 |
TOPOLOGICAL METHODS
IN
NONLINEAR ANALYSIS |
| Vol. 22, No. 2 December 2003 |
TABLE OF CONTENTS
| Title and Author(s) |
Page |
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Conley Index Continuation for Singularly Perturbed Hyperbolic Equations
K. P. Rybakowski
ABSTRACT.
Let $\Omega\subset \R^N$, $N\le 3$, be a bounded domain with smooth boundary,
$\gamma\in L^2(\Omega)$ be arbitrary and $\phi\co \R\to \R$ be a
$C^1$-function satisfying a subcritical growth condition.
For every $\eps\in\oi0,\infty..$ consider the semiflow $\pi_\eps$ on
$H^1_0(\Omega)\times L^2(\Omega)$ generated by the damped wave equation
\eps \partial_{tt}u+\partial_t u&\,=\Delta u+\phi(u)+\gamma(x) &\quad& x\in\Omega,&\ &t>0,
u(x,t)&\,=0&\quad& x\in \partial \Omega,&\ &t>0.
Moreover, let $\pi'$ be the semiflow on $H^1_0(\Omega)$ generated
by the parabolic
equation
\partial_t u&\,=\Delta u+\phi(u)+\gamma(x) &\quad& x\in\Omega,&\ &t>0,
u(x,t)&\,=0&\quad& x\in \partial \Omega,&\ &t>0.
Let
$\Gamma\co H^2(\Omega)\to H^1_0(\Omega)\times L^2(\Omega)$ be the imbedding $u\mapsto (u,\Delta u+\phi(u)+\gamma)$.
We prove in this paper that every compact isolated $\pi'$-invariant set
$K'$ lies in $H^2(\Omega)$ and the imbedded set $K_0=\Gamma(K')$
continues to a family $K_\eps$, $\eps\ge0$ small, of isolated
$\pi_\eps$-invariant sets having the same Conley index as $K'$.
This family is upper-semicontinuous at $\eps=0$.
Moreover, any (partially ordered) Morse-decomposition of $K'$,
imbedded into $H^1_0(\Omega)\times L^2(\Omega)$ via $\Gamma$,
continues to a family of Morse decompositions of $K_\eps$, for
$\eps\ge 0$ small. This family is again
upper-semicontinuous at $\eps=0$.
These results extend and refine some upper semicontinuity results for
attractors obtained previously by Hale and Raugel.
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203
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Periodic Solutions of a Class of Integral Equations
S. Kang, G. Zhang and S. S. Cheng
| ABSTRACT.
Based on the fixed point index theory for a Banach space, nontrivial
periodic solutions are found for a class of integral equation of the form
\phi (x)=\int_{[x,x+\omega ]\cap \Omega }K(x,y)f(y,\phi (y-\tau (y)))\,dy,
\quad
x\in \Omega ,
where $\Omega $ is a closed subset of $\R^{N}$ with perioidc structure.
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245 |
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Some Remarks on Degree Theory for SO(2)-Equivariant Transversal Maps
N. Hirano and S. Rybicki
| ABSTRACT.
The aim of this article is to introduce a new class $\sone$-equivariant
transversal maps $\TR$ and to define
degree theory for such maps. We define degree for $\sone$-equivariant transversal maps and prove some properties
of this invariant. Moreover, we characterize $\sone$-equivariant transversal isomorphisms and derive formula
for degree of such isomorphisms.
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253
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Existence and Multiplicity Results for Wave Equations with Time-Independent Nonlinearity
J. Berkovits, H. Leinfelder and V. Mustonen
| ABSTRACT.
We shall study the existence of time-periodic solutions for a semilinear
wave equation with a given time-independent nonlinear perturbation and
small forcing. Since the distribution of eigenvalues of the linear part
varies with the period, the solvability of the problem depends essentially
on the frequency. The main idea of this paper is to consider the situation
where the period is not prescribed and hence treated as a parameter.
The description of the distribution of eigenvalues as a function of
the period enables us to show that under certain conditions the
interaction between the nonlinearity and the spectrum of the wave
operator induces multiple solutions.
