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volume 7, issue 2, article 72, 2006.

Received 05 November, 2005;

accepted 28 December, 2005.

Communicated by:R.U. Verma

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Journal of Inequalities in Pure and Applied Mathematics

ITERATIVE ALGORITHM FOR A NEW SYSTEM OF NONLINEAR SET-VALUED VARIATIONAL INCLUSIONS INVOLVING

(H, η)-MONOTONE MAPPINGS

MAO-MING JIN

Department of Mathematics Fuling Normal University Fuling, Chongqing 40800 P. R. China

EMail:mmj1898@163.com

c

2000Victoria University ISSN (electronic): 1443-5756 012-06

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Iterative Algorithm for A New System of Nonlinear Set-Valued Variational Inclusions Involving

(H, η)-monotone Mappings Mao-Ming Jin

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J. Ineq. Pure and Appl. Math. 7(2) Art. 72, 2006

Abstract

In this paper, a new system of nonlinear set-valued variational inclusions involving(H, η)-monotone mappings in Hilbert spaces is introduced and studied.

By using the resolvent operator method associated with(H, η)-monotone mappings, an existence theorem of solutions for this kind of system of nonlinear set-valued variational inclusion is established and a new iterative algorithm is suggested and discussed. The results presented in this paper improve and generalize some recent results in this field.

2000 Mathematics Subject Classification:49J40; 47H10.

Key words:(H, η)-monotone mapping; System of nonlinear set-valued variational inclusions; Resolvent operator method; Iterative algorithm.

This work was supported by the National Natural Science Foundation of China(10471151) and the Educational Science Foundation of Chongqing, Chongqing of China (KJ051307).

1. Introduction

Variational inclusions are an important generalization of classical variational inequalities and thus, have wide applications to many fields including, for ex- ample, mechanics, physics, optimization and control, nonlinear programming, economics, and the engineering sciences. For these reasons, various variational inclusions have been intensively studied in recent years. For details, we refer the reader to [1] – [21], [23] – [31] and the references therein.

Verma [24,25] introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solutions of a system of variational inequalities in Hilbert spaces. Recently, Kim and Kim [21] introduced a new system of generalized nonlinear mixed variational inequalities and obtained some existence and uniqueness results for solutions of the system of generalized nonlinear mixed variational inequalities in Hilbert spaces. Very recently, Fang, Huang and Thompson [9] introduced a system of variational inclusions and developed a Mann iterative algorithm to approximate the unique solution of the system.

On the other hand, monotonicity techniques were extended and applied in recent years because of their importance in the theory of variational inequalities, complementarity problems, and variational inclusions. In 2003, Huang and Fang [16] introduced a class of generalized monotone mappings, maximal η-monotone mappings, and defined an associated resolvent operator. Using resolvent operator methods, they developed some iterative algorithms to approximate the solution of a class of general variational inclusions involving maximal η-monotone operators. Huang and Fang’s method extended the resolvent operator method associated with anη-subdifferential operator due to Ding and Luo

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[6]. In [7], Fang and Huang introduced another class of generalized monotone operators, H-monotone operators, and defined an associated resolvent operator. They also established the Lipschitz continuity of the resolvent operator and studied a class of variational inclusions in Hilbert spaces using the resolvent operator associated with H-monotone operators. In a recent paper [9], Fang, Huang and Thompson further introduced a new class of generalized monotone operators,(H, η)-monotone operators, which provide a unifying framework for classes of maximal monotone operators, maximal η-monotone operators, and H-monotone operators. They also studied a system of variational inclusions using the resolvent operator associated with(H, η)-monotone operators.

Inspired and motivated by recent research works in this field, in this paper, we shall introduce and study a new system of nonlinear set-valued variational inclusions involving (H, η)-monotone mappings in Hilbert spaces. By using the resolvent operator method associated with (H, η)-monotone mappings, an existence theorem for solutions for this type of system of nonlinear set-valued variational inclusion is established and a new iterative algorithm is suggested and discussed. The results presented in this paper improve and generalize some recent results in this field.

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2. Preliminaries

LetX be a real Hilbert space endowed with a norm k · kand an inner product h·,·i, respectively. 2^X andC(X)denote the family of all the nonempty subsets of X and the family of all closed subsets of X, respectively. Let us recall the following definitions and some known results.

