Oscillation criteria for first-order delay equations

This paper is concerned with the oscillatory behaviour of first-order delay differential equations of the form x'(t) + p(t)x(tau(t)) = 0, tgreater than or equal to t(o), (1) where p,tau is an element of C([t(o),infinity),R+),R+ = [0, infinity), tau(t) is non-decreasing, tau(t) < t for t ≥ t(o) and lim(t-->infinity)(t) = infinity. Let the numbers k and L be defined by [GRAPHICS] It is proved here that when L < 1 and 0 < k less than or equal to 1/e all solutions of equation (1) oscillate in several cases in which the condition L > ln lambda(l)-1+root5-2lambda(1)+2klambda(1)/lambda(1) holds, where lambda(1) is the smaller root of the equation lambda = e(klambda).