Positive Periodic Solutions of Nonlinear First-Order Functional Difference Equations

Abstract

We consider the existence, multiplicity, and nonexistence of positive 𝑇-periodic solutions for the difference equations Δ𝑥(𝑛)=𝑎(𝑛)𝑔(𝑥(𝑛))𝑥(𝑛)−𝜆𝑏(𝑛)𝑓(𝑥(𝑛−𝜏(𝑛))), and Δ𝑥(𝑛)+𝑎(𝑛)𝑔(𝑥(𝑛))𝑥(𝑛)=𝜆𝑏(𝑛)𝑓(𝑥(𝑛−𝜏(𝑛))), where 𝑎,𝑏∶ℤ→[0,∞) are 𝑇-periodic, 𝜏∶ℤ→ℤ is 𝑇-periodic.

1. Introduction

In the recent years, there has been considerable interest in the existence of periodic solutions of the following equation: 𝑥^′(𝑡)=𝑎(𝑡)𝑔(𝑥(𝑡))𝑥(𝑡)−𝜆𝑏̃(𝑡)𝑓(𝑥(𝑡−𝜏(𝑡))), (1.1) where 𝑎,𝑏̃∈𝐶(ℝ,[0,∞)) are 𝜔-periodic functions, ∫^𝜔₀𝑎(𝑡)𝑑𝑡>0, ∫^𝜔₀𝑏̃(𝑡)𝑑𝑡>0, 𝜏 is a continuous 𝜔-periodic function. Equation (1.1) has been proposed as a model for a variety of physiological processes and conditions including production of blood cells, respiration, and cardiac arrhythmias. See, for example, [1–13] and the references therein.

Let ℤ be the set of all integers. In the present paper, we study the existence of positive 𝑇-periodic solutions of discrete analogues to (1.1) of the form Δ𝑥(𝑛)=𝑎(𝑛)𝑔(𝑥(𝑛))𝑥(𝑛)−𝜆𝑏(𝑛)𝑓(𝑥(𝑛−𝜏(𝑛))),𝑛∈ℤ, (1.2) and the difference equation Δ𝑥(𝑛)+𝑎(𝑛)𝑔(𝑥(𝑛))𝑥(𝑛)=𝜆𝑏(𝑛)𝑓(𝑥(𝑛−𝜏(𝑛))),𝑛∈ℤ, (1.3) where ℤ is the set of integer numbers, 𝑇∈ℕ is a fixed integer, 𝑎,𝑏∶ℤ→[0,+∞) are 𝑇-periodic, 𝑎(𝑛)≢0, 𝑏(𝑛)≢0 on [0,𝑇−1]=∶{0,1,…,𝑇−1}, 𝜏∶ℤ→ℤ is 𝑇-periodic, and 𝑓,𝑔∈𝐶([0,+∞),[0,+∞)), 𝜆>0 is a parameter.

So far, relatively little is known about the existence of positive periodic solutions of (1.2) and (1.3). To our best knowledge, only Raffoul [14] dealt with the special equations of (1.2) and (1.3) of the form Δ𝑥(𝑛)=𝛼(𝑛)𝑥(𝑛)−𝜆𝑏(𝑛)𝑓(𝑥(𝑛−𝜏(𝑛))), (1.4) with 𝛼(𝑛)=𝑎(𝑛)−1>0, and Δ𝑥(𝑛)+𝛼(𝑛)𝑥(𝑛)=𝜆𝑏(𝑛)𝑓(𝑥(𝑛−𝜏(𝑛))), (1.5) with 𝛼(𝑛)=1−𝑎(𝑛)>0, and determining values of 𝜆, for which there exist positive 𝑇-periodic solutions of (1.4) and (1.5), respectively.

It is the purpose of this paper to study more general equations (1.2) and (1.3) and generalize the main results of Raffoul [14]. We establish some existence, multiplicity, and nonexistence results of positive periodic solutions for (1.2) and (1.3), respectively. The main tools we will use are fixed point theorem in cones and fixed point index theory [15, 16]. Throughout this paper, we denote the product of 𝑦(𝑛) from 𝑛=𝑎 to 𝑛=𝑏 by ∏^𝑏_𝑛=𝑎𝑦(𝑛) with the understanding that ∏^𝑏_𝑛=𝑎𝑦(𝑛)=1 for all 𝑎>𝑏.

The rest of the paper is arranged as follows: in Section 2, we give some preliminary results. Section 3 is devoted to generalize the main results of Raffoul [14]. Finally, in Section 4, we state and prove some existence, multiplicity, and nonexistence results of positive periodic solutions for (1.2) and (1.3). For related results on the associated differential equations, see Wang [11].

2. Preliminaries

In this paper, we make the following assumptions.(H1)𝑎,𝑏∶ℤ→[0,+∞) are 𝑇-periodic, and 𝑎(𝑛)≢0, 𝑏(𝑛)≢0 on 𝑛∈[0,𝑇−1], 𝜏∶ℤ→ℤ is 𝑇-periodic.(H2)𝑓,𝑔∶[0,+∞)→[0,+∞) are continuous, 0<𝑙≤𝑔(𝑥)≤𝐿<∞ with 𝑙,𝐿 are positive constants, and 𝑓(𝑥)>0, for 𝑥>0. Also, 𝜎_𝑙=^𝑇−1_𝑠=0(1+𝑎(𝑠)𝑙),𝜎_𝐿=^𝑇−1_𝑠=0(1+𝑎(𝑠)𝐿), (2.1) it is clear that 1<𝜎_𝑙≤𝜎_𝐿. Let 𝑀(𝑟)𝑓(𝑥)(𝑟)𝑓(𝑥)∣𝜎=max∣0≤𝑥≤𝑟,𝑚=min^𝑙−1𝜎_𝐿−1𝜎_𝐿⋅𝑟≤𝑥≤𝑟. (2.2)

The following well-known result of fixed point index is crucial to our arguments.

