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volume 7, issue 3, article 114, 2006.

Received 27 July, 2004;

accepted 09 May, 2006.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

GENERALIZED NONLINEAR MIXED QUASI-VARIATIONAL

INEQUALITIES INVOLVING MAXIMALη-MONOTONE MAPPINGS

MAO-MING JIN

Department of Mathematics Fuling Teachers College Fuling, Chongqing 408003 People’s Republic of China.

EMail:mmj1898@163.com

c

2000Victoria University ISSN (electronic): 1443-5756 144-04

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Generalized Nonlinear Mixed Quasi-Variational Inequalities Involving Maximalη-Monotone

Mappings Mao-Ming Jin

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J. Ineq. Pure and Appl. Math. 7(3) Art. 114, 2006

Abstract

In this paper, we introduce and study a new class of generalized nonlinear mixed quasi-variational inequalities involving maximal η-monotone mapping.

Using the resolvent operator technique for maximalη-monotone mapping, we prove the existence of solution for this kind of generalized nonlinear multi-valued mixed quasi-variational inequalities without compactness and the convergence of iterative sequences generated by the new algorithm. We also discuss the convergence and stability of the iterative sequence generated by the perturbed iterative algorithm for solving a class of generalized nonlinear mixed quasi- variational inequalities.

2000 Mathematics Subject Classification:49J40; 47H10.

Key words: Mixed quasi-variational inequality, Maximalη-monotone mapping, Re- solvent operator, Convergence, stability.

This work was supported by the National Natural Science Foundation of China (69903012) and the Educational Science Foundation of Chongqing, Chongqing of China (021301).

1. Introduction

In recent years, variational inequalities have been generalized and extended in many different directions using novel and innovative techniques. These have been used to study wider classes of unrelated problems arising in optimization and control, economics and finance, transportation and electrical networks, op- erations research and engineering sciences in a general and unified framework, see [1] – [15], [18] – [27] and the references therein. An important and use- ful generalization of variational inequality is called the variational inclusion.

It is well known that one of the most important and interesting problems in the theory of variational inequalities is the development of an efficient and im- plementable algorithm for solving variational inequalities. For the past years, many numerical methods have been developed for solving various classes of variational inequalities, such as the projection method and its variant forms, linear approximation, descent, and Newton’s methods.

Recently, Huang and Fang [10] introduced a new class of maximalη-monotone mappings and proved the Lipschitz continuity of the resolvent operator for max- imalη-monotone mappings in Hilbert spaces. They also introduced and studied a new class of generalized variational inclusions involving maximalη-monotone mappings and constructed a new algorithm for solving this class of generalized variational inclusions by using the resolvent operator technique for maximal η-monotone mappings.

The main purpose of this paper is to introduce and study a new class of generalized nonlinear mixed quasi-variational inequalities involving maximal η-monotone mappings. Using the resolvent operator technique for maximal η-monotone mappings, we prove the existence of a solution for this kind of

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generalized nonlinear multivalued mixed quasi-variational inequalities without compactness and the convergence of iterative sequences generated by the new algorithm. We also discuss the convergence and stability of the iterative sequence generated by the perturbed iterative algorithm for solving a class of generalized nonlinear mixed quasi-variational inequalities. The results shown in this paper improve and extend the previously known results in this area.

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2. Preliminaries

Let H be a real Hilbert space endowed with a normk·k and an inner product h·,·i, respectively. Let 2^H, CB(H), and H(·,·) denote the family of all the nonempty subsets ofH, the family of all the nonempty closed bounded subsets of H, and the Hausdorff metric on CB(H), respectively. Let η, N : H × H → H be two single-valued mappings with two variables and g : H → H be a single-valued mapping. Let S, T, G : H → CB(H)be three multivalued mappings andM :H×H →2^H be a multivalued mapping such that for each t ∈H,M(·, t)is maximalη-monotone withRan(g)T

DomM(·, t)6=∅. Now we consider the following problem:

Findu∈ H, x ∈Su, y ∈ T u, and z ∈ Gusuch thatg(u) ∈ Dom(M(·, z)) and

(2.1) 0∈N(x, y) +M(g(u), z)).

