Convergence analysis for proximal split feasibility problems and fixed point problems

Abstract

Our aim in this paper is to introduce iterative algorithms and prove their strong convergence for solving proximal split feasibility problems and fixed point problems for $$k$$ -strictly pseudocontractive mappings in Hilbert spaces. The sequence generated by our first iterative scheme converges strongly to an approximate common solution of convex minimization feasibility problem and fixed point problem. Furthermore, our second algorithm generates a strongly convergent sequence to an approximate common solution of non-convex minimization feasibility problem and fixed point problem. In all our results in this work, our iterative schemes are proposed with a way of selecting the step-sizes such that their implementation does not need any prior information about the operator norm because the calculation or at least an estimate of the operator norm $$||A||$$ is very difficult, if not an impossible task. Our result complements many recent and important results in this direction.