The Existence of Positive Solutions to Neutral Differential Equations

In this paper, we shall consider a class of neutral differential equations of the form

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where τ (0, ∞), σ 0, ∞), Q(t) C(t₀, ∞), R ⁺ ), r(t) C(t₀, ∞), (0, ∞)) with r(t) nondecreasing on t₀ − τ, ∞). 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).