Let C be a closed convex subset of a Banach space E. A mapping T on C is called a nonexpansive mapping if ‖Tx− Ty‖ ≤ ‖x− y‖ for all x, y ∈ C. We denote by F (T ) the set of fixed points of T . Kirk [21] proved that F (T ) is nonempty in the case that C is weakly compact and has normal structure. See also [3, 4, 5, 14] and others. If C is weakly compact and E has the Opial property, then C has n...

Throughout this paper, by R we denote the set of all nonnegative numbers, while N is the set of all natural numbers. Let A and B be nonempty subsets of a metric space X, d . Consider a mapping f : A ∪ B → A ∪ B, f is called a cyclic map if f A ⊆ B and f B ⊆ A. A point x in A is called a best proximity point of f in A if d x, fx d A,B is satisfied, where d A,B inf{d x, y : x ∈ A,y ∈ B}, and x ∈ ...

Recently the author [Proc. Amer. Math. Soc. 103(1988), 11291135] proved random versions of an interesting theorem of Ky Fan [Theorem 2, Math. Z. 112 (1969), 234-240] for continuous condensing random maps and nonexpansive random maps defined on a closed convex bounded subset in a separable Hilbert space. In this paper, we prove that it is still true for (more general) continuous 1-set-contractiv...

Manuscript received January 01, 2014; revised March 28, 2014. This work was supported in part by King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. Cyril Dennis Enyi is with King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. (corresponding author: +966550782390; e-mail: cenyi@ kfupm.edu.sa). Mukiawa Edwin Soh is with King Fahd University of Petroleum and ...

Let E be a real reflexive Banach space which has uniformly Gâteaux differentiable norm. Let K be aclosed convex subset of E which is also a sunny nonexpansive retract of E, and T : K → E be nonexpansive mapping satisfying the weakly inward condition and F (T ) = {x ∈ K, Tx = x} 6= ∅, and f : K → K be a contractive mapping. Suppose that x0 ∈ K, {xn} is defined by { xn+1 = αnf(xn) + (1− αn)((1− δ...

Recently, S. Reich and S. Simons provided a novel proof of the Kirszbraun-Valentine extension theorem using Fenchel duality and Fitzpatrick functions. In the same spirit, we provide a new proof of an extension result for firmly nonexpansive mappings with an optimally localized range. Throughout this paper, we assume that X is a real Hilbert space, with inner product p = 〈· | ·〉 and induced norm...

Some random versions of deterministic fixed point theorems for condensing and nonexpansive operators are obtained.

Abstract Some results extending the Goebel–Kirk fixed point theorem for asymptotically nonexpansive mappings are presented.