Antonio Jiménez-Melado

• Ph.D. in Mathematics
• Ph.D. in Mathematics at University of Malaga

In this paper we prove a norm inequality in James’ space J, and use it to show that the fixed point property for nonexpansive mappings is passed on from J to those Banach spaces X whose Banach–Mazur distance to J satisfies d(X,J)<17+9712\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb...

In this paper we propose the study of a scalar integral equation of the type(formula Presented) and give conditions on g, a and f that ensure the existence of solutions on [0, ∞) which are asymptotically equal to g(y) at ∞. As a consequence, we obtain results on the existence of solutions for a problem of the type y″(t) = a(t)f (y(t)), y(∞) = g(y),...

Orthogonal convexity is a geometric property of Banach spaces which implies the fixed point property for nonexpansive mappings. In this paper we examine a way to check it for Banach spaces with Schauder basis through a coefficient ρ(X) associated to the basis. We also examine the relationship of ρ(X) with the coefficient R(X) introduced by Garcfa-F...

In this work we consider a discrete nonlocal problem of the following type: {Δ(qΔx)(n)+f(n+1,x(n+1))=0,x(∞)=g(x),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textst...

During the last three decades, two geometric properties implying the fixed point property for nonexpansive mappings were named 'Orthogonal Convexity'. In this paper, we examine the relationship between them.

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

In this paper we prove some norm inequalities in the classical James sequence space J, and use them to prove that orthogonal convexity is a stable property in J.

In this paper we propose the study of an integral equation, with deviating arguments, of the type \[ y(t) =\om(t) - \int_0^\infty f\bigl(t,s,y\bigl(\g_1(s)\bigr), \dotsc, y\bigl(\g_N(s)\bigr)\bigr)\,ds, \quad t\geq0, \] in the context of Banach spaces, with the intention of giving sufficient conditions that ensure the existence of solutions with th...

Suzuki has recently proved a fixed point theorem that generalizes the classical Banach-Caccioppoli fixed point theorem. A somewhat different generalization of this classical result was given by Dugundji and Granas, who extended Banach-Caccioppoli theorem to the class of maps which are contractive in a certain weak sense. It turns out that the class...

A famous open question in metric fixed point theory is whether every Banach space which is isomorphic to the Hilbert space l2 has the fixed point property for nonexpansive mappings. In this paper, we give a fixed point theorem for a class of renormings of l2 which generalizes some previous results. We also show that some spaces of this class lack o...

We introduce a new class of selfmaps T of metric spaces, which generalizes the weakly Zamfirescu maps (and therefore weakly contraction maps, weakly Kannan maps, weakly Chatterjea maps and quasi-contraction maps with constant h < 1/2). We give an explicit Cauchy rate for the Picard iteration sequences {Tnx0}n∈ℕ for this type of maps, and show that...

In [5], Zamfirescu (1972) gave a fixed point theorem that generalizes the classical fixed point theorems by Banach, Kannan, and Chatterjea. In this paper, we follow the ideas of Dugundji and Granas to extend Zamfirescu’s fixed point theorem to the class of weakly Zamfirescu maps. A continuation method for this class of maps is also given.

The first continuation method for contractive maps in the setting of a metric space was given by Granas. Later, Frigon extended Granas theorem to the class of weakly contractive maps, and recently Agarwal and O'Regan have given the corresponding result for a certain type of quasicontractions which includes maps of Kannan type. In this note we intro...

Recently, Frigon proved that, for weakly contractive maps, the property of having a fixed point is invariant by a certain class of homotopies, obtaining as a consequence a Leray-Schauder alternative for this class of maps in a Banach space. We prove here that the Leray-Schauder condition in the aforementioned result can be replaced by a modificatio...

In this paper we show that the well-known Mönch fixed point theorem for non-self mappings remains valid if we replace the Leray–Schauder boundary condition by the interior condition. As a consequence, we obtain a partial generalization of Petryshyn's result for nonexpansive mappings.

In this paper we propose the study of an integral equation of the type y(t)=ω(t)−∫0∞f(t,s,y(s))ds,t≥0. We investigate which conditions give existence, and which ones uniqueness, of solutions behaving like the function ω(t) at ∞.In applying our results to second-order nonlinear differential equations, we are able to recover the previous results and...

In this paper we exhibit some connections between the Dunkl–Williams constant and some other well-known constants and notions. We establish bounds for the Dunkl–Williams constant that explain and quantify a characterization of uniformly nonsquare Banach spaces in terms of the Dunkl–Williams constant given by M. Baronti and P.L. Papini. We also stud...

Starting from results of Dubé and Mingarelli, Wahlén, and Ehrström, 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...

We consider the difference equation Delta(2)x(n) + f(n,x(n+tau)) = 0, tau = 0, 1,..., in the context of a Hilbert space. In this setting, we propose a concept of oscillation with respect to a direction and give sufficient conditions so that all its solutions be directionally oscillatory, as well as conditions which guarantee the existence of direct...

We show a fixed point theorem for condensing mappings under a new condition of the Leray-Schauder type. We call it the Interior Condition. We also discuss examples that demonstrate the independence of these two conditions.

In this paper we consider the Volterra difference equation xn−fn=−∑k=0nan−kGk(xk), in the context of Hilbert spaces and give sufficient conditions so that the solutions exist and show a bounded behavior.

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 are computed, and are used to show that our results are sharp.

In this paper, we consider the difference equation on an arbitrary Banach space (X, ∥·∥x), Δ(qnΔxn + fn(xn) = 0, where {qn} is a positive sequence and fn is X-valued. We shall give conditions so that for a given xϵX, there exists a solution of this equation asymptotically equal to x.

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.

In this paper we study the asymptotic behavior of solutions of the difference equations Δxn = ∞Σi=0 ∫ⁱn(xn+i) + yn, and Δxn = ∞Σi=0 aⁱn ∫ (xn+1) + yn, and improve various results of J. Popenda and E. Schmeidel.

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].

In this paper we consider the first order difference equationΔxn=∑i=0∞ainfxn+i+∑i=0∞bingxn+i+ynand the second order difference equationΔqnΔxn+rnfxn+sngxn+zn=0,where f is a Lipschitz mapping and g is a compact operator, both defined on a Banach space X. We give sufficient conditions so that there exist solutions which are asymptotically constant. Th...

The paper establishes that a Banach space (X, |. |) retains the weak fixed point property provided that (X, ||.||) has this property and ||.|| \le |.| \le B ||.||, for 1\le B < K , where K is a constant which depends on the so called Opial modulus of (X,|.|).

In this paper we consider the first order difference equation Δ xn = ∑i=0 ∞an if(x $_{ n+i}$ ), 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 sec...

Roughly speaking, we show that a Banach space X has the fixed point property for nonexpansive mappings whenever X has the WORTH property and the unit sphere of X does not contain a triangle with sides of length larger than 2.

We show that for any renorming kk of '2, the well known xed point free mappings by Kakutani, Baillon and others are not nonexpansive.

Sufficient conditions for normal structure of a Banach space are given. One of them implies reflexivity for Banach spaces with an unconditional basis, and also for Banach lattices.

It is shown that the space E has the fixed point property for nonexpansive mappings on nonempty, convex, weakly compact subsets if E satisfies the condition OC.

We introduce a coefficient on general Banach spaces which allows us to derive the weak normal structure for those Banach spaces whose Banach-Mazur distance to James quasi reflexive space is less than .

It is shown that every Banach space E such that the Banach-Mazur distance from E to a Hilbert space H is d(E,H)<2·61⋯ has the fixed point property.