(4.1)

where {(jn, j′n)}n≥0 is a sequence in H1 × H2 introduced to take into account possible inexact computation which satisfieslimn→∞ ‖jn‖ = limn→∞ ‖j′n‖ = 0, and the sequence {an}n≥0 satisfies the following conditions

0 < a < an ≤ 1, ∀n ≥ 0,

where a is a constant. Let {(un, vn)}n≥0 be any sequence in H1 × H2 and define εn = ε1n + ε2n for n ≥ 0 by

ε1n = ‖un+1 − {(1− an)un + an{un − (f1 − g1)un + RG1n,η1nM1n(vn,·),ρ1

[un + G1((f1 − g1)un)+ ρ1h1(vn)

− ρ1N1(A1un, B1vn, C1un,D1vn)]} + anjn}‖,

ε2n = ‖vn+1 − {(1− an)vn + an{vn − (f2 − g2)vn + RG2n,η2nM2n(un,·),ρ2

[vn + G2((f2 − g2)vn)+ ρ2h2(un)

− ρ2N2(A2vn, B2un, C2vn,D2un)]} + anj′n}‖.

(4.2)

Definition 4.1. LetH be a real Hilbert space. For n ≥ 0, letMn,M : H → 2H be sn-(Gn, ηn)-maximalmonotone and s-(G, η)-maximal monotone, respectively. The sequence {Mn}n≥0 is said to be graph convergence to M , noted by Mn gc

−→M , if for each

(x, y) ∈ graph(M), there’s a sequence {(xn, yn)}n≥0 ⊆ graph(Mn) such that xn → x, yn → y as n→∞.

68 Z. Liu et al. / Mathematical and Computer Modelling 51 (2010) 63–71

Lemma 4.1. Let H be a real Hilbert space. For n ≥ 0, let ηn, η : H × H → H be σn-Lipschitz continuous and σ -Lipschitzcontinuous, respectively, Gn : H → H be dn-ηn-strongly monotone and tn-Lipschitz continuous, G : H → H be d-η-stronglymonotone and t-Lipschitz continuous, Mn,M : H → 2H be sn − (Gn, ηn)-maximal monotone and s-(G, η)-maximal monotone,respectively. Assume that

{σn

dn−ρsn

}n≥0 and

{σntndn−ρsn

}n≥0 are bounded and limn→∞ Gn(x) = G(x) for each x ∈ H. If Mn gc−→

M, then

limn→∞

RGn,ηnMn,ρ (x) = RG,ηM,ρ(x), ∀x ∈ H,

where d > ρs > 0.

Proof. It follows from H = (G + ρM)H that for each u ∈ H , there exists (x, y) ∈ graphM such that u = G(x) + ρy. SinceMn gc−→M , it follows that there exists a sequence {(xn, yn)}n≥0 ⊆ graph(Mn) such that limn→∞ xn = x and limn→∞ yn = y. Put

un = Gn(xn)+ρyn. Note that RG,ηM,ρ(G(x)+ρy) = x and R

Gn,ηnMn,ρ (Gn(xn)+ρyn) = xn, keeping in mind Lemma 2.2, we infer that∥∥∥RGn,ηnMn,ρ (u)− R

G,ηM,ρ(u)

∥∥∥ ≤ ∥∥∥RGn,ηnMn,ρ (un)− RG,ηM,ρ(u)

∥∥∥+ ∥∥∥RGn,ηnMn,ρ (un)− RGn,ηnMn,ρ (u)

∥∥∥≤

∥∥∥RGn,ηnMn,ρ (Gn(xn)+ ρyn)− RG,ηM,ρ(G(x)+ ρy)

∥∥∥+ σn

dn − ρsn‖un − u‖

≤ ‖xn − x‖ +σn

dn − ρsn[‖Gn(xn)− G(x)‖ + ρ‖yn − y‖]

≤

(1+

σntndn − ρsn

)‖xn − x‖ +

σn

dn − ρsn[‖Gn(x)− G(x)‖ + ρ‖yn − y‖]

→ 0 as n→∞.

