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

Jaroslav Jaroš and Kusano Takaŝi

J. Jaroš, Comenius University, Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, 842 48 Bratislava, Slovakia
T. Kusano, Professor Emeritus at: Hiroshima University, Department of Mathematics, Faculty of Science, Higashi-Hiroshima 739-8526, Japan


Abstract: 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.

AMSclassification: primary 34C11; secondary 26A12.

Keywords: systems of nonlinear differential equations, positive solutions, asymptotic behavior, regularly varying functions.

DOI: 10.5817/AM2014-3-131