Research Article A New System of Nonlinear Fuzzy Variational Inclusions Involving A, η -Accretive Mappings in Uniformly Smooth Banach Spaces

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1 Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009, Article ID , 33 pages doi:0.55/2009/ Research Article A New System of Nonlinear Fuzzy Variational Inclusions Involving A, η -Accretive Mappings in Uniformly Smooth Banach Spaces M. Alimohammady,, 2 J. Balooee, Y. J. Cho, 3 and M. Roohi Department of Mathematics, Faculty of Basic Sciences, University of Mazandaran, Babolsar , Iran 2 Department of Mathematics, King s College London Strand, London WC2R 2LS, UK 3 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju , South Korea Correspondence should be addressed to Y. J. Cho, yjcho@gnu.ac.kr Received 4 October 2009; Accepted 4 November 2009 Recommended by Charles E. Chidume A new system of nonlinear fuzzy variational inclusions involving A, η -accretive mappings in uniformly smooth Banach spaces is introduced and studied many fuzzy variational and variational inequality inclusion problems as special cases of this system. By using the resolvent operator technique associated with A, η -accretive operator due to Lan et al. and Nadler s fixed points theorem for set-valued mappings, an existence theorem of solutions for this system of fuzzy variational inclusions is proved. We also construct some new iterative algorithms for the solutions of this system of nonlinear fuzzy variational inclusions in uniformly smooth Banach spaces and discuss the convergence of the sequences generated by the algorithms in uniformly smooth Banach spaces. Our results extend, improve, and unify many known results in the recent literatures. Copyright 009 M. Alimohammady 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.. Introduction Variational inequality was initially studied by Stampacchia in 964. In order to study many kinds of problems arising in industrial, physical, regional, economical, social, pure, and applied sciences, the classical variational inequality problems have been extended and generalized in many directions. Among these generalizations, variational inclusion introduced and studied by Hassouni and Moudafi 2 is of interest and importance. It provides us with a unified, natural, novel innovative, and general technique to study a wide class of the problems arising in different branches of mathematical and engineering sciences see, e.g., 3 7.

2 2 Journal of Inequalities and Applications Next, the development of variational inequality is to design efficient iterative algorithms to compute approximate solutions for variational inequalities and their generalizations. Up to now, many authors have presented implementable and significant numerical methods such as projection method, and its variant forms, linear approximation, descent method, Newton s method and the method based on the auxiliary principle technique. In particular, the method based on the resolvent operator technique is a generalization of the projection method and has been widely used to solve variational inclusions. Some new and interesting problems, which are called the systems of variational inequality problems, were introduced and studied. Pang 8, Cohen and Chaplais 9, Bianchi 0, and Ansari and Yao considered some systems of scalar variational inequalities and Pang showed that the traffic equilibrium problem, the spatial equilibrium problem, the Nash equilibrium, and the general equilibrium programming problems can be modelled as variational inequalities. He decomposed the original variational inequality into a system of variational inequalities which are easy to solve and studied the convergence of such methods. Ansari et al. 2 introduced and studied a system of vector variational inequalities by a fixed point theorem. Allevi et al. 3 considered a system of generalized vector variational inequalities and established some existence results under relative pseudomonotonicity. Kassay and Kolumbán 4 introduced a system of variational inequalities and proved an existence theorem by the Ky Fan lemma. Kassay et al. 5 studied Minty and Stampacchia variational inequality systems with the help of the Kakutani-Fan-Glicksberg fixed point theorem. Peng 6, 7 Peng and Yang 8 introduced a system of quasivariational inequality problems and proved its existence theorem by maximal element theorems. Verma 9 23 introduced and studied some systems of variational inequalities and developed some iterative algorithms for approximating the solution for this system of generalized nonlinear quasivariational inequalities in Hilbert spaces. J. K. Kim and D. S. Kim 24 introduced a new system of generalized nonlinear quasivariational inequalities and obtained some existence and uniqueness results of solution for this system of generalized nonlinear quasivariational inequalities in Hilbert spaces. Cho et al. 25 introduced a new system of nonlinear variational inequalities and proved some existence and uniqueness theorems of solutions for the system of nonlinear variational inequalities in Hilbert spaces. As generalizations of system of variational inequalities, Agarwal et al. 26 introduced a system of generalized nonlinear mixed quasivariational inclusions and investigated the sensitivity of solutions for this system of generalized nonlinear mixed quasivariational inclusions in Hilbert spaces. Kazmi and Bhat 27 introduced a system of nonlinear variational-like inclusions and gave an iterative algorithm for finding its approximate solution. It is known that accretivity of the underlying operator plays indispensable roles in the theory of variational inequality and its generalizations. In 200, Huang and Fang 28 were the first to introduce generalized m-accretive mapping and give the definition of the resolvent operator for generalized m-accretive mappings in Banach spaces. They also proved some properties of the resolvent operator for generalized m-accretive mappings in Banach spaces. Subsequently, Fang and Huang 29, Yan et al. 30, Fang et al. 3, Lan et al. 32, 33, Fang and Huang 34, and Peng et al. 35 introduced and investigated many new systems of variational inclusions involving H-monotone operators and H, η -monotone operators in Hilbert spaces, generalized m- accretive mappings, H-accretive mappings and H, η -accretive mappings in Banach spaces, respectively.

3 Journal of Inequalities and Applications 3 In 2004, Verma in 36, 37 introduced new notions of A-monotone and A, η monotone operators and studied some properties of A-monotone and A, η -monotone operators in Hilbert spaces. In 38, Lan et al. first introduced a new concept of A, η -accretive mappings, which generalizes the existing monotone or accretive operators and studied some properties of A, η -accretive mappings and defined resolvent operators associated with A, η -accretive mappings. They also investigated a class of variational inclusions using the resolvent operator associated with A, η -accretive mappings. Subsequently, Lan 39, by using the concept of A, η -accretive mappings and the new resolvent operator technique associated with A, η -accretive mappings, introduced and studied a system of general mixed quasivariational inclusions involving A, η -accretive mappings in Banach spaces and constructed a perturbed iterative algorithm with mixed errors for this system of nonlinear A, η -accretive variational inclusions in q-uniformly smooth Banach spaces. On the other hand, the fuzzy set theory introduced by Zadeh 40 has emerged as an interesting and fascinating branch of pure and applied sciences. The application of the fuzzy set theory can be found in many branches of regional, physical, mathematical, and engineering sciences see 4 45 and the references therein. In 989, Chang and Zhu 46 first introduced the classes of variational inequalities for fuzzy mappings. In subsequent years, several classes of variational inequalities, variational inclusions, and complementarity problems for fuzzy mappings were investigated by many authors, in particular, by Chang and Haung 47, 48, Lan et al. 49, Noor 50 52, Noor and Al-said 53, and many others. Recently, Lan and Verma 54, by using the concept of A, η -accretive mappings, the resolvent operator technique associated with A, η -accretive mappings, introduced and studied a new class of nonlinear fuzzy variational inclusion systems with A, η -accretive mappings in Banach spaces and construct some new iterative algorithms to approximate the solutions of the nonlinear fuzzy variational inclusion systems. Inspired and motivated by recent research works in these fields, in this paper, we introduce and study a new system of nonlinear fuzzy variational inclusions with A, η -accretive mappings in Banach spaces. By using the resolvent operator associated with A, η -mappings due to Lan et al. and Nadler s fixed points theorem, we construct some new iterative algorithms for approximating the solutions of this system of nonlinear fuzzy variational inclusions in Banach spaces and prove the existence of solutions and the convergence of the sequences generated by the algorithms in q-uniformly smooth Banach spaces. The results presented in this paper improve and extend the corresponding results of 29 35, 38, 39, and many other recent works. 2. Preliminaries Let X be a real Banach space with dual space X,, be the dual pair between X and X, 2 X denote the family of all nonempty subsets of X, andletcb X denote the family of all nonempty closed bounded subsets of X. The generalized duality mapping J q : X 2 X is defined by J q x { f X : x, f x q, f x q }, x X, 2. where q> is a constant. In particular, J 2 is the usual normalized duality mapping.

