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.