The purpose of this paper is to study a new class of **fuzzy** **nonlinear** **set**-**valued** **variational** **inclusions** in real Banach spaces. By using the **fuzzy** resolvent operator techniques for m-accretive mappings, we establish the equivalence between **fuzzy** **nonlinear** **set**-**valued** **variational** **inclusions** and **fuzzy** resolvent operator equation problem. Applying this equivalence and Nadler’s theorem, we suggest some iterative algorithms for solving **fuzzy** **nonlinear** **set**-**valued** **variational** **inclusions** in real Banach spaces. By using the inequality of Petryshyn, the existence of solutions for these kinds of **fuzzy** **nonlinear** **set**-**valued** **variational** **inclusions** without compactness is proved and convergence criteria of iterative sequences generated by the algorithm are also discussed.

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Recently, some systems of **variational** inequalities, **variational** **inclusions**, complementar- ity problems, and equilibrium problems have been studied by many authors because of their close relations to some problems arising in economics, mechanics, engineering sci- ence and other pure and applied sciences. Among these methods, the resolvent opera- tor technique is very important. Huang and Fang [] introduced a system of order com- plementarity problems and established some existence results for the system using ﬁxed point theory. Verma [] introduced and studied some systems of **variational** inequalities and developed some iterative algorithms for approximating the solutions of the systems of **variational** inequalities. Cho et al. [] introduced and studied a new system of nonlin- ear **variational** inequalities in Hilbert spaces. Further, the authors proved some existence and uniqueness theorems of solutions for the systems, and also constructed some iterative algorithms for approximating the solution of the systems of **nonlinear** **variational** inequal- ities, respectively.

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Next, the development of **variational** inequality is to design eﬃcient iterative algorithms to compute approximate solutions for **variational** inequalities and their gen- eralizations. 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**.

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i If X X ∗ is a Hilbert space, N 0 is the zero operator in X, Q I is the identity operator in X, and u 0, then problem 2.5 becomes the parametric usual **variational** inclusion 0 ∈ Mx with a A, η-maximal monotone mapping M, which was studied by Verma 12.

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The quasi **variational** inequalities have been intro- duced by Bensoussan and Lions [1] and closely related to contact problems with friction in electrostatics and non- linear random equations frequently arise in biological, physical and system sciences [4,5]. With the emergence of probabilistics functional analysis, the study of random operators became a central topic of this discipline [4,5]. The theory of resolvent operators introduced by Brezis [2] is closely related to the **variational** inequality problems; for applications we refer to [6-8].

[20] N.-J. Huang, Y.-P. Fang, and C. X. Deng, “A new class of generalized **nonlinear** **variational** in- clusions in Banach spaces,” in Proceedings of International Conference on Mathematical Program- ming, M. Y. Yue, J. Y. Han, L. S. Zhang, and S. Z. Zhang, Eds., pp. 207–214, Shanghai University Press, Shanghai, China, 2004.

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Meanwhile, it is known that accretivity of the underlying operator plays indispensable roles in the theory of **variational** inequality and its generalizations. In 2001, Huang and Fang [41] were the first to introduce generalized m-accretive mapping and gave the definition of the resolvent operator for generalized m-accretive mappings in Banach spaces. Subsequently, Verma [59,60] introduced and studied new notions of A-mono- tone and (A, h)-monotone operators and studied some properties of them in Hilbert spaces. In [52], Lan et al. first introduced the concept of (A, h)-accretive mappings, which generalizes the existing h-subdifferential operators, maximal h-monotone opera- tors, H-monotone operators, A-monotone operators, (H, h)-monotone operators, (A, h)-monotone operators in Hilbert spaces, H-accretive mapping, generalized m-accretive mappings and (H, h)-accretive mappings in Banach spaces.

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solutions for the cases of a convex and of a nonconvex **valued** perturbation term, which is new for nonlocal problems. Our approach will be based on the techniques and results of the theory of monotone operators, **set**-**valued** analysis and the Leray-Schauder ﬁxed point theorem.

