We consider the singularly perturbed system 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 normed space and $\mathcal{M}$ is a smooth manifold. We assume that there is a reduced manifold of $(E_\varepsilon)$ given by the graph of a function $\phi\colon \mathcal{M}\to Y$ and satisfying an appropriate hyperbolicity assumption with unstable dimension $k\in{\mathbb N}_0$. We prove that every Morse decomposition $(M_p)_{p\in P}$ of a compact isolated invariant set $S_0$ of the reduced equation $$ \dot x=h(\phi(x),x,0) $$ gives rises, for $\varepsilon> 0$ small, to a Morse decomposition $(M_{p,\varepsilon})_{p\in P}$ of an isolated invariant set $S_\varepsilon$ of $(E_\varepsilon)$ such that $(S_\varepsilon,(M_{p,\varepsilon})_{p\in P})$ is close to $(\{0\}\times S_0,(\{0\}\times M_p)_{p\in P})$ and the (co)homology index braid of $(S_\varepsilon,(M_{p,\varepsilon})_{p\in P})$ is isomorphic to the (co)homology index braid of $(S_0,(M_{p})_{p\in P})$ shifted by $k$ to the left.

In this paper we deal with impulsive Cauchy problems in Banach spaces governed by a delay semilinear differential inclusion $y'\in A(t)y + F(t,y_t)$. The family $\{A(t)\}_{t\in [0,b]}$ of linear operators is supposed to generate an evolution operator and $F$ is an upper Carathéodory type multifunction. We first provide the existence of mild solutions on a compact interval and the compactness of the solution set. Then we apply this result to obtain the existence of mild solutions for the impulsive Cauchy problem on non-compact intervals.

In this note we investigate in Banach spaces the existence of mild solutions for initial problems, also in presence of impulses, governed by semilinear differential inclusions where the non-linear part is a Scorza-Dragoni multifunction. All the results are obtained via a generalization of Artstein-Prikry selection theorem that we obtain in the first part of the paper.

In this paper, we first present an impulsive version of Filippov's Theorem for first-order semilinear functional differential inclusions of the form: $$ \begin{cases} (y'-Ay) \in F(t,y_t) &\text{a.e. } t\in J\setminus \{t_{1},\ldots,t_{m}\}, \\ y(t^+_{k})-y(t^-_k)=I_{k}(y(t_{k}^{-})) &\text{for } k=1,\ldots,m, \\ y(t)=\phi(t) &\text{for } t\in[-r,0], \end{cases} $$ where $J=[0,b]$, $A$ is the infinitesimal generator of a $C_0$-semigroup on a separable Banach space $E$ and $F$ is a set-valued map. The functions $I_k$ characterize the jump of the solutions at impulse points $t_k$ ($k=1,\ldots,m$). Then the convexified problem is considered and a Filippov-Ważewski result is proved. Further to several existence results, the topological structure of solution sets - closeness and compactness - is also investigated. Some results from topological fixed point theory together with notions of measure on noncompactness are used. Finally, some geometric properties of solution sets, AR, $R_\delta$-contractibility and acyclicity, corresponding to Aronszajn-Browder-Gupta type results, are obtained.

We give a sufficient condition for the existence of an isolating block $B$ for an isolated invariant set $S$ such that the inclusion induced map in cohomology $H^* (B)\to H^*(S)$ is an isomorphism. We discuss the Easton's result concerning the special case of flows on a $3$-manifold. We prove that if $S$ is an isolated invariant set for a flow on a $3$-manifold and $S$ is of finite type, then each isolating neighbourhood of $S$ contains an isolating block $B$ such that $B$ and $B^-$ are manifolds with boundary and the inclusion induced map in cohomology is an isomorphism.

It is known that the semigroup generated by the initial-boundary value problem for the perturbed viscous Cahn-Hilliard equation with $\varepsilon> 0$ as a parameter admits a global attractor $\mathcal{A}_{\varepsilon}$ in the phase space $X^{{1}/{2}} =(H^2(\Omega)\cap H^{1}_{0}(\Omega))\times L^2(\Omega)$, $\Omega\subset \mathbb{R}^n$, $n\leq 3$ (see [M. B. Kania, Global attractor for the perturbed viscous Cahn–Hilliard equation, Colloq. Math. 109 (2007), 217–229]). In this paper we show that the family $\{\mathcal{A}_{\varepsilon}\}_{\varepsilon\in[0,1]}$ is upper semicontinuous at $0$, which means that the Hausdorff semidistance $$ d_{X^{{1}/{2}}}(\mathcal{A}_{\varepsilon},\mathcal{A}_0)\equiv \sup_{\psi\in \mathcal{A}_{\varepsilon}}\inf_{\phi\in\mathcal{A}_{0}}\| \psi-\phi\|_{X^{{1}/{2}}}, $$ tends to $0$ as $\varepsilon\to 0^{+}$.

In this paper we prove the existence and some time regularity of weak solutions to a three-dimensional (3-D) Cahn-Hilliard system coupled with nonstationary elasticity. Such nonlinear parabolic-hyperbolic system arises as a model of phase separation in deformable alloys. The regularity result is based on the analysis of time differentiated problem by means of the Faedo-Galerkin method. The obtained regularity provides a first step to the proof of strong solvability of the problem to be presented in a forthcoming paper [I. Pawłow, W. M. Zajączkowski, Strong solvability of 3-D Cahn–Hilliard system in elastic solids, Math. Methods Appl. Sci.].

A solid combinatorial theory is presented. The generalized Sperner lemma for chains is derived from the combinatorial Stokes formula. Many other generalizations follow from applications of an $n$-index of a labelling defined on chains with values in primoids. Primoids appear as the most general structure for which Sperner type theorems can be formulated. Their properties and various examples are given. New combinatorial theorems for primoids are proved. Applying them to different primoids the well-known classic results of Sperner, Fan, Shapley, Lee and Shih are obtained.

In the article: “A nonlinear problem for age-structured population dynamics with spatial diffusion”, Lemma 4.2 is not correct under its present form.

Theorem 1.2, page 142, is wrong as stated. We give a counterexample and present a convenient hypothesis on the nonlinearity $f$ under which the theorem and its proof are correct. The hypothesis is satisfied by all examples in the rest of the paper.