This completes the proof. �

Lemma 4.2 ([18]). Let {an}n≥0, {bn}n≥0, {cn}n≥0 be nonnegative sequences satisfying

an+1 ≤ (1− λn)an + λnbn + cn, ∀n ≥ 0,

where

{λn}∞

n=0 ⊂ [0, 1],∞∑n=0

λn = +∞,

∞∑n=0

cn < +∞, limn→∞

bn = 0.

Then limn→∞ an = 0.

Definition 4.2 ([13]). Let n ≥ 0, T be a self-mapping of H, x0 ∈ H , xn+1 = f (T , xn) be an iteration procedure which yieldsa sequence of points {xn}n≥0 ⊂ H , where f is a continuous mapping. Suppose that {x ∈ H : Tx = x} 6= ∅ and {xn}n≥0converges to a fixed-point x∗ of T . Let {un}n≥0 ⊂ H, En = ‖un+1− f (T , un)‖. If limn→∞ En = 0 implies that limn→∞ un = x∗,then the iteration procedure defined by xn+1 = f (T , xn) is said to be T -stable or stablewith respect to T .

Theorem 4.1. Let all the conditions of Theorem 3.1 hold. For i ∈ {1, 2}, j ∈ {1, 2} \ {i}, n ≥ 0, let ηin : Hi × Hi → Hi beσin-Lipschitz continuous, Gin : Hi → Hi be din-ηin-strongly monotone and tin-Lipschitz continuous and Min : Hj × Hi → 2Hisatisfy that(C4) Min(xi, ·) : Hi → 2Hi is sin − (Gin, ηin)-maximal monotone for xi ∈ Hj;(C5) Min(xi, ·)gc

−→Mi(xi, ·) and Gin(x)→ Gi(x) as n→∞ for each xi ∈ Hj, x ∈ Hi;

(C6) ‖RGin,ηinMin(y,·),ρi(x)− RGin,ηinMin(z,·),ρi

(x)‖i ≤ cn‖y− z‖j, ∀x ∈ Hi, y, z ∈ Hj;(C7) din → di, sin → si, σin → σi, tin → ti, cn → c as n→∞.

If there exist positive constants ρ1 and ρ2 such that (C2), (C3) and(C8) din > ρisin, ∀n ≥ 0, i ∈ {1, 2} then

(a) the iterative sequence {(xn, yn)}n≥0 generated by Algorithm 4.1 converges to the unique solution (x, y) of the system ofgeneralized nonlinear variational-like inclusions;

(b) for any sequences {(un, vn)}n≥0 in H1×H2, limn→∞(un, vn) = (x, y) if and only if limn→∞ εn = 0, where εn is definedby (4.2).

Proof. It follows fromTheorem3.1 that the systemof generalized nonlinear variational-like inclusions has a unique solution(x, y) ∈ H1 × H2. By Lemma 3.1, we gain that

x = (1− an)x+ an{x− (f1 − g1)(x)+ RG1,η1M1(y,·),ρ1

[x+ G1((f1 − g1)x)+ ρ1h1(y)

− ρ1N1(A1x, B1y, C1x,D1y)]},

y = (1− an)y+ an{y− (f2 − g2)(y)+ RG2,η2M2(x,·),ρ2

[y+ G2((f2 − g2)y)+ ρ2h2(x)

− ρ2N2(A2y, B2x, C2y,D2x)]}.

(4.3)

Z. Liu et al. / Mathematical and Computer Modelling 51 (2010) 63–71 69

Let

kn = max{m1 +

σ1n

d1n − ρ1s1n(r1 + ρ1l1)+

σ2nρ2

d2n − ρ2s2nχ2,m2 +

σ2n

d2n − ρ2s2n(r2 + ρ2l2)

+σ1nρ1

d1n − ρ1s1nχ1

}+ cn, ∀n ≥ 0.