Definition 2.1. LetT, H : X → X be two single-valued mappings. T is said to be:

(i) monotone, if

hT x−T y, x−yi ≥0 for all x, y ∈X;

(ii) strictly monotone, ifT is monotone and

hT x−T y, x−yi= 0 if and only ifx=y;

(iii) r-strongly monotone, if there exists a constantr >0such that hT(x)−T(y), x−yi ≥rkx−yk² for all x, y ∈X;

(iv) s-strongly monotone with respect to H, if there exists a constant s > 0 such that

hT(x)−T(y), H(x)−H(y)i ≥Skx−yk² for all x, y ∈X;

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(v) t-Lipschitz continuous, if there exists a constantt >0such that kT(x)−T(y)k ≤tkx−yk for all x, y ∈X.

Definition 2.2. A single-valued mappingη:X×X →Xis said to be:

(i) monotone, if

hx−y, η(x, y)i ≥0 for all x, y ∈X;

(ii) strictly monotone, if

hx−y, η(x, y)i ≥0 for all x, y ∈X and equality holds if and only ifx=y;

(iii) δ-srongly monotone, if there exists a constantδ >0such that hx−y, η(x, y)i ≥δkx−yk² for all x, y ∈X;

(iv) τ-Lipschitz continuous, if there exists a constantτ >0such that kη(x, y)k ≤τkx−yk, for all x, y ∈X.

Definition 2.3. Let η : X×X → X and H : X → X be two single-valued mappings. A set-valued mappingM :X →2^X is said to be:

(i) monotone, if

hu−v, x−yi ≥0, ∀x, y ∈X, u∈M x, v ∈M y;

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(ii) η-monotone, if

hu−v, η(x, y)i ≥0 ∀x, y ∈X, u∈M x, v ∈M y;

(iii) strictly η-monotone, ifM isη-monotone and equality holds if and only if x=y;

(iv) r-stronglyη-monotone, if there exists a constantr >0such that hu−v, η(x, y)i ≥rkx−yk² ∀x, y ∈X, u∈M x, v∈M y;

(v) maximal monotone, if M is monotone and (I +λM)(X) = X, for all λ >0, whereI denotes the identity mapping onX;

(vi) maximalη-monotone, ifM isη-monotone and(I+λM)(X) = X, for all λ >0;

(vii) H-monotone, ifM is monotone and(H+λM)(X) = X, for allλ >0;

(viii) (H, η)-monotone, if M is η-monotone and (H +λM)(X) = X, for all λ >0.

Remark 1. Maximalη-monotone mappings,H-monotone mappings, and(H, η)- monotone mappings were first introduced in Huang and Fang [16], Fang and Huang [7, 9], respectively. Obviously, the class of (H, η)- monotone map- pings provides a unifying framework for classes of maximal monotone map- pings, maximalη-monotone mappings, andH-monotone mappings. For details about these mappings, we refer the reader to [6, 7, 9, 16] and the references therein.

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Lemma 2.1 ([9]). Letη:X×X →X be a single-valued mapping,H :X → X be a strictly η-monotone mapping and M : X → 2^X an (H, η)-monotone mapping. Then the mapping(H+λM)⁻¹ is single-valued.

By Lemma2.1, we can define the resolvent operatorR^H,η_M,λas follows.

Definition 2.4 ([9]). Let η : X ×X → X be a single-valued mapping, H : X →Xa strictlyη-monotone mapping andM :X →2^X an(H, η)-monotone mapping. The resolvent operatorR^H,η_M,λ :X→X is defined by

R^H,η_M,λ(z) = (H+λM)⁻¹(z) for all z ∈X, whereλ >0is a constant.

Remark 2.

(i) When H = I, Definition 2.4 reduces to the definition of the resolvent operator of a maximalη-monotone mapping, see [16].

(ii) When η(x, y) = x− y for all x, y ∈ X, Definition 2.4 reduces to the definition of the resolvent operator of aH-monotone mapping, see [7].

(iii) WhenH =Iandη(x, y) =x−yfor allx, y ∈X, Definition2.4reduces to the definition of the resolvent operator of a maximal monotone mapping, see [31].

