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Abstract and Applied Analysis
VolumeΒ 2012Β (2012), Article IDΒ 707319, 17 pages
http://dx.doi.org/10.1155/2012/707319
Research Article

Periodic Solutions in Shifts 𝜹± for a Nonlinear Dynamic Equation on Time Scales

Department of Mathematics, Ege University, Bornova, 35100 Izmir, Turkey

Received 16 March 2012; Accepted 27 June 2012

Academic Editor: ElenaΒ Braverman

Copyright Β© 2012 Erbil Γ‡etin and F. Serap Topal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let π•‹βŠ‚β„ be a periodic time scale in shifts 𝛿±. We use a fixed point theorem due to Krasnosel'skiΔ­ to show that nonlinear delay in dynamic equations of the form π‘₯Ξ”(𝑑)=βˆ’π‘Ž(𝑑)π‘₯𝜎(𝑑)+𝑏(𝑑)π‘₯Ξ”(π›Ώβˆ’(π‘˜,𝑑))π›ΏΞ”βˆ’(π‘˜,𝑑)+π‘ž(𝑑,π‘₯(𝑑),π‘₯(π›Ώβˆ’(π‘˜,𝑑))),π‘‘βˆˆπ•‹, has a periodic solution in shifts 𝛿±. We extend and unify periodic differential, difference, β„Ž-difference, and π‘ž-difference equations and more by a new periodicity concept on time scales.

1. Introduction

The time scales approach unifies differential, difference, β„Ž-difference, and π‘ž-differences equations and more under dynamic equations on time scales. The theory of dynamic equations on time scales was introduced by Hilger in this Ph.D. thesis in 1988 [1]. The existence problem of periodic solutions is an important topic in qualitative analysis of ordinary differential equations. There are only a few results concerning periodic solutions of dynamic equations on time scales such as in [2, 3]. In these papers, authors considered the existence of periodic solutions for dynamic equations on time scales satisfying the condition "thereexistsaπœ”>0suchthatπ‘‘Β±πœ”βˆˆπ•‹βˆ€π‘‘βˆˆπ•‹.”(1.1) Under this condition all periodic time scales are unbounded above and below. However, there are many time scales such as π‘žβ„€={π‘žπ‘›βˆΆπ‘›βˆˆβ„€}βˆͺ{0} and βˆšβˆšβ„•={π‘›βˆΆπ‘›βˆˆβ„•} which do not satisfy condition (1.1). AdΔ±var and Raffoul introduced a new periodicity concept on time scales which does not oblige the time scale to be closed under the operation π‘‘Β±πœ” for a fixed πœ”>0. He defined a new periodicity concept with the aid of shift operators 𝛿± which are first defined in [4] and then generalized in [5].

Let 𝕋 be a periodic time scale in shifts 𝛿± with period π‘ƒβˆˆ(𝑑0,∞)𝕋 and 𝑑0βˆˆπ•‹ is nonnegative and fixed. We are concerned with the existence of periodic solutions in shifts 𝛿± for the nonlinear dynamic equation with a delay function π›Ώβˆ’(π‘˜,𝑑): π‘₯Ξ”(𝑑)=βˆ’π‘Ž(𝑑)π‘₯𝜎(𝑑)+𝑏(𝑑)π‘₯Ξ”ξ€·π›Ώβˆ’ξ€Έπ›Ώ(π‘˜,𝑑)Ξ”βˆ’ξ€·ξ€·π›Ώ(π‘˜,𝑑)+π‘žπ‘‘,π‘₯(𝑑),π‘₯βˆ’(π‘˜,𝑑)ξ€Έξ€Έ,π‘‘βˆˆπ•‹,(1.2) where π‘˜ is fixed if 𝕋=ℝ and π‘˜βˆˆ[𝑃,∞)𝕋 if 𝕋 is periodic in shifts 𝛿± with period 𝑃.

Kaufmann and Raffoul in [2] used Krasnosel'skiΔ­ fixed point theorem and showed the existence of a periodic solution of (1.2) and used the contraction mapping principle to show that the periodic solution is unique when 𝕋 satisfies condition (1.1). Similar results were obtained concerning (1.2) in [6, 7] in the case 𝕋=ℝ, 𝕋=β„€, respectively. Currently, AdΔ±var and Raffoul used Lyapunov's direct method to obtain inequalities that lead to stability and instability of delay dynamic equations of (1.2) when π‘ž=0 on a time scale having a delay function π›Ώβˆ’ in [8] and also using the topological degree method and Schaefers fixed point theorem, they deduce the existence of periodic solutions of nonlinear system of integrodynamic equations on periodic time scales in [9].

Hereafter, we use the notation [π‘Ž,𝑏]𝕋 to indicate the time scale interval [π‘Ž,𝑏]βˆ©π•‹. The intervals [π‘Ž,𝑏)𝕋, (π‘Ž,𝑏]𝕋 and (π‘Ž,𝑏)𝕋 are similarly defined.

In Section 2, we will state some facts about the exponential function on time scales, the new periodicity concept for time scales, and some important theorems which will be needed to show the existence of a periodic solution in shifts 𝛿±. In Section 3, we will give some lemmas about the exponential function and the graininess function with shift operators. Finally, we present our main result in Section 4 by using Krasnosel'skiΔ­ fixed point theorem and give an example.

2. Preliminaries

In this section, we mention some definitions, lemmas, and theorems from calculus on time scales which can be found in [10, 11]. Next, we state some definitions, lemmas, and theorems about the shift operators and the new periodicity concept for time scales which can be found in [12].

Definition 2.1 (see [10]). A function π‘βˆΆπ•‹β†’β„ is said to be regressive provided 1+πœ‡(𝑑)𝑝(𝑑)β‰ 0 for all π‘‘βˆˆπ•‹πœ…, where πœ‡(𝑑)=𝜎(𝑑)βˆ’π‘‘. The set of all regressive rd-continuous functions πœ‘βˆΆπ•‹β†’β„ is denoted by β„› while the set β„›+ is given by β„›+={πœ‘βˆˆβ„›βˆΆ1+πœ‡(𝑑)πœ‘(𝑑)>0forallπ‘‘βˆˆπ•‹}.

Let πœ‘βˆˆβ„› and πœ‡(𝑑)>0 for all π‘‘βˆˆπ•‹. The exponential function on 𝕋 is defined by π‘’πœ‘(ξ‚΅ξ€œπ‘‘,𝑠)=expπ‘‘π‘ πœπœ‡(π‘Ÿ)(ξ‚Ά,πœ‘(π‘Ÿ))Ξ”π‘Ÿ(2.1) where πœπœ‡(𝑠) is the cylinder transformation given by πœπœ‡(π‘Ÿ)ξƒ―1(πœ‘(π‘Ÿ))∢=πœ‡(π‘Ÿ)Log(1+πœ‡(π‘Ÿ)πœ‘(π‘Ÿ)),ifπœ‡(π‘Ÿ)>0,πœ‘(π‘Ÿ),ifπœ‡(π‘Ÿ)=0.(2.2) Also, the exponential function 𝑦(𝑑)=𝑒𝑝(𝑑,𝑠) is the solution to the initial value problem 𝑦Δ=𝑝(𝑑)𝑦,𝑦(𝑠)=1. Other properties of the exponential function are given in the following lemma [10, Theorem 2.36].

Lemma 2.2 (see [10]). Let 𝑝,π‘žβˆˆβ„›. Then(i)𝑒0(𝑑,𝑠)≑1 and 𝑒𝑝(𝑑,𝑑)≑1;(ii)𝑒𝑝(𝜎(𝑑),𝑠)=(1+πœ‡(𝑑)𝑝(𝑑))𝑒𝑝(𝑑,𝑠); (iii)1/𝑒𝑝(𝑑,𝑠)=π‘’βŠ–(𝑑,𝑠), where, βŠ–π‘(𝑑)=βˆ’(𝑝(𝑑))/(1+πœ‡(𝑑)𝑝(𝑑));(iv)𝑒𝑝(𝑑,𝑠)=1/𝑒𝑝(𝑠,𝑑)=π‘’βŠ–π‘(𝑠,𝑑); (v)𝑒𝑝(𝑑,𝑠)𝑒𝑝(𝑠,π‘Ÿ)=𝑒𝑝(𝑑,π‘Ÿ); (vi)𝑒𝑝(𝑑,𝑠)π‘’π‘ž(𝑑,𝑠)=π‘’π‘βŠ•π‘ž(𝑑,𝑠); (vii)𝑒𝑝(𝑑,𝑠)/π‘’π‘ž(𝑑,𝑠)=π‘’π‘βŠ–π‘ž(𝑑,𝑠); (viii)(1/𝑒𝑝(β‹…,𝑠))Ξ”=βˆ’π‘(𝑑)/π‘’πœŽπ‘(β‹…,𝑠).

