Some geometric conditions which imply the fixed point property for multivalued nonexpansive mappings

Abstract

We show some geometric conditions on a Banach space X concerning the modulus of smoothness, the coefficient of weak orthogonality, the coefficient R(a, X), the James constant and the Jordan-von Neumann constant, which imply the existence of fixed points for multivalued nonexpansive mappings. These fixed point theorems improve some previous results and give affirmative answers to some open questions.

Edelstein's method and fixed point theorems for some generalized nonexpansive mappings

“…showed in[8, Corollary 2] that , and let E be a nonempty bounded closed convex subset of X . If T : E → E is a mapping satisfying condition (C), then T has a fixed point.…”

Hölder’s means and fixed points for multivalued nonexpansive mappings

“…are called the von Neumann-Jordan [7] and James constants [19], respectively. The following result regarding the relationship between the James and von Neumann-Jordan constants was proved in [20]. Theorem 2.6.…”

“…Theorem 2.6. ( [20]) If X is a Banach space such that C NJ (X) < 1 + 1 (J(X)) 2 , then X satisfies the (DL)-condition.…”

On von Neumann-Jordan type constant and sufficient conditions for fixed points of multivalued nonexpansive mappings

“…Both the James constant J(X) and the von Neumann-Jordan constant C N J (X) play an important role in the description of various geometric structures. Therefore, many recent studies have focused on these constants (see [3,4,5,6,7,8,10,11,15,16,20,23,25,26]). S. Dhompongsa et al [5] introduced the Domínguez-Lorenzo condition ((DL)-condition, in short) which implies weak normal structure of a Banach space and the fixed point property for multivalued nonexpansive mappings.…”