Lemma 2.2 ([9]). Letη :X ×X → X be aτ-Lipschtiz continuous mapping, H : X → X be an(r, η)-strongly monotone mapping and M : X → 2^X be

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an (H, η)-monotone mapping. Then the resolvent operatorR^H,η_M,λ : X → X is τ /r-Lipschitz continuous, that is,

R^H,η_M,λ(x)−R^H,η_M,λ(y) ≤ τ

rkx−yk for all x, y ∈X.

We define a Hausdorff pseudo-metricD: 2^X×2^X →(−∞,+∞)∪ {+∞}

by

D(·,·) = max

sup

u∈A

v∈Binf ku−vk,sup

u∈B

v∈Ainf ku−vk

for any given A, B ∈ 2^X. Note that if the domain ofD is restricted to closed bounded subsets, thenDis the Hausdorff metric.

Definition 2.5. A set-valued mapping A : X → 2^X is said to be D-Lipschitz continuous if there exists a constantη >0such that

D(A(u), A(v))≤ηku−vk, for all u, v ∈X.

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3. System of Variational Inclusions

In this section, we shall introduce a new system of set-valued variational inclusions involving (H, η)-monotone mappings in Hilbert spaces. In what follows, unless other specified, we shall suppose that X1 and X2 are two real Hilbert spaces,K₁ ⊂X₁ andK₂ ⊂X₂are two nonempty, closed and convex sets. Let F :X₁×X₂ → X₁, G: X₁ ×X₂ → X₂, H_i : X_i → X_i, η_i : X_i ×X_i → X_i (i= 1,2)be nonlinear mappings. LetA:X1 →2^X¹ andB :X2 →2^X² be set- valued mappings, M_i : X_i → 2^Xⁱ be(H_i, η_i)-monotone mappings (i = 1,2).

The system of nonlinear set-valued variational inclusions is formulated as follows. Find(a, b)∈X1×X2, u∈A(a)andv ∈B(b)such that

(3.1)







0∈F(a, v) +M₁(a) 0∈G(u, b) +M2(b) Special Cases

Case 1. If M1(x) = ∂ϕ(x) and M2 = ∂φ(y) for all x ∈ X1 and y ∈ X2, whereϕ :X₁ →R∪ {+∞}andφ:X₂ →R∪ {+∞}are two proper, convex and lower semi-continuous functionals, ∂ϕ and ∂φ denote the subdifferential operators ofϕ andφ, respectively, then problem (3.1) reduces to the following problem: find(a, b)∈X₁×X₂,u∈A(a), andv ∈B(v)such that

(3.2)







hF(a, v), x−ai+ϕ(x)−ϕ(a)≥0, ∀x∈X₁, hG(u, b), y −ai+φ(y)−φ(b)≥0, ∀y ∈X₂,

which is called a system of set-valued mixed variational inequalities. Some special cases of problem (3.2) can be found in [26].

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Case 2. If AandB are both identity mappings, then problem (3.2) reduces to the following problem: find(a, b)∈X1×X2 such that

(3.3)







hF(a, b), x−ai+ϕ(x)−ϕ(a)≥0, ∀x∈X₁, hG(a, b), y−ai+φ(y)−φ(b)≥0, ∀y∈X₂,

which is called system of nonlinear variational inequalities considered by Cho, Fang, Huang and Hwang [5]. Some special cases of problem (3.3) were studied by Kim and Kim [21], and Verma [24].

Case 3. If M₁(x) = ∂δ_K₁(x) and M₂(y) = ∂δ_K₂(y), for all x ∈ K₁ and y ∈ K2, where K1 ⊂ X1 andK2 ⊂ X2 are two nonempty, closed, and con- vex subsets, and δ_K₁ and δ_K₂ denote the indicator functions of K₁ and K₂, respectively. Then problem (3.2) reduces to the following system of variational inequalities: find(a, b)∈K1×K2 such that

(3.4)







hF(a, b), x−ai ≥0, ∀x∈K₁, hG(a, b), y−ai ≥0, ∀y∈K₂,

which is the problem in [20] with bothF andGbeing single-valued.