The following definitions, lemmas, corollaries, and examples are about the shift operators and new periodicity concept for time scales which can be found in [12].

Definition 2.3 (see [12]). Let π•‹βˆ— be a nonempty subset of the time scale 𝕋 including a fixed number 𝑑0βˆˆπ•‹βˆ— such that there exist operators π›ΏΒ±βˆΆ[𝑑0,∞)π•‹Γ—π•‹βˆ—β†’π•‹βˆ— satisfying the following properties.(𝑃.1) The function 𝛿± are strictly increasing with respect to their second arguments, that is, if 𝑇0ξ€Έ,𝑇,𝑑0ξ€Έ,π‘’βˆˆπ’ŸΒ±ξ€½ξ€Ίπ‘‘βˆΆ=(𝑠,𝑑)∈0ξ€Έ,βˆžπ•‹Γ—π•‹βˆ—βˆΆπ›Ώβˆ“(𝑠,𝑑)βˆˆπ•‹βˆ—ξ€Ύ,(2.3) then 𝑇0≀𝑑<𝑒implies𝛿±𝑇0ξ€Έ,𝑑<𝛿±𝑇0ξ€Έ.,𝑒(2.4)(𝑃.2) If (𝑇1,𝑒),(𝑇2,𝑒)βˆˆπ’Ÿ with 𝑇1<𝑇2, then π›Ώβˆ’(𝑇1,𝑒)>π›Ώβˆ’(𝑇2,𝑒), and if (𝑇1,𝑒),(𝑇2,𝑒)∈𝐷+ with 𝑇1<𝑇2, then 𝛿+(𝑇1,𝑒)<𝛿+(𝑇2,𝑒).(𝑃.3) If π‘‘βˆˆ[𝑑0,∞)𝕋, then (𝑑,𝑑0)∈𝐷+ and 𝛿+(𝑑,𝑑0)=𝑑. Moreover, if π‘‘βˆˆπ•‹βˆ—, then (𝑑0,𝑑)∈𝐷+ and 𝛿+(𝑑0,𝑑)=𝑑 holds. (𝑃.4) If (𝑠,𝑑)∈𝐷±, then (𝑠,𝛿±(𝑠,𝑑))βˆˆπ·βˆ“ and π›Ώβˆ“(𝑠,𝛿±(𝑠,𝑑))=𝑑, respectively.(𝑃.5) If (𝑠,𝑑)∈𝐷± and (𝑒,𝛿±(𝑠,𝑑))∈𝐷±, then (𝑠,π›Ώβˆ“(𝑒,𝑑))∈𝐷± and π›Ώβˆ“(𝑒,𝛿±(𝑠,𝑑))=𝛿±(𝑠,π›Ώβˆ“(𝑒,𝑑)), respectively.
Then the operators π›Ώβˆ’ and 𝛿+ associated with 𝑑0βˆˆπ•‹βˆ— (called the initial point) are said to be backward and forward shift operators on the set π•‹βˆ—, respectively. The variable π‘ βˆˆ[𝑑0,∞)𝕋 in 𝛿±(𝑠,𝑑) is called the shift size. The values 𝛿+(𝑠,𝑑) and π›Ώβˆ’(𝑠,𝑑) in π•‹βˆ— indicate 𝑠 units translation of the term π‘‘βˆˆπ•‹βˆ— to the right and left, respectively. The sets π’ŸΒ± are the domains of the shift operator 𝛿±, respectively. Hereafter, π•‹βˆ— is the largest subset of the time scale 𝕋 such that the shift operators π›ΏΒ±βˆΆ[𝑑0,∞)π•‹Γ—π•‹βˆ—β†’π•‹βˆ— exist.

Example 2.4 (see [12]). (i)𝕋=ℝ, 𝑑0=0, π•‹βˆ—=ℝ, π›Ώβˆ’(𝑠,𝑑)=π‘‘βˆ’π‘ , and 𝛿+(𝑠,𝑑)=𝑑+𝑠.(ii)𝕋=β„€, 𝑑0=0, π•‹βˆ—=β„€, π›Ώβˆ’(𝑠,𝑑)=π‘‘βˆ’π‘ , and 𝛿+(𝑠,𝑑)=𝑑+𝑠.(iii)𝕋=π‘žβ„€βˆͺ{0}, 𝑑0=1, π•‹βˆ—=π‘žβ„€, π›Ώβˆ’(𝑠,𝑑)=𝑑/𝑠, and 𝛿+(𝑠,𝑑)=𝑑𝑠.(iv)𝕋=β„•1/2, 𝑑0=0, π•‹βˆ—=β„•1/2, π›Ώβˆ’βˆš(𝑠,𝑑)=𝑑2βˆ’π‘ 2, and 𝛿+√(𝑠,𝑑)=𝑑2+𝑠2.

Definition 2.5 (periodicity in shifts [12]). Let 𝕋 be a time scale with the shift operators 𝛿± associated with the initial point 𝑑0βˆˆπ•‹βˆ—. The time scale 𝕋 is said to be periodic in shift 𝛿± if there exists a π‘βˆˆ(𝑑0,∞)π•‹βˆ— such that (𝑝,𝑑)∈𝐷± for all π‘‘βˆˆπ•‹βˆ—. Furthermore, if ξ€½ξ€·π‘‘π‘ƒβˆΆ=infπ‘βˆˆ0ξ€Έ,βˆžπ•‹βˆ—βˆΆ(𝑝,𝑑)∈𝐷±,βˆ€π‘‘βˆˆπ•‹βˆ—ξ€Ύβ‰ π‘‘0,(2.5) then 𝑃 is called the period of the time scale 𝕋.

Example 2.6 (see [12]). The following time scales are not periodic in the sense of condition (1.1) but periodic with respect to the notion of shift operators given in Definition 2.5:(i)𝕋1={±𝑛2βˆΆπ‘›βˆˆβ„€},𝛿±(𝑃,𝑑)=(βˆšβˆšπ‘‘Β±π‘ƒ)2√,𝑑>0;±𝑃,𝑑=0;βˆ’(βˆšβˆ’π‘‘Β±π‘ƒ)2,𝑑<0;, 𝑃=1,𝑑0=0,(ii)𝕋2=π‘žβ„€, 𝛿±(𝑃,𝑑)=𝑃±1𝑑, 𝑃=π‘ž,𝑑0=1,(iii)𝕋3=βˆͺπ‘›βˆˆβ„€[22𝑛,22𝑛+1],  𝛿±(𝑃,𝑑)=𝑃±1𝑑,  𝑃=4,𝑑0=1,(iv)𝕋4={π‘žπ‘›/(1+π‘žπ‘›)βˆΆπ‘ž>1isconstantandπ‘›βˆˆβ„€}βˆͺ{0,1}, 𝛿±(π‘žπ‘ƒ,𝑑)=((ln(𝑑/(1βˆ’π‘‘))Β±ln(𝑃/(1βˆ’π‘ƒ)))/lnπ‘ž)1+π‘ž((ln(𝑑/(1βˆ’π‘‘))Β±ln(𝑃/(1βˆ’π‘ƒ)))/lnπ‘ž)π‘ž,𝑃=.1βˆ’π‘ž(2.6)
Notice that the time scale 𝕋4 in Example 2.6 is bounded above and below and π•‹βˆ—4={π‘žπ‘›/(1+π‘žπ‘›)βˆΆπ‘ž>1 is constant and π‘›βˆˆβ„€}.

Remark 2.7 (see [12]). Let 𝕋 be a time scale, that is, periodic in shifts with the period 𝑃. Thus, by (𝑃.4) of Definition 2.3 the mapping 𝛿𝑃+βˆΆπ•‹βˆ—β†’π•‹βˆ— defined by 𝛿𝑃+(𝑑)=𝛿+(𝑃,𝑑) is surjective. On the other hand, by (𝑃.1) of Definition 2.3 shift operators 𝛿± are strictly increasing in their second arguments. That is, the mapping 𝛿𝑃+(𝑑)=𝛿+(𝑃,𝑑) is injective. Hence, 𝛿𝑃+ is an invertible mapping with the inverse (𝛿𝑃+)βˆ’1=π›Ώπ‘ƒβˆ’ defined by π›Ώπ‘ƒβˆ’(𝑑)∢=π›Ώβˆ’(𝑃,𝑑).
We assume that 𝕋 is a periodic time scale in shift 𝛿± with period 𝑃. The operators π›Ώπ‘ƒΒ±βˆΆπ•‹βˆ—β†’π•‹βˆ— are commutative with the forward jump operator πœŽβˆΆπ•‹β†’π•‹ given by 𝜎(𝑑)∢=inf{π‘ βˆˆπ•‹βˆΆπ‘ >𝑑}. That is, (π›Ώπ‘ƒΒ±βˆ˜πœŽ)(𝑑)=(πœŽβˆ˜π›Ώπ‘ƒΒ±)(𝑑) for all π‘‘βˆˆπ•‹βˆ—.

