Conditions are given for a class of nonlinear ordinary differential equations $x^{\text{'}\text{'}}+a\left(t\right)w\left(x\right)=0$, $t\ge t_0\ge 1$, which includes the linear equation to possess solutions $x\left(t\right)$ with prescribed oblique asymptote that have an oscillatory pseudo-wronskian $x^{\text{'}}\left(t\right)-\frac{x\left(t\right)}{t}$.

For an arbitrary analytic system which has a linear center at the origin we compute recursively all its Poincare-Lyapunov constants in terms of the coefficients of the system, giving an answer to the classical center problem. We also compute the coefficients of the Poincare series in terms of the same coefficients. The algorithm for these computations has an easy implementation. Our method does not need the computation of any definite or indefinite integral. We apply the algorithm to some polynomial...

The main objective of this paper is to study the boundedness character, the periodicity character, the convergence and the global stability of positive solutions of the difference equation $$x_{n+1}=\frac{{\alpha }_0{x}_n+{\alpha }_1{x}_{n-l}+{\alpha }_2{x}_{n-k}}{{\beta }_0{x}_n+{\beta }_1{x}_{n-l}+{\beta }_2{x}_{n-k}},n=0,1,2,\cdots$$ where the coefficients ${\alpha }_i,{\beta }_i\in (0,\infty )$ for $i=0,1,2,$ and $l$, $k$ are positive integers. The initial conditions $x_{-k},\cdots ,x_{-l},\cdots ,x_{-1},x_0$ are arbitrary positive real numbers such that $l<k$. Some numerical experiments are presented.

${}^{**}$ In the paper we will be concerned with the topological structure of the set of solutions of the initial value problem of a semilinear multi-valued system on a closed and convex set. Assuming that the linear part of the system generates a $C_0$-semigroup we show the $R_{\delta }$-structure of this set under certain natural boundary conditions. Using this result we obtain several criteria for the existence of periodic solutions for the semilinear system. As an application the problem of controlled heat transfer...

We prove that the solutions of a sweeping process make up an $R_{\delta }$-set under the following assumptions: the moving set C(t) has a lipschitzian retraction and, in the neighbourhood of each point x of its boundary, it can be seen as the epigraph of a lipschitzian function, in such a way that the diameter of the neighbourhood and the related Lipschitz constant do not depend on x and t. An application to the existence of periodic solutions is given.