Lemma 2.1 (see [15, 16]). Let 𝐸 be a Banach space, and let 𝐾 be a cone in 𝐸. For 𝑟>0, define 𝐾_𝑟={𝑢∈𝐾∶‖𝑢‖<𝑟}. Assume that 𝑇∶𝐾_𝑟→𝐾 is completely continuous such that 𝑇𝑢≠𝑢 for 𝑢∈𝜕𝐾_𝑟={𝑢∈𝐾∶‖𝑢‖=𝑟}.(i)If ‖𝑇𝑢‖≥‖𝑢‖ for 𝑢∈𝜕𝐾_𝑟, then 𝑖𝑇,𝐾_𝑟,𝐾=0. (2.3)(ii)If ‖𝑇𝑢‖≤‖𝑢‖ for 𝑢∈𝜕𝐾_𝑟, then 𝑖𝑇,𝐾_𝑟,𝐾=1. (2.4)

Let 𝑋 be the set of all real 𝑇-periodic sequences. This set, endowed with the maximum norm ‖𝑥‖=max_{𝑛∈[0,𝑇−1]}|𝑥(𝑛)|, is a Banach space. The next lemma is essential in obtaining our results.

Lemma 2.2. Assume that (H1)-(H2) hold. Then, 𝑥∈𝑋 is a solution of (1.2) if and only if 𝑥(𝑛)=𝜆^{𝑛+𝑇−1}_𝑢=𝑛𝐺_𝑥(𝑛,𝑢)𝑏(𝑢)𝑓(𝑥(𝑢−𝜏(𝑢))),𝑛∈ℤ, (2.5) where 𝐺_𝑥()=∏𝑛,𝑢^{𝑛+𝑇−1𝑠=𝑢+1}1+𝑎(𝑠)𝑔(𝑥(𝑠))∏^𝑇−1_𝑠=0(s)𝑔(𝑥(𝑠[]1+𝑎))−1,𝑢∈𝑛,𝑛+𝑇−1. (2.6)

Proof. If 𝑥∈𝑋 and satisfies (2.5), then 𝑥(𝑛+1)=𝜆^𝑛+𝑇_𝑢=𝑛+1𝐺_𝑥(𝑛+1,𝑢)𝑏(𝑢)𝑓(𝑥(𝑢−𝜏(𝑢)))=𝜆^{𝑛+𝑇−1}_𝑢=𝑛+1𝐺_𝑥(𝑛+1,𝑢)𝑏(𝑢)𝑓(𝑥(𝑢−𝜏(𝑢)))+𝜆𝐺_𝑥(𝑛+1,𝑛+𝑇)𝑏(𝑛+𝑇)𝑓(𝑥(𝑛+𝑇−𝜏(𝑛+𝑇)))=𝜆^{𝑛+𝑇−1}_𝑢=𝑛+1𝐺_𝑥(𝑛+1,𝑢)𝑏(𝑢)𝑓(𝑥(𝑢−𝜏(𝑢)))+𝜆𝐺_𝑥()𝑏(𝑛)𝑓(𝑥((𝑛,(𝑛)𝑔(𝑥(𝑛𝑥(𝑛)𝑛+1,𝑛+𝑇𝑛−𝜏)))1+𝑎))=𝜆^{𝑛+𝑇−1}_𝑢=𝑛+11+𝑎(𝑛)𝑔(𝑥(𝑛))𝐺_𝑥()𝑏(𝑢)𝑓(𝑥((𝑢(𝑛)𝑔(𝑥(𝑛𝐺𝑛,𝑢𝑢−𝜏)))+𝜆1+𝑎))_𝑥(𝑛,𝑛)𝑏(𝑛)𝑓(𝑥(𝑛−𝜏(𝑛))). (2.8) From (2.2), we know that 𝐺_𝑥()=∏𝑛+1,𝑛+𝑇^{𝑛+𝑇𝑠=𝑛+𝑇+1}1+𝑎(𝑠)𝑔(𝑥(𝑠))∏^𝑇−1_𝑠=0(𝑠)𝑔(𝑥(𝑠=11+𝑎))−1∏^𝑇−1_𝑠=0(𝑠)𝑔(𝑥(𝑠,(𝑛)𝑔(𝑥(𝑛𝐺1+𝑎))−11+𝑎))_𝑥()=∏𝑛,𝑛^{𝑛+𝑇−1𝑠=𝑛}1+𝑎(𝑠)𝑔(𝑥(𝑠))∏^𝑇−1_𝑠=0(𝑠)𝑔(𝑥(𝑠=∏1+𝑎))−1_{𝑇−1𝑠=0}1+𝑎(𝑠)𝑔(𝑥(𝑠))∏^𝑇−1_𝑠=0(𝑠)𝑔(𝑥(𝑠,𝐺1+𝑎))−1_𝑥()=∏𝑛+1,𝑢^{𝑛+𝑇𝑠=𝑢+1}1+𝑎(𝑠)𝑔(𝑥(𝑠))∏^𝑇−1_𝑠=0(𝑠)𝑔(𝑥(𝑠=∏1+𝑎))−1_{𝑛+𝑇−1𝑠=𝑢+1}1+𝑎(𝑠)𝑔(𝑥(𝑠))1+𝑎(𝑛+𝑇)𝑔(𝑥(𝑛+𝑇))∏^𝑇−1_𝑠=0(𝑠)𝑔(𝑥(𝑠=(𝑛)𝑔(𝑥(𝑛𝐺1+𝑎))−11+𝑎))_𝑥(𝑛,𝑢). (2.9) So, by (2.8)-(2.9), we can conclude that 𝑥(𝑛+1)−1+𝑎(𝑛)𝑔(𝑥(𝑛))𝑥(𝑛)=−𝜆𝑏(𝑛)𝑓(𝑥(𝑛−𝜏(𝑛))), (2.10) thus, 𝑥 is a 𝑇-periodic solutions of (1.2). On the other hand, if 𝑥∈𝑋 and satisfies (1.2), then (1.2) is equivalent to Δ^𝑛−1_𝑠=−∞1+𝑎(𝑠)𝑔(𝑥(𝑠))⁻¹𝑥(𝑛)=−𝜆𝑏(𝑛)𝑓(𝑥(𝑛−𝜏(𝑛)))^𝑛_𝑠=−∞1+𝑎(𝑠)𝑔(𝑥(𝑠))⁻¹, (2.11) by summing the above equation from 𝑢=𝑛 to 𝑢=𝑛+𝑇−1, we obtain (2.5).