Lemma 2.8 (see [12]). The mapping 𝛿𝑃+βˆΆπ•‹βˆ—β†’π•‹βˆ— preserves the structure of the points in π•‹βˆ—. That is, 𝜎(𝑑)=𝑑impliesπœŽξ€·π›Ώ+ξ€Έ(𝑃,𝑑)=𝛿+(𝑃,𝑑)and𝜎(𝑑)>𝑑impliesπœŽξ€·π›Ώ+ξ€Έ(𝑃,𝑑)>𝛿+(𝑃,𝑑).(2.7)

Corollary 2.9 (see [12]). 𝛿+(𝑃,𝜎(𝑑))=𝜎(𝛿+(𝑃,𝑑)) and π›Ώβˆ’(𝑃,𝜎(𝑑))=𝜎(π›Ώβˆ’(𝑃,𝑑)) for all π‘‘βˆˆπ•‹βˆ—.

Definition 2.10 (periodic function in shift 𝛿± [12]). Let 𝕋 be a time scale that is periodic in shifts 𝛿± with the period 𝑃. We say that a real value function 𝑓 defined on π•‹βˆ— is periodic in shifts 𝛿± if there exists a π‘‡βˆˆ[𝑃,∞)π•‹βˆ— such that (𝑇,𝑑)βˆˆπ·Β±ξ€·π›Ώ,𝑓𝑇±(𝑑)=𝑓(𝑑)βˆ€π‘‘βˆˆπ•‹βˆ—,(2.8) where π›Ώπ‘‡Β±βˆΆ=𝛿±(𝑇,𝑑). The smallest number π‘‡βˆˆ[𝑃,∞)π•‹βˆ— such that (5) holds is called the period of 𝑓.

Definition 2.11 (Ξ”-periodic function in shifts 𝛿± [12]). Let 𝕋 be a time scale that is periodic in shifts 𝛿± with the period 𝑃. We say that a real value function 𝑓 defined on π•‹βˆ— is Ξ”-periodic in shifts 𝛿± if there exists a π‘‡βˆˆ[𝑃,∞)π•‹βˆ— such that (𝑇,𝑑)βˆˆπ·Β±βˆ€π‘‘βˆˆπ•‹βˆ—,theshifts𝛿𝑇±areΞ”-differentiablewithπ‘Ÿπ‘‘-continuousderivatives,𝑓𝛿𝑇±𝛿(𝑑)±Δ𝑇=𝑓(𝑑)βˆ€π‘‘βˆˆπ•‹βˆ—,(2.9) where π›Ώπ‘‡Β±βˆΆ=𝛿±(𝑇,𝑑). The smallest number π‘‡βˆˆ[𝑃,∞)π•‹βˆ— such that (2.9) hold is called the period of 𝑓.

Notice that Definitions 2.10 and 2.11 give the classic periodicity definition on time scales whenever π›Ώπ‘‡Β±βˆΆ=𝑑±𝑇 are the shifts satisfying the assumptions of Definitions 2.10 and 2.11.

Now, we give two theorems concerning the composition of two functions. The first theorem is the chain rule on time scales [10, Theorem 1.93].

Theorem 2.12 (chain rule [10]). Assume that πœβˆΆπ•‹β†’β„ is strictly increasing and ξ‚π•‹βˆΆ=𝜐(𝕋) is a time scale. Let ξ‚π‘€βˆΆπ•‹β†’β„. If πœˆΞ”(𝑑) and 𝑀Δ exist for π‘‘βˆˆπ•‹πœ…, then (π‘€βˆ˜π‘£)Ξ”=ξ‚΅π‘€ξ‚Ξ”ξ‚Άπœˆβˆ˜πœˆΞ”.(2.10)

Let 𝕋 be a time scale that is periodic in shifts 𝛿±. If one takes 𝜈(𝑑)=𝛿±(𝑇,𝑑), then one has 𝜈(𝕋)=𝕋 and [𝑓(𝜈(𝑑))]Ξ”=(π‘“Ξ”βˆ˜πœˆ(𝑑))πœˆΞ”(𝑑).

The second theorem is the substitution rule on periodic time scales in shifts 𝛿± which can be found in [12].

Theorem 2.13 (see [12]). Let 𝕋 be a time scale that is periodic in shifts 𝛿± with period π‘ƒβˆˆ[𝑑0,∞)π•‹βˆ— and 𝑓 a Ξ”-periodic function in shifts 𝛿± with the period π‘‡βˆˆ[𝑃,∞)π•‹βˆ—. Suppose that π‘“βˆˆπ’žπ‘Ÿπ‘‘(𝕋), then ξ€œπ‘‘π‘‘0ξ€œπ‘“(𝑠)Δ𝑠=𝛿𝑇±𝛿(𝑑)𝑇±𝑑0𝑓(𝑠)Δ𝑠.(2.11)

This work is mainly based on the following theorem [13].

Theorem 2.14 (Krasnosel'skiΔ­). Let 𝑀 be a closed convex nonempty subset of a Banach space (𝔹,β€–.β€–). Suppose that 𝐴 and 𝐡 map 𝑀 into 𝔹 such that (i)π‘₯,π‘¦βˆˆπ‘€ imply 𝐴π‘₯+π΅π‘¦βˆˆπ‘€,(ii)𝐴 is completely continuous, (iii)𝐡 is a contraction mapping.
Then there exists π‘§βˆˆπ‘€ with 𝑧=𝐴𝑧+𝐡𝑧.

3. Some Lemmas

In this section, we show some interesting properties of the exponential functions 𝑒𝑝(𝑑,𝑑0) and shift operators on time scales.

Lemma 3.1. Let 𝕋 be a time scale that is periodic in shifts 𝛿± with the period 𝑃 and the shift 𝛿𝑇± is Ξ”- differentiable on π‘‘βˆˆπ•‹βˆ— where π‘‡βˆˆ[𝑃,∞)π•‹βˆ—. Then the graininess function πœ‡βˆΆπ•‹β†’[0,∞) satisfies πœ‡ξ€·π›Ώπ‘‡Β±ξ€Έ(𝑑)=𝛿±Δ𝑇(𝑑)πœ‡(𝑑).(3.1)

Proof. Since 𝛿𝑇± is Ξ”-differentiable at 𝑑 we can use Theorem 1.16 (iv) in [10]. Then we have πœ‡(𝑑)𝛿±Δ𝑇(𝑑)=𝛿𝑇±(𝜎(𝑑))βˆ’π›Ώπ‘‡Β±(𝑑).(3.2)
Then by using Corollary 2.9 we have πœ‡(𝑑)𝛿±Δ𝑇𝛿(𝑑)=πœŽπ‘‡Β±ξ€Έ(𝑑)βˆ’π›Ώπ‘‡Β±ξ€·π›Ώ(𝑑)=πœ‡π‘‡Β±ξ€Έ.(𝑑)(3.3)
Thus, the proof is complete.

Lemma 3.2. Let 𝕋 be a time scale, that is, periodic in shifts 𝛿± with the period 𝑃 and the shift 𝛿𝑇± is Ξ”-differentiable on π‘‘βˆˆπ•‹βˆ—, where π‘‡βˆˆ[𝑃,∞)π•‹βˆ—. Suppose that π‘βˆˆβ„› is Ξ”-periodic in shifts 𝛿± with the period π‘‡βˆˆ[𝑃,∞)π•‹βˆ—. Then, 𝑒𝑝𝛿𝑇±(𝑑),𝛿𝑇±𝑑0ξ€Έξ€Έ=𝑒𝑝𝑑,𝑑0ξ€Έπ‘“π‘œπ‘Ÿπ‘‘,𝑑0βˆˆπ•‹βˆ—.(3.4)