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In this paper, we extend the auxiliary principle technique to study the generalized **set**- **valued** strongly **nonlinear** mixed implicit quasi-**variational**-like inequalities problem (.) in Hilbert spaces. First, we establish the existence of solutions of the corresponding system of auxiliary **variational** inequalities (.). Then, using the existence result, we construct a new iterative algorithm. Finally, both the existence of solutions of the original problem and the convergence of iterative sequences generated by the algorithm are proved. Our results improve and extend some known results.

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In this paper, we ﬁrst introduce a new class of completely generalized multivalued **nonlinear** quasi-**variational** **inclusions** for multivalued mappings. Motivated and in- spired by the methods of Aldy [1], Huang [4], M. A. Noor [10], and Shim et al. [14], we construct two new iterative algorithms for solving the completely generalized multi- **valued** **nonlinear** quasi-**variational** **inclusions** with bounded closed **valued** mappings. We also establish four existence theorems of solutions for the class of completely gen- eralized multivalued **nonlinear** quasi-**variational** **inclusions** involving strongly mono- tone, relaxed Lipschitz and generalized pseudocontractive multivalued mappings, and give some convergence results of iterative sequences generated by the algorithms. Our results extend, improve and unify a lot of results due to Adly [1], Huang [2, 3, 4], Jou and Yao [5], Kazmi [6], M. A. Noor [8, 9, 10], M. A. Noor and Al-Said [11], M. A. Noor and K. I. Noor [12], M. A. Noor et al. [13], Shim et al. [14], Siddiqi and Ansari [15, 16], Verma [18, 19], Yao [20], and Zhang [21].

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One of the most common methods for solving the **variational** problem is to transfer the **variational** inequality into an operator equation, and then transfer the operator equa- tion into the ﬁxed point problems. In the present paper, we introduce and study a class of new systems of generalized **set**-**valued** **nonlinear** quasi-**variational** inequalities in a Hilbert space. We prove that the system of generalized **set**-**valued** **nonlinear** quasi-**variational** in- equalities is equivalent to the ﬁxed point problem and the system of Wiener-Hopf equa- tions. By using the projection operator technique and the system of Wiener-Hopf equa- tions technique, we suggest several new iterative algorithms to ﬁnd the approximate so-

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in [4], Chang and Huang introduced and studied some new **nonlinear** complementarity problems for compact-**valued** **fuzzy** mappings and **set**-**valued** mappings which include many kinds of complementarity problems, considered by Chang [1], Cottle et al. [7], Isac [9], and Noor [13, 14], as special cases.

As generalizations of system of **variational** inequalities, Agarwal et al. 18 introduced a system of generalized **nonlinear** mixed quasivariational **inclusions** and investigated the sensitivity analysis of solutions for this system of generalized **nonlinear** mixed quasivariational **inclusions** in Hilbert spaces. Peng and Zhu 19 introduce a new system of generalized **nonlinear** mixed quasivariational **inclusions** in q-uniformly smooth Banach spaces and prove the existence and uniqueness of solutions and the convergence of several new two-step iterative algorithms with or without errors for this system of generalized **nonlinear** mixed quasivariational **inclusions**. Kazmi and Bhat 20 introduced a system of **nonlinear** **variational**-like **inclusions** and proved the existence of solutions and the convergence of a new iterative algorithm for this system of **nonlinear** **variational**-like **inclusions**. Fang and Huang 21, Verma 22, and Fang et al. 23 introduced and studied a new system of **variational** **inclusions** involving H-monotone operators, A-monotone operators and H, η-monotone operators, respectively. Yan et al. 24 introduced and studied a system of **set**-**valued** **variational** **inclusions** which is more general than the model in 21. Peng and Zhu 25 introduced and studied a system of generalized mixed quasivariational **inclusions** involving H, η-monotone operators which contains those mathematical models in 11–16, 21–24 as special cases.