Case 4. If X₁ = X₂ = X, K₁ = K₂ = K, F(X, y) = ρT(y) +x−y, and G(x, y) = γT(x) +y−x, for allx, y ∈ X, whereT :K →X is a nonlinear mapping,ρ >0andγ >0are two constants, then problem (3.4) reduces to the following system of variational inequalities: find(a, b)∈K ×K such that (3.5)







hρT(b) +a−b, x−ai ≥0, ∀x∈K, hγT(a) +b−a, x−bi ≥0, ∀x∈K,

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which is the system of nonlinear variational inequalities considered by Verma [25].

Case 5. IfAandBare both identity mappings, the problem (3.1) reduces to the following problem: (a, b)∈X₁×X₂ such that

(3.6)







0∈F(a, b) +M₁(a) 0∈G(a, b) +M₂(b)

which is the system of variational inclusions considered by Fang, Huang and Thompson [9].

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4. Iterative Algorithm and Convergence

In this section, by using the resolvent operator method associated with (H, η)- monotone mappings, a new iterative algorithm for solving problem (3.1) is suggested. The convergence of the iterative sequence generated by the algorithm is proved.

Theorem 4.1. For given(a, b) ∈ X₁ ×X₂, u∈ A(a), v ∈ B(b),(a, b, u, v)is a solution of problem (3.1) if and only if(a, b, u, v)satisfies the relation

(4.1)







a=R^H_M¹^,η¹

1,ρ1[H₁(a)−ρ₁F(a, v)], b=R^H_M²₂^,η_,ρ²₂[H₂(b)−ρ₂G(u, b)], whereρi >0are two constants fori= 1,2.

Proof. This directly follows from Definition2.4.

The relation (4.1) and Nadler [22] allows us to suggest the following iterative algorithm.

Algorithm 1.

Step 1. Choose(a0, b0)∈X1×X2and chooseu0 ∈A(a0)andv0 ∈B(b0).

Step 2. Let

(4.2)







a_n+1 = (1−λ)a_n+λR^H_M¹^,η¹

1,ρ1[H₁(a_n)−ρ₁F(a_n, v_n)], b_n+1 = (1−λ)b_n+λR^H_M²^,η²

2,ρ2[H₂(b_n)−ρ₂G(u_n, b_n)], where0< λ≤1is a constant.

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Step 3. Chooseu_n+1 ∈A(a_n+1)andv_n+1 ∈B(b_n+1)such that

(4.3)







ku_n+1−u_nk ≤(1 + (1 +n)⁻¹)D₁(A(a_n+1), A(a_n)), kv_n+1−v_nk ≤(1 + (1 +n)⁻¹)D₂(B(b_n+1), B(b_n)), whereD_i(·,·)is the Hausdorff pseudo-metric on2^Xⁱ fori= 1,2.

Step 4. If a_n+1, b_n+1, u_n+1 andv_n+1 satisfy (4.2) to sufficient accuracy, stop; oth- erwise, setn:=n+ 1and return to Step 2.

Theorem 4.2. Let η_i : X_i ×X_i → X_i be τ_i-Lipschitz continuous mappings, H_i : X_i → X_i (r_i, η)-strongly monotone and β_i-Lipschitz continuous map- pings, M_i : X_i → 2^X_i be (H_i, η_i)-monotone mappings for i = 1,2. Let A : X₁ → C(X₁) be D₁-γ₁-Lipschitz continuous and B : X₂ → C(X₂) be D₂- γ₂-Lipschitz continuous. Let F : X₁ × X₂ → X₁ be a nonlinear mapping such that for any given(a, b) ∈X₁ ×X₂, F(·, b)isµ₁-strongly monotone with respect to H₁ andα₁-Lipschitz continuous andF(a,·)isζ₁-Lipschitz continu- ous. Let G : X₁×X₂ → X₂ be a nonlinear mapping such that for any given (x, y) ∈ X₁ × X₂, G(x,·) is µ₂-strongly monotone with respect to H₂ and α₂-Lipschitz continuous and G(·, y) is ζ₂-Lipschitz continuous. If there exist constantsρ_i >0fori= 1,2such that

(4.4)







τ₁r₂p

β₁²−2ρ₁µ₁ +ρ²₁α²₁+τ₂r₁ζ₂γ₁ < r₁r₂, τ₂r₁p

β₂²−2ρ₂µ₂ +ρ²₂α²₂+τ₁r₂ζ₁γ₂ < r₁r₂,

then problem (3.1) admits a solution (a, b, u, v)and iterative sequences {an},

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{b_n}, {u_n} and {v_n} converge strongly to a, b, u and v, respectively, where {an},{bn},{un}and{vn}are the sequences generated by Algorithm1.

