We prove the existence of at least one sign-changing solution for a third-order nonlocal boundary value problem by applying Leray-Schauder Continuation Principle. To illustrate the applicability of the obtained results, we consider an example.

The main goal of the present paper is to prove the existence of saddle-type solutions for the following class of quasilinear problems $$ -\Delta_{\Phi}u + V'(u)=0\quad \text{in }\mathbb{R}^2, $$% where $$ \Delta_{\Phi}u=\text{div}(\phi(|\nabla u|)\nabla u), $$% $\Phi\colon \mathbb{R}\rightarrow [0,+\infty)$ is an N-function and the potential $V$ satisfies some technical condition and we have as an example $ V(t)=\Phi(|t^2-1|)$.

In this paper we characterize the sequentially weakly lower semicontinuity of the parameter-depending energy functional associated with the critical Kirchhoff problem in context of (sub)Riemannian manifolds. We also present some spectral gap and convexity results.

In a 1961 paper by E. Rakotch it was shown that a uniformly locally contractive mapping has a fixed point. In the present paper we show that for such a mapping, the fixed point problem is well posed and that inexact iterates of such a mapping converge to its unique fixed point, uniformly on bounded sets. Using the porosity notion, we also show that most uniformly locally nonexpansive mappings are, in fact, uniformly locally contractive.

The index $\mathrm{ind}(\mathbf{F})$ of a fixed point class $\mathbf{F}$ is a classical invariant in the Nielsen fixed point theory. In the recent paper \cite{ZZ}, the authors introduced a new invariant $\mathrm{ichr}(\mathbf{F})$ called the improved characteristic, and proved that $\mathrm{ind}(\mathbf{F})\leq \mathrm{ichr}(\mathbf{F})$ for all fixed point classes of $\pi_1$-injective selfmaps of connected finite graphs. In this note, we show that the two homotopy invariants mentioned above are exactly the same. Since the improved characteristic is totally determined by the endomorphism of the fundamental group, we give a group-theoretical approach to compute indices of fixed point classes of graph selfmaps. As a consequence, we give a new criterion of a fixed point, which extends the one due to Gaboriau, Jaeger, Levitt and Lustig.

We are concerned with of existence of solutions to the semilinear elliptic problem $$ \begin{cases} - \Delta u=\lambda_{k}u+u^3 &\text{in } \Omega, \\ u= 0 &\text{on }\partial \Omega, \end{cases} $$% in a bounded domain $\Omega \subset \mathbb{R}^{4}$. Here $\lambda_k$ is an eigenvalue of the $-\Delta$ in $H_0^1(\Omega)$. We prove that this problem has a nontrivial solution.

A theorem on computation of the homological Conley index of an isolated invariant set of the Poincaré map associated to a section in a rotating local dynamical system $ϕ$ is proved. Let $(N,L)$ be an index pair for a discretization $ϕ^h$ of $ϕ$, where $h>0$, and let $S$ denote the invariant part of $N\setminus L$; it follows that the section $S_0$ of $S$ is an isolated invariant set of the Poincaré map. The theorem asserts that if the sections $N_0$ of $N$ and $L_0$ of $L$ are ANRs, the homology classes $[u_j]$ of some cycles $u_j$ form a basis of $H(N_0,L_0)$, and for some scalars $a_{ij}$, the cycles $u_j$ and $\sum a_{ij}u_i$ are homologous in the covering pair $(\widetilde N,\widetilde L)$ of $(N,L)$ and the homology relation is preserved in $(\widetilde N,\widetilde L)$ under the transformation induced by $ϕ^t$ for $t\in [0,h]$ then the homological Conley index of $S_0$ is equal to the Leray reduction of the matrix $[a_{ij}]$. In particular, no information on the values of the Poincaré map or its approximations is required. In a special case of the system generated by a $T$-periodic non-autonomous ordinary differential equation with rational $T/h>1$, the theorem was proved in the paper M.\,Mrozek, R.\,Srzednicki, and F.\,Weilandt, SIAM J. Appl. Dyn. Syst. 14 (2015), 1348-1386, and it motivated a construction of an algorithm for determining the index.