On split inclusion problem and fixed point problem for multi-valued mappings

Abstract

Many of the iterative schemes for solving split inclusion and fixed point problems involve step-sizes that depend on the norm of a bounded linear operator. The implementation of such algorithms are usually difficult to handle. This is because they require the computation of the operator norm. In this paper, we propose an algorithm involving a step-size selected in such a way that its implementation does not require the computation or an estimate of some spectral radius. Using our algorithm we proved strong convergence theorem for split inclusion problem and fixed point problem for multi-valued quasi-nonexpansive mappings in real Hilbert spaces. Our result generalizes some important and recent results in the literature. Some applications of our main result to game theory and variational inequality problem are also presented.