This paper is devoted to study the existence of periodic solutions to the second-order differential equation $u''+f(u)u'+g(u)=h(t,u)$, where $h$ is a Carathéodory function and $f,g$ are continuous functions on $(0,\infty)$ which may have singularities at zero. The repulsive case is considered. By using Schaefer's fixed point theorem, new conditions for existence of periodic solutions are obtained. Such conditions are compared with those existent in the related literature and applied to the Rayleigh-Plesset equation, a physical model for the oscillations of a spherical bubble in a liquid under the influence of a periodic acoustic field. Such a model has been the main motivation of this work.

We discuss the class of $u_0$-positive linear operators relative to two cones and use a comparison theorem for this class to give some short proofs of new fixed point index results for some nonlinear operators that arise from boundary value problems. In particular, for some types of boundary conditions, especially nonlocal ones, we obtain a new existence result for multiple positive solutions under conditions which depend solely on the positive eigenvalue of a linear operator. We also treat some problems where the nonlinearity $f(t,u)$ is singular at $u=0$.

In this paper, we establish the existence of many rotationally non-equivalent and nonradial solutions for the following class of quasilinear problems \begin{equation} \begin{cases} -\Delta_{N} u = \lambda f(|x|,u) &x\in \Omega_r,\\ u > 0 &x\in \Omega_r,\\ u=0 &x\in \partial\Omega_r, \end{cases} \tag{$\textrm{P}$} \end{equation} where $\Omega_r = \{ x \in \mathbb{R}^{N}: r < |x| < r+1\}$, $N \geq 2$, $N\neq 3$, $r > 0$, $\lambda > 0$, $\Delta_{N}u= {\rm div}(|\nabla u|^{N-2}\nabla u ) $ is the $N$-Laplacian operator and $f$ is a continuous function with exponential critical growth.

We consider the nonlinear system of partial differential equations describing the thermoviscoelastic medium ocupied a bounded domain $\Omega\subset\mathbb{R}^3$. We proved the global existence (in time) of solution for the nonlinear thermoviscoelasticity system for the initial-boundary value problem with the Dirichlet boundary conditions for the displacement vector and the heat flux at the boundary. In the proof we assume some growth conditions on nonlinearity and some smallness conditions on data in some norms.

We consider a class of minimal Cantor systems that up to conjugacy contains all systems strong orbit equivalent to a given system. We define a metric on this strong orbit equivalence class and prove several properties about the resulting metric space including that the space is complete and separable but not compact. If the strong orbit equivalence class contains a finite rank system, we show that the set of finite rank systems is residual in the metric space. The final result shown is that the set of systems with zero entropy is residual in every strong orbit equivalence class of this type.

We construct the Conley index over a phase space for flows. Our definition is an alternative for the Conley index over a base defined in [M. Mrozek, J.F. Reineck and R. Srzednicki, The Conley index over a base, Trans. Amer. Math. Soc. 352 (2000), 4171-4194]. We also compare it to other Conley-type indices and prove its continuation property.

We present the properties of three Green functions for:

1. general complex "clamped beam" \begin{equation}\tag{${\textrm{BC}}$} \begin{split} D_{\alpha ,\beta }[y] &\equiv y'''' -(\alpha ^{2}+\beta ^{2}) y''+\alpha ^{2}\beta^{2}y=f, \\ y(0)& =y(1) =y'(0) =y'(1) =0. \end{split} \end{equation}

2. Timoshenko clamped beam $D_{\alpha ,\overline{\alpha }}[y] \equiv f$ with (BC).

3. Euler-Bernoulli clamped beam $D_{k(1+i) ,k(1-i)} [ y] \equiv f$ with (BC).

In case 1. we represent solution via a Green operator expressed in terms of Kourensky type system of fundamental solutions for homogeneous case. This condense form is, up-to our knowledge, new even for the Euler-Bernoulli clamped beam and it allows to recognize the set of $\alpha's$ for which the Pointwise Comparison Principle for the Timoshenko beam holds. The presented approach to positivity of the Green function is much straightforward then ones known in the literature for the case 3 (see [J. Schröder, Zusammenhängende mengen inverspositiver differentialoperatoren vierter ordnung, Math. Z. 96 (1967) 89-110]).

In the paper we study the existence of constrained graph approximations of set-valued maps with non-convex values. We prove, in particular, that any open neighbourhood of the graph of a map satisfying the so-called topological tangency assumptions contains a graph of constrained continuous single-valued map provided that the domain is finite-dimensional.