Ishikawa iteration convergence to fixed points of a multi-valued mapping in modular function spaces

Abstract

We prove the ρ-convergence of Ishikawa iterative algorithm to fixed points of a multi-valued mapping T:C->P_ρ(C), where C is a nonempty ρ-bounded ρ-closed subset of Lρ, ρ is a convex function modular satisfying ∆₂-type condition, and P_ρ(C) is the family of nonempty ρ-bounded ρ-proximinal subsets of C, such that the mapping PT is ρ-nonexpansive. This is the modular version of approximating fixed points of multi-valued nonexpansive mapping in Banach spaces by Ishikawa iterative algorithm and it generalizes some results in the literature.