UMK Logo

TOPOLOGICAL METHODS

IN

NONLINEAR ANALYSIS



Vol. 17, No. 1           March 2001



TABLE OF CONTENTS

Title and Author(s) Page
item Heteroclinic Solutions Between Stationary Points at Different Energy Levels
V. Coti Zelati and P. H. Rabinowitz
ABSTRACT. Consider the system of equations
-\ddot{q} = a(t)V'(q).
The main goal of this paper is to present a simple minimization method to find heteroclinic connections between isolated critical points of V, say 0 and \xi, which are local maxima but do not necessarily have the same value of V. In particular we prove that there exist heteroclinic solutions from 0 to \xi and from \xi to 0 for a class of positive slowly oscillating periodic functions a provided \delta = |V(0) - V(\xi)| is sufficiently small (and another technical condition is satisfied). Note that when V(0) \neq V(\xi), a cannot be constant be conservation of energy. Existence of "multi-bump" solutions is also proved.
1
item On the Existence of Positive Entire Solutions of Nonlinear Elliptic Equations
M. Squassina
ABSTRACT. Via non-smooth critical point theory, we prove existence of entire positive solutions for a class of nonlinear elliptic problems with asymptotic p-Laplacian behaviour and subjected to natural growth conditions.
23
item Morse Theory Applied to a T2-Equivariant Problem
G. Vannella
ABSTRACT. The following T2-equivariant problem of periodic type is considered:
(P) \cases u\in C^2({\Bbb R}^2,{\Bbb R}),\cr -\varepsilon\Delta u(x,y)+F^{\prime }(u(x,y))=0 & \text{in ${\Bbb R}^{2}$,}\cr u(x,y)=u(x+T,y)=u(x,y+S) &\text{for all $(x,y)\in {\Bbb R}^2$,}\cr \nabla u(x,y)=\nabla u(x+T,y)=\nabla u(x,y+S) &\text{for all $(x,y)\in {\Bbb R}^{2}$.}
Using a suitable version of Morse theory for equivariant problems, it is proved that an arbitrarily great number of orbits of solutions to (P) is founded, choosing \varepsilon< 0 suitably small. Each orbit is homeomorphic to S1 or to T2.
41
item On the Second Deformation Lemma
J.-N. Corvellec
ABSTRACT. In the framework of critical point theory for continuous functionals defined on metric spaces, we give a new, simpler proof of the so-called Second Deformation Lemma, a basic tool of Morse theory.
55
item On a Generalized Critical Point Theory on Gauge Spaces and Applications to Elliptic Problems on RN
M. Frigon
ABSTRACT. In this paper, we introduce some aspects of a critical point theory for multivalued functions
\Phi : E \to {\Bbb R}^{\Bbb N} \cup \{\infty\}
defined on E a complete gauge space and with closed graph. The existence of a critical point is established in presence of linking. Finally, we present applications of this theory to semilinear elliptic problems on RN.
67
item Global Existence and Blow-Up Results for an Equation of Kirchhoff Type on RN
P. G. Papadopoulos and N. M. Stavrakakis
ABSTRACT. We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type
u_{tt}-\phi (x)\Vert\nabla u(t)\Vert^{2}\Delta u+\delta u_{t} = |u|^{a}u,\quad x\in {\Bbb R}^N,\ t\geq 0,
with initial conditions
u(x,0)=u_0(x) and u_t(x,0)=u_1 (x),
in the case where
N\geq 3, \delta\geq 0 and (\phi(x))^{-1}=g(x)
is a positive function lying in L^{N/2}(\Bbb R^N)\cap L^{\infty}(\Bbb R^N ). When the initial energy E(u0,u1), which corresponds to the problem, is non-negative and small, there exists a unique global solution in time. When the initial energy E(u0,u1) is negative, the solution blows-up in finite time. A combination of the modified potential well method and the concavity method is widely used.
91
item Nonlinear Riemann-Hilbert Problems for Doubly Connected Domains and Closed Boundary Data
M. A. Efendiev and W. L. Wendland
ABSTRACT. In this paper, for nonlinear Riemann-Hilbert problems in doubly connected domains with smooth as well as Lipschitz continuous boundary data, existence of at least two topologically different solutions is established. The main tools are the topological degree of quasi-ruled Fredholm mappings, Montel's theorem, a priori estimates and the employment of classical modular function theory.
111
item A Fixed Point Theorem for Multivalued Mappings with Nonacyclic Values
D. Miklaszewski
ABSTRACT. The aim of this paper is to prove that every Borsuk continuous set-valued map of the closed ball in the 3-dimensional Euclidean space, taking values which are one point sets or knots, has a fixed point. This result is a special case of the Górniewicz Conjecture.
125
item On Stability of Fixed Point of Multivalued Maps
V. Obukhovskii and T. Starova
ABSTRACT. The criterion for the stability of a fixed point of a compact or condensing multimap in a Banach space with respect to a small perturbation is expressed in terms of its topological index.
133
item On Some Classes of Operator Inclusions with Lower Semicontinuous Nonlinearities
R. Bader, M. Kamenskii and V. Obukhovskii
ABSTRACT. We consider a class of multimaps which are the composition of a superposition multioperator PF generated by a nonconvex-valued almost lower semicontinuous nonlinearity F and an abstract solution operator S. We prove that under some suitable conditions such multimaps are condensing with respect to a special vector-valued measure of noncompactness and construct a topological degree theory for this class of multimaps yielding some fixed point principles. It is shown how abstract results can be applied to semilinear inclusions, inclusions with m-accretive operators and time-dependent subdifferentials, nonlinear evolution inclusions and integral inclusions in Banach spaces.
143
item Sets of Solutions of Nonlinear Initial-Boundary Value Problems
V. Durikovic and M. Durikovicova
ABSTRACT. In this paper we deal with the general initial-boundary value problem for a second order nonlinear nonstationary evolution equation. The associated operator equation is studied by the Fredholm and Nemitskii operator theory. Under local Holder conditions for the nonlinear member we observe quantitative and qualitative properties of the set of solutions of the given problem. These results can be applied for the different mechanical and natural science models.
157
item Inequalities in Metric Spaces with Applications
I. Beg
ABSTRACT. We prove the parallelogram inequalities in metric spaces and use them to obtain the fixed points of involutions.
183



go to vol-17.2 go home archives go to vol-16.2