The purpose of this paper is to study a new class of fuzzynonlinearset-valuedvariationalinclusions in real Banach spaces. By using the fuzzy resolvent operator techniques for m-accretive mappings, we establish the equivalence between fuzzynonlinearset-valuedvariationalinclusions and fuzzy resolvent operator equation problem. Applying this equivalence and Nadler’s theorem, we suggest some iterative algorithms for solving fuzzynonlinearset-valuedvariationalinclusions in real Banach spaces. By using the inequality of Petryshyn, the existence of solutions for these kinds of fuzzynonlinearset-valuedvariationalinclusions without compactness is proved and convergence criteria of iterative sequences generated by the algorithm are also discussed.
Recently, some systems of variational inequalities, variationalinclusions, 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 nonlinearvariational inequal- ities, respectively.
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 variationalinclusions.
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.
The quasi variational inequalities have been intro- duced by Bensoussan and Lions  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  is closely related to the variational inequality problems; for applications we refer to [6-8].
 N.-J. Huang, Y.-P. Fang, and C. X. Deng, “A new class of generalized nonlinearvariational 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.
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  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 , 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.
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.
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.
In this paper, we ﬁrst introduce a new class of completely generalized multivalued nonlinear quasi-variationalinclusions for multivalued mappings. Motivated and in- spired by the methods of Aldy , Huang , M. A. Noor , and Shim et al. , we construct two new iterative algorithms for solving the completely generalized multi- valuednonlinear quasi-variationalinclusions with bounded closed valued mappings. We also establish four existence theorems of solutions for the class of completely gen- eralized multivalued nonlinear quasi-variationalinclusions 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 , Huang [2, 3, 4], Jou and Yao , Kazmi , M. A. Noor [8, 9, 10], M. A. Noor and Al-Said , M. A. Noor and K. I. Noor , M. A. Noor et al. , Shim et al. , Siddiqi and Ansari [15, 16], Verma [18, 19], Yao , and Zhang .
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-valuednonlinear quasi-variational inequalities in a Hilbert space. We prove that the system of generalized set-valuednonlinear 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-
in , Chang and Huang introduced and studied some new nonlinear complementarity problems for compact-valuedfuzzy mappings and set-valued mappings which include many kinds of complementarity problems, considered by Chang , Cottle et al. , Isac , 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 nonlinearvariational-like inclusions and proved the existence of solutions and the convergence of a new iterative algorithm for this system of nonlinearvariational-like inclusions. Fang and Huang 21, Verma 22, and Fang et al. 23 introduced and studied a new system of variationalinclusions involving H-monotone operators, A-monotone operators and H, η-monotone operators, respectively. Yan et al. 24 introduced and studied a system of set-valuedvariationalinclusions 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.
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 nonlinearvariationalinclusions 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
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- valuedvariational inclusion systems involving A, η-accretive mappings in Banach spaces. Lan and Verma 15 studied a class of nonlinearFuzzyvariational 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 variationalinclusions.
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
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.
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.
constructed some approximation algorithms for some nonlinearvariationalinclusions 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.