Consider the system of equations $$ -\ddot{q} = a(t)V'(q). $$ The main goal of this paper is to present a simple minimization method to find heteroclinic connections between isolated critical points of $V$, say $0$ and $\xi$, which are local maxima but do not necessarily have the same value of $V$. In particular we prove that there exist heteroclinic solutions from $0$ to $\xi$ and from $\xi$ to $0$ for a class of positive slowly oscillating periodic functions $a$ provided $\delta = |V(0) - V(\xi)|$ is sufficiently small (and another technical condition is satisfied). Note that when $V(0) \neq V(\xi)$, $a$ cannot be constant be conservation of energy. Existence of "multi-bump" solutions is also proved.

Via non-smooth critical point theory, we prove existence of entire positive solutions for a class of nonlinear elliptic problems with asymptotic $p$-Laplacian behaviour and subjected to natural growth conditions.

The following $T^2$-equivariant problem of periodic type is considered: $$ \begin{cases} u\in C^2({\mathbb R}^2,{\mathbb R}),\\ -\varepsilon\Delta u(x,y)+F'(u(x,y))=0 & \text{in ${\mathbb R}^{2}$,}\\ u(x,y)=u(x+T,y)=u(x,y+S) &\text{for all $(x,y)\in {\mathbb R}^2$,}\\ \nabla u(x,y)=\nabla u(x+T,y)=\nabla u(x,y+S) &\text{for all $(x,y)\in {\mathbb R}^{2}$.} \end{cases}\tag{\text{P}} $$ Using a suitable version of Morse theory for equivariant problems, it is proved that an arbitrarily great number of orbits of solutions to (P) is founded, choosing $\varepsilon> 0$ suitably small. Each orbit is homeomorphic to $S^1$ or to $T^2$.

In the framework of critical point theory for continuous functionals defined on metric spaces, we give a new, simpler proof of the so-called Second Deformation Lemma, a basic tool of Morse theory.

In this paper, we introduce some aspects of a critical point theory for multivalued functions $\Phi : E \to {\mathbb R}^{\mathbb N} \cup \{\infty\}$ defined on $E$ a complete gauge space and with closed graph. The existence of a critical point is established in presence of linking. Finally, we present applications of this theory to semilinear elliptic problems on ${\mathbb R}^N$.

We discuss the asymptotic behaviour of solutions for the nonlocal quasilinear hyperbolic problem of Kirchhoff Type $$ u_{tt}-\phi (x)\Vert\nabla u(t)\Vert^{2}\Delta u+\delta u_{t} = |u|^{a}u,\quad x\in {\mathbb R}^N,\ t\geq 0, $$ with initial conditions $ u(x,0) = u_0 (x)$ and $u_t(x,0) = u_1 (x)$, in the case where $N \geq 3$, $\delta \geq 0$ and $(\phi (x))^{-1} =g (x)$ is a positive function lying in $L^{N/2}(\mathbb R^N)\cap L^{\infty}(\mathbb R^N )$. When the initial energy $ E(u_{0},u_{1})$, which corresponds to the problem, is non-negative and small, there exists a unique global solution in time. When the initial energy $E(u_{0},u_{1})$ is negative, the solution blows-up in finite time. A combination of the modified potential well method and the concavity method is widely used.

In this paper, for nonlinear Riemann–Hilbert problems in doubly connected domains with smooth as well as Lipschitz continuous boundary data, existence of at least two topologically different solutions is established. The main tools are the topological degree of quasi-ruled Fredholm mappings, Montel’s theorem, a priori estimates and the employment of classical modular function theory.

The aim of this paper is to prove that every Borsuk continuous set-valued map of the closed ball in the 3-dimensional Euclidean space, taking values which are one point sets or knots, has a fixed point. This result is a special case of the Górniewicz Conjecture.

The criterion for the stability of a fixed point of a compact or condensing multimap in a Banach space with respect to a small perturbation is expressed in terms of its topological index.

We consider a class of multimaps which are the composition of a superposition multioperator ${\mathcal P}_F$ generated by a nonconvex-valued almost lower semicontinuous nonlinearity $F$ and an abstract solution operator $S$. We prove that under some suitable conditions such multimaps are condensing with respect to a special vector-valued measure of noncompactness and construct a topological degree theory for this class of multimaps yielding some fixed point principles. It is shown how abstract results can be applied to semilinear inclusions, inclusions with $m$-accretive operators and time-dependent subdifferentials, nonlinear evolution inclusions and integral inclusions in Banach spaces.

In this paper we deal with the general initial-boundary value problem for a second order nonlinear nonstationary evolution equation. The associated operator equation is studied by the Fredholm and Nemitskiĭ operator theory. Under local Hölder conditions for the nonlinear member we observe quantitative and qualitative properties of the set of solutions of the given problem. These results can be applied for the different mechanical and natural science models.

We prove the parallelogram inequalities in metric spaces and use them to obtain the fixed points of involutions.