Antonio Jiménez-Melado

The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces

We give some sufficient conditions for normal structure in terms of the von Neumann-Jordan constant, the James constant and the weak orthogonality coefficient introduced by B. Sims. In the rest of the paper, the von Neumann-Jordan constant and the James constant for the Bynum space l 2,∞ are computed, and are used to show that our results are sharp.

Stability of the Fixed Point Property for Nonexpansive Mappings

In 1971 Zidler [Zi 71] showed that every separable Banach space (X, ‖·‖) admits an equivalent renorming, (X, ‖·‖0), which is uniformly convex in every direction (UCED), and consequently it has weak normal structure and so the weak fixed point property (WFPP) [D-J-S 71].

Asymptotic behavior of solutions of difference equations in Banach spaces

In this paper we consider the first order difference equation = Zan f (Xn+i), i=O and give necessary and sufficient conditions so that there exist solutions which are asymptotically constant. These results generalize those given earlier by Popenda and Schmeidel. As an application we give necessary and sufficient conditions for the second order difference equation A (qnAxn) + pnf (Xn) = 0 to have asymptotically constant solutions.

EXISTENCE OF MONOTONIC ASYMPTOTICALLY CONSTANT SOLUTIONS FOR SECOND ORDER DIFFERENTIAL EQUATIONS

Starting from results of Dube and Mingarelli, Wahlen, and Ehrstrom, who give conditions that ensure the existence and uniqueness of nonnegative nondecreasing solutions asymptotically constant of the equation we have been able to reduce their hypotheses in order to obtain the same existence results, at the expense of losing the uniqueness part. The main tool they used is the Banach Fixed Point Theorem, while ours has been the Schauder Fixed Point Theorem together with one version of the Arzela-Ascoli Theorem.

A RENORMING OF 2, RARE BUT WITH THE FIXED-POINT PROPERTY

We give an example of a renorming of ℓ2 with the fixed-point property (FPP) for nonexpansive mappings, but which seems to fall out of the scope of all the commonly known sufficient conditions for FPP.

Rakotch type contractive maps, square roots and uniform convexity

In this paper we introduce a family of weakly contractive maps on the space B(S)

Asymptotic Behavior of Solutions to a Vector Integral Equation with Deviating Arguments

B(S)

Oscillatory Solutions for Second-Order Difference Equations in Hilbert Spaces

of bounded real valued functions and use it to show that a fundamental step in the proof of the well-known Stone-Weierstrass approximation theorem can be achieved via Rakotch’s fixed point theorem for weakly contractive maps. With the same technique, we obtain Zemanek’s theorem on the existence of square roots in certain Banach subalgebras of B(S)

A sufficient condition for the fixed point property

B(S)

Opial modulus and stability of the fixed point property

, and, finally, in the context of abstract Banach algebras, we exhibit some relationship between weakly contractive maps on the closed unit ball and the geometry of the spheres.