
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS

Vol. 43, No. 1 March 2014 
TABLE OF CONTENTS
Title and Author(s) 
Page 

Caratheodory convex selections of setvalued functions in Banach lattices
Jerzy Motyl
ABSTRACT.
Let $T$ be a measurable space, $X$ a Banach space while $Y$ a Banach lattice. We consider the
class of ``upper separated'' setvalued functions $F\colon T\times X \rightarrow 2^{Y}$ and investigate the problem of
the existence of Carath\'eodory type selection, that is, measurable in the first variable and orderconvex in the second variable.


1


Existence of positive solutions for a second order periodic boundary value problem with impulsive
effects
Jiafa Xu, Zhongli Wei and Youzheng Ding
ABSTRACT.
In this paper, we are mainly concerned with the existence and
multiplicity of positive solutions for the following second order
periodic boundary value problem involving impulsive effects
\cases
u''+\rho^2u=f(t,u), & t\in J',\\
\Delta u'_{t=t_k}=I_k(u(t_k)), & k=1,\ldots,m,
u(0)u(2\pi)=0,\quad u'(0)u'(2\pi)=0.
\endcases
Here $J'=J\setminus \{t_1,\ldots, t_m\}$, $f\in C(J\times
\Bbb R^+, \Bbb R^+)$, $I_k\in C( \Bbb R^+, \Bbb R^+)$, where $
\Bbb R^+=[0,\infty)$, $J=[0,2\pi]$. The proof of our main results
relies on the fixed point theorem on cones. The paper
extends some previous results and reports some new results about
impulsive differential equations.


11


Generalizations of Krasnoselskii's fixed point theorem in cones and applications
Sorin Budisan
ABSTRACT.
We give some generalizations of Krasnosel'ski{\u\i}'s fixed point theorem in cones,
replacing norms with functionals. We will apply these theorems to obtain at
least one positive solution for the boundary value problems for secondorder
differential equations.
Two positive solution results are also obtained.


23


Heteroclinics for some non autonomous third order differential equations
Denis Bonheure, Jose Angel Cid, Colette De Coster and Luis Sanchez
ABSTRACT.
We study the existence of heteroclinics connecting the two equilibria $\pm1$ of the third order differential equation
u'''=f(u)+p(t)u'
where $f$ is a continuous function such that $f(u)(u^21)>0$ if $u\neq\pm1$ and
$p$ is a bounded non negative function.
Uniqueness is also addressed.


53


Weighted pseudo almost automorphic solutions to nonautonomous semilinear evolution equations with delay and $S^p$weighted pseudo almost automorphic coefficients
YongKui Chang, Rui Zhang and G. M. N'Guerekata
ABSTRACT.
By some new properties of Stepanovlike weighted pseudo almost automorphic functions established by Chang, Zhang and N'Gu\'{e}r\'{e}kata recently, we shall deal with weighted pseudo almost automorphic solutions to the nonautonomous semilinear evolution equations with a constant delay: $u'(t)=A(t)u(t)+f(t,u(th))$, $ t\in\mathbb{R}$ in
a Banach space $\mathbb{X}$, where $A(t),t\in\mathbb{R}$ generates an exponentially stable evolution family $\{U(t,s)\}$ and $f \colon\mathbb{R}\times\mathbb{X} \rightarrow \mathbb{X}$ is a $\S^{p}$weighted pseudo almost automorphic function satisfying some suitable conditions. We obtain our main results by the LerayShauder Alternative theorem.


69


Holder continuous retractions and amenable semigroups of uniformly Lipschitzian mappings in Hilbert spaces
Andrzej Wisnicki
ABSTRACT.
Suppose that $S$ is a left amenable semitopological semigroup. We prove that
if $\mathcal{S}=\{ T_{t}:t\in S\} $ is a uniformly $k$Lipschitzian semigroup on a bounded closed and convex subset $C$ of a
Hilbert space and $k<\sqrt{2}$, then the set of fixed points of $\mathcal{S}$
is a H\"{o}lder continuous retract of $C$. This gives a qualitative
complement to the IshiharaTakahashi fixed point existence theorem.


89


Limit sets in impulsive semidynamical systems
Changming Ding
ABSTRACT.
In this paper, we establish several
fundamental properties in impulsive semidynamical systems. First, we
formulate a counterpart of the continuous dependence on the initial
conditions for impulsive dynamical systems, and also establish some
equivalent properties. Second, we present several theorems similar
to the Poincar\'eBendixson theorem for twodimensional impulsive
systems, i.e\. if the omega limit set of a bounded infinite
trajectory (with an infinite number of impulses) contains no rest
points, then there exists an almost recurrent orbit in the limit
set. Further, if the omega limit set contains an interior point,
then it is a chaotic set; otherwise, if the limit set contains no
interior points, then the limit set contains a periodic orbit or
a Cantortype minimal set in which each orbit is almost recurrent.


97


A criterion for bijectivity of mappings of euclidean spaces
Albeta Mafra and Marcelo Tavares
ABSTRACT.
We study the following problem introduced by J. Hadamard in 1906: to
find sufficient conditions for a local diffeomorphism of an
Euclidean space to be a global diffeomorphism. J. Hadamard
introduced a celebrated integral condition which is a sufficient
condition for the bijectivity of a local diffeomorphism. In this
paper we improve the classical result of Hadamard giving a new
sufficient condition for a $C^2$ mapping to be bijective.