Proof. Assume that πœ‡(𝜏)β‰ 0. Set 𝑓(𝜏)=(1/πœ‡(𝜏))Log(1+𝑝(𝜏)πœ‡(𝜏)). Using Lemma 3.1 and Ξ”-periodicity of 𝑝 in shifts 𝛿± we get 𝑓𝛿𝑇±(ξ€Έπ›Ώπœ)±Δ𝑇(π›Ώπœ)=±Δ𝑇(𝜏)πœ‡ξ€·π›Ώπ‘‡Β±ξ€Έ(𝜏)Log𝛿1+𝑝𝑇±(ξ€Έπœ‡ξ€·π›Ώπœ)𝑇±(=π›Ώπœ)±Δ𝑇(𝜏)πœ‡ξ€·π›Ώπ‘‡Β±ξ€Έ(𝜏)Log𝛿1+𝑝𝑇±𝛿(𝜏)±Δ𝑇1π›ΏΒ±Ξ”π‘‡πœ‡ξ€·π›Ώπ‘‡Β±ξ€Έξƒͺ=1(𝜏)πœ‡(𝜏)Log(1+𝑝(𝜏)πœ‡(𝜏))=𝑓(𝜏).(3.5)
Thus, 𝑓 is Ξ”βˆ’periodic in shifts 𝛿± with the period 𝑇. By using Theorem 2.13 we have 𝑒𝑝𝛿𝑇±(𝑑),𝛿𝑇±𝑑0=⎧βŽͺ⎨βŽͺβŽ©ξ‚΅βˆ«ξ€Έξ€Έexp𝛿𝑇±𝛿(𝑑)𝑇±𝑑0ξ€Έ1πœ‡(𝜏)Logξ‚Ά,(1+𝑝(𝜏)πœ‡(𝜏))Ξ”πœforξ‚€βˆ«πœ‡(𝜏)β‰ 0,exp𝛿𝑇±𝛿(𝑑)𝑇±𝑑0,𝑝(𝜏)Ξ”πœfor=⎧βŽͺ⎨βŽͺβŽ©ξ‚΅βˆ«πœ‡(𝜏)=0,exp𝑑𝑑01πœ‡(𝜏)Logξ‚Ά,(1+𝑝(𝜏)πœ‡(𝜏))Ξ”πœforξ‚€βˆ«πœ‡(𝜏)β‰ 0,exp𝑑𝑑0,𝑝(𝜏)Ξ”πœforπœ‡(𝜏)=0,=𝑒𝑝𝑑,𝑑0ξ€Έ.(3.6)
The proof is complete.

Lemma 3.3. Let 𝕋 be a time scale, that is, periodic in shifts 𝛿± with the period 𝑃 and the shift 𝛿𝑇± is Ξ”-differentiable on π‘‘βˆˆπ•‹βˆ— where π‘‡βˆˆ[𝑃,∞)π•‹βˆ—. Suppose that π‘βˆˆβ„› is Ξ”-periodic in shifts 𝛿± with the period π‘‡βˆˆ[𝑃,∞)π•‹βˆ—. Then 𝑒𝑝𝛿𝑇±𝛿(𝑑),πœŽπ‘‡Β±(𝑠)ξ€Έξ€Έ=𝑒𝑝𝑒(𝑑,𝜎(𝑠))=𝑝(𝑑,𝑠)1+πœ‡(𝑑)𝑝(𝑑)for𝑑,π‘ βˆˆπ•‹βˆ—.(3.7)

Proof. From Corollary 2.9, we know 𝜎(𝛿𝑇±(𝑠))=𝛿𝑇±(𝜎(𝑠)). By Lemmas 3.2 and 2.2 we obtain 𝑒𝑝𝛿𝑇±𝛿(𝑑),πœŽπ‘‡Β±(𝑠)ξ€Έξ€Έ=𝑒𝑝𝛿𝑇±(𝑑),𝛿𝑇±(𝜎(𝑠))=𝑒𝑝𝑒(𝑑,𝜎(𝑠))=𝑝(𝑑,𝑠).1+πœ‡(𝑑)𝑝(𝑑)(3.8)
The proof is complete.

4. Main Result

We will state and prove our main result in this section. We define 𝑃𝑇=𝛿π‘₯βˆˆπ’ž(𝕋,ℝ)∢π‘₯𝑇+ξ€Έξ€Ύ,(𝑑)=π‘₯(𝑑)(4.1) where π’ž(𝕋,ℝ) is the space of all real valued continuous functions. Endowed with the norm β€–π‘₯β€–=maxπ‘‘βˆˆ[𝑑0,𝛿𝑇+(𝑑0)]𝕋||||,π‘₯(𝑑)(4.2)𝑃𝑇 is a Banach space.

Lemma 4.1. Let π‘₯βˆˆπ‘ƒπ‘‡. Then β€–π‘₯πœŽβ€– exists and β€–π‘₯πœŽβ€–=β€–π‘₯β€–.

Proof. Since π‘₯βˆˆπ‘ƒπ‘‡, then π‘₯(𝛿𝑇+(𝑑0))=π‘₯(𝑑0), and by Corollary 2.9, we have π‘₯(𝜎(𝛿𝑇+(𝑑0)))=π‘₯(𝜎(𝑑0)). For all π‘‘βˆˆ[𝑑0,𝛿𝑇+(𝑑0)]𝕋,|π‘₯(𝜎(𝑑))|≀‖π‘₯β€–. Hence β€–π‘₯πœŽβ€–β‰€β€–π‘₯β€–. Since π‘₯βˆˆπ’ž(𝕋,ℝ), there exists 𝑑1∈[𝑑0,𝛿𝑇+(𝑑0)] such that β€–π‘₯β€–=|π‘₯(𝑑1)|. If 𝑑1 is left scattered, then 𝜎(𝜌(𝑑1))=𝑑1. And so, β€–π‘₯πœŽβ€–β‰₯|π‘₯𝜎(𝜌(𝑑1))|=π‘₯(𝑑1)=β€–π‘₯β€–. Thus, we have β€–π‘₯πœŽβ€–=β€–π‘₯β€–. If 𝑑1 is dense, 𝜎(𝑑1)=𝑑1 and β€–π‘₯πœŽβ€–=β€–π‘₯β€–.
Assume that 𝑑1 is left dense and right scattered. Note that if 𝑑1=𝑑0 then we work 𝑑1=𝛿𝑇+(𝑑0). Fix πœ–>0 and consider a sequence {π‘Žπ‘›} such that π‘Žπ‘›β†‘π‘‘1. Note that 𝜎(π‘Žπ‘›)≀𝑑1 for all 𝑛. By the continuity of π‘₯, there exists 𝑁 such that for all 𝑛>𝑁, |π‘₯(𝑑1)βˆ’π‘₯𝜎(π‘Žπ‘›)|<πœ–. This implies that β€–π‘₯β€–βˆ’πœ–β‰€β€–π‘₯πœŽβ€–. Since πœ–>0 was arbitrary, then β€–π‘₯β€–=β€–π‘₯πœŽβ€– and the proof is complete.

In this paper we assume that π‘Ž(𝑑)βˆˆβ„›+ is a continuous function with π‘Ž(𝑑)>0 for all π‘‘βˆˆπ•‹ and π‘Žξ€·π›Ώπ‘‡+𝛿(𝑑)+Δ𝑇𝛿(𝑑)=π‘Ž(𝑑),𝑏𝑇+ξ€Έ(𝑑)=𝑏(𝑑),(4.3) where 𝑏Δ(𝑑) is continuous. We further assume that π‘ž(𝑑,π‘₯,𝑦) is continuous and periodic with 𝛿± in 𝑑 and Lipschitz continuous in π‘₯ and 𝑦. That is, π‘žξ€·π›Ώπ‘‡+𝛿(𝑑),π‘₯,𝑦+Δ𝑇(𝑑)=π‘ž(𝑑,π‘₯,𝑦),(4.4) and there are some positive constants 𝐿 and 𝐸 such that ||||π‘ž(𝑑,π‘₯,𝑦)βˆ’π‘ž(𝑑,𝑧,𝑀)≀𝐿‖π‘₯βˆ’π‘§β€–+πΈβ€–π‘¦βˆ’π‘€β€–.(4.5)

Lemma 4.2. Suppose that (4.3)–(4.5) hold. If π‘₯(𝑑)βˆˆπ‘ƒπ‘‡, then π‘₯(𝑑) is a solution of (1.2) if and only if 𝛿π‘₯(𝑑)=𝑏(𝑑)π‘₯βˆ’ξ€Έ+1(π‘˜,𝑑)1βˆ’π‘’βŠ–π‘Ž(𝑑)𝑑,π›Ώπ‘‡βˆ’ξ€ΈΓ—ξ€œ(𝑑)π‘‘π›Ώπ‘‡βˆ’(𝑑)ξ€Ίβˆ’π‘Ÿ(𝑠)π‘₯πœŽξ€·π›Ώβˆ’ξ€Έξ€·ξ€·π›Ώ(π‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»βŠ–π‘Ž(𝑠)(𝑑,𝑠)Δ𝑠,(4.6) where π‘Ÿ(𝑠)=π‘Ž(𝑠)π‘πœŽ(𝑠)+𝑏Δ(𝑠).(4.7)

