 |
TOPOLOGICAL METHODS
IN
NONLINEAR ANALYSIS
|
| Vol. 30, No. 2 December 2007 |
TABLE OF CONTENTS
|
Title and Author(s) |
Page |
|
Relative homological linking in critical point theory
Alexandre Girouard
| ABSTRACT.
A relative homological linking of pairs is proposed. It is shown to
imply homotopical linking, as well as earlier non-relative notion of
homological linkings. Using Morse theory we prove a simple
``homological linking principle'', thereby generalizing and
simplifying many well known results in critical point theory.
|
|
211
|
|
|
Some remarks on the critical point theory
Chong Li
ABSTRACT.
In this paper we discuss some problems about critical point
theory. In the first part of the paper we study existence and
multiplicity results of semilinear second order elliptic equation:
\cases
-\Delta u=f(x,u) &\text{for } x\in \Omega,
u=0 &\text{for } x\in \partial \Omega,
\endcases
In \cite{4}, the authors study the contractibility of level
sets of functionals associated with some elliptic boundary value
problems. In this paper by using Morse theory and minimax method
we give a more precise description of topological construction of
level set of critical value of energy functional for mountain pass
type critical point. It is well known that nondegenerate critical
point is isolated, so if a critical point is not isolated, it must
be a degenerate critical point. In the second part we will give an
example that all the critical points of functional of a class of
oscillating equation with Neumann boundary condition are isolated
and the equation has only constant solutions. Moreover, critical
groups of each critical point of the functional are trivial. The
elliptic sine-Gordon equation originates from the static case of
the hyperbolic sine-Gordon equation modelling the Josephson
junction in superconductivity, which is of contemporary interest
to physicists. The problem is similar to the elliptic sine-Gordon
equation so we believe that it derives from profound physical
backdrop.
|
|
223
|
|
|
Existence of solutions for $p(x)$-Laplacian problem on an unbounded domain
Fu Yongqiang
ABSTRACT.
In this paper we study the following $p(x)$-Laplacian
problem:
\alignat 2
-\div(a(x)|\nabla u|^{p(x)-2}\nabla
u)+b(x)|u|^{p(x)-2}u&\,=f(x,u) &\quad& x\in \Omega,
u&\,=0 &\quad&\text{on }\partial\Omega,
\endalignat
where $1&lq; p_{1}&lq; p(x)&lq; p_{2}&lq; n$, $\Omega\subset {\Bbb R}^{n}$
is an exterior domain. Applying Mountain Pass Theorem we obtain
the existence of solutions in $W_{0}^{1,p(x)}(\Omega)$ for the
$p(x)$-Laplacian problem in the superlinear case.
|
|
235
|
|
|
Attractors for reaction-diffusion equations on arbitrary unbounded domains
Martino Prizzi and Krzysztof P. Rybakowski
ABSTRACT.
We prove existence of global attractors for parabolic
equations of the form
\alignedat2
u_t+\beta(x)u-\sum_{ij}\partial_i(a_{ij}(x)\partial_j u)&=f(x,u),&\quad &x\in \Omega,\ t\in\ro0,\infty..,
u(x,t)&=0,&\quad &x\in \partial \Omega,\ t\in\ro0,\infty...
\endalignedat
on an arbitrary unbounded domain $\Omega$ in $\R^3$, without
smoothness assumptions on $a_{ij}(\,\cdot\,)$ and $\partial\Omega$.
|
|
251
|
|
|
Cutting surfaces and applications to periodic points and chaotic-like dynamics
Marina Pireddu and Fabio Zanolin
| ABSTRACT.
In this paper we propose an elementary topological
approach which unifies and extends various different results
concerning fixed points and periodic points for maps defined on
sets homeomorphic to rectangles embedded in euclidean
spaces. We also investigate the associated discrete semidynamical
systems in view of detecting the presence of chaotic-like
dynamics.
|
|
279
|
|
|
Ergodic cocycles for Gaussian actions
Dariusz Skrenty
| ABSTRACT.
Ergodic Gaussian cocycles for rigid Gaussian actions are constructed. It is
also shown when any isomorphism between Gaussian actions is Gaussian.
|
|
321
|
|
|
Locally expanding mappings and hyperbolicity
Jacek Tabor
| ABSTRACT.
The aim of the following paper is to propose and investigate the
partial generalization of hyperbolicity to metric spaces. The
locally expanding mappings, as we call them, possess many similar
behaviour to that characteristic to hyperbolic mappings, in
particular, they have lipschitz shadowing property. As a direct
corollary we obtain for example shadowing on the Julia set.
|
|
335
|
|
|
On the structure of fixed-point sets of uniformly lipschitzian mappings
Ewa Sędłak and Andrzej Wiśnicki
| ABSTRACT.
It is shown that the set of fixed points of any $k$-uniformly lipschitzian
mapping in a uniformly convex space is a retract of a domain if $k$ is close
to $1$.
|
|
345
|
|
|
Asymptotically critical points and multiple elastic bounce trajectories
Antonio Marino and Claudio Saccon
| ABSTRACT.
We study multiplicity of elastic bounce trajectories (e.b.t.'s) with fixed
end points $A$ and $B$ on a nonconvex ``billiard table'' $\Omega$.
As well known, in general, such trajectories might not exist at all.
Assuming the existence of a ``bounce free'' trajectory $\gamma_0$ in $\Omega$
joining $A$ and $B$ we prove the existence of multiple families of e.b.t.'s
$\gamma_{\lambda}$ bifurcating from $\gamma_0$ as a suitable parameter $\lambda$
varies.
Here $\lambda$ appears in the dynamics equation as a multiplier of the potential
term.
We use a variational approach and look for solutions as the critical points of
the standard Lagrange integrals on the space $X(A,B)$ of curves joining
$A$ and $B$.
Moreover, we adopt an approximation scheme to obtain the elastic response of the walls
as the limit of a sequence of repulsive potentials fields which vanish inside $\Omega$
and get stronger and stronger outside.
To overcome the inherent difficulty of distinct solutions for the approximating
problems covering to a single solutions to the limit one, we use the notion
of ``asymptotically critical points'' (a.c.p.'s) for a sequence of functional.
Such a notion behaves much better than the simpler one of ``limit of critical
points'' and allows to prove multliplicity theorems in a quite natural way.
A remarkable feature of this framework is that, to obtain the e.b.t.'s as
a.c.p.'s for the approximating Lagrange integrals, we are lead to consider
the $L^2$ metric on $X(A,B)$.
So we need to introduce a nonsmooth version of the definition of a.c.p\. and
prove nonsmooth versions of the multliplicity theorems, in particular of the
``$\nabla$-theorems'' used for the bifurcation result.
To this aim we use several results from the theory of $\varphi$-convex functions.
|
|
351
|
|
|
Asymtotically stable one-dimensional compact minimal sets
Konstantin Athanassopoulos
| ABSTRACT.
It is proved that an asymptotically stable, $1$-dimensional,
compact minimal set $A$
of a continuous flow on a locally compact, metric space $X$ is
a periodic orbit, if $X$ is locally
connected at every point of $A$.
So, if the intrinsic topology of the region of attraction of an isolated,
$1$-dimensional, compact minimal set $A$ of a continuous flow on a locally
compact, metric space is locally
connected at every point of $A$, then $A$ is a periodic orbit.
|
|
397
|
|