Problem (2.1) is called a generalized nonlinear multivalued mixed quasi-variational inequality.

Some special cases of the problem (2.1):

(I) If η(x, y) = x−y for allx, y in H and G is the identity mapping, then problem (2.1) reduces to finding u ∈ H, x ∈ Su, y ∈ T u such that g(u)∈Dom(M(·, u))and

(2.2) 0∈N(x, y) +M(g(u), u).

Problem (2.2) is called the multivalued quasi-variational inclusion, which was studied by Noor [18] – [22].

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(II) If S, T are single-valued mappings and G is the identity mapping, then problem (2.1) is equivalent to findingu∈Hsuch thatg(u)∈Dom(M(·, u)) and

(2.3) 0∈N(Su, T u) +M(g(u), u)).

Problem (2.3) is called a generalized nonlinear mixed quasi-variational inequality.

(III) If M(·, t) = ∂ϕ(·, t), where ϕ : H ×H → RS

{+∞} is a functional such that for each fixed t in H, ϕ(·, t) : H → RS

{+∞} is lower semicontinuous and η-subdifferentiable on H, and ∂ϕ(·, t) denotes the η-subdifferential of ϕ(·, t), then problem (2.1) reduces to the following problem:

Findu∈H, x∈Suandy ∈T usuch that

(2.4) hN(x, y), η(v, g(u))i ≥ϕ(g(u), z)−ϕ(v, z)

for all v in H, which which appears to be a new one. Furthermore, if N(x, y) =x−yfor allx, y inH,S, T are single-valued mappings andG is the identity mapping, then problem (2.4) reduces to the general quasi- variational-like inclusion considered by Ding and Luo [3].

(IV) If S, T : H → H are single-valued mappings, g is an identity mapping, N(x, y) =x−yfor allx, yinH, andM(·, t) =∂ϕfor alltinH, where

∂ϕdenotes theη-subdifferential of a proper convex lower semicontinuous functionϕ :H→RS

{+∞}, then problem (2.1) reduces to the following problem:

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Findu∈Hsuch that

(2.5) hSu−T u, η(v, u)i ≥ϕ(u)−ϕ(v)

for allvinH, which is called the strongly nonlinear variational-like inclusion problem considered by Lee et al. [15].

(V) IfGis an identity mapping,η(x, y) =x−yandM(·, t) =∂ϕfor eacht∈ H, whereϕ :H → RS

{+∞}is a proper convex lower semicontinuous function onHandg(H)TDom(∂ϕ(·, t))6=∅for eacht∈Hand∂ϕ(·, t) denotes the subdifferential of functionϕ(·, t), then problem (2.1) reduces to finding u ∈ H, x ∈ Su andy ∈ T usuch that g(u) ∈ Dom(∂ϕ(·, t)) and

(2.6) hN(x, y), v−g(u)i ≥ϕ(g(u))−ϕ(v)

for all v in H. Furthermore, if N(x, y) = x−y for all x, y in H, and g is an identity mapping, then the problem (2.6) is equivalent to the set- valued nonlinear generalized variational inclusion considered by Huang [6] and, in turn, includes the variational inclusions studied by Hassouni and Moudafi [5] and Kazmi [14] as special cases.

For a suitable choice of N, η, M, S, T, G, g, and for the space H, one can obtain a number of known and new classes of variational inclusions, variational inequalities, and corresponding optimization problems from the general set-valued variational inclusion problem (2.1). Furthermore, these types of variational inclusions enable us to study many important problems arising in the mathematical, physical, and engineering sciences in a general and unified framework.

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Definition 2.1. Let T be a selfmap of H, x₀ ∈ H and let x_n+1 = f(T, x_n) define an iteration procedure which yields a sequence of points{xn}^∞_n=0 inH.

Suppose that{x ∈ H : T x = x} 6= ∅ and{x_n}^∞_n=0 converges to a fixed point x^∗ ofT. Let{y_n} ⊂H and let_n =||y_n+1−f(T, y_n)||. If lim

n→∞_n= 0implies that lim

n→∞y_n = x^∗, then the iteration procedure defined by x_n+1 = f(T, x_n)is said to beT-stable or stable with respect toT.