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Now, we explore some basic properties derived from the notion of (A,η)-monotonicity. Let H denote a real Hilbert space with the norm · and inner product · , · . Let η : H × H : → H be a single-**valued** mapping. The mapping η is called τ-Lipschitz continuous if there is a constant τ > 0 such that η(u,v) ≤ τ y − v for all u, v ∈ H.

In this section, we will introduce a new system of **nonlinear** **variational** **inclusions** in Hilbert spaces. In what follows, unless other specified, for each i 1, 2, . . . , p, we always suppose that H i is a Hilbert space with norm denoted by · i , A i : H i → H i , F i : p j1 H j → H i are single-**valued** mappings, and M i : H i → 2 H i is a **nonlinear** mapping. We consider the

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enriched and improved the class of generalized resolvent operators. Lan 10 studied a system of general mixed quasivariational **inclusions** involving A, η-accretive mappings in q-uniformly smooth Banach spaces. Lan et al. 14 constructed some iterative algorithms for solving a class of **nonlinear** A, η-monotone operator inclusion systems involving nonmonotone **set**-**valued** mappings in Hilbert spaces. Lan 9 investigated the existence of solutions for a class of A, η-accretive **variational** inclusion problems with nonaccretive **set**- **valued** mappings. Lan 11 analyzed and established an existence theorem for **nonlinear** parametric multivalued **variational** inclusion systems involving A, η-accretive mappings in Banach spaces. By using the random resolvent operator technique associated with A, η- accretive mappings, Lan 13 established an existence result for **nonlinear** random multi- **valued** **variational** inclusion systems involving A, η-accretive mappings in Banach spaces. Lan and Verma 15 studied a class of **nonlinear** **Fuzzy** **variational** inclusion systems with A, η-accretive mappings in Banach spaces. On the other hand, some interesting and classical techniques such as the Banach contraction principle and Nalder’s fixed point theorems have been considered by many researchers in studying **variational** **inclusions**.

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The usefulness and importance of limits of sequences of crisp sets, and limits (continuity) and derivatives of crisp **set**-**valued** mappings have been recognized in many areas, for ex- ample, **variational** analysis, **set**-**valued** optimization, stability theory, sensitivity analysis, etc. For details, see, for example, [–]. The concept of limits of sequences of crisp sets is interesting and important for itself, and it is necessary to introduce the concepts of limits and derivatives of crisp **set**-**valued** mappings. Typical and important applications of them are (i) **set**-**valued** optimization and (ii) stability theory and sensitivity analysis for mathe- matical models. For the case (ii), consider the following system. Some mathematical model outputs the **set** of optimal values W ∗ (u) ⊂ R and the **set** of optimal solutions S ∗ (u) ⊂ R n

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It is not a surprise that many practical situations occur by chance and so **variational** inequalities with random variables/mappings have also been widely studied in the past decade. For instance, some random **variational** inequalities and random quasivariational inequalities problems have been introduced and studied by Chang 8, Chang and Huang 9, 10, Chang and Zhu 11, Huang 12, 13, Husain et al. 14, Tan et al. 15, Tan 16, and Yuan 7.

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A(x) − B(x), η y,g (x) ≥ φ g (x), x − φ(y, x), ∀ y ∈ H. (2.18) We remark that for the appropriate and suitable choices of the mappings η, M, N, A, B, C, D, G, g, φ and the space E, one can obtain from problem (2.14) many known and new classes of generalized **variational** and quasivariational inequalities (**inclusions**) and complementarity problems, studied previously by many authors as special cases, see [2, 4, 5, 7, 12] and the references therein.

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constructed some approximation algorithms for some **nonlinear** **variational** **inclusions** in Hilbert spaces or Banach spaces. Verma has developed a hybrid version of the Eckstein- Bertsekas 11 proximal point algorithm, introduced the algorithm based on the A, η- maximal monotonicity framework 12, and studied convergence of the algorithm. For the past few years, many existence results and iterative algorithms for various **variational** inequalities and **variational** inclusion problems have been studied. For details, please see 1– 37 and the references therein.

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