4 4 Journal of Inequalities and Applications It is known that, in general, J q x x q 2 J 2 x for all x / 0andJ q is single valued if X is strictly convex. In the sequel, we always assume that X is a real Banach space such that J q is single-valued. If X is a Hilbert space, then J 2 becomes the identity mapping on X. The modulus of smoothness of X is the function ρ X : 0, 0, defined by { } ρ X t sup x y x y : x, y t A Banach space X is said to be uniformly smooth if ρ X t lim 0. t 0 t 2.3 X is called q-uniformly smooth if there exists a constant c>0 such that ρ X t ct q, q >. 2.4 Note that J q is single-valued if X is uniformly smooth. Concerned with the characteristic inequalities in q-uniformly smooth Banach spaces, Xu 6 proved the following result. Lemma 2.. A real Banach space X is q-uniformly smooth if and only if there exists a constant c q > 0 such that, for all x, y X, x y q x q q y, J q x c q y q. 2.5 Definition 2.2. A set-valued mapping T : X 2 X is said to be ξ-ĥ-lipschitz continuous if there exists a constant ξ>0 such that Ĥ T x,t y ξ x y, x, y X, 2.6 where Ĥ :2 X 2 X R { } is the Hausdorff pseudo-metric, that is, Ĥ A, B max { sup x A d x, B, sup d y, A }, A, B 2 X, 2.7 y B where d u, K inf v K u v. It should be pointed that if domain of Ĥ is restricted to closed bounded subsets CB X, then Ĥ is the Hausdorff metric.

5 Journal of Inequalities and Applications 5 Lemma 2.3 see 62. Let X, d be a complete metric space and let T : X CB X be a set-valued mapping satisfying Ĥ T x,t y kd x, y, x, y X, 2.8 where k 0, is a constant. Then the mapping T has a fixed point in X. Lemma 2.4 see 62. Let X, d be a complete metric space and let T : X CB X ba a set-valued mapping. Then for any ε>0 and any x, y X, u T x, there exists v T y such that d u, v ε Ĥ T x,t y. 2.9 Definition 2.5. Let X be a q-uniformly smooth Banach space, T, A : X X and let η : X X X be single-valued mappings. i T is said to be accretive if T x T y,jq x y 0, x, y X; 2.0 ii T is said to be strictly accretive if T is accretive and T x T y,jq x y 0 2. if and only if x y; iii T is said to be r-strongly accretive if there exists a constant r>0such that T x T y,jq x y r x y q, x, y X; 2.2 iv T is said to be m-relaxed accretive if there exists a constant m>0 such that T x T y,jq x y m x y q, x, y X; 2.3 v T is said to be ζ, ς -relaxed cocoercive if there exist constants ζ, ς > 0 such that T x T y,jq x y ζ T x T y q ς x y q, x, y X; 2.4 vi T is said to be γ-lipschitz continuous if there exists a constant γ>0such that T x T y γ x y, x, y X; 2.5 vii η is said to be τ-lipschitz continuous if there exists a constant τ such that η x, y τ x y, x, y X; 2.6

6 6 Journal of Inequalities and Applications viii η, is said to be ɛ-lipschitz continuous in the first variable if there exists a constant ɛ>0such that η x, u η y, u ɛ x y, x, y, u X; 2.7 ix η,u is said to be ρ, ξ -relaxed cocoercive with respect to A if there exist constants ρ, ξ > 0 such that η x, u η y, u,jq A x A y ρ η x, u η y, u q ξ x y q, x, y, u X. 2.8 In a similar way to viii and ix, we can define the Lipschitz continuity of the mapping η, in the second variable and relaxed cocoercivity of η u, with respect to A. Definition 2.6. Let X be a q-uniformly smooth Banach space, η : X X X and let H, A : X X be three single-valued mappings. Set-valued mapping M : X 2 X is said to be i accretive if u v, Jq x y 0, x, y X, u Mx, v My; 2.9 ii η-accretive if u v, Jq η x, y 0, x, y X, u Mx, v My; 2.20 iii strictly η-accretive if M is η-accretive and the equality holds if and only if x y; iv r-strongly η-accretive if there exists a constant r>0such that u v, Jq η x, y r x y q, x, y X, u Mx, v My; 2.2 v α-relaxed η-accretive if there exists a constant α>0 such that u v, Jq η x, y α x y q, x, y X, u Mx, v My; 2.22 vi m-accretive if M is accretive and I λm X X for all λ>0, where I denotes the identity operator on X; vii generalized m-accretive if M is η-accretive and I λm X X for all λ>0; viii H-accretive if M is accretive and H λm X X for all λ>0; ix H, η -accretive if M is η-accretive and H λm X X for all λ>0. Remark 2.7. The following should be noticed. The class of generalized m-accretive operators was first introduced by Huang and Fang 28 and includes that of m-accretive operators as a special case. The class of

7 Journal of Inequalities and Applications 7 H-accretive operators was first introduced and studied by Fang and Huang 63 and also includes that of m-accretive operators as a special case. 2 When X H is a Hilbert space, i ix of Definition 2.6 reduce to the definitions of monotone operators, η-monotone operators, strictly η-monotone operators, strongly η-monotone operators, relaxed η-monotone operators, maximal monotone operators, maximal η-monotone operators, H-monotone operators, and H, η monotone operators, respectively. Definition 2.8. Let A : X X, η : X X X be two single-valued mappings and let M : X 2 X be a set-valued mapping. Then M is said to be A, η -accretive with constant m if M is m-relaxed η-accretive and A λm X X for all λ>0. Remark 2.9. For appropriate and suitable choices of m, A, η, and the space X, itiseasyto see that Definition 2.8 includes a number of definitions of monotone operators and accretive operators see 38. In 38, Lan et al. showed that A ρm is a single-valued operator if M : X 2 X is an A, η -accretive mapping and A : X X an r-strongly η-accretive mapping. Based on this fact, we can define the resolvent operator R η,a M,ρ associated with an A, η -accretive mapping M as follows. Definition 2.0. Let A : X X be a strictly η-accretive mapping and let M : X 2 X be an A, η -accretive mapping. The resolvent operator R η,a M,ρ : X X associated with A and M is defined by R η,a M,ρ x A ρm x, x X Proposition 2. see 38. Let X be a q-uniformly smooth Banach space, let η : X X X be τ-lipschitz continuous, let A : X X be a r-strongly η-accretive mapping and let M : X 2 X be an A, η -accretive mapping with constant m. Then the resolvent operator R η,a M,ρ : X X is τ q / r ρm -Lipschitz continuous, that is, R η,a M,ρ τ q x Rη,A M,ρ y r ρm x y, x, y X, 2.24 where ρ 0,r/m is a constant. In what follows, we denote the collection of all fuzzy sets on X by F X {A A : X 0, }. A mapping S from X to F X is called a fuzzy mapping. IfS : X F X is a fuzzy mapping, then the set S x for any x X is a fuzzy set on F X in the sequel we denote S x by S x and S x y for any y X is the degree of membership of y in S x. For any A F X and α 0,, theset A α {x X : A x α} 2.25 is called a α-cut set of A.

