Three Positive Periodic Solutions to Nonlinear Neutral Functional Differential Equations with Parameters on Variable Time Scales

Abstract

Using two successive reductions: B-equivalence of the system on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation and by Leggett-Williams fixed point theorem, we investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equation on variable time scales with a transition condition between two consecutive parts of the scale (𝑑/𝑑𝑡)(𝑥(𝑡)+𝑐(𝑡)𝑥(𝑡−𝛼))=𝑎(𝑡)𝑔(𝑥(𝑡))𝑥(𝑡)−∑^𝑛𝑗=1𝜆_𝑗𝑓_𝑗(𝑡,𝑥(𝑡−𝑣_𝑗(𝑡))), (𝑡,𝑥)∈𝕋₀(𝑥),Δ𝑡|_{(𝑡,𝑥)∈𝒮2𝑖}=Π^1𝑖(𝑡,𝑥)−𝑡, Δ𝑥|_{(𝑡,𝑥)∈𝒮2𝑖}=Π^2𝑖(𝑡,𝑥)−𝑥, where Π^1𝑖(𝑡,𝑥)=𝑡_2𝑖+1+𝜏_2𝑖+1(Π^2𝑖(𝑡,𝑥)) and Π^2𝑖(𝑡,𝑥)=𝐵_𝑖𝑥+𝐽_𝑖(𝑥)+𝑥,𝑖=1,2,….𝜆_𝑗(𝑗=1,2,…,𝑛) are parameters, 𝕋₀(𝑥) is a variable time scale with (𝜔,𝑝)-property, 𝑐(𝑡),𝑎(𝑡), 𝑣_𝑗(𝑡), and 𝑓_𝑗(𝑡,𝑥)(𝑗=1,2,…,𝑛) are 𝜔-periodic functions of 𝑡, 𝐵_𝑖+𝑝=𝐵_𝑖,𝐽_𝑖+𝑝(𝑥)=𝐽_𝑖(𝑥) uniformly with respect to 𝑖∈ℤ.

1. Introduction

In the last several decades, the theory of dynamic equations on time scales (DETS) has been developed very intensively. For the full description of the equations we refer to the nicely written books [1, 2] and papers [3, 4]. The equations have a very special transition condition for adjoint elements of time scales. To enlarge the field of applications of the DETS, Akhmet and Turan proposed to generalize the transition operator [5], correspondingly to investigate differential equations on variable time scales with transition condition (DETC). In [6], Akhmet and Turan proposed some basic theory of dynamic equations on variable time scales; the method of investigation is by means of two successive reductions: 𝐵-equivalence of the system [7–9] on a variable time scale to a system on a time scale and a reduction to an impulsive differential equation [5, 7]. Consequently, these results are very effective to develop methods of investigation of mechanical models with impacts.

Also, neutral differential equations arise in many areas of applied mathematics, and for this reason these equations have received much attention in the last few decades; they are not only an extension of functional differential equations but also provide good models in many fields including biology, mechanics, and economics. In particular, qualitative analysis such as periodicity and stability of solutions of neutral functional differential equations has been studied extensively by many authors. We refer to [10–19] for some recent work on the subject of periodicity and stability of neutral equations. In [20], the authors discussed a class of neutral functional differential equations with impulses and parameters on nonvariable time scales𝑥(𝑡)+𝑐(𝑡)𝑥𝑡−𝑟₁^Δ=𝑎(𝑡)𝑔(𝑡𝑥≠(𝑡𝑡)_𝑗)𝑥(𝑡)−^𝑛_𝑖=1𝜆_𝑖𝑓_𝑖𝑡,𝑥𝑡−𝜏_𝑖(𝑡),𝑥𝑡,𝑡∈𝕋,𝑗=1,2,…,𝑞,_−𝑗−𝑥𝑡^+𝑗=𝐼^𝑗𝑥𝑡_𝑗,𝑡=𝑡_𝑗,𝑗=1,2,…,𝑞,(1.1) where 𝜆_𝑖, 𝑖=1,2,…,𝑛 are parameters, 𝕋 is an 𝜔-periodic nonvariable time scale, 𝑎∈𝐶(𝕋,ℝ⁺),𝑐∈𝐶(𝕋,[0,1)) and both of them are 𝜔-periodic functions, 𝜏_𝑖∈𝐶(𝕋,ℝ),𝑖=1,2,…,𝑛 are 𝜔-periodic functions, 𝑓_𝑖∈𝐶(𝕋×ℝ⁺,ℝ⁺),𝑖=1,2,…,𝑛 are nondecreasing with respect to their second arguments and 𝜔-periodic with respect to their first arguments, respectively; 𝑔∈𝐶(ℝ,ℝ⁺) and there exist two positive constants 𝑙,𝐿 such that 0<𝑙≤𝑔(𝑥)≤𝐿<∞ for all 𝑥>0,𝐼_𝑗∈𝐶(ℝ,ℝ⁺)(𝑗=1,2,…,𝑞) and is bounded, 𝑟₁ is a constant.

To the best of authors’ knowledge, there has been no paper published on the existence of solutions to neutral functional differential equations on variable time scales. Our main purpose of this paper is by using theory of dynamic equations on variable time scales to investigate the existence of three positive periodic solutions to the nonlinear neutral functional differential equations on variable time scales with a transition condition between two consecutive parts of the scaledd𝑡(𝑥(𝑡)+𝑐(𝑡)𝑥(𝑡−𝛼))=𝑎(𝑡)𝑔(𝑥(𝑡))𝑥(𝑡)−^𝑛_𝑗=1Δ𝑡|_{(𝑡,𝑥)∈𝒮2𝑖}𝜆_𝑗𝑓_𝑗𝑡,𝑥𝑡−𝑣_𝑗(𝑡),(𝑡,𝑥)∈𝕋₀(𝑥),Δ𝑥∣_{(𝑡,𝑥)∈𝒮2𝑖}=Π^1𝑖(𝑡,𝑥)−𝑡,=Π^2𝑖(𝑡,𝑥)−𝑥,(1.2) where Π^1𝑖(𝑡,𝑥)=𝑡_2𝑖+1+𝜏_2𝑖+1(Π^2𝑖(𝑡,𝑥)) and Π^2𝑖(𝑡,𝑥)=𝐵_𝑖𝑥+𝐽_𝑖(𝑥)+𝑥,𝑖=1,2,…𝜆_𝑗(𝑗=1,2,…,𝑛) are parameters, 𝕋₀(𝑥) is a variable time scale with (𝜔,𝑝)-property, 𝑐(𝑡),𝑎(𝑡), 𝑣_𝑗(𝑡), and 𝑓_𝑗(𝑡,𝑥)(𝑗=1,2,…,𝑛) are 𝜔-periodic functions of 𝑡, 𝐵_𝑖+𝑝=𝐵_𝑖,𝐽_𝑖+𝑝(𝑥)=𝐽_𝑖(𝑥) uniformly with respect to 𝑖∈ℤ.

For convenience, we introduce the notation𝑎=max_{[]𝑡∈0,𝜔}𝑎(𝑡),𝑎=min_t∈[]0,𝜔𝑎(𝑡),𝑐=min_{[]𝑡∈0,𝜔}𝑐(𝑡),𝑐=max_{[]𝑡∈0,𝜔}𝑐(𝑡),𝑟₀=exp^0𝜔𝑎(𝑠)d𝑠.(1.3)

Throughout this paper, we assume the following.(H₁) 𝕋₀(𝑥) is a variable time scale with (𝜔,𝑝)-property, 𝑐∈𝐶(ℝ,[0,1)),𝑎∈𝐶(ℝ,ℝ⁺),𝑣_𝑗∈𝐶(ℝ,ℝ), and 𝑓_𝑗(𝑡,𝑥)∈𝐶(𝕋₀(𝑥),ℝ⁺),(𝑗=1,2,…,𝑛) are 𝜔-periodic functions of 𝑡, 𝐽_𝑖∈𝐶(ℝ,ℝ),𝐵_𝑖+𝑝=𝐵_𝑖,𝐽_𝑖+𝑝(𝑥)=𝐽_𝑖(𝑥) uniformly with respect to 𝑖∈ℤ,𝐽_𝑖(0)=0,𝐵_𝑖≥0 for each 𝑖∈ℤ.(H₂) 𝑓_𝑗(𝑡,𝑥)∈𝐶(𝕋₀(𝑥),ℝ⁺) is nondecreasing with respect to 𝑥 and 𝑓_𝑗(𝑡,0)=0 for each 𝑗∈{1,2,…,𝑛}.(H₃)𝑔∈𝐶(ℝ,ℝ⁺) and there exist two positive constants 𝑙, 𝐿 such that 0<𝑙≤𝑔(𝑥)≤𝐿<∞ for all 𝑥>0.(H₄)There exists a number 𝛿>0 such that (𝑐+𝑐)/2≤𝛿<1,0≤((2𝛿−𝑐)(1−𝑐²))/(1−(2𝛿−𝑐)²)<𝑟^𝐿0(1−𝑟^𝑙0)/(1−𝑟^𝐿0).

2. Preliminaries

Let 𝐸 be a real Banach space and 𝑃 be a cone in 𝐸. A map 𝜌 is said to be a nonnegative continuous concave functional on 𝑃 if 𝜌∶𝑃→[0,∞) is continuous and 𝜌(𝑡𝑥+(1−𝑡)𝑦)≥𝑡𝜌(𝑥)+(1−𝑡)𝜌(𝑦)∀𝑥,𝑦∈𝑃and𝑡∈[0,1].(2.1)

For numbers 𝛽₁,𝛽₄ such that 0<𝛽₁<𝛽₄ and 𝜌 is a nonnegative continuous concave function on 𝑃, we define the following sets: 𝑃_𝛽1={𝑥∈𝑃∶‖𝑥‖<𝛽₁},𝑃_𝛽1={𝑥∈𝑃∶‖𝑥‖≤𝛽₁},𝑃(𝜌,𝛽₁,𝛽₄)={𝑥∈𝑃∶𝛽₁≤𝜌(𝑥),‖𝑥‖≤𝛽₄}.

Now, we state the following Leggett-Williams fixed-point theorem, which is critical to the proof of our main results.

Lemma 2.1 (see [21]). Let 𝑇∶𝑃_𝛽4→𝑃_𝛽4 be completely continuous and 𝜌 nonnegative continuous concave functional on 𝑃 such that 𝜌(𝑢)≤‖𝑢‖ for all 𝑢∈𝑃_𝛽4. Suppose that there exist positive constants 𝛽₁,𝛽₂,𝛽₃,𝛽₄ with 0<𝛽₁<𝛽₂<𝛽₃≤𝛽₄ such that(1){𝑢∈𝑃(𝜌,𝛽₂,𝛽₃)∶𝜌(𝑥)>𝛽₂}≠𝜙 and 𝜌(𝑇𝑢)>𝛽₂ for 𝑢∈𝑃(𝜌,𝛽₂,𝛽₃);(2)‖𝑇𝑢‖<𝛽₁ for 𝑢∈𝑃_𝛽1;(3)𝜌(𝑇𝑢)>𝛽₂ for 𝑢∈𝑃(𝜌,𝛽₂,𝛽₄) with ‖𝑇𝑢‖>𝛽₃.

