Existence Of Periodic Solutions For First Order Functional Differential Equations

The functional differential equations with periodic delays represent a naturalframework for mathematical modeling of many real world phenomena such as bi-ology, economy, ecology ,the population dynamic system and so on. For example,animal blood red cells existence model, population dynamic system model and othermodels. Therefore, the researches on existence of periodic solutions for functional dif-ferential equations with periodic delays have practical significance. In recent years,many scholars have made thoroughly research on it and obtained quite rich results.In this paper, we mainly consider the existence of one or multiple periodic solu-tions for first order functional differential equations. This paper has two sections. Itis organized as follows.In the first chapter, we used the partial ordering theory and the topological de-gree theory to establish the existence of nontrivial periodic solutions for first orderfunctional differential equations y'(t)=-a(t)y(t)+h(t)f(t,y(t-Ï„₁(t)),y(t-Ï„₂(t)),…,y(t-Ï„_n(t)))where a(t), h(t) andÏ„_i(t)(i=1, 2,…, n) are continuous T-periodic functions (T＞0 isfixed constant) and integral from n=0 to T a(t)dt＞0,h(t)＞0 for any t∈R, f∈C(Rⁿ⁺¹, R)is T-periodicwith respect to the first variable. We extend functional differential equations withsingle delay in the corresponding paper to multiple delays and obtain the main resultsas the following.Theorem 1.3.1 Let f(t,μ)=f₁(t,μ)-f₂(t,μ), where f_i(t,μ) are non-negativecontinuous functions which satisfy f_i(t, 0)=0(i=1, 2). Assume thatuniformly with respect to all t∈R, whereμ=(μ₁,μ₂,…,μ_n)∈Rⁿ, |μ|=max|μ_i|.Then Eq (1.2.1) has at least a nontrivial T-periodic solution.In the second chapter, we discuss multiple positive periodic solutions of the fol-lowing first order functional differential equations y'(t)=-a(t)y(t)+f(t,y(t-Ï„₁(t)),y(t-Ï„₂(t)),…,y(t-Ï„_n(t)))we assume that a(t),Ï„_i(t)(i=1,2,…, n) are continuous T-periodic functions, andintegral from n=0 to T a(t)dt＞0 (T＞0 is fixed constant), f∈C(R×[0,∞)ⁿ, [0,∞))is T-periodic with respect to the first variable. In this section, existence of multiple positive periodicsolutions is considered by using the fixed point index theory at variance with Kras-noselskii fixed point theorem in the related literature. In addition, the obtained resultsimprove and extend the corresponding results in this literature. The main results canbe stated as the follows.Theorem 2.3.1 Suppose the hypotheses (H₁)-(H₃) hold, then Eq (2.2.1) hasat least two positive T-periodic solutions y₁ and y₂ such that 0＜‖y₁‖＜ρ₁＜‖y₂‖.Corollary 2.3.1 Let the hypotheses (H₇) and (H)₈ hold, assume that theprevious hypothesis (H₃) hold as well. Then Eq (2.2.1) has at least two positiveT-periodic solutions y₁ and y₂ such that 0＜‖y₁‖＜ρ₁＜‖y₂‖.Theorem 2.3.2 Suppose the hypotheses (H₄)-(H₆) hold, then Eq (2.2.1) hasat least two positive T-periodic solutions y₁ and y₂ such that 0＜‖y₁‖＜ρ₂＜‖y₂‖.Corollary 2.3.2 Let the hypotheses (H₉) and (H)₁₀ hold, assume that theprevious hypothesis (H₆) hold as well. Then Eq (2.2.1) has at least two positiveT-periodic solutions y₁ and y₂ such that 0＜‖y₁‖＜ρ₂＜‖y₂‖.