8 8 Journal of Inequalities and Applications A fuzzy mapping S : X F X is said to satisfy the condition if there exists a function a : X 0, such that for each x X the set S x a x : { y X : S x y a x } 2.26 is a nonempty closed and bounded subset of X, thatis, S x a x CB X. By using the fuzzy mapping S satisfying the condition with corresponding function a : X 0,, we can define a set-valued mapping S as follows: S : X CB X, x S x a x In the sequel, S, T, L, D, G, W, andk are called the set-valued mappings induced by the fuzzy mappings S, T, L, D, G, W,andK, respectively. 3. A New System of Fuzzy Variational Inclusions In this section, we introduce some systems of fuzzy variational inclusions q-uniformly smooth Banach spaces X and their relations. Let X be a q -uniformly smooth Banach space with q >, let X 2 be a -uniformly smooth Banach space with >, let E, P : X X 2 X, F, Q : X X 2 X 2, A : X X, A 2 : X 2 X 2, f, p, l : X X, g,h,k : X 2 X 2, η : X X X, η 2 : X 2 X 2 X 2 be single-valued mappings, and let S, T, L, D : X F X and G, W, K : X 2 F X 2 be fuzzy mappings. Further, suppose that M : X X 2 X and N : X 2 X 2 2 X 2 are any nonlinear operators such that for all z X, M,z : X 2 X is an A,η accretive with f x y dom M,z for all x, y X and for all t X 2, N,t : X 2 2 X 2 is an A 2,η 2 -accretive with g u dom N,t for all u X 2. Now, for given mappings ã, b, c, d : X 0, and ẽ, f, g : X 2 0,, we consider the following system. System 3.. For any given a X, b X 2, > 0, > 0, our problem is as follows: Find x, z, u, v, m X and y, w, t, s X 2 such that S x u ã x, T x v b x, L x z c x, D x m d x, G y w ẽ y, W y t f y, K y s g y,and a E p x,w P l z,t M f x v, x, b F u, h y Q m, k s N g y,y. 3. This system is called a system of nonlinear fuzzy variational inclusions involving A, η -accretive mappings in uniformly smooth Banach spaces. Remark 3.2. For appropriate and suitable choices of X, X 2, q,, E, P, F, Q, A, A 2, f, g, h, k, l, p, η, η 2, S, T, L, D, G, W, K, M, N, ã, b, c, d, ẽ, f, and g one can obtain many known and new classes of fuzzy variational inequalities and fuzzy variational inclusions as special cases of System 3.. Now, we consider some special cases of System 3..

9 Journal of Inequalities and Applications 9 System 3.3. Let S, T, L, D : X CB X and G, W, K : X 2 CB X 2 be classical set-valued mappings and let M, N, f, g, E, P, F, Q, p, l, h, k be the mappings as in System 3.. Now, by using S, T, L, D, G, W, andk, we define fuzzy mappings S, T, L, D : X 2 X and G, W, K : X 2 2 X 2 as follows: S x χ S x, T x χ T x, L x χ L x D x χ D x, G x χ G x, W x χ W x, K x χ K x, 3.2 where χ S x, χ T x, χ L x, χ D x, χ G x, χ W x,andχ K x are the characteristic functions of the sets S x, T x, L x, D x, G x, W x,andk x, respectively. It is easy to see that S, T, L, andd are fuzzy mappings satisfying the condition with constant functions ã x, b x, c x, d x for all x X, respectively, and G, W, andk are fuzzy mappings satisfying the condition with constant functions ẽ y, f y, g y for all y X 2, respectively. Also S ã x χ S x { r X : χ S x r } S x, T b x χ T x { r X : χ T x r } T x, L c x χ L x { r X : χ L x r } L x, D d x χ D x { r X : χ D x r } D x, 3.3 G ẽ y χ G y { t X 2 : χ G y t } G y, W f y χ W y { t X 2 : χ W y t } W y, K g y χ K y { t X 2 : χ K y t } K y. Then System 3. is equivalent to the following: Find x, z, u, v, m X, y, w, t, s X 2 such that u S x, v T x, z L x, m D x, w G y, t W y, s K y, and a E p x,w P l z,t M f x v, x, b F u, h y Q m, k s N g y,y 3.4. System 3.3 is called a system of nonlinear set-valued variational inclusions with A, η accretive mappings. System 3.4. If T : X X is a single-valued mapping, then System 3.3 collapses to the following system of nonlinear variational inclusions: Find x, z, u, m X, y, w, t, s X 2 such that u S x, z L x, m D x, w G y, t W y, s K y, and a E p x,w P l z,t M f x T x,x, b F u, h y Q m, k s N g y,y 3.5.

10 0 Journal of Inequalities and Applications System 3.5. If X i H i i, 2 are two Hilbert spaces, S : H H and G : H 2 H 2 are two single-valued mappings, p h l k I : the identity mapping,, T 0 : the zero mapping, a b 0, M x, y M x for all x, y H H, N x, y N x for all x, y H 2 H 2, then System 3.4 reduces to the following system: Find x, y, z, m, t, s such that x, y H H 2, z L x, m D x, t W y, s K y, and 0 E x, y P z, t M f x, 0 F x, y Q m, s N g y. 3.6 System 3.5 was introduced and studied by Peng and Zhu in 59. System 3.6. If a b 0,, and P Q 0, then System 3.3 can be replaced by the following: Find u S x, v T x and w G y such that 0 E p x,w M f x v, x, 0 F u, h y N g y,y, 3.7 which is studied by Lan and Verma 54. System 3.7. If T : X X is a single-valued mapping, then System 3.6 collapses to the following system of nonlinear variational inclusions: Find x, u X, y, w X 2 such that u S x, w G y and 0 E p x,w M f x T x,x, 0 F u, h y N g y,y, 3.8 which is studied by Lan and Verma 54. System 3.8. If p h I, T 0, M x, y M x for all x, y X X, N x, y N x for all x, y X 2 X 2, S : X X,andG : X 2 X 2 are identity mappings, then System 3.7 reduces to the following system: Find x, y X X 2 such that 0 E x, G y M f x, 0 F S x,y N g y. 3.9 System 3.8 is investigated by Jin 55 when S and G are the identity mappings. System 3.9. If f T g I, P Q 0,, S : X X,andG : X 2 X 2 are two single-valued mappings, then System 3.4 is equivalent to the following:

11 Journal of Inequalities and Applications Find x, y X X 2 such that a E p x,g y M x, x, b F S x,h y N y, y, 3.0 which is introduced and studied by Lan 39 when S and G are identity mappings. System 3.0. When p h S G I, a b 0, System 3.9 can be replaced to the following: Find x, y X X 2 such that 0 E x, y M x, x, 0 F x, y N y, y, 3. which is studied by Jin 56. System 3.. If X i H i i, 2 is two Hilbert spaces, M x, y M x for all x, y H H and N x, y N x for all x, y H 2 H 2, then System 3.7 reduces to the following generalized system of set-valued variational inclusions: Find x, u H, y, w H 2 such that u S x, w G y, and 0 E p x,w M f x T x, 0 F u, h y N g y, 3.2 which is studied by Lan et al. 57 when M, N are A-monotone mappings and there exists single-valued mapping ψ : H H such that ψ x f x T x for all x H. System 3.2. If g p h f T I, then System 3. collapses to the following system of nonlinear variational inclusions: Find x, y H H 2, u S x, w G y such that 0 E x, w M x, 0 F u, y N y, 3.3 which is considered by Huang and Fang 28. System 3.3. If S : H H and G : H 2 H 2 are two single-valued mappings, then System 3.2 is equivalent to the following: Find x, y H H 2 such that 0 E x, G y M x, 0 F S x,y N y, 3.4

