For an arbitrary unbounded domain $\Omega\subset\mathbb R^3$ and for $\varepsilon> 0$, we consider the damped hyperbolic equations \begin{equation} \varepsilon u_{tt}+ u_t+\beta(x)u- \sum_{ij}(a_{ij}(x) u_{x_j})_{x_i}=f(x,u), \tag{$(\text{\rm H}_\varepsilon)$} \end{equation} with Dirichlet boundary condition on $\partial\Omega$, and their singular limit as $\varepsilon\to0$. Under suitable assumptions, (H$_\varepsilon)$ possesses a compact global attractor $\mathcal A_\varepsilon$ in $H^1_0(\Omega)\times L^2(\Omega)$, while the limiting parabolic equation possesses a compact global attractor $\widetilde{\mathcal A_0}$ in $H^1_0(\Omega)$, which can be embedded into a compact set ${\mathcal A_0}\subset H^1_0(\Omega)\times L^2(\Omega)$. We show that, as $\varepsilon\to0$, the family $({\mathcal A_\varepsilon})_{\varepsilon\in[0,\infty[}$ is upper semicontinuous with respect to the topology of $H^1_0(\Omega)\times H^{-1}(\Omega)$.

We consider the $m$-point boundary value problem consisting of the equation \begin{equation} -\phi_p (u')'=f(u), \quad \text{on $(0,1)$},\tag{$1$} \end{equation} together with the boundary conditions \begin{equation} u(0) = 0,\quad u(1) = \sum^{m-2}_{i=1}\alpha_i u(\eta_i) ,\tag{$2$} \end{equation} where $p> 1$, $\phi_p(s) := |s|^{p-1} \text{\rm sgn} s$, $s \in {\mathbb R}$, $m \ge 3$, $\alpha_i , \eta_i \in (0,1)$, for $i=1,\dots,m-2$, and $\sum^{m-2}_{i=1} \alpha_i < 1$. We assume that the function $f \colon {\mathbb R} \to{\mathbb R}$ is continuous, satisfies $sf(s) > 0$ for $s \in {\mathbb R} \setminus \{0\}$, and that $f_0 := \lim_{\xi \rightarrow 0}{f(\xi)}/{\phi_p(\xi)} > 0$.

Closely related to the problem (1), (2), is the spectral problem consisting of the equation \begin{equation} -\phi_p (u')' = \lambda \phi_p(u) , \tag{$3$} \end{equation} together with the boundary conditions (2). It will be shown that the spectral properties of (2), (3), are similar to those of the standard Sturm-Liouville problem with separated (2-point) boundary conditions (with a minor modification to deal with the multi-point boundary condition). The topological degree of a related operator is also obtained. These spectral and degree theoretic results are then used to prove a Rabinowitz-type global bifurcation theorem for a bifurcation problem related to the problem (1), (2). Finally, we use the global bifurcation theorem to obtain nodal solutions of (1), (2), under various conditions on the asymptotic behaviour of $f$.

We prove the existence of a large positive solution for the boundary value problems $$ \begin{alignat}{2} -\Delta u &=\lambda (-h(u)+g(x,u))&\quad& \text{in }\Omega , \\ u &=0 &\quad &\text{on }\partial \Omega , \end{alignat} $$ where $\Omega $ is a bounded domain in ${\mathbb R}^{N}$, $\lambda $ is a positive parameter, $g(x,\cdot)$ is sublinear at $\infty$, and $h$ is allowed to become $\infty $ at $u=0$. Uniqueness is also considered.

Multiple nodal solutions are obtained for the elliptic problem $$ \begin{alignat}{2} -\Delta u&=f(x, u)+\varepsilon g(x, u)&\quad& \text{in } \Omega,\\ u&=0&\quad& \text{on } \partial \Omega , \end{alignat} $$ where $\varepsilon $ is a parameter, $\Omega $ is a smooth bounded domain in ${{\mathbb R}}^{N}$, $f\in C(\overline{\Omega }\times {{\mathbb R}})$, and $g\in C(\overline{\Omega }\times {{\mathbb R}})$. For a superlinear $C^{1}$ function $f$ which is odd in $u$ and for any $C^{1}$ function $g$, we prove that for any $j\in {\mathbb N}$ there exists $\varepsilon _{j}> 0$ such that if $|\varepsilon |\leq \varepsilon _{j}$ then the above problem possesses at least $j$ distinct nodal solutions. Except $C^{1}$ continuity no further condition is needed for $g$. We also prove a similar result for a continuous sublinear function $f$ and for any continuous function $g$. Results obtained here refine earlier results of S. J. Li and Z. L. Liu in which the nodal property of the solutions was not considered.

We prove the large time existence of solutions to the Navier-Stokes equations with slip boundary conditions in a cylindrical domain. Assuming smallness of $L_2$-norms of derivatives of initial velocity with respect to variable along the axis of the cylinder, we are able to obtain estimate for velocity in $W^{2,1}_2$ without restriction on its magnitude. Then existence follows from the Leray-Schauder fixed point theorem.