Then 𝑇 has at least three fixed points 𝑢₁,𝑢₂,𝑢₃ satisfying𝑢₁∈𝑃_𝛽1,𝑢₂∈𝑢∈𝑃𝜌,𝛽₂,𝛽₄∶𝜌(𝑢)>𝛽₂,𝑢₃∈𝑃_𝛽4⧵𝑃𝜌,𝛽₂,𝛽₄∪𝑃_𝛽1.(2.2)

Let 𝕋 be a periodic time scale, and let 𝐸={𝑥∈𝐶(𝕋,ℝ)∶𝑥(𝑡)=𝑥(𝑡+𝜔)} be a Banach space with the norm ‖𝑥‖=sup_{𝑡∈[0,𝜔]∩𝕋}{|𝑥(𝑡)|∶𝑥∈𝐸}, and let Φ∶𝐸→𝐸 be defined by(Φ𝑥)(𝑡)=𝑥(𝑡)+𝑐(𝑡)𝑥(𝑡−𝜏).(2.3)

Lemma 2.2 (see [20]). If 0≤𝑐(𝑡)<1 and 𝐸 is a Banach space, then Φ has a bounded inverse Φ⁻¹ on 𝐸, and for all 𝑥∈𝐸, Φ⁻¹𝑥(𝑡)=_𝑗≥0_{0≤𝑖≤𝑗−1}(−1)^𝑘𝑐(𝑡−𝑖𝜏)𝑥(𝑡−𝑗𝜏)(2.4) and ‖Φ⁻¹𝑥‖≤‖𝑥‖/(1−𝑐).

Definition 2.3 (see [6]). A nonempty closed set 𝕋₀(𝑥) in ℝ×ℝ^𝑛 is said to be a variable time scale if for any 𝑥₀∈ℝ^𝑛 the projection of 𝕋₀(𝑥₀) on time axis, that is, the set {𝑡∈ℝ∶(𝑡,𝑥₀)∈𝕋₀(𝑥₀)} is a time scale in Hilger sense.
Fix a sequence {𝑡_𝑖}⊂ℝ such that 𝑡_𝑖<𝑡_𝑖+1 for all 𝑖∈ℤ, and |𝑡_𝑖|→∞ as |𝑖|→∞. Denote 𝛿_𝑖=𝑡_2𝑖−1−𝑡_2𝑖, 𝜅_𝑖=𝑡_2𝑖−𝑡_2𝑖−1 and take a sequence of functions {𝜏_𝑖(𝑥)}⊂𝐶(ℝ^𝑛,ℝ). Assume that(C₁) for some positive numbers 𝜃^′,𝜃∈ℝ,𝜃^≤𝑡_𝑖+1−𝑡_𝑖≤𝜃;(C₂)there exists 𝑙₀, 0<2𝑙₀<𝜃^ such that ‖𝜏_𝑖(𝑥)‖≤𝑙₀ for all 𝑥∈ℝ^𝑛,𝑖∈ℤ.
Denote 𝑙_𝑖∶=inf_{𝑥∈ℝ𝑛}𝑡_𝑖+𝜏_𝑖(𝑥),𝑟_𝑖∶=sup_{𝑥∈ℝ𝑛}𝑡_𝑖+𝜏_𝑖(𝑥).(2.5)
We set ℰ𝒮_𝑖=(𝑡,𝑥)∈ℝ×ℝ^𝑛∶𝑡_2𝑖+𝜏_2𝑖(𝑥)<𝑡<𝑡_2𝑖+1+𝜏_2𝑖+1(𝑥)_𝑖,𝒟=(𝑡,𝑥)∈ℝ×ℝ^𝑛∶𝑡=𝑡_𝑖+𝜏_𝑖(𝑥)_𝑖,=(𝑡,𝑥)∈ℝ×ℝ^𝑛∶𝑡_2𝑖−1+𝜏_2𝑖−1(𝑥)≤𝑡≤𝑡_2𝑖+𝜏_2𝑖(𝑥).(2.6)
Following [6], we denote 𝕋₀(𝑥)∶=⋃^{∞𝑖=−∞}𝒟_𝑖 and 𝕋_𝑐=⋃^{∞𝑖=−∞}[𝑡_2𝑖−1,𝑡_2𝑖].

A transition operator Π_𝑖∶𝒮_2𝑖→𝒮_2𝑖+1, for all 𝑖∈ℤ, such that Π_𝑖(𝑡,𝑦)=(Π^1𝑖(𝑡,𝑦),Π^2𝑖(𝑡,𝑦)) where Π^1𝑖∶𝒮_2𝑖→ℝ and Π^2𝑖∶𝒮_2𝑖→ℝ^𝑛, andΠ^1𝑖(𝑡,𝑦)=𝑡_2𝑖+1+𝜏_2𝑖+1Π^2𝑖(𝑡,𝑦),Π^2𝑖(𝑡,𝑦)=𝐼_𝑖(𝑦)+𝑦,(2.7) where 𝐼_𝑖∶ℝ^𝑛→ℝ^𝑛 is a function. One can easily see that Π^1𝑖(𝑡,𝑦) is the time coordinate of (𝑡⁺,𝑦⁺)∶=Π_𝑖(𝑡,𝑦), the image of (𝑡,𝑦)∈𝒮_2𝑖 under the operator Π_𝑖, and Π^2𝑖(𝑡,𝑦) is the space coordinate of the image.

Let 𝑡=𝛼_𝑖 and 𝑡=𝛽_𝑖 be the moments that the graph of 𝑦=𝜑(𝑡) intersects the surface 𝒮_2𝑖−1 and 𝒮_2𝑖, respectively, where the surfaces are defined previously. Then, we set the nonvariable time scale𝕋^𝜑𝑐∶=^∞_𝑖_=−∞𝛼_𝑖,𝛽_𝑖,(2.8) which is the domain of 𝜑, and define the Δ-derivative as in the introduction. That is, for 𝑡=𝛽_𝑖, we have𝜑^Δ𝛽_𝑖=𝜑𝛼^𝑖+1−𝜑𝛽_𝑖𝛼_𝑖+1−𝛽_𝑖𝜑_Δ,(𝑡)=lim_𝑠→𝑡𝜑(𝑠)−𝜑(𝑡)𝑠−𝑡,(2.9) for any other 𝑡∈𝕋^𝜑𝑐, whenever the limit exists.

Consider the system on variable time scales:𝑦^=𝐴(Δ𝑡)𝑡|𝑦+𝑓(_{(𝑡,𝑦)∈𝒮2𝑖}𝑡,𝑦),(𝑡,𝑦)∈𝕋₀(𝑦),Δ𝑦|_{(𝑡,𝑦)∈𝒮2𝑖}=Π^1𝑖(𝑡,𝑦)−𝑡,=Π^2𝑖(𝑡,𝑦)−𝑦,(2.10) where 𝐴(𝑡)∶ℝ→ℝ^𝑛×𝑛 is an 𝑛×𝑛 continuous real-valued matrix function, 𝐵_𝑖 is an 𝑛×𝑛 matrix, functions 𝑓(𝑡,𝑦)∶𝕋₀(𝑦)→ℝ^𝑛 and 𝐽_𝑖(𝑦)∶ℝ^𝑛→ℝ^𝑛 are continuous, Π^1𝑖(𝑡,𝑦)=𝑡_2𝑖+1+𝜏_2𝑖+1(Π^2𝑖(𝑡,𝑦)), and Π^2𝑖(𝑡,𝑦)=𝐵_𝑖𝑦+𝐽_𝑖(𝑦)+𝑦.

For any 𝛼,𝛽∈ℝ we define the oriented interval [𝛼,𝛽] as𝛼,𝛽=[[𝛼𝛽,,𝛽𝛼]],,iofth𝛼e≤rw𝛽is,e.(2.11)

Consider the nonvariable time scale𝕋^0𝑐=^∞_𝑖_=−∞𝑙_2𝑖−1,𝑟_2𝑖,(2.12) where 𝑙_𝑖,𝑟_𝑖,𝑖∈ℤ are defined by (2.5) for the variable time scale 𝕋₀(𝑦), and take a continuation 𝑓∶𝕋^0𝑐×ℝ^𝑛→ℝ^𝑛 of 𝑓∶𝕋₀(𝑦)→ℝ^𝑛 which is Lipschitzian with the same Lipschitz constant 𝑙; furthermore, if 𝑓 is a monotone function, a continuation 𝑓 can also have the same monotony with 𝑓. Set 𝕋_𝑐∶=⋃^{∞𝑖=−∞}[𝑡_2𝑖−1,𝑡_2𝑖].(C₃)‖𝜏_𝑖(𝑥)−𝜏_𝑖(𝑦)‖+‖𝐽_𝑖(𝑥)−𝐽_𝑖(𝑦)‖+‖𝑓(𝑡,𝑥)−𝑓(𝑡,𝑦)‖≤𝑙‖𝑥−𝑦‖ for arbitrary 𝑥,𝑦∈ℝ^𝑛, where 𝑙 is a Lipschitz constant.

By (C₃), in [6], one can see the following important lemma.

Lemma 2.4 (see [6]). Assume (C₃) is satisfied. Then there are mappings 𝑊_𝑖(𝑧)∶ℝ^𝑛→ℝ^𝑛,𝑖∈ℤ, such that, corresponding to each solution 𝑦(𝑡) of (2.10), there is a solution 𝑧(𝑡) of the system 𝑧𝑡_2𝑖+1𝑧^′=𝐴(𝑡)𝑧+𝑓(𝑡,𝑧),𝑡≠𝑡_2𝑖,=𝐵_𝑖𝑧𝑡_2𝑖+𝑊_𝑖𝑧𝑡_2𝑖+𝑧𝑡_2𝑖,(2.13) such that 𝑦(𝑡)=𝑧(𝑡) for all 𝑡∈𝕋_𝑐 except possibly on [𝑡_2𝑖−1,𝛼_𝑖] and [𝛽_𝑖,𝑡_2𝑖], where 𝛼_𝑖 and 𝛽_𝑖 are the moments that 𝑦(𝑡) meets the surfaces 𝒮_2𝑖−1 and 𝒮_2𝑖, respectively.
Furthermore, the functions 𝑊_𝑖 satisfy the inequality ‖‖𝑊_𝑖(𝑧)−𝑊_𝑖(𝑦)‖‖≤𝑘(𝑙)𝑙‖𝑧−𝑦‖,(2.14) uniformly with respect to 𝑖∈ℤ for all 𝑧,𝑦∈ℝ^𝑛 such that ‖𝑧‖≤ℎ and ‖𝑦‖≤ℎ; here 𝑘(𝑙₀)=𝑘(𝑙₀,ℎ) is a bounded function. Under the sense of Lemma 2.4, we say that systems (2.10) and (2.13) are 𝐵-equivalent.

