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

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.