Positive Periodic Solutions for First-Order Neutral Functional Differential Equations with Periodic Delays

Abstract

In this paper, two classes of first-order neutral functional differential equations with periodic delays are considered. Some results on the existence of positive periodic solutions for the equations are obtained by using the Krasnoselskii fixed point theorem. Four examples are included to illustrate our results.

1. Introduction and Preliminaries

In recent years, there have been a few papers written on the existence of periodic solutions, nontrivial periodic solutions, maximal and minimal periodic solutions and positive periodic solutions for several classes of functional differential equations with periodic delays, which arise from a number of mathematical ecological models, economical and control models, physiological and population models, and other models, see, for example, [1–5] and the references therein.

In 2004, Wan et al. [5] studied the first-order functional differential equation with periodic delays𝑥^′(𝑡)=−𝑎(𝑡)𝑥(𝑡)+𝑓(𝑡,𝑥(𝑡−𝜏(𝑡))),∀𝑡∈ℝ, (1.1) where 𝑎∈𝐶(ℝ,ℝ⁺⧵{0}),𝜏∈𝐶(ℝ,ℝ) are 𝜔-periodic, and 𝑓∈𝐶(ℝ×ℝ⁺,ℝ⁺) is 𝜔-periodic with respect to the first variable. By using a fixed point theorem in cones, they proved the existence of a periodic solution and a positive periodic solution of (1.1), respectively, under certain conditions. In 2005, Kang and Zhang [2] used the partial ordering and topological degree theory to establish the existence of a nontrivial periodic solution of (1.1). In 2010, Kang et al. [1] gave the existence of maximal and minimal periodic solutions of (1.1) by utilizing the method of lower and upper solutions. By means of the continuation theorem of coincidence degree principle, Serra [4] discussed the existence of periodic solutions for the following neutral functional differential equation[𝑥(𝑡)+𝑐𝑥(𝑡−𝜏)]^′=𝑓(𝑡,𝑥(𝑡)),∀𝑡∈ℝ, (1.2) where |𝑐|≤1 and 𝜏>0 are constants. In 2008, Luo et al. [3] employed the Krasnoselskii fixed point theorem to prove the existence of positive periodic solutions for two kinds of neutral functional differential equations with periodic delays[𝑥(𝑡)−𝑐𝑥(𝑡−𝜏(𝑡))]^′(𝑡)𝑥(𝑡)(((𝑡𝑥(𝑡)=−𝑎+𝑓𝑡,𝑥𝑡−𝜏))),∀𝑡∈ℝ,−𝑐⁰_−∞𝑄(𝑟)𝑥(𝑡+𝑟)𝑑𝑟^′=−𝑎(𝑡)𝑥(𝑡)+𝑏(𝑡)⁰_−∞𝑄(𝑟)𝑓(𝑡,𝑥(𝑡+𝑟))𝑑𝑟,∀𝑡∈ℝ, (1.3) where 𝜔∈ℝ⁺⧵{0} and |𝑐|<1 are constants, 𝜏∈𝐶(ℝ,ℝ), 𝑎,𝑏∈𝐶(ℝ,ℝ⁺⧵{0}), 𝑓∈𝐶(ℝ²,ℝ), 𝜏,𝑎, and 𝑏 are 𝜔-periodic and 𝑓 is 𝜔-periodic with respect to the first variable, 𝑄∈𝐶(ℝ₋,ℝ⁺), and ∫⁰_−∞𝑄(𝑟)𝑑𝑟=1.

Motivated by the papers [1–5] and the references therein, we consider two new kinds of first-order neutral functional differential equations with periodic delays:𝑔(𝑡)(𝑥(𝑡)+𝑐(𝑡)𝑥(𝑡−𝜏(𝑡)))^′(𝑡)𝑥(𝑡)(((𝑡𝑔(𝑡)𝑥(𝑡)(𝑡)=−𝑎+𝑓𝑡,𝑥𝑡−𝜏))),∀𝑡∈ℝ,+𝑐⁰_−∞𝑄(𝑟)𝑥(𝑡+ℎ(𝑟))𝑑𝑟^′=−𝑎(𝑡)𝑥(𝑡)+𝑏(𝑡)⁰_−∞𝑄(𝑟)𝑓(𝑡,𝑥(𝑡+ℎ(𝑟)))𝑑𝑟,∀𝑡∈ℝ, (1.4)(1.5) where 𝜔∈ℝ⁺⧵{0} is a constant, 𝜏,𝑎,𝑏,𝑐∈𝐶(ℝ,ℝ), 𝑓∈𝐶(ℝ²,ℝ), ℎ∈𝐶(ℝ₋,ℝ), 𝑔∈𝐶¹(ℝ,ℝ⁺⧵{0}), 𝜏, 𝑎, 𝑏, 𝑐, and 𝑔 are 𝜔-periodic functions and 𝑓 is 𝜔-periodic with respect to the first variable, 𝑄∈𝐶(ℝ₋,ℝ⁺), and ∫⁰_−∞𝑄(𝑟)𝑑𝑟=1. It is evident that (1.4) and (1.5) include, respectively, (1.1)–(1.3) as special cases. To the best of our knowledge, the existence of periodic solutions for (1.4) and (1.5) have not been investigated till now. The aim of this paper is, by applying the Krasnoselskii fixed point theorem and some new techniques, to establish a set of sufficient conditions which guarantee the existence of positive periodic solutions of (1.4) and (1.5). Four examples are given to show the efficiency and applications of our results.