Proof. Fix 𝑖∈ℤ. Let 𝑧(𝑡) be the solution of (2.10) such that 𝑧(𝑡_2𝑖)=𝑧, and assume that 𝛼_i and 𝛽_𝑖 are solutions of 𝛼=𝑡_2𝑖−1+𝜏_2𝑖−1(𝑧(𝛼)) and 𝛽=𝑡_2𝑖+𝜏_2𝑖(𝑧(𝛽)), respectively. Let 𝑧₁(𝑡) be the solution of the system 𝑧^′=𝐴(𝑡)𝑧+𝑓(𝑡,𝑧)(2.15) with the initial condition 𝑧₁(𝛼_𝑖+1)=Π^2𝑖(𝛽_𝑖,𝑧(𝛽_𝑖)).
We first note that 𝑧₁(𝛼_𝑖+1)=(𝐼+𝐵_𝑖)𝑧(𝛽_𝑖)+𝐽_𝑖(𝑧(𝛽_𝑖)). Moreover, for 𝑡∈[𝑡_2𝑖,𝛽_𝑖],𝑧(𝑡)=𝑧𝑡_2𝑖+^𝑡𝑡_2𝑖𝐴(𝑠)𝑧(𝑠)+𝑓(𝑠,𝑧(𝑠))d𝑠,(2.16) and for 𝑡∈[𝛼_𝑖+1,𝑡_2𝑖+1], 𝑧₁(𝑡)==𝑧𝐼₁+𝛼𝐵_𝑖+1𝑖+^𝑡𝛼_𝑖+1𝐴(𝑠)𝑧₁(𝑠)+𝑓𝑠,𝑧₁(𝑠)d𝑠=𝐼+𝐵_𝑖𝑧𝛽_𝑖+𝐽_𝑖𝑧𝛽_𝑖+^𝑡𝛼_𝑖+1𝐴(𝑠)𝑧¹(𝑠)+𝑓𝑠,𝑧₁(𝑠)d𝑠+𝑧𝑡_2𝑖+^𝛽𝑖_𝑡2𝑖𝐴(𝑠)𝑧(𝑠)+𝑓(𝑠,𝑧(𝑠))d𝑠+𝐽^𝑖𝑧𝛽_𝑖_{𝑡𝛼𝑖+1}𝐴(𝑠)𝑧¹(𝑠)+𝑓𝑠,𝑧₁(𝑠)d𝑠.(2.17) Thus, we have 𝑊_𝑖(𝑧)=+𝐼+𝐵_𝑖^𝛽𝑖_𝑡2𝑖𝐴(𝑠)𝑧(𝑠)+𝑓(𝑠,𝑧(𝑠))d𝑠+𝐽^𝑖𝑧𝛽_𝑖_{𝑡2𝑖+1𝛼𝑖+1}𝐴(𝑠)𝑧¹(𝑠)+𝑓𝑠,𝑧₁(𝑠)d𝑠.(2.18)
Substituting (2.18) in (2.13), we see that 𝑊_𝑖(𝑧) satisfies the first conclusion of the lemma.
Next, we prove (2.14). Let ‖𝑧(𝑡_2𝑖)‖≤ℎ. By employ integrals (2.16) and (2.17), we find that the solutions 𝑧(𝑡) and 𝑧₁(𝑡) determined above satisfy the inequalities ‖𝑧(𝑡)‖≤𝐻 and ‖𝑧₁(𝑡)‖≤𝐻 on [𝛽_𝑖,𝑡_2𝑖] and[𝛼_𝑖+1,𝑡_2𝑖+1], where 𝐻=𝑀(1+𝑙)+(1+𝑁+𝑙)(ℎ+𝑀𝑙)𝑒^{𝑁𝑙+𝑙2}𝑒^{𝑁𝑙+𝑙2}.(2.19) Let 𝑦(𝑡) be the solution of (2.10) such that 𝑦(𝑡_2𝑖)=𝑦, and assume that𝛼_𝑖 and𝛽_𝑖 are solutions of𝛼=𝑡_2𝑖−1+𝜏_2𝑖−1(𝑦(𝛼)) and𝛽=𝑡_2𝑖+𝜏_2𝑖(𝑦(𝛽)), respectively. Let 𝑦₁(𝑡) be the solution of (2.15) with initial condition 𝑦₁(𝛼_𝑖+1)=Π^2𝑖(𝛽_𝑖,𝑦(𝛽_𝑖)). Without loss of any generality, we assume that𝛽_𝑖≥𝛽_𝑖 and 𝛼_𝑖+1≤𝛼_𝑖+1. Application of the Gronwall-Bellman lemma shows that, for 𝑡∈[𝛽_𝑖,𝑡_2𝑖], ‖𝑧(𝑡)−𝑦(𝑡)‖≤𝑒^{(𝑁+𝑙)𝑙}‖𝑧−𝑦‖.(2.20) The equation 𝑦𝛽_𝑖=𝑦𝛽^𝑖+^𝛽𝑖_𝛽𝑖𝐴(𝑠)𝑦(𝑠)+𝑓(𝑠,𝑦(𝑠))d𝑠(2.21) gives us ‖‖𝑦𝛽_𝑖−𝑦𝛽^𝑖‖‖≤(𝑁𝐻+𝑙𝐻+𝑀)𝛽_𝑖−𝛽_𝑖.(2.22) Thus, we obtain ‖‖𝑧𝛽_𝑖−𝑦𝛽_𝑖‖‖≤𝑒^{(𝑁+𝑙)𝑙}‖𝑧−𝑦‖+(𝑁𝐻+𝑙𝐻+𝑀)𝛽_𝑖−𝛽_𝑖.(2.23) Now condition (C₃) together with (2.23) leads to 𝛽_𝑖−𝛽_𝑖≤𝑙𝑒^{(𝑁+𝑙)𝑙}1−𝑙(𝑁𝐻+𝑙𝐻+𝑀)‖𝑧−𝑦‖.(2.24) Hence (2.23) becomes ‖‖𝑧𝛽_𝑖−𝑦𝛽_𝑖‖‖≤1−𝑙(𝑁𝑒𝐻+𝑙^{(𝑁+𝑙)𝑙}𝐻+𝑀)‖𝑧−𝑦‖.(2.25) On the other hand, 𝑦₁𝛼_𝑖+1=𝑦₁𝛼_𝑖+1+^𝛼𝑖+1_𝛼𝑖+1𝐴(𝑠)𝑦¹(𝑠)+𝑓𝑠,𝑦₁(𝑠)d𝑠(2.26) gives us ‖‖𝑦₁𝛼_𝑖+1−𝑦₁𝛼_𝑖+1‖‖≤(𝑁𝐻+𝑙𝐻+𝑀)𝛼_𝑖+1−𝛼_𝑖+1.(2.27) Using the transition operators and (2.25) we get ‖‖𝑧₁𝛼_𝑖+1−𝑦₁𝛼_𝑖+1‖‖≤(1+𝑁+𝑙)𝑒^{(𝑁+𝑙)𝑙}1−𝑙(𝑁𝐻+𝑙𝐻+𝑀)‖𝑧−𝑦‖.(2.28) Condition (C₃) and (2.28) imply that 𝛼_𝑖+1−𝛼_𝑖+1≤𝑙(1+𝑁+𝑙)𝑒^{(𝑁+𝑙)𝑙}1−𝑙(𝑁𝐻+𝑙𝐻+𝑀)‖𝑧−𝑦‖.(2.29) From (2.27)–(2.29) we obtain ‖‖𝑧₁𝛼_𝑖+1−𝑦₁𝛼_𝑖+1‖‖≤𝐻₁𝑒^{(𝑁+𝑙)𝑙}‖𝑧−𝑦‖,(2.30) where 𝐻₁=(1+𝑁+𝑙)[1+𝑙(𝑁𝐻+𝑙𝐻+𝑀)]/[1−𝑙(𝑁𝐻+𝑙𝐻+𝑀)]. Solutions 𝑧₁(𝑡) and 𝑦₁(𝑡) on [𝛼_𝑖+1,𝑡_2𝑖+1] satisfy the inequality ‖‖𝑧₁(𝑡)−𝑦₁(𝑡)‖‖≤𝐻₁𝑒^{2(𝑁+𝑙)𝑙}‖𝑧−𝑦‖.(2.31) Now subtracting the expression 𝑊_𝑖(𝑦)=+𝐼+𝐵_𝑖^𝛽𝑖_𝑡2𝑖𝐴(𝑠)𝑦(𝑠)+𝑓(𝑠,𝑦(𝑠))d𝑠+𝐽^𝑖𝑦𝛽_𝑖_{𝑡2𝑖+1𝛼𝑖+1}𝐴(𝑠)𝑦¹(𝑠)+𝑓𝑠,𝑦₁(𝑠)d𝑠(2.32) from (2.18) and using (2.20), (2.24), (2.29), and (2.31), we conclude that (2.14) holds. The proof is complete.

A special transformation called 𝜓-substitution [5], which is change of the independent variable and defined for 𝑡∈⋃^{∞𝑖=−∞}(𝑡_2𝑖−1,𝑡_2𝑖] as 𝜓(𝑡)=⎧⎪⎪⎨⎪⎪⎩𝑡−^{0<𝑡2𝑘<𝑡}𝑡+_{𝑡≤𝑡2𝑘<0}𝛿_𝑘,𝑡≥0,𝛿_𝑘,𝑡<0,(2.33) where 𝛿_𝑘=𝑡_2𝑘+1−𝑡_2𝑘. Setting 𝑠_𝑖=𝜓(𝑡_2𝑖), we see that this transformation has an inverse given by𝜓⁻¹(𝑠)=⎧⎪⎪⎨⎪⎪⎩𝑠+^{0<𝑠𝑘<𝑠}𝑠−_{𝑠≤𝑠𝑘<0}𝛿_𝑘,𝑠≥0,𝛿_𝑘,𝑠<0.(2.34)

Lemma 2.5 (see [5, 6]). 𝜓^′(𝑡)=1 if 𝑡∈⋃^{∞𝑖=−∞}(𝑡_2𝑖−1,𝑡_2𝑖].

Proof. Assume that 𝑡≥0. Then, 𝜓^(𝑡)=lim_ℎ→0𝜓(𝑡+ℎ)−𝜓(𝑡)=lim_ℎ→0ℎ1ℎ⎡⎢⎢⎣⎛⎜⎜⎝𝑡+ℎ−_{0<𝑡2𝑘<𝑡+ℎ}𝛿_𝑘⎞⎟⎟⎠−⎛⎜⎜⎝𝑡−_{0<𝑡2𝑘<𝑡}𝛿_𝑘⎞⎟⎟⎠⎤⎥⎥⎦=1.(2.35) The assertion for 𝑡<0 can be proved in the same way. The proof is complete.

Definition 2.6 (see [5]). The time scale 𝕋₀ is said to have an 𝜔-property if there exists a number 𝜔∈ℝ⁺ such that 𝑡+𝜔∈𝕋₀ whenever 𝑡∈𝕋₀.

Definition 2.7 (see [5]). A sequence {𝑎_𝑖}⊂ℝ is said to satisfy an (𝜔,𝑝)-property if there exist numbers 𝜔∈ℝ⁺ and 𝑝∈ℕ such that 𝑎_𝑖+𝑝=𝑎_𝑖+𝜔 for all 𝑖∈ℤ.

