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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 consider a system of general variational inclusions in q-uniformly smooth Banach spaces. Using proximal-point mapping technique, we prove the existence and uniqueness of solution and suggest a Mann type perturbed iterative algorithm for the system of general variational inclusions. We also discuss the convergence criteria and stability of Mann type perturbed iterative algorithm. The techniques and results presented here improve the corresponding techniques and results for the variational inequalities and inclusions 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 generalized variational-like inequality problems (SGVLIP) involving nondifferentiable terms, and its related auxiliary problems in real Hilbert spaces. An existence theorem for auxiliary problems is established. Further, by exploiting this theorem, an algorithm for the SGVLIP is constructed. Furthermore, we prove the existence of a unique solution of SGVLIP and discuss the convergence analysis of the algorithm. The technique and results presented in this paper generalize and unify the corresponding techniques and results given in [M.A. Noor, General nonlinear variational inequalities, J. Math. Anal. Appl. 126 (1987) 78–84; M.A. Noor, General variational inequalities, Appl. Math. Lett. 1 (1988) 119–122; M.A. Noor, Mixed variational-like inequalities, Commun. Appl. Nonlinear Anal. 1 (4) (1994) 63–75]. [Copyright &y& Elsevier]

Abstract: In this paper, we give the notion of P-η-proximal mapping, an extension of P-proximal mapping given by Ding and Xia [J. Comput. Appl. Math. 147 (2002) 369], for a nonconvex lower semicontinuous η-subdifferentiable proper functional on Banach space and prove its existence and Lipschitz continuity. Further, we consider a class of generalized set-valued variational-like inclusions in Banach space and show its equivalence with a class of implicit Wiener–Hopf equations using the concept of P-η-proximal mapping. Using this equivalence, we propose a new class of iterative algorithms for the class of generalized set-valued variational-like inclusions. Furthermore, we prove the existence of solution of generalized set-valued variational-like inclusions and discuss the convergence criteria and the stability of the iterative algorithm. [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]