Oscillation of first-order nonlinear differential equations with deviating arguments

Abstract:

This paper is devoted to the study of the oscillatory behavior of solutions of the first-order nonlinear functional differential equations \[ \begin {array}{*{20}{c}} x’(t) = \sum \limits _{i = 1}^N {{q_i}(t){f_i}(x({g_i}(t)))} \\ + F(t,x(t),x({g_1}(t)), \ldots ,x({g_N}(t))),\tag {$A$} \\ \end {array} \] \[ \begin {array}{*{20}{c}} x’(t) + \sum \limits _{i = 1}^N {{q_i}(t){f_i}(x({g_i}(t)))} \\ + F(t,x(t),x({g_1}(t)), \ldots ,x({g_N}(t))) = 0.\tag {$B$} \\ \end {array} \] First, without assuming that the deviating arguments ${g_i}(t),1 \leqslant i \leqslant N$, are retarded or advanced, sufficient conditions are established for all solutions of (A) and (B) to be oscillatory. Secondly, a characterization of oscillation of all solutions is obtained for equation (A) with $F \equiv 0$ and ${g_i}(t) > t,1 \leqslant i \leqslant N$, as well as for equation (B) with $F \equiv 0$ and ${g_i}(t) < t,1 \leqslant i \leqslant N$.