Note that since 𝑎(𝑛)≢0 for 𝑛∈[0,𝑇−1] and 0<𝑙≤𝑔(𝑥)≤𝐿<∞, we have 1𝜎_𝐿−1≤𝐺_𝑥()≤𝜎𝑛,𝑢^𝐿𝜎_𝑙−1,𝑛≤𝑢≤𝑛+𝑇−1, (2.12) and 0<(𝜎_𝑙−1)/(𝜎_𝐿−1)𝜎_𝐿<1.

Define 𝐾 as a cone in 𝑋 by (𝑛)(𝑛)≥𝜎𝐾=𝑥∈𝑋∶𝑥≥0,𝑛∈ℤ,𝑥^𝑙−1𝜎_𝐿−1𝜎_𝐿‖𝑥‖. (2.13) Also, define, for 𝑟 a positive number, Ω_𝑟 by Ω_𝑟={𝑥∈𝐾∶‖𝑥‖<𝑟}, (2.14) note that 𝜕Ω_𝑟={𝑥∈𝐾∶‖𝑥‖=𝑟}.

Define 𝑇_𝜆∶𝑋→𝑋 by 𝑇_𝜆𝑥(𝑛)=𝜆^{𝑛+𝑇−1}_𝑢=𝑛𝐺_𝑥(𝑛,𝑢)𝑏(𝑢)𝑓(𝑥(𝑢−𝜏(𝑢))),𝑛∈ℤ, (2.15) where 𝐺_𝑥(𝑛,𝑢) is given by (2.6).

Lemma 2.3. Assume that (H1)-(H2) hold. Then, 𝑇_𝜆(𝐾)⊂𝐾 and 𝑇_𝜆∶𝐾→𝐾 is compact and continuous.

Proof. In the view of the definition of 𝐾, for 𝑥∈𝐾, we have 𝑇_𝜆𝑥(𝑛+𝑇)=𝜆^{𝑛+2𝑇−1}_{𝑢=𝑛+𝑇}𝐺_𝑥(𝑛+𝑇,𝑢)𝑏(𝑢)𝑓(𝑥(𝑢−𝜏(𝑢)))=𝜆^{𝑛+𝑇−1}_𝑠=𝑛𝐺_𝑥(𝑛+𝑇,𝑠+𝑇)𝑏(𝑠+𝑇)𝑓(𝑥(𝑠+𝑇−𝜏(𝑠+𝑇)))=𝜆^{𝑛+𝑇−1}_𝑢=𝑛𝐺_𝑥()𝑏(𝑢)𝑓(𝑥((𝑢=𝑇𝑛+𝑇,𝑢+𝑇𝑢−𝜏)))_𝜆𝑥(𝑛). (2.16) In fact, 𝐺_𝑥()=∏𝑛+𝑇,𝑢+𝑇^{𝑛+2𝑇−1𝑠=𝑢+𝑇+1}1+𝑎(𝑠)𝑔(𝑥(𝑠))∏^𝑇−1_𝑠=0(𝑠)𝑔(𝑥(𝑠=∏1+𝑎))−1_{𝑛+𝑇−1𝑠=𝑢+1}1+𝑎(𝑠)𝑔(𝑥(𝑠))∏^𝑇−1_𝑠=01+𝑎(𝑠)𝑔(𝑥(𝑠))−1=𝐺_𝑥(𝑛,𝑢). (2.17) So, 𝑇_𝜆𝑥∈𝑋. One can show that for 𝑥∈𝐾 and 𝑛∈[0,𝑇−1], 𝑇_𝜆𝑥(𝑛)≥𝜆𝜎_𝐿−1^{𝑛+𝑇−1}_𝑢=𝑛𝑏(𝑢)𝑓(𝑥((𝑢=𝜎𝑢−𝜏)))_𝑙−1𝜎_𝐿−1𝜎_𝐿⋅𝜎_𝐿𝜎_𝑙−1⋅𝜆^{𝑛+𝑇−1}_𝑢=𝑛𝑏(𝑢)𝑓(𝑥((𝑢≥𝜎𝑢−𝜏)))_𝑙−1𝜎_𝐿−1𝜎_𝐿𝑇_𝜆𝑥. (2.18) Thus, 𝑇_𝜆(𝐾)⊂𝐾, and it is easy to show that 𝑇_𝜆∶𝐾→𝐾 is compact and continuous.

Consequently, 𝑥∈𝐾 is a solution of (1.2) if and only if 𝑥 is a fixed point of 𝑇_𝜆 in 𝐾.

Lemma 2.4. Assume that (H1)-(H2) hold. Let 𝜂>0, if 𝑥∈𝐾 and 𝑓(𝑥(𝑛))≥𝑥(𝑛)𝜂 for 𝑛∈[0,𝑇−1], then 𝑇_𝜆𝑥𝜂𝜎≥𝜆^𝑙−1𝜎_𝐿−1²𝜎_𝐿⋅^𝑇−1_𝑢=0𝑏(𝑢)⋅‖𝑥‖. (2.19)

Proof. Since 𝑥∈𝐾 and 𝑓(𝑥(𝑛))≥𝑥(𝑛)𝜂 for 𝑛∈[0,𝑇−1], we have 𝑇_𝜆𝑥(𝑛)≥𝜆𝜎_𝐿−1^{𝑛+𝑇−1}_𝑢=𝑛𝑏(𝑢)𝑓(𝑥((𝑢=𝑢−𝜏)))𝜆𝜂𝜎_𝐿⋅𝜎−1^𝑙−1𝜎_𝐿−1𝜎_𝐿^𝑇−1_𝑢=0𝑏(𝑢)⋅‖𝑥‖𝜂𝜎=𝜆^𝑙−1𝜎_𝐿−1²𝜎_𝐿⋅^𝑇−1_𝑢=0𝑏(𝑢)⋅‖𝑥‖. (2.20) Thus, 𝑇_𝜆𝑥𝜂𝜎≥𝜆^𝑙−1𝜎_𝐿−1²𝜎_𝐿⋅^𝑇−1_𝑢=0𝑏(𝑢)⋅‖𝑥‖. (2.21)