Definition 2.8 (see [6]). The variable time scale 𝕋₀(𝑦) is said to satisfy an (𝜔,𝑝)-property if (𝑡±𝜔,𝑦) is in 𝕋₀(𝑦) whenever (𝑡,𝑦) is. In this case, there exists 𝑝∈ℕ such that the sequences {𝑡_2𝑖−1} and {𝑡_2𝑖} satisfy the (𝜔,𝑝)-property and 𝜏_𝑖+𝑝(𝑦)=𝜏_𝑖(𝑦) for all 𝑖∈ℤ.
Suppose now that (2.10) is 𝜔-periodic; that is, 𝕋₀(𝑦) satisfies the (𝜔,𝑝)-property, 𝐴(𝑡) and 𝑓(𝑡,𝑦) are 𝜔-periodic functions of 𝑡, and 𝐵_𝑖+𝑝=𝐵_𝑖,𝐽_𝑖+𝑝(𝑦)=𝐽_𝑖(𝑦) uniformly with respect to 𝑖∈ℤ.

Lemma 2.9 (see [6]). If (2.10) is 𝜔-periodic, then the sequence 𝑊_𝑖(𝑧) is 𝑝-periodic uniformly with respect to 𝑧∈ℝ^𝑛.

Proof. Since the variable time scale 𝕋₀(𝑦) satisfies an (𝜔,𝑝)-property, by (2.18), one can easily see that 𝑊_𝑖(𝑧) is 𝑝-periodic uniformly with respect to 𝑧∈ℝ^𝑛. The proof is complete.

Lemma 2.10 (see [5]). If 𝕋₀ has an 𝜔-property, then the sequence {𝑠_𝑖},𝑠_𝑖=𝜓(𝑡_2𝑖), is (𝜔,𝑝₀)-periodic with 𝜔=𝜔−_{0<𝑡2𝑘<𝜔}𝛿_𝑘=𝜓(𝜔).(2.36)

Proof. In order to prove this lemma, we only need to verify that 𝑠_𝑖+𝑝0=𝑠_𝑖+𝜔 for all 𝑖. Assume that 𝑖≥0,𝑖=𝑛𝑝₀+𝑗 for some 𝑛∈ℤ,0≤𝑗<𝑝₀ and 0<𝑡₀<⋅⋅⋅<𝑡_2(𝑝0−1)<𝜔. Then 𝑠_𝑖+𝑝0=𝜓𝑡_{2(𝑖+𝑝0)}=𝑡_{2(𝑖+𝑝0)}−_{0<𝑡2𝑘<𝑡0)2(𝑖+𝑝}𝛿_𝑘=𝑡_2𝑖+𝜔−_{0<𝑡2𝑘<𝑡2i}𝛿_𝑘−_{𝑡2𝑖≤𝑡2𝑘<𝑡0)2(𝑖+𝑝}𝛿_𝑘=𝜓𝑡_2𝑖+𝜔−^{𝑖+𝑝0−1}_𝑘=𝑖𝛿_𝑘=𝑠_𝑖+𝜔−^{𝑗+𝑝0−1}_𝑘=𝑗𝛿_{𝑘+𝑛𝑝0}=𝑠_𝑖+𝜔−^{𝑗+𝑝0−1}_𝑘=𝑗𝛿_𝑘=𝑠_𝑖+𝜔−^𝑝0−1_𝑘=0𝛿_𝑘=𝑠_𝑖+𝜔−_{0<𝑡2𝑘<𝜔}𝛿_𝑘=𝑠_𝑖+𝜔,(2.37) where we have used the fact that ^𝑗+^𝑝0−1_𝑘=𝑗𝛿_𝑘=^𝑝0−1_𝑘=𝑗𝛿_𝑘+^{𝑗+𝑝0−1}_𝑘_=𝑝0𝛿_𝑘=^𝑝0−1_𝑘=𝑗𝛿_𝑘+^𝑗−1_𝑘=0𝛿_𝑘+𝑝0=^𝑝0−1_𝑘=𝑗𝛿_𝑘+^𝑗−1_𝑘=0𝛿_𝑘=^𝑝0−1_𝑘=0𝛿_𝑘.(2.38) All other cases can be verified similarly. The proof is complete.

Lemma 2.11 (see [5, 6]). If 𝕋₀(𝑦) satisfies an (𝜔,𝑝)-property, then 𝜓(𝑡+𝜔)=𝜓(𝑡)+𝜓(𝜔).Proof. Assume that 𝑡≥0. By Lemma 2.10, we have 𝜓(𝑡+𝜔)=𝑡+𝜔−_{0<𝑡2𝑘<𝑡+𝜔}𝛿_𝑘=𝑡+𝜔−_{0<𝑡2𝑘<𝜔}𝛿_𝑘−_{𝜔≤𝑡2𝑘<𝑡+𝜔}𝛿_𝑘=𝑡−_{𝜔≤𝑡2𝑘<𝑡+𝜔}𝛿_𝑘+𝜓(𝜔)=𝜓(𝑡)+𝜓(𝜔).(2.39) The assertion for 𝑡<0 can be proved in the same way. The proof is complete.

Lemma 2.12 (see [5, 6]). A function 𝜙(𝑡) is an 𝜔-periodic function on 𝕋_𝑐, if and only if 𝜙(𝜓⁻¹(𝑠)) is an 𝜔-periodic function on ℝ, where 𝜔=𝜓(𝜔).

Proof. By Lemma 2.11, 𝑠+𝜔=𝜓(𝑡+𝜔). Then the equality 𝜙𝜓⁻¹𝑠+𝜔=𝜙(𝑡+𝜔)=𝜙(𝑡)=𝜙𝜓⁻¹(𝑠)(2.40) completes the proof.

For any fixed 𝑥₀∈ℝ, we set 𝑃𝐶=𝕋𝑥𝑥₀∈𝐶𝑡_2𝑖−1+𝜏_2𝑖−1𝑥₀,𝑡_2𝑖+𝜏_2𝑖𝑥₀,ℝ∶Π^2𝑖(𝑡,𝑥)=𝐵_𝑖𝑥+𝐽_𝑖(𝑥)+𝑥,𝑖=1,2,…(2.41) and consider the Banach space 𝐸=𝑥∶𝑥∈𝑃𝐶𝕋𝑥₀∶𝑥(𝑡)=𝑥(𝑡+𝜔).(2.42) Let Φ∶𝐸→𝐸 be defined by(Φ𝑥)(𝑡)=𝑥(𝑡)+𝑐(𝑡)𝑥(𝑡−𝛼)∶=𝑦(𝑡).(2.43)

Using the inverse transformation of Φ, we can obtainΠ^1𝑖𝑡,Φ⁻¹𝑦=𝑡_2𝑖+1+𝜏_2𝑖+1Π^2𝑖𝑡,Φ⁻¹𝑦,Π^2𝑖𝑡,Φ⁻¹𝑦=𝐵_𝑖Φ⁻¹𝑦+𝐽_𝑖Φ⁻¹𝑦+Φ⁻¹𝑦.(2.44) From the second equation, it is easy to getΦΠ^2𝑖𝑡,Φ⁻¹𝑦=Φ𝐵_𝑖Φ⁻¹𝑦+Φ𝐽_𝑖Φ⁻¹𝑦+𝑦∶=Π^2𝑖(𝑡,𝑦),(2.45) that is, Π^2𝑖(𝑡,Φ⁻¹𝑦)=Φ⁻¹(Π^2𝑖(𝑡,𝑦)). HenceΠ^1𝑖𝑡,Φ⁻¹𝑦∶==𝑡𝑡_{2𝑖+12𝑖+1}+𝜏_2𝑖+1Φ⁻¹Π^2𝑖(𝑡,𝑦)=𝑡_2𝑖+1+𝜏_2𝑖+1Φ⁻¹Π^2𝑖(𝑡,𝑦)+𝜏̃_2𝑖+1Π^2𝑖(𝑡,𝑦)∶=Π^1𝑖(𝑡,𝑦).(2.46)

Therefore, we can obtain the other variable time scale plane 𝑇₀(𝑦) by the inverse transformation of Φ and𝒮ℰ^′_𝑖=(𝑡,𝑦)∈ℝ×ℝ∶𝑡_2𝑖+𝜏̃_2𝑖(𝑦)<𝑡<𝑡_2𝑖+1+𝜏̃_2𝑖+1(𝑦)_𝑖,𝒟=(𝑡,𝑦)∈ℝ×ℝ∶𝑡=𝑡_𝑖+𝜏̃_𝑖(𝑦)_𝑖,=(𝑡,𝑦)∈ℝ×ℝ∶𝑡_2𝑖−1+𝜏̃_2𝑖−1(𝑦)≤𝑡≤𝑡_2𝑖+𝜏̃_2𝑖(𝑦).(2.47)

Hence, (1.2) can be changed into the following form:𝑦^=𝑎(𝑡)𝑔Φ⁻¹𝑦(𝑡)𝑦(𝑡)−𝑎(𝑡)Δ𝐻𝑡|(𝑦(𝑡))_{(𝑡,𝑦)∈𝒮′2𝑖}−∑^𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝑡,Φ⁻¹𝑦𝑡−𝑣_𝑗(𝑡),(𝑡,𝑦)∈𝕋₀(𝑦),Δ𝑦|_{(𝑡,𝑦)∈𝒮′2𝑖}=Π^1𝑖(𝑡,𝑦)−𝑡,=Π^2𝑖(𝑡,𝑦)−𝑦,(2.48) where 𝐻(𝑦(𝑡))=𝑐(𝑡)𝑔((Φ⁻¹𝑦)(𝑡))(Φ⁻¹𝑦)(𝑡−𝛼).

Define a cone in 𝐸 by𝑃₀={𝑦(𝑡)∈𝐸∶𝑦(𝑡)≥𝑘‖𝑦‖},(2.49) where 𝑘∈(((2𝛿−𝑐)(1−𝑐²))/(1−(2𝛿−𝑐)²),𝑟^𝐿0(1−𝑟^𝑙0)/(1−𝑟^𝐿0)].

Lemma 2.13 (see [20]). Suppose that conditions (H₁)–(H₄) hold and 0≤𝑐(𝑡)<1 and 𝑦∈𝑃₀, then 𝛼‖𝑦‖≤Φ⁻¹𝑦(𝑡)≤1𝑙𝑐1−𝑐‖𝑦‖,(2.50)𝛼‖𝑦‖≤𝐻(𝑦(𝑡))≤𝐿𝑐1−𝑐‖𝑦‖,(2.51) where 𝛼=(𝑘/(1−𝑐²))−(2𝛿−𝑐)/(1−(2𝛿−𝑐)²).

(H₅)‖𝜏_𝑖(𝑥)−𝜏_𝑖(𝑦)‖+‖𝐽_𝑖(𝑥)−𝐽_𝑖(𝑦)‖+∑^𝑛𝑗=1‖𝑓_𝑗(𝑡,𝑥)−𝑓_𝑗(𝑡,𝑦)‖≤𝑙₀‖𝑥−𝑦‖ for arbitrary 𝑥,𝑦∈ℝ^𝑛, where 𝑙₀ is a Lipschitz constant.

