Iterative methods for convex proximal split feasibility problems and fixed point problems

In this paper we prove strong convergence result for a problem of finding a point which minimizes a proper convex lower-semicontinuous function f which is also a fixed point of a total asymptotically strict pseudocontractive mapping such that its image under a bounded linear operator A minimizes another proper convex lower-semicontinuous function g in real Hilbert spaces. In our result in this work, our iterative scheme is proposed with a way of selecting the step-size such that its implementation does not need any prior information about the operator norm ||A|| because the calculation or at least an estimate of the operator norm ||A|| is very difficult, if it is not an impossible task. Our result complements many recent and important results in this direction. © 2015, African Mathematical Union and Springer-Verlag Berlin Heidelberg.