It is clear that for k′ = 12 (k + 1) ∈ (k, 1), there exists N0 ≥ 1 such that kn < k

′, ∀n ≥ N0. In terms of (4.1) and (4.3), (C4),(C6) and (C8), we obtain that for any n ≥ N0

‖xn+1 − x‖1 ≤ (1− an)‖xn − x‖1 + an‖xn − x− [(f1 − g1)xn − (f1 − g1)x]‖1+ an‖R

G1n,η1nM1n(yn,·),ρ1

(xn + G1((f1 − g1)xn)+ ρ1h1(yn)− ρ1N1(A1xn, B1yn, C1xn,D1yn))

− RG1n,η1nM1n(yn,·),ρ1(x+ G1((f1 − g1)x)+ ρ1h1(y)− ρ1N1(A1x, B1y, C1x,D1y))‖1

+ an‖RG1n,η1nM1n(yn,·),ρ1

(x+ G1((f1 − g1)x)+ ρ1h1(y)− ρ1N1(A1x, B1y, C1x,D1y))

− RG1n,η1nM1n(y,·),ρ1(x+ G1((f1 − g1)x)+ ρ1h1(y)− ρ1N1(A1x, B1y, C1x,D1y))‖1

+ an‖RG1n,η1nM1n(y,·),ρ1

(x+ G1((f1 − g1)x)+ ρ1h1(y)− ρ1N1(A1x, B1y, C1x,D1y))

− RG1,η1M1(y,·),ρ1(x+ G1((f1 − g1)x)+ ρ1h1(y)− ρ1N1(A1x, B1y, C1x,D1y))‖1 + an‖jn‖1

≤ (1− an)‖xn − x‖1 + an‖xn − x− [(f1 − g1)xn − (f1 − g1)x]‖1

+ anσ1n

d1n − ρ1s1n(‖xn − x+ G1((f1 − g1)xn)− G1((f1 − g1)x)‖1

+ ρ1‖h1(yn)− h1(y)− [N1(A1xn, B1yn, C1xn,D1yn)−N1(A1x, B1y, C1x,D1y)]‖1)+ ancn‖yn − y‖2 + anbM1n + an‖jn‖1

≤

[(1− an)+ anm1 + an

σ1n

d1n − ρ1s1n(r1 + ρ1l1)

]‖xn − x‖1

+

[an

σ1nρ1

d1n − ρ1s1nχ1 + ancn

]‖yn − y‖2 + anbM1n + an‖jn‖1, (4.4)

where

bM1n = ‖RG1n,η1nM1n(y,·),ρ1

(x+ G1((f1 − g1)x)+ ρ1h1(y)− ρ1N1(A1x, B1y, C1x,D1y))

− RG1,η1M1(y,·),ρ1(x+ G1((f1 − g1)x)+ ρ1h1(y)− ρ1N1(A1x, B1y, C1x,D1y))‖1.

Similarly, we obtain that

‖yn+1 − y‖2 ≤[(1− an)+ anm2 + an

σ2n

d2n − ρ2s2n(r2 + ρ2l2)

]‖yn − y‖2

+

[an

σ2nρ2

d2n − ρ2s2nχ2 + ancn

]‖xn − x‖1 + anbM2n + an‖j

′

n‖2, ∀n ≥ 0, (4.5)

where

bM2n = ‖RG2n,η2nM2n(x,·),ρ1

(y+ G2((f2 − g2)y)+ ρ2h2(x)− ρ2N2(A2y, B2x, C2y,D2x))

− RG2,η2M2(x,·),ρ2(y+ G2((f2 − g2)y)+ ρ2h2(x)− ρ2N2(A2y, B2x, C2y,D2x))‖2.

It follows from (4.4) and (4.5) that

‖xn+1 − x‖1 + ‖yn+1 − y‖2 ≤ [1− a(1− k′)](‖xn − x‖1 + ‖yn − y‖2)

+ bM1n + bM2n + ‖jn‖1 + ‖j

′

n‖2, ∀n ≥ N0, (4.6)

which means that (xn, yn)→ (x, y) as n→∞ by Algorithm 4.1 and Lemma 4.2.

