We establish a resolvent convergence result for the Laplace operator on certain classes of unbounded curved squeezed domains $\Omega_\varepsilon$ as $\varepsilon\to0$. 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 [F. Antoci and M. Prizzi, Reaction-diffusion equations on unbounded thin domains, Topol. Methods Nonlinear Anal. 18 (2001), 283-302] in the flat squeezing case.

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.

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.

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 Fučík 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.

Given an integer $m> 1$ we consider $\mathbb{Z}_m$-equivariant and orientation preserving homeomorphisms in $\mathbb{R}^2$ 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 $\mathbb{Z}_m$-equivariant Denjoy maps of the circle.

We consider an autonomous dynamical system coming from a coupled system in cascade where the uncoupled part of the system satisfies that the solution 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 Łojasiewicz-Simon condition is satisfied. Indeed, we illustrate the usefulness of the theory with several examples.

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.

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.

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 [Periodic solutions of the forced relavitistic pendulum, Differential Integral Equations 23 (2010), 801-810], [Periodic solutions of Lagrangian systems of relatvitistic oscillations, Comm. Appl. Anal. 15 (2011), 235-250] by Brezis and Mawhin recently.

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.

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.

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 ${\rm dim}\,{\rm Ker}\,L=2$, which will cause some difficulties in constructing the projector $Q$. An example is given to illustrate our result.