Proof. It follows from (4.2) and Lemma2.2that ka_n+1−a_nk

=

(1−λ)a_n+λR^H_M¹^,η¹

1,ρ1(H₁(a_n)−ρ₁F(a_n, v_n))

−h

(1−λ)a_n−1+λR^H_M¹^,η¹

1,ρ1(H₁(a_n−1)−ρ₁F(a_n−1, v_n−1))i

≤(1−λ)ka_n−an−1k+λ

R^H_M¹₁^,η_,ρ¹₁(H₁(a_n)−ρ₁F(a_n, v_n))

−R^H_M¹^,η¹

1,ρ1(H₁(an−1)−ρ₁F(an−1, vn−1))

≤(1−λ)ka_n−an−1k +λτ₁

r₁kH₁(a_n)−H₁(an−1)−ρ₁[F(a_n, v_n)−F(an−1, vn−1)]k

≤(1−λ)ka_n−an−1k +λτ₁

r₁(kH1(an)−H1(an−1)−ρ1[F(an, vn)−F(an−1, vn)]k +kF(an−1, v_n)−F(an−1, vn−1)k).

(4.5)

Similarly, we can prove that

(4.6) kbn+1−bnk ≤(1−λ)kbn−bn−1k +λτ₂

r₂(kH₂(b_n)−H₂(bn−1)−ρ₂[G(u_n, b_n)−G(u_n, bn−1)]k +kG(u_n, bn−1)−G(un−1, bn−1)k).

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SinceH_i areβ_i-Lipschitz continuous fori = 1,2, F(·, b)isµ₁-strongly monotone with respect to H1 and α1-Lipschitz continuous, G(x,·) is µ2-strongly monotone with respect toH₂ andα₂-Lipschitz continuous, we obtain

kH₁(a_n)−H₁(an−1)−ρ₁[F(a_n, v_n)−F(an−1, v_n)]k²

=kH₁(a_n)−H₁(an−1)k²

−2ρ₁hF(a_n, v_n)−F(an−1, v_n), H₁(a_n)−H₁(an−1)i +ρ²₁kF(a_n, v_n)−F(an−1, v_n)k²

≤(β₁²−2ρ₁µ₁+ρ²₁α²₁)ka_n−a_n−1k² (4.7)

and

kH2(bn)−H2(bn−1)−ρ2[G(un, bn)−G(un, bn−1)]k²

=kH₂(b_n)−H₂(bn−1)k²

−2ρ₂hG(u_n, b_n)−G(u_n, bn−1), H₂(b_n)−H₂(bn−1)i +ρ²₂kG(u_n, b_n)−G(u_n, bn−1)k²

≤(β₂²−2ρ₂µ₂+ρ²₂α²₂)kb_n−bn−1k². (4.8)

Further, from the assumptions, we have

kF(a_n−1, v_n)−F(a_n−1, v_n−1)k ≤ζ₁kv_n−v_n−1k (4.9)

≤ζ1γ2(1 +n⁻¹)kbn−bn−1k, kG(u_n, bn−1)−G(un−1, bn−1)k ≤ζ₂ku_n−un−1k

(4.10)

≤ζ₂γ₁(1 +n⁻¹)ka_n−an−1k.

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It follows from (4.5) – (4.10) that

(4.11)











ka_n+1−a_nk

≤

1−λ+λ^τ_r¹

1

pβ₁²−2ρ₁µ₁+ρ²₁α²₁

ka_n−an−1k +λ^τ_r¹

1ζ₁γ₂(1 +n⁻¹)kb_n−b_n−1k, kb_n+1−b_nk

≤

1−λ+λ^τ_r²

2

pβ₂²−2ρ₂µ₂+ρ²₂α²₂

kb_n−bn−1k +λ^τ_r²

2ζ₂γ₁(1 +n⁻¹)ka_n−an−1k.

