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Abstract and Applied Analysis
VolumeΒ 2010Β (2010), Article IDΒ 234015, 26 pages
http://dx.doi.org/10.1155/2010/234015
Research Article

Boundary Value Problems for Systems of Second-Order Dynamic Equations on Time Scales with Ξ”-CarathΓ©odory Functions

1DΓ©partement de MathΓ©matiques et de Statistique, UniversitΓ© de MontrΓ©al, CP 6128, Succursale Centre-Ville, MontrΓ©al, QC, H3C 3J7, Canada
2Département de Mathématiques, Collège Édouard-Montpetit, 945 Chemin de Chambly, Longueuil, QC, J4H 3M6, Canada

Received 31 August 2010; Accepted 30 November 2010

Academic Editor: J.Β Mawhin

Copyright Β© 2010 M. Frigon and H. Gilbert. 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

We establish the existence of solutions to systems of second-order dynamic equations on time scales with the right member 𝑓, a Ξ”-CarathΓ©odory function. First, we consider the case where the nonlinearity 𝑓 does not depend on the Ξ”-derivative, π‘₯Ξ”(𝑑). We obtain existence results for Strum-Liouville and for periodic boundary conditions. Finally, we consider more general systems in which the nonlinearity 𝑓 depends on the Ξ”-derivative and satisfies a linear growth condition with respect to π‘₯Ξ”(𝑑). Our existence results rely on notions of solution-tube that are introduced in this paper.

1. Introduction

In this paper, we establish existence results for the following systems of second-order dynamic equations on time scales:π‘₯ΔΔ(𝑑)=𝑓(𝑑,π‘₯(𝜎(𝑑))),Ξ”-a.e.π‘‘βˆˆπ•‹πœ…20,π‘₯∈(BC),(1.1)π‘₯ΔΔ(𝑑)=𝑓𝑑,π‘₯(𝜎(𝑑)),π‘₯Ξ”ξ€Έ(𝑑),Ξ”-a.e.π‘‘βˆˆπ•‹πœ…20,π‘Ž0π‘₯(π‘Ž)βˆ’π‘₯Ξ”(π‘Ž)=π‘₯0,π‘Ž1π‘₯(𝑏)+𝛾1π‘₯Ξ”(𝜌(𝑏))=π‘₯1.(1.2) Here, 𝕋 is a compact time scale where π‘Ž=min𝕋, 𝑏=max𝕋, and π•‹πœ…20 is defined in (2.4). The map π‘“βˆΆπ•‹πœ…0×ℝ𝑛→ℝ𝑛 is Ξ”-CarathΓ©odory (see Definition 2.9), and (BC) denotes one of the following boundary conditions:π‘₯π‘₯(π‘Ž)=π‘₯(𝑏),Ξ”(π‘Ž)=π‘₯Ξ”(𝜌(𝑏)),(1.3)π‘Ž0π‘₯(π‘Ž)βˆ’π›Ύ0π‘₯Ξ”(π‘Ž)=π‘₯0,π‘Ž1π‘₯(𝑏)+𝛾1π‘₯Ξ”(𝜌(𝑏))=π‘₯1,(1.4) where π‘Ž0,π‘Ž1,𝛾0,𝛾1β‰₯0, max{π‘Ž0,𝛾0}>0, and max{π‘Ž1,𝛾1}>0.

Problem (1.1) was mainly treated in the case where it has only one equation (𝑛=1) and 𝑓 is continuous. In particular, the existence of a solution of (1.1) was established by AkΔ±n [1] for the Dirichlet boundary condition and by StehlΓ­k [2] for the periodic boundary condition. Equation (1.1) with nonlinear boundary conditions was studied by Peterson et al. [3]. In all those results, the method of lower and upper solutions was used. See also [4, 5] and the references therein for other results on the problem (1.1) when 𝑛=1.

Very few existence results were obtained for the system (1.1) when 𝑛>1. Recently, Henderson et al. [6] and Amster et al. [7] established the existence of solutions of (1.1) with Sturm-Liouville and nonlinear boundary conditions, respectively, assuming that 𝑓 is a continuous function satisfying the following condition:βˆƒπ‘…>0suchthat⟨π‘₯,𝑓(𝑑,π‘₯)⟩>0ifβ€–π‘₯β€–=𝑅.(1.5)

The fact that the right member in the system (1.2) depends also on the Ξ”-derivative, π‘₯Ξ”, increases considerably the difficulty of this problem. So, it is not surprising that there are almost no results for this problem in the literature. Atici et al. [8] studied this problem in the particular case, where there is only one equation (𝑛=1) and 𝑓 is positive, continuous and satisfies a monotonicity condition. Assuming a growth condition of Wintner type and using the method of lower and upper solutions, they obtained the existence of a solution.

The system (1.2) with the Dirichlet boundary condition was studied by Henderson and Tisdell [9] in the general case where 𝑛>1. They considered 𝑓 a continuous function and 𝕋 a regular time scale (i.e., 𝜌(𝑑)<𝑑<𝜎(𝑑) or 𝕋=[π‘Ž,𝑏]). They established the existence of a solution of (1.2) under the following assumptions: (A1) there exists 𝑅>π‘šπ‘Žπ‘₯{β€–π‘₯0β€–,β€–π‘₯1β€–} such that 2⟨π‘₯,𝑓(𝑑,π‘₯,𝑦)⟩+‖𝑦‖2>0 if β€–π‘₯β€–=𝑅, βˆ’2⟨π‘₯,π‘¦βŸ©β‰€πœ‡(𝑑)‖𝑦‖2,(A2) there exist 𝑐,𝑑β‰₯0 such that 𝑑(𝜌(𝑏)βˆ’π‘Ž)<1 and ‖𝑓(𝑑,π‘₯,𝑦)‖≀𝑐+𝑑‖𝑦‖ if β€–π‘₯‖≀𝑅.

In the third section of this paper, we establish an existence theorem for the system (1.1). To this aim, we introduce a notion of solution-tube of (1.1) which generalizes to systems the notions of lower and upper solutions introduced in [1, 2]. This notion generalizes also condition (1.5) used by Henderson et al. [6] and Amster et al. [7]. Our notion of solution-tube is in the spirit of the notion of solution-tube for systems of second-order differential equations introduced in [10]. Our notion is new even in the case of systems of second-order difference equations.

In the last section of this paper, we study the system (1.2). Again, we introduce a notion of solution-tube of (1.2) which generalizes the notion of lower and upper solutions used by Atici et al. [8]. This notion generalizes also condition (1.5) and the notion of solution-tube of systems of second-order differential equations introduced in [10]. In addition, we assume that 𝑓 satisfies a linear growth condition. It is worthwhile to mention that the time scale 𝕋 does not need to be regular, and we do not require the restriction 𝑑(𝜌(𝑏)βˆ’π‘Ž)<1 as in assumption (A2) used in [9].

Moreover, we point out that the right members of our systems are not necessarily continuous. Indeed, we assume that the weaker condition: 𝑓 is a Ξ”-CarathΓ©odory function. This condition is interesting in the case where the points of 𝕋 are not all right scattered. We obtain the existence of solutions to (1.1) and to (1.2) in the Sobolev space π‘ŠΞ”2,1(𝕋,ℝ𝑛). To our knowledge, it is the first paper applying the theory of Sobolev spaces with topological methods to obtain solutions to (1.1) and (1.2). Solutions of second-order Hamiltonian systems on time scales were obtained in a Sobolev space via variational methods in [11]. Finally, let us mention that our results are new also in the continuous case and for systems of second-order difference equations.

2. Preliminaries and Notations

For sake of completeness, we recall some notations, definitions, and results concerning functions defined on time scales. The interested reader may consult [12, 13] and the references therein to find the proofs and to get a complete introduction to this subject.