Lemma 2.5. Assume that (H1)-(H2) hold. Let 𝑟>0, if 𝑥∈𝜕Ω_𝑟, and there exists an 𝜀>0 such that 𝑓(𝑥(𝑛))≤𝑥(𝑛)𝜀 for 𝑛∈[0,𝑇−1], then 𝑇_𝜆𝑥𝜎≤𝜆𝜀^𝐿𝜎_𝑙⋅−1^𝑇−1_𝑢=0𝑏(𝑢)⋅‖𝑥‖. (2.22)

Proof. From the definition of 𝑇_𝜆, for 𝑥∈𝜕Ω_𝑟, we have 𝑇_𝜆𝑥𝜎≤𝜆^𝐿𝜎_𝑙⋅−1^𝑇−1_𝑢=0𝑏(𝑢)𝑓(𝑥((𝑢𝜎𝑢−𝜏)))≤𝜆𝜀^𝐿𝜎_𝑙⋅−1^𝑇−1_𝑢=0𝑏(𝑢)⋅‖𝑥‖. (2.23)

The following two lemmas are weak forms of Lemmas 2.4 and 2.5.

Lemma 2.6. Assume that (H1)-(H2) hold. If 𝑥∈𝜕Ω_𝑟, 𝑟>0, then 𝑇_𝜆𝑥∑≥𝜆^{𝑇−1𝑢=0}𝑏(𝑢)𝜎_𝐿(𝑟).−1⋅𝑚 (2.24)

Proof. Since 𝑓(𝑥(𝑛))≥𝑚(𝑟) for 𝑛∈[0,𝑇−1], it is easy to see that this lemma can be achieved in a similar manner as in Lemma 2.4.

Lemma 2.7. Assume that (H1)-(H2) hold. If 𝑥∈𝜕Ω_𝑟, 𝑟>0, then 𝑇_𝜆𝑥𝜎≤𝜆^𝐿𝑀(𝑟)𝜎_𝑙⋅−1^𝑇−1_𝑢=0𝑏(𝑢). (2.25)

Proof. Since 𝑓(𝑥(𝑛))≤𝑀(𝑟) for 𝑛∈[0,𝑇−1], it is easy to see that this lemma can be obtained in a similar manner as in Lemma 2.5.

3. Generalization of the Main Results in [14]

Let 𝑓₀=lim_𝑥→0+𝑓(𝑥)𝑥,𝑓_∞=lim_𝑥→∞𝑓(𝑥)𝑥. (3.1) In this Section, we make the following assumptions.(L1)𝑓₀=∞. (L2)𝑓_∞=∞. (L3)𝑓₀=0. (L4)𝑓_∞=0. (L5)𝑓₀=𝑚 with 0<𝑚<∞.(L6)𝑓₀=𝑀 with 0<𝑀<∞. And let 𝐴=max_{0≤𝑛≤𝑇−1}^𝑇−1_𝑢=0𝐺_𝑥(𝑛,𝑢)𝑏(𝑢),𝐵=min_{0≤𝑛≤𝑇−1}^𝑇−1_𝑢=0𝐺_𝑥(𝑛,𝑢)𝑏(𝑢). (3.2)

Theorem 3.1. Assume that (H1), (H2), (L5), and (L6) hold. Then, for each 𝜆 satisfying 𝜎_𝐿𝜎_𝐿−1𝜎_𝑙⋅1−11𝐵𝑀<𝜆<,𝐴𝑚 (3.3) or 𝜎_𝐿𝜎_𝐿−1𝜎_𝑙⋅1−11𝐵𝑚<𝜆<,𝐴𝑀 (3.4) equation (1.2) has at least one positive 𝑇-periodic solutions.

Proof. Using the same method to prove[14, Theorem 2.3] with obvious changes, we can prove Theorem 3.1. The process of the proof is omitted.

Theorem 3.2. Assume that (H1) and (H2) hold. If (L1) and (L4) hold, or (L2) and (L3) hold, then (1.2) has at least one positive 𝑇-periodic solution for any 𝜆>0.

Proof. The proof is similar to arguments to prove [14, Theorem 2.4].

The next two corollaries are consequences of the previous two theorems.

Corollary 3.3. Assume that (H1) and (H2) hold. If (L1) and (L6) hold, or (L2) and (L5) hold, then (1.2) has at least one positive 𝑇-periodic solution if 𝜆 satisfies either 0<𝜆<1/𝐴𝑀 or 0<𝜆<1/𝐴𝑚.

Corollary 3.4. Assume that (H1) and (H2) hold. Also, if (L3) and (L6) hold, or (L4) and (L5) hold, then (1.2) has at least one positive 𝑇-periodic solution if 𝜆 satisfies either 𝜎_𝐿(𝜎_𝐿−1)/(𝜎_𝑙−1)⋅(1/𝐵𝑀)<𝜆<∞ or 𝜎_𝐿(𝜎_𝐿−1)/(𝜎_𝑙−1)⋅(1/𝐵𝑚)<𝜆<∞.

4. Existence, Multiplicity, and Nonexistence Results of PositivePeriodic Solution(s) for (1.2) and (1.3)

In this section, we use the notations 𝑖₀=numberofzerosintheset𝑓₀,𝑓_∞,𝑖_∞=numberofinfinitiesintheset𝑓₀,𝑓_∞. (4.1) It is clear that 𝑖₀,𝑖_∞=0,1 or 2. Then, we will show that (1.2) has 𝑖₀ or 𝑖_∞ positive 𝑇-periodic solution(s) for sufficiently large or small 𝜆.

