The Jordan–von Neumann constants and fixed points for multivalued nonexpansive mappings

Abstract

The purpose of this paper is to study the existence of fixed points for nonexpansive multivalued mappings in a particular class of Banach spaces. Furthermore, we demonstrate a relationship between the weakly convergent sequence coefficient WCS(X) and the Jordan-von Neumann constant C NJ (X) of a Banach space X. Using this fact, we prove that if C NJ (X) is less than an appropriate positive number, then every multivalued nonexpansive mapping T : E → KC(E) has a fixed point where E is a nonempty weakly compact convex subset of a Banach space X, and KC(E) is the class of all nonempty compact convex subsets of E.

A common fixed point theorem for a commuting family of nonexpansive mappings one of which is multivalued

“…A more general space than spaces satisfying the Kirk-Massa condition is a space having property (D). Property (D) introduced in [19] 19], Theorem 3.6] Let E be a nonempty weakly compact convex subset of a Banach space X that has property (D). Assume that T : E KC(E) is a nonexpansive mapping.…”

“…The class of spaces having property (D) contains several well-known ones including k-uniformly rotund, nearly uniformly convex, uniformly convex in every direction spaces, and spaces satisfying Opial condition (see [3,[19][20][21][22][23] and references therein).…”

On Some Geometric Constants and the Fixed Point Property for Multivalued Nonexpansive Mappings

“…It was observed that property D is weaker than the DL condition and stronger than weak normal structure, and Dhompongsa et al 4 proved that property D implies the w-MFPP. Before going to the results, let us recall some more definitions.…”

Some Sufficient Conditions for Fixed Points of Multivalued Nonexpansive Mappings

“…Dhompongsa et al 4 introduced the property D if there exists λ ∈ 0, 1 such that for any nonempty weakly compact convex subset C of X, any sequence {x n } ⊂ C which is regular asymptotically uniform relative to C, and any sequence {y n } ⊂ A C, {x n } which is regular asymptotically uniform relative to X we have r C, y n ≤ λr C, {x n } .…”

“…Dhompongsa et al 3,4 introduced the DL condition and property D which imply the FPP for multivalued nonexpansive mappings. A possible approach to the above problem is to look for geometric conditions in a Banach space X which imply either the DL condition or property D .…”