Let 𝕋 be a compact time scale with π‘Ž=min𝕋<𝑏=max𝕋. The forward jump operator πœŽβˆΆπ•‹β†’π•‹ (resp., the backward jump operator πœŒβˆΆπ•‹β†’π•‹) is defined by ξƒ―πœŽ(𝑑)=inf{π‘ βˆˆπ•‹βˆΆπ‘ >𝑑}if𝑏𝑑<𝑏,if𝑑=𝑏,resp.ξƒ―,𝜌(𝑑)=sup{π‘ βˆˆπ•‹βˆΆπ‘ <𝑑}ifπ‘Žπ‘‘>π‘Ž,ifξƒͺ.𝑑=π‘Ž(2.1) We say that 𝑑<𝑏 is right scattered (resp., 𝑑>π‘Ž is left scattered) if 𝜎(𝑑)>𝑑 (resp., 𝜌(𝑑)<𝑑) otherwise, we say that 𝑑 is right dense (resp., left dense). The set of right-scattered points of 𝕋 is at most countable, see [14]. We denote it by π‘…π•‹ξ€½π‘‘βˆΆ={π‘‘βˆˆπ•‹βˆΆπ‘‘<𝜎(𝑑)}=𝑖,βˆΆπ‘–βˆˆπΌ(2.2) for some πΌβŠ‚β„•. The graininess function πœ‡βˆΆπ•‹β†’[0,∞) is defined by πœ‡(𝑑)=𝜎(𝑑)βˆ’π‘‘. We denote π•‹πœ…]=𝕋⧡(𝜌(𝑏),𝑏,𝕋0=𝕋⧡{𝑏}.(2.3) So, π•‹πœ…=𝕋 if 𝑏 is left dense, otherwise π•‹πœ…=𝕋0. Since π•‹πœ… is also a time scale, we denoteπ•‹πœ…2=(π•‹πœ…)πœ…,π•‹πœ…20=π•‹πœ…2⧡{𝑏}.(2.4)

In 1990, Hilger [15] introduced the concept of dynamic equations on time scales. This concept provides a unified approach to continuous and discrete calculus with the introduction of the notion of delta-derivative π‘₯Ξ”(𝑑). This notion coincides with π‘₯β€²(𝑑) (resp., Ξ”π‘₯(𝑑)) in the case where the time scale 𝕋 is an interval (resp., the discrete set {0,1,…,𝑁}).

Definition 2.1. A map π‘“βˆΆπ•‹β†’β„π‘› is Ξ”-differentiable at π‘‘βˆˆπ•‹πœ… if there exists 𝑓Δ(𝑑)βˆˆβ„π‘› (called the Ξ”-derivative of 𝑓 at 𝑑) such that for all πœ€>0, there exists a neighborhood π‘ˆ of 𝑑 such that ‖‖𝑓(𝜎(𝑑))βˆ’π‘“(𝑠)βˆ’π‘“Ξ”ξ€Έβ€–β€–||||(𝑑)(𝜎(𝑑)βˆ’π‘ )β‰€πœ€πœŽ(𝑑)βˆ’π‘ βˆ€π‘ βˆˆπ‘ˆ.(2.5) We say that 𝑓 is Ξ”-differentiable if 𝑓Δ(𝑑) exists for every π‘‘βˆˆπ•‹πœ….
If 𝑓 is Ξ”-differentiable and if 𝑓Δ is Ξ”-differentiable at π‘‘βˆˆπ•‹πœ…2, we call 𝑓ΔΔ(𝑑)=(𝑓Δ)Ξ”(𝑑) the second Ξ”-derivative of 𝑓 at 𝑑.

Proposition 2.2. Let π‘“βˆΆπ•‹β†’β„π‘› and π‘‘βˆˆπ•‹πœ…. (i)If 𝑓 is Ξ”-differentiable at 𝑑, then 𝑓 is continuous at 𝑑.(ii)If 𝑓 is continuous at π‘‘βˆˆπ‘…π•‹, then 𝑓Δ(𝑑)=𝑓(𝜎(𝑑))βˆ’π‘“(𝑑).πœ‡(𝑑)(2.6)(iii)The map 𝑓 is Ξ”-differentiable at π‘‘βˆˆπ•‹πœ…β§΅π‘…π•‹ if and only if 𝑓Δ(𝑑)=lim𝑠→𝑑𝑓(𝑑)βˆ’π‘“(𝑠).π‘‘βˆ’π‘ (2.7)

Proposition 2.3. If π‘“βˆΆπ•‹β†’β„π‘› and π‘”βˆΆπ•‹β†’β„π‘š are Ξ”-differentiable at π‘‘βˆˆπ•‹πœ…, then (i)if 𝑛=π‘š, (𝛼𝑓+𝑔)Ξ”(𝑑)=𝛼𝑓Δ(𝑑)+𝑔Δ(𝑑) for every π›Όβˆˆβ„,(ii)if π‘š=1, (𝑓𝑔)Ξ”(𝑑)=𝑔(𝑑)𝑓Δ(𝑑)+𝑓(𝜎(𝑑))𝑔Δ(𝑑)=𝑓(𝑑)𝑔Δ(𝑑)+𝑔(𝜎(𝑑))𝑓Δ(𝑑),(iii)if π‘š=1 and 𝑔(𝑑)𝑔(𝜎(𝑑))β‰ 0, then 𝑓𝑔Δ(𝑑)=𝑔(𝑑)𝑓Δ(𝑑)βˆ’π‘“(𝑑)𝑔Δ(𝑑),𝑔(𝑑)𝑔(𝜎(𝑑))(2.8)(iv)if π‘ŠβŠ‚β„π‘› is open and β„ŽβˆΆπ‘Šβ†’β„ is differentiable at 𝑓(𝑑)βˆˆπ‘Š and π‘‘βˆ‰π‘…π•‹, then (β„Žβˆ˜π‘“)Ξ”(𝑑)=βŸ¨β„Žβ€²(𝑓(𝑑)),𝑓Δ(𝑑)⟩.

We denote 𝐢(𝕋,ℝ𝑛) the space of continuous maps on 𝕋, and we denote 𝐢1(𝕋,ℝ𝑛) the space of continuous maps on 𝕋 with continuous Ξ”-derivative on π•‹πœ…. With the norm β€–π‘₯β€–0=max{β€–π‘₯(𝑑)β€–βˆΆπ‘‘βˆˆπ•‹} (resp., β€–π‘₯β€–1=max{β€–π‘₯β€–0,max{β€–π‘₯Ξ”(𝑑)β€–βˆΆπ‘‘βˆˆπ•‹πœ…}}), 𝐢(𝕋,ℝ𝑛) (resp., 𝐢1(𝕋,ℝ𝑛)) is a Banach space.

We study the second Ξ”-derivative of the norm of a map.