Throughout this paper, we assume that ℝ=(−∞,+∞), ℝ⁺=[0,+∞), ℝ₋=(−∞,0], ℕ denotes the set of all positive integers, 𝑃=min_{𝑡∈[0,𝜔]}𝑔(𝑡),𝐺()=∫𝑡,𝑠exp^𝑠_𝑡𝑔^′(𝑟)+𝑎(𝑟)/𝑔(𝑟)𝑑𝑟𝑔(𝑠)exp∫^𝜔₀𝑔^′(𝑟)(𝑟)(𝑟)()+𝑎/𝑔𝑑𝑟−1,∀𝑡,𝑠∈ℝ²,𝑋={𝑥∈𝐶(ℝ,ℝ)∶𝑥(𝑡)=𝑥(𝑡+𝜔),∀𝑡∈ℝ}. (1.6) It is well known that 𝑋 is a Banach space with the norm‖𝑥‖=sup_{[]𝑡∈0,𝜔}|𝑥(𝑡)|,foreach𝑥∈𝑋. (1.7) Let𝐴(𝑁,𝑀)={𝑥∈𝑋∶𝑁≤𝑥(𝑡)≤𝑀,∀𝑡∈[0,𝜔]},forany𝑀>𝑁≥0. (1.8) It is easy to see that 𝐴(𝑁,𝑀) is a bounded closed and convex subset of the Banach space 𝑋.

Lemma 1.1 (the Krasnoselskii fixed point theorem). Let 𝑌 be a nonempty bounded closed convex subset of a Banach space 𝑍 and 𝑓,𝑔 mappings from 𝑌 into 𝑍 such that 𝑓𝑥+𝑔𝑦∈𝑌 for every pair 𝑥,𝑦∈𝑌. If 𝑓 is a contraction mapping and 𝑔 is completely continuous, then the equation 𝑓𝑥+𝑔𝑥=𝑥 has at least one solution in 𝑌.

2. Main Results

Now we use the Krasnoselskii fixed point theorem to show the existence of positive solutions for (1.4) and (1.5).

Theorem 2.1. Assume that there exist constants 𝑁, 𝑀, 𝐺₁, 𝐺₂, 𝑐₁, and 𝑐₂ satisfying 0<𝑁<𝑀,𝑐₁≥0,𝑐₂≥0,𝑐₁+𝑐₂<1,−𝑐₁≤𝑐(𝑡)≤𝑐₂,∀𝑡∈[0,𝜔],0<𝐺₁≤𝑔^′(𝑡)+𝑎(𝑡)≤𝐺₂,∀𝑡∈[0,𝜔],𝑁+𝑐₂𝑀𝐺₂≤𝑓(𝑡,𝑠)+𝑎(𝑡)𝑐(𝑡)𝑠≤1−𝑐₁𝑀𝐺₁,∀(𝑡,𝑠)∈[0,𝜔]×[𝑁,𝑀]. (2.1)(2.2)(2.3) Then (1.5) has at least one positive 𝜔-periodic solution in 𝐴(𝑁,𝑀).