12 2 Journal of Inequalities and Applications which is investigated by Fang et al. 3 and Peng et al. 35, Fang and Huang 34, and Verma 36 with S G I. System 3.4. If M x ϕ x and N y φ y for all x H and y H 2, where ϕ : H R { } and φ : H 2 R { } are two proper, convex, and lower semicontinuous functionals, and ϕ and φ denote subdifferential operators of ϕ and φ, respectively, then System 3.3 reduces to the following system: Find x, y H H 2 such that E x, G y,s x ϕ s ϕ x 0, s H, F S x,y,t y φ t φ y 0, t H2, 3.5 which is called a system of nonlinear mixed variational inequalities. Some special cases of System 3.7 can be found in 22. Further, if S G I, then System 3.7 reduces to the system of nonlinear variational inequalities considered by Cho et al. 25. System 3.5. If M x δ K x and N y δ K2 y for all x K and y K 2, where K and K 2 are nonempty closed convex subsets of H and H 2, respectively, and δ K and δ K2 denote indicator functions of K and K 2, respectively, then System 3.4 becomes the following problem: Find x, y K K 2 such that E x, G y,s x 0, s K, F S x,y,t y 0, t K 2, 3.6 which is the just system in 24 when S and G are singlevalued and S G I. System 3.6. If H H 2 H, K K 2 K, E x, G y ρ G y x y, andf S x,y ρ 2 S x y x for all x, y H, where ρ > 0andρ 2 > 0 are two constants, then System 3.5 is equivalent to the following: Find an element x, y K K such that ρ G y x y, s x 0, s K, ρ 2 S x y x, t y 0, t K, 3.7 which is the system of nonlinear variational inequalities considered by Verma 22 with S G. Remark 3.7. If x y, S G and ρ ρ 2, then System 3.6 reduces to the following classical nonlinear variational inequality problem: Find an element x K such that S x,z x 0, z K. 3.8

13 Journal of Inequalities and Applications 3 4. Existence Theorems In this section, we prove the existence theorem for solutions of Systems 3.. For our main results, we have the following lemma which offers a good approach to solve System 3.. Lemma 4.. Let X i, A i, η i, λ i i, 2, E, F, P, Q, S, T, L, D, G, W, K, M, N, f, g, h, p, l, k, a, and b be the same as in System 3.. Then, for any given x, z, u, v, m X and y, w, t, s X 2, x, y, z, t, m, s, u, v, w is a solution of System 3. if and only if ] f x v R η,a ρ M,x,ρ A f x v E p x,w P l z,t a, g y R η 2,A 2 N,y,ρ 2 A 2 g y ρ 2 F u, h y Q m, k s b ], 4. where ρ > 0 and ρ 2 > 0 are two constants. Proof. The conclusion follows directly from Definition 2.0 and some simple arguments. From Lemma 4., wehavethefollowing. Theorem 4.2. Let X and X 2 bethesameasinlemma 4., lets, T, L, D : X F X and G, W, K : X 2 F X 2 be fuzzy mappings satisfying the condition with the corresponding functions ã, b, c, d, ẽ, f and g, respectively, S, T, L, D : X CB X, and let G, W, K : X 2 CB X 2 be ξ-ĥ -Lipschitz continuous, ζ-ĥ -Lipschitz continuous, γ-ĥ -Lipschitz continuous, ϖ-ĥ -Lipschitz continuous, ξ -Ĥ 2 -Lipschitz continuous, ζ -Ĥ 2 -Lipschitz continuous, and γ -Ĥ 2 - Lipschitz continuous, respectively, where Ĥ i is the Hausdorff pseudometric on 2 X i for i, 2. Assume that η i : X i X i X i is τ i -Lipschitz continuous, A i : X i X i is r i -strongly η i -accretive and β i - Lipschitz continuous for i, 2, p, l : X X are δ -Lipschitz continuous, and δ 2 -Lipschitz continuous, respectively, h, k : X 2 X 2 are π -Lipschitz continuous and π 2 -Lipschitz continuous, respectively, f : X X is κ, e -relaxed cocoercive, μ-lipschitz continuous and g : X 2 X 2 is σ, e 2 -relaxed cocoercive, ɛ-lipschitz continuous. Suppose that M,z : X 2 X is an A,η accretive operator with constant m for all z X and N,t : X 2 2 X 2 is an A 2,η 2 -accretive operator with constant m 2 for all t X 2, and E, P : X X 2 X are two single-valued mappings such that E,y and P,y are ν -Lipschitz continuous and ν 2 -Lipschitz continuous in the first variable, respectively, E x,, P x, are ι -Lipschitz continuous ι 2 -Lipschitz continuous in the second variable, respectively, for all x, y X X 2, and E p,y is θ,s -relaxed cocoercive with respect to f,wheref : X X is defined by f x A f x v A f x v for all x X, b : X 0, and T x v b x. Further, suppose that F, Q : X X 2 X 2 are two nonlinear mappings such that F,y, Q,y are ρ -Lipschitz continuous and ρ 2 -Lipschitz continuous in the first variable, respectively, F x, and Q x, are υ -Lipschitz continuous and υ 2 - Lipschitz continuous in the second variable, respectively, and F x, h is θ 2,s 2 -relaxed cocoercive with respect to g,whereg : X 2 X 2 is defined by g x A 2 g x A 2 g x for all x X 2. In addition, if there exist constants ρ 0,r /m and ρ 2 0,r 2 /m 2 such that R η,a M,x,ρ z R η,a R η 2,A 2 N,x,ρ 2 z R η 2,A 2 M,y,ρ z N,y,ρ 2 z ς x y, x, y, z X, 4.2 ϑ x y, x, y, z X 2, 4.3

14 4 Journal of Inequalities and Applications ς ζ q q e c q q κ μ q <, ϑ e 2 c q2 σ ɛ <, q β q q q ρ μ ζ q θ ν q δ q c q ρ q ν q δ s β q ρ ɛ q2 θ 2 υ π s2 c ρ 2 q υ π < τ q r ρ m χ ν 2 δ 2 γ, ρ < τ 2 r2 ρ 2 m 2 χ2 υ 2 π 2 γ, ρ where χ ς ζ q q e c q q κ μ q ρ 2τ 2 ρ ξ ρ 2 ϖ, r2 ρ 2 m 2 χ 2 ϑ e 2 c q2 σ ɛ ρ τ q ι ξ ι 2 ζ, r ρ m 4.5, are the same as in System 3., and c q, c q2 System 3. admits a solution. are two constants guaranteed by Lemma 2., then Proof. For any given ρ > 0andρ 2 > 0, define mappings Φ ρ : X X X X 2 X 2 X and Ψ ρ2 : X X X 2 X 2 X 2 as follows: Φ ρ x, z, v, t, w ] x f x v R η,a ρ M,x,ρ A f x v E p x,w P l z,t a, Ψ ρ2 u, m, s, y y g y ] R η 2,A 2 ρ 2 N,y,ρ 2 A 2 g y F u, h y Q m, k s b 4.6 for all x, y, z, t, m, s, u, v, w X X 2 X X 2 X X 2 X X X 2, where a X and b X 2 are the same as in System 3.,andletã, b, c, d : X 0, and ẽ, f, g : X 2 0, be mappings such that S x u ã x, T x v b x, L x z c x, D x m d x, G y w ẽ y, W y t f y, andk y s g y. Now, define a norm on X X 2 by u, v u v, u, v X X It is easy to see that X X 2, is a Banach space see 34. For any given ρ > 0and ρ 2 > 0, define a mapping Q ρ,ρ 2 : X X 2 X X X X X 2 X 2 X 2 X X 2 by Q ρ,ρ 2 x, y, z, u, v, m, s, t, w Φρ x, z, v, t, w, Ψ ρ2 u, m, s, y 4.8