Lemma 2.4. Let π‘₯βˆΆπ•‹β†’β„π‘› be Ξ”-differentiable. (1)On {π‘‘βˆˆπ•‹πœ…2βˆΆβ€–π‘₯(𝜎(𝑑))β€–>0 and π‘₯ΔΔ(𝑑) exists}, β€–π‘₯(𝑑)‖ΔΔβ‰₯⟨π‘₯(𝜎(𝑑)),π‘₯ΔΔ(𝑑)⟩.β€–π‘₯(𝜎(𝑑))β€–(2.9)(2) On {π‘‘βˆˆπ•‹πœ…2β§΅π‘…π•‹βˆΆβ€–π‘₯(𝜎(𝑑))β€–>0 and π‘₯ΔΔ(𝑑) exists}, β€–π‘₯(𝑑)‖ΔΔ=⟨π‘₯(𝑑),π‘₯ΔΔ(𝑑)⟩+β€–π‘₯Ξ”(𝑑)β€–2βˆ’β€–π‘₯(𝑑)β€–βŸ¨π‘₯(𝑑),π‘₯Ξ”(𝑑)⟩2β€–π‘₯(𝑑)β€–3.(2.10)

Proof. Denote 𝐴={π‘‘βˆˆπ•‹πœ…2βˆΆβ€–π‘₯(𝜎(𝑑))β€–>0 and π‘₯ΔΔ(𝑑) exists}. By Proposition 2.3, on the set 𝐴⧡𝑅𝕋, we have β€–π‘₯(𝑑)β€–Ξ”=⟨π‘₯(𝑑),π‘₯Ξ”(𝑑)⟩,β€–π‘₯(𝑑)β€–β€–π‘₯(𝑑)‖ΔΔ=⟨π‘₯(𝑑),π‘₯ΔΔ(𝑑)⟩+β€–π‘₯Ξ”(𝑑)β€–2βˆ’β€–π‘₯(𝑑)β€–βŸ¨π‘₯(𝑑),π‘₯Ξ”(𝑑)⟩2β€–π‘₯(𝑑)β€–3β‰₯⟨π‘₯(𝜎(𝑑)),π‘₯ΔΔ(𝑑)⟩.β€–π‘₯(𝜎(𝑑))β€–(2.11) If π‘‘βˆˆπ΄ is such that 𝑑<𝜎(𝑑)=𝜎2(𝑑), then by Propositions 2.2 and 2.3, we have β€–π‘₯(𝑑)‖ΔΔ=β€–π‘₯(𝜎(𝑑))β€–Ξ”βˆ’β€–π‘₯(𝑑)β€–Ξ”=πœ‡(𝑑)⟨π‘₯(𝜎(𝑑)),π‘₯Ξ”(𝜎(𝑑))βŸ©βˆ’πœ‡(𝑑)β€–π‘₯(𝜎(𝑑))β€–β€–π‘₯(𝜎(𝑑))β€–βˆ’β€–π‘₯(𝑑)β€–πœ‡(𝑑)2=⟨π‘₯(𝜎(𝑑)),π‘₯Ξ”(𝑑)+πœ‡(𝑑)π‘₯ΔΔ(𝑑)βŸ©βˆ’πœ‡(𝑑)β€–π‘₯(𝜎(𝑑))β€–βŸ¨π‘₯(𝜎(𝑑)),π‘₯(𝑑)+πœ‡(𝑑)π‘₯Ξ”(𝑑)βŸ©πœ‡(𝑑)2+β€–π‘₯(𝜎(𝑑))β€–β€–π‘₯(𝑑)β€–πœ‡(𝑑)2=⟨π‘₯(𝜎(𝑑)),π‘₯ΔΔ(𝑑)βŸ©βˆ’β€–π‘₯(𝜎(𝑑))β€–βŸ¨π‘₯(𝜎(𝑑)),π‘₯(𝑑)βŸ©πœ‡(𝑑)2+β€–π‘₯(𝜎(𝑑))β€–β€–π‘₯(𝑑)β€–πœ‡(𝑑)2β‰₯⟨π‘₯(𝜎(𝑑)),π‘₯ΔΔ(𝑑)⟩.β€–π‘₯(𝜎(𝑑))β€–(2.12) If π‘‘βˆˆπ΄ is such that 𝑑<𝜎(𝑑)<𝜎2(𝑑), then β€–π‘₯(𝑑)‖ΔΔ=β€–π‘₯(𝜎(𝑑))β€–Ξ”βˆ’β€–π‘₯(𝑑)β€–Ξ”=ξ€·πœŽπœ‡(𝑑)β€–π‘₯2ξ€Έ(𝑑)β€–βˆ’β€–π‘₯(𝜎(𝑑))β€–βˆ’πœ‡(𝜎(𝑑))πœ‡(𝑑)β€–π‘₯(𝜎(𝑑))β€–βˆ’β€–π‘₯(𝑑)β€–πœ‡(𝑑)2β‰₯ξ€·πœŽβŸ¨π‘₯(𝜎(𝑑)),π‘₯2ξ€Έ(𝑑)βˆ’π‘₯(𝜎(𝑑))βŸ©βˆ’πœ‡(𝜎(𝑑))πœ‡(𝑑)β€–π‘₯(𝜎(𝑑))β€–β€–π‘₯(𝜎(𝑑))β€–βˆ’β€–π‘₯(𝑑)β€–πœ‡(𝑑)2=⟨π‘₯(𝜎(𝑑)),π‘₯Ξ”(𝜎(𝑑))βŸ©βˆ’πœ‡(𝑑)β€–π‘₯(𝜎(𝑑))β€–β€–π‘₯(𝜎(𝑑))β€–βˆ’β€–π‘₯(𝑑)β€–πœ‡(𝑑)2,(2.13) and we conclude as in the previous case.

Let πœ–>0. The exponential function π‘’πœ–(β‹…,𝑑0) is defined byπ‘’πœ–ξ€·π‘‘,𝑑0ξ€Έξ‚΅ξ€œ=exp[𝑑0,𝑑)βˆ©π•‹πœ‰πœ–ξ‚Ά,(πœ‡(𝑠))Δ𝑠(2.14) where πœ‰πœ–βŽ§βŽͺ⎨βŽͺ⎩(β„Ž)=πœ–,ifβ„Ž=0,log(1+β„Žπœ–)β„Ž,ifβ„Ž>0.(2.15) It is the unique solution to the initial value problem π‘₯Δ𝑑(𝑑)=πœ–π‘₯(𝑑),π‘₯0ξ€Έ=1.(2.16)

Here is a result on time scales, analogous to Gronwall's inequality. The reader may find the proof of this result in [13].

Theorem 2.5. Let 𝛼>0, πœ–>0, and π‘¦βˆˆπΆ(𝕋,ℝ). If ξ€œπ‘¦(𝑑)≀𝛼+[π‘Ž,𝑑)βˆ©π•‹πœ–π‘¦(𝑠)Δ𝑠foreveryπ‘‘βˆˆπ•‹,(2.17) then 𝑦(𝑑)β‰€π›Όπ‘’πœ–(𝑑,π‘Ž)foreveryπ‘‘βˆˆπ•‹.(2.18)

We recall some notions and results related to the theory of Ξ”-measure and Ξ”-Lebesgue integration introduced by Bohner and Guseinov in [12]. The reader is also referred to [14] for expressions of the Ξ”-measure and the Ξ”-integral in terms of the classical Lebesgue measure and the classical Lebesgue integral, respectively.

Definition 2.6. A set π΄βŠ‚π•‹ is said to be Ξ”-measurable if for every set πΈβŠ‚π•‹, π‘šβˆ—1(𝐸)=π‘šβˆ—1(𝐸∩𝐴)+π‘šβˆ—1(𝐸∩(𝕋⧡𝐴)),(2.19) where π‘šβˆ—1⎧βŽͺ⎨βŽͺβŽ©ξƒ―(𝐸)=infπ‘šξ“π‘˜=1ξ€·π‘‘π‘˜βˆ’π‘π‘˜ξ€ΈβˆΆπΈβŠ‚π‘šξšπ‘˜=1ξ€Ίπ‘π‘˜,π‘‘π‘˜ξ€Έwithπ‘π‘˜,π‘‘π‘˜ξƒ°βˆˆπ•‹ifβˆžπ‘βˆ‰πΈ,ifπ‘βˆˆπΈ.(2.20) The Ξ”-measure on β„³(π‘šβˆ—1)∢={π΄βŠ‚π•‹βˆΆπ΄isΞ”-measurable}, denoted by πœ‡Ξ”, is the restriction of π‘šβˆ—1 to β„³(π‘šβˆ—1). So, (𝕋,β„³(π‘šβˆ—1),πœ‡Ξ”) is a complete measurable space.