Lemma 2.1 ([16]). Let{a_n},{b_n}, and{c_n}be three sequences of nonnegative numbers satisfying the following conditions: there existsn0such that

an+1 ≤(1−tn)an+bntn+cn, for alln ≥ n0, wheretn ∈ [0,1],P∞

n=0tn = ∞, lim

n→∞bn = 0andP∞

n=0cn <

∞. Then lim

n→∞a_n= 0.

Definition 2.2. A mappingg :H→H is said to be

(i) α-strongly monotone if there exists a constantα >0such that hg(u1)−g(u2), u1−u2i ≥αku1−u2k², for allui ∈H, i= 1,2;

(ii) β-Lipschitz continuous if there exists a constantβ >0such that kg(u₁)−g(u₂)k ≤βku₁−u₂k,

for allu_i ∈H, i= 1,2.

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Definition 2.3. A multivalued mappingS :H→CB(H)is said to be (i) H-Lipschitz continuous if there exists a constantγ >0such that

H(Su₁, Su₂)≤γku₁ −u₂k, for allu_i ∈H, i= 1,2;

(ii) strongly monotone with respect to the first argument ofN(·,·) :H×H→ H, if there exists a constantµ > 0such that

hN(x₁,·)−N(x₂,·), u₁−u₂i ≥µku₁−u₂k², for allx_i ∈Su_i, u_i ∈H, i= 1,2.

Definition 2.4. A mappingN(·,·) : H ×H → H is said to be Lipschitz con- tinuous with respect to the first argument if there exists a constant ν > 0such that

kN(u₁,·)−N(u₂,·)k ≤νku₁−u₂k, for allu_i ∈H, i = 1,2.

In a similar way, we can define Lipschitz continuity of N(·,·)with respect to the second argument.

Definition 2.5. Letη :H×H →Hbe a single-valued mapping. A multivalued mappingM :H →2^H is said to be

(i) η-monotone if

hx−y, η(u, v)i ≥0 for allu, v ∈H, x∈M u, y ∈M v;

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(ii) strictlyη-monotone if

hx−y, η(u, v)i ≥0 for allu, v ∈H, x∈M u, y ∈M v and equality holds if and only ifu=v;

(iii) stronglyη-monotone if there exists a constantr >0such that

hx−y, η(u, v)i ≥rku−vk² for allu, v ∈H, x∈M u, y ∈M v;

(iv) maximalη-monotone ifM isη-monotone and(I+λM)(H) = Hfor any λ >0.

Remark 1.

1. Ifη(u, v) = u−vfor allu, vinH, then (i)-(iv) of Definition2.5reduce to the classical definitions of monotonicity, strict monotonicity, strong mono- tonicity, and maximal monotonicity, respectively.

2. Huang and Fang gave one example of maximal η-monotone mapping in [10].

Lemma 2.2 ([10]). Letη:H×H →Hbe strictly monotone andM :H →2^H be a maximalη-monotone mapping. Then the following conclusions hold:

1. hx− y, η(u, v)i ≥ 0 for all (v, y) ∈ Graph(M) implies that (u, x) ∈ Graph(M), whereGraph(M) = {(u, x)∈H×H :x∈M u};

2. the inverse mapping(I+λM)⁻¹ is single-valued for anyλ >0.

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Based on Lemma2.2, we can define the resolvent operator for a maximal η-monotone mappingM as follows:

(2.7) J_ρ^M(z) = (I +ρM)⁻¹(z) for allz ∈H,

whereρ >0is a constant andη:H×H →H is a strictly monotone mapping.

Lemma 2.3 ([10]). Let η : H×H → H be strongly monotone and Lipschtiz continuous with constants δ > 0and τ > 0, respectively. Let M : H → 2^H be a maximalη-monotone mapping. Then the resolvent operatorJ_ρ^M forM is Lipschitz continuous with constantτ /δ, i.e.,

kJ_ρ^M(u)−J_ρ^M(v)k ≤ τ

δku−vk for allu, v ∈H.

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3. Iterative Algorithms

We first transfer problem (2.1) into a fixed point problem.