Now (4.11) implies that

ka_n+1−a_nk+kb_n+1−b_nk

≤

1−λ+λτ₁ r₁

q

β₁² −2ρ₁µ₁+ρ²₁α²₁ +λτ₂

r₂ζ₂γ₁(1 +n⁻¹)

ka_n−a_n−1k +

1−λ+λτ₂ r₂

q

β₂²−2ρ₂µ₂+ρ²₂α²₂ +λτ₁

r₁ζ₁γ₂(1 +n⁻¹)

kb_n−b_n−1k

≤(1−λ+λθn)(kan−an−1k+kbn−bn−1k), (4.12)

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where θ_n= max

τ₁ r1

q

β₁²−2ρ₁µ₁+ρ²₁α²₁+τ₂ r2

ζ₂γ₁(1 +n⁻¹), τ₂

r2

q

β₂²−2ρ₂µ₂+ρ²₂α²₂+ τ₁ r1

ζ₁γ₂(1 +n⁻¹)

. Letting

θ = max τ₁

r1

q

β₁²−2ρ₁µ₁+ρ²₁α²₁+ τ₂ r2

ζ₂γ₁ , τ₂

r2

q

β₂²−2ρ₂µ₂+ρ²₂α²₂+ τ₁ r1

ζ₁γ₂

, we have thatθ_n→θasn → ∞. It follows from condition (4.4) that0< θ <1.

Therefore, by (4.12) and0< λ≤1,{a_n}and{b_n}are both Cauchy sequences and so there exista∈X₁ andb ∈X₂ such thata_n→aandb_n→basn → ∞.

Now we prove thatu_n→u∈A(u)andv_n→v ∈ B(b)asn → ∞. In fact, it follows from (4.9) and (4.10) that{u_n}and{v_n}are also Cauchy sequences.

Therefore, there exist u ∈ X₁ andv ∈ X₂ such that u_n → u and v_n → v as n → ∞. Further,

d(u, A(u)) = inf{ku−tk:t∈A(a)}

≤ ku−u_nk+d(u_n, A(a))

≤ ku−u_nk+D₁(A(a_n), A(a))

≤ ku−u_nk+ζ₁ka_n−ak →0.

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Hence, since A(a)is closed, we haveu ∈ A(a). Similarly, we can prove that v ∈B(b).

By continuity,a, b, uandv satisfy the following relation







a=R^H_M¹₁^,η_,ρ¹₁[H₁(a)−ρ₁F(a, v)], b=R^H_M²^,η²

2,ρ2[H₂(b)−ρ₂G(u, b)].

By Theorem 4.1, we know that(a, b, u, v)is a solution of problem (3.1). This completes the proof.

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References

[1] S. ADLY, Perturbed algorithm and sensitivity analysis for a general class of variational inclusions, J. Math. Anal. Appl., 201 (1996), 609–630.

[2] R.P. AGARWAL, N.J. HUANG AND Y.J. CHO, Generalized nonlinear mixed implicit quasi-variational inclusions with set-valued mappings, J.

Inequal. Appl., 7(6) (2002), 807–828.

[3] R. AHMAD AND Q.H. ANSARI, An iterative algorithm for generalized nonlinear variational inclusions, Appl. Math. Lett., 13(5) (2002), 23–26.

[4] S.S. CHANG, Y.J. CHO ANDH.Y. ZHOU, Iterative Methods for Nonlin- ear Operator Equations in Banach Spaces, Nova Sci. Publ., New York, (2002).

[5] Y.J. CHO, Y.P. FANG, N.J. HUANG ANDH.J. HWANG, Algorithms for systems of nonlinear variational inequalities, J. Korean Math Soc., 41 (2004), 489–499.

[6] X.P. DINGANDC.L. LUO, Perturbed proximal point algorithms for gen- eralized quasi-variational-like inclusions, J. Comput. Appl. Math., 210 (2000), 153–165.

[7] Y.P. FANGANDN.J. HUANG,H-Monotone operator and resolvent oper- ator technique for variational inclusions, Appl. Math. Comput., 145 (2003), 795–803.

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JJ II

J I

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Quit Page21of23

[8] Y.P. FANGANDN.J. HUANG, H-monotone operators and system of vari- ational inclusions, Communications on Applied Nonlinear Analysis, 11(1) (2004), 93–101.

