We consider the variational system $-\Delta u = \lambda ( \nabla F)(u)$ in $\Omega$, $u = 0$ on $\partial \Omega$, where $\Omega$ is a bounded region in $\mathbb R^m$ ($m \geq 1$) with $C^1$ boundary, $\lambda$ is a positive parameter, $u\colon\Omega \rightarrow \mathbb R^N$ ($N > 1$), and $\Delta$ denotes the Laplace operator. Here $F\colon \mathbb R^N \rightarrow \R$ is a function of class $C^2$. Using variational methods, we show how changes in the sign of $F$ lead to multiple positive solutions.

We study the Hausdorff distance between a set and its convex hull. Let $X$ be a Banach space, define the CHD-constant of the space $X$ as the supremum of this distance over all subsets of the unit ball in $X$. In the case of finite dimensional Banach spaces we obtain the exact upper bound of the CHD-constant depending on the dimension of the space. We give an upper bound for the CHD-constant in $L_p$ spaces. We prove that the CHD-constant is not greater than the maximum of Lipschitz constants of metric projection operators onto hyperplanes. This implies that for a Hilbert space the CHD-constant equals $1$. We prove a characterization of Hilbert spaces and study the contractibility of proximally smooth sets in a uniformly convex and uniformly smooth Banach space.

We prove the existence of infinitely many solutions to a class of non-symmetric Dirichlet problems with exponential nonlinearities. Here the domain $\Omega \Subset \mathbb{R}^{2l}$ where $2l$ is also the order of the equation. Considered are the problem with no symmetry requirements, the radial problem on an annulus, and the radial problem on a ball with a Hardy potential term of critical Hardy exponent. These generalize results obtained by Sugimura in [Nonlinear Anal. 22 (1994), 277-293].

We state existence theorems for almost periodic solutions of evolution problems, namely, quasi-autonomous problems and more generally, time dependent evolution equations. We apply these theorems firstly, to a boundary value quasilinear hyperbolic equation of first order, and secondly, to a boundary value quasi-parabolic equation.

We consider a large class of geodesic metric spaces, including Banach spaces, hyperbolic spaces and geodesic $\mathrm{CAT}(\kappa)$-spaces, and investigate the space of nonexpansive mappings on either a convex or a star-shaped subset in these settings. We prove that the strict contractions form a negligible subset of this space in the sense that they form a $\sigma$-porous subset. For certain separable and complete metric spaces we show that a generic nonexpansive mapping has Lipschitz constant one at typical points of its domain. These results contain the case of nonexpansive self-mappings and the case of nonexpansive set-valued mappings as particular cases.

We consider continuous maps of compact metric spaces. It is proved that every pseudotrajectory with sufficiently small errors contains a subsequence of positive density that is point-wise close to a subsequence of an exact trajectory with the same indices. Also, we study homeomorphisms such that any pseudotrajectory can be shadowed by a finite number of exact orbits. In terms of numerical methods this property (we call it multishadowing) implies possibility to calculate minimal points of the dynamical system. We prove that for the non-wandering case multishadowing is equivalent to density of minimal points. Moreover, it is equivalent to existence of a family of $\varepsilon$-networks ($\varepsilon > 0$) whose iterations are also $\varepsilon$-networks. Relations between multishadowing and some ergodic and topological properties of dynamical systems are discussed.

The aim of this paper is the study of existence of solutions for nonlinear $2n^{\rm th}$-order difference equations involving $p$-Laplacian. In the first part, the existence of a nontrivial homoclinic solution for a discrete $p$-Laplacian problem is proved. The proof is based on the mountain-pass theorem of Brezis and Nirenberg. Then, we study the existence of multiple solutions for a discrete $p$-Laplacian boundary value problem. In this case the proof is based on the three critical points theorem of Averna and Bonanno.

We study the global solvability and dynamical behaviour of the modified Cahn-Hilliard equation with biological applications in the Sobolev space $H^1(\mathbb{R}^N)$.

We consider the existence of almost periodic solutions to differential equations by using coincidence degree theory. A new equivalent spectral condition for the compactness of integral operators on almost periodic function spaces is established. It is shown that semigroup conditions are crucial in applications.