Proof. It is obvious that (1.4) has a solution 𝑥(𝑡) if and only if the integral equation 𝑥(𝑡)=_𝑡^𝑡+𝜔𝐺(𝑡,𝑠)𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)))+𝑎(𝑠)𝑐(𝑠)𝑥(𝑠−𝜏(𝑠))𝑑𝑠−𝑐(𝑡)𝑥(𝑡−𝜏(𝑡)),∀𝑡∈ℝ, (2.4) has a solution 𝑥(𝑡). Define two mappings 𝑇 and 𝑆∶𝐴(𝑁,𝑀)→𝑋 by (𝑇𝑥)(𝑡)=_𝑡^𝑡+𝜔𝐺(𝑓(((𝑠(𝑠)𝑐(𝑠)𝑥((𝑠(𝑡)(𝑡)𝑥((𝑡𝑡,𝑠)𝑠,𝑥𝑠−𝜏)))+𝑎𝑠−𝜏))𝑑𝑠,∀𝑡∈ℝ,𝑆𝑥)(=−𝑐𝑡−𝜏)),∀𝑡∈ℝ, (2.5) for each 𝑥∈𝐴(𝑁,𝑀). It follows from (2.5) that for any 𝑥∈𝐴(𝑁,𝑀) and 𝑡∈ℝ(𝑇𝑥)(𝑡+𝜔)=^𝑡+2𝜔_𝑡+𝜔𝐺(𝑓(((𝑠(𝑠)𝑐(𝑠)𝑥((𝑠=𝑡+𝜔,𝑠)𝑠,𝑥𝑠−𝜏)))+𝑎𝑠−𝜏))𝑑𝑠_{𝑡𝑡+𝜔}𝐺(𝑓(((()𝑐()𝑥((=𝑡+𝜔,𝑢+𝜔)𝑢+𝜔,𝑥𝑢+𝜔−𝜏𝑢+𝜔)))+𝑎𝑢+𝜔𝑢+𝜔𝑢+𝜔−𝜏𝑢+𝜔))]𝑑𝑢_{𝑡𝑡+𝜔}𝐺(𝑓(((𝑢(𝑢)𝑐(𝑢)𝑥((𝑢(𝑡),()()𝑥(((𝑡)𝑥((𝑡=(𝑡),𝑡,𝑢)𝑢,𝑥𝑢−𝜏)))+𝑎𝑢−𝜏))𝑑𝑢=𝑇𝑥)(𝑆𝑥)(𝑡+𝜔=−𝑐𝑡+𝜔𝑡+𝜔−𝜏𝑡+𝜔))=−𝑐𝑡−𝜏))𝑆𝑥)( (2.6) which mean that 𝑇(𝐴(𝑁,𝑀))⊆𝑋,𝑆(𝐴(𝑁,𝑀))⊆𝑋. (2.7)
Using (2.1)–(2.3) and (2.5), we infer that for all 𝑥,𝑦∈𝐴(𝑁,𝑀) and 𝑡∈ℝ(𝑇𝑥)(𝑡)+𝑆𝑦(𝑡)=_𝑡^𝑡+𝜔𝐺(𝑓(((𝑠(𝑠)𝑐(𝑠)𝑥((𝑠(𝑡)𝑦((𝑡≤𝑡,𝑠)𝑠,𝑥𝑠−𝜏)))+𝑎𝑠−𝜏))𝑑𝑠−𝑐𝑡−𝜏))1−𝑐₁𝑀𝐺_1𝑡^𝑡+𝜔𝐺(𝑡,𝑠)𝑑𝑠+𝑐₁𝑀≤1−𝑐₁𝑀_𝑡^𝑡+𝜔𝐺(𝑡,𝑠)𝑔^′(𝑠)+𝑎(𝑠)𝑑𝑠+𝑐₁𝑀=1−𝑐₁𝑀exp∫^𝜔₀𝑔^′(𝑟)(𝑟(𝑟×+𝑎)/𝑔)𝑑𝑟−1_{𝑡𝑡+𝜔}exp^𝑠_𝑡𝑔^′(𝑟)+𝑎(𝑟)𝑔(𝑟)𝑔𝑑𝑟^′(𝑠)+𝑎(𝑠)𝑔(𝑠)𝑑𝑠+𝑐₁𝑀=1−𝑐₁𝑀exp∫^𝜔₀𝑔^′(𝑟)(𝑟(𝑟×+𝑎)/𝑔)𝑑𝑟−1exp_𝑡^𝑡+𝜔𝑔^′(𝑟)+𝑎(𝑟)𝑔(𝑟)𝑑𝑟−1+𝑐₁𝑀=1−𝑐₁𝑀+𝑐₁(𝑡)+(𝑡)≥𝑀=𝑀,𝑇𝑥)(𝑆𝑦𝑁+𝑐₂𝑀𝐺_2𝑡^𝑡+𝜔𝐺(𝑡,𝑠)𝑑𝑠−𝑐₂𝑀≥𝑁+𝑐₂𝑀−𝑐₂𝑡)−(𝑡=|𝑐(𝑡𝑥((𝑡((𝑡≤𝑐𝑀=𝑁,(𝑆𝑥)(𝑆𝑦))|𝑡−𝜏))−𝑦𝑡−𝜏))₁+𝑐₂𝑥−𝑦, (2.8)(2.9)(2.10) which imply that 𝑇𝑥+𝑆𝑦∈𝐴(𝑁,𝑀),𝑆𝑥−𝑆𝑦≤𝑐₁+𝑐₂𝑥−𝑦,∀𝑥,𝑦∈𝐴(𝑁,𝑀). (2.11)
Now we show that 𝑇 is a completely continuous mapping in 𝐴(𝑁,𝑀). First, we claim that 𝑇 is continuous in 𝐴(𝑁,𝑀). Let {𝑦_𝑘}_𝑘∈𝑁⊂𝐴(𝑁,𝑀) and 𝑦∈𝐴(𝑁,𝑀) with lim_𝑘→∞𝑦_𝑘=𝑦. Note that 𝑓∈𝐶(ℝ²,ℝ). It follows from the uniform continuity of 𝑓 in [0,𝜔]×[𝑁,𝑀] that for