Electron. J. Diff. Eqns., Vol. 2003(2003), No. 12, pp. 1-21.

Oscillation for equations with positive and negative coefficients and with distributed delay I: General results

Leonid Berezansky & Elena Braverman

Abstract:
We study a scalar delay differential equation with a bounded distributed delay,
$$ \dot{x}(t)+ \int_{h(t)}^t x(s)\,d_s R(t,s) - \int_{g(t)}^t x(s)\,d_s T(t,s)=0, $$
where $R(t,s)$, $T(t,s)$ are nonnegative nondecreasing in $s$ for any $t$,
$$ R(t,h(t))=T(t,g(t))=0, \quad R(t,s) \geq T(t,s). $$
We establish a connection between non-oscillation of this differential equation and the corresponding differential inequalities, and between positiveness of the fundamental function and the existence of a nonnegative solution for a nonlinear integral inequality that constructed explicitly. We also present comparison theorems, and explicit non-oscillation and oscillation results.
In a separate publication (part II), we will consider applications of this theory to differential equations with several concentrated delays, integrodifferential, and mixed equations.

Submitted October 29, 2002. Published February 11, 2003.
Math Subject Classifications: 34K11, 34K15.
Key Words: Oscillation, non-oscillation, distributed delay, comparison theorems.

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A related article has been published by the same authors in this journal: Oscillation for equations with positive and negative coefficients and with distributed delay II: Applications, Vol. 2003(2003), No. 47, pp. 1-25.

Leonid Berezansky
Department of Mathematics
Ben-Gurion University of the Negev
Beer-Sheva 84105, Israel
e-mail: brznsky@cs.bgu.ac.il
Elena Braverman
Department of Mathematics and Statistics
University of Calgary
2500 University Drive N. W.,
Calgary, Alberta, Canada, T2N 1N4
Fax: (403)-282-5150, phone: (403)-220-3956
email: maelena@math.ucalgary.ca

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