[9] Y.P. FANG, N.J. HUANGANDTHOMPSON, A new system of variational inclusions with(H, η)-monotone operators in Hilbert spaces, Computers.

Math. Applic., 49 (2005), 365–374.

[10] F. GIANNESSIANDA. MAUGERI, Variational Inequalities and Network Equilibrium Problems, New York (1995).

[11] A. HASSOUNI AND A. MOUDAFI, A perturbed algorithms for varia- tional inequalities, J. Math. Anal. Appl., 185 (1994), 706–712.

[12] N.J. HUANG, Generalized nonlinear variational inclusions with noncom- pact valued mapping, Appl. Math. Lett., 9(3) (1996), 25–29.

[13] N.J. HUANG, Nonlinear Implicit quasi-variational inclusions involving generalizedm-accretive mappings, Arch. Inequal. Appl., 2 (2004), 403–

416.

[14] N.J. HUANG, Mann and Ishikawa type perturbed iterative algorithms for generalized nonlinear implicit quasi-variational inclusions, Computers Math. Applic., 35(10) (1998), 1–7.

[15] N.J. HUANG, A new completely general class of variational inclusions with noncompact valued mappings, Computers Math. Applic., 35(10) (1998), 9–14.

(22)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page22of23

[16] N.J. HUANG AND Y.P. FANG, A new class of general variational inclusions involving maximalη-monotone mappings, Publ Math. Debrecen, 62 (2003), 83–98.

[17] M.M. JIN, A new proximal point algorithm with errors for nonlinear variational inclusions involving generalizedm-accretive mappings, Nonlinear Anal. Forum, 9(1) (2004), 87–96.

[18] M.M. JIN AND Q.K. LIU, Nonlinear quasi-variational inclusions involving generalizedm-accretive mappings, Nonlinear Funct. Anal. Appl., 9(3) (2004), 485–494.

[19] M.M. JIN, Generalized nonlinear implicit quasi-variational inclusions with relaxed monotone mappings, Adv. Nonlinear Var. Inequal., 7(2) (2004), 173–181.

[20] G. KASSAY AND J. KOLUMBAN, System of multi-valued variational inequalities, Publ. Math. Debrecen, 56 (2000), 185–195.

[21] J.K. KIM ANDD.S. KIM, A new system of generalized nonlinear mixed variational inequalities in Hilbert spaces, J. Convex Anal., 11 (2004), 117–

124.

[22] S.B. NADLER, Muliti-valued contraction mappings, Pacific J. Math., 30 (1969), 475–488.

[23] K.R. KAZMI, Mann and Ishikawa type perturbed iterative algorithms for generalized quasivariational inclusions, J. Math. Anal. Appl., 209 (1997), 572–584.

(23)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page23of23

[24] R.U. VERMA, Iterative algorithms and a new system of nonlinear qua- sivariational inequalities, Adv. Nonlinear Var. Inequal., 4(1) (2001), 117–

124.

[25] R.U. VERMA, Projection methods, Algorithms, and a new system of nonlinear quasivariational inequalities, Comput. Math. Appl., 41 (2001), 1025–1031.

[26] R.U. VERMA, Generalized system for relaxed coercive variational in- equalities and projection methods, J. Optim. Theory Appl., 121 (2004), 203–210.

[27] R.U. VERMA, Generalized variational inequalities involving multivalued relaxed monotone operators, Appl. Math. Lett., 10(4) (1997), 107–109.

[28] R.U. VERMA, A-monotonicity and applications to nonlinear variational inclusions, Journal of Applied Mathematics and Stochastic Analysis, 17(2) (2004), 193–195.

[29] R.U. VERMA, Approximation-solvability of a class of A-monotone vari- ational inclusion problems, J. KSIAM, 8(1) (2004), 55–66.

[30] GEORGE X.Z. YUAN, KKM Theory and Applications in Nonlinear Anal- ysis, Marcel Dekker, New York, 1999.

[31] D.L. ZHU AND P. MARCOTTE, Co-coercivity and its roll in the conver- gence of iterative schemes for solving variational inequalities, SIAM J.

Optimization, 6 (1996), 714–726.