70 Z. Liu et al. / Mathematical and Computer Modelling 51 (2010) 63–71

Now we demonstrate (b). By using (4.2) and (4.3), and as in the proof of (4.6), we deduce that

‖un+1 − x‖1 + ‖vn+1 − y‖2 ≤ ‖un+1 − {(1− an)un + an{un + (f1 − g1)un+ [RG1n,η1nM1n(vn,·),ρ1

(un + G1((f1 − g1)un))+ ρ1h1(vn)− ρ1N1(A1un, B1vn, C1un,D1vn)]} + anjn}‖1

+‖(1− an)un + an{un + (f1 − g1)un + [RG1n,η1nM1n(vn,·),ρ1

(un + G1((f1 − g1)un))+ ρ1h1(vn)

− ρ1N1(A1un, B1vn, C1un,D1vn)]} + anjn − x‖1 + ‖vn+1 − {(1− an)vn + an{vn + (f2 − g2)vn+ [RG2n,η2nM2n(un,·),ρ2

(vn + G2((f2 − g2)vn))+ ρ2h2(un)− ρ2N2(A2vn, B2un, C2vn,D2un)]} + anj′n}‖2

+‖(1− an)vn + an{vn + (f2 − g2)vn + [RG2n,η2nM2n(un,·),ρ2

(vn + G2((f2 − g2)vn))

+ ρ2h2(un)− ρ2N2(A2vn, B2un, C2vn,D2un)]} + anj′n − y‖2

≤ [1− a(1− k′)](‖un − x‖1 + ‖vn − y‖2)+ an(hM1n + hM2n + ‖jn‖1 + ‖j

′

n‖2)+ εn, ∀n ≥ N0. (4.7)

Suppose that limn→∞ εn = 0. By (4.7) and Lemma 4.2, we get that (un, vn)→ (x, y) as n→∞.Conversely, suppose that (un, vn)→ (x, y) as n→∞. In view of (4.7), we infer that

εn = ε1n + ε

2n ≤ ‖un+1 − x‖1 + ‖vn+1 − y‖2 + [1− a(1− k

′)](‖un − x‖1 + ‖vn − y‖2)

+ hM1n + hM2n + ‖jn‖1 + ‖j

′

n‖2, ∀n ≥ N0.

In the light of Lemma 4.2, we gain that limn→∞ εn = 0. This completes the proof. �

Remark 4.1. We have the same conclusions as in Theorems 3.1 and 4.1 if the (θi, ϕi, %i)-hi-relaxed cocoercivity of Ni in thefourth argument and

χi =

√(νiβi + υiκi)2 − 2(ξiιiλi − θiυiκi − ϕiϑhi + %i)+ ϑ

2hi

are replaced by the θi − hi-relaxed monotonicity of Ni with respect to Di in the fourth argument and χi =√(νiβi + υiκi)2 − 2(ξiιiλi − θi)+ ϑ2hi for i ∈ {1, 2}, respectively.

Remark 4.2. Theorem4.1 extend, improve and unify Theorem3.1 of Fang andHuang [3], Theorem4.1 of Fang andHuang [4],Theorem 4.1 of Huang [5], Theorem 2.1 of He, Lou and He [8], Theorem 3.1 of Kazmi and Bhat [11], Theorem 3.5–3.8 ofLiu, Kang and Ume [16]; the class of s-(G, η)-maximal monotone operators in this paper is much wider and more generalthan those of maximal monotone operators in [5], maximal η-monotone operators in [6], H-monotone operators in [3,8,16]and (H, η)-monotone operators in [6]; the class of (θ, ϕ, %)-h-relaxed cocoercive operators includes that of (α, ξ)-relaxedcocoercive operators of Lan [12] as a special case.

Acknowledgement

This work was supported by the Science Research Foundation of Educational Department of Liaoning Province(2009A419).

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