Lemma 3.1. For given u ∈ H, x ∈Su, y ∈ T u, andz ∈ Gu, (u, x, y, z)is a solution of the problem (2.1) if and only if

(3.1) g(u) =J_ρ^M(·,z)(g(u)−ρN(x, y)), whereJρ^M^(·,z) = (I +ρM(·, z))⁻¹ andρ >0is a constant.

Proof. This directly follows from the definition ofJρ^M^(·,u). Remark 2. Equality (3.1) can be written as

u= (1−λ)u+λ[u−g(u) +J_ρ^M(·,z)(g(u)−ρN(x, y))],

where 0 < λ ≤ 1 is a parameter and ρ > 0 is a constant. This fixed point formulation enables us to suggest the following iterative algorithm for problem (2.1) as follows:

Algorithm 1. Letη, N :H×H →H, g :H →H be single-valued mappings andS, T, G :H → CB(H)be multivalued mappings. LetM : H×H → 2^H be such that for each fixedt ∈H,M(·, t) :H →2^H is a maximalη-monotone mapping satisfying g(u) ∈ Dom(M(·, z)). For given λ ∈ [0,1], u0 ∈ H, x₀ ∈Su₀, y₀ ∈T u₀ andz₀ ∈Gu₀, let

u₁ = (1−λ)u₀+λ

u₀−g(u₀) +J_ρ^M(·,z⁰⁾(g(u₀)−ρN(x₀, y₀)) .

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By Nadler [17], there existx₁ ∈Su₁, y₁ ∈T u₁ andz₁ ∈Gu₁ such that kx₀−x₁k ≤(1 + 1)H(Su₀, Su₁),

ky0−y1k ≤(1 + 1)H(T u0, T u1), kz₀ −z₁k ≤(1 + 1)H(Gu₀, Gu₁).

Let

u₂ = (1−λ)u₁+λ

u₁−g(u₁) +J_ρ^M(·,z¹⁾(g(u₁)−ρN(x₁, y₁)) . By induction, we can obtain sequences{un},{xn},{yn}and{zn}satisfying

(3.2)











u_n+1 = (1−λ)u_n +λh

u_n−g(u_n) +Jρ^M(·,zⁿ⁾(g(u_n)−ρN(x_n, y_n))i , xn∈Sun,kxn−xn+1k ≤(1 + (1 +n)⁻¹)H(Sun, Sun+1), y_n ∈T u_n, ky_n−y_n+1k ≤(1 + (1 +n)⁻¹)H(T u_n, T u_n+1), z_n ∈gu_n, kz_n−z_n+1k ≤(1 + (1 +n)⁻¹)H(Gu_n, Gu_n+1), forn= 1,2,3, . . . ,where0< λ≤1andρ >0are both constants.

Now we construct a new pertured iterative algorithm for solving the generalized nonlinear mixed quasi-variational inequality (2.3) as follows:

Algorithm 2. Let η, N : H×H → H and S, T : H → H be single-valued mappings. LetM : H×H → 2^H be such that for each fixed t ∈ H,M(·, t) : H → 2^H is a maximal η-monotone mapping satisfyingg(u)∈ Dom(M(·, u)).

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For givenu₀ ∈H, the perturbed iterative sequence{u_n}is defined by

(3.3)











u_n+1 = (1−α_n)u_n+α_n[v_n−g(v_n)

+Jρ^M(·,vⁿ⁾(g(vn)−ρN(Svn, T vn))] +αnen, v_n = (1−β_n)u_n+β_n[u_n−g(u_n)

+Jρ^M(·,uⁿ⁾(g(un)−ρN(Sun, T un))] +βnfn,

for n = 0,1,2, . . . , where {e_n} and {f_n} are two sequences of the elements of H introduced to take into account possible inexact computation and the se- quences{α_n},{β_n}satisfy the following conditions:

0≤αn, βn ≤1(n ≥0), and

∞

X

n=0

αn=∞.

Let{y_n}be any sequence inHand define{_n}by

(3.4)









 _n=

y_n+1−n

(1−α_n)y_n+α_nh

x_n−g(x_n) +Jρ^M(·,xⁿ⁾(g(x_n)−ρN(Sx_n, T x_n))i

+α_ne_no , x_n= (1−β_n)y_n+β_nh

y_n−g(y_n)

+Jρ^M^(·,yⁿ⁾(g(y_n)−ρN(Sy_n, T y_n))i

+β_nf_n, forn= 0,1,2, . . . .

