Iterative approximation of a unique solution of a system of variational-like inclusions in real image-uniformly smooth Banach spaces Original Research Article

In this paper, we give the notion of PP-ηη-proximal-point mapping, an extension of ηη-mm-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 PP-proximal-point mappings [Y.-P. Fang, N.-J. Huang, HH-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 PP-ηη-accretive mapping. We prove that PP-ηη-proximal-point mapping is single-valued and Lipschitz continuous. Further, we consider a system of variational-like inclusions involving PP-ηη-accretive mappings in real qq-uniformly smooth Banach spaces. Using PP-ηη-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.