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Oscillation of first-order nonlinear differential equations with deviating arguments

Authors: Yuichi Kitamura and Takaŝi Kusano
Journal: Proc. Amer. Math. Soc. 78 (1980), 64-68
MSC: Primary 34K15; Secondary 34K25
MathSciNet review: 548086
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Abstract: This paper is devoted to the study of the oscillatory behavior of solutions of the first-order nonlinear functional differential equations

\begin{displaymath}\begin{array}{*{20}{c}} x'(t) = \sum\limits_{i = 1}^N {{q_i}(... ...(t),x({g_1}(t)), \ldots ,x({g_N}(t))),\tag{$A$} \\ \end{array} \end{displaymath}

\begin{displaymath}\begin{array}{*{20}{c}} x'(t) + \sum\limits_{i = 1}^N {{q_i}(... ...x({g_1}(t)), \ldots ,x({g_N}(t))) = 0.\tag{$B$} \\ \end{array} \end{displaymath}

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$.

References [Enhancements On Off] (What's this?)

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Keywords: First-order differential equation, nonlinear differential equation, deviating argument, oscillatory solution, oscillation
Article copyright: © Copyright 1980 American Mathematical Society

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