Search

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, using the concept of --proximal-point mapping introduced by Kazmi and Bhat [11], we study the existence and sensitivity analysis of the solution set of a system of parametric general quasi-variational-like inequality problems in uniformly smooth Banach spaces. Further under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to the parameters. The approach used in this paper may be treated as an extension and unification of approaches for studying sensitivity analysis for various important classes of variational inequalities given in . [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 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]