Proof. Let π‘₯(𝑑)βˆˆπ‘ƒπ‘‡ be a solution of (1.2). We can rewrite (1.2) as π‘₯Ξ”(𝑑)+π‘Ž(𝑑)π‘₯𝜎(𝑑)=𝑏(𝑑)π‘₯Ξ”ξ€·π›Ώβˆ’ξ€Έπ›Ώ(π‘˜,𝑑)Ξ”βˆ’ξ€·ξ€·π›Ώ(π‘˜,𝑑)+π‘žπ‘‘,π‘₯(𝑑),π‘₯βˆ’.(π‘˜,𝑑)ξ€Έξ€Έ(4.8) Multiply both sides of the above equation by π‘’π‘Ž(𝑑)(𝑑,𝑑0) and then integrate from π›Ώπ‘‡βˆ’(𝑑) to 𝑑 to obtain ξ€œπ‘‘π›Ώπ‘‡βˆ’(𝑑)ξ€Ίπ‘₯(𝑠)π‘’π‘Ž(𝑠)𝑠,𝑑0ξ€Έξ€»Ξ”=ξ€œΞ”π‘ π‘‘π›Ώπ‘‡βˆ’(𝑑)𝑏(𝑠)π‘₯Ξ”ξ€·π›Ώβˆ’ξ€Έπ›Ώ(π‘˜,𝑠)Ξ”βˆ’ξ€·ξ€·π›Ώ(π‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»π‘Ž(𝑠)𝑠,𝑑0Δ𝑠.(4.9) We arrive at π‘₯𝑒(𝑑)π‘Ž(𝑑)𝑑,𝑑0ξ€Έβˆ’π‘’π‘Ž(𝑑)ξ€·π›Ώπ‘‡βˆ’(𝑑),𝑑0=ξ€œξ€Έξ€»π‘‘π›Ώπ‘‡βˆ’(𝑑)𝑏(𝑠)π‘₯Ξ”ξ€·π›Ώβˆ’ξ€Έπ›Ώ(π‘˜,𝑠)Ξ”βˆ’ξ€·ξ€·π›Ώ(π‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»π‘Ž(𝑠)𝑠,𝑑0Δ𝑠.(4.10) Dividing both sides of the above equation by π‘’π‘Ž(𝑑)(𝑑,𝑑0) and using π‘₯(𝛿𝑇+(𝑑))=π‘₯(𝑑) and Lemma 2.2, we have π‘₯ξ€·(𝑑)1βˆ’π‘’π‘Ž(𝑑)ξ€·π›Ώπ‘‡βˆ’=ξ€œ(𝑑),π‘‘ξ€Έξ€Έπ‘‘π›Ώπ‘‡βˆ’(𝑑)𝑏(𝑠)π‘₯Ξ”ξ€·π›Ώβˆ’ξ€Έπ›Ώ(π‘˜,𝑠)Ξ”βˆ’ξ€·ξ€·π›Ώ(π‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»π‘Ž(𝑠)(𝑠,𝑑)Δ𝑠.(4.11)
Now, we consider the first term of the integral on the right-hand side of (4.11) ξ€œπ‘‘π›Ώπ‘‡βˆ’(𝑑)𝑏(𝑠)π‘₯Ξ”ξ€·π›Ώβˆ’(ξ€Έπ›Ώπ‘˜,𝑠)Ξ”βˆ’(π‘˜,𝑠)π‘’π‘Ž(𝑠)(𝑠,𝑑)Δ𝑠.(4.12) Using integration by parts from rule [10] we obtain ξ€œπ‘‘π›Ώπ‘‡βˆ’(𝑑)𝑏(𝑠)π‘₯Ξ”ξ€·π›Ώβˆ’(ξ€Έπ›Ώπ‘˜,𝑠)Ξ”βˆ’(π‘˜,𝑠)π‘’π‘Ž(𝑠)(=ξ€œπ‘ ,𝑑)Ξ”π‘ π‘‘π›Ώπ‘‡βˆ’(𝑑)𝑏(𝑠)π‘’π‘Ž(𝑠)𝛿(𝑠,𝑑)π‘₯βˆ’(π‘˜,𝑠)ξ€Έξ€»Ξ”ξ€œΞ”π‘ βˆ’π‘‘π›Ώπ‘‡βˆ’(𝑑)𝑏(𝑠)π‘’π‘Ž(𝑠)ξ€»(𝑠,𝑑)Δ𝑠π‘₯πœŽξ€·π›Ώβˆ’ξ€Έ(π‘˜,𝑠)Δ𝑠(4.13)ξ€œπ‘‘π›Ώπ‘‡βˆ’(𝑑)𝑏(𝑠)π‘₯Ξ”ξ€·π›Ώβˆ’ξ€Έπ›Ώ(π‘˜,𝑠)Ξ”βˆ’(π‘˜,𝑠)π‘’π‘Ž(𝑠)(𝑠,𝑑)Δ𝑠=𝑏(𝑑)π‘’π‘Ž(𝑑)(𝛿𝑑,𝑑)π‘₯βˆ’(ξ€Έξ€·π›Ώπ‘˜,𝑑)βˆ’π‘π‘‡βˆ’(𝑒𝑑)π‘Ž(𝑠)ξ€·π›Ώπ‘‡βˆ’(ξ€Έπ‘₯𝛿𝑑),π‘‘βˆ’ξ€·π‘˜,π›Ώπ‘‡βˆ’(βˆ’ξ€œπ‘‘)ξ€Έξ€Έπ‘‘π›Ώπ‘‡βˆ’(𝑑)ξ€Ίπ‘πœŽ(𝑠)π‘Ž(𝑠)π‘’π‘Ž(𝑠)(𝑠,𝑑)+𝑏Δ(𝑠)π‘’π‘Ž(𝑠)ξ€»π‘₯(𝑠,𝑑)πœŽξ€·π›Ώβˆ’ξ€Έ(π‘˜,𝑠)Δ𝑠.(4.14) Since 𝑏(π›Ώπ‘‡βˆ’(𝑑))=𝑏(𝑑) and π‘₯(π›Ώπ‘‡βˆ’(𝑑))=π‘₯(𝑑), the above equality reduces to ξ€œπ‘‘π›Ώπ‘‡βˆ’(𝑑)𝑏(𝑠)π‘₯Ξ”ξ€·π›Ώβˆ’(ξ€Έπ›Ώπ‘˜,𝑠)Ξ”βˆ’(π‘˜,𝑠)π‘’π‘Ž(𝑠)(𝛿𝑠,𝑑)Δ𝑠=𝑏(𝑑)π‘₯βˆ’(π‘˜,𝑑)ξ€Έξ€·1βˆ’π‘’π‘Ž(𝑠)ξ€·π›Ώπ‘‡βˆ’βˆ’ξ€œ(𝑑),π‘‘ξ€Έξ€Έπ‘‘π›Ώπ‘‡βˆ’(𝑑)ξ€Ίπ‘Ž(𝑠)π‘πœŽ(𝑠)+𝑏Δπ‘₯(𝑠)πœŽξ€·π›Ώβˆ’ξ€Έπ‘’(π‘˜,𝑠)π‘Ž(𝑠)(𝑠,𝑑)Δ𝑠.(4.15)
Substituting (4.15) into (4.11) we get π‘₯𝛿(𝑑)=𝑏(𝑑)π‘₯βˆ’ξ€Έ+1(π‘˜,𝑑)1βˆ’π‘’βŠ–π‘Ž(𝑑)𝑑,π›Ώπ‘‡βˆ’ξ€ΈΓ—ξ€œ(𝑑)π‘‘π›Ώπ‘‡βˆ’(𝑑)ξ€Ίβˆ’π‘Ÿ(𝑠)π‘₯πœŽξ€·π›Ώβˆ’ξ€Έξ€·ξ€·π›Ώ(π‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»βŠ–π‘Ž(𝑠)(𝑑,𝑠)Δ𝑠.(4.16)
Thus the proof is complete.