We aim to introduce the generalized recurrence into the theory of impulsive semidynamical systems. Similarly to Auslander's construction in [J. Auslander, Generalized recurrence in dynamical systems, Contrib. Differential Equations 3 (1964), 65-74], we present two different characterizations, respectively, by Lyapunov functions and higher prolongations. In fact, we show that if the phase space is a locally compact separable metric space, then the generalized recurrent set is the same as the quasi prolongational recurrent set. Also, we see that many new phenomena appear for the impulse effects in the semidynamical system.

We are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem: \begin{equation*} \begin{cases} \displaystyle - \bigg(\varepsilon ^2a + \varepsilon b\int_{{\mathbb{R}^3}}{|\nabla u{|^2}} dx \bigg)\Delta u + u = Q(x)|u|^{p-2}u,& x\in \mathbb{R}^3, \hfill \\ u \in H^1(\mathbb{R}^3), \quad u > 0, & x\in \mathbb{R}^3 , \end{cases} \end{equation*} where $\varepsilon> 0 $ is a parameter, $a, b> 0$ are constants, $p\in (2, 6)$, and $Q\in C(\mathbb{R}^3)$ is a nonnegative function. We show how the profile of $Q$ affects the number of positive solutions when $\varepsilon $ is sufficiently small.

Van der Pol equation (in short, vdP) as well as many its non-symmetric generalizations (the so-called van der Pol-like oscillators (in short, vdPl)) serve as nodes in coupled networks modeling real-life phenomena. Symmetric properties of periodic regimes of networks of vdP/vdPl depend on symmetries of coupling. In this paper, we consider $N^3$ identical vdP/vdPl oscillators arranged in a cubical lattice, where opposite faces are identified in the same way as for a $3$-torus. Depending on which nodes impact the dynamics of a given node, we distinguish between $\mathbb D_N \times \mathbb D_N \times \mathbb D_N$-equivariant systems and their $\mathbb Z_N \times \mathbb Z_N \times \mathbb Z_N$-equivariant counterparts. In both settings, the local equivariant Hopf bifurcation together with the global existence of periodic solutions with prescribed period and symmetry, are studied. The methods used in the paper are based on the results rooted in both equivariant degree theory and (equivariant) singularity theory.

We consider a semilinear Robin problem driven by Laplacian plus an indefinite and unbounded potential. The reaction function contains a concave term and a perturbation of arbitrary growth. Using a variant of the symmetric mountain pass theorem, we show the existence of smooth nodal solutions which converge to zero in $C^1(\overline{\Omega})$. If the coefficient of the concave term is sign changing, then again we produce a sequence of smooth solutions converging to zero in $C^1(\overline{\Omega})$, but we cannot claim that they are nodal.

Topological complexity for spaces was introduced by M. Farber as a minimal number of continuity domains for motion planning algorithms. It turns out that this notion can be extended to the case of not necessarily commutative $C^*$-algebras. Topological complexity for spaces is closely related to the Lusternik-Schnirelmann category, for which we do not know any noncommutative extension, so there is no hope to generalize the known estimation methods, but we are able to evaluate the topological complexity for some very simple examples of noncommutative $C^*$-algebras.

We consider the following quasilinear elliptic equation with convex and concave nonlinearities: \begin{equation*} -\Delta_p u-(\Delta_pu^2)u+V(x)|u|^{p-2}u=\lambda K(x) |u|^{q-2}u+\mu g(x,u),\quad \text{in }\mathbb{R}^N, \end{equation*} where $2\leq p< N$, $1< q< p$, $\lambda,\mu\in\mathbb{R}$, $V$ and $K$ are potential functions, and $g\in C(\mathbb{R}^N\times\mathbb{R},\mathbb{R})$ is a continuous function. Under some suitable conditions on $V,K$ and $g$, the existence of infinitely many solutions is established.

We deal with the existence of solutions to a critical elliptic system involving the fractional Laplacian operator. We consider the primitive of the nonlinearity interacting with the spectrum of the operator. The one side resonant case is also considered. Variational methods are used to obtain the existence, and our result improves earlier results of the authors.