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4. Main Results

In this section, we first prove the existence of a solution of problem (2.1) and the convergence of an iterative sequence generated by Algorithm1.

Theorem 4.1. Let η : H ×H → H be strongly monotone and Lipschitz con- tinuous with constants δ and τ, respectively. Let S, T, G : H → CB(H)be H-Lipschitz continuous with constants α, β, γ, respectively, g : H → H be µ-Lipschitz continuous and ν-strongly monotone. Let N : H ×H → H be Lipschitz continuous with respect to the first and second arguments with con- stantsξ andζ, respectively, and S : H → CB(H)be strongly monotone with respect to the first argument ofN(·,·)with constantr. LetM : H×H → 2^H be a multivalued mapping such that for each fixedt ∈ H, M(·, t)is maximal η-monotone. Suppose that there exist constantsρ >0andκ >0such that for eachx, y, z ∈H,

(4.1)

J_ρ^M^(·,x)(z)−J_ρ^M^(·,y)(z)

≤κkx−yk, and

(4.2)











ρ− τ r−δ(1−h)ζβ τ(ξ²α²−ζ²β²)

<

√[τ r−δ(1−h)ζβ]²−(ξ²α²−ζ²β²)(τ²−δ²(1−h)²) τ(ξ²α²−ζ²β²) , τ r > δ(1−h)ζβ

+p

(ξ²α²−ζ²β²)(τ²−δ²(1−h)²), ξα > ζβ, h = (1 +δτ⁻¹)p

1−2ν+µ²

+κγ, ρτ ζβ < δ(1−h), h <1.

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Then the iterative sequences{u_n},{x_n},{y_n}and{z_n}generated by Algorithm 1 converge strongly to u^∗, x^∗, y^∗ and z^∗, respectively and (u^∗, x^∗, y^∗, z^∗) is a solution of problem (2.1).

Proof. It follows from (3.2), (4.1) and Lemma2.3that ku_n+1−u_nk

=

(1−λ)(u_n−un−1) +λ

[u_n−un−1−(g(u_n)−g(un−1)) +J_ρ^M(·,zⁿ⁾(g(un)−ρN(xn, yn))

−J_ρ^M(·,zⁿ⁻¹⁾(g(un−1)−ρN(xn−1, yn−1))

≤(1−λ)kun−un−1k+λkun−un−1−(g(un+1)−g(un))k +λ

J_ρ^M^(·,zⁿ⁾(g(u_n)−ρN(x_n, y_n))

−J_ρ^M(·,zⁿ⁾(g(u_n−1)−ρN(x_n−1, y_n−1)) +λ

J_ρ^M^(·,zⁿ⁾(g(un−1)−ρN(xn−1, yn−1))

−J_ρ^M(·,zⁿ⁻¹⁾(g(un−1)−ρN(xn−1, yn−1))

≤(1−λ)kun−un−1k+λkun−un−1−(g(un)−g(un−1))k +λτ

δkg(u_n)−g(un−1)−ρ(N(x_n, y_n)−N(xn−1, yn−1))k +λκkz_n−z_n−1k

≤(1−λ)kun−un−1k +λ

1 + τ

δ

kun−un−1−(g(un)−g(un−1))k

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+λτ

δku_n−un−1−ρ(N(x_n, y_n)−N(xn−1, y_n))k +λρτ

δkN(xn−1, y_n)−N(xn−1, yn−1)k+λκkz_n−zn−1k.

