Oscillation criterion for certain second order differential equations

Abstract

The purpose of this paper is to introduce a new oscillation criteria for second order nonlinear differential equations with retarded argument. Consider the delay differential equation x(t) + g(t)f(x(t))g(i(t)) = O (1) where q : [t0,+oo) —» (0,+oo), /, g : R —*• R are continuous and not identically zero on any ray of the form [T, +00), T > t0 > 0 and i(t) lirn^ sup -p 1 = c (2) where c is a constant. Some qualitative behavior of differential equations of the form (1) has been studied by many authors. Grace and Lalli [2] prove a theorem with strong assumptios that equation (1) is oscillatory. The aim of this paper is using the ideas of Hamedani [3] to find a new oscillation criterion with weaker (or different) set of assumptions for Equation(l).