Our basic new result states that the autonomous equation admits
at least two nontrivial solutions (free vibrations) for a restricted
(but infinite) set of periods such that the nonlinearity interacts with
one simple eigenvalue. As a corollary we prove that the semilinear wave
equation with time-independent nonlinearity and small forcing admits an
infinite sequence of pairs of periodic solutions with corresponding period
tending to zero. The results are obtained via generalized topological
degree theory.
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273
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Forced Singular Oscillators and the Method of Lower and Upper Solutions
D. Bonheure and C. De Coster
| ABSTRACT.
In this note, we study the existence of positive
periodic solutions of the second order differential equation
u''+g(u)u'+f(t,u)=h(t)
where $f(t,\,\cdot\,)$ has a singularity of
repulsive type at the origin. We use the method of lower and upper
solutions.
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297
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Impulsive Hyperbolic Differential Inclusions with Variable Times
M. Benchohra, L. Górniewicz, S. K. Ntouyas and A. Ouahab
| ABSTRACT.
In this paper the nonlinear alternative of Leray--Schauder type is
used to investigate the existence of solutions for second order impulsive hyperbolic differential
inclusions with variable times.
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319
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Completely Squashable Smooth Ergodic Cocycles over Irrational Rotations
D. Volny
| ABSTRACT.
Let $\alpha$ be an irrational number and the trasformation
Tx \mapsto x+\alpha \bmod 1, \quad x\in [0,1),
represent an irrational
rotation of the unit circle. We construct an ergodic and completely
squashable smooth real extension, i.e\. we find a real analytic or $k$
time continuously differentiable real function $F$ such that for every
$\lambda\neq 0$ there exists a commutor $S_\lambda$ of $T$ such that
$F\circ S_\lambda$ is $T$-cohomologous to $\lambda\,\v$ and the skew product
$T_F(x,y) = (Tx, y+F(x))$ is ergodic.
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331
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Nonlinear Submeans on Semigroups
A. To-Ming Lau and W. Takahashi
| ABSTRACT.
The purpose of this paper is to study some algebraic structure
of submeans on certain spaces $X$ of bounded real valued functions
on a semigroup and to find local conditions on $X$ in terms of submean
for the existence of a left invariant mean.
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345
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Multiple Solutions of Compact $H$-Surfaces in Euclidean Space
Y. Ge and F. Zhou
| ABSTRACT.
We prove here the multiplicity results for the solutions of compact
$H$-surfaces in Euclidean space.
Some minimax methods and topological arguments are used for the existence
of such solutions in multiply
connected domains.
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355
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On a Multivalued Version of the Sharkovskii Theorem and its Application to Differential Inclusions, III
J. Andres and K. Pastor
| ABSTRACT.
An extension of the celebrated Sharkovski{\u\i} cycle coexisting
theorem (see [14]) is given for (strongly) admissible multivalued
self-maps in the sense of [8], on a Cartesian product of linear
continua. Vectors of admissible self-maps have a triangular
structure as in [10]. Thus, we make a joint generalization of the
results in [2], [5], [6] (a multivalued case), in [10] (a
multidimensional case), and in [15] (a linear continuum case).
The obtained results can be applied, unlike in the single-valued
case, to differential equations and inclusions.
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369
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Combinatorial Lemmas for Nonoriented Pseudomanifolds
A. Idzik and K. Junosza-Szaniawski
| ABSTRACT.
Sperner lemma type theorems are proved for nonoriented primoids and
pseudomanifolds. A rank function of a primoid is defined. Applications of
these theorems to the geometric simplex are given. Also
Knaster--Kuratowski--Mazurkiewicz type theorems on covering of the geometric
simplex are presented.
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387
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Periodic Points of Multi-Valued $\varepsilon $-Contractive Maps
S. B. Nadler Jr.
| ABSTRACT.
Let $(X,d)$ be a nonempty metric space, and let $(2^{X},H_{d})$
be the hyperspace of all nonempty compact subsets of $X$ with
the Hausdorff metric. Let $F\colon X\rightarrow 2^{X}$ be an $\varepsilon$-contractive
map. A general condition is given that guarantees
the existence of a periodic point of $F$ (the theorem extends a result of
Edelstein to multi-valued maps). The condition holds
when $X$ is compact; hence, $F$ has a periodic point when $X$ is compact. It
is shown that $F$ has a fixed point (a point $p\in F(p)$) if $X$ is a
continuum. Applications to single-valued $\varepsilon$-expansive maps are given.
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399
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