Proposition 2.7 (see [14]). Let π΄βŠ‚π•‹, then 𝐴 is Ξ”-measurable if and only if 𝐴 is Lebesgue measurable. Moreover, if π‘βˆ‰π΄, πœ‡Ξ”ξ“(𝐴)=π‘š(𝐴)+π‘‘π‘–βˆˆπ΄βˆ©π‘…π•‹ξ€·πœŽξ€·π‘‘π‘–ξ€Έβˆ’π‘‘π‘–ξ€Έ,(2.21) where π‘š is the Lebesgue measure.

The notions of Ξ”-measurable and Ξ”-integrable functions π‘“βˆΆπ•‹β†’β„π‘› can be defined similarly to the general theory of Lebesgue integral.

Let πΈβŠ‚π•‹ be a Ξ”-measurable set and π‘“βˆΆπ•‹β†’β„π‘› a Ξ”-measurable function. We say that π‘“βˆˆπΏ1Ξ”(𝐸,ℝ𝑛) provided ξ€œπΈβ€–π‘“(𝑠)‖Δ𝑠<∞.(2.22) The set 𝐿1Ξ”(𝕋0,ℝ𝑛) is a Banach space endowed with the norm ‖𝑓‖𝐿1Ξ”ξ€œβˆΆ=𝕋0‖𝑓(𝑠)‖Δ𝑠.(2.23) Here is an analog of the Lebesgue dominated convergence Theorem which can be proved as in the general theory of Lebesgue integration theory.

Theorem 2.8. Let {π‘“π‘˜}π‘˜βˆˆβ„• be a sequence of functions in 𝐿1Ξ”(𝕋0,ℝ𝑛). Assume that there exists a function π‘“βˆΆπ•‹0→ℝ𝑛 such that π‘“π‘˜(𝑑)→𝑓(𝑑)Ξ”-a.e. π‘‘βˆˆπ•‹0, and there exists a function π‘”βˆˆπΏ1Ξ”(𝕋0) such that β€–π‘“π‘˜(𝑑)‖≀𝑔(𝑑)Ξ”-a.e. π‘‘βˆˆπ•‹0 and for every π‘˜βˆˆβ„•, then π‘“π‘˜β†’π‘“ in 𝐿1Ξ”(𝕋0,ℝ𝑛).

In our existence results, we will consider Ξ”-CarathΓ©odory functions.

Definition 2.9. A function π‘“βˆΆπ•‹0Γ—β„π‘šβ†’β„π‘› is Ξ”-CarathΓ©odory if the following conditions hold: (i)𝑑↦𝑓(𝑑,π‘₯) is Ξ”-measurable for every π‘₯βˆˆβ„π‘š,(ii)π‘₯↦𝑓(𝑑,π‘₯) is continuous for Ξ”-almost every π‘‘βˆˆπ•‹0,(iii)for every π‘Ÿ>0, there exists β„Žπ‘ŸβˆˆπΏ1Ξ”(𝕋0,[0,∞)) such that ‖𝑓(𝑑,π‘₯)β€–β‰€β„Žπ‘Ÿ(𝑑) for Ξ”-almost every π‘‘βˆˆπ•‹0 and for every π‘₯βˆˆβ„π‘š such that β€–π‘₯β€–β‰€π‘Ÿ.

In this context, there is also a notion of absolute continuity introduced in [16].

Definition 2.10. A function π‘“βˆΆπ•‹β†’β„π‘› is said to be absolutely continuous on 𝕋 if for every πœ€>0, there exists a 𝛿>0 such that if {[π‘Žπ‘˜,π‘π‘˜)}π‘šπ‘˜=1 with π‘Žπ‘˜,π‘π‘˜βˆˆπ•‹ is a finite pairwise disjoint family of subintervals satisfying π‘šξ“π‘˜=1ξ€·π‘π‘˜βˆ’π‘Žπ‘˜ξ€Έ<𝛿,thenπ‘šξ“π‘˜=1ξ€·π‘β€–π‘“π‘˜ξ€Έξ€·π‘Žβˆ’π‘“π‘˜ξ€Έβ€–<πœ€.(2.24)

Proposition 2.11 (see [17]). If π‘“βˆΆπ•‹β†’β„π‘› is an absolutely continuous function, then the Ξ”-measure of the set {π‘‘βˆˆπ•‹0β§΅π‘…π•‹βˆΆπ‘“(𝑑)=0and𝑓Δ(𝑑)β‰ 0} is zero.

Proposition 2.12 (see [17]). If π‘”βˆˆπΏ1Ξ”(𝕋0,ℝ𝑛) and π‘“βˆΆπ•‹β†’β„π‘› is the function defined by ξ€œπ‘“(𝑑)∢=[π‘Ž,𝑑)βˆ©π•‹π‘”(𝑠)Δ𝑠,(2.25) then 𝑓 is absolutely continuous and 𝑓Δ(𝑑)=𝑔(𝑑)Ξ”-almost everywhere on 𝕋0.

Proposition 2.13 (see [16]). A function π‘“βˆΆπ•‹β†’β„ is absolutely continuous on 𝕋 if and only if 𝑓 is Ξ”-differentiable Ξ”-almost everywhere on 𝕋0, π‘“Ξ”βˆˆπΏ1Ξ”(𝕋0) and ξ€œ[π‘Ž,𝑑)βˆ©π•‹π‘“Ξ”(𝑠)Δ𝑠=𝑓(𝑑)βˆ’π‘“(π‘Ž),foreveryπ‘‘βˆˆπ•‹.(2.26)

We also recall a notion of Sobolev space, see [18],π‘ŠΞ”1,1(𝕋,ℝ𝑛)=π‘₯∈𝐿1Δ𝕋0,β„π‘›ξ€ΈβˆΆβˆƒπ‘”βˆˆπΏ1Δ𝕋0,ℝ𝑛suchthatξ€œπ•‹0π‘₯(𝑠)πœ‘Ξ”ξ€œ(𝑠)Δ𝑠=βˆ’π•‹0𝑔(𝑠)πœ‘(𝜎(𝑠))Δ𝑠foreveryπœ‘βˆˆπΆ10,rdξ‚Ό,(𝕋)(2.27) where 𝐢10,rd(𝕋)={πœ‘βˆΆπ•‹β†’β„βˆΆπœ‘(π‘Ž)=0=πœ‘(𝑏),πœ‘isΞ”-differentiableandπœ‘Ξ”iscontinuousatright-densepointsof𝕋anditsleft-sidedlimitsexistatleft-densepointsof𝕋}.(2.28) A function π‘₯βˆˆπ‘ŠΞ”1,1(𝕋,ℝ𝑛) can be identified to an absolutely continuous map.

Proposition 2.14 (see [18]). Suppose that π‘₯βˆˆπ‘ŠΞ”1,1(𝕋,ℝ) with some π‘”βˆˆπΏ1Ξ”(𝕋0,ℝ) satisfying (2.27), then there exists π‘¦βˆΆπ•‹β†’β„ absolutely continuous such that 𝑦=π‘₯,𝑦Δ=𝑔Δ-a.e.on𝕋0.(2.29) Moreover, if 𝑔 is 𝐢rd(π•‹πœ…,ℝ), then there exists π‘¦βˆˆπΆ1rd(𝕋,ℝ) such that 𝑦=π‘₯Ξ”-a.e.on𝕋0,𝑦Δ=𝑔Δ-a.e.onπ•‹πœ….(2.30)

Sobolev spaces of higher order can be defined inductively as follows: π‘ŠΞ”2,1(𝕋,ℝ𝑛)=π‘₯βˆˆπ‘ŠΞ”1,1(𝕋,ℝ𝑛)∢π‘₯Ξ”βˆˆπ‘ŠΞ”1,1(π•‹πœ…,ℝ𝑛)ξ€Ύ.(2.31) With the norm β€–π‘₯β€–π‘ŠΞ”1,1∢=β€–π‘₯‖𝐿1Ξ”+β€–π‘₯Δ‖𝐿1Ξ” (resp., β€–π‘₯β€–π‘ŠΞ”2,1∢=β€–π‘₯‖𝐿1Ξ”+β€–π‘₯Δ‖𝐿1Ξ”+β€–π‘₯ΔΔ‖𝐿1Ξ”), π‘ŠΞ”1,1(𝕋,ℝ𝑛) (resp., π‘ŠΞ”2,1(𝕋,ℝ𝑛)) is a Banach space.