Theorem 4.1. Assume that (H1)-(H2) hold.(a) If 𝑖₀=1 or 2, then (1.2) has 𝑖₀ positive 𝑇-periodic solutions for 𝜆>(𝜎_𝐿−1)/𝑚(1)∑^𝑇−1_𝑢=0𝑏(𝑢)>0.(b) If 𝑖_∞=1 or 2, then (1.2) has 𝑖_∞ positive 𝑇-periodic solutions for 0<𝜆<(𝜎_𝑙−1)/𝜎_𝐿𝑀(1)∑^𝑇−1_𝑢=0𝑏(𝑢).(c) If 𝑖₀=0 or 𝑖_∞=0, then (1.2) has no positive 𝑇-periodic solution for sufficiently large or small 𝜆>0, respectively.

Proof. (a) Choose 𝑟₁=1. By Lemma 2.6, we infer that there exists a 𝜆₀=(𝜎_𝐿−1)/𝑚(𝑟₁)∑^𝑇−1_𝑢=0𝑏(𝑢)>0, such that 𝑇_𝜆𝑥>‖𝑥‖for𝑥∈𝜕Ω_𝑟1,𝜆>𝜆₀. (4.2)
If 𝑓₀=0, we can choose 0<𝑟₂<𝑟₁ so that 𝑓(𝑥)≤𝜀𝑥 for 0≤𝑥≤𝑟₂, where the constant 𝜀>0 satisfies 𝜎𝜆𝜀^𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)𝜎_𝑙−1<1. (4.3) Thus, 𝑓(𝑥(𝑛))≤𝜀𝑥(𝑛) for 𝑥∈𝜕Ω_𝑟2 and 𝑛∈[0,𝑇−1]. We have by Lemma 2.5 that 𝑇_𝜆𝑥𝜎≤𝜆𝜀^𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)𝜎_𝑙‖𝑥‖<‖𝑥‖−1for𝑥∈𝜕Ω_𝑟2. (4.4) It follows from Lemma 2.1 that 𝑖𝑇_𝜆,Ω_𝑟1,𝐾=0,𝑖𝑇_𝜆,Ω_𝑟2,𝐾=1. (4.5) Thus, 𝑖(𝑇_𝜆,Ω_𝑟1⧵Ω_𝑟2,𝐾)=−1 and 𝑇_𝜆 has a fixed point in Ω_𝑟1⧵Ω_𝑟2, which is a positive 𝑇-periodic solution of (1.2) for 𝜆>𝜆₀.
If 𝑓_∞=0, there is an 𝐻>0 such that 𝑓(𝑥)≤𝜀𝑥 for 𝑥≥𝐻, where 𝜀>0 satisfies (4.3). Let 𝑟₃=max{2𝑟₁,𝐻/((𝜎_𝑙−1)/𝜎_𝐿(𝜎_𝐿−1))}, and it follows that 𝑥(𝑛)≥((𝜎_𝑙−1)/𝜎_𝐿(𝜎_𝐿−1))‖𝑥‖≥𝐻 for 𝑥∈𝜕Ω_𝑟3 and 𝑛∈[0,𝑇−1]. Thus, 𝑓(𝑥(𝑛))≤𝜀𝑥(𝑛) for 𝑥∈𝜕Ω_𝑟3 and 𝑛∈[0,𝑇−1]. In view of Lemma 2.5, we have 𝑇_𝜆𝑥𝜎≤𝜆𝜀^𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)𝜎_𝑙‖𝑥‖<‖𝑥‖−1for𝑥∈𝜕Ω_𝑟3. (4.6) Again, it follows from Lemma 2.1 that 𝑖𝑇_𝜆,Ω_𝑟1,𝐾=0,𝑖𝑇_𝜆,Ω_𝑟3,𝐾=1. (4.7) Thus, 𝑖(𝑇_𝜆,Ω_𝑟3⧵Ω_𝑟1,𝐾)=1 and (1.2) has a positive 𝑇-periodic solution for 𝜆>𝜆₀.
If 𝑓₀=𝑓_∞=0, it is easy to see from the above proof that 𝑇_𝜆 has a fixed point 𝑥₁ in Ω_𝑟1⧵Ω_𝑟2 and a fixed point 𝑥₂ in Ω_𝑟3⧵Ω_𝑟1 such that 𝑟₂<𝑥₁<𝑟₁<𝑥₂<𝑟₃. (4.8) Consequently, (1.2) has two positive 𝑇-periodic solutions for 𝜆>𝜆₀ if 𝑓₀=𝑓_∞=0.
(b) Choose 𝑟₁=1. By Lemma 2.7, we infer that there exists 𝜆₀=𝜎_𝑙−1𝜎_𝐿𝑀𝑟₁∑^𝑇−1_𝑢=0𝑏(𝑢)>0, (4.9) such that 𝑇_𝜆𝑥<‖𝑥‖for𝑥∈𝜕Ω_𝑟1,0<𝜆<𝜆₀. (4.10) If 𝑓₀=∞, there is a positive number 𝑟₂<𝑟₁ such that 𝑓(𝑥)≥𝜂𝑥 for 0≤𝑥≤𝑟₂, where 𝜂>0 is chosen so that 𝜆𝜂𝜎_𝑙−1𝜎_𝐿−1²𝜎_𝐿^𝑇−1_𝑢=0𝑏(𝑢)>1. (4.11) Then, 𝑓(𝑥(𝑛))≥𝜂𝑥(𝑛)for𝑥∈𝜕Ω_𝑟2,𝑛∈[0,𝑇−1]. (4.12) By Lemma 2.4, we have that 𝑇_𝜆𝑥𝜎≥𝜆𝜂^𝑙−1𝜎_𝐿−1²𝜎_𝐿^𝑇−1_𝑢=0𝑏(𝑢)⋅‖𝑥‖>‖𝑥‖for𝑥∈𝜕Ω_𝑟2. (4.13) It follows from Lemma 2.1 that 𝑖𝑇_𝜆,Ω_𝑟1,𝐾=1,𝑖𝑇_𝜆,Ω_𝑟2,𝐾=0. (4.14) Thus, 𝑖(𝑇_𝜆,Ω_𝑟1⧵Ω_𝑟2,𝐾)=1 and 𝑇_𝜆 has a fixed point in Ω_𝑟1⧵Ω_𝑟2 for 0<𝜆<𝜆₀, which is a positive 𝑇-periodic solution of (1.2).
