Modified viscosity implicit rules for proximal split feasibility and fixed point problems

The purpose of this paper is to present a modified implicit rules for finding a common element of the set of solutions of proximal split feasibility problem and the set of fixed point problems for ϑ-strictly pseudo-contractive mappings in Hilbert spaces. First, we prove strong convergence results for finding a point which minimizes a convex function such that its image under a bounded linear operator minimizes another convex function which is also a solution to fixed point of ϑ-strictly pseudo-contractive mapping. Our second algorithm generates a strong convergent sequence to approximate common solution of non-convex minimization feasibility problem and fixed point problem. In all our results in this work, our iterative scheme is proposed by a way of selecting the step size such that their implementation does not need any prior information about the operator norm because the calculation or at least an estimate of an operator norm is not an easy task. Finally, we gave numerical example to study the efficiency and implementation of our schemes.