Electronic Journal of Differential Equations,
Vol. 2003(2003), No. 12, pp. 1-21.
Title: Oscillation for equations with positive and negative coefficients
and with distributed delay I: General results
Authors: Leonid Berezansky (Ben-Gurion Univ. of the Negev, Israel)
Elena Braverman (Univ. of Calgary, Alberta, Canada)
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.