Asymptotic properties of nonoscillatory solutions of differential equations with deviating argument

Abstract

Recently, there is an increasing interest in studying the nth order differential equations involving the so called nth order r -derivative of x (r n -,(0(r n -2 (0( (r,(O('o(O*(O)7 )')')' which causes damped terms. Here, the asymptotic behavior of nonoscillatory solutions of such general differential equations with deviating argument is studied and, more precisely, sufficient conditions which guarantee that lim x(t) = 0 for the bounded nonoscillatory solutions x(t) are established. A basic theorem is obtained for the general case and then it is specialized into four corollaries concerning the particular case r, = 1 for jn-N and r n _ N = r (1 ^ N ^ n -1) which is of special interest. Finally, some examples are given to illustrate the significance of the results. In this paper we consider the nth order (n > 1) differential equation with deviating argument of the form (E) + a(t)F(x[(t)])=b(t), t^to