A diffeomorphism on a $C^1$-smooth manifold is studied possessing a hyperbolic fixed point. If the stable and unstable manifolds of the hyperbolic fixed point have a nontrivial local topological crossing then a chaotic behaviour of the diffeomorphism is shown. A perturbed problem is also studied by showing the relationship between a corresponding Melnikov function and the nontriviality of a local topological crossing of invariant manifolds for the perturbed diffeomorphism.

We consider the motion of $n$ particles under the action of a potential $F$. Imposing appropriate conditions on $F$ we obtain quasiperiodic solutions using variational methods. A Diophantine condition on the frequency similar to those encountered in KAM theory allows us to establish the necessary properties of the corresponding functional. The solutions are then obtained by means of the mountain pass theorem on a suitable convex subset.

We consider a family of non-autonomous reaction-diffusion equations $$ u_t=\sum_{i,j=1}^N a_{ij}(\omega t)\partial_i\partial_j u+f(\omega t,u)+ g(\omega t,x), \quad x\in\mathbb R^N \tag{$\text{\rm E}_\omega$} $$ with almost periodic, rapidly oscillating principal part and nonlinear interactions. As $\omega\to \infty$, we prove that the solutions of $(\text{\rm E}_\omega)$ converge to the solutions of the averaged equation $$ u_t=\sum_{i,j=1}^N \overline a_{ij}\partial_i\partial_j u+\overline f(u)+ \overline g(x), \quad x\in\mathbb R^N. \tag{$\text{\rm E}_\infty$} $$ If $f$ is dissipative, we prove existence and upper-semicontinuity of attractors for the family (E$_\omega$) as $\omega\to\infty$.

In this paper we formulate and prove a theorem, which provides the conserved quantities of a system described by the generalized variational principle of Herglotz. This new theorem contains as a special case the classical first Noether theorem. It reduces to it when the generalized variational principle of Herglotz reduces to the classical variational principle. Several examples for applications to physics are given.

In this paper we consider a Neumann problem of the type $$ \begin{cases} -\Delta u = \alpha (x) (|u|^{q-2}u-u)+\lambda f(x,u) &\text{in } \Omega, \\ \displaystyle {\partial u\over \partial \nu}=0 &\text{on } \partial\Omega. \end{cases} \tag \hbox{$\text{\rm P}_{\lambda}$} $$ Applying the theory developed in [On a three critical points theorem, Arch. Math. (Basel) 75 (2000), 220–226], we establish, under suitable assumptions, the existence of an open interval $\Lambda\subseteq \mathbb R$ and of a positive real number $\varrho$, such that, for each $\lambda\in\Lambda$, problem (P$_{\lambda}$) admits at least three weak solutions in $W^{1,2}(\Omega)$ whose norms are less than $\varrho$.

In this work we show that, generically in the set of $\mathcal{C}^2$ bounded regions of $\mathbb R^n$, $n \geq 2$, the inequality $ \int_{\Omega} \phi^3 \neq 0$ holds for any eigenfunction of the Laplacian with either Dirichlet or Neumann boundary conditions.

The Lefschetz Fixed Point Theorem for compact absorbing contraction morphisms (${\mathbb{CAC}}$-morphisms) of retracts of open subsets in admissible spaces in the sense of Klee is proved. Moreover, the relative version of the Lefschetz Fixed Point Theorem and the Lefschetz Periodic Theorem are considered. Additionally, a full classification of morphisms with compact attractors in the non-metric case is obtained.

The reduced equivariant degree for $G=SO(3)\times S^1$ is introduced and studied in the case of one free parameter equivariant maps. Computational and multiplication tables for the reduced $SO(3)\times S^1$-equivariant degree are presented together with an application to an $SO(3)$-symmetric Hopf bifurcation problem. A method for classification of $SO(3)$-symmetric bifurcations is established.

In this paper we prove an alternative existence theorem for variational inequalities defined on an unbounded set in a Hilbert space. This theorem is based on the concept of exceptional family of elements (EFE) for a mapping and on the concept of $(0,k)$-epi mapping which is similar to the topological degree. We show that when a $k$-set field is without (EFE) then the variational inequality has a solution. Based on this result we present several classes of mappings without (EFE).

In this article, we use Palais-Smale approaches to describe the achieved and nonachieved domains. We characterizes the achieved domain by the existence of a ground state solution for the energy functional $J$ in $\Omega$.