In view of (H₅) by Lemma 2.4, it is easy to get the following lemma.

Lemma 2.14. Assume that (H₅) is satisfied. Then there are mappings 𝑊_𝑖(𝑧)∶ℝ→ℝ,𝑖∈ℤ such that, corresponding to each solution 𝑦(𝑡) of (2.48), there is a solution 𝑧(𝑡) of the system 𝑧^=𝑎(𝑧𝑡)𝑔𝑡_2𝑖+1Φ⁻¹𝑧(𝑡)𝑧(𝑡)−𝑎(𝑡)𝐻(𝑧(𝑡))−∑^𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝑡,Φ⁻¹𝑧𝑡−𝑣_𝑗(𝑡),𝑡≠𝑡_2𝑖,=Φ𝐵_𝑖Φ⁻¹𝑧𝑡_2𝑖+Φ𝑊_𝑖Φ⁻¹𝑧𝑡_2𝑖+𝑧𝑡_2𝑖,𝑡=𝑡_2𝑖(2.52) such that 𝑦(𝑡)=𝑧(𝑡) for all 𝑡∈𝕋_𝑐 except possibly on [𝑡_2𝑖−1,𝛼_𝑖] and [𝛽_𝑖,𝑡_2𝑖] where 𝛼_𝑖 and 𝛽_𝑖 are the moments that 𝑦(𝑡) meets the surfaces 𝒮^2𝑖−1 and 𝒮^2𝑖, respectively.
Furthermore, the functions 𝑊_𝑖 satisfy the inequality ‖‖𝑊_𝑖(𝑧)−𝑊_𝑖(𝑦)‖‖≤𝑘𝑙₀𝑙₀‖𝑧−𝑦‖,(2.53) uniformly with respect to 𝑖∈ℤ for all 𝑧,𝑦∈ℝ^𝑛 such that ‖𝑧‖≤ℎ and ‖𝑦‖≤ℎ; here 𝑘(𝑙₀)=𝑘(𝑙₀,ℎ) is a bounded function. Under the sense of lemma 2.14, we say that systems (2.48) and (2.52) are 𝐵-equivalent.

Proof. For fixed 𝑖∈ℤ. Let 𝑧(𝑡) be the solution of (2.48) such that 𝑧(𝑡_2𝑖)=𝑧, and assume that 𝛼_𝑖 and 𝛽_𝑖 are solutions of 𝛼=𝑡_2𝑖−1+𝜏̃_2𝑖−1(𝑧(𝛼)) and 𝛽=𝑡_2𝑖+𝜏̃_2𝑖(𝑧(𝛽)), respectively. Let 𝑧₁(𝑡) be the solution of the system 𝑧^=𝑎(𝑡)𝑔Φ⁻¹𝑧(𝑡)z(𝑡)−𝑎(𝑡)𝐻(𝑧(𝑡))−^𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝑡,Φ⁻¹𝑧𝑡−𝑣_𝑗(𝑡)(2.54) with the initial condition 𝑧₁(𝛼_𝑖+1)=Π^2𝑖(𝛽_𝑖,𝑧(𝛽_𝑖)).
We first note that 𝑧₁(𝛼_𝑖+1)=(Φ𝐵_𝑖Φ⁻¹)(𝑧(𝛽_𝑖))+(Φ𝐽_𝑖Φ⁻¹)(𝑧(𝛽_𝑖))+𝑧(𝛽_𝑖). Moreover, for 𝑡∈[𝑡_2𝑖,𝛽_𝑖], 𝑧(𝑡)=𝑧𝑡_2𝑖+^𝑡2𝑖𝑎(𝑠)𝑔Φ⁻¹−𝑧(𝑠)𝑧(𝑠)−𝑎(𝑠)𝐻(𝑧(𝑠))_𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝑠,Φ⁻¹𝑧𝑠−𝑣_𝑗(𝑠)d𝑠,(2.55) and for 𝑡∈[𝛼_𝑖+1,𝑡_2𝑖+1], 𝑧₁(𝑡)=𝑧₁𝛼_𝑖+1+^𝑡𝛼_𝑖+1𝑎(𝑠)𝑔Φ⁻¹−𝑎(𝑠)𝐻𝑧₁𝑧₁(𝑠)𝑧₁(𝑠)(𝑠)−^𝑛_𝑗=1=Φ𝐵_𝑖𝜆_𝑗𝑓_𝑗𝑠,Φ⁻¹𝑧₁𝑠−𝑣_𝑗(𝑠)d𝑠+_{𝑡𝛼𝑖+1}Φ⁻¹𝑧𝛽_𝑖+Φ𝐽_𝑖Φ⁻¹𝑧𝛽_𝑖+𝑧𝛽_𝑖𝑎(𝑠)𝑔Φ⁻¹−𝑎(𝑠)𝐻𝑧₁𝑧₁(𝑠)𝑧₁(𝑠)(𝑠)−^𝑛_𝑗=1=Φ𝐵𝜆_𝑗𝑓_𝑗𝑠,Φ⁻¹𝑧₁𝑠−𝑣_𝑗(𝑠)d𝑠_𝑖+𝐼Φ⁻¹𝑧𝑡_2𝑖+^𝛽𝑖_𝑡2𝑖𝑎(𝑠)𝑔Φ⁻¹−𝑧(𝑠)𝑧(𝑠)−𝑎(𝑠)𝐻(𝑧(𝑠))_𝑛_𝑗=1+Φ𝐽_𝑖𝜆_𝑗𝑓_𝑗𝑠,Φ⁻¹𝑧𝑠−𝑣_𝑗(𝑠)d𝑠Φ⁻¹𝑧𝛽_𝑖+^𝑡𝛼_𝑖+1𝑎(𝑠)𝑔Φ⁻¹−𝑎(𝑠)𝐻𝑧₁𝑧₁(𝑠)𝑧₁(𝑠)(𝑠)−^𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝑠,Φ⁻¹𝑧₁𝑠−𝑣_𝑗(𝑠)d𝑠.(2.56) Thus, we set 𝑊_𝑖(𝑧)=Φ𝐵_𝑖+𝐼Φ⁻¹^𝛽𝑖_𝑡2𝑖𝑎(𝑠)𝑔Φ⁻¹−𝑧(𝑠)𝑧(𝑠)−𝑎(𝑠)𝐻(𝑧(𝑠))_𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝑠,Φ⁻¹𝑧𝑠−𝑣_𝑗(𝑠)d𝑠+Φ𝐽_𝑖Φ⁻¹𝑧𝛽_𝑖_{𝑡2𝑖+1𝛼𝑖+1}𝑎(𝑠)𝑔Φ⁻¹−𝑎(𝑠)𝐻𝑧₁𝑧₁(𝑠)𝑧₁(𝑠)(𝑠)−^𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝑠,Φ⁻¹𝑧₁𝑠−𝑣_𝑗(𝑠)d𝑠.(2.57) Substituting (2.57) in (2.52), we see that 𝑊_𝑖(𝑧) satisfies the first conclusion of the lemma.
The rest of the proof is similar to that of Lemma 2.4, and we can use Gronwall-Bellman lemma to show that 𝑊_𝑖(𝑧) satisfies (2.53) and it will be omitted here. This completes the proof.

Next, we will use 𝜓-substitution, reducing (2.52) to an impulsive differential equation. Letting 𝑚(𝑠)=𝑧(𝜓⁻¹(𝑠)), we obtain, for 𝑡≠𝑡_2𝑖, Φ⁻¹𝑧𝑡−𝑣_𝑗(𝑡)==ΦΦ⁻¹𝑧𝜓⁻¹(𝑠)−𝑣_𝑗𝜓⁻¹(𝑠)₋₁=Φ𝑧𝜓⁻¹𝜓𝜓⁻¹(𝑠)−𝑣_𝑗𝜓⁻¹(𝑠)₋₁𝑚𝜓𝜓⁻¹(𝑠)−𝑣_𝑗𝜓⁻¹(𝑠)∶=𝜈(𝑠),(2.58) hence 𝑚^′=𝑎𝜓⁻¹(𝑠)𝑔Φ⁻¹𝑚(𝑠)𝑚(𝑠)−𝑎𝜓⁻¹(𝑠)𝐻(𝑚(𝑠))−^𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑠),𝜈(𝑠),(2.59) and for 𝑡=𝑡_2𝑖, we get 𝑚𝑠^−𝑖=𝑧𝑡_2𝑖+1=Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖+𝑚𝑠_𝑖.(2.60) Thus, the second equation in (2.23) leads to Δ𝑚|_{𝑠=𝑠𝑖}=Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖,(2.61) where Δ𝑚|_{𝑠=𝑠𝑖}=𝑚(𝑠^−𝑖)−𝑚(𝑠_𝑖). Hence, 𝑚(𝑠) is a solution of the impulsive differential equation: 𝑚^=𝑎𝜓⁻¹(𝑠)𝑔Φ⁻¹𝑚(𝑠)𝑚(𝑠)−𝑎𝜓⁻¹(𝑠)𝐻(𝑚(𝑠))−^𝑛_𝑗=1Δ𝑚∣_{𝑠=𝑠𝑖}𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑠),𝜈(𝑠),𝑠≠𝑠_𝑖,=Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖,𝑠=𝑠_𝑖.(2.62)

In the following, we set 𝑃𝐶=𝑚∶𝑚∣_{(𝑠𝑖,𝑠𝑖+1)}∈𝐶𝑠_𝑖,𝑠_𝑖+1,ℝ,𝑚𝑠^+𝑖=𝑚𝑠_𝑖,𝑖=1,2,…(2.63) and consider the Banach space 𝐸=𝑚∶𝑚∈𝑃𝐶,𝑚(𝑠)=𝑚𝑠+𝜔(2.64) with the norm ‖𝑚‖=sup_𝑠∈[0,𝜔_]{|𝑚(𝑠)|∶𝑚∈𝐸}, where 𝜔=𝜓(𝜔). Define a cone in𝐸 by 𝑃=𝑚(𝑠)∈𝐸∶𝑚(𝑠)≥𝑘‖𝑚‖,(2.65) where 𝑘∈((2𝛿−𝑐)(1−𝑐²)/(1−(2𝛿−𝑐)²),𝑟^𝐿0(1−𝑟^𝑙0)/(1−𝑟^𝐿0)].

Let the operator Ψ∶𝑃→𝐸 be defined by (Ψ𝑚)(𝑠)=^𝑠+_𝑠𝜔𝐺(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖,(2.66) where 𝐺(𝑠,𝜃)=𝑒^{∫𝑠𝜃𝑎(𝜓−1(𝑟))𝑔((Φ−1𝑚)(𝑟))d𝑟}1−𝑒^{∫0𝜔𝑎(𝜓−1(𝑟))𝑔((Φ−1𝑚)(𝑟))d𝑟},𝜃∈𝑠,𝑠+𝜔.(2.67) By the assumptions, we have 𝑟^𝐿01−𝑟^𝐿0≤𝐺(𝑠,𝜃)≤11−𝑟^𝑙0.(2.68)

Lemma 2.15. 𝑚 is an 𝜔-periodic solution of (2.62) if and only if 𝑚 is a fixed point of the operator Ψ.