If 𝑓_∞=∞, there is an 𝐻>0 such that 𝑓(𝑥)≥𝜂𝑥 for 𝑥≥𝐻, where 𝜂>0 satisfies (4.11). Let 𝑟₃=max{2𝑟₁,𝐻/((𝜎_𝑙−1)/𝜎_𝐿(𝜎_𝐿−1))}. If 𝑥∈𝜕Ω_𝑟3, then 𝑥(𝑛)≥𝜎_𝑙−1𝜎_𝐿𝜎_𝐿‖𝑥‖≥−1𝐻, (4.15) and hence, 𝑓(𝑥(𝑛))≥𝜂𝑥(𝑛)for𝑛∈[0,𝑇−1]. (4.16) Again, it follows from Lemma 2.4 that 𝑇_𝜆𝑥𝜂𝜎≥𝜆^𝑙−1𝜎_𝐿−1²𝜎_𝐿^𝑇−1_𝑢=0𝑏(𝑢)‖𝑥‖>‖𝑥‖for𝑥∈𝜕Ω_𝑟3. (4.17) It follows from Lemma 2.1 that 𝑖𝑇_𝜆,Ω_𝑟1,𝐾=1,𝑖𝑇_𝜆,Ω_𝑟3,𝐾=0, (4.18) and hence, 𝑖(𝑇_𝜆,Ω_𝑟3⧵Ω_𝑟1,𝐾)=−1. Thus, 𝑇_𝜆 has a fixed point in Ω_𝑟3⧵Ω_𝑟1 for 0<𝜆<𝜆₀, which is a positive 𝑇-periodic solution of (1.2).
If 𝑓₀=𝑓_∞=∞, it is easy to see from the above proof that 𝑇_𝜆 has a fixed point 𝑥₁ in Ω_𝑟1⧵Ω_𝑟2 and a fixed point 𝑥₂ in Ω_𝑟3⧵Ω_𝑟1 such that 𝑟₂<𝑥₁<𝑟₁<𝑥₂<𝑟₃. (4.19) Consequently, (1.2) has two positive 𝑇-periodic solutions for 0<𝜆<𝜆₀ if 𝑓₀=𝑓_∞=∞.
(c) If 𝑖₀=0, then 𝑓₀>0 and 𝑓_∞>0. It follows that there exist positive numbers 𝜂₁, 𝜂₂, 𝑟₁ and 𝑟₂, such that 𝑟₁<𝑟₂ and 𝑓(𝑥)≥𝜂₁𝑥for𝑥∈0,𝑟₁,𝑓(𝑥)≥𝜂₂𝑥for𝑥∈𝑟₂,∞. (4.20) Let 𝑐₁=min{𝜂₁,𝜂₂,min_{𝑟1≤𝑥≤𝑟2}{𝑓(𝑥)/𝑥}}>0. Thus, we have 𝑓(𝑥)≥𝑐₁𝑥for𝑥∈[0,∞). (4.21) Assume 𝑣 is a positive 𝑇-periodic solution of (1.2). We will show that this leads to a contradiction for 𝜆>𝜆₀, where 𝜆₀=(𝜎_𝐿−1)²𝜎_𝐿/𝑐₁(𝜎_𝑙−1)∑^𝑇−1_𝑢=0𝑏(𝑢). Since 𝑇_𝜆𝑣(𝑛)=𝑣(𝑛) for 𝑛∈[0,𝑇−1], it follows from Lemma 2.4 that for 𝜆>𝜆₀, ‖𝑣‖=𝑇_𝜆𝑣𝜎≥𝜆^𝑙−1∑^𝑇−1_𝑢=0𝑏(𝑢)𝜎_𝐿−1²𝜎_𝐿𝑐₁‖𝑣‖>‖𝑣‖, (4.22) which is a contradiction.
If 𝑖_∞=0, then 𝑓₀<∞ and 𝑓_∞<∞. It follows that there exist positive numbers 𝜀₁, 𝜀₂, 𝑟₁ and 𝑟₂ such that 𝑟₁<𝑟₂, 𝑓(𝑥)≤𝜀₁𝑥for𝑥∈0,𝑟₁,𝑓(𝑥)≤𝜀₂𝑥for𝑥∈𝑟₂,∞. (4.23) Let 𝑐₂=max{𝜀₁,𝜀₂,max_{𝑟1≤𝑥≤𝑟2}{𝑓(𝑥)/𝑥}}>0. Thus, we have 𝑓(𝑥)≤𝑐₂𝑥for𝑥∈[0,∞). (4.24) Assume 𝑣 is a positive 𝑇-periodic solution of (1.2). We will show that this leads to a contradiction for 0<𝜆<𝜆₀, where 𝜆₀=(𝜎_𝑙−1)/𝑐₂𝜎_𝐿∑^𝑇−1_𝑢=0𝑏(𝑢). Since 𝑇_𝜆𝑣(𝑛)=𝑣(𝑛) for 𝑛∈[0,𝑇−1], it follows from Lemma 2.5 that for 0<𝜆<𝜆₀, ‖𝑣‖=𝑇_𝜆𝑣𝜎≤𝜆^𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)𝜎_𝑙𝑐−1²‖𝑣‖<‖𝑣‖, (4.25) which is a contradiction.

The following result is a direct consequence of Theorem 4.1(c).