(4.3)

Sincegis strongly monotone and Lipschitz continuous, we obtain ku_n−u_n−1−(g(u_n)−g(u_n−1))k²

=kun−un−1k²

−2hu_n−un−1, g(u_n)−g(un−1)i+kg(u_n)−g(un−1)k²

≤(1−2ν+µ²)ku_n−un−1k². (4.4)

Since S is H-Lipschitz continuous and strongly monotone with respect to the first argument ofN(·,·)andN is Lipschitz continuous with respect to the first argument, we have

ku_n−u_n−1−ρ(N(x_n, y_n)−N(x_n−1, y_n))k²

=kun−un−1k²−2ρhun−un−1, N(xn, yn)−N(xn−1, yn)i +ρ²kN(x_n, y_n)−N(xn−1, y_n)k²

≤(1−2ρr+ρ²ξ²(1 +n⁻¹)²α²)ku_n−un−1k². (4.5)

Further, sinceT, G areH-Lipschitz continuous andN is Lipschitz continuous with respect to the second argument, we get

kN(xn−1, y_n)−N(xn−1, yn−1)k ≤ζky_n−yn−1k (4.6)

≤ζβ(1 +n⁻¹)ku_n−un−1k,

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(4.7) kz_n−z_n−1k ≤γ(1 +n⁻¹)ku_n−u_n−1k.

By (4.3) – (4.7), we obtain

ku_n−un−1k ≤(1−λ+λ(1 +τ δ⁻¹)p

1−2ν+µ² +λτ δ⁻¹p

1−2ρr+ρ²ξ²(1 +n⁻¹)²α² +λρτ δ⁻¹ζβ(1 +n⁻¹) +λκγ(1 +n⁻¹)

= (1−λ+λh_n+λt_n(ρ))ku_n−un−1k

=θ_nku_n−un−1k, (4.8)

where

h_n= (1 +τ δ⁻¹)p

1−2ν+µ²+κγ(1 +n⁻¹), t_n(ρ) = τ δ⁻¹p

1−2ρr+ρ²ξ²(1 +n⁻¹)²α²+ρτ δ⁻¹ζβ(1 +n⁻¹) and θ_n= 1−λ+λh_n+λt_n(ρ).

Lettingθ= 1−λ+λh+λt(ρ),where h= (1 +τ δ⁻¹)p

1−2ν+µ²+κγ and t(ρ) =τ δ⁻¹p

1−2ρr+ρ²ξ²α²+ρτ δ⁻¹ζβ,

we have that hn → h, tn(ρ) → t(ρ) andθn → θ asn → ∞. It follows from condition (4.2) that θ < 1. Hence θ_n < 1forn sufficiently large. Therefore,

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(4.8) implies that{u_n}is a Cauchy sequence in H and so we can assume that un →u^∗ ∈Hasn→ ∞. By the Lipschitz continuity ofS, T andGwe obtain

kxn−xn−1k ≤(1 + (1 +n)⁻¹)H(Sun, Sun−1)

≤α(1 + (1 +n)⁻¹)ku_n−un−1k, ky_n−yn−1k ≤(1 + (1 +n)⁻¹)H(T u_n, T un−1)

≤β(1 + (1 +n)⁻¹)ku_n−un−1k, kz_n−zn−1k ≤(1 + (1 +n)⁻¹)H(Gu_n, Gun−1)

≤γ(1 + (1 +n)⁻¹)ku_n−u_n−1k.

It follows that {xn},{yn} and{zn}are also Cauchy sequences in H. We can assume that x_n → x^∗, y_n → y^∗ and z_n → z^∗, respectively. Note that for x_n ∈Su_n, we have

d(x^∗, Su^∗)≤ kx^∗−x_nk+d(x_n, Su^∗)

≤ kx^∗−x_nk+H(Su_n, Su^∗)

≤ kx^∗−x_nk+αku_n−u^∗k →0,

as n → ∞. Hence we must have x^∗ ∈ Su^∗. Similarly, we can show that y^∗ ∈T u^∗ andz^∗ ∈Gu^∗. From

u_n+1 = (1−λ)u_n+λ

u_n−g(u_n) +J_ρ^M(·,zⁿ⁾(g(u_n)−ρN(x_n, y_n)) , it follows that

g(u^∗) = J_ρ^M^(·,z^∗⁾(g(u^∗)−ρN(x^∗, y^∗)).

By Lemma 3.1, (u^∗, x^∗, y^∗, z^∗)is a solution of problem (2.1). This completes the proof.

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Remark 3. For an appropriate and suitable choice of the mappings η, N, S, T, G, g, M and the space H, we can obtain several known results in [1], [3], [5] – [8], [14], [18] – [22], [24] – [26] as special cases of Theorem4.1.

