The boundedness and asymptotic behavior of solutions of differential system of second-order with variable coefficients

Abstract

In this paper, the differential system of second-order with variable coefficients is studied, and some criteria of the boundedness and asymptotic behavior for solutions are given. Consider a system of differential equations

$$\left. {\begin{array}{*{20}c} {\frac{{dx_1 }}{{dt}} = p_{11} (t)x_1 + p_{12} (i)x_2 } \\ {\frac{{dx_2 }}{{dt}} = p_{21} (t)x_1 + p_{22} (t)x_2 } \\ \end{array} } \right\}$$

((0.1))

Now we study the boundedness and asymptotic behavior of its solutions. In the case of p_ij ^(t)being periodic functions, it was investigated by Burdina^[1]; in the case of p_ij ^(t) being arbitrary functions, it has not been investigated yet. Besides, the method used by Burdina is only appropriate for the former but not for the latter case. In this paper we shall give a method which is appropriate for both cases.