Proof. If 𝑚(𝑠) is an 𝜔-periodic solution of (2.62), for any 𝑠∈ℝ, there exists 𝑖∈ℤ such that 𝑠_𝑖 is the first impulsive point after 𝑠. Hence, for 𝜃∈[𝑠,𝑠_𝑖], we have 𝑚(𝜃)=𝑒^{∫𝜃𝑠𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}−^𝜃𝑠𝑒^{∫𝜃𝑟𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑎𝜓⁻¹(𝑟)𝐻(𝑚(𝑟))+^𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑟),𝜈(𝑟)d𝑟,(2.69) then 𝑚𝑠_𝑖=−𝑒^{∫𝑠𝑖𝑠𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑚(𝑠)_𝑠𝑖𝑠𝑒^{∫𝑠𝑖r𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑎𝜓⁻¹(𝑟)𝐻(𝑚(𝑟))+^𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑟),𝜈(𝑟)d𝑟.(2.70) Again, for 𝜃∈(𝑠_𝑖,𝑠_𝑖+1], then 𝑚(𝑠)=−𝑒^{∫𝜃𝑠𝑖𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}_𝜃𝑠𝑖𝑚𝑠^+𝑖𝑒^{∫𝜃𝑟𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑎𝜓⁻¹(𝑟)𝐻(𝑚(𝑟))+^𝑛_𝑗=1=𝑒_{∫𝜃𝑠𝑖𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑟),𝜈(𝑟)d𝑟Φ𝐵^𝑖−_𝜃𝑠𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖+𝑚𝑠_𝑖𝑒^{∫𝜃𝑟𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑎𝜓⁻¹(𝑟)𝐻(𝑚(𝑟))+^𝑛_𝑗=1=𝑒_{∫𝜃𝑠𝑖𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑟),𝜈(𝑟)d𝑟−Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖_𝜃𝑠𝑖𝑒^{∫𝜃𝑟𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑎𝜓⁻¹(𝑟)𝐻(𝑚(𝑟))+^𝑛_𝑗=1+𝑒_{∫𝜃𝑠𝑖𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑟),𝜈(𝑟)d𝑟𝑒_{∫𝜃𝑠𝑖𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑚𝑠_𝑖,=𝑒𝑚𝑠_𝑖_{∫𝜃𝑠𝑖𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}⎧⎪⎨⎪⎩𝑒^{∫𝑠𝑖𝑠𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}−𝑚(𝑠)_𝑠𝑖𝑠𝑒^{∫𝑠𝑖𝑟𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}×𝑎𝜓⁻¹(𝑟)𝐻(𝑚(𝑟))+^𝑛_𝑗=1=𝑒_{∫𝜃𝑠𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑟),𝜈(𝑟)d𝑟−𝑚(𝑠)_𝑠𝑖𝑠𝑒^{∫𝜃𝑟𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑎𝜓⁻¹(𝑟)𝐻(𝑚(𝑟))+^𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑟),𝜈(𝑟)d𝑟.(2.71) So we can obtain 𝑚(𝑠)=−𝑒^{∫𝜃𝑠𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑚(𝑠)_𝜃𝑠𝑒^{∫𝜃𝑟𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑎𝜓⁻¹(𝑟)𝐻(𝑚(𝑟))+^𝑛_𝑗=1+𝑒_{∫𝜃𝑠𝑖𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑟),𝜈(𝑟)d𝑟Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖.(2.72) Repeating the above process for 𝜃∈[𝑠,𝑠+𝜔], we obtain 𝑚(𝑠)=−𝑒^{∫𝜃𝑠𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑚(𝑠)_𝜃𝑠𝑒^{∫𝜃𝑟𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}𝑎𝜓⁻¹(𝑟)𝐻(𝑚(𝑟))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑟),𝜈(𝑟)d𝑟_{𝑖∶𝑠𝑖∈[𝑠,𝜃)}𝑒^{∫𝜃𝑠𝑖𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}Φ⁻¹𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖.(2.73) Noticing that 𝑚(𝑠)=𝑚(𝑠+𝜔) and 𝑒^∫𝑠𝑠+𝜔^{𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}=𝑒^{∫0𝜔𝑎(𝜓−1(𝜏))𝑔((Φ−1𝑚)(𝜏))d𝜏}, we find that 𝑚 is a fixed point of Ψ.
Let 𝑚 be a fixed point of Ψ. If 𝑠≠𝑠_𝑖,𝑖∈ℤ, we have 𝑚^′(𝑠)=𝐺𝑠,𝑠+𝜔𝑎𝜓⁻¹𝑠+𝜔𝐻𝑚𝑠+𝜔+^𝑛_𝑗=1−𝐺(𝑠,𝑠)𝑎𝜓₋₁𝜆_𝑗𝑓_𝑗𝜓⁻¹𝑠+𝜔,𝜈𝑠+𝜔(𝑠)𝐻(𝑚(𝑠))+^𝑛_𝑗=1+𝑎𝜓₋₁𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑠),𝜈(𝑠)(𝑠)𝑔Φ⁻¹𝑚(𝑠)𝑚(𝑠).(2.74) By Lemmas 2.11 and 2.12, we have 𝜓(𝑡+𝜔)=𝜓(𝑡)+𝜓(𝜔)=𝜓(𝑡)+𝜔, so it is easy to have 𝑡+𝜔=𝜓⁻¹(𝜓(𝑡)+𝜔)=𝜓⁻¹(𝑠+𝜔) and 𝜈(𝑠+𝜔)=𝜈(𝑠). Therefore, we can obtain 𝑚^(𝑠)=𝑎𝜓⁻¹(𝑠)𝑔Φ⁻¹𝑚(𝑠)𝑚(𝑠)−𝑎𝜓⁻¹(𝑠)𝐻(𝑚(𝑠))+^𝑛_𝑗=1𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑠),𝜈(𝑠).(2.75) If 𝑠=𝑠_𝑖,𝑖∈ℤ, we can get 𝑚𝑠^+𝑖−𝑚𝑠^−𝑖=_{𝑗∶𝑠𝑗∈𝑠+𝑖,𝑠+𝑖+}𝜔_−𝐺𝑠_𝑖,𝑠_𝑗Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑗+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑗_{𝑗∶𝑠𝑗∈𝑠−𝑖,𝑠−𝑖+}𝜔_=𝐺𝑠_𝑖𝐺𝑠_𝑖,𝑠_𝑗Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑗+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑗−𝐺𝑠,𝑠_𝑖𝑖+𝜔Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑗+𝜔+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑗+𝜔=−Φ𝐵,𝑠_𝑖𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑗+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑗Φ⁻¹𝑚𝑠_𝑗+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑗.(2.76) Therefore, 𝑚 is 𝜔-periodic solution of (2.62). The proof is complete.

Lemma 2.16. Assume that (H₁)–(H₅) hold, then Ψ(𝑃)⊂𝑃, and Ψ∶𝑃→𝑃 is compact and continuous.

Proof. By the definition of 𝑃, for 𝑚∈𝑃, we have (Ψ𝑚)𝑠+𝜔=^𝑠+2_𝑠+𝜔𝜔𝐺𝑠+𝜔,𝜃𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠+}𝜔_,𝑠+2𝜔_=𝐺𝑠+𝜔,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖_𝑠+𝑠𝜔𝐺𝑠+𝜔+,𝜃+𝜔𝑎𝜓⁻¹𝜃+𝜔𝐻𝑚𝜃+𝜔_𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹𝜃+𝜔,𝜈𝜃+𝜔d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_=𝐺𝑠+𝜔,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖_𝑠+𝑠𝜔𝐺(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_=(Ψ𝑚)(𝑠).𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖(2.77) Thus, (Ψ𝑚)(𝑠+𝜔)=(Ψ𝑚)(𝑠),𝑠∈ℝ. So in view of (2.66), (2.68), for 𝑚∈𝑃,𝑠∈[0,𝜔], we have (Ψ𝑚)(𝑠)=^𝑠+_𝑠𝜔𝐺(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≥𝑟𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖_𝐿01−𝑟^𝐿0^𝑠+_𝑠𝜔𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≥𝑘1Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖⎫⎪⎬⎪⎭1−𝑟^𝑙0^𝑠+_𝑠𝜔𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≥𝑘‖Ψ𝑚‖.Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖⎫⎪⎬⎪⎭(2.78) Therefore, Ψ𝑚⊂𝑃. Next, we will show that Ψ is continuous and compact. Firstly, we will consider the continuity of Ψ. Let 𝑚_𝑛∈𝑃 and ‖𝑚_𝑛−𝑚‖→0 as 𝑛→+∞, then 𝑚∈𝑃 and |𝑚_𝑛(𝑠)−𝑚(𝑠)|→0 as 𝑛→+∞ for any 𝑠∈[0,𝜔]. By the continuity of 𝑓_𝑗(𝑗=1,2,…,𝑛),𝑔,Φ,Φ⁻¹,𝑊_𝑖(𝑖=1,2,…,𝑝), for any 𝑠∈[0,𝜔] and 𝜀>0, we have ||𝐻𝑚_𝑛(𝑠)−𝐻(𝑚(𝑠))||≤1−𝑟^𝑙03𝜔𝑎𝜀,(2.79) and denote 𝜈_𝑛(𝑠)∶=(Φ⁻¹𝑚_𝑛)(𝜓(𝜓⁻¹(𝑠)−𝑣_𝑗(𝜓⁻¹(𝑠)))), and it is easy to see that ‖𝑚_𝑛−𝑚‖→0 as 𝑛→+∞ implies ‖𝜈_𝑛−𝜈‖→0 as 𝑛→+∞, thus ||𝑓_𝑗𝜓⁻¹(𝑠),𝜈_𝑛(𝑠)−𝑓_𝑗𝜓⁻¹(𝑠),𝜈(𝑠)||≤1−𝑟^𝑙0||Φ𝐵_𝑖3𝜔𝜆_𝑗𝑛𝜀,𝑗=1,2,…,𝑛,−Φ𝐵_𝑖Φ⁻¹𝑚_𝑛𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚_𝑛𝑠_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖||≤1−𝑟^𝑙03𝜔𝑝𝜀,(2.80) where 𝑛 is sufficiently large. For 𝑠∈[0,𝜔], we have ‖‖Ψ𝑚_𝑛−Ψ𝑚‖‖=sup_𝑠∈0,𝜔_⎧⎪⎪⎨⎪⎪⎩|||||||_𝑠+𝑠𝜔𝐺(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻𝑚_𝑛(𝜃)+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈_𝑛(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_−𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚_𝑛𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚_𝑛𝑠_𝑖_𝑠+𝑠𝜔𝐺(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1−𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_=sup_𝑠∈0,𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖|||||||⎫⎪⎪⎬⎪⎪⎭𝜔_⎧⎪⎨⎪⎩|||||^𝑠+𝜔_𝑠𝐺+(𝑠,𝜃)𝑎𝜓⁻¹(𝑠)𝐻𝑚_𝑛(𝑠)−𝐻(𝑚(𝑠))_𝑛_𝑗=1+𝜆_𝑗𝑓^𝑗𝜓⁻¹(𝜃),𝜈_𝑛(𝜃)−𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_−Φ𝐵_𝑖𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚_𝑛𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚_𝑛𝑠_𝑖≤𝜔Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖|||||⎫⎪⎬⎪⎭1−𝑟^𝑙0𝑎1−𝑟^𝑙03𝜔𝑎𝜀+^𝑛_𝑗=1𝜆_𝑗1−𝑟^𝑙03𝜔𝜆_𝑗𝑛𝜀+𝑝13−𝜔𝑝𝑟^𝑙0𝜀=𝜀.(2.81) Therefore, Ψ is continuous on 𝑃.
Next, we prove that Ψ is a compact operator. Let 𝑆⊂𝑃 be an arbitrary bounded set in 𝑃, then there exists a number 𝐿₀>0 such that ‖𝑚‖<𝐿₀ for any 𝑚∈𝑆. We prove that Ψ𝑆 is compact. In fact, from (H₁), one has 𝑊_𝑖(0)=0,𝑖=1,2,…; by (2.53), it is easy to see that for any 𝑧∈ℝ, one has the following: ‖‖𝑊_𝑖(𝑧)‖‖≤𝑘𝑙₀𝑙₀‖𝑧‖+‖‖𝑊_𝑖(0)‖‖=𝑘𝑙₀𝑙₀‖𝑧‖,𝑖=1,2,….(2.82) So for any {𝑚_𝑛}_𝑛∈ℕ⊂𝑆 and 𝑠∈[0,𝜔], we have ‖‖Ψ𝑚_𝑛‖‖=sup_𝑠∈0,𝜔_⎧⎪⎨⎪⎩|||||^𝑠+𝜔_𝑠(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻𝑚_𝑛(𝜃)+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈_𝑛(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≤𝜔𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚_𝑛𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚_𝑛𝑠_𝑖|||||||⎫⎪⎪⎬⎪⎪⎭1−𝑟^𝑙0⎧⎪⎨⎪⎩𝑎𝐿𝑐1−𝑐+^𝑛_𝑗=1max_{𝜈∈−𝐿′0,𝐿′0𝑠∈0,}𝜔_+𝐿_0𝑝𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑠),𝜈(𝑠)_𝑖=1𝐵_𝑖‖‖Ψ𝑚_𝑛+𝑘𝑙₀𝑙₀∶=𝐾,^‖‖=sup_𝑠∈0,𝜔_⎧⎪⎨⎪⎩|||||𝑎𝜓₋₁−𝑎𝜓(𝑠)𝑔Φ⁻¹𝑚_𝑛(𝑠)𝑚_𝑛(𝑠)₋₁(𝑠)𝐻𝑚_𝑛(𝑠)+^𝑛_𝑗=1≤𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑠),𝜈_𝑛(𝑠)|||||⎫⎪⎬⎪⎭𝑎𝐿₀‖‖Ψ𝑚_𝑛‖‖+𝑎𝐿𝑐1−𝑐+^𝑛_𝑗=1max_{𝜈∈−𝐿′0,𝐿′0𝑠∈0,}𝜔_≤𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑠),𝜈(𝑠)𝑎𝐿₀𝐾+𝑎𝐿𝑐1−𝑐+^𝑛_𝑗=1max_{𝜈∈−𝐿′0,𝐿′0𝑠∈0,}𝜔_𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝑠),𝜈(𝑠)∶=𝑄,(2.83) where 𝐿^0=𝐿₀/(1−𝑐), which implies that {Ψ𝑚_𝑛}_𝑛∈ℕ and {(Ψ𝑚_𝑛)^}_𝑛∈ℕ are uniformly bounded on[0,𝜔]. Therefore, there exists a subsequence of {Ψ𝑚_𝑛}_𝑛∈ℕ which converges uniformly on [0,𝜔]; namely,Ψ𝑆 is compact. The proof is complete.