15 Journal of Inequalities and Applications 5 for all x, y, z, u, v, m, s, t, w X X 2 X X X X X 2 X 2 X 2 and let R ρ,ρ 2 x, y { Qρ,ρ 2 x, y, z, u, v, m, s, t, w : S x u ã x, T x v b x, L x z c x, D x m d x, G y w ẽ y, W y t f y, K y s g y, where ã, b, c, d : X 0,, ẽ, f, } g : X 2 0, 4.9 for all x, y X X 2. Then, for any given x, y, x,y X X 2, ε > 0 and Q ρ,ρ 2 x, y, z, u, v, m, s, t, w R ρ,ρ 2 x, y, there exists z, u, v, m, s, t, w X X X X X 2 X 2 X 2 such that S x u ã x, T x v b x, L x z c x, D x m d x, G y w ẽ y, W y t f y, K y s g y, 4.0 where ã, b, c, d : X 0,, ẽ, f, g : X 2 0, and 4.6 holds. Since S x u ã x, T x v b x, L x z c x, D x m d x, G y w ẽ y, W y t f y, K y s g y, thatis,u S x CB X, v T x CB X, z L x CB X, m D x CB X, w G y CB X 2, t W y CB X 2, s K y CB X 2, it follows from Lemma 2.4 that there exist u S x, v T x, z L x, m D x, w G y, t W y, s K y,that is, S x u ã x, T x v b x, L x z c x, D x m d x, G y w ẽ y, W y t f y, K y s g y such that u u ε Ĥ S x,s x, z z ε Ĥ L x,l x, w w ε Ĥ 2 G y,g y, s s ε Ĥ2 K y,k y. v v ε Ĥ T x,t x, m m ε Ĥ D x,d x, t t ε Ĥ 2 W y,w y, 4. Letting Φ ρ x,z,v,t,w x f x v R η,a M,x,ρ A f x v ρ E p x,w P l z,t a ], Ψ ρ2 u,m,s,y y g y R η 2,A 2 N,y,ρ 2 A 2 g y ρ 2 F u,h y Q m,k s b ], 4.2

16 6 Journal of Inequalities and Applications we have Φρ x,z,v,t,w, Ψ ρ2 u,m,s,y Q ρ,ρ 2 x,y,z,u,v,m,s,t,w. 4.3 Now, it follows from 4.2 and Proposition 2. that Φρ x, z, v, t, w Φ ρ x,z,v,t,w x x f x f x v v ] Rη,A ρ M,x,ρ A f x v E p x,w P l z,t a R η,a M,x,ρ A f x v ρ E p x,w P l z,t a ] x x f x f x v v ] Rη,A ρ M,x,ρ A f x v E p x,w P l z,t a R η,a M,x,ρ A f x v ρ E p x,w P l z,t a ] Rη,A M,x,ρ A f x v ρ E p x,w P l z,t a ] R η,a M,x,ρ A f x v ρ E p x,w P l z,t a ] x x f x f x v v ς x x τ q r ρ m A ρ f x v E p x,w P l z,t a A f x v ρ E p x,w P l z,t a x x f x f x v v ς x x τ q { ρ E p x,w E p x,w P l z,t P l z,t r ρ m P l z,t P l z,t ] A f x v A f x v ρ E p x,w E p x,w } λ. 4.4

17 Journal of Inequalities and Applications 7 Thus, by Lemma 2., we have x x f x f x q x x q q f x f x,j q x x c q f x f x q. 4.5 Since f is κ, e -relaxed cocoercive and μ-lipschitz continuous, we conclude that x x f x f x q x x q q e x x q c q q κ μ q x x q q e c q q κ μ q x x q. 4.6 By 4. and ζ-ĥ -Lipschitz continuity of T, v v ε Ĥ T x,t x ζ ε x x. 4.7 Since E x, is ι -Lipschitz continuous in the second variable and G is ξ -Ĥ 2 -Lipschitz continuous, by 4., we have E p x,w E p x,w ι w w ι ε Ĥ 2 G y,g y ι ξ ε y y. 4.8 Since P x, is ι 2 -Lipschitz continuous in the second variable, W is ζ -Ĥ 2 -Lipschitz continuous, p 2 is δ 2 -Lipschitz continuous, P,y is ν 2 -Lipschitz continuous in the first variable, and L is γ-ĥ -Lipschitz continuous, using 4., we deduce that P l z,t P l z,t ι2 t t ι2 ε Ĥ 2 W y,w y ι 2 ζ ε y y, 4.9 and also P l z,t P l z,t ν2 l z l z ν2 δ 2 z z ν 2 δ 2 ε Ĥ L x,l x 4.20 ν 2 δ 2 γ ε x x.

18 8 Journal of Inequalities and Applications Again, by Lemma 2., it follows that A f x v A f x v ρ E p x,w E p x,w λ q A f x v A f x v q q ρ E p x,w E p x,w,j q A f x v A f x v 4.2 c q ρ q q E p x,w E p x,w q. Since A is β -Lipschitz continuous, f is μ-lipschitz continuous, and T is ζ-lipschitz continuous, by 4., weget A f x v A f x v β f x f x v v β f x f x v v 4.22 β μ ζ ε x x. Since E p,y is θ,s -relaxed cocoercive with respect to f, where f x A f x v A f x v, E,y is ν -Lipschitz continuous in the first variable and p is δ -Lipschitz continuous, we have E p x,w E p x,w,j q A f x v A f x v θ E p x,w E p x,w q s x x q θ ν q p x p x q s x x q 4.23 θ ν q δ q s x x q, and also E p x,w E p x,w ν p x p x ν δ x x Hence, using , we have A f x v A f x v ρ q E p x,w E p x,w λ q q β q ρ μ ζ ε q q θ ν q δ c q ρ q ν q δ x s x q, q 4.25 where c q is the constant as in Lemma 2.. Using ,and 4.25, it follows that Φρ x, z, v, t, w Φ ρ x,z,v,t,w ϕ ε x x φ ε y y, 4.26

19 Journal of Inequalities and Applications 9 where ϕ ε ς ζ ε q q e c q q κ ρ μ q τ q ν2 δ 2 γ ε ψ ε, r ρ m ψ ε q q q β q ρ μ ζ ε q θ ν q δ q c q ρ q ν q δ s q, 4.27 φ ε ρ τ q ι ξ ι 2 ζ ε r ρ m. Similarly, for any u, m, s, y, u,m,s,y X X X 2 X 2, it follows from 4.3 and Proposition 2. that Ψρ2 u, m, s, y Ψρ2 u,m,s,y y y g y g y Rη 2,A 2 N,y,ρ 2 ] ρ 2 A 2 g y F u, h y Q m, k s b R η 2,A 2 N,y,ρ 2 A 2 g y ρ 2 F u,h y Q m,k s b ] y y g y g y Rη 2,A 2 N,y,ρ 2 ] ρ 2 A 2 g y F u, h y Q m, k s b R η 2,A 2 N,y,ρ 2 A 2 g y ρ 2 F u,h y Q m,k s b ] Rη 2,A 2 N,y,ρ 2 A 2 g y ρ 2 F u,h y Q m,k s b ] R η 2,A 2 N,y,ρ 2 A 2 g y ρ 2 F u,h y Q m,k s b ] y y g y g y ϑ y y τ 2 r 2 ρ 2 m A ρ 2 2 g y F u, h y Q m, k s b 2 A 2 g y ρ 2 F u,h y Q m,k s b

20 20 Journal of Inequalities and Applications y y g y g y ϑ y y τ 2 r 2 ρ 2 m 2 { ρ2 F u, h y F u,h y Q m, k s Q m,k s Q m,k s Q m,k s A 2 g y A2 g y ρ 2 F u,h y F u,h y } λ Thus, by Lemma 2., we have y y g y g y y y g y g y,j q2 y y c q2 g y g y Since g is σ, e 2 -relaxed cocoercive and ɛ-lipschitz continuous, we have y y g y g y y y e 2 y y c q2 σ ɛ y y e 2 c q2 σ ɛ q 2 y y Since F,y is ρ -Lipschitz continuous in the first variable and S is ξ-ĥ -Lipschitz continuous, by 4., weobtain F u, h y F u,h y ρ u u ρ ε Ĥ S x,s x ρ ξ ε x x. 4.3 Since Q x, is υ 2 -Lipschitz continuous in the second variable, k is π 2 -Lipschitz continuous, Q,y is ρ 2 -Lipschitz continuous in the first variable, D is ϖ-ĥ -Lipschitz continuous, and K is γ -Ĥ 2 -Lipschitz continuous, using 4., we conclude that Q m, k s Q m,k s ρ2 m m ρ 2 ε Ĥ D x,d x 4.32 ρ 2 ϖ ε x x, Q m,k s Q m,k s υ 2 k s k s υ 2 π 2 s s υ 2 π 2 ε Ĥ 2 K y,k y 4.33 υ 2 π 2 γ ε y y.

