Oscillations of integro-differential equations

We establish sufficient and also necessary conditions for certain linear and some nonlinear integra-differential inequalities and/ or equations to have positive solutions.The following is a special corollary of our results.COROLLARY.Suppose that K E C[~+, ~+] and let T > 0 be such that J( is not identically zero on [0, T].Then -,\ + 1= e-'s K(s) ds :=::; 0 for some,\> 0 is a necessary and sufficient condition for the integra-differential equation x(t) + 1t K(t-s)x(s) ds = 0, t > T to have a solution x which is positive on [0, oo ).