In this paper, sharp a priori estimate of the periodic solutions is obtained for the discrete analogue of the continuous time ratio-dependent predator-prey system, which is governed by nonautonomous difference equations, modelling the dynamics of the $n-1$ competing preys and one predator having nonoverlapping generations. Based on more precise a priori estimate and the continuation theorem of the coincidence degree, an easily verifiable sufficient criterion of the existence of positive periodic solutions...

In this paper, we employ some new techniques to study the existence of positive periodic solution of $n$-species neutral delay system $$N_i^{\text{'}}\left(t\right)=N_i\left(t\right)\left[a_i\left(t\right)-\sum _{j=1}^n{\beta }_{ij}\left(t\right)N_j\left(t\right)-\sum _{j=1}^n{b}_{ij}\left(t\right)N_j(t-{\tau }_{ij}\left(t\right))-\sum _{j=1}^n{c}_{ij}\left(t\right)N_j^{\text{'}}(t-{\tau }_{ij}\left(t\right))\right].$$ As a corollary, we answer an open problem proposed by Y. Kuang.

We study the existence and uniqueness of a positive solution to the problem $$u^{\text{'}\text{'}}=p\left(t\right)u+q(t,u)u+f\left(t\right);u\left(0\right)=u\left(\omega \right),u^{\text{'}}\left(0\right)=u^{\text{'}}\left(\omega \right)$$ with a super-linear nonlinearity and a nontrivial forcing term $f$. To prove our main results, we combine maximum and anti-maximum principles together with the lower/upper functions method. We also show a possible physical motivation for the study of such a kind of periodic problems and we compare the results obtained with the facts well known for the corresponding autonomous case.

We study solutions tending to nonzero constants for the third order differential equation with the damping term $${\left(a_1\left(t\right){\left(a_2\left(t\right)x^{\text{'}}\left(t\right)\right)}^{\text{'}}\right)}^{\text{'}}+q\left(t\right)x^{\text{'}}\left(t\right)+r\left(t\right)f\left(x\left(\varphi \left(t\right)\right)\right)=0$$ in the case when the corresponding second order differential equation is oscillatory.

In this paper we consider positive unbounded solutions of second order quasilinear ordinary differential equations. Our objective is to determine the asymptotic forms of unbounded solutions. An application to exterior Dirichlet problems is also given.

Sufficient conditions are given which guarantee that the linear transformation converting a given linear Hamiltonian system into another system of the same form transforms principal (antiprincipal) solutions into principal (antiprincipal) solutions.

In this paper two sequences of oscillation criteria for the self-adjoint second order differential equation ${\left(r\left(t\right)u^{\text{'}}\left(t\right)\right)}^{\text{'}}+p\left(t\right)u\left(t\right)=0$ are derived. One of them deals with the case ${\int }^{\infty }\frac{\mathrm{d}t}{r\left(t\right)}=\infty$, and the other with the case ${\int }^{\infty }\frac{\mathrm{d}t}{r\left(t\right)}<\infty$.