21 Journal of Inequalities and Applications 2 Again, by Lemma 2., it follows that A 2 g y A 2 g y ρ 2 F u,h y F u,h y λ 2 A 2 g y A 2 g y ρ 2 F u,h y F u,h y,j q2 A2 g y A2 g y 4.34 c q2 ρ 2 F u,h y F u,h y. Since A 2 is β 2 -Lipschitz continuous and g is ɛ-lipschitz continuous, we have A 2 g y A2 g y β 2 g y g y β 2 ɛ y y Since F u, h is θ 2,s 2 -relaxed cocoercive with respect to g A 2 g, F x, is υ -Lipschitz continuous in the second variable, and h is π -Lipschitz continuous, we get F u,h y F u,h y,j q2 A2 g y A2 g y θ 2 F u,h y F u,h y s 2 y y θ 2 υ h y h y s 2 y y q θ 2 υ 2 q π 2 s 2 y y, F u,h y F u,h y υ h y h y υ π y y Therefore, it follows from that A 2 g y A 2 g y ρ 2 F u,h y F u,h y λ 2 q β 2 2 ɛ ρ 2 q θ 2 υ 2 q π 2 s 2 c q ρ 2 υ 2 q π 2 y y, 4.38 where c q2 is the constant as in Lemma 2.. From , and 4.38, it follows that Ψρ2 u, m, s, y Ψρ2 u,m,s,y ϕ2 ε x x φ2 ε y y, 4.39

22 22 Journal of Inequalities and Applications where ϕ 2 ε ρ 2τ 2 ρ ξ ρ 2 ϖ ε, r2 ρ 2 m 2 φ 2 ε ϑ e 2 c q2 σ ρ ɛ τ 2 2 υ2 π 2 γ ε ψ 2, r2 ρ 2 m 2 ψ 2 q β 2 ρ 2 ɛ 2 q2 θ 2 υ π s2 c q ρ 2 υ 2 q π 2 q It follows from 4.26 and 4.39 that Φρ x, z, v, t, w Φ ρ x,z,v,t,w Ψρ2 u, m, s, y Ψρ2 u,m,s,y ω ε x x y y, 4.4 where ω ε max{ϕ ε ϕ 2 ε,φ ε φ 2 ε }.Using 4.8 and 4.4, we deduce that Qρ,ρ 2 x, y, z, u, v, m, s, t, w Q ρ,ρ 2 x,y,z,u,v,m,s,t,w ω ε x, y x,y, 4.42 that is, sup d Q ρ,ρ 2 x, y, z, u, v, m, s, t, w, Rρ,ρ 2 x,y Q ρ,ρ 2 x,y,z,u,v,m,s,t,w R ρ,ρ 2 x,y ω ε x, y x,y Similarly, we have sup d Q ρ,ρ 2 x,y,z,u,v,m,s,t,w, R ρ,ρ 2 x, y Q ρ,ρ 2 x,y,z,u,v,m,s,t,w R ρ,ρ 2 x,y ω ε x, y x,y By 4.43, 4.44, and the definition of Hausdorff pseudo-metric, we have Ĥ R ρ,ρ 2 x, y, Rρ,ρ 2 x,y ω ε x, y x,y, x, y, x,y X X Letting ε 0, one has Ĥ R ρ,ρ 2 x, y, Rρ,ρ 2 x,y ω x, y x,y, x, y, x,y X X 2, 4.46

23 Journal of Inequalities and Applications 23 where ω max { ϕ ϕ 2,φ φ 2 }, ϕ ς ζ q q e c q q κ ρ μ q τ q ν2 δ 2 γ ψ, r ρ m q q q ψ β q ρ μ ζ q θ ν q δ q c q ρ q ν q δ s q, φ 2 ϑ e 2 c q2 σ ρ ɛ τ 2 2 υ2 π 2 γ ψ 2, r2 ρ 2 m 2 ϕ 2 ρ 2τ 2 ρ ξ ρ 2 ϖ, r2 ρ 2 m φ ρ τ q ι ξ ι 2 ζ r ρ m and ψ 2 is the constant as in From 4.4, we know that 0 ω< and so it follows from 4.46 that R ρ,ρ 2 : X X 2 X X 2 is a contractive mapping. Hence Lemma 2.3 implies that R ρ,ρ 2 has a fixed point in X X 2 ; that is, there exists a point x,y X X 2 such that x,y R ρ,ρ 2 x,y. Now, it follows from 4.6, 4.8, andlemma 4. that x,y,z,u,v,m,n,t,w is a solution of System 3. and this is the desired result. This completes the proof. By using Theorem 4.2, we can derive the following. Theorem 4.3. Let X i, A i, η i i, 2, S, T, L, D, G, W, K, S, T, L, D, G, W, K, M, N, E, P, F, Q, p, l, h, k, f, and g bethesameasintheorem 4.2. Assume that f : X X is κ-strongly accretive μ-lipschitz continuous and g : X 2 X 2 is σ-strongly accretive ɛ-lipschitz continuous. Further, if there exist constants ρ 0,r /m and ρ 2 0,r 2 /m 2 such that 4.2 and 4.3 hold and q ς ζ q q κ c q μ q <, ϑ σ c q2 ɛ <, q q β q ρ μ ζ q θ ν q δ q c q ρ q ν q δ s q q β 2 ρ 2 ɛ 2 q2 θ 2 υ π s2 c q ρ 2 υ 2 q π 2 < τ q r ρ m χ ν 2 δ 2 γ, ρ < τ 2 r2 ρ 2 m 2 χ2 υ 2 π 2 γ, ρ