given 𝜀>0, there exists 𝛿>0 satisfying 𝑓𝑡₁,𝑠₁−𝑓𝑡₂,𝑠₂<𝐺₁𝜀2,∀𝑡₁,𝑡₂,𝑠₁,𝑠₂∈[0,𝜔]²×[𝑁,𝑀]²,withmax𝑡₁−𝑡₂,𝑠₁−𝑠₂<𝛿. (2.12) Since lim_𝑘→∞𝑦_𝑘=𝑦, it follows that there exists 𝑁₁∈ℕ satisfying 𝑦_𝑘<𝐺−𝑦¹min{𝜀,𝛿}21+𝐺₂(1+‖𝑎‖),∀𝑘≥𝑁₁. (2.13) In view of (2.1), (2.2), (2.5), (2.12), and (2.13), we get that 𝑇𝑦_𝑘−𝑇𝑦=sup_{𝑡∈[0,𝜔]𝑡}^𝑡+𝜔𝐺(𝑡,𝑠)𝑓𝑠,𝑦_𝑘(𝑠−𝜏(𝑠))+𝑎(𝑠)𝑐(𝑠)𝑦_𝑘((𝑠−𝑠−𝜏))𝑑𝑠_{𝑡𝑡+𝜔}𝐺(𝑡,𝑠)𝑓𝑠,𝑦(𝑠−𝜏(𝑠))+𝑎(𝑠)𝑐(𝑠)𝑦(𝑠−𝜏(𝑠))𝑑𝑠≤sup_{[]𝑡∈0,𝜔𝑡}^𝑡+𝜔𝐺(𝑡,𝑠)𝑓𝑠,𝑦_𝑘((𝑠((𝑠+|𝑎(𝑠)𝑐(𝑠𝑦𝑠−𝜏))−𝑓𝑠,𝑦𝑠−𝜏)))|_𝑘(𝑠−𝜏(𝑠))−𝑦(𝑠−𝜏(𝑠))𝑑𝑠<𝐺₁𝜀2+‖𝑎‖𝑐₁+𝑐₂min{𝜀,𝛿}21+𝐺₂(1+‖𝑎‖)sup_{[]𝑡∈0,𝜔𝑡}^𝑡+𝜔𝐺(𝑡,𝑠)𝑑𝑠<𝜀,∀𝑘≥𝑁₁, (2.14) which yields that lim_𝑘→∞𝑇𝑦_𝑘=𝑇𝑦, that is, 𝑇 is continuous in 𝐴(𝑁,𝑀).
Second, we claim that 𝑇(𝐴(𝑁,𝑀)) is relatively compact. It is sufficient to show that 𝑇(𝐴(𝑁,𝑀)) is uniformly bounded and equicontinuous in [0,𝜔]. Notice that (2.1)–(2.3) and (2.5) ensure that ‖𝑇𝑥‖=sup_{[]𝑡∈0,𝜔𝑡}^𝑡+𝜔𝐺(𝑓(((𝑠(𝑠)𝑐(𝑠)𝑥((𝑠≤𝑡,𝑠)𝑠,𝑥𝑠−𝜏)))+𝑎𝑠−𝜏))𝑑𝑠1−𝑐₁𝑀𝐺₁sup_{[]𝑡∈0,𝜔𝑡}^𝑡+𝜔𝐺(𝑡,𝑠)𝑑𝑠≤1−𝑐₁(),)𝑀,∀𝑥∈𝐴𝑁,𝑀(𝑇𝑥^′(𝑡=−𝑔)^′(𝑡)+𝑎(𝑡)𝑔(𝑡)(𝑡)()×𝑓(((()𝑐()𝑥(((𝑓(((𝑡(𝑡)𝑐(𝑡)𝑥((𝑡≤𝑔𝑇𝑥)(+𝐺𝑡,𝑡+𝜔𝑡+𝜔,𝑥𝑡+𝜔−𝜏𝑡+𝜔)))+𝑎𝑡+𝜔𝑡+𝜔𝑡+𝜔−𝜏𝑡+𝜔))−𝐺𝑡,𝑡)𝑡,𝑥𝑡−𝜏)))+𝑎𝑡−𝜏))_′(𝑡)+𝑎(𝑡)𝑔(𝑡)𝑡+|𝐺()(𝑓(((𝑡(𝑡)𝑐(𝑡)𝑥((𝑡≤𝐺|(𝑇𝑥)()|𝑡,𝑡+𝜔−𝐺𝑡,𝑡)|𝑡,𝑥𝑡−𝜏)))+𝑎𝑡−𝜏))₂𝑃1−𝑐₁∫𝑀+exp^𝜔₀𝑔^′(𝑟)+𝑎(𝑟)/𝑔(𝑟)𝑑𝑟−1𝑔(𝑡)exp∫^𝜔₀𝑔^′(𝑟)+𝑎(𝑟)/𝑔(𝑟)𝑑𝑟−11−𝑐₁𝑀𝐺₁≤1−𝑐₁𝑀𝐺₁+𝐺₂𝑃,∀(𝑥,𝑡)∈𝐴(𝑁,𝑀)×[0,𝜔], (2.15) which give that 𝑇(𝐴(𝑁,𝑀)) is uniformly bounded and equicontinuou sin [0,𝜔], which together with (2.7), (2.11), and Lemma 1.1 yields that there is 𝑥₀∈𝐴(𝑁,𝑀) with 𝑇𝑥₀+𝑆𝑥₀=𝑥₀. It follows from (2.4) and (2.5) that 𝑥₀ is a positive 𝜔-periodic solution of (1.4). This completes the proof.

