In this paper we study the first order nonautonomous Hamiltonian system $$ \dot{z}=\mathcal J H_{z}(t,z), $$ where $H(t,z)$ depends periodically on $t$. By using a generalized linking theorem for strongly indefinite functionals, we prove that the system has infinitely many homoclinic orbits for weak superlinear cases.

In this paper, we investigate the existence and multiplicity of positive solutions for the following $2n$th-order $p$-Laplacian boundary value problem $$ \begin{cases} -(((-1)^{n-1}x^{(2n-1)})^{p-1})'\\ =f(t,x,-x'',\ldots,(-1)^{n-1}x^{(2n-2)}) &\text{for } t\in [0,1], \\ x^{(2i)}(0)=x^{(2i+1)}(1)=0 & \text{for } i=0,\ldots,n-1, \end{cases} $$ where $n\ge 1$ and $f\in C([0,1]\times \mathbb{R}_+^{n}, \mathbb{R}_+)(\mathbb{R}_+:=[0,\infty))$ depends on $x$ and all derivatives of even orders. Based on a priori estimates achieved by utilizing properties of concave functions and Jensen's integral inequalities, we use fixed point index theory to establish our main results. Moreover, our nonlinearity $f$ is allowed to grow superlinearly and sublinearly.

In this paper we find existence results for elliptic and parabolic nonlinear variational inequalities involving a multivalued map. Both cases of a lower semicontinuous multivalued map and an upper semicontinuous one are considered.

The global attractor of a gradient-like semigroup has a Morse decomposition. Associated to this Morse decomposition there is a Lyapunov function (differentiable along solutions) - defined on the whole phase space - which proves relevant information on the structure of the attractor. In this paper we prove the continuity of these Lyapunov functions under perturbation. On the other hand, the attractor of a gradient-like semigroup also has an energy level decomposition which is again a Morse decomposition but with a total order between any two components. We claim that, from a dynamical point of view, this is the optimal decomposition of a global attractor; that is, if we start from the finest Morse decomposition, the energy level decomposition is the coarsest Morse decomposition that still produces a Lyapunov function which gives the same information about the structure of the attractor. We also establish sufficient conditions which ensure the stability of this kind of decomposition under perturbation. In particular, if connections between different isolated invariant sets inside the attractor remain under perturbation, we show the continuity of the energy level Morse decomposition. The class of Morse-Smale systems illustrates our results.

We prove the existence of positive solutions for the singular boundary value problems $$ \begin{cases} \displaystyle -\Delta u=\frac{p(x)}{u^{\beta }}+\lambda f(u) & \text{in }\Omega , \\ u=0 &\text{on }\partial \Omega , \end{cases} $$ where $\Omega $ is a bounded domain in $\mathbb{R}^n$ with smooth boundary $\partial \Omega$, $0< \beta < 1$, $\lambda > 0$ is a small parameter, $f\colon (0,\infty )\rightarrow \mathbb{R}$ is asymptotically linear at $\infty$ and is possibly singular at $0$.

In this paper, by a space decomposition we will study the existence and multiplicity for the $p$-Laplacian equation with resonance at the first two eigenvalues.

We construct two retracts in Banach spaces and compute the topological degree for completely continuous operator by means of semi-concave functional. The results extend and complement the previous conclusions.

Sharkovskiĭ proved that the existence of a periodic orbit of period which is not a power of 2 in a one-dimensional dynamical system implies existence of infinitely many periodic orbits. We obtain an analog of Sharkovskiĭ's theorem for periodic orbits of shear homeomorphisms of the torus. This is done by obtaining a dynamical order relation on the set of simple orbits and simple pairs. We then use this order relation for a global analysis of a quantum chaotic physical system called the kicked accelerated particle.

We study elliptic Neumann problems in which the reaction term at infinity is resonant with respect to any pair $\{ \widehat{\lambda}_m, \widehat{\lambda}_{m+1}\}$ of distinct consecutive eigenvalues. Using variational methods combined with Morse theoretic techniques, we show that when the double resonance occurs in a ``nonprincipal'' spectral interval $[\widehat{\lambda}_m, \widehat{\lambda}_{m+1}]$, $ m\geq 1$, we have at least three nontrivial smooth solutions, two of which have constant sign. If the double resonance occurs in the ``principal'' spectral $[\widehat{\lambda}_0=0,\widehat{\lambda}_1]$, then we show that the problem has at least one nontrivial smooth solution.

In this paper we consider the nonlinear Dirac equation $$ -i\partial_t\psi=ic\hbar\sum^3_{k=1}\alpha_k\partial_k\psi-mc^2\beta\psi+ G_\psi(x,\psi). $$ Under suitable superlinear assumptions on the nonlinearities we can obtain the existence of at least one stationary state for the equation by applying a generalized linking theorem.

We show that if $(X,T)$ is a topological dynamical system which is deterministic in the sense of Kamiński, Siemaszko and Szymański then $(X,T^{-1})$ and $(X\times X,T\times T)$ need not be deterministic in this sense. However if $(X\times X,T\times T)$ is deterministic then $(X,T^{n})$ is deterministic for all $n\in{\mathbb{N}}\setminus\{0\}$.