We study the existence of solutions of a nonlinear quadratic Volterra integral equation of fractional order. In our considerations we apply the technique of measures of noncompactness in conjunction with the classical Schauder fixed point principle. The mentioned equation is considered in the Banach space of real functions defined, continuous and bounded on an unbounded interval. We will show that solutions of the investigated integral equation are locally attractive.

We are interested in the existence of weak solutions to steady non-Newtonian fluids with nonstandard growth conditions of the Cauchy stress tensor. Since the $L^p$ framework is not suitable to capture the description of strongly inhomogeneous fluids, we formulate the problem in generalized Orlicz spaces. The existence proof consists in showing that for Galerkin approximations the sequence of symmetric gradients of the flow velocity converges modularly. As an example of motivation for considering non-Newtonian fluids in generalized Orlicz spaces we recall the smart fluids.

In this paper, we shall establish sufficient conditions for the existence of solutions for a class of boundary value problem for fractional differential inclusions involving the Caputo fractional derivative of order $\alpha\in (1,2]$. The both cases of convex valued and nonconvex valued right hand side are considered.

The second order Hamiltonian system $\ddot{q}+V_{q}(t,q)=f(t)$, where $t\in\mathbb R$ and $q\in\mathbb R^n$, is considered. We assume that a potential $V\in C^{1}(\mathbb R\times\mathbb R^n,\mathbb R)$ is of the form $V(t,q)=-K(t,q)+W(t,q)$, where $K$ satisfies the pinching condition and $W_{q}(t,q)=o(|q|)$, as $|q|\to 0$ uniformly with respect to $t$. It is also assumed that $f\in C(\mathbb R,\mathbb R^n)$ is non-zero and sufficiently small in $L^{2}(\mathbb R,\mathbb R^n)$. In this case $q\equiv 0$ is not a solution. Therefore there are no orbits homoclinic to $0$ in a classical sense. However, we show that there is a solution emanating from $0$ and terminating at $0$. We are to call such a solution almost homoclinic to $0$. It is obtained here as a weak limit in $W^{1,2}(\mathbb R,\mathbb R^n)$ of a sequence of almost critical points.

We obtain several results on shadowing and inverse shadowing for set-valued dynamical systems that have a contractive property. Applications to $T$-flows of differential inclusions are discussed.

We introduce a new hyperbolicity condition for set-valued dynamical systems and show that this condition implies the shadowing and inverse shadowing properties.

Let ${\mathcal P}_\varepsilon\in C^0({\mathbb R}^n,{\mathbb R}^n)$ be the Poincaré-Andronov operator over period $T> 0$ of $T$-periodically perturbed autonomous system $\dot x=f(x)+\varepsilon g(t,x,\varepsilon)$, where $\varepsilon> 0$ is small. Assuming that for $\varepsilon=0$ this system has a $T$-periodic limit cycle $x_0$ we evaluate the topological degree $d(I-{\mathcal P}_\varepsilon,U)$ of $I-{\mathcal P}_\varepsilon$ on an open bounded set $U$ whose boundary $\partial U$ contains $x_0([0,T])$ and ${\mathcal P}_0(v)\not=v$ for any $v\in \partial U\setminus x_0([0,T])$. We give an explicit formula connecting $d(I-{\mathcal P}_\varepsilon,U)$ with the topological indices of zeros of the associated Malkin's bifurcation function. The goal of the paper is to prove the Mawhin's conjecture claiming that $d(I-{\mathcal P}_\varepsilon,U)$ can be any integer in spite of the fact that the measure of the set of fixed points of ${\mathcal P}_0$ on $\partial U$ is zero.

In the present paper we obtain a new homological version of the implicit function theorem and some versions of the Darboux theorem. Such results are proved for continuous maps on topological manifolds. As a consequence, some versions of these classic theorems are proved when we consider differenciable (not necessarily $C^{1}$) maps.

It is shown that if ${\mathcal N}$ and ${\mathcal N}'$ are natural families of factors (in the sense of [E. Glasner, M. K. Mentzen and A. Siemaszko, A natural family of factors for minimal flows, Contemp. Math. 215 (1998), 19–42]) for minimal flows $(X,T)$ and $(X',T')$, respectively, then $\{R\otimes R'\colon R\in{\mathcal N},R'\in{\mathcal N}'\}$ is a natural family of factors for the product $\mathbb{Z}^2$-action on $X\times X'$ generated by $T$ and $T'$. An example is given showing the existence of topologically disjoint minimal flows $(X,T)$ and $(X',T')$ for which the family of factors of the flow $(X\times X',T\times T')$ is strictly bigger than the family of factors of the product $\mathbb{Z}^2$-action on $X\times X'$ generated by $T$ and $T'$. There is also an example of a minimal distal system with no nontrivial compact subgroups in the group of its automorphisms.