Approximation of a common element of the set of fixed points of multi-valued type-one demicontractive-type mappings and the set of solutions of an equilibrium problem in Hilbert spaces (Q2821994)

The paper is devoted to the study of fixed points of multivalued hemicontractive-type and demicontractive-type mappings. The concept of demicontractive mapping, which was introduced independently by \textit{St. Maruster} [Proc. Am. Math. Soc. 63, 69--73 (1977; Zbl 0355.47037)] and \textit{T. L. Hicks} and \textit{J. D. Kubicek} [J. Math. Anal. Appl. 59, 498--504 (1977; Zbl 0361.65057)] for the case of single-valued mappings, is adapted to the case of multi-valued mappings.NEWLINENEWLINEAmong the most important additional tools used in establishing convergence theorems for iterative schemes that approximate fixed points of nonexpansive type single-valued mappings is Browder's demiclosedness principle [\textit{F. E. Browder}, Bull. Am. Math. Soc. 74, 660--665 (1968; Zbl 0164.44801)], which states that, if \(E\) is a uniformly convex Banach space, \(C\) is a closed convex subset of \(E\), and \(T : C\to E\) is nonexpansive, then \(I-T\) is demiclosed at \(0\).NEWLINENEWLINEIn the paper under review, the authors are using an adequate concept of demiclosedness at zero adapted to the case of multi-valued mappings. By exploiting many other concepts and results, they establish some convergence theorems, both weak and strong, for an iterative scheme designed to approximate a fixed point of a demicontractive-type multivalued mapping (Theorem 3.2), a common element of the set of fixed points of two demicontractive-type multivalued mappings, and the set of solutions of an equilibrium problem (Theorem 3.6), to mention just a few of the obtained results.