
TOPOLOGICAL METHODS IN NONLINEAR ANALYSIS

Vol. 44, No. 2 December 2014 
TABLE OF CONTENTS
Title and Author(s) 
Page 

The splitting lemmas for nonsmooth functional on Hilbert spaces II. The case at infinity.
Guangcun Lu
ABSTRACT.
We generalize the BartschLi's splitting lemma at
infinity for $C^2$functionals in \cite{BaLi} and some later
variants of it to a class of continuously directional differentiable
functionals on Hilbert spaces. Different from the previous flow
methods our proof is to combine the ideas of the MorsePalais lemma
due to DucHungKhai \cite{DHK} with some techniques from \cite{JM},
\cite{Skr}, \cite{Va1}. A simple application is also presented.


277


Periodic solutions of a kind of Liénard equations with two deviating arguments
Tiantian Ma
ABSTRACT.
In this paper, we deal with the existence of periodic solutions of
a kind of Li\'enard equations with two deviating arguments
x''+f(t, x(t\sigma(t)))x'(t)+g(t, x(t\tau (t)))=p(t).
Some new results on the existence of periodic solutions of the given
equations are proved by using the continuation theorem.


337


Existence principle for BVPs with statedependent impulses
Irena Rachůnková and Jan Tomeček
ABSTRACT.
The paper provides an existence principle for the SturmLiouville boundary value problem with statedependent impulses
z''(t) = f(t,z(t),z'(t)) \quad \text{for a.e. } t \in [0,T] \subset \re,
z(0)  az'(0) = c_1, \quad z(T) + bz'(T) = c_2,
z(\x{\tau}{i}+)  z(\x{\tau}{i}) = J_i(\x{\tau}{i},z(\x{\tau}{i})), z'(\x{\tau}{i}+)  z'(\x{\tau}{i}) = \m_i(\x{\tau}{i},z(\x{\tau}{i})),
where the points $\x{\tau}{1}, \ldots, \x{\tau}{p}$ depend on $z$ through the equations
\x{\tau}{i} = \gamma(z(\x{\tau}{i})), \quad i = 1,\ldots,p, \ p \in \en.
Provided $a$, $b \in [0,\infty)$, $c_j \in \re$, $j = 1,2$, and the data functions $f$, $J_i$, $\m_i$, $i=1,\ldots,p$, are bounded, transversality conditions for barriers $\gamma_i$, $i = 1,\ldots,p$, which yield the solvability of the problem, are delivered. An application to the problem with unbounded data functions is demonstrated.


349


Some remarks on Park's abstract convex spaces
Władysław Kulpa and Andrzej Szymanski
ABSTRACT.
We discuss S. Park's abstract convex spaces and their relevance to classical
convexieties and $L^{\ast }$operators. We construct an example of a space
satisfying the partial KKM principle that is not a KKM space. The existence
of such a space solves a problem by S. Park.


369


On the solvability of nonlinear impulsive boundary value problems
Daniel Maroncelli and Jesús Rodríguez
ABSTRACT.
In this paper we provide sufficient conditions for the existence of solutions to twopoint boundary value problems for nonlinear ordinary differential equations subject to impulses. Our results depend on properties of the nonlinearities as well as on the solution space of the associated linear problem. Our approach is based on topological degree arguments in conjunction with the LyapunovSchmidt procedure.


381


On a class of nonhomogeneous elliptic problems involving exponential critical growth
Manassés de Souza, Everaldo Souto de Medeiros and Uberlandio Severo
ABSTRACT.
In this paper we establish the existence of solutions for elliptic
equations of the form $\text{div}(\nabla u^{n2}\nabla u) +
V(x)u^{n2}u=g(x,u)+\lambda h$ in $\mathbb{R}^n$ with $n\geq2$.
Here the potential $V(x)$ can change sign and the nonlinearity
$g(x,u)$ is possibly discontinuous and may exhibit exponential
growth. The proof relies on the application of a fixed point
result and a version of the TrudingerMoser inequality.