Now we prove the convergence and stability of the iterative sequence generated by the Algorithm2.

Theorem 4.2. Let η : H ×H → H be strongly monotone and Lipschitz con- tinuous with constants δ andτ, respectively. LetS, T : H → H be Lipschitz continuous with constantsα, β, respectively,g :H →Hbeµ-Lipschitz contin- uous andν-strongly monotone. LetN : H ×H →H be Lipschitz continuous with respect to the first and second arguments with constants ξ andζ, respec- tively, andS :H → Hbe strongly monotone with respect to the first argument of N(·,·)with constant r. Let M : H×H → 2^H be a multivalued mapping such that for each fixedt ∈ H, M(·, t)is maximal η-monotone. Suppose that there exist constantsρ >0andκ >0such that for eachx, y, z∈H,

(4.9)

J_ρ^M^(·,x)(z)−J_ρ^M^(·,y)(z)

≤κkx−yk, and

(4.10)











ρ− τ r−δ(1−h)ζβ τ(ξ²α²−ζ²β²)

<

√[τ r−δ(1−h)ζβ]²−(ξ²α²−ζ²β²)(τ²−δ²(1−h)²) τ(ξ²α²−ζ²β²) , τ r > δ(1−h)ζβ

+p

(ξ²α²−ζ²β²)(τ²−δ²(1−h)²), ξα > ζβ, h= (1 +δτ⁻¹)p

1−2ν+µ²+κ, ρτ ζβ < δ(1−h), h <1.

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If lim

n→∞ke_nk= 0, lim

n→∞kf_nk= 0, then

(I) The sequence{u_n}defined by Algorithm2converges strongly to the unique solutionu^∗ of problem (2.3).

(II) IfP∞

n=0n<∞, then lim

n→∞yn=u^∗. (III) If lim

n→∞y_n=u^∗, then lim

n→∞_n= 0.

Proof. (I) It follows from Theorem 4.1 that there exists u^∗ ∈ H which is a solution of problem (2.3) and so

(4.11) g(u^∗) = J_ρ^M^(·,u^∗⁾(g(u^∗)−ρN(Su^∗, T u^∗)).

From (4.9), (4.11) and Algorithm2, it follows that ku_n+1−u^∗k

=

(1−α_n)(u_n−u^∗)−α_nh

v_n−u^∗−(g(v_n)−g(u^∗)) +J_ρ^M(·,vⁿ⁾(g(v_n)−ρN(Sv_n, T v_n))

−J_ρ^M^(·,u^∗⁾(g(u^∗)−ρN(Su^∗, T u^∗))i

+α_ne_n

≤(1−α_n)ku_n−u^∗k+α_nkv_n−u^∗−(g(v_n)−g(u^∗))k+α_nke_nk +α_n

J_ρ^M^(·,vⁿ⁾(g(v_n)−ρN(Sv_n, T v_n))

−J_ρ^M^(·,vⁿ⁾(g(u^∗)−ρN(Su^∗, T u^∗)) +αn

J_ρ^M^(·,vⁿ⁾(g(u^∗)−ρN(Su^∗, T u^∗))

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−J_ρ^M(·,u^∗⁾(g(u^∗)−ρN(Su^∗, T u^∗))

≤(1−α_n)ku_n−u^∗k+α_nkv_n−u^∗−(g(v_n)−g(u^∗))k+α_nke_nk +αn

τ

δkg(vn)−g(u^∗)−ρ(N(Svn, T vn)−N(Su^∗, T u^∗))k +α_nκkv_n−u^∗k

≤(1−α_n)ku_n−u^∗k +α_n

1 + τ δ

ku_n−u^∗ −(g(v_n)−g(u^∗))k+α_nke_nk +α_nτ

δkv_n−u^∗−ρ(N(Sv_n, T v_n)−N(Su^∗, T v_n))k +αnρτ

δkN(Su^∗, T vn)−N(Su^∗, T u^∗)k+αnκkvn−u^∗k.

