On Split Fixed Point Problems for Multi-Valued Mappings and Designing a Self-Adaptive Method

Abstract

Over the past decades, the split inverse problem has been widely studied, and one of the objectives of those researches is to invent some efficient algorithms for approximating solutions. Most of those algorithms depend on the norms of bounded linear operators; however, the calculation of the operator norms is not an easy task in general practice. In this article, we study and investigate the split fixed point problem for multi-valued mappings in Hilbert spaces. We introduce a self-adaptive algorithm without prior knowledge of the operator norm for two demicontractive multi-valued mappings, and establish a strong convergence theorem of the proposed method under some suitable conditions. Our main result in this paper generalizes and improves many results in the literature.