The existence of positive solutions to neutral differential equations

In this paper, we shall consider a class of neutral differential equations of the form [x(t)-r(t)/r(t-tau)x(t-tau)]' + Q(t)x(t-sigma) = 0, t greater than or equal to t(0), (*) where tau is an element of (0, infinity), sigma is an element of [0, infinity), Q(t) is an element of C([t(0), infinity), R+), r(t) is an element of C([t(0), infinity), (0, infinity)) with r(t) nondecreasing on [t(0) - tau, infinity). We shall show that all positive solutions of (*) can be classified into four types, A, B, C, and D, and we shall obtain sufficient and necessary conditions for the existence of A-type, B-type, and D-type positive solutions of (*), respectively. A sufficient condition for the existence of C-type positive solutions of (*) is also given. Finally, we shall offer a sharp oscillation result for all solutions of(*). Our results generalize and improve those established in B. Yang and B. G. Zhang (Funkcial. Ekvac. 39 (1996), 347-362). (C) 1999 Academic Press.