OSCILLATION CRITERIA FOR FIRST-ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH SEVERAL DELAYS

Abstract

In this article, we study a first order nonlinear delay differential
equation

x'(t) +mXi=1pi(t)fi(x(to_i(t)))= 0; t>= t0;

where pi(t) and p_i(t) are the functions of nonnegative of real numbers and to_i(t) are not necessarily monotone for 1 i m. Also, we present new sufficient conditions for the oscillatory solutions of this equation. Our results essentially improve the all conditions in the literature. Finally, we give an example to illustrate our results.