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TOPOLOGICAL METHODS

IN

NONLINEAR ANALYSIS



Vol. 18, No. 2           December 2001



TABLE OF CONTENTS

Title and Author(s) Page
item Morse Decompositions in the Absence of Uniqueness
M. C. Carbinatto and K. P. Rybakowski
ABSTRACT. In this paper we define attractors and Morse decompositions in an abstract framework of curves in a metric space. We establish some basic properties of these concepts including their stability under perturbations. This extends results known for flows and semiflows on metric spaces to large classes of ordinary or partial differential equations with possibly nonunique solutions of the Cauchy problem. As an application, we first prove a Morse equation in the context of a Conley index theory which was recently defined in [IR] for problems without uniqueness, and then apply this equation to give an elementary proof of two multiplicity results for strongly indefinite elliptic systems previously obtained in [AV] using Morse-Floer homology.
205
item Index Bundle, Leray-Schauder Reduction and Bifurcation of Solutions of Nonlinear Elliptic Boundary Value Problems
J. Pejsachowicz
ABSTRACT. We show that a family Fp; p in P of nonlinear elliptic boundary value problems of index 0 parametrized by a compact manifold admits a reduction to a family of compact vector fields parametrized by P if and only if its index bundle IndF vanishes. Our second conclusion is that, in the presence of bounds for the solutions of the boundary value problem, the non vanishing of the image of the index bundle under generalized J-homomorphism produces restrictions on the possible values of the degree of Fp. The most striking manifestation of this arises when the first Stiefel-Whitney class of the index bundle is nontrivial. In this case, the degree of Fp must vanish! From this we obtain a number of corollaries about bifurcation from infinity for solutions of nonlinear elliptic boundary value problems.
243
item Multiplicity of Solutions for~Nonhomogeneuous Nonlinear Elliptic Equations with Critical Exponents
N. Hirano
ABSTRACT. Let N \geq 3, \Omega\subset R^{N} be a bounded domain with a smooth boundary \partial\Omega. We consider a semilinear boundary value problem of the form
(P)\cases -\Delta u = \vert u\vert ^{2^*-2}u+f &\text{\rm in }\Omega,\\ u> 0 &\text{\rm in }\Omega,\\ u= 0 &\text{\rm on }\partial\Omega
where f\in C(\ol\Omega) and 2^*=2N/(N-2). We show the effect of topology of \Omega on the multiple existence of solutions.
269
item Reaction-Diffusion Equations on Unbounded Thin Domains
F. Antoci and M. Prizzi
ABSTRACT. We prove existence and upper semicontinuity of attractors for a reaction-diffusion equation on a family of thin unbounded domains collapsing onto a lower dimensional subspace.
283
item Nontrivial Solutions of Variational Inequalities. The Degenerate Case
S. Lancelotti
ABSTRACT. We consider a class of asymptotically linear variational inequalities. We show the existence of a nontrivial solution under assumptions which allow the problem to be degenerate at the origin.
303
item On H. Friedrich's Formulation of the Einstein Equations with Fluid Sources
Y. Choquet-Bruhat and J. W. York
ABSTRACT. We establish a variant of the symmetric quasi linear first order system given by H. Friedrich for the evolution equations of gravitating fluid bodies in General Relativity which can be important to solve realistic problems. Our version has the advantage of introducing only physical characteristics. We state explicitly the conditions under which the system hyperbolic and admits a well posed Cauchy problem.
321
item Almost-Periodicity Problem as a Fixed-Point Problem for Evolution Inclusions
J. Andres and A. M. Bersani
ABSTRACT. Existence of almost-periodic solutions to quasi-linear evolution inclusions under a Stepanov almost-periodic forcing is nontraditionally examined by means of the Banach-like and the Schauder-Tikhonov-like fixed-point theorems. These multivalued fixed-point principles concern condensing operators in almost-periodic function spaces or their suitable closed subsets. The Bohr-Neugebauer-type theorem jointly with the Bochner transform are employed, besides another, for this purpose. Obstructions related to possible generalizations are discussed.
337
item Recursive Coboundary Formula for Cycles in Acyclic Chain Complexes
T. Kaczynski
ABSTRACT. Given an (m - 1)-dimensional cycle z in a finitely generated acyclic chain complex, we want to explicitly construct an m-dimensional chain Cob(z) whose algebraic boundary is z. The acyclicity of the chain complex implies that a solution exists (it is not unique) but the traditional linear algebra methods of finding it lead to a high complexity of computation. We are searching for more efficient algorithms based on geometric considerations. The main motivation for studying this problem comes from the topological and computational dynamics, namely, from designing general algorithms computing the homomorphism induced in homology by a continuous map. This, for turn, is an essential step in computing such invariants of dynamical properties of nonlinear systems as Conley index or Lefschetz number. Another potential motivation is in the relationship of our problem to the problem of finding minimal surfaces of closed curves.
351
item F-epi Maps
J. Appel, M. Vath and A. Vignoli
ABSTRACT. The concept of 0-epi maps is a known homotopic analogue to maps with nonzero degree. There exist various related notions on unbounded sets and for multivalued maps. We introduce a concept which unifies these definitions. We also compare the various concepts. In particular, we prove that proper 0-epi maps are also 0-multiepi.
373
item Well-Posedness and Porosity in Best Approximation Problems
S. Reich and A. J. Zaslavski
ABSTRACT. Given a nonempty closed subset A of a Banach space X and a point x in X, we consider the problem of finding a nearest point to x in A. We define an appropriate complete metric space M of all pairs (A,x) and construct a subset \Omega of M which is the countable intersection of open everywhere dense sets such that for each pair in \Omega this problem is well-posed. As a matter of fact, we show that the complement of \Omega is not only of the first category, but also sigma-porous.
395
item Author Index for Volumes 17 and 18
 409



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