Theorem 2.2. Assume that there exist constants 𝑁, 𝑀, 𝐺₁, 𝐺₂, 𝑐₁, 𝑐₂, and 𝑡₀∈[0,𝜔] satisfying (2.2), (2.3): 0≤𝑁<𝑀,𝑐₁≥0,𝑐₂≥0,𝑐₁+𝑐₂<1,−𝑐₁≤𝑐(𝑡)≤𝑐₂,∀𝑡∈[0,𝜔], (2.16) and either 𝑓𝑡₀,𝑠+𝑎𝑡₀𝑐𝑡₀𝑠>𝑁+𝑐₂𝑀𝐺₂,∀𝑠∈[𝑁,𝑀], (2.17) or 𝑔^′𝑡₀+𝑎𝑡₀<𝐺₂. (2.18) Then (1.4) has at least one positive 𝜔-periodic solution 𝑥∈𝐴(𝑁,𝑀) with 𝑁<𝑥(𝑡)≤𝑀 for each 𝑡∈[0,𝜔].

Proof. As in the proof of Theorem 2.1, we conclude similarly that (1.4) has an 𝜔-periodic solution 𝑥∈𝐴(𝑁,𝑀). Now we assert that 𝑥(𝑡)>𝑁 for all 𝑡∈[0,𝜔]. Otherwise, there exists 𝑡^∗∈[0,𝜔] satisfying 𝑥(𝑡^∗)=𝑁. In view of (2.4), (2.5), and (2.16), we have 𝑁=^𝑡∗_𝑡^+𝜔_∗𝐺𝑡^∗,𝑠𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)))+𝑎(𝑠)𝑐(𝑠)𝑥(𝑠−𝜏(𝑠))𝑑𝑠−𝑐𝑡^∗𝑥𝑡^∗−𝜏𝑡^∗≥^𝑡∗_𝑡^+𝜔_∗𝐺𝑡^∗,𝑠𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)))+𝑎(𝑠)𝑐(𝑠)𝑥(𝑠−𝜏(𝑠))𝑑𝑠−𝑐₂𝑀, (2.19) which implies that 0≥^𝑡∗_𝑡^+𝜔_∗𝐺𝑡^∗,𝑠𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)))+𝑎(𝑠)𝑐(𝑠)𝑥(𝑠−𝜏(𝑠))𝑑𝑠−𝑁+𝑐₂𝑀=^𝑡∗_𝑡^+𝜔_∗𝐺𝑡^∗,𝑠𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)))+𝑎(𝑠)𝑐(𝑠)𝑥(𝑠−𝜏(𝑠))−𝑁+𝑐₂𝑀𝑔^′(𝑠)+𝑎(𝑠)𝑑𝑠. (2.20)
Assume that (2.17) holds. By means of (2.2), (2.3), (2.17), and the continuity of 𝐺, 𝑓, 𝑎, 𝑐, 𝑔, 𝑔^′, 𝜏, and 𝑥, we get that ^𝑡∗_𝑡^+𝜔_∗𝐺𝑡^∗,𝑠𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)))+𝑎(𝑠)𝑐(𝑠)𝑥(𝑠−𝜏(𝑠))−𝑁+𝑐₂𝑀𝑔^′(𝑠)(𝑠)≥+𝑎𝑑𝑠_{𝑡∗𝑡+𝜔∗}𝐺𝑡^∗,𝑠𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)))+𝑎(𝑠)𝑐(𝑠)𝑥(𝑠−𝜏(𝑠))−𝑁+𝑐₂𝑀𝐺₂𝑑𝑠>0, (2.21) which contradicts (2.20).
Assume that (2.18) holds. In light of (2.2), (2.3), (2.18), and the continuity of 𝐺, 𝑓, 𝑎, 𝑐, 𝑔, 𝑔^′,𝜏, and 𝑥, we infer that ^𝑡∗_𝑡^+𝜔_∗𝐺𝑡^∗,𝑠𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)))+𝑎(𝑠)𝑐(𝑠)𝑥(𝑠−𝜏(𝑠))−𝑁+𝑐₂𝑀𝑔^′(𝑠)(𝑠)>+𝑎𝑑𝑠_{𝑡∗𝑡+𝜔∗}𝐺𝑡^∗,𝑠𝑓(𝑠,𝑥(𝑠−𝜏(𝑠)))+𝑎(𝑠)𝑐(𝑠)𝑥(𝑠−𝜏(𝑠))−𝑁+𝑐₂𝑀𝐺₂𝑑𝑠≥0, (2.22) which contradicts (2.20). This completes the proof.

