Vol. 44, No. 2           December 2014


Title and Author(s) Page
item The splitting lemmas for nonsmooth functional on Hilbert spaces II. The case at infinity.
Guangcun Lu
ABSTRACT. We generalize the Bartsch-Li'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 Morse-Palais lemma due to Duc-Hung-Khai \cite{DHK} with some techniques from \cite{JM}, \cite{Skr}, \cite{Va1}. A simple application is also presented.
item 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.
item Existence principle for BVPs with state-dependent impulses
Irena Rachůnková and Jan Tomeček
ABSTRACT. The paper provides an existence principle for the Sturm-Liouville boundary value problem with state-dependent 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.
item 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.
item 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 two-point 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 Lyapunov-Schmidt procedure.
item 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 $-\text{div}(|\nabla u|^{n-2}\nabla u) + V(x)|u|^{n-2}u=g(x,u)+\lambda h$ 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 Trudinger-Moser inequality.
item The trajectory attractor and its limiting behavior for the convective Brinkman-Forchheimer equations
Caidi Zhao, Lei Kong and Min Zhao
ABSTRACT. This paper studies the trajectory behavior of the convective Brinkman-Forchheimer equations in three-dimensional (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 Navier-Stokes equations in a proper topological space.
item Strauss and Lions type results for a class of Orlicz-Sobolev 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 Lions-type results for Orlicz-Sobolev spaces. After, we use these results to study the existence of solutions for a class of quasilinear problems in $\mathbb{R}^{N}$.
item 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.
item Connection matrices for Morse-Bott flows
Dahisy V. de S. Lima and Ketty A. de Rezende
ABSTRACT. A Connection Matrix Theory approach is presented for Morse-Bott flows $\varphi$ on smooth closed $n$-manifolds by characterizing the set of connection matrices in terms of Morse-Smale 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 $S$.
item 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.).
item Saddle point solutions for non-local elliptic operators
Alessio Fiscella
ABSTRACT. The paper deals with equations driven by a non-local 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.
item 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)=\alpha|u|^{p-2}u+\beta|u|^{q-2}u,\quad x\in \Omega, \leqno(\rom{P})
where $\Omega\subset\mathbb{R}^N$ is a bounded smooth domain, $1

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