Corollary 4.2. Assume that (H1)-(H2) hold.(a) If there exists a 𝑐₁>0 such that 𝑓(𝑥)≥𝑐₁𝑥 for 𝑥∈[0,∞), then there is a 𝜆₀=(𝜎_𝐿−1)²𝜎_𝐿/𝑐₁(𝜎_𝑙−1)∑^𝑇−1_𝑢=0𝑏(𝑢) such that for all 𝜆>𝜆₀, (1.2) has no positive 𝑇-periodic solutions.(b) If there exists a 𝑐₂>0 such that 𝑓(𝑥)≤𝑐₂𝑥 for 𝑥∈[0,∞), then there is a 𝜆₀=(𝜎_𝑙−1)/𝑐₂𝜎_𝐿∑^𝑇−1_𝑢=0𝑏(𝑢) such that for all 0<𝜆<𝜆₀, (1.2) has no positive 𝑇-periodic solutions.

Theorem 4.3. Assume that (H1)-(H2) hold and 𝑖₀=𝑖_∞=0. If 𝜎_𝐿−1²𝜎_𝐿𝜎_𝑙−1∑^𝑇−1_𝑢=0𝑏(𝑢)⋅1max𝑓₀,𝑓_∞𝜎<𝜆<^𝑙−1𝜎_𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)⋅1min𝑓₀,𝑓_∞, (4.26) then (1.2) has a positive 𝑇-periodic solutions.

Proof. If 𝑓_∞>𝑓₀, then (𝜎_𝐿−1)²𝜎_𝐿/(𝜎_𝑙−1)∑^𝑇−1_𝑢=0𝑏(𝑢)𝑓_∞<𝜆<(𝜎_𝑙−1)/𝜎_𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)𝑓₀. It is easy to see that there exists 𝜀∶0<𝜀<𝑓_∞ such that 𝜎_𝐿−1²𝜎_𝐿𝜎_𝑙−1∑^𝑇−1_𝑢=0𝑏(𝑢)𝑓_∞𝜎−𝜀<𝜆<^𝑙−1𝜎_𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)𝑓₀.+𝜀 (4.27) Now, turning to 𝑓₀ and 𝑓_∞, there is an 𝑟₁>0, such that 𝑓(𝑥)≤(𝑓₀+𝜀)𝑥 for 0≤𝑥≤𝑟₁. Thus, 𝑓(𝑥(𝑛))≤(𝑓₀+𝜀)𝑥(𝑛) for 𝑥∈𝜕Ω_𝑟1 and 𝑛∈[0,𝑇−1]. We have by Lemma 2.5 that 𝑇_𝜆𝑢≤𝜆𝑓₀𝜎+𝜀^𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)𝜎_𝑙‖𝑥‖<‖𝑥‖−1for𝑥∈𝜕Ω_𝑟1. (4.28) On the other hand, there is an 𝐻>𝑟₁ such that 𝑓(𝑥)≥(𝑓_∞−𝜀)𝑥 for 𝑥≥𝐻. Let 𝑟₂=max{2𝑟₁,𝐻/((𝜎_𝑙−1)/𝜎_𝐿(𝜎_𝐿−1))}, and it follows that 𝑥(𝑛)≥((𝜎_𝑙−1)/𝜎_𝐿(𝜎_𝐿−1))‖𝑥‖≥𝐻 for 𝑥∈𝜕Ω_𝑟2 and 𝑛∈[0,𝑇−1]. In view of Lemma 2.4, we have 𝑇_𝜆𝑢≥𝜆𝑓_∞𝜎−𝜀^𝑙−1∑^𝑇−1_𝑢=0𝑏(𝑢)𝜎_𝐿−1²𝜎_𝐿‖𝑥‖>‖𝑥‖for𝑥∈𝜕Ω_𝑟2. (4.29) It follows from Lemma 2.1 that 𝑖𝑇_𝜆,Ω_𝑟1,𝐾=1,𝑖𝑇_𝜆,Ω_𝑟2,𝐾=0. (4.30) Thus, 𝑖(𝑇_𝜆,Ω_𝑟2⧵Ω_𝑟1,𝐾)=−1. Hence, 𝑇_𝜆 has a fixed point in Ω_𝑟2⧵Ω_𝑟1. Consequently, (1.2) has a positive 𝑇-periodic solution.
If 𝑓_∞<𝑓₀, then (𝜎_𝐿−1)²𝜎_𝐿/(𝜎_𝑙−1)∑^𝑇−1_𝑢=0𝑏(𝑢)𝑓₀<𝜆<(𝜎_𝑙−1)/𝜎_𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)𝑓_∞. It is easy to see that there exists an 𝜀∶0<𝜀<𝑓₀ such that 𝜎_𝐿−1²𝜎_𝐿𝜎_𝑙−1∑^𝑇−1_𝑢=0𝑏(𝑢)𝑓₀𝜎−𝜀<𝜆<^𝑙−1𝜎_𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)𝑓_∞.+𝜀 (4.31) Now, turning to 𝑓₀ and 𝑓_∞, there is an 𝑟₁>0 such that 𝑓(𝑥)≥(𝑓₀−𝜀)𝑥 for 0≤𝑥≤𝑟₁. Thus, 𝑓(𝑥(𝑛))≥(𝑓₀−𝜀)𝑥(𝑛) for 𝑥∈𝜕Ω_𝑟1 and 𝑛∈[0,𝑇−1]. We have by Lemma 2.4 that 𝑇_𝜆𝑢≥𝜆𝑓₀𝜎−𝜀^𝑙−1∑^𝑇−1_𝑢=0𝑏(𝑢)𝜎_𝐿−1²𝜎_𝐿‖𝑥‖>‖𝑥‖for𝑥∈𝜕Ω_𝑟1. (4.32) On the other hand, there is an 𝐻>𝑟₁ such that 𝑓(𝑥)≤(𝑓_∞+𝜀)𝑥 for 𝑥≥𝐻. Let 𝑟₂=max{2𝑟₁,𝐻/((𝜎_𝑙−1)/𝜎_𝐿(𝜎_𝐿−1))}, and it follows that 𝑥(𝑛)≥((𝜎_𝑙−1)/𝜎_𝐿(𝜎_𝐿−1))‖𝑥‖≥𝐻 for 𝑥∈𝜕Ω_𝑟2 and 𝑛∈[0,𝑇−1]. Thus, 𝑓(𝑥(𝑛))≤(𝑓_∞+𝜀)𝑥(𝑛) for 𝑥∈𝜕Ω_𝑟2 and 𝑛∈[0,𝑇−1]. In view of Lemma 2.5, we have 𝑇_𝜆𝑢≤𝜆𝑓_∞𝜎+𝜀^𝐿∑^𝑇−1_𝑢=0𝑏(𝑢)𝜎_𝑙‖𝑥‖<‖𝑥‖−1for𝑥∈𝜕Ω_𝑟2. (4.33) It follows from Lemma 2.1 that 𝑖𝑇_𝜆,Ω_𝑟1,𝐾=0,𝑖𝑇_𝜆,Ω_𝑟2,𝐾=1. (4.34) Thus, 𝑖(𝑇_𝜆,Ω_𝑟2⧵Ω_𝑟1,𝐾)=1. Hence, 𝑇_𝜆 has a fixed point in Ω_𝑟2⧵Ω_𝑟1. Consequently, (1.2) has a positive 𝑇-periodic solution.