24 24 Journal of Inequalities and Applications where χ ς ζ q q κ c q μ q ρ 2τ 2 ρ ξ ρ 2 ϖ, r2 ρ 2 m 2 χ 2 ϑ σ c q2 ɛ ρ τ q ι ξ ι 2 ζ, r ρ m 4.50, are the same as in System 3., and c q, c q2 System 3. admits a solution. are two constants guaranteed by Lemma 2., then Theorem 4.4. Let X i, A i, η i i, 2, p, l, h, k, f, g, F, Q, M, N, and P bethesameasin Theorem 4.2. Assume that T : X X is ζ-lipschitz continuous, and S, L, D : X CB X and G, W, K : X 2 CB X 2 are ξ-ĥ -Lipschitz continuous, γ-ĥ -Lipschitz continuous, ϖ-ĥ - Lipschitz continuous, ξ -Ĥ 2 -Lipschitz continuous, ζ -Ĥ 2 -Lipschitz continuous, and γ -Ĥ 2 -Lipschitz continuous, respectively. Suppose that E : X X 2 X is a single-valued mapping such that E,y is ν -Lipschitz continuous in the first variable and E x, is ι -Lipschitz continuous in the second variable for all x, y X X 2, and E p,y is a θ,s -relaxed cocercive mapping with respect to f A f T defined by f x A f x T x A f x T x for all x X. If there exist constants ρ 0,r /m and ρ 2 0,r 2 /m 2 such that conditions hold, then System 3.4 has a solution x,y,z,u,m,s,t,w. Theorem 4.5. Let X i, A i, η i i, 2, p, l, h, k, E, F, P, Q, M, N, S, T, L, D, G, W, and K bethesameasintheorem 4.4. Suppose that f : X X is κ-strongly accretive and μ-lipschitz continuous and g : X 2 X 2 is σ-strongly accretive and ɛ-lipschitz continuous. If there exist constants ρ 0,r /m and ρ 2 0,r 2 /m 2 such that conditions 4.2, 4.3, and 4.49 hold, then System 3.3 has a solution x,y,z,u,m,s,t,w. 5. Iterative Algorithm and Convergence In this section, motivated by Theorems 4.2 and 4.4, Lemmas 4. and 2.4, we construct the following iterative algorithms for approximating solutions of Systems 3. and 3.3 and discuss the convergence analysis of the algorithms. Algorithm 5.. Let X i, A i, η i, λ i i, 2, E, P, F, Q, p, l, h, k, f, g, M, N, S, T, L, D, G, W, K, S, T, L, D, G, W, K, a and b be the same as in System 3.. For any given x 0,y 0 X X 2, ã, b, c, d : X 0, and ẽ, f, g : X 2 0, for all n 0 and an element x, y, z, u, v, m, s, t, w X X 2 X X X X X 2 X 2 X 2, define the iterative sequence { x n,y n,z n,u n,v n,m n,s n,t n,w n } n 0 by x n α n x n α n x n f x n v n R η,a M,x n,ρ Θ n α n e n r n, y n α n y n α n y n g η y n R 2,A 2 N,y n,ρ 2 Ω n α n f n k n, S xn u n ã x n, u n u Ĥ S x n,s x, n

25 Journal of Inequalities and Applications 25 T xn v n b x n, L xn z n c x n, D xn m n d x n, G yn w n ẽ y n, W yn t n f y n, K yn s n g y n, v n v z n z m n m wn w tn t sn s n n n n n n Ĥ T x n,t x, Ĥ L x n,l x, Ĥ D x n,d x, Ĥ 2 G yn,g y, Ĥ 2 W yn,w y, Ĥ 2 K yn,k y, 5. where Θ n A f xn v n ρ E p xn,w n P l zn,t n a, Ω n A 2 g yn ρ 2 F un,h y n Q mn,k s n b, 5.2 ρ and ρ 2 are constants, {α n } is a sequence in 0, with n 0 α n, and{ e n,f n } n 0 and { r n,k n } n 0 are two sequences in X X 2 to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions: e n e n e n, f n f n f n, lim e n,f n 0, n e n,f n <, n 0 r n,k n <. n Algorithm 5.2. Assume that X i, A i, η i, λ i i, 2, E, P, F, Q, p, l, h, k, f, g, M, N, S, T, L, D, G, W, K, a and b are the same as in System 3.4. For any given x 0,y 0 X X 2, n 0and an element x, y, z, u, m, s, t, w X X 2 X X X X 2 X 2 X 2, define the iterative sequence { x n,y n,z n,u n,m n,s n,t n,w n } n 0 by x n α n x n α n x n f x n T x n R η,a M,x n,ρ Θ n α n e n r n, y n α n y n α n y n g η y n R 2,A 2 N,y n,ρ 2 Ω n α n f n k n, u n S x n, u n u Ĥ S x n,s x, n

26 26 Journal of Inequalities and Applications z n L x n, m n D x n, w n G y n, t n W y n, s n K y n, z n z m n m wn w tn t sn s n n n n n Ĥ L x n,l x, Ĥ D x n,d x, Ĥ 2 G yn,g y, Ĥ 2 W yn,w y, Ĥ 2 K yn,k y, 5.4 where Θ n A f x n T x n ρ / E p x n,w n P l z n,t n a, Ω n, ρ, ρ 2, {α n }, { e n,f n } n 0 and { r n,k n } n 0 are the same as in Algorithm 5.. Remark 5.3. If e n f n 0 for all n 0, L D W K 0, P Q 0, a b 0, and, then Algorithms 5. and 5.2 reduce to Algorithms 4. and 4.2 of 38. In particular, when we choose suitable α n, e n, f n, r n, k n, A i, η i i, 2, E, P, F, Q, p, l, h, k, f, g, M, N, S, T, L, D, G, W, K, S, T, L, D, G, W, K, and the spaces X, X 2, then Algorithms 5. and 5.2 can be degenerated to a number of algorithms involving many known algorithms due to classes of variational inequalities and variational inclusions see, e.g., 38, 55, 56, and the references therein. Lemma 5.4. Let {a n }, {b n }, and {c n } be three nonnegative real sequences satisfying the following condition: there exists a natural number n 0 such that a n t n a n b n t n c n, n n 0, 5.5 where t n 0,, n 0 t n, lim n b n 0, n 0 c n <.Thenlim n 0 a n 0. Proof. The proof directly follows from Liu 64, Lemma 2. Theorem 5.5. Let X i, A i, η i i, 2, E, P, F, Q, p, l, h, k, f, g, M, N, S, T, L, D, G, W, K, S, T, L, D, G, W, and K be the same as in Theorem 4.2. Suppose that all the conditions of Theorem 4.2 hold. Then the iterative sequence { x n,y n,z n,u n,v n,m n,s n,t n,w n } n 0 generated by Algorithm 5. converges strongly to a solution x,y,z,u,v,m,s,t,w of System 3.. Proof. It follows from Theorem 4.2 that System 3. has a solution x,y,z,u,v,m,s,t,w. Hence, by Lemma 4., we have f x v R η,a M,x,ρ A f x v ρ E p x,w P l z,t a ], g y R η 2,A 2 M,y,ρ 2 A 2 g y ρ 2 F u,h y Q m,k s b ]. 5.6

27 Journal of Inequalities and Applications 27 Using 5., 5.6, and our assumptions, it follows that x n x α n x n x α n xn x f x n f x vn v Rη,A M,x n,ρ α n e n r n ρ A f xn v n E p xn,w n P l zn,t n a ] R η,a M,x,ρ A f x v ρ E p x,w P l z,t a ] α n x n x α n xn x f x n f x vn v Rη,A M,x n,ρ ρ A f xn v n E p xn,w n P l zn,t n a ] R η,a M,x n,ρ A f x v ρ E p x,w P l z,t a ] Rη,A M,x n,ρ A f x v ρ E p x,w P l z,t a ] R η,a M,x,ρ A f x v ρ E p x,w P l z,t a ] α n e n e r n α n x n x α n n xn x f x n f x vn v ς x n x τ q r ρ m ρ E p xn,w n E p xn,w P l zn,t n P l z n,t P l z n,t P l z,t A f xn v n A f x v ρ E p xn,w E p x,w λ α n e n e n rn α n x n x α n ϕ n x n x φ n y n y α n e n e n r n, 5.7

28 28 Journal of Inequalities and Applications where ϕ n ς ζ n ψ n q q e c q q κ ρ μ q τ q ν2 δ 2 γ / n ψ n, r ρ m q β q μ ζ q ρ q θ ν q n λ δq s c q ρ q νq λ q δq, 5.8 φ n ρ τ q ι ξ ι 2 ζ / n r ρ m. Similarly, we have yn y αn yn y αn yn y g y n g y Rη 2,A 2 N,y n,ρ 2 α n fn kn ρ 2 A 2 g yn F un,h y n Q mn,k s n b ] R η 2,A 2 N,y,ρ 2 A 2 g y ρ 2 F u,h y Q m,k s b ] α n yn y αn y n y g y n g y Rη 2,A 2 N,y n,ρ 2 ρ 2 A 2 g yn F un,h y n Q mn,k s n b ] R η 2,A 2 N,y n,ρ 2 A 2 g y ρ 2 F u,h y Q m,k s b ] Rη 2,A 2 N,y,ρ 2 A 2 g y ρ 2 F u,h y Q m,k s b ] R η 2,A 2 N,y,ρ 2 A 2 g y ρ 2 F u,h y Q m,k s b ] α n f n f n k n

