A common solution of variational inclusion problem involving maximal η-relaxed monotone mapping and fixed point problem of nonlinear mapping on Banach spaces

Abstract

Let \(E\) be a real uniformly convex and uniformly smooth Banach space with the dual space \(E^*\). Let \(J:E\to E^*\) be the normalized duality mapping and \(A:E\to 2^{E^*}\) a maximal \(\eta\)-relaxed monotone mapping. The purpose of this article is two fold. First we present the characterizations of \(\eta\)-relaxed monotone mappings on uniformly convex and uniformly smooth Banach spaces, and also some properties of the resolvent mapping associated with maximal \(\eta\)-relaxed monotone mapping are proved. Secondly, iterative algorithms for approximating a common element of the set of fixed points of nonlinear mapping and the set of solution of variational inclusion problems involving maximal \(\eta\)-relaxed mappings is proposed, and then strong convergence theorems are proved. The results improve and extend some recent results in the literature.