Next, we pay our attention to (1.3), that is 𝑥(𝑛+1)=1−𝑎(𝑛)𝑔(𝑥(𝑛))𝑥(𝑛)+𝜆𝑏(𝑛)𝑓(𝑥(𝑛−𝜏(𝑛))),𝑛∈ℤ, (4.35) where 𝜆, 𝑎, 𝑏, 𝑓(𝑥), and 𝑔(𝑥) satisfy the same assumptions stated for (1.2) except that 0≤𝑙𝑎(𝑛)≤𝐿𝑎(𝑛)<1 for all 𝑛∈[0,𝑇−1]. In view of (1.3), we have that 𝑥(𝑛)=𝜆^{𝑛+𝑇−1}_𝑢=𝑛𝐾_𝑥(𝑛,𝑢)𝑏(𝑢)𝑓(𝑥(𝑢−𝜏(𝑢))), (4.36) where 𝐾_𝑥()=∏𝑛,𝑢^{𝑛+𝑇−1𝑠=𝑢+1}1−𝑎(𝑠)𝑔(𝑥(𝑠))1−∏^𝑇−1_𝑠=0(𝑠)𝑔(𝑥(𝑠[]1−𝑎)),𝑢∈𝑛,𝑛+𝑇−1. (4.37)

Note that since 0≤𝑙𝑎(𝑛)≤𝐿𝑎(𝑛)<1 for all 𝑛∈[0,𝑇−1], we have 𝜌_𝐿1−𝜌_𝐿≤𝐾_𝑥()≤1𝑛,𝑢1−𝜌_𝑙,𝑛≤𝑢≤𝑛+𝑇−1, (4.38) here 𝜌_𝑙=^T−1_𝑠=0(1−𝑎(𝑠)𝑙),𝜌_𝐿=^𝑇−1_𝑠=0(1−𝑎(𝑠)𝐿), (4.39) and 0<𝜌_𝐿(1−𝜌_𝑙)/(1−𝜌_𝐿)<1.

Similarly, we can get the following theorems.

Theorem 4.4. Assume that (H1)-(H2) hold and 0≤𝑙𝑎(𝑛)≤𝐿𝑎(𝑛)<1 for 𝑛∈[0,𝑇−1]. (a) If 𝑖₀=1 or 2, then (1.3) has 𝑖₀ positive 𝑇-periodic solutions for 𝜆>(1−𝜌_𝐿)/𝜌_𝐿𝑚(1)∑^𝑇−1_𝑢=0𝑏(𝑢)>0.(b) If 𝑖_∞=1 or 2, then (1.3) has 𝑖_∞ positive 𝑇-periodic solutions for 0<𝜆<(1−𝜌_𝑙)/𝑀(1)∑^𝑇−1_𝑢=0𝑏(𝑢).(c) If 𝑖₀=0 or 𝑖_∞=0, then (1.3) has no positive 𝑇-periodic solution for sufficiently large or small 𝜆>0, respectively.

The following result is a direct consequence of Theorem 4.4(c).

Corollary 4.5. Assume that (H1)-(H2) hold and 0≤𝑙𝑎(𝑛)≤𝐿𝑎(𝑛)<1 for 𝑛∈[0,𝑇−1]. (a) If there exists a 𝑐₁>0 such that 𝑓(𝑥)≥𝑐₁𝑥 for 𝑥∈[0,∞), then there is a 𝜆₀=((1−𝜌_𝐿)/𝜌_𝐿)²⋅(1/𝑐₁(1−𝜌_𝑙)∑^𝑇−1_𝑢=0𝑏(𝑢)) such that for all 𝜆>𝜆₀ (1.3) has no positive 𝑇-periodic solutions.(b) If there exists a 𝑐₂>0 such that 𝑓(𝑥)≤𝑐₂𝑥 for 𝑥∈[0,∞), then there is a 𝜆₀=(1−𝜌_𝑙)/𝑐₂∑^𝑇−1_𝑢=0𝑏(𝑢) such that for all 0<𝜆<𝜆₀ (1.3) has no positive 𝑇-periodic solutions.

Theorem 4.6. Assume that (H1)-(H2) hold and 𝑖₀=𝑖_∞=0. Let 0≤𝑙𝑎(𝑛)≤𝐿𝑎(𝑛)<1 for 𝑛∈[0,𝑇−1]. If 1−𝜌_𝐿𝜌_𝐿²⋅11−𝜌_𝑙∑^𝑇−1_𝑢=0𝑏(𝑢)⋅1max𝑓₀,𝑓_∞<𝜆<1−𝜌_𝑙∑^𝑇−1_𝑢=0𝑏(𝑢)⋅1min𝑓₀,𝑓_∞, (4.40) then (1.3) has a positive 𝑇-periodic solution.

Acknowledgment

This work was supported by the NSFC (no. 11061030 and no. 11026060), the Fundamental Research Funds for the Gansu Universities.