Let $\Omega\subset \mathbb R^N$, $N\le 3$, be a bounded domain with smooth boundary, $\gamma\in L^2(\Omega)$ be arbitrary and $\phi\colon \mathbb R\to \mathbb R$ be a $C^1$-function satisfying a subcritical growth condition. For every $\varepsilon\in]0,\infty[$ consider the semiflow $\pi_\varepsilon$ on $H^1_0(\Omega)\times L^2(\Omega)$ generated by the damped wave equation $$ \begin{alignedat}{3} \varepsilon \partial_{tt}u+\partial_t u&=\Delta u+\phi(u)+\gamma(x) &\quad& x\in\Omega,&\ &t> 0,\\ u(x,t)&=0&\quad& x\in \partial \Omega,&\ &t> 0. \end{alignedat} $$ Moreover, let $\pi'$ be the semiflow on $H^1_0(\Omega)$ generated by the parabolic equation $$ \begin{alignedat}{3} \partial_t u&=\Delta u+\phi(u)+\gamma(x) &\quad& x\in\Omega,&\ &t> 0,\\ u(x,t)&=0&\quad& x\in \partial \Omega,&\ &t> 0. \end{alignedat} $$ Let $\Gamma\colon H^2(\Omega)\to H^1_0(\Omega)\times L^2(\Omega)$ be the imbedding $u\mapsto (u,\Delta u+\phi(u)+\gamma)$. We prove in this paper that every compact isolated $\pi'$-invariant set $K'$ lies in $H^2(\Omega)$ and the imbedded set $K_0=\Gamma(K')$ continues to a family $K_\varepsilon$, $\varepsilon\ge0$ small, of isolated $\pi_\varepsilon$-invariant sets having the same Conley index as $K'$. This family is upper-semicontinuous at $\varepsilon=0$. Moreover, any (partially ordered) Morse-decomposition of $K'$, imbedded into $H^1_0(\Omega)\times L^2(\Omega)$ via $\Gamma$, continues to a family of Morse decompositions of $K_\varepsilon$, for $\varepsilon\ge 0$ small. This family is again upper-semicontinuous at $\varepsilon=0$.

These results extend and refine some upper semicontinuity results for attractors obtained previously by Hale and Raugel.

Based on the fixed point index theory for a Banach space, nontrivial periodic solutions are found for a class of integral equation of the form $$ \phi (x)=\int_{[x,x+\omega ]\cap \Omega }K(x,y)f(y,\phi (y-\tau (y)))\,dy, \quad x\in \Omega , $$ where $\Omega $ is a closed subset of $\mathbb R^{N}$ with perioidc structure.

The aim of this article is to introduce a new class ${\rm SO}(2)$-equivariant transversal maps $\mathcal{TR}({\rm cl}(\Omega),\partial \Omega)$ and to define degree theory for such maps. We define degree for ${\rm SO}(2)$-equivariant transversal maps and prove some properties of this invariant. Moreover, we characterize ${\rm SO}(2)$-equivariant transversal isomorphisms and derive formula for degree of such isomorphisms.

We shall study the existence of time-periodic solutions for a semilinear wave equation with a given time-independent nonlinear perturbation and small forcing. Since the distribution of eigenvalues of the linear part varies with the period, the solvability of the problem depends essentially on the frequency. The main idea of this paper is to consider the situation where the period is not prescribed and hence treated as a parameter. The description of the distribution of eigenvalues as a function of the period enables us to show that under certain conditions the interaction between the nonlinearity and the spectrum of the wave operator induces multiple solutions. Our basic new result states that the autonomous equation admits at least two nontrivial solutions (free vibrations) for a restricted (but infinite) set of periods such that the nonlinearity interacts with one simple eigenvalue. As a corollary we prove that the semilinear wave equation with time-independent nonlinearity and small forcing admits an infinite sequence of pairs of periodic solutions with corresponding period tending to zero. The results are obtained via generalized topological degree theory.

In this note, we study the existence of positive periodic solutions of the second order differential equation $$ u''+g(u)u'+f(t,u)=h(t) $$ where $f(t,\cdot)$ has a singularity of repulsive type at the origin. We use the method of lower and upper solutions.

In this paper the nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions for second order impulsive hyperbolic differential inclusions with variable times.

Let $\alpha$ be an irrational number and the trasformation $$ Tx \mapsto x+\alpha\,{\rm mod}\,1, \quad x\in [0,1), $$ represent an irrational rotation of the unit circle. We construct an ergodic and completely squashable smooth real extension, i.e. we find a real analytic or $k$ time continuously differentiable real function $F$ such that for every $\lambda\neq 0$ there exists a commutor $S_\lambda$ of $T$ such that $F\circ S_\lambda$ is $T$-cohomologous to $\lambda\varphi$ and the skew product $T_F(x,y) = (Tx, y+F(x))$ is ergodic.

The purpose of this paper is to study some algebraic structure of submeans on certain spaces $X$ of bounded real valued functions on a semigroup and to find local conditions on $X$ in terms of submean for the existence of a left invariant mean.

We prove here the multiplicity results for the solutions of compact $H$-surfaces in Euclidean space. Some minimax methods and topological arguments are used for the existence of such solutions in multiply connected domains.

An extension of the celebrated Sharkovskiĭ cycle coexisting theorem (see [Coexistence of cycles of a continuous map of a line into itself, Ukrain. Math. J. 16 (1964), 61–71]) is given for (strongly) admissible multivalued self-maps in the sense of [L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Kluwer, Dordrecht, 1999], on a Cartesian product of linear continua. Vectors of admissible self-maps have a triangular structure as in [P. E. Kloeden, On Sharkovsky’s cycle coexisting ordering, Bull. Austral. Math. Soc. 20 (1979), 171–177]. Thus, we make a joint generalization of the results in [J. Andres, J. Fišer and L. Jüttner, On a multivalued version of the Sharkovskiĭ theorem and its application to differential inclusions, Set-Valued Anal. 10 (2002), 1–14], [J. Andres and L. Jüttner, Period three plays a negative role in a multivalued version of Sharkovskiĭ’s theorem, Nonlinear Anal. 51 (2002), 1101–1104], [J. Andres, L. Jüttner and K. Pastor, On a multivalued version of the Sharkovskiĭ theorem and its application to differential inclusions II] (a multivalued case), in [P. E. Kloeden, On Sharkovsky’s cycle coexisting ordering, Bull. Austral. Math. Soc. 20 (1979), 171–177] (a multidimensional case), and in [H. Schirmer, A topologist’s view of Sharkovskiĭ’s theorem, Houston, J. Math. 11 (1985), 385–395] (a linear continuum case). The obtained results can be applied, unlike in the single-valued case, to differential equations and inclusions.

Sperner lemma type theorems are proved for nonoriented primoids and pseudomanifolds. A rank function of a primoid is defined. Applications of these theorems to the geometric simplex are given. Also Knaster-Kuratowski-Mazurkiewicz type theorems on covering of the geometric simplex are presented.

Let $(X,d)$ be a nonempty metric space, and let $(2^{X},H_{d})$ be the hyperspace of all nonempty compact subsets of $X$ with the Hausdorff metric. Let $F\colon X\rightarrow 2^{X}$ be an $\varepsilon$-contractive map. A general condition is given that guarantees the existence of a periodic point of $F$ (the theorem extends a result of Edelstein to multi-valued maps). The condition holds when $X$ is compact; hence, $F$ has a periodic point when $X$ is compact. It is shown that $F$ has a fixed point (a point $p\in F(p)$) if $X$ is a continuum. Applications to single-valued $\varepsilon$-expansive maps are given.