We define the concept of a Conley index and a homology index braid class for ordinary differential equations of the form \begin{equation} \dot x= F_1(x), \tag{$E$} \end{equation} where $\mathcal{M}$ is a $C^2$-manifold and $F_1$ is the principal part of a continuous vector field on $\mathcal{M}$. This allows us to extend our previously obtained results from [M.C. Carbinatto and K.P. Rybakowski, On the suspension isomorphism for index braids in a singular perturbation problem, Topological Methods in Nonl. Analysis 32 (2008), 199-225] on singularly perturbed systems of ordinary differential equations \begin{equation} \begin{split} \varepsilon\dot y&=f(y,x,\varepsilon),\\ \dot x&=h(y,x,\varepsilon) \end{split} \tag($E_\varepsilon$) \end{equation} on $Y\times \mathcal{M}$, where $Y$ is a finite dimensional Banach space and $\mathcal{M}$ is a $C^2$-manifold, to the case where the vector field in $(E_\varepsilon)$ is continuous, but not necessarily locally Lipschitzian.

In this paper we prove the existence of a ground state solution for the nonlinear Klein-Gordon-Maxwell equations in the electrostatic case.

Existence of one dimensional traveling wave solutions $u( x,t):=\phi ( x-ct) $ at the stationary equilibria, for the nonlinear degenerate reaction-diffusion equation $u_{t}=[K( u)u_{x}]_{x}+F( u) $ is studied, where $K$ is the density coefficient and $F$ is the reactive part. We use the Conley index theory to show that there is a traveling front solutions connecting the critical points of the reaction-diffusion equations. We consider the nonlinear degenerate generalized Fisher-KPP and Nagumo equations.

We perturb the mean curvature operator and find multiple critical points of functionals that are not even. As a consequence we find infinitely many solutions for a quasilinear elliptic equation. The generality of our results are also reflected in the relaxed hypotheses related to the behavior of the functions around zero and at infinity.

Given a local diffeomorphism $f\colon \mathbb{R}^n\to \mathbb{R}^n$, we consider certain incompressibility conditions on the parallelepiped $Df(x)([0,1]^n)$ which imply that the pre-image of an affine subspace is non-empty and has trivial homotopy groups. These conditions are then used to establish criteria for $f$ to be globally invertible, generalizing in all dimensions the previous results of M. Sabatini.

In random differential equations with bounded noise minimal forward invariant (MFI) sets play a central role since they support stationary measures. We study the stability and possible bifurcations of MFI sets. In dimensions 1 and 2 we classify all minimal forward invariant sets and their codimension one bifurcations in bounded noise random differential equations.

We present a method to study twice degenerate at infinity asymptotically linear vector fields, i.e. the fields with degenerate principal linear parts and next order bounded terms. The main features of the method are sharp asymptotic expansions for projections of nonlinearities onto the kernel of the linear part. The method includes theorems in abstract Banach spaces, the expansions which are the main assumptions of these abstract theorems, and lemmas on the exact form of the expansions for generic functional nonlinearities with saturation. The method leads to several new results on solvability and bifurcations for various classic BVPs.

If the leading terms in the expansions are of order $0$, then solvability conditions (and conditions for the index at infinity to be non-zero) coincide with Landesman-Lazer conditions, traditional for the BVP theory. If the terms of order $0$ vanish (the Landesman-Lazer conditions fail), then it is necessary to determine and to take into account nonlinearities that are smaller at infinity. The presented method uses such nonlinearities and makes it possible to obtain the expansions with the leading terms of arbitrary possible orders.

The method is applicable if the linear part has simple degeneration, if the corresponding eigenfunction vanishes, and if the small nonlinearities decrease at infinity sufficiently fast. The Dirichlet BVP for a second order ODE is the main model example, scalar and vector cases being considered separately. Other applications (the Dirichlet problem for the Laplace PDE and the Neumann problem for the second order ODE) are given rather schematically.

In this paper, we study the existence of non-collision periodic solutions for the second order singular dynamical systems. We consider the systems where the potential has a repulsive or attractive type behavior near the singularity. The proof is based on Schauder's fixed point theorem involving a new type of cone. The so-called strong force condition is not needed and the nonlinearity could have sign changing behavior. We allow that the Green function is non-negative, so the critical case for the repulsive case is covered. Recent results in the literature are generalized and improved.

This paper considers the existence of periodic solutions for a class of non-autonomous differential delay equations \begin{equation} x'(t)=-\sum_{i=1}^{n-1}f(t,x(t-i\tau)), \tag{$*$} \end{equation} where $\tau> 0$ is a given constant. It is shown that under some conditions on $f$ and by using symplectic transformations, Floquet theory and some results in critical point theory, the existence of single periodic solution of the differential delay equation $(*)$ is obtained. These results generalize previous results on the cases that the equations are autonomous.

We consider a singular perturbation of the generalized viscous Cahn-Hilliard equation based on constitutive equations introduced by M. E. Gurtin and we establish the existence of a family of inertial manifolds which is continuous with respect to the perturbation parameter $\varepsilon> 0$ as $\varepsilon$ goes to 0. In a recent paper, we proved a similar result for the singular perturbation of the standard viscous Cahn-Hilliard equation, applying a construction due to X. Mora and J. Solà-Morales for equations involving linear self-adjoint operators only. Here we extend the result to the singularly perturbed Cahn-Hilliard-Gurtin equation which contains a non-self-adjoint operator. Our method can be applied to a larger class of nonlinear dynamical systems.

For a bounded domain $Z\subseteq{\mathbb{R}}^N$ with a $C^2$-boundary, we prove the existence of an ordered pair of smooth positive strong solutions for the nonlinear Dirichlet problem $$ -\Delta_p x(z) = \beta(z)x(z)^{-\eta}+f(z,x(z)) \quad \text{a.e. on } Z \text{ with } x\in W^{1,p}_0(Z), $$ which exhibits the combined effects of a singular term ($\eta\geq 0$) and a $(p-1)$-linear term $f(z,x)$ near $+\infty$, by using a combination of variational methods, with upper-lower solutions and with suitable truncation techniques.