Define the mapping π»βˆΆπ‘ƒπ‘‡β†’π‘ƒπ‘‡ by 𝛿𝐻π‘₯(𝑑)∢=𝑏(𝑑)π‘₯βˆ’ξ€Έ+1(π‘˜,𝑑)ξ€·1βˆ’π‘’βŠ–π‘Ž(𝑑)𝑑,π›Ώπ‘‡βˆ’Γ—ξ€œ(𝑑)ξ€Έξ€Έπ‘‘π›Ώπ‘‡βˆ’(𝑑)ξ€Ίβˆ’π‘Ÿ(𝑠)π‘₯πœŽξ€·π›Ώβˆ’ξ€Έξ€·ξ€·π›Ώ(π‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»βŠ–π‘Ž(𝑠)(𝑑,𝑠)Δ𝑠.(4.17) To apply Theorem 2.14 we need to construct two mappings: one map is a contraction and the other map is compact and continuous. We express (4.17) as 𝐻π‘₯(𝑑)=𝐡π‘₯(𝑑)+𝐴π‘₯(𝑑),(4.18) where 𝐴, 𝐡 are given 𝛿𝐡π‘₯(𝑑)=𝑏(𝑑)π‘₯βˆ’ξ€Έ,(π‘˜,𝑑)(4.19)1𝐴π‘₯(𝑑)=1βˆ’π‘’βŠ–π‘Ž(𝑑)𝑑,π›Ώπ‘‡βˆ’ξ€ΈΓ—ξ€œ(𝑑)π‘‘π›Ώπ‘‡βˆ’(𝑑)ξ€Ίβˆ’π‘Ÿ(𝑠)π‘₯πœŽξ€·π›Ώβˆ’ξ€Έξ€·ξ€·π›Ώ(π‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»βŠ–π‘Ž(𝑠)(𝑑,𝑠)Δ𝑠,(4.20) and π‘Ÿ(𝑠) is defined in (4.7).

Lemma 4.3. Suppose that (4.3)–(4.5) hold. Then π΄βˆΆπ‘ƒπ‘‡β†’π‘ƒπ‘‡, as defined by (4.20), is compact and continuous.

Proof. We show that π΄βˆΆπ‘ƒπ‘‡β†’π‘ƒπ‘‡. Evaluate (4.20) at 𝛿𝑇+(𝑑), 𝛿𝐴π‘₯𝑇+ξ€Έ=1(𝑑)1βˆ’π‘’βŠ–π‘Ž(𝑑)𝛿𝑇+(𝑑),π›Ώπ‘‡βˆ’ξ€·π›Ώπ‘‡+Γ—ξ€œ(𝑑)𝛿𝑇+𝛿(𝑑)π‘‡βˆ’ξ€·π›Ώπ‘‡+ξ€Έ(𝑑)ξ€Ίβˆ’π‘Ÿ(𝑠)π‘₯πœŽξ€·π›Ώβˆ’ξ€Έξ€·ξ€·π›Ώ(π‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»βŠ–π‘Ž(𝑠)𝛿𝑇+ξ€Έ=1(𝑑),𝑠Δ𝑠1βˆ’π‘’π‘(𝑑)𝛿𝑇+ξ€ΈΓ—ξ€œ(𝑑),𝑑𝛿𝑇+𝛿(𝑑)𝑇+ξ€·π›Ώπ‘‡βˆ’ξ€Έ(𝑑)ξ€Ίβˆ’π‘Ÿ(𝑠)π‘₯πœŽξ€·π›Ώβˆ’ξ€Έξ€·ξ€·π›Ώ(π‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»βŠ–π‘Ž(𝑠)𝛿𝑇+ξ€Έ(𝑑),𝑠Δ𝑠.(4.21) Now, since (4.3) and Corollary 2.9 hold, then we have π‘Ÿξ€·π›Ώπ‘‡+𝛿(𝑠)+𝑇Δ𝛿(𝑠)=π‘Žπ‘‡+𝛿(𝑠)+𝑇Δ(𝑠)π‘πœŽξ€·π›Ώπ‘‡+ξ€Έ(𝑠)+𝑏Δ𝛿𝑇+𝛿(𝑠)+𝑇Δ(𝑠)=π‘Ž(𝑠)π‘πœŽ(𝑠)+𝑏Δ(𝑠)=π‘Ÿ(𝑠).(4.22) That is, π‘Ÿ(𝑠) is Ξ”-periodic in 𝛿± with period 𝑇. Using the periodicity of π‘Ÿ,π‘₯,π‘ž, and Lemma 3.2 we get ξ€Ίξ€·π›Ώβˆ’π‘Ÿπ‘‡+ξ€Έπ‘₯(𝑠)πœŽξ€·π›Ώβˆ’ξ€·π‘˜,𝛿𝑇+𝛿(𝑠)ξ€Έξ€Έ+π‘žπ‘‡+𝛿(𝑠),π‘₯𝑇+𝛿(𝑠),π‘₯βˆ’ξ€·π‘˜,𝛿𝑇+𝛿(𝑠)ξ€Έξ€Έξ€Έξ€»+𝑇Δ(𝑠)Γ—π‘’βŠ–π‘Ž(𝑠)𝛿𝑇+(𝑑),𝛿𝑇+ξ€Έ=ξ€Ί(𝑠)βˆ’π‘Ÿ(𝑠)π‘₯πœŽξ€·π›Ώβˆ’(ξ€Έξ€·ξ€·π›Ώπ‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’(π‘’π‘˜,𝑠)ξ€Έξ€Έξ€»βŠ–π‘Ž(𝑠)(𝑑,𝑠).(4.23) That is, inside the integral of (4.21) is Ξ”-periodic in 𝛿± with period 𝑇. By Theorem 2.13 and Lemma 3.2 we have 𝛿𝐴π‘₯𝑇+ξ€Έ=1(𝑑)1βˆ’π‘’βŠ–π‘Ž(𝑑)𝑑,π›Ώπ‘‡βˆ’ξ€ΈΓ—ξ€œ(𝑑)π‘‘π›Ώπ‘‡βˆ’(𝑑)ξ€Ίβˆ’π‘Ÿ(𝑠)π‘₯πœŽξ€·π›Ώβˆ’ξ€Έξ€·ξ€·π›Ώ(π‘˜,𝑠)+π‘žπ‘ ,π‘₯(𝑠),π‘₯βˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»βŠ–π‘Ž(𝑠)(𝑑,𝑠)Δ𝑠=𝐴π‘₯(𝑑).(4.24)
That is, π΄βˆΆπ‘ƒπ‘‡β†’π‘ƒπ‘‡.
To see that 𝐴 is continuous, we let πœ‘,πœ“βˆˆπ‘ƒπ‘‡ with β€–πœ‘β€–β‰€πΆ and β€–πœ“β€–β‰€πΆ and define πœ‚βˆΆ=maxξ€Ίπ‘‘π‘‘βˆˆ0,𝛿𝑇+𝑑0𝕋|||ξ€·1βˆ’π‘’βŠ–π‘Ž(𝑑)𝑑,π›Ώπ‘‡βˆ’(𝑑)ξ€Έξ€Έβˆ’1|||,π›ΎβˆΆ=maxξ€Ίπ›Ώπ‘’βˆˆπ‘‡βˆ’ξ€»(𝑑),π‘‘π•‹π‘’βŠ–π‘Ž(𝑑)(𝑑,𝑒),π›½βˆΆ=maxξ€Ίπ‘‘π‘‘βˆˆ0,𝛿𝑇+𝑑0𝕋||||.π‘Ÿ(𝑑)(4.25)
Given that πœ–>0, take 𝛿=πœ–/𝑀 such that β€–πœ‘βˆ’πœ“β€–<𝛿. By making use of the Lipschitz inequality (4.5) in (4.20), we get ξ€œβ€–π΄πœ‘βˆ’π΄πœ“β€–β‰€π›Ύπœ‚π‘‘π›Ώπ‘‡βˆ’(𝑑)[]ξ€·π‘‘π›½β€–πœ‘βˆ’πœ“β€–+πΏβ€–πœ‘βˆ’πœ“β€–+πΈβ€–πœ‘βˆ’πœ“β€–Ξ”π‘ =πœ‚π›Ύπ›½+𝐿+𝐸0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0ξ€Έξ€Έβ€–πœ‘βˆ’πœ“β€–β‰€π‘€β€–πœ‘βˆ’πœ“β€–<πœ–,(4.26) where 𝐿,𝐸 are given by (4.5) and 𝑀=πœ‚π›Ύ[𝛽+𝐿+𝐸](𝑑0βˆ’π›Ώπ‘‡βˆ’(𝑑0)). This proves that 𝐴 is continuous.
We need to show that 𝐴 is compact. Consider the sequence of periodic functions in 𝛿±{πœ‘π‘›}βŠ‚π‘ƒπ‘‡ and assume that the sequence is uniformly bounded. Let 𝑅>0 be such that β€–πœ‘π‘›β€–β‰€π‘…, for all π‘›βˆˆβ„•. In view of (4.5) we arrive at ||||=||||≀||||+||||π‘ž(𝑑,π‘₯,𝑦)π‘ž(𝑑,π‘₯,𝑦)βˆ’π‘ž(𝑑,0,0)+π‘ž(𝑑,0,0)π‘ž(𝑑,π‘₯,𝑦)βˆ’π‘ž(𝑑,0,0)π‘ž(𝑑,0,0)≀𝐿‖π‘₯β€–+𝐸‖𝑦‖+𝛼,(4.27) where π›ΌβˆΆ=maxπ‘‘βˆˆ[𝑑0,𝛿𝑇+(𝑑0)]𝕋|π‘ž(𝑑,0,0)|. Hence, ||π΄πœ‘π‘›||=||||11βˆ’π‘’βŠ–π‘Ž(𝑑)𝑑,π›Ώπ‘‡βˆ’ξ€Έξ€œ(𝑑)π‘‘π›Ώπ‘‡βˆ’(𝑑)ξ€Ίβˆ’π‘Ÿ(𝑠)πœ‘πœŽπ‘›ξ€·π›Ώβˆ’ξ€Έξ€·(π‘˜,𝑠)+π‘žπ‘ ,πœ‘π‘›(𝑠),πœ‘π‘›ξ€·π›Ώβˆ’π‘’(π‘˜,𝑠)ξ€Έξ€Έξ€»βŠ–π‘Ž(𝑠)||||ξ€Ίβ€–β€–πœ‘(𝑑,𝑠)Ξ”π‘ β‰€πœ‚π›Ύ(𝛽+𝐿+𝐸)𝑛‖‖𝑑+𝛼0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0[]ξ€·π‘‘ξ€Έξ€Έβ‰€πœ‚π›Ύ(𝛽+𝐿+𝐸)𝑅+𝛼0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0ξ€Έξ€ΈβˆΆ=𝐷.(4.28) Thus, the sequence {π΄πœ‘π‘›} is uniformly bounded. If we find the derivative of π΄πœ‘π‘›, we have ξ€·π΄πœ‘π‘›ξ€ΈΞ”(𝑑)=π‘Ž(𝑑)π΄πœ‘π‘›+(𝑑)βˆ’π‘Ž(𝑑)+βŠ–π‘Ž(𝑑)1βˆ’π‘’βŠ–π‘Ž(𝑑)ξ€·πœŽ(𝑑),π›Ώπ‘‡βˆ’ξ€ΈΓ—ξ€œ(𝜎(𝑑))π‘‘π›Ώπ‘‡βˆ’(𝑑)ξ€Ίβˆ’π‘Ÿ(𝑠)πœ‘πœŽπ‘›ξ€·π›Ώβˆ’(ξ€Έξ€·π‘˜,𝑠)+π‘žπ‘ ,πœ‘π‘›(𝑠),πœ‘π‘›ξ€·π›Ώβˆ’(π‘’π‘˜,𝑠)ξ€Έξ€Έξ€»βŠ–π‘Ž(𝑠)(+1𝑑,𝑠)Δ𝑠1+πœ‡(𝑑)π‘Ž(𝑑)βˆ’π‘Ÿ(𝑑)πœ‘πœŽξ€·π›Ώβˆ’ξ€Έξ€·(π‘˜,𝑠)+π‘žπ‘ ,πœ‘π‘›(𝑠),πœ‘π‘›ξ€·π›Ώβˆ’.(π‘˜,𝑠)ξ€Έξ€Έξ€»(4.29)
Consequently, |||ξ€·π΄πœ‘π‘›ξ€ΈΞ”|||[]𝑑(𝑑)β‰€π·β€–π‘Žβ€–+(𝛽+𝐸+𝐿)𝑅+𝛼2β€–π‘Žβ€–π›Ύπœ‚0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0ξ€Έξ€Έξ€»βˆΆ=𝐹.(4.30) for all 𝑛. That is, |(π΄πœ‘π‘›)Ξ”(𝑑)|≀𝐹, for some positive constant 𝐹. Thus the sequence {π΄πœ‘π‘›} is uniformly bounded and equicontinuous. The Arzela-Ascoli theorem implies that {π΄πœ‘π‘›π‘˜} uniformly converges to a continuous 𝑇-periodic function πœ‘βˆ— in 𝛿±. Thus 𝐴 is compact.

