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Abstract: In this paper, we give the notion of M-η-proximal mapping for a nonconvex, proper, lower semicontinuous and subdifferentiable functional on Banach space, which is an extension of proximal mappings studied in [X.P. Ding, F.Q. Xia, A new class of completely generalized quasi-variational inclusions in Banach spaces, J. Comput. Appl. Math. 147 (2002) 369–383; K.R. Kazmi, M.I. Bhat, Convergence and stability of iterative algorithms of generalized set-valued variational-like inclusions in Banach spaces, Appl. Math. Comput. 113 (2005) 153–165; K.R. Kazmi, M.I. Bhat, N. Ahmad, An iterative algorithm based on M-proximal mappings for a system of generalized implicit variational inclusions in Banach spaces, J. Comput. Appl. Math. 233 (2009) 361–371]. We prove its existence and Lipschitz continuity in reflexive Banach space. Further, we consider a system of generalized implicit variational-like inclusions in Banach spaces and show its equivalence with a system of implicit equations using the concept of M-η-proximal mappings. Using this equivalence, we propose a new iterative algorithm for the system of generalized implicit variational-like inclusions. Furthermore, we prove the existence of solution of the system of generalized implicit variational-like inclusions and discuss the convergence and stability analysis of the iterative algorithm in the setting of uniformly smooth Banach spaces. [Copyright &y& Elsevier]

Abstract: In this paper, we give a hybrid extragradient iterative method for finding the approximate element of the common set of solutions of a generalized equilibrium problem, a system of variational inequality problems, a variational inequality problem and a fixed point problem for a strictly pseudocontractive mapping in a real Hilbert space. Further we establish a strong convergence theorem based on this method. The results presented in this paper improves and generalizes the results given in Yao et al. and Ceng et al. , and some known corresponding results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a class of accretive mappings called generalized H(·,·)-accretive mappings in Banach spaces. We prove that the proximal-point mapping of the generalized H(·,·)-accretive mapping is single-valued and Lipschitz continuous. Further, we consider a system of generalized variational inclusions involving generalized H(·,·)-accretive mappings in real q-uniformly smooth Banach spaces. Using proximal-point mapping method, we prove the existence and uniqueness of solution and suggest an iterative algorithm for the system of generalized variational inclusions. Furthermore, we discuss the convergence criteria of the iterative algorithm under some suitable conditions. Our results can be viewed as a refinement and improvement of some known results in the literature. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a generalized mixed equilibrium problem involving non-monotone set-valued mappings in real Hilbert space. We extend the notions of the Yosida approximation and its corresponding regularized operator given by Moudafi and Thera [A. Moudafi, M. Thera, Proximal and dynamical approaches to equilibrium problems, in: Lecture Notes in Econom. and Math. System, vol. 477, Springer-Verlag, Berlin, 2002, pp. 187–201] and discuss some of their properties. Further, we consider a generalized Wiener–Hopf equation problem and show its equivalence with the generalized mixed equilibrium problem. Using a fixed point formulation of the generalized Wiener–Hopf equation problem, we construct an iterative algorithm. Furthermore, we extend the notion of stability given by Harder and Hick [A.M. Harder, T.L. Hicks, Stability results for fixed-point iteration procedures, Math. Japonica 33 (5) (1998) 693–706]. We prove the existence of a solution and discuss the convergence and stability of the iterative algorithm for the generalized Wiener–Hopf equation problem. Since the generalized mixed equilibrium problems include variational inequalities as special cases, the results presented in this paper continue to hold for these problems. [Copyright &y& Elsevier]

Abstract: In this paper, we consider a system of nonlinear variational-like inclusions (SNVLI) in Hilbert spaces. In particular, SNVLI reduces to a variational inclusion, an extension of variational inclusion studied by Hassouni and Moudafi [1]. Using fixed-point method, we suggest an iterative algorithm for finding an approximate solution to SNVLI. Further, we prove the existence of solution and discuss convergence criteria for the approximate solution of SNVLI. The theorems presented in this paper improve and unify many known results of variational inclusions and variational inequalities, see for example [1–3]. [Copyright &y& Elsevier]