Theorem 2.3. Assume that there exist constants 𝑁, 𝑀, 𝐺₁, 𝐺₂, 𝑐₁, and 𝑐₂ satisfying (2.1), (2.2), and 𝑁+𝑐₂𝑀𝐺₂≤𝑏(𝑡)𝑓(𝑡,𝑠)+𝑎(𝑡)𝑐(𝑡)𝑠≤1−𝑐₁𝑀𝐺₁,∀(𝑡,𝑠)∈[0,𝜔]×[𝑁,𝑀]. (2.23) Then (1.5) has at least one positive 𝜔-periodic solution in 𝐴(𝑁,𝑀).

Proof. It is obvious that (1.5) has a solution 𝑥(𝑡) if and only if the integral equation 𝑥(𝑡)=_𝑡^𝑡+𝜔𝐺(𝑡,𝑠)𝑏(𝑠)⁰_−∞𝑄(𝑟)𝑓(𝑠,𝑥(𝑠+ℎ(𝑟)))𝑑𝑟+𝑎(𝑠)𝑐(𝑠)⁰_−∞𝑄(𝑟)𝑥((𝑟(𝑡)𝑠+ℎ))𝑑𝑟𝑑𝑠−𝑐⁰_−∞𝑄(𝑟)𝑥(𝑡+ℎ(𝑟))𝑑𝑟,∀𝑡∈ℝ, (2.24) has a solution 𝑥(𝑡). Define two mappings 𝑇 and 𝑆∶𝐴(𝑁,𝑀)→𝑋 by (𝑇𝑥)(𝑡)=_𝑡^𝑡+𝜔𝐺(𝑡,𝑠)𝑏(𝑠)⁰_−∞𝑄(𝑟)𝑓(𝑠,𝑥(𝑠+ℎ(𝑟)))𝑑𝑟+𝑎(𝑠)𝑐(𝑠)⁰_−∞𝑄(𝑟)𝑥((𝑟(𝑡)(𝑡)𝑠+ℎ))𝑑𝑟𝑑𝑠,𝑆𝑥)(=−𝑐⁰_−∞𝑄(𝑟)𝑥(𝑡+ℎ(𝑟))𝑑𝑟, (2.25) for each (𝑥,𝑡)∈𝐴(𝑁,𝑀)×ℝ. The rest of the proof is similar to that of Theorem 2.1, and is omitted. This completes the proof.

Theorem 2.4. Assume that there exist constants 𝑁, 𝑀, 𝐺₁, 𝐺₂, 𝑐₁, 𝑐₂, and 𝑡₀∈[0,𝜔] satisfying (2.2), (2.16), (2.23), and either (2.18) or 𝑏𝑡₀𝑓𝑡₀,𝑠+𝑎𝑡₀𝑐𝑡₀𝑠>𝑁+𝑐₂𝑀𝐺₂,∀𝑠∈[𝑁,𝑀]. (2.26) Then (1.5) has at least one positive 𝜔-periodic solution 𝑥∈𝐴(𝑁,𝑀) with 𝑁<𝑥(𝑡)≤𝑀 for each 𝑡∈[0,𝜔].
The proof of Theorem 2.4 is similar to that of Theorems 2.2 and 2.3 and is omitted.

