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 $H^1_0(\Omega)$ generatedby the parabolicequation$$\begin{alignedat}{3}\partial_t u&=\Delta u+\phi(u)+\gamma(x) &\quad& x\in\Omega,&\ &t> 0,\\u(x,t)&=0&\quad& x\in \partial \Omega,&\ &t> 0.\end{alignedat}$$Let$\Gamma\colon H^2(\Omega)\toH^1_0(\Omega)\timesL^2(\Omega)$ 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 againupper-semicontinuous at $\varepsilon=0$.

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

Based on the fixed point index theory for a Banach space, nontrivialperiodic solutions are found for a class of integral equation of the form $$\phi (x)=\int_{[x,x+\omega ]\cap \Omega }K(x,y)f(y,\phi (y-\tau (y)))\,dy,\quadx\in \Omega , $$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 definedegree theory for such maps. We define degree for ${\rm SO}(2)$-equivariant transversal maps and prove some propertiesof this invariant. Moreover, we characterize ${\rm SO}(2)$-equivariant transversal isomorphisms and derive formulafor 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 positiveperiodic solutions of the second order differential equation $$u''+g(u)u'+f(t,u)=h(t) $$ where $f(t,\cdot)$ has a singularity ofrepulsive type at the origin. We use the method of lower and uppersolutions.

In this paper the nonlinear alternative of Leray-Schauder type is used to investigate theexistence of solutions for second order impulsive hyperbolic differentialinclusions 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 irrationalrotation of the unit circle. We construct an ergodic and completelysquashable 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$F\circS_\lambda$ 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 structureof submeans on certain spaces $X$ of bounded real valued functionson a semigroup and to find local conditions on $X$ in terms of submeanfor 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 existenceof such solutions in multiplyconnected domains.

An extension of the celebrated Sharkovskiĭ cycle coexistingtheorem (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 multivaluedself-maps in the sense of [L. Górniewicz, Topological Fixed Point Theory of Multivalued Mappings, Kluwer, Dordrecht,1999], on a Cartesian product of linearcontinua. Vectors of admissible self-maps have a triangularstructure 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 theresults in [J. Andres, J. Fišer and L. Jüttner, On a multivalued version of the Sharkovskiĭtheorem 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 versionof Sharkovskiĭ’s theorem, Nonlinear Anal. 51 (2002), 1101–1104], [J. Andres, L. Jüttner and K. Pastor, On a multivalued version of the Sharkovskiĭtheorem 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] (amultidimensional 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-valuedcase, 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 ofthese theorems to the geometric simplex are given. AlsoKnaster-Kuratowski-Mazurkiewicz type theorems on covering of the geometricsimplex 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 $X$ withthe Hausdorff metric. Let $F\colon X\rightarrow 2^{X}$ be an $\varepsilon$-contractive map. A general condition is given that guaranteesthe existence of a periodic point of $F$ (the theorem extends a result ofEdelstein to multi-valued maps). The condition holdswhen $X$ is compact; hence, $F$ has a periodic point when $X$ is compact. Itis shown that $F$ has a fixed point (a point $p\in F(p)$) if $X$ is acontinuum. Applications to single-valued $\varepsilon$-expansive maps are given.