UMK Logo TOPOLOGICAL METHODS

IN

NONLINEAR ANALYSIS


Vol. 42, No. 2           December 2013


TABLE OF CONTENTS


Title and Author(s) Page
item Resolvent convergence for Laplace operators on unbounded curved squeezed domains
Maria C. Carbinatto and Krzysztof P. Rybakowski
ABSTRACT. We establish a resolvent convergence result for the Laplace operator on certain classes of unbounded curved squeezed domains $\Omega_\eps$ as $\eps\to0$. As a consequence, we obtain Trotter-Kato-type convergence results for the corresponding family of $C^0$-semigroups. This extends previous results obtained by Antoci and Prizzi in \cite{\rfa{AP}} in the flat squeezing case.
233
item Study on a quadratic Hadamard type fractional integral equation on an unbounded interval
JinRong Wang, Chun Zhu and Yong Zhou
ABSTRACT. In this paper, a quadratic Hadamard type fractional integral equations on an unbounded interval is studied. By applying a technique of measure of noncompactness and Schauder fixed point theorem, existence and uniform local attractivity of solutions are presented after overcoming some difficulty from the Hadamard type singular kernel. Moreover, three new solutions sets who tend to zero at infinity are constructed to obtain local stability of solutions. Finally, two examples are made to illustrate our theory results.
257
item Multiple solutions to a Dirichlet eigenvalue problem with p-Laplacian
Salvatore A. Marano, Dumitru Motreanu and Daniele Puglisi
ABSTRACT. The existence of a greatest negative, a smallest positive, and a nodal weak solution to a homogeneous Dirichlet problem with $p$-Laplacian and reaction term depending on a positive parameter is investigated via variational as well as topological methods, besides truncation techniques.
277
item Generalized Sturm-Liouville boundary conditions for first order differential systems in the plane
Alessandro Fonda and Maurizio Garrione
ABSTRACT. We study asymptotically positively homogeneous first order systems in the plane, with boundary conditions which are positively homogeneous, as well. Defining a generalized concept of Fuèik spectrum which extends the usual one for the scalar second order equation, we prove existence and multiplicity of solutions. In this way, on one hand we extend to the plane some known results for scalar second order equations (with Dirichlet, Neumann or Sturm-Liouville boundary conditions), while, on the other hand, we investigate some other kinds of boundary value problems, where the boundary points are chosen on a polygonal line, or in a cone. Our proofs rely on the shooting method.
293
item Rotation numbers for planar attractors of equivariant homeomorphisms
Begona Alarcón
ABSTRACT. Given an integer $m>1$ we consider $\Z_m$-equivariant and orientation preserving homeomorphisms in $\R^2$ with an asymptotically stable fixed point at the origin. We present examples without periodic points and having some complicated dynamical features. The key is a preliminary construction of $\Z_m$-equivariant Denjoy maps of the circle.
327
item Gradient-like nonlinear semigroups with infinitely many equilibria and applications to cascade systems
Alexandre N. Carvalho, Eder R. Aragao-Costa, Pedro Marín-Rubio and Gabriela Planas
ABSTRACT. We consider an autonomous dynamical system coming from a coupled system in cascade where the uncoupled part of the system satisfies that the solutions comes from $-\infty $ and goes to $\infty $ to equilibrium points, and where the coupled part generates asymptotically a gradient-like nonlinear semigroup. Then, the complete model is proved to be also gradient-like. The interest of this extension comes, for instance, in models where a continuum of equilibrium points holds, and for example a £ojasiewicz-Simon condition is satisfied. Indeed, we illustrate the usefulness of the theory with several examples.
345
item Controllability for systems governed by second-order differential inclusions with nonlocal conditions
Tran Dinh Ke and Valeri Obukhovskii
ABSTRACT. We study a controllability problem for a system governed by a semilinear second-order differential inclusion involving control perturbations and nonlocal conditions in a Hilbert space. By using the fixed point theory for condensing multimaps, the $(E_0,X_0)$-controllability result for the mentioned problem is proved under the assumption that the corresponding linear system is $(E_0,X_0)$-controllable.
377
item On parametric equilibrium problems
Marcel Bogdan and József Kolumbán
ABSTRACT. The goal of this article is to study a kind of stability property of a sequence of solutions to parametric equilibrium problems. The main result gives sufficient conditions for this purpose, in the presence of the topological pseudomonotonicity in the limit problem.
405
item Note on periodic solutions of relativistic pendulum type systems
Kazuya Hata, Jiaquan Liu and Zhi-Qiang Wang
ABSTRACT. We establish multiplicity results of periodic solutions for relativistic pendulum type systems of ordinary differential equations. We provide a different approach to the problems and answer some questions raised in \cite{6}, \cite{7} by Brezis and Mawhin recently.
417
item Two positive solutions for one-dimensional p-Laplacian with a singular weight
Ryuji Kajikiya, Yong-Hoon Lee and Inbo Sim
ABSTRACT. We investigate a bifurcation problem for one-dimensional $p$-Laplace equation with a singular weight under Dirichlet boundary condition. Using super-subsolution method and mountain pass lemma, we prove the existence of at least two positive solutions, at least one positive solution and no positive solution according to the range of a bifurcation parameter.
427
item Existence and nonexistence of positive periodic solutions to a differential inclusion
Yuqiang Feng and Ping Tong
ABSTRACT. In this paper, the existence and nonexistence of positive periodic solutions for a second-order differential inclusion are considered. Some existence and nonexistence results are established by the use of Bohnenblust-Karlin's fixed-point theorem for multivalued operators and Sobolev constant.
449
item Solvability of fractional differential equations with integral boundary conditions at resonance
Yude Ji and Weihua Jiang
ABSTRACT. By using the coincidence degree theory due to Mawhin and constructing suitable operators, some sufficient conditions for the existence of solution for a class of fractional differential equations with integral boundary conditions at resonance are established, which are complement of previously known results. The interesting point is that we shall deal with the case $\text{\rm dim}\,\text{\rm Ker}\,L=2$, which will cause some difficulties in constructing the projector $Q$. An example is given to illustrate our result.
461



go to vol-43.1 go home archives go to vol-42.1