Oscillation tests for delay equations

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form (1) x'(t) + p(t)x(tau(t)) = 0, t greater than or equal to T, where p, tau is an element of C([T, infinity),R+), R+ = [0, infinity), tau( t) is nondecreasing, tau(t) < t for t greater than or equal to T and lim(t- ->infinity) tau(t) = infinity. Let the numbers k and L be defined by k = lim inf(t-->infinity) integral(tau(t)) (t) p(s) ds and L = lim sup(t-->infinity) integral(tau(t)) (t) p(s) ds. It is proved that, when L < 1 and 0 < Ic less than or equal to 1/e, all sol utions of Equation (1) oscillate if the condition L > ln lambda(1) + 1/lambda(1) - /1 - k - root 1 - 2k - k(2)/2, where lambda(1) is smaller root of the equation lambda = e(k lambda), is sa tisified.