3. Main Results

Our main results of this paper are as follows.

Theorem 3.1. Assume that (H₁)–(H₅) hold, 0≤𝑐(𝑡)<1, for a sufficiently small Lipschitz constant 𝑙₀; suppose that the following conditions hold:(H₆)𝛼₀=(1−𝑐)(1−𝑟^𝑙0)−𝜔𝑎𝐿𝑐−(1−𝑐)∑^𝑝𝑖=1(𝐵_𝑖+𝑘(𝑙₀)𝑙₀)>0. (H₇)There exist positive constants 𝛽₁,𝛽₂, and 𝛽₄ with 0<𝛽₁<𝛽₂<𝛽₄ such that sup_𝑠∈[0,𝜔_]𝑓_𝑗𝜓⁻¹(𝑠),𝛽₁/1−𝑐𝛽₁/1−𝑐𝛼₀<sup^𝑠∈[0,𝜔_]𝑓_𝑗𝜓⁻¹(𝑠),𝛽₄/1−𝑐𝛽₄/1−𝑐𝛼₀<inf^𝑠∈[0,𝜔_]𝑓_𝑗𝜓⁻¹(𝑠),𝛼𝛽₂𝛼𝛽₂𝛽̃₀,(3.1) where 𝛽̃₀=((1−𝑟^𝐿0)/𝛼𝑟^𝐿0)−𝜔𝑎𝑙𝑐 and 𝜔=𝜓(𝜔).Then for all 𝑗=1,2,…,𝑛,𝜆_𝑗∈(𝜆_𝑗1,𝜆_𝑗2], (1.2) has at least three positive 𝜔-periodic solutions, where 𝜆_𝑗1=𝛼𝛽²𝛽̃₀𝜔𝑛inf_𝑠∈[0,𝜔_]𝑓_𝑗𝑡,𝛼𝛽₂,𝜆^𝑗2=𝛽⁴/1−𝑐𝛼₀𝜔𝑛sup_𝑠∈[0,𝜔_]𝑓_𝑗𝑡,𝛽₄/1−𝑐,𝑗=1,2,…,𝑛.(3.2)