29 Journal of Inequalities and Applications 29 α n yn y αn yn y g y n g y ϑ yn y τ 2 r 2 ρ 2 m 2 ρ2 F un,h y n F u,h y n Q mn,k s n Q m,k s n Q m,k s n Q m,k s A 2 g yn A2 g y ρ 2 F u,h y n F u,h y } λ 2 α n f n f n kn α n yn y αn ϕ2 n x n x φ 2 n yn y α n f n f n kn, 5.9 where φ 2 n ϑ e 2 c q2 σ ρ ɛ τ 2 2 υ2 π 2 γ / n ψ 2, r2 ρ 2 m 2 ϕ 2 n ρ 2τ 2 ρ ξ ρ 2 ϖ / n, r2 ρ 2 m and ψ 2 is the same as By 5.7 and 5.9, weobtain xn,y n x,y x n x yn y αn xn,y n x,y α n max { ϕ n ϕ 2 n,φ n φ 2 n } x n,y n x,y α n e n,f n e n,f n r n,k n ω n α n x n,y n x,y α n e n,f n e n,f n r n,k n, 5. where ω n max { ϕ n ϕ 2 n,φ n φ 2 n }. 5.2 Now, ω n ω max{ϕ ϕ 2,φ φ 2 } as n, where ϕ, ϕ 2, φ,andφ 2 are the constants as in 4.48.

30 30 Journal of Inequalities and Applications Since ω /2 ω ω,, deduce that there exists n 0 such that ω n < ω,for all n n 0. Accordingly, it follows from 5. that for all n n 0, xn,y n x,y ω α n xn,y n x,y α n e n,f n e n,f n r n,k n. 5.3 Letting a n xn,y n x,y, t n ω α n, e n,f n b n ω, c n e n,f n r n,k n, 5.4 then 5.3 can be written as a n t n a n b n t n c n, n Therefore, it follows from Lemma 5.4 that lim n a n 0 and so the sequence { x n,y n,z n,u n,v n,m n,s n,t n,w n } n 0 defined by Algorithm 5. converges strongly to a solution x,y,z,u,v,m,s,t,w of System 3.. Theorem 5.6. Suppose that X i, A i, η i i, 2, E, P, F, Q, p, l, h, k, f, g, M, N, S, T, L, D, G, W, K, S, T, L, D, G, W, and K are the same as in Theorem 4.3. Assume that all the conditions of Theorem 4.3 hold. Then the iterative sequence { x n,y n,z n,u n,v n,m n,s n,t n,w n } n 0 generated by Algorithm 5. converges strongly to the solution x,y,z,u,v,m,s,t,w of System 3.. Theorem 5.7. Assume that X i, A i, η i i, 2, E, P, F, Q, p, l, h, k, f, g, M, N, S, T, L, D, G, W, and K are the same as in Theorem 4.4. Suppose that all the conditions of Theorem 4.4 hold. Then the iterative sequence { x n,y n,z n,u n,m n,s n,t n,w n } n 0 generated by Algorithm 5.2 converges strongly to a solution x,y,z,u,m,s,t,w of System 3.3. Theorem 5.8. Let X i, A i, η i i, 2, E, P, F, Q, p, l, h, k, f, g, M, N, S, T, L, D, G, W, and K be the same as in Theorem 4.5. Suppose that all the conditions of Theorem 4.5 hold. Then the iterative sequence { x n,y n,z n,u n,m n,s n,t n,w n } n 0 generated by Algorithm 5.2 converges strongly to a solution x,y,z,u,m,s,t,w of System 3.3. Remark 5.9. The following should be noticed. Theorem 3. in 54 is a special case of the Theorems 4.2 and 4.3. Moreover, Theorems 4.4 and 4.5 improve and extend Theorem In view of Remark 5.3, Theorems 5.5 and 5.6 improve and generalize Theorem 4. in 54. Also, Theorems 5.7 and 5.8 are extensions of Theorem 4.2 in 54. Remark 5.0. When M and N are A, η -monotone operators, A-accretive mappings, A- monotone operators, H, η -accretive mappings, H, η -monotone operators, or H-monotone operators, respectively, from Theorems and , we can obtain the existence and convergence results of solutions for Systems 3. and 3.4. In brief, for a suitable and

31 Journal of Inequalities and Applications 3 appropriate choice of the mappings A i, η i i, 2, E, P, F, Q, p, l, h, k, f, g, M, N, S, T, L, D, G, W, K, S, T, L, D, G, W, K, and the spaces X, X 2, Theorems and include many known results of the generalized variational inclusions as special cases see 29 35, 38, 39, and the references therein. Acknowledgment This paper was written while the first author visiting the KCL. This work was supported by the Korea Research Foundation Grant funded by the Korean Government KRF C Also, this paper is partially supported by the Research Center in Algebraic Hyperstructures and Fuzzy Mathematics, University of Mazandaran, Babolsar, Iran. References G. Stampacchia, Formes bilinéaires coercitives sur les ensembles convexes, Comptes Rendus de l Académie des Sciences, vol. 258, pp , A. Hassouni and A. Moudafi, A perturbed algorithm for variational inclusions, Journal of Mathematical Analysis and Applications, vol. 85, no. 3, pp , S. Adly, Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, Journal of Mathematical Analysis and Applications, vol. 20, no. 2, pp , C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities, A Wiley-Interscience Publications, John Wiley & Sons, New York, NY, USA, V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhaff, Leyden, The Netherlands, K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, X. P. Ding, Perturbed proximal point algorithms for generalized quasivariational inclusions, Journal of Mathematical Analysis and Applications, vol. 20, no., pp. 88 0, J.-S. Pang, Asymmetric variational inequality problems over product sets: applications and iterative methods, Mathematical Programming, vol. 3, no. 2, pp , G. Cohen and F. Chaplais, Nested monotony for variational inequalities over product of spaces and convergence of iterative algorithms, Journal of Optimization Theory and Applications, vol. 59, no. 3, pp , M. Bianchi, Pseudo P-monotone operators and variational inequalities, Report 6, Istituto di econometria e Matematica per le desisioni economiche, Universita Cattolica del sacro Cuore, Milan, Italy, 993. Q. H. Ansari and J.-C. Yao, A fixed point theorem and its applications to a system of variational inequalities, Bulletin of the Australian Mathematical Society, vol. 59, no. 3, pp , Q. H. Ansari, S. Schaible, and J. C. Yao, System of vector equilibrium problems and its applications, Journal of Optimization Theory and Applications, vol. 07, no. 3, pp , E. Allevi, A. Gnudi, and I. V. Konnov, Generalized vector variational inequalities over product sets, Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no., pp , G. Kassay, J. Kolumbán, and Z. Páles, On Nash stationary points, Publicationes Mathematicae Debrecen, vol. 54, no. 3-4, pp , G. Kassay, J. Kolumbán, and Z. Páles, Factorization of Minty and Stampacchia variational inequality systems, European Journal of Operational Research, vol. 43, no. 2, pp , J.-W. Peng, System of generalised set-valued quasi-variational-like inequalities, Bulletin of the Australian Mathematical Society, vol. 68, no. 3, pp , J.-W. Peng, Set-valued variational inclusions with T-accretive operators in Banach spaces, Applied Mathematics Letters, vol. 9, no. 3, pp , J.-W. Peng and X. Yang, On existence of a solution for the system of generalized vector quasi-equilibrium problems with upper semicontinuous set-valued maps, International Journal of Mathematics and Mathematical Sciences, vol. 2005, no. 5, pp , 2005.