Lemma 4.4. Let 𝐡 be defined by (4.19) and ‖𝑏(𝑑)β€–β‰€πœ‰<1.(4.31)
Then π΅βˆΆπ‘ƒπ‘‡β†’π‘ƒπ‘‡ is a contraction.

Proof. Trivially, π΅βˆΆπ‘ƒπ‘‡β†’π‘ƒπ‘‡. For πœ‘,πœ“βˆˆπ‘ƒπ‘‡, we have β€–π΅πœ‘βˆ’π΅πœ“β€–=maxξ€Ίπ‘‘π‘‘βˆˆ0,𝛿𝑇+𝑑0𝕋||||π΅πœ‘(𝑑)βˆ’π΅πœ“(𝑑)=maxξ€Ίπ‘‘π‘‘βˆˆ0,𝛿𝑇+𝑑0𝕋||||||πœ‘ξ€·π›Ώπ‘(𝑑)βˆ’ξ€Έξ€·π›Ώ(π‘˜,𝑠)βˆ’πœ“βˆ’ξ€Έ||ξ€Ύ(π‘˜,𝑠)β‰€πœ‰β€–πœ‘βˆ’πœ“β€–.(4.32)
Hence 𝐡 defines a contraction mapping with contraction constant πœ‰.

Theorem 4.5. Let π›ΌβˆΆ=π‘šπ‘Žπ‘₯π‘‘βˆˆ[𝑑0,𝛿𝑇+(𝑑0)]𝕋|π‘ž(𝑑,0,0)|. Let 𝛽,πœ‚, and 𝛾 be given by (4.39). Suppose that (4.3)–(4.5) and (4.31) hold and that there is a positive constant 𝐺 such that all solutions π‘₯(𝑑) of (1.2), π‘₯βˆˆπ‘ƒπ‘‡, satisfy |π‘₯(𝑑)|≀𝐺, the inequality ξ€½ξ€·π‘‘πœ‰+π›Ύπœ‚(𝛽+𝐿+𝐸)0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0𝑑𝐺+π›Ύπœ‚π›Ό0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0≀𝐺(4.33) holds. Then (1.2) has a 𝑇-periodic solution in 𝛿±.

Proof. Define π‘€βˆΆ={π‘₯βˆˆπ‘ƒπ‘‡βˆΆβ€–π‘₯‖≀𝐺}. Then Lemma 4.3 implies that π΄βˆΆπ‘ƒπ‘‡β†’π‘ƒπ‘‡ is compact and continuous. Also, from Lemma 4.4, the mapping π΅βˆΆπ‘ƒπ‘‡β†’π‘ƒπ‘‡ is contraction.
We need to show that if πœ‘,πœ“βˆˆπ‘€, we have β€–π΄πœ‘βˆ’π΅πœ“β€–β‰€πΊ. Let πœ‘,πœ“βˆˆπ‘€ with β€–πœ‘β€–,β€–πœ“β€–β‰€πΊ. From (4.19) and (4.20) and the fact that |π‘ž(𝑑,π‘₯,𝑦)|≀𝐿‖π‘₯β€–+𝐸‖𝑦‖+𝛼, we have ξ€œβ€–π΄πœ‘+π΅πœ“β€–β‰€π›Ύπœ‚π‘‘π›Ώπ‘‡βˆ’(𝑑)[]β‰€ξ€½ξ€·π‘‘πΏβ€–πœ‘β€–+πΈβ€–πœ‘β€–+π›½β€–πœ‘β€–+𝛼Δ𝑠+πœ‰β€–πœ“β€–πœ‰+π›Ύπœ‚(𝛽+𝐿+𝐸)0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0𝑑𝐺+π›Ύπœ‚π›Ό0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0≀𝐺.(4.34) We see that all the conditions of Krasnosel'skiΔ­ theorem are satisfied on the set 𝑀. Thus there exists a fixed point 𝑧 in 𝑀 such that 𝑧=𝐡𝑧+𝐴𝑧. By Lemma 4.2, this fixed point is a solution of (2) has a 𝑇-periodic solution in 𝛿±.

Theorem 4.6. Suppose that (4.3)–(4.5) and (4.31) hold. Let 𝛽,πœ‚, and 𝛾 be given by (4.39). If ξ€·π‘‘πœ‰+π›Ύπœ‚(𝛽+𝐿+𝐸)0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0≀1,(4.35) then (1.2) has a unique 𝑇-periodic solution in 𝛿±.