399


The trajectory attractor and its limiting behavior for the convective BrinkmanForchheimer equations
Caidi Zhao, Lei Kong and Min Zhao
ABSTRACT.
This paper studies the trajectory behavior of the convective
BrinkmanForchheimer equations in threedimensional (3D) bounded
domains. We first prove the existence of the trajectory attractor
${\mathcal A}^{\rm tr}_\alpha$ for the natural translation semigroup
in the trajectory space. Then we establish that the trajectory
attractor $\mathcal{A}^{\rm tr}_\alpha$ converges, as
$\alpha\rightarrow 0^+$, to the trajectory attractor
$\mathcal{A}^{\rm tr}_0$ of the 3D NavierStokes equations in
a proper topological space.


413


Strauss and Lions type results for a class of OrliczSobolev spaces and applications
Claudianor O. Alves, Giovany M. Figueiredo and Jefferson A. Santos
ABSTRACT.
The main goal of this work is to prove Strauss and Lionstype results for OrliczSobolev
spaces. After, we use these results to study the existence of solutions for a class
of quasilinear problems in $\mathbb{R}^{N}$.


435


On the Schrodinger equations with a nonlinearity in the critical growth
Jian Zhang
ABSTRACT.
In this paper, we consider the Schr\"{o}dinger equation with a nonlinearity in the critical growth. The purpose of this paper is to establish the existence of ground states via variational methods.


457


Connection matrices for MorseBott flows
Dahisy V. de S. Lima and Ketty A. de Rezende
ABSTRACT.
A Connection Matrix Theory approach is presented for MorseBott flows $\varphi$ on smooth closed $n$manifolds by characterizing the set of connection matrices in terms of MorseSmale perturbations. Further results are obtained on the effect on the set of connection matrices $\mathcal{CM}(S)$ caused by changes in the partial orderings and in the Morse decompositions of an isolated invariant set $S$.


471


A completion construction for continuous dynamical systems
José Manuel García Calcines, Luis Javier Hernández Paricio and María Teresa Rivas Rodríguez
ABSTRACT.
In this work we use the theory of exterior spaces to construct a $\Co^{\r}$completion and a $\Co^{\l}$completion of a dynamical
system. If $X$ is a flow, we construct canonical maps $X\to
\Co^{\r}(X)$ and $X\to \Co^{\l}(X)$ and when these maps are
homeomorphisms we have the class of $\Co^{\r}$complete and
$\Co^{\l}$complete flows, respectively. In this study we find
out many relations between the topological properties of the
completions and the dynamical properties of a given flow. In the
case of a complete flow this gives interesting relations between
the topological properties (separability properties, compactness,
convergence of nets, etc.) and dynamical properties (periodic
points, omega limits, attractors, repulsors, etc.).


497


Saddle point solutions for nonlocal elliptic operators
Alessio Fiscella
ABSTRACT.
The paper deals with equations driven by a nonlocal integrodifferential operator $\mathcal L_K$ with homogeneous Dirichlet boundary conditions.
These equations have a variational structure and we find a solution for them
using the Saddle Point Theorem. We prove this result for a general
integrodifferential operator of fractional type
and from this, as a particular case, one can derive an
existence theorem for the fractional Laplacian, finding
solutions of the equation
\begin{cases}
(\Delta)^s u=f(x,u) & {\mbox{in }} \Omega,
u=0 & {\mbox{in }} \mathbb{R}^n\setminus \Omega,
\end{cases}
where the nonlinear term $f$ satisfies a linear growth condition.


527


Infinitely many solutions to quasilinear elliptic equation with concave and convex terms
Leran Xia, Minbo Yang and Fukun Zhao
ABSTRACT.
In this paper, we are concerned with the following quasilinear elliptic equation with concave and convex terms
\Delta u{\frac12}\,u\Delta(u^2)=\alphau^{p2}u+\betau^{q2}u,\quad x\in \Omega,
\leqno(\rom{P})
where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain,
$1


539

