Iterative approximation of a unique solution of a system of variational-like inclusions in real [formula omitted]-uniformly smooth Banach spaces

Abstract

In this paper, we give the notion of P - η -proximal-point mapping, an extension of η - m -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 P -proximal-point 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], associated with a new accretive mapping named P - η -accretive mapping. We prove that P - η -proximal-point mapping is single-valued and Lipschitz continuous. Further, we consider a system of variational-like inclusions involving P - η -accretive mappings in real q -uniformly smooth Banach spaces. Using P - η -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.