Proof. Let the mapping 𝐻 be given by (4.17). For πœ‘,πœ“βˆˆπ‘ƒπ‘‡ we have ξ€œβ€–π»πœ‘βˆ’π»πœ“β€–β‰€πœ‰β€–πœ‘βˆ’πœ“β€–+π›Ύπœ‚π‘‘π›Ώπ‘‡βˆ’(𝑑)[]β‰€ξ€Ίξ€·π‘‘πΏβ€–πœ‘βˆ’πœ“β€–+πΈβ€–πœ‘βˆ’πœ“β€–+π›½β€–πœ‘βˆ’πœ“β€–Ξ”π‘ πœ‰+π›Ύπœ‚(𝛽+𝐿+𝐸)0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0ξ€Έξ€Έξ€»β€–πœ‘βˆ’πœ“β€–.(4.36) This completes the proof.

Example 4.7. Let 𝕋={2𝑛}π‘›βˆˆπ‘0βˆͺ{1/4,1/2} be a periodic time scale in shift 𝛿±(𝑃,𝑑)=𝑃±1𝑑 with period 𝑃=2. We consider the dynamic equation (1.2) with π‘Ž(𝑑)=1/5𝑑, 𝑏(𝑑)=(1/500)(βˆ’1)ln𝑑/lnπ‘ž and π‘ž(𝑑,π‘₯,𝑦)=(sinπ‘₯+arctanπ‘₯+1)/1000𝑑.
The operators π›Ώβˆ’(𝑠,𝑑)=𝑑/𝑠 and 𝛿+(𝑠,𝑑)=𝑠𝑑 are backward and forward shift operators for (𝑠,𝑑)∈𝐷±. Here π•‹βˆ—=𝕋, the initial point 𝑑0=1 and π›Ώβˆ’(π‘˜,𝑑)=𝑑/π‘˜ for π‘˜βˆˆ[2,∞)𝕋. If we consider conditions (4.3)-(4.4) we find 𝑇=4. Then π‘Ž(𝑑), 𝑏(𝑑) satisfy condition (4.3), π‘Ž(𝑑)βˆˆβ„›+ and π‘ž(𝑑,π‘₯,𝑦) satisfies the condition (4.4) for all π‘‘βˆˆπ•‹. Also, π‘ž(𝑑,π‘₯,𝑦) is Lipschitz continuous in π‘₯ and 𝑦 for 𝐿=𝐸=1/250. Since ‖𝑏(𝑑)β€–=β€–(1/500)(βˆ’1)ln𝑑/lnπ‘žβ€–=(1/500)=πœ‰<1, then the condition (4.31) holds.
If we compute πœ‚,𝛾, and 𝛽, we have πœ‚=maxξ€Ίπ‘‘π‘‘βˆˆ0,𝛿𝑇+𝑑0𝕋|||ξ€·1βˆ’π‘’βŠ–π‘Ž(𝑑)𝑑,π›Ώπ‘‡βˆ’(𝑑)ξ€Έξ€Έβˆ’1|||β‰…3,45,𝛼=maxξ€Ίπ‘‘π‘‘βˆˆ0,𝛿𝑇+𝑑0𝕋||||=1π‘ž(𝑑,0,0)250(4.37)𝛾=maxξ€Ίπ›Ώπ‘’βˆˆπ‘‡βˆ’ξ€»(𝑑),π‘‘π•‹π‘’βŠ–π‘Ž(𝑑)(𝑑,𝑒)β‰…1,5,𝛽=maxξ€Ίπ‘‘π‘‘βˆˆ0,𝛿𝑇+𝑑0𝕋||||=π‘Ÿ(𝑑)11.2500(4.38) If we take 𝐺=1, then inequality (4.33) satisfies.
Let π‘₯(𝑑)βˆˆπ‘ƒπ‘‡. We show that β€–π‘₯(𝑑)‖≀𝐺=1. Integrate (1.2) from 1 to 4, we get ξ€œπ‘₯(4)βˆ’π‘₯(1)=41ξ‚ƒβˆ’π‘Ž(𝑑)π‘₯𝜎(𝑑)+𝑏(𝑑)π‘₯Ξ”ξ‚€π‘‘π‘˜ξ‚1π‘˜ξ‚€ξ‚€π‘‘+π‘žπ‘‘,π‘₯(𝑑),π‘₯π‘˜ξ‚ξ‚ξ‚„Ξ”π‘‘.(4.39)
Since π‘₯(𝑑)βˆˆπ‘ƒπ‘‡, then π‘₯(4)=π‘₯(1) and so after integration by parts (23) becomes ξ€œ41π‘Ž(𝑑)π‘₯πœŽξ€œ(𝑑)Δ𝑑=41π‘žξ‚€ξ‚€π‘‘π‘‘,π‘₯(𝑑),π‘₯π‘˜ξ‚ξ‚βˆ’π‘Ξ”ξ‚€π‘‘(𝑑)π‘₯π‘˜ξ‚Ξ”π‘‘.(4.40)

Claim 1. There exist π‘‘βˆ—βˆˆ[1,4]𝕋 such that 3π‘Ž(π‘‘βˆ—)π‘₯𝜎(π‘‘βˆ—βˆ«)≀41π‘Ž(𝑑)π‘₯𝜎(𝑑)Δ𝑑.
Suppose that the claim is false. Define βˆ«π‘†βˆΆ=41π‘Ž(𝑑)π‘₯𝜎(𝑑)Δ𝑑. Then there exists πœ–>0 such that 3π‘Ž(𝑑)π‘₯𝜎(𝑑)>𝑆+πœ–,(4.41) for all π‘‘βˆˆ[1,4]𝕋. So, ξ€œπ‘†=41π‘Ž(𝑑)π‘₯𝜎1(𝑑)Δ𝑑>3ξ€œ41(𝑆+πœ–)Δ𝑇=𝑆+πœ–.(4.42) That is, 𝑆>𝑆+πœ–, a contradiction.
As a consequence of the claim, we have 3||π‘Žξ€·π‘‘βˆ—ξ€Έ||||π‘₯πœŽξ€·π‘‘βˆ—ξ€Έ||β‰€ξ€œ41|||π‘žξ‚€ξ‚€π‘‘π‘‘,π‘₯(𝑑),π‘₯π‘˜|||+|||𝑏Δ𝑑(𝑑)π‘₯π‘˜ξ‚|||β‰€ξ€œΞ”π‘‘41[]2(𝐿+𝐸)β€–π‘₯β€–+𝛼+𝛿‖π‘₯‖Δ𝑑=3+1250‖1250π‘₯β€–+3250=3β€–1250π‘₯β€–+ξ‚„,250(4.43) where 𝛿=max[1,4]𝕋|𝑏Δ(𝑑)|=1/250.
So, |π‘Ž(π‘‘βˆ—)||π‘₯𝜎(π‘‘βˆ—)|≀(3/250)β€–π‘₯β€–+(1/250), which implies |π‘₯𝜎(π‘‘βˆ—)|≀20[(3/250)β€–π‘₯β€–+(1/250)]. Since for all π‘‘βˆˆ[1,4]𝕋, π‘₯𝜎(𝑑)=π‘₯πœŽξ€·π‘‘βˆ—ξ€Έ+ξ€œπ‘‘π‘‘βˆ—π‘₯Ξ”(𝜎(𝑠))Δ𝑠,(4.44) we have ||π‘₯𝜎(||≀||π‘₯𝑑)πœŽξ€·π‘‘βˆ—ξ€Έ||+ξ€œπ‘‘1||π‘₯Ξ”(||3𝜎(𝑠))Δ𝑠≀201250β€–π‘₯β€–+ξ‚„β€–β€–π‘₯250+3Ξ”β€–β€–.(4.45)
This implies that 2β€–π‘₯‖≀+191500β€–β€–π‘₯19Ξ”β€–β€–.(4.46)
Taking the norm in (1.2) yields β€–β€–π‘₯Δ‖‖≀()β€–π‘Žβ€–+𝐿+𝐸‖π‘₯β€–+𝛼=(1βˆ’β€–π‘β€–1/5+2/250)β€–π‘₯β€–+1/250=1βˆ’1/500104β€–π‘₯β€–+2.250.499(4.47) Substitution of (4.47) into (4.46) yields that for all π‘₯(𝑑)βˆˆπ‘ƒπ‘‡,β€–π‘₯(𝑑)‖≀𝐺=1. Then by Theorem 4.5, (1.2) has a 4-periodic solution in shifts 𝛿±.
In this example, if we take π‘ž(𝑑,π‘₯,𝑦)=(sinπ‘₯+arctanπ‘₯)/1000𝑑, we have ξ€·π‘‘πœ‰+π›Ύπœ‚(𝛽+𝐿+𝐸)0βˆ’π›Ώπ‘‡βˆ’ξ€·π‘‘0=1ξ€Έξ€Έ+500962552.106<1.(4.48)
So, all the conditions of Theorem 4.6 are satisfied. Therefore, (1.2) has a unique 4-periodic solution in shifts 𝛿±.

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