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TOPOLOGICAL METHODS

IN

NONLINEAR ANALYSIS



Vol. 18, No. 1           September 2001



TABLE OF CONTENTS

Title and Author(s) Page
item On the Existence of Three Solutions for Jumping Problems Involving Quasilinear Operators
A. Canino
ABSTRACT. A jumping problem for quasilinear elliptic equations is considered. A local saddle argument in the framework of nonsmooth critical point theory is applied.
1
item Multiple Positive Symmetric Solutions of a Singularly Perturbed Elliptic Equation
M. Clapp and G. Izquierdo
ABSTRACT. This paper is concerned with the multiplicity of positive solutions of the Dirichlet problem
-\varepsilon ^{2}\Delta u+u=K( x) \vert u\vert ^{p-2}u\quad\text{in }\Omega,
where \Omega is a smooth domain in RN which is either bounded or has bounded complement (including the case \Omega =\Bbb{R}^{N}$), $N\geq 3$ K is continuous and p is subcritical. It is known that critical points of K give rise to multibump solutions of this type of problems. It is also known that, in general, the presence of symmetries has the effect of producing many additional solutions. So, we consider domains \Omega which are invariant under the action of a group G of orthogonal transformations of RN, we assume that K is G-invariant, and study the combined effect of symmetries and the nonautonomous term K on the number of positive solutions of this problem. We obtain multiplicity results which extend previous results of Benci and Cerami (1994), Cingolani and Lazzo (1997) and Qiao and Wang (1999).
17
item Structure of Large Positive Solutions of Some Semilinear Elliptic Problems where the Nonlinearity Changes Sign
Z. Guo
ABSTRACT. Existence and uniqueness of large positive solutions are obtained for some semilinear elliptic Dirichlet problems in bounded smooth domains \Omega with a large parameter \lambda. It is shown that the large positive solution has flat core. The distance of its flat core to the boundary \partial\Omega is exactly measured as \lambda\to\infty.
41
item Asymptotic Behavior of Solutions of Some Nonlinearly Damped Wave Equations on RN
N. Karachalios and N. Stavrakakis
ABSTRACT. We discuss the asymptotic behavior of solutions of the nonlinearly amped wave equation
u_{tt} +\delta \vert u_t\vert ^{m-1}u_t -\phi (x)\Delta u = \lambda u\vert u\vert ^{\beta -1}, \quad x \in \Ren, \ t \geq 0,
ith the initial conditions u(x,0) = u0(x) and ut(x,0) = u1(x), in the case where $N \geq 3$, $ \delta > 0$ and $(\phi x))^{-1} =g (x)$ is a positive function lying in L^{p}(\Ren)\cap L^{\infty}(\Ren) for some p. We prove blow-up of solutions when the source term dominates over the damping, and the initial energy is assumed to be positive. We also discuss global existence energy decay of solutions.
73
item Resolution de Problemes de Rafle et Application a un Probleme de Frottement
M. Chraibi Kaadoud
ABSTRACT. In this paper we study the sweeping processes by convex sets depending on time and the solution. We do some application to a dry friction's problem.
RESUME. Dans ce travail, nous etudions des problemes de rafle par des multifonctions qui dependent du temps et de la solution. Nous donnons une application a un probleme de frottement.
89
item Variational and boundary value problems with perturbations
S. Walczak
ABSTRACT. In the paper an optimization problem with parameters is considered. Some sufficient conditions under which the solutions of the problem continuosly depend on parameters (in the weak or the strong topology of a Banach space) are proved. Moreover, some applications to the eigenvalue and boundary value problems for differential operators are given.
106
item Symbolic Representations of Iterated Maps
X.-C. Fu, W. Lu, P. Ashwin and J. Duan
ABSTRACT. This paper presents a general and systematic discussion of various symbolic representations of iterated maps through subshifts. A unified model for all continuous maps on a metric space is given. It is shown that at most the second order representation is enough for a continuous map. By introducing distillations, partial representations of some general continuous maps are obtained. Finally, partitions and representations of a class of discontinuous maps and some examples are discussed.
119
item Weak Compactness of Solution Sets to Stochastic Differential Inclusions with Convex Right-Hand Sides
M. Kisielewicz
ABSTRACT. Necessary and sufficient conditions for the existence of weak solutions to stochastic differential inclusions with convex right-hand sides are given. The main results of the paper deal with the weak compactness with respect to the convergence in distribution of solution sets to such inclusions.
149
item Continuous Selections via Geodesics
G. Colombo and V. V. Goncharov
ABSTRACT. Some continuous selection results for a class of nonconvex-valued maps are obtained. One of them contains Michael's theorem, in the case of a Hilbert codomain. Methods of nonsmooth analysis and $\Gamma$-convergence are used.
171
item Computer Assisted Proof of Chaotic Dynamics in the Rossler Map
D. Wilczak
ABSTRACT. In this paper we present the proof of the existence of symbolic dynamics for third iterate of the Rossler map. We combine an abstract topological results based on the fixed point index and covering relations with computer assisted rigorous computations.
183
item Leray-Schauder Type Alternatives and the Solvability of Complementarity Problems
G. Isac
ABSTRACT. We present in this paper several existence theorems for nonlinear complementarity problems in Hilbert spaces. Our results are based on the concept of "exceptional family of elements" and on Leray-Schauder type altrenatives.
191



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