(4.12)

By the Lipschitz continuity ofN, S, T, gand the strong monotonicity ofS and g, we obtain

(4.13) kv_n−u^∗ −(g(v_n)−g(u^∗))k² ≤(1−2ν+µ²)kv_n−u^∗k², (4.14) kv_n−u^∗ −ρ(N(Sv_n, T v_n)−N(Su^∗, T v_n))k²

≤(1−2ρr+ρ²ξ²α²)kv_n−u^∗k²,

(4.15) kN(Su^∗, T v_n)−N(Su^∗, T u^∗))k ≤ζβkv_n−u^∗k.

It follows from (4.12) – (4.15) that

(4.16) kun+1−u^∗k ≤(1−αn)kun−u^∗k+θαnkvn−u^∗k+αnkenk,

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where

θ =κ+ (1 +τ δ⁻¹)p

1−2ν+µ²+τ δ⁻¹p

1−2ρr+ρ²ξ²α²+ρτ δ⁻¹ζβ.

Similarly, we have

(4.17) kv_n−u^∗k ≤(1−β_n)ku_n−u^∗k+θβ_nku_n−u^∗k+β_nkf_nk.

From (4.16) and (4.17), we have

ku_n+1−u^∗k ≤[1−α_n(1−θ)(1 +θβ_n)]ku_n−u^∗k+α_nβ_nθkf_nk+α_nke_nk Condition (4.10) implies that0< θ <1, and so

(4.18) ku_n+1−u^∗k ≤[1−α_n(1−θ)]ku_n−u^∗k+α_n(1−θ)d_n,

whered_n = (β_nθkf_nk+ke_nk)(1−θ)⁻¹ →0, asn→ ∞. It follows from (4.18) and Lemma2.1thatu_n →u^∗asn→ ∞.

Now we prove thatu^∗ is a unique solution of problem (2.3). In fact, ifuis also a solution of problem (2.3), then

g(u) =J_ρ^M^(·,u)(g(u)−ρN(Su, T u)), and, as the proof of (4.16), we have

ku^∗−uk ≤θku^∗−uk,

where0< θ <1and sou^∗ =u. This completes the proof of Conclusion (I).

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Next we prove Conclusion (II). Using (3.4) we obtain ky_n+1−u^∗k

≤

y_n+1−

(1−α_n)y_n+α_n

x_n−g(x_n) + J_ρ^M^(·,xⁿ⁾(g(x_n)−ρN(Sx_n, T x_n))

+α_ne_n +

(1−α_n)y_n+α_n

x_n−g(x_n) + J_ρ^M^(·,xⁿ⁾(g(x_n)−ρN(Sx_n, T x_n))

+α_ne_n−u^∗

=

(1−α_n)y_n+α_n

x_n−g(x_n) (4.19)

+ J_ρ^M^(·,xⁿ⁾(g(x_n)−ρN(Sx_n, T x_n))

+α_ne_n−u^∗ +_n. As the proof of inequality (4.18), we have

(4.20)

(1−α_n)y_n+α_n

x_n−g(x_n)

+J_ρ^M(·,xⁿ⁾(g(x_n)−ρN(Sx_n, T x_n))

+α_ne_n−u^∗

≤(1−α_n(1−θ))ky_n−u^∗k+α_n(1−θ)d_n. It follows from (4.19) and (4.20) that

(4.21) ky_n+1−u^∗k ≤(1−α_n(1−θ))ky_n−u^∗k+α_n(1−θ)d_n+_n. Since P∞

n=0_n < ∞, d_n → 0, and P∞

n=0α_n < ∞. It follows that (4.21) and Lemma2.1that lim

n→∞yn=u^∗.

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Now we prove Conclusion (III). Suppose that lim

n→∞y_n =u^∗. Then we have _n =

y_n+1−(1−α_n)y_n+α_n x_n

−g(x_n) + J_ρ^M(·,xⁿ⁾(g(x_n)−ρN(Sx_n, T x_n))

+α_ne_n

≤ ky_n+1−u^∗k+

(1−α_n)y_n+α_n x_n

−g(x_n) + J_ρ^M(·,xⁿ⁾(g(x_n)−ρN(Sx_n, T x_n))

+α_ne_n−u^∗

≤ ky_n+1−u^∗k+ (1−α_n(1−θ))ky_n−u^∗k+α_n(1−θ)d_n→0 asn→ ∞. This completes the proof.

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