One-Signed Periodic Solutions of First-Order Functional Differential Equations with a Parameter

and Applied Analysis 3 H4 f ∈ C R,R ; there exist two constants s2 < 0 < s1 such that f s2 f 0 f s1 0, f s > 0 for s ∈ 0, s1 ∪ s1,∞ , and f s < 0 for s ∈ −∞, s2 ∪ s2, 0 . The rest of this paper is organized as follows. In Section 2, we give some notations and the main results. Section 3 is devoted to proving the main results. 2. Statement of the Main Results Let Y {u ∈ C R,R : u t u t ω }with the norm ‖u‖∞ max t∈ 0,ω |u t |. 2.1 Then Y, ‖ · ‖∞ is a Banach space. Let E { u ∈ C1 R,R : u t u t ω } 2.2 be the Banach space with the norm ‖u‖ max{‖u‖∞, ‖u‖∞}. It is well known that 1.1 is equivalent to