Remark 2.5. Even if 𝑔(𝑡)≡1, 𝑐(𝑡)≡𝑐 and ℎ(𝑟)=𝑟 for all 𝑟∈ℝ₋, the conditions of Theorems 2.2 and 2.4 in this paper are different from these conditions of Theorems 2.1–2.4 in [3], respectively.

3. Examples

Now we construct four examples which illustrate the results obtained in Section 2. Note that none of the known results can be applied to the examples.

Example 3.1. Consider the first-order neutral functional differential equation with periodic delays 1+cos𝑡𝑥(𝑡)+1001+2sin𝑡𝑥()100𝑡−3sin𝑡−2cos𝑡^′=−1+sin𝑡𝑥(𝑡)50+20+cos²𝑡+sin²𝑥⁵(𝑡−3sin𝑡−2cos𝑡)cos𝑡,∀𝑡∈ℝ. (3.1) Let 𝜔=2𝜋, 𝑀=100, 𝑁=1, 𝑐₁=1/100, 𝑐₂=3/100, 𝐺₁=99/100, 𝐺₂=101/100, and 𝑔(𝑡)=1+cos𝑡(𝑡)=100,𝑐1+2sin𝑡(𝑡)100,𝑎=1+sin𝑡(𝑡)𝑓()50,𝜏=3sin𝑡+2cos𝑡,𝑡,𝑠=20+cos²𝑡+sin²𝑠⁵cos𝑡,∀(𝑡,𝑠)∈ℝ². (3.2) It is easy to see that (2.1) and (2.2) hold. Notice that 𝑁+𝑐₂𝑀𝐺₂1=4.04<20+1+501−2()(𝑡)𝑐(𝑡)𝑠1100⋅100≤𝑓𝑡,𝑠+𝑎≤22+1+501+2100⋅100<98.01=1−𝑐₁𝑀𝐺₁,∀(𝑡,𝑠)∈[0,𝜔]×[𝑁,𝑀], (3.3) that is, (2.3) is satisfied. Thus Theorem 2.1 yields that (3.1) has a positive 𝜔-periodic solution in 𝐴(𝑁,𝑀).

Example 3.2. Consider the first-order neutral functional differential equation with periodic delays 3+2cos𝑡+sin𝑡𝑥(𝑡)+1002+2sin𝑡+cos𝑡𝑥20𝑡−sin³𝑡^′=−100+2sin𝑡+3cos𝑡𝑥(𝑡)𝑥100+60+𝑡−sin³𝑡sin𝑡+𝑥⁸𝑡−sin³𝑡+150+10cos𝑡−𝑥⁵𝑡−sin³𝑡,∀𝑡∈ℝ. (3.4) Let 𝜔=2𝜋, 𝑀=100, 𝑁=0, 𝑐₁=1/20, 𝑐₂=1/4, 𝐺₁=24/25, 𝐺₂=26/25, 𝑡₀=𝜋/2, and 𝑔(𝑡)=3+2cos𝑡+sin𝑡(𝑡)=100,𝑐2+2sin𝑡+cos𝑡(𝑡)20,𝜏=sin³𝑎(𝑡)=𝑡,100+2sin𝑡+3cos𝑡()100,𝑓𝑡,𝑠=60+𝑠sin𝑡+𝑠⁸+150+10cos𝑡−𝑠⁵,∀(𝑡,𝑠)∈ℝ². (3.5) It is clear that (2.2), (2.16), and (2.18) hold. It follows that 𝑁+𝑐₂𝑀𝐺₂()=26<60+100−1+50−10105⋅100−1()(𝑡)𝑐(𝑡)𝑠20⋅100≤𝑓𝑡,𝑠+𝑎≤60+10050−10+26.25<91.2=1−𝑐₁𝑀𝐺₁,∀(𝑡,𝑠)∈[0,𝜔]×[𝑁,𝑀], (3.6) that is, (2.3) holds. Obviously (2.17) follows from the above inequalities. Hence, Theorem 2.2 ensures that (3.4) has a positive 𝜔-periodic solution 𝑥∈𝐴(𝑁,𝑀) with 𝑁<𝑥(𝑡)≤𝑀 for all 𝑡∈[0,𝜔].

