Oscillation of first-order nonlinear differential equations with deviating arguments

Kitamura, Yuichi; Kusano, Takaŝi

doi:10.1090/S0002-9939-1980-0548086-5

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
DOI: https://doi.org/10.1090/S0002-9939-1980-0548086-5
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 ${g_i}(t) < t,1 \leqslant i \leqslant N$ .

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