The Existence of Positive Solutions to Neutral Differential Equationsʹ

In this paper, we shall consider a class of neutral differential equations of the form r ( t ) X(t - T )I + Q ( t ) X ( t - a ) = 0, t 2 t o , ( * ) whereTE (O,m), a € [ O,m), Q ( t ) E C ( [ t , , m ) , R + ) , r ( tE) C([t,,m),(O,m))withr ( t ) nondecreasing on [ t o - T , m). 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 (19961, 347-3621,