117


Existence of periodic travelingwave solutions for a nonlinear Schrodinger system: A topological approach
Nghiem V. Nguyen
ABSTRACT.
In this paper, the existence of periodic travelingwave solutions for
a nonlinear Schr\"odinger system is established using the topological degree theory for positive operators. The method guarantees existence of periodic solutions in a parameter region in the period and phase speed plane.


129


Existence of periodic solutions for some singular elliptic equations with strong resonant data
Laura Gonella
ABSTRACT.
We prove the existence of at least one $T$periodic solution $(T\nobreak>\nobreak0)$ for differential equations of the form
\left(\frac{u'(t)}{\sqrt{1{u'}^2(t)}}\right)' =f(u(t))+h(t),\quad \text{in } (0,T),
where
$f$ is a continuous function defined on $\mathbb{R}$ that satisfies a {\it strong resonance condition}, $h$ is continuous and with zero mean value. Our method uses variational techniques for nonsmooth functionals.


157


Conley index orientations
Axel Janig
ABSTRACT.
The homotopy Conley index along heteroclinic
solutions of certain parabolic evolution equations is zero
under appropriate assumptions.
This result implies that the socalled connecting homomorphism
associated with a heteroclinic solution is an isomorphism. Hence, using $\IZ$coefficients
it can be viewed as either $1$ or $1$  depending on the choice of generators
for the homology Conley index. We develop a method to choose such generators,
and compute the connecting homomorphism
relative to these generators.


171


Properness and topological degree for nonlocal integrodifferential systems
Narcisia C. Apreutesei and Vitaly A. Volpert
ABSTRACT.
Reactiondiffusion systems of equations with integral terms are
studied. Essential spectrum of the corresponding linear operators
is determined and the Fredholm property is studied. Properness of
nonlinear operators is proved and topological degree is
constructed.


215


Lyapunov functions, shadowing and topological stability
Alexey A. Petrov and Sergei Yu. Pilyugin
ABSTRACT.
We use Lyapunov type functions to give new conditions
under which a homeomorphism of a compact metric space has the
shadowing property. These conditions are applied to establish
the topological stability of some homeomorphisms with nonhyperbolic
behavior.


231


A relaxed Halpern's type iteration for an infinite family of nonexpansive mappings in CAT(0) spaces
WeiQi Deng
ABSTRACT.
Under weaker conditions on parameters, we prove
strong convergence theorems of Halpern type iteration schemes for
sequences of nonexpansive mappings in CAT(0) spaces. Since there is
no assumption of the AKTTcondition imposed on the involved
mappings, the results improve those of the authors with related
researches.


241


A critica fractional Laplace equation in the resonant case
Raffaella Servadei
ABSTRACT.
In this paper we complete the study of the following nonlocal fractional equation involving critical nonlinearities
\cases
(\Delta)^s u\lambda u=u^{2^*2}u & {\text{in }} \Omega,
u=0 & {\text{in }} \RR^n\setminus \Omega,
\endcases
started in the recent papers \cite{13}, \cite{17}\cite{19}.
Here $s\in (0,1)$ is a fixed parameter, $(\Delta )^s$ is the fractional Laplace operator, $\lambda$ is a positive constant, $2^*=2n/(n2s)$ is the fractional critical Sobolev exponent and $\Omega$ is an open bounded subset of $\RR^n$, $n>2s$, with Lipschitz boundary. Aim of this paper is to study this critical problem in the special case when $n\not=4s$ and $\lambda$ is an eigenvalue of the operator $(\Delta)^s$ with homogeneous Dirichlet boundary datum. In this setting we prove that this problem admits a nontrivial solution, so that with the results obtained in \cite{13}, \cite{17}\cite{19}, we are able to show that this critical problem admits a nontrivial solution provided
 $n>4s$ and $\lambda>0$,
 $n=4s$ and $\lambda>0$ is different from the eigenvalues of $(\Delta)^s$,
 $2s0$ is sufficiently large.
In this way we extend completely the famous result of Brezis and Nirenberg (see \cite{4}, \cite{5}, \cite{9}, \cite{23}) for the critical Laplace equation to the nonlocal setting of the fractional Laplace equation.


251


Radial symmetry of nmode positive solutions for semilinear elliptic equations in a disc and its application to the Henon equation
Naoki Shioji and Kohtaro Watanabe
ABSTRACT.
Let
$f \in C((0,1)\times (0,\infty),\R)$ and $n \in \N$ with $n \geq 2$
such that for each $u \in (0,\infty)$,
$r\mapsto r^{22n}f(r,u)\colon (0,1)\rightarrow \R$ is nonincreasing
and let $D=\{x=(x_1,x_2)\in\R^2: x<1\}$.
We show that each positive solution of
\Delta u + f(x,u) =0 \quad\text{in $D$,}
\qquad u=0 \quad\text{on $\partial D$}
which satisfies
$u(r,\theta)= u(r,\theta+2\pi/n)$ by the polar coordinates
is radially symmetric and $u_r(x)<0$ for each $r=x \in (0,1)$.
We apply our result to the H\'enon equation.


269