Proof. First of all, since 0<𝛼<1/(1−𝑐) and 0<𝑟₀<1, we have 𝛼₀>0, so 𝛽̃₀=1−𝑟^𝐿0𝛼𝑟^𝐿0−𝜔𝑎𝑙𝑐>1−𝑟^𝐿0𝛼𝑟^𝐿0−𝜔𝑎𝑙𝑐>1−𝑟^𝐿0𝛼−1−+1−𝑐1−𝑟^𝑙0𝑐^𝑝_𝑖=1𝐵_𝑖>1−+𝑘𝑙₀𝑙₀𝑐𝑟^𝑙0−𝑟^𝐿0+1−𝑐^𝑝_𝑖=1𝐵_𝑖+𝑘𝑙₀𝑙₀>0.(3.3) Furthermore, 0<𝜆_𝑗1<𝜆_𝑗2 in view of (3.1).
Now, define for each 𝜆_𝑗∈(𝜆_𝑗1,𝜆_𝑗2] and 𝑚∈𝑃 a mapping Ψ∶𝑃→𝑃 by (Ψ𝑚)(𝑠)=^𝑠+_𝑠𝜔𝐺(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_𝐺𝑠,𝑠_iΦ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖,(3.4) and a function 𝜌∶𝑃→[0,∞) by 𝜌(𝑚)=min_𝑠∈[0,𝜔_]𝑚(𝑠).(3.5) For 𝑚∈𝑃_𝛽4, by Lemma 2.13, we have 0<Φ⁻¹𝑚(𝑠)<𝛽⁴1−𝑐.(3.6) It follows from (2.51), (2.68), (3.6), and (H₂), for all 𝑗=1,2,…,𝑛,𝜆_𝑗∈(𝜆_𝑗1,𝜆_𝑗2] and 𝑚∈𝑃_𝛽4 that (Ψ𝑚)(𝑠)=^𝑠+_𝑠𝜔𝐺(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≤1𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖1−𝑟^𝑙0^𝑠+_𝑠𝜔𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≤1Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖⎫⎪⎬⎪⎭1−𝑟^𝑙0⎧⎪⎨⎪⎩𝑎𝜔𝐿𝑐1−𝑐𝛽⁴+𝜔^𝑛_𝑗=1𝜆_𝑗2sup_𝑠∈0,𝜔_𝑓_𝑗𝜓⁻¹(𝑠),𝛽⁴1−𝑐+𝛽^4𝑝_𝑖=1=1𝑏_𝑖+𝑘𝑙₀𝑙₀⎫⎪⎬⎪⎭1−𝑟^𝑙0⎧⎪⎨⎪⎩𝑎𝜔𝐿𝑐1−𝑐𝛽⁴+^𝑛_𝑗=1𝛽₄/1−𝑐𝛼₀𝜔𝑛sup_𝑠∈[0,𝜔_]𝑓_𝑗𝜓⁻¹(𝑠),𝛽₄/1−𝑐sup_𝑠∈0,𝜔_𝑓_𝑗𝜓⁻¹(𝑠),𝛽⁴+𝛽_4𝑝1−𝑐_𝑖=1𝐵_𝑖+𝑘𝑙₀𝑙₀≤𝛽₄.(3.7) By Lemma 2.16, we know that Ψ is completely continuous on𝑃_𝛽4.
We now assert that the condition (2) of Lemma 2.1 holds. Indeed, if 𝑚∈𝑃_𝛽1, then similar to above argument, by (3.1), we have (Ψ𝑚)(𝑠)=^𝑠+_𝑠𝜔𝐺(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≤1𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖1−𝑟^𝑙0^𝑠+_𝑠𝜔𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≤1Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖⎫⎪⎬⎪⎭1−𝑟^𝑙0⎧⎪⎨⎪⎩_𝑠+𝜔_𝑠𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≤1Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖⎫⎪⎬⎪⎭1−𝑟^𝑙0⎧⎪⎨⎪⎩𝑎𝜔𝐿𝑐1−𝑐𝛽¹+𝜔^𝑛_𝑗=1𝜆_𝑗2sup_𝑠∈0,𝜔_𝑓_𝑗𝜓⁻¹(𝑠),𝛽¹1−𝑐+𝛽^1𝑝_𝑖=1=1𝐵_𝑖+𝑘𝑙₀𝑙₀⎫⎪⎬⎪⎭1−𝑟^𝑙0⎧⎪⎨⎪⎩𝑎𝜔𝐿𝑐1−𝑐𝛽¹+^𝑛_𝑗=1𝛽₄/1−𝑐𝛼₀𝜔𝑛sup_𝑠∈[0,𝜔_]𝑓_𝑗𝜓⁻¹(𝑠),𝛽₄/1−𝑐sup_𝑠∈0,𝜔_𝑓_𝑗𝜓⁻¹(𝑠),𝛽¹+𝛽_1𝑝1−𝑐_𝑖=1𝐵_𝑖<1+𝑘𝑙₀𝑙₀⎫⎪⎬⎪⎭1−𝑟^𝑙0𝑎𝜔𝐿𝑐1−𝑐𝛽¹+𝛽⁴1−𝑐𝛼⁰𝛽₁𝛽₄+𝛽_1𝑝_𝑖=1𝐵_𝑖+𝑘𝑙₀𝑙₀=𝛽₁.(3.8) Hence, ‖Ψ𝑚‖<𝛽₁ holds.
Choose a positive constant 𝛽₃ such that 0<𝛽₂<𝑘𝛽₃<𝛽₃≤𝛽₄. Next, we show that the condition (1) of Lemma 2.1 holds. Obviously, 𝜌 is a concave continuous function on 𝑃 with 𝜌(𝑚)≤‖𝑚‖ for 𝑚∈𝑃_𝛽4. We notice that if 𝑚(𝑠)=(2/5)𝛽₂+(3/5)𝛽₃ for 𝑠∈[0,𝜔], then 𝑚∈{𝑚∈𝑃(𝜌,𝛽₂,𝛽₃)∶𝜌(𝑚)>𝛽₂} which implies {𝑚∈𝑃(𝜌,𝛽₂,𝛽₃)∶𝜌(𝑚)>𝛽₂}≠∅. For 𝑚∈𝑃(𝜌,𝛽₂,𝛽₃), we have 𝛽₂≤𝜌(𝑚)=min_𝑠∈[0,𝜔_]𝑚(𝑠)≤‖𝑚‖≤𝛽₃,(3.9) which implies, from (2.50), that Φ⁻¹𝑚(𝑠)≥𝛼‖𝑚‖≥𝛼𝛽₂.(3.10) And it is also clear that Φ(𝑥) is nondecreasing for 𝑥>0 and 𝐵_𝑖,𝑊_𝑖∈𝐶(ℝ,ℝ⁺), and we can easily have Φ𝐵_𝑖Φ⁻¹,Φ𝑊_𝑖Φ⁻¹∈𝐶(ℝ,ℝ⁺). Hence 𝜌(Ψ𝑚)=min_𝑠∈[0,𝜔_]=min_𝑠∈[0,(Ψ𝑚)(𝑠)𝜔_]^𝑠+_𝑠𝜔𝐺(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≥𝑟𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖_𝐿0⎫⎪⎬⎪⎭1−𝑟^𝐿0min_𝑠∈[0,𝜔_]^𝑠+_𝑠𝜔𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝛼𝛽₂d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_>𝑟Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖_𝐿0⎫⎪⎬⎪⎭1−𝑟^𝐿0⎧⎪⎨⎪⎩𝜔𝑎𝑙𝑐𝛼𝛽₂+𝜔^𝑛_𝑗=1𝜆_𝑗1inf_𝑠∈0,𝜔_+min_{𝛽2≤𝑚≤𝛽3𝑝}𝑓_𝑗𝜓⁻¹(𝑠),𝛼𝛽₂_𝑖=1Φ𝐵_𝑖≥𝑟Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖_𝐿01−𝑟^𝐿0⎧⎪⎪⎨⎪⎪⎩𝜔𝑎𝑙𝑐𝛼𝛽₂+𝜔^𝑛_𝑗=1𝛼𝛽₂𝛽̃₀𝜔𝑛inf_𝑠∈[0,𝜔_]𝑓_𝑗𝜓⁻¹(𝑠),𝛼𝛽₂inf^𝑠∈0,𝜔_+min_{𝛽2≤𝑚≤𝛽3𝑝}𝑓_𝑗𝜓⁻¹(𝑠),𝛼𝛽₂_𝑖=1Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖⎫⎪⎪⎬⎪⎪⎭>𝛽₂(3.11) for all 𝑚∈𝑃(𝜌,𝛽₂,𝛽₃).
Finally, we prove that the condition (3) of Lemma 2.1 holds. Let 𝑚∈𝑃(𝜌,𝛽₂,𝛽₄) and ‖Ψ𝑚‖>𝛽₃, then 𝜌(Ψ𝑚)>𝛽₂. We notice that (3.4) implies that ‖Ψ𝑚‖≤11−𝑟^𝑙0⎧⎪⎪⎨⎪⎪⎩_𝑠+𝑠𝜔𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝛼𝛽₂d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖⎫⎪⎬⎪⎭.(3.12) Thus 𝜌(Ψ𝑚)=min_𝑠∈[0,𝜔_]^𝑠+_𝑠𝜔𝐺(𝑠,𝜃)𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≥𝑟𝐺𝑠,𝑠_𝑖Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖_𝐿0⎫⎪⎬⎪⎭1−𝑟^𝐿0min_𝑠∈[0,𝜔_]⎧⎪⎪⎨⎪⎪⎩_𝑠+𝑠𝜔𝑎𝜓⁻¹(𝜃)𝐻(𝑚(𝜃))+^𝑛_𝑗=1+𝜆_𝑗𝑓_𝑗𝜓⁻¹(𝜃),𝜈(𝜃)d𝜃_{𝑖∶𝑠𝑖∈𝑠,𝑠+}𝜔_≥𝑟Φ𝐵_𝑖Φ⁻¹𝑚𝑠_𝑖+Φ𝑊_𝑖Φ⁻¹𝑚𝑠_𝑖_𝐿0⎫⎪⎬⎪⎭1−𝑟^𝑙01−𝑟^𝑙0‖Ψ𝑚‖≥𝑘𝛽₃>𝛽₂.(3.13) To sum up, all the hypotheses of Lemma 2.1 are satisfied. Hence Ψ has at least three positive fixed points. That is, (1.2) has at least three positive 𝜔-periodic solutions. This completes the proof.

Corollary 3.2. Suppose (H₁)–(H₆) hold. If lim_𝑥→∞sup_𝑠∈[0,𝜔_]𝑓_𝑗𝜓⁻¹(𝑠),𝑥lim_𝑥→0𝑥=0,(3.14)sup_𝑠∈[0,𝜔_]𝑓_𝑗𝜓⁻¹(𝑠),𝑥𝑥=0,(3.15) where 𝜔=𝜓(𝜔); then (1.2) has at least three positive 𝜔-periodic solutions.

Proof. In view of (3.14), we can choose 𝛽₄>𝛽₂>0 such that the second inequality in (3.1) holds, and in view of (3.15), we can choose 𝛽₁∈(0,𝛽₂) such that the first inequality in (3.1) holds. Therefore, the conclusion of Theorem 3.1 holds. This completes the proof.

4. An Example

Let us consider the variable time scale 𝕋₀(𝑥) constructed by 𝑡_𝑖=𝑖,𝜏_𝑖(𝑥)=(−1)^𝑖𝑙₀sin𝑥, where |𝑥|>1 for all 𝑡∈𝕋₀(𝑥), 0<𝑙₀<(1/2) and consider 𝜋-periodic system:𝑥(𝑡)+19|cos𝑡|𝑥(𝑡−𝛼)^′=𝜋1||sin𝑡||31+13𝑒^−𝑥𝑥(𝑡)−3𝑙⁰𝑛^𝑛_𝑗=1𝜆_𝑗𝑥(^1/2𝑡,(𝑥𝑡))l∈n𝕋𝑥₀𝑡−𝑒^{(1/𝑗)|sin𝑡|}+1,𝑥(𝑥),₊=0.0323^𝑖𝑡𝑥+0.02𝑙₀sin𝑥+𝑥,₊=2𝑖+1−𝑙₀sin𝑥,(4.1) where 𝛼 is a constant, 𝜆_𝑗,𝑗=1,2,…,𝑛 are nonnegative parameters. In this case, 𝑒𝑐(𝑡)=(1/9)|cos𝑡|,𝑎(𝑡)=1/𝜋,𝑔(𝑥(𝑡))=(1/3)+(1/3)𝑒^−𝑥,𝐵_𝑖=0.03(2/3)^𝑖,𝐽_𝑖(𝑥)=0.02𝑙₀sin𝑥,𝑣_𝑗(𝑡)=_{(1/𝑗)|sin𝑡|} and 𝑓_𝑗(𝑡,𝑥(𝑡−𝑣_𝑗(𝑡)))=(𝑙₀/3𝑛)𝑥^1/2(𝑡)ln(𝑥(𝑡−𝑣_𝑗(𝑡))+1),𝑗=1,2,…,𝑛,𝑖∈ℕ. Obviously, (H₁)–(H₃) are satisfied, and it is easy to see that (3.14) and (3.15) hold.

By the formula of 𝜓-substitution and 𝛿_𝑘=1, one can find𝜔=𝜓(𝜔)=𝜔−_0<2𝑘<𝜔𝛿_𝑘=𝜋−1.(4.2)

Clearly, 𝐿=(2/3),𝑙=(1/3),0≤𝑐(𝑡)≤(1/9)<1,𝑟₀=𝑒^{−(𝜋−1)(1/𝜋)}≈0.5058 and 𝑐=1/9,𝑐=0, so we can find 𝛿=(𝑐+𝑐)/2=1/18, and it is easy to check that ((2𝛿−𝑐)(1−𝑐²))/(1−(2𝛿−𝑐)²)=9/80≈0.1125<(𝑟^𝐿0(1−𝑟^𝑙0))/(1−𝑟^𝐿0)=0.3532. Thus, (H₄) holds. Furthermore, we also have‖‖‖𝜕𝑓_𝑗𝜕𝑥‖‖‖=3𝑙⁰𝑛‖‖‖ln2(𝑥𝑥+^1/21)+𝑥^1/2𝑥+1‖‖‖≤3𝑙⁰𝑛‖‖‖ln(𝑥𝑥+^1/21)^1/2‖‖‖+‖‖‖𝑥^1/2≤𝑙𝑥+1₀‖‖‖3𝑛‖‖‖(𝑥+𝑥1^1/2)^1/2‖‖‖+‖‖‖𝑥^1/2𝑥+1‖‖‖=3𝑙⁰𝑛‖‖‖1+𝑥1^1/2‖‖‖+‖‖‖𝑥^1/2<𝑙𝑥+1₀‖‖‖3𝑛(2+1)=𝑙𝑛⁰.(4.3) Hence, for any 𝑥₁,𝑥₂∈ℝ, one can get^𝑛_𝑗=1||𝑓_𝑗𝑡,𝑥₁−𝑓_𝑗𝑡,𝑥₂||≤||||𝜕𝑓^𝑗𝜕𝑥||||||𝑥¹−𝑥₂||<𝑛⋅𝑙⁰𝑛||𝑥¹−𝑥₂||=𝑙₀||𝑥₁−𝑥₂||.(4.4) So (H₅) is satisfied. For a sufficiently small 𝑙₀, one can also have𝛼₀=1−𝑐1−𝑟^𝑙0−𝜔𝑎𝐿𝑐−1−𝑐^𝑝_𝑖=1𝐵_𝑖+𝑘𝑙₀𝑙₀=0.0768−89^𝑝_𝑖=1𝑘𝑙₀𝑙₀,(4.5) since 𝑘(𝑙₀) is a bounded function; for a sufficiently small 𝑙₀, one can have 𝛼₀>0 such that (H₆) holds. Therefore, according to Corollary 3.2, (4.1) has at least three positive 𝜋-periodic solutions.

Acknowledgment

This work is supported by the National Natural Sciences Foundation of China under Grant no. 10971183.