Example 3.3. Consider the first-order neutral functional differential equation with periodic delays 5+cos𝑡+3sin𝑡𝑥(𝑡)+50+sin𝑡2+3sin𝑡6(50−sin𝑡)²⁰_−∞exp(𝑟)𝑥(𝑡−𝑟cos𝑟)𝑑𝑟^′=−151+50sin𝑡(50+sin𝑡)²𝑥(𝑡)+13+2sin𝑡×⁰_−∞exp(𝑟)4.53+3sin𝑡+3+2sin𝑡𝑥()10000+cos𝑡𝑡−𝑟cos𝑟cos²𝑡+cos𝑡+𝑥¹⁰⁰(𝑡−𝑟cos𝑟)𝑑𝑟,∀𝑡∈ℝ. (3.7) Let 𝜔=2𝜋, 𝑀=1440.6, 𝑁=1, 𝑐₁=1/14406, 𝑐₂=5/14406, 𝐺₁=5/2601, 𝐺₂=295/2401, and 𝑔(𝑡)=5+cos𝑡+3sin𝑡(𝑡)=50+sin𝑡,𝑐2+3sin𝑡6(50−sin𝑡)²,𝑎(𝑡)=151+50sin𝑡(50+sin𝑡)²(𝑡)=1,𝑏,𝑓()(3+2sin𝑡𝑡,𝑠=4.53+3sin𝑡+3+2sin𝑡)𝑠cos²𝑡+cos𝑡+𝑠¹⁰⁰()10000+cos𝑡,∀𝑡,𝑠∈ℝ²,𝑄(𝑟)=exp(𝑟),ℎ(𝑟)=−𝑟cos𝑟,∀𝑟∈ℝ₋. (3.8) Clearly, (2.1) and (2.2) hold. Note that 𝑁+𝑐₂𝑀𝐺₂=885<48024.53+3+3+2−1+10000−1302−553(𝑡)𝑓()(𝑡)𝑐(𝑡)𝑠≤37470006⋅1440.6≤𝑏𝑡,𝑠+𝑎4.53−3+3−21441.6+10000−1100537470006⋅1440.6<14404=52821−𝑐₁𝑀𝐺₁,∀(𝑡,𝑠)∈[0,𝜔]×[𝑁,𝑀], (3.9) that is, (2.23) is fulfilled. Thus Theorem 2.3 yields that (3.7) has a positive 𝜔-periodic solution in 𝐴(𝑁,𝑀).

Example 3.4. Consider the first-order neutral functional differential equation with periodic delays ln(100+2sin𝑡)𝑥(𝑡)+2+3cos𝑡100−2sin𝑡⁰_−∞exp(𝑟)𝑥(𝑡+𝑟)𝑑𝑟^′3=−𝑥(𝑡)+100+2sin𝑡1+cos𝑡+sin𝑡100⁰_−∞(𝑟()exp)4+2𝑥𝑡+𝑟sin𝑡+cos²𝑡5000+5sin²𝑡−ln1+𝑥²(𝑡+𝑟)𝑑𝑟,∀𝑡∈ℝ. (3.10) Let 𝜔=2𝜋, 𝑀=1000, 𝑁=1, 𝑐₁=1/98, 𝑐₂=5/98, 𝐺₁=1/102, 𝐺₂=5/98, 𝑡₀=𝜋/2, and 𝑔(𝑡)=ln(100+2sin𝑡),𝑐(𝑡)=2+3cos𝑡(𝑡)=3100−2sin𝑡,𝑎(𝑡)100+2sin𝑡,𝑏=1+cos𝑡+sin𝑡,𝑓()100𝑡,𝑠=4+2𝑠sin𝑡+cos²𝑡5000+5sin²𝑡−ln1+𝑠²,∀(𝑡,𝑠)∈ℝ²,𝑄(𝑟)=exp(𝑟),ℎ(𝑟)=𝑟,∀𝑟∈ℝ₋. (3.11) Obviously, (2.2), (2.16), and (2.18) hold. A simple calculation yields that 𝑁+𝑐₂𝑀𝐺₂=12745<4802336103=1041251+−1−11004+−2000+0+50006−9(𝑡)𝑓()(𝑡)𝑐(𝑡)10000−4⋅1000≤𝑏𝑡,𝑠+𝑎𝑠≤1+1+11004+2000+1+50006+9=10000−4⋅1000623563<10412524250=24991−𝑐₁𝑀𝐺₁,∀(𝑡,𝑠)∈[0,𝜔]×[𝑁,𝑀]; (3.12) that is, (2.23) holds. Clearly (2.26) follows from the above inequalities. Thus Theorem 2.4 ensures that (3.10) has a positive 𝜔-periodic solution 𝑥∈𝐴(𝑁,𝑀) with 𝑁<𝑥(𝑡)≤𝑀 for all 𝑡∈[0,𝜔].

Acknowledgment

This study was supported by research funds from the Dong-A University.