Search

Abstract: In this paper, using proximal-point mapping technique of P-η-accretive mapping and the property of the fixed-point set of set-valued contractive mappings, we study the behavior and sensitivity analysis of the solution set of a parametric generalized implicit quasi-variational-like inclusion involving P-η-accretive mapping in real uniformly smooth Banach space. Further, under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to the parameter. The technique and results presented in this paper can be viewed as extension of the techniques and corresponding results given in [R.P. Agarwal, Y.-J. Cho, N.-J. Huang, Sensitivity analysis for strongly nonlinear quasi-variational inclusions, Appl. Math. Lett. 13 (2002) 19–24; S. Dafermos, Sensitivity analysis in variational inequalities, Math. Oper. Res. 13 (1988) 421–434; X.-P. Ding, Sensitivity analysis for generalized nonlinear implicit quasi-variational inclusions, Appl. Math. Lett. 17 (2) (2004) 225–235; X.-P. Ding, Parametric completely generalized mixed implicit quasi-variational inclusions involving h-maximal monotone mappings, J. Comput. Appl. Math. 182 (2) (2005) 252–269; X.-P. Ding, C.L. Luo, On parametric generalized quasi-variational inequalities, J. Optim. Theory Appl. 100 (1999) 195–205; Z. Liu, L. Debnath, S.M. Kang, J.S. Ume, Sensitivity analysis for parametric completely generalized nonlinear implicit quasi-variational inclusions, J. Math. Anal. Appl. 277 (1) (2003) 142–154; R.N. Mukherjee, H.L. Verma, Sensitivity analysis of generalized variational inequalities, J. Math. Anal. Appl. 167 (1992) 299–304; M.A. Noor, Sensitivity analysis framework for general quasi-variational inclusions, Comput. Math. Appl. 44 (2002) 1175–1181; M.A. Noor, Sensitivity analysis for quasivariational inclusions, J. Math. Anal. Appl. 236 (1999) 290–299; J.Y. Park, J.U. Jeong, Parametric generalized mixed variational inequalities, Appl. Math. Lett. 17 (2004) 43–48]. [Copyright &y& Elsevier]

Abstract: In this paper, we give the notion of --proximal-point mapping, an extension of --accretive mapping [C.E. Chidume, K.R. Kazmi, H. Zegeye, Iterative approximation of a solution of a general variational-like inclusion in Banach spaces, Int. J. Math. Math. Sci. 22 (2004) 1159–1168] and -proximal-point mappings [Y.-P. Fang, N.-J. Huang, -accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (2004) 647–653], associated with a new accretive mapping named --accretive mapping. We prove that --proximal-point mapping is single-valued and Lipschitz continuous. Further, we consider a system of variational-like inclusions involving --accretive mappings in real -uniformly smooth Banach spaces. Using --proximal-point mapping technique, we prove the existence and uniqueness of solution and suggest a Mann-type iterative algorithm for the system of variational-like inclusions. Furthermore, we discuss the convergence criteria and stability of Mann-type iterative algorithm. [Copyright &y& Elsevier]

Abstract: In this paper, we introduce a class of P-η-accretive mappings, an extension of η-m-accretive mappings [C.E. Chidume, K.R. Kazmi, H. Zegeye, Iterative approximation of a solution of a general variational-like inclusion in Banach spaces, Int. J. Math. Math. Sci. 22 (2004) 1159–1168] and P-accretive mappings [Y.-P. Fang, N.-J. Huang, H-accretive operators and resolvent operator technique for solving variational inclusions in Banach spaces, Appl. Math. Lett. 17 (2004) 647–653], in real Banach spaces. We prove some properties of P-η-accretive mappings and give the notion of proximal-point mapping, termed as P-η-proximal-point mapping, associated with P-η-accretive mapping. Further, using P-η-proximal-point mapping technique, we prove the existence of solution and discuss the convergence analysis of iterative algorithm, for multi-valued variational-like inclusions in real Banach space. The theorems presented in this paper extend and improve many known results in the literature. [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]