We study descent-like approximation methods and proximal methods of the retraction type for solving fixed-point problems with
nonself-mappings in Hilbert and Banach spaces. We prove strong
and weak convergences for weakly contractive and nonexpansive
maps, respectively. We also establish the stability of these
methods with respect to perturbations of the operators and the
constraint sets.

Yakov Alber

Simeon Reich

Jen-Chih Yao

Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces

We prove that every Banach space containing an isomorphic copy of
<mml:math alttext="$c_0$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> fails to have the fixed-point property for asymptotically
nonexpansive mappings with respect to some locally convex topology
which is coarser than the weak topology. If the copy  of <mml:math alttext="$c_0$" id="E3" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>c</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> is asymptotically isometric,  this result can be improved, because
we can prove the failure of the fixed-point property for
nonexpansive mappings.

Maria A. Japón Pineda

The fixed-point property in Banach spaces containing a copy of <svg style="vertical-align:-3.3907pt" height="11.9625" version="1.1" viewBox="0 0 13.3625 11.9625" width="13.3625" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.017,-0,0,-.017,.062,7.675)"><path id="x1D450" d="M383 397q0 -32 -35 -49q-12 -6 -23 8q-37 45 -84 45t-90 -71q-40 -65 -40 -167q0 -57 22 -86t59 -29q38 0 81.5 24.5t69.5 51.5l16 -21q-44 -53 -104 -84t-109 -31q-56 0 -89.5 41t-33.5 117q0 61 30 124t79 105q33 28 81 50.5t86 22.5q34 0 59 -15.5t25 -35.5z" /></g><g transform="matrix(.012,-0,0,-.012,6.962,11.762)"><path id="x30" d="M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105&#xA;q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z" /></g></svg> 


The paper is devoted to general elliptic operators in H&#246;lder spaces in bounded or unbounded domains. We discuss the Fredholm property of linear operators and properness of nonlinear operators. We construct a topological degree for Fredholm and
proper operators of index zero.

V.  Volpert

A.  Volpert

Properness and topological degree for general elliptic operators

We study the topology of a subspace of the function space of
continuous self-mappings of a given manifold: the subspace
determined by maps having the least number of fixed points in its
homotopy class. In the case that the manifold is a closed disk of
finite dimension, we prove that this subspace is both globally
and locally path connected. We also prove this result when the
manifold is a sphere of dimension 1, 3, or 7.

Daciberg L. Gonçalves

Michael R. Kelly

Connectivity properties for subspaces of function spaces
determined by fixed points

The aim of this paper is to work with the measure of
nonhyperconvexity in a similar way as J. Cano (1990) does with the
index of nonconvexity. We apply this measure to obtain different
extensions of the famous Schauder fixed-point theorem in
hyperconvex spaces.

Dariusz Bugajewski

Rafael Espínola

Measure of nonhyperconvexity and fixed-point theorems

We provide some new estimates of the smallest possible Lipschitz
constant for retractions of the unit ball <mml:math alttext="$B$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>B</mml:mi></mml:math> onto the unit
sphere <mml:math alttext="$S$" id="E2" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>S</mml:mi></mml:math> in infinite-dimensional Banach spaces.

Kazimierz Goebel

On the problem of retracting balls onto their boundary

A technique, based on the investigations of the images of maps,
for obtaining fixed-point and coincidence results in a new class of maps and domains is described. In particular, we show that the problem concerning the existence of fixed points of expansive set-valued maps and inner set-valued maps on not necessarily convex or compact sets in Hausdorff topological vector spaces has a solution. As a consequence, we prove a new intersection theorem concerning not necessarily convex or compact sets and its applications. We also give new coincidence and section theorems for maps defined on not necessarily convex sets in Hausdorff topological vector spaces. Examples and counterexamples show a fundamental difference between our results and the well-known ones.

Kazimierz  W&#322;odarczyk

Dorota  Klim

Fixed-point and coincidence theorems for set-valued maps with nonconvex or noncompact domains in topological vector spaces

About this journal

<p><em>Abstract and Applied Analysis</em> publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.</p>


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<p>Chief Editor, Dr Wong, is an associate professor at Nanyang Technological University, Singapore. Her research interests include differential equations, difference equations, integral equations, and numerical mathematics.</p>


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Abstract and Applied Analysis - Table of contents 2003

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Abstract and Applied Analysis publishes research with an emphasis on important developments in classical analysis, linear and nonlinear functional analysis, ordinary and partial differential equations, optimisation theory, and control theory.

Abstract and Applied Analysis