Existence and sharp asymptotic behavior of positive decreasing solutions of a class [4pt] of differential systems with power-type nonlinearities

Jaroš, Jaroslav ; Takaŝi, Kusano

Archivum mathematicum, Tome 50 (2014) no. 3, pp. 131-150.

Résumé

The system of nonlinear differential equations \begin{equation*} x^{\prime } + p_1(t)x^{\alpha _1} + q_1(t)y^{\beta _1} = 0\,, \qquad y^{\prime } + p_2(t)x^{\alpha _2} + q_2(t)y^{\beta _2} = 0\,, A \end{equation*} is under consideration, where $\alpha _i$ and $\beta _i$ are positive constants and $p_i(t)$ and $q_i(t)$ are positive continuous functions on $[a,\infty )$. There are three types of different asymptotic behavior at infinity of positive solutions $(x(t),y(t))$ of (). The aim of this paper is to establish criteria for the existence of solutions of these three types by means of fixed point techniques. Special emphasis is placed on those solutions with both components decreasing to zero as $t \rightarrow \infty $, which can be analyzed in detail in the framework of regular variation.

Détail

DOI : 10.5817/AM2014-3-131

Classification : 26A12, 34C11
Keywords: systems of nonlinear differential equations; positive solutions; asymptotic behavior; regularly varying functions