Iterative approximation of a solution of multi-valued variational-like inclusion in Banach spaces: A P-η-proximal-point mapping approach

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.