Oscillatory Periodic Solutions for Two Differential-Difference Equations Arising in Applications

and Applied Analysis 3 The previous work mainly focuses on the autonomous differential-difference equation 1.2 . However, some papers 13, 24 contain some interesting nonautonomous differential difference equations arising in economics and population biology where the delay r of 1.2 depends on time t instead of a positive constant. Motivated by the lack of more results on periodic solutions for nonautonomous differential-difference equations, in the present paper, we study the following equations: ẋ t −f t, x t − r , 1.7 ẋ t −f t, x t − s − f t, x t − 2s , 1.8 where f t, x ∈ C × , is odd with respect to x and r π/2, s π/3. Here, we borrow the terminology “oscillatory periodic solution” for 1.7 and 1.8 since f t, x is odd with respect to x. Now, we state our main results as follows. Theorem 1.1. Suppose that f t, x ∈ C × , is odd with respect to x and r-periodic with respect to t. Suppose that lim x→ 0 f t, x x ω0 t , lim x→∞ f t, x x ω∞ t 1.9 exist. Write α0 1/r ∫ r 0 ω0 t dt and α∞ 1/r ∫ r 0 ω∞ t dt. Assume that (H1) α0 / ± k, α∞ / ± k, for all k ∈ , (H2) there exists at least an integer k0 with k0 ∈ such that min{α0, α∞} < ±k0 < max{α0, α∞}, 1.10 then 1.7 has at least one nontrivial oscillatory periodic solution x satisfying x t −x t − π . Theorem 1.2. Suppose that f t, x ∈ C × , is odd with respect to x and s-periodic with respect to t. Let ω0 t and ω∞ t be the two functions defined in Theorem 1.1. Write β0 1/s ∫s 0 ω0 t dt and β∞ 1/s ∫s 0 ω∞ t dt. Assume that (H3) β0, 3β0 / ± k, β∞, 3β∞ / ± k, for all k ∈ , (H4) there exists at least an integer k0 with k0 ∈ such that min { β0, β∞ } < ±k0 < max { β0, β∞ } 1.11