Remark 2.15. By Proposition 2.7, we know that πœ‡Ξ”({𝑑})>0 for every π‘‘βˆˆπ‘…π•‹. From this fact and the previous proposition, one realizes that there is no interest to look for solutions to (1.1) and (1.2) in π‘ŠΞ”2,1(𝕋,ℝ𝑛) and to consider Ξ”-CarathΓ©odory maps 𝑓 in the case where the time scale is such that πœ‡Ξ”(𝕋0⧡𝑅𝕋)=0. In particular, this is the case for difference equations. Let us point out that we consider more general time scales. Nevertheless, the results that we obtained are new in both cases.

As in the classical case, some embeddings have nice properties.

Proposition 2.16 (see [18]). The inclusion 𝑗1βˆΆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)→𝐢1(𝕋,ℝ𝑛) is continuous.

Proposition 2.17. The inclusion 𝑗0βˆΆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)β†ͺ𝐢(𝕋,ℝ𝑛) is a continuous, compact, linear operator.

Proof. Arguing as in the proof of the ArzelΓ -Ascoli Theorem, we can show that the inclusion π‘–βˆΆπΆ1(𝕋,ℝ𝑛)β†ͺ𝐢(𝕋,ℝ𝑛) is linear, continuous, and compact. The conclusion follows from the previous proposition since 𝑗0=π‘–βˆ˜π‘—1.

We obtain a maximum principle in this context. To this aim, we use the following result.

Lemma 2.18 (see [19]). Let π‘“βˆΆπ•‹β†’β„ be a function with a local maximum at 𝑑0∈(π‘Ž,𝑏)βˆ©π•‹. If 𝑓ΔΔ(𝜌(𝑑0)) exists, then 𝑓ΔΔ(𝜌(𝑑0))≀0 provided 𝑑0 is not at the same time left dense and right scattered.

Theorem 2.19. Let π‘Ÿβˆˆπ‘ŠΞ”2,1(𝕋) be a function such that π‘ŸΞ”Ξ”(𝑑)>0Ξ”-almost everywhere on {π‘‘βˆˆπ•‹πœ…20βˆΆπ‘Ÿ(𝜎(𝑑))>0}. If one of the following conditions holds: (i)π‘Ž0π‘Ÿ(π‘Ž)βˆ’π›Ύ0π‘ŸΞ”(π‘Ž)≀0 and π‘Ž1π‘Ÿ(𝑏)+𝛾1π‘ŸΞ”(𝜌(𝑏))≀0 (where π‘Ž0, π‘Ž1, 𝛾0, and 𝛾1 are defined as in (1.4)),(ii)π‘Ÿ(π‘Ž)=π‘Ÿ(𝑏) and π‘ŸΞ”(π‘Ž)β‰₯π‘ŸΞ”(𝜌(𝑏)), then π‘Ÿ(𝑑)≀0, for every π‘‘βˆˆπ•‹.

Proof. If the conclusion is false, there exists 𝑑0βˆˆπ•‹ such that π‘Ÿξ€·π‘‘0ξ€Έ=maxπ‘‘βˆˆπ•‹π‘Ÿ(𝑑)>0.(2.32)
In the case where π‘Žβ‰€πœŒ(𝑑0)<𝑑0<𝑏, then π‘ŸΞ”Ξ”(𝜌(𝑑0)) exists since πœ‡Ξ”({𝜌(𝑑0)})=𝑑0βˆ’πœŒ(𝑑0)>0 and π‘Ÿβˆˆπ‘ŠΞ”2,1(𝕋). By Lemma 2.18, π‘ŸΞ”Ξ”(𝜌(𝑑0))≀0, which is a contradiction since π‘Ÿ(𝜎(𝜌(𝑑0)))=π‘Ÿ(𝑑0)>0.
If π‘Ž<𝜌(𝑑0)=𝑑0=𝜎(𝑑0)<𝑏, there exists 𝑑1>𝑑0 such that π‘Ÿ(𝜎(𝑑))>0 for every π‘‘βˆˆ(𝑑0,𝑑1)βˆ©π•‹. On the other hand, since π‘Ÿ(𝑑0) is a maximum, π‘ŸΞ”(𝑑0)=0, and there exists π‘ βˆˆ(𝑑0,𝑑1) such that π‘ŸΞ”(𝑠)≀0, thus, 0β‰₯π‘ŸΞ”(𝑠)βˆ’π‘ŸΞ”ξ€·π‘‘0ξ€Έ=ξ€œ[𝑑0,𝑠)βˆ©π•‹π‘ŸΞ”Ξ”(𝜏)Ξ”πœ>0,(2.33) by hypothesis and Proposition 2.13, which is a contradiction. Observe that the same argument applies if π‘Ž=𝜌(𝑑0)=𝑑0=𝜎(𝑑0) and π‘ŸΞ”(𝑑0)=0. Notice also that if 𝜌(𝑑0)=𝑑0=𝜎(𝑑0)=𝑏 and π‘ŸΞ”(𝑑0)=0, we get a contradiction with an analogous argument for a suitable 𝑑1<𝑑0.
If π‘Žβ‰€πœŒ(𝑑0)=𝑑0<𝜎(𝑑0)<𝑏 and π‘Ÿ(𝑑0)=π‘Ÿ(𝜎(𝑑0)), we argue as in the first case replacing 𝑑0 by 𝜎(𝑑0) to obtain a contradiction.
In the case where π‘Ž<𝜌(𝑑0)=𝑑0<𝜎(𝑑0)≀𝑏 and π‘Ÿ(𝑑0)>π‘Ÿ(𝜎(𝑑0)), then π‘ŸΞ”(𝑑0)<0. Since π‘‘β†¦π‘ŸΞ”(𝑑) is continuous, there exists 𝛿>0 such that π‘ŸΞ”(𝑑)<0 on an interval (𝑑0βˆ’π›Ώ,𝑑0), which contradicts the maximality of π‘Ÿ(𝑑0).
Observe that π‘Ž=𝑑0 and π‘ŸΞ”(π‘Ž)<0 could happen if π‘Ž=𝑑0=𝜎(𝑑0) or if π‘Ž=𝑑0<𝜎(𝑑0) and π‘Ÿ(π‘Ž)>π‘Ÿ(𝜎(π‘Ž)). In this case, we get a contradiction if π‘Ÿ satisfies condition (i). On the other hand, if π‘Ÿ satisfies condition (ii), then π‘Ÿ(𝑏)=π‘Ÿ(π‘Ž)>0 and π‘ŸΞ”(𝜌(𝑏))<0. So 𝜌(𝑏)=𝑏 since π‘Ÿ(𝑏)β‰₯π‘Ÿ(𝜌(𝑏)). This contradicts the maximality of π‘Ÿ(𝑏).
On the other hand, the case where 𝑑0=𝑏 and π‘Ÿ satisfies condition (i) can be treated similarly to the previous case.

Finally, we define the Ξ”-differential operator associated to the problems that we will consider πΏβˆΆπ‘Š2,1Ξ”,𝐡(𝕋,ℝ𝑛)⟢𝐿1Ξ”ξ€·π•‹πœ…0,ℝ𝑛definedby𝐿(π‘₯)(𝑑)∢=π‘₯ΔΔ(𝑑)βˆ’π‘₯(𝜎(𝑑)),(2.34) where π‘Š2,1Ξ”,𝐡(𝕋,ℝ𝑛)∢={π‘₯βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)∢π‘₯∈(BC)} with (BC) denoting the periodic boundary condition (1.3) or the Sturm-Liouville boundary condition (1.4).

Proposition 2.20. The operator 𝐿 is invertible and πΏβˆ’1 is affine and continuous.

Proof. If (BC) denotes the Sturm-Liouville boundary condition (1.4), consider the associated homogeneous boundary condition π‘Ž0π‘₯(π‘Ž)βˆ’π›Ύ0π‘₯Ξ”π‘Ž(π‘Ž)=0,1π‘₯(𝑏)+𝛾1π‘₯Ξ”(𝜌(𝑏))=0.(2.35) Denote π‘Š2,1Ξ”,𝐡0(𝕋,ℝ𝑛)=π‘₯βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)∢π‘₯satisfiesξ€Ύ(1.3)if(BC)denotesξ€½(1.3),π‘₯βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)∢π‘₯satisfiesξ€Ύ(2.35)if(BC)denotes(1.4).(2.36) Notice that π‘Š2,1Ξ”,𝐡0(𝕋,ℝ𝑛) is a Banach space. Define 𝐿0βˆΆπ‘Š2,1Ξ”,𝐡0(𝕋,ℝ𝑛)⟢𝐿1Ξ”ξ€·π•‹πœ…0,ℝ𝑛by𝐿0(π‘₯)(𝑑)∢=π‘₯ΔΔ(𝑑)βˆ’π‘₯(𝜎(𝑑)).(2.37) It is obvious that 𝐿0 is linear and continuous. It follows directly from Theorem 2.19 that 𝐿0 is injective.
If (BC) denotes (1.4) (resp., (1.3)), let 𝐺(𝑑,𝑠) be the Green function given in [13, Theorem  4.70] (resp., [13, Theorem  4.89]). Arguing as in [13, Theorem  4.70] (resp., [13, Theorem  4.89]), one can verify that for any β„ŽβˆˆπΏ1Ξ”(π•‹πœ…0,ℝ𝑛), ξ€œπ‘₯(𝑑)=[π‘Ž,𝜌(𝑏))βˆ©π•‹πΊ(𝑑,𝑠)β„Ž(𝑠)Δ𝑠(2.38) is a solution of 𝐿0(π‘₯)=β„Ž. So, 𝐿0 is bijective and, hence, invertible with 𝐿0βˆ’1 continuous by the inverse mapping theorem.
Finally, if (BC) denotes (1.3), since 𝐿=𝐿0, we have the conclusion. On the other hand, if (BC) denotes (1.4), let 𝑦 be given in [13, Theorem  4.67] such that 𝑦ΔΔ(𝑑)βˆ’π‘¦(𝜎(𝑑))=0onπ•‹πœ…2,π‘Ž0𝑦(π‘Ž)βˆ’π›Ύ0𝑦Δ(π‘Ž)=π‘₯0,π‘Ž1𝑦(𝑏)+𝛾1𝑦Δ(𝜌(𝑏))=π‘₯1,(2.39) then πΏβˆ’1=𝑦+𝐿0βˆ’1.

Remark 2.21. We could have considered the operator ξ‚πΏβˆΆπ‘Š2,1Ξ”,𝐡(𝕋,ℝ𝑛)⟢𝐿1Ξ”ξ€·π•‹πœ…0,ℝ𝑛definedby𝐿(π‘₯)(𝑑)∢=π‘₯ΔΔ(𝑑),(2.40) when the boundary condition is (1.4) with suitable constants π‘Ž0, π‘Ž1, 𝛾0, 𝛾1 such that 𝐿 is injective. For simplicity, we prefer to use only the operator 𝐿.

3. Nonlinearity Not Depending on the Delta-Derivative

In this section, we establish existence results for the problem π‘₯ΔΔ(𝑑)=𝑓(𝑑,π‘₯(𝜎(𝑑))),Ξ”-a.e.π‘‘βˆˆπ•‹πœ…20,π‘₯∈(BC),(3.1) where (BC) denotes the periodic boundary condition π‘₯π‘₯(π‘Ž)=π‘₯(𝑏),Ξ”(π‘Ž)=π‘₯Ξ”(𝜌(𝑏)),(3.2) or the Sturm-Liouville boundary condition π‘Ž0π‘₯(π‘Ž)βˆ’π›Ύ0π‘₯Ξ”(π‘Ž)=π‘₯0,π‘Ž1π‘₯(𝑏)+𝛾1π‘₯Ξ”(𝜌(𝑏))=π‘₯1,(3.3) where π‘Ž0,π‘Ž1,𝛾0,𝛾1β‰₯0, max{π‘Ž0,𝛾0}>0, and max{π‘Ž1,𝛾1}>0. We look for solutions in π‘ŠΞ”2,1(𝕋,ℝ𝑛).

We introduce the notion of solution-tube for the problem (3.1).

Definition 3.1. Let (𝑣,𝑀)βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)Γ—π‘ŠΞ”2,1(𝕋,[0,∞)). We say that (𝑣,𝑀) is a solution-tube of (3.1) if (i)⟨π‘₯βˆ’π‘£(𝜎(𝑑)),𝑓(𝑑,π‘₯)βˆ’π‘£Ξ”Ξ”(𝑑)⟩β‰₯𝑀(𝜎(𝑑))𝑀ΔΔ(𝑑) for Ξ”-almost every π‘‘βˆˆπ•‹πœ…20 and for every π‘₯βˆˆβ„π‘› such that β€–π‘₯βˆ’π‘£(𝜎(𝑑))β€–=𝑀(𝜎(𝑑)),(ii)𝑣ΔΔ(𝑑)=𝑓(𝑑,𝑣(𝜎(𝑑))) and 𝑀ΔΔ(𝑑)≀0 for Ξ”-almost every π‘‘βˆˆπ•‹πœ…20 such that 𝑀(𝜎(𝑑))=0,(iii)(a)if (BC) denotes (3.2), then 𝑣(π‘Ž)=𝑣(𝑏), 𝑀(π‘Ž)=𝑀(𝑏), and ‖𝑣Δ(𝜌(𝑏))βˆ’π‘£Ξ”(π‘Ž)‖≀𝑀Δ(𝜌(𝑏))βˆ’π‘€Ξ”(π‘Ž), (b)if (BC) denotes (3.3), β€–π‘₯0βˆ’(π‘Ž0𝑣(π‘Ž)βˆ’π›Ύ0𝑣Δ(π‘Ž))β€–β‰€π‘Ž0𝑀(π‘Ž)βˆ’π›Ύ0𝑀Δ(π‘Ž), β€–π‘₯1βˆ’(π‘Ž1𝑣(𝑏)+𝛾1𝑣Δ(𝜌(𝑏)))β€–β‰€π‘Ž1𝑀(𝑏)+𝛾1𝑀Δ(𝜌(𝑏)). We denote 𝑇(𝑣,𝑀)=π‘₯βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛.)βˆΆβ€–π‘₯(𝑑)βˆ’π‘£(𝑑)‖≀𝑀(𝑑)βˆ€π‘‘βˆˆπ•‹(3.4)

We state the main theorem of this section.

Theorem 3.2. Let π‘“βˆΆπ•‹πœ…0×ℝ𝑛→ℝ𝑛 be a Ξ”-CarathΓ©odory function. If (𝑣,𝑀)βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)Γ—π‘ŠΞ”2,1(𝕋,[0,∞)) is a solution-tube of (3.1), then the system (3.1) has a solution π‘₯βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)βˆ©π‘‡(𝑣,𝑀).

In order to prove this result, we consider the following modified problem:π‘₯ΔΔ(𝑑)βˆ’π‘₯(𝜎(𝑑))=𝑓𝑑,ξ€Έβˆ’π‘₯(𝜎(𝑑))π‘₯(𝜎(𝑑)),Ξ”-a.e.π‘‘βˆˆπ•‹πœ…20,π‘₯∈(BC),(3.5) where⎧βŽͺ⎨βŽͺ⎩π‘₯(𝑠)=𝑀(𝑠)β€–π‘₯βˆ’π‘£(𝑠)β€–(π‘₯βˆ’π‘£(𝑠))+𝑣(𝑠)ifπ‘₯β€–π‘₯βˆ’π‘£(𝑠)β€–>𝑀(𝑠),otherwise.(3.6)

We define the operator 𝐹∢𝐢(𝕋,ℝ𝑛)→𝐿1Ξ”(π•‹πœ…0,ℝ𝑛) by 𝐹(π‘₯)(𝑑)∢=𝑓𝑑,ξ€Έβˆ’π‘₯(𝜎(𝑑))π‘₯(𝜎(𝑑)).(3.7)

Proposition 3.3. Under the assumptions of Theorem 3.2, the operator 𝐹 defined above is continuous and bounded.

Proof. First of all, we show that the set 𝐹(𝐢(𝕋,ℝ𝑛)) is bounded. Fix 𝑅>‖𝑣‖0+‖𝑀‖0. Let β„Žπ‘…βˆˆπΏ1Ξ”(π•‹πœ…0,[0,∞)) be given by Definition 2.9(iii). Thus, for every π‘₯∈𝐢(𝕋,ℝ𝑛), ‖𝐹(π‘₯)(𝑠)β€–β‰€β„Žπ‘…(𝑠)+𝑅=βˆΆβ„Ž(𝑠)Ξ”-a.e.π‘ βˆˆπ•‹πœ…0.(3.8)
To prove the continuity of 𝐹, we consider {π‘₯π‘˜}π‘˜βˆˆβ„• a sequence of 𝐢(𝕋,ℝ𝑛) converging to π‘₯∈𝐢(𝕋,ℝ𝑛). We already know that for every π‘˜βˆˆβ„•, ξ€·π‘₯β€–πΉπ‘˜ξ€Έ(𝑠)β€–β‰€β„Ž(𝑠)Ξ”-a.e.π‘ βˆˆπ•‹πœ…0.(3.9) One can easily check that π‘₯𝑛(𝑑)β†’π‘₯(𝑑) for all π‘‘βˆˆπ•‹. It follows from Definition 2.9(ii) that 𝐹π‘₯π‘˜ξ€Έ(𝑠)⟢𝐹(π‘₯)(𝑠)Ξ”-a.e.π‘ βˆˆπ•‹πœ…0.(3.10) Theorem 2.8 implies that 𝐹(π‘₯π‘˜)→𝐹(π‘₯) in 𝐿1Ξ”(π•‹πœ…0,ℝ𝑛).

Lemma 3.4. Under the assumptions of Theorem 3.2, every solution π‘₯ of (3.5) is in 𝑇(𝑣,𝑀).

Proof. Since π‘₯βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛) (resp., π‘£βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛), π‘€βˆˆπ‘ŠΞ”2,1(𝕋,ℝ)), π‘₯ΔΔ(𝑑) (resp., 𝑣ΔΔ(𝑑), 𝑀ΔΔ(𝑑)), there exists Ξ”-almost everywhere on π•‹πœ…2. Denote 𝐴=π‘‘βˆˆπ•‹πœ…20.βˆΆβ€–π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–>𝑀(𝜎(𝑑))(3.11) By Lemma 2.4(1), (β€–π‘₯(𝑑)βˆ’π‘£(𝑑)β€–βˆ’π‘€(𝑑))ΔΔβ‰₯⟨π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑)),π‘₯ΔΔ(𝑑)βˆ’π‘£Ξ”Ξ”(𝑑)βŸ©β€–π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–βˆ’π‘€Ξ”Ξ”(𝑑)Ξ”-a.e.on𝐴.(3.12) We claim that (β€–π‘₯(𝑑)βˆ’π‘£(𝑑)β€–βˆ’π‘€(𝑑))ΔΔ>0Ξ”-a.e.on𝐴.(3.13) Indeed, we deduce from the fact that (𝑣,𝑀) is a solution-tube of (3.5) and from (3.12) that Ξ”-almost everywhere on {π‘‘βˆˆπ΄βˆΆπ‘€(𝜎(𝑑))>0}, (β€–π‘₯(𝑑)βˆ’π‘£(𝑑)β€–βˆ’π‘€(𝑑))ΔΔβ‰₯ξ€·βŸ¨π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑)),𝑓𝑑,ξ€Έβˆ’π‘₯(𝜎(𝑑))π‘₯(𝜎(𝑑))+π‘₯(𝜎(𝑑))βˆ’π‘£Ξ”Ξ”(𝑑)βŸ©β€–π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–βˆ’π‘€Ξ”Ξ”=⟨(𝑑)ξ€·π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑)),𝑓𝑑,ξ€Έπ‘₯(𝜎(𝑑))βˆ’π‘£Ξ”Ξ”(𝑑)βŸ©π‘€(𝜎(𝑑))+β€–π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–βˆ’π‘€(𝜎(𝑑))βˆ’π‘€Ξ”Ξ”>(𝑑)𝑀(𝜎(𝑑))𝑀ΔΔ(𝑑)𝑀(𝜎(𝑑))βˆ’π‘€Ξ”Ξ”(𝑑)=0,(3.14) and Ξ”-almost everywhere on {π‘‘βˆˆπ΄βˆΆπ‘€(𝜎(𝑑))=0}, (β€–π‘₯(𝑑)βˆ’π‘£(𝑑)β€–βˆ’π‘€(𝑑))ΔΔβ‰₯ξ€·βŸ¨π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑)),𝑓𝑑,ξ€Έβˆ’π‘₯(𝜎(𝑑))π‘₯(𝜎(𝑑))+π‘₯(𝜎(𝑑))βˆ’π‘£Ξ”Ξ”(𝑑)βŸ©β€–π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–βˆ’π‘€Ξ”Ξ”=(𝑑)⟨π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑)),𝑓(𝑑,𝑣(𝜎(𝑑)))βˆ’π‘£Ξ”Ξ”(𝑑)βŸ©β€–π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–+β€–π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–βˆ’π‘€Ξ”Ξ”(𝑑)>βˆ’π‘€Ξ”Ξ”(𝑑)β‰₯0.(3.15)
Observe that if β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–βˆ’π‘€(π‘Ž)>0, β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–Ξ”β‰₯⟨π‘₯(π‘Ž)βˆ’π‘£(π‘Ž),π‘₯Ξ”(π‘Ž)βˆ’π‘£Ξ”(π‘Ž)⟩.β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–(3.16) Indeed, this follows from Proposition 2.3 when π‘Ž=𝜎(π‘Ž). For π‘Ž<𝜎(π‘Ž), β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–Ξ”=β€–π‘₯(𝜎(π‘Ž))βˆ’π‘£(𝜎(π‘Ž))β€–βˆ’β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–β‰₯πœ‡(π‘Ž)⟨π‘₯(π‘Ž)βˆ’π‘£(π‘Ž),π‘₯(𝜎(π‘Ž))βˆ’π‘£(𝜎(π‘Ž))βˆ’(π‘₯(π‘Ž)βˆ’π‘£(π‘Ž))⟩=πœ‡(π‘Ž)β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–βŸ¨π‘₯(π‘Ž)βˆ’π‘£(π‘Ž),π‘₯Ξ”(π‘Ž)βˆ’π‘£Ξ”(π‘Ž)⟩.β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–(3.17) Similarly, if β€–π‘₯(𝑏)βˆ’π‘£(𝑏)β€–βˆ’π‘€(𝑏)>0, β€–π‘₯(𝜌(𝑏))βˆ’π‘£(𝜌(𝑏))β€–Ξ”β‰€βŸ¨π‘₯(𝑏)βˆ’π‘£(𝑏),π‘₯Ξ”(𝜌(𝑏))βˆ’π‘£Ξ”(𝜌(𝑏))⟩.β€–π‘₯(𝑏)βˆ’π‘£(𝑏)β€–(3.18)
If (BC) denotes (3.2), β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–βˆ’π‘€(π‘Ž)=β€–π‘₯(𝑏)βˆ’π‘£(𝑏)β€–βˆ’π‘€(𝑏).(3.19) We deduce from (3.16), (3.18), and Definition 3.1 that β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–βˆ’π‘€(π‘Ž)=β€–π‘₯(𝑏)βˆ’π‘£(𝑏)β€–βˆ’π‘€(𝑏)≀0,(3.20) or ξ€·β€–π‘₯(𝜌(𝑏))βˆ’π‘£(𝜌(𝑏))β€–Ξ”βˆ’π‘€Ξ”ξ€Έβˆ’ξ€·(𝜌(𝑏))β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–Ξ”βˆ’π‘€Ξ”ξ€Έβ‰€(π‘Ž)⟨π‘₯Ξ”(𝜌(𝑏))βˆ’π‘£Ξ”(𝜌(𝑏)),π‘₯(𝑏)βˆ’π‘£(𝑏)βŸ©βˆ’β€–π‘₯(𝑏)βˆ’π‘£(𝑏)β€–βŸ¨π‘₯Ξ”(π‘Ž)βˆ’π‘£Ξ”(π‘Ž),π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)βŸ©βˆ’ξ€·π‘€β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–Ξ”(𝜌(𝑏))βˆ’π‘€Ξ”ξ€Έ=(π‘Ž)βŸ¨π‘£Ξ”(π‘Ž)βˆ’π‘£Ξ”(𝜌(𝑏)),π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)βŸ©βˆ’ξ€·π‘€β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–Ξ”(𝜌(𝑏))βˆ’π‘€Ξ”ξ€Έ(π‘Ž)≀‖𝑣Δ(π‘Ž)βˆ’π‘£Ξ”ξ€·π‘€(𝜌(𝑏))β€–βˆ’Ξ”(𝜌(𝑏))βˆ’π‘€Ξ”ξ€Έ(π‘Ž)≀0.(3.21)
If (BC) denotes (3.3), we deduce from (3.16), (3.18), and Definition 3.1 that β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–βˆ’π‘€(π‘Ž)≀0,(3.22) or π‘Ž0(β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–βˆ’π‘€(π‘Ž))βˆ’π›Ύ0ξ€·β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–Ξ”βˆ’π‘€Ξ”ξ€Έβ‰€(π‘Ž)⟨π‘₯(π‘Ž)βˆ’π‘£(π‘Ž),π‘Ž0(π‘₯(π‘Ž)βˆ’π‘£(π‘Ž))βˆ’π›Ύ0ξ€·π‘₯Ξ”(π‘Ž)βˆ’π‘£Ξ”ξ€ΈβŸ©(π‘Ž)β€–π‘₯(π‘Ž)βˆ’π‘£(π‘Ž)β€–βˆ’π‘Ž0𝑀(π‘Ž)+𝛾0𝑀Δ≀‖‖π‘₯(π‘Ž)0βˆ’ξ€·π‘Ž0𝑣(π‘Ž)βˆ’π›Ύ0𝑣Δ‖‖(π‘Ž)βˆ’π‘Ž0𝑀(π‘Ž)+𝛾0𝑀Δ(π‘Ž)≀0,(3.23) and similarly β€–π‘₯(𝑏)βˆ’π‘£(𝑏)β€–βˆ’π‘€(𝑏)≀0,(3.24) or π‘Ž1(β€–π‘₯(𝑏)βˆ’π‘£(𝑏)β€–βˆ’π‘€(𝑏))+𝛾1ξ€·β€–π‘₯(𝜌(𝑏))βˆ’π‘£(𝜌(𝑏))β€–Ξ”βˆ’π‘€Ξ”ξ€Έ(𝜌(𝑏))≀0.(3.25)
Finally, it follows from (3.13), (3.20), (3.21), (3.22), (3.23), (3.24), (3.25), and Theorem 2.19 applied to π‘Ÿ(𝑑)=β€–π‘₯(𝑑)βˆ’π‘£(𝑑)β€–βˆ’π‘€(𝑑) that β€–π‘₯(𝑑)βˆ’π‘£(𝑑)‖≀𝑀(𝑑) for every π‘‘βˆˆπ•‹.

Now, we can prove the main theorem of this section.

Proof of Theorem 3.2. A solution of (3.5) is a fixed point of the operator π‘‡βˆΆπΆ(𝕋,ℝ𝑛)⟢𝐢(𝕋,ℝ𝑛)definedbyπ‘‡βˆΆ=𝑗0βˆ˜πΏβˆ’1∘𝐹,(3.26) where 𝐿 and 𝑗0 are defined in (2.34) and Proposition 2.17, respectively. By Propositions 2.17, 2.20, and 3.3, the operator 𝑇 is compact. So, the Schauder fixed point theorem implies that 𝑇 has a fixed point and, hence, Problem (3.5) has a solution π‘₯. By Lemma 3.4, this solution is in 𝑇(𝑣,𝑀). Thus, π‘₯ is a solution of (3.1).

In the particular case where 𝑛=1, as corollary of Theorem 3.2, we obtain a generalization of results of AkΔ±n [1] and StehlΓ­k [2] for the Dirichlet and the periodic boundary conditions, respectively.

Corollary 3.5. Let π‘“βˆΆπ•‹πœ…0×ℝ→ℝ be a Ξ”-CarathΓ©odory function. Assume that there exists 𝛼,π›½βˆˆπ‘ŠΞ”2,1(𝕋,ℝ) such that (i)𝛼(𝑑)≀𝛽(𝑑) for Ξ”-almost every π‘‘βˆˆπ•‹,(ii)𝛼ΔΔ(𝑑)β‰₯𝑓(𝑑,𝛼(𝜎(𝑑))) and 𝛽ΔΔ(𝑑)≀𝑓(𝑑,𝛽(𝜎(𝑑))) for Ξ”-almost every π‘‘βˆˆπ•‹πœ…2,(iii)(a)if (BC) denotes (3.2), then 𝛼(π‘Ž)=𝛼(𝑏), 𝛼Δ(π‘Ž)β‰₯𝛼Δ(𝜌(𝑏)), 𝛽(π‘Ž)=𝛽(𝑏), and 𝛽Δ(π‘Ž)≀𝛽Δ(𝜌(𝑏)),(b)if (BC) denotes (3.3), then π‘Ž0𝛼(π‘Ž)βˆ’π›Ύ0𝛼Δ(π‘Ž)≀π‘₯0β‰€π‘Ž0𝛽(π‘Ž)βˆ’π›Ύ0𝛽Δ(π‘Ž) and π‘Ž1𝛼(𝑏)+𝛾1𝛼Δ(𝜌(𝑏))≀π‘₯1β‰€π‘Ž1𝛽(𝑏)+𝛾1𝛽Δ(𝜌(𝑏)), then (3.1) has a solution π‘₯βˆˆπ‘ŠΞ”2,1(𝕋,ℝ) such that 𝛼(𝑑)≀π‘₯(𝑑)≀𝛽(𝑑) for every π‘‘βˆˆπ•‹.

Proof. Observe that ((𝛼+𝛽)/2,(π›½βˆ’π›Ό)/2) is a solution-tube of (3.1). The conclusion follows from Theorem 3.2.

Theorem 3.2 generalizes also a result established by Henderson et al. [6] for systems of second-order dynamic equations on time scales. Let us mention that they considered a continuous map 𝑓 and they assumed a strict inequality in (3.27).

Corollary 3.6. Let π‘“βˆΆπ•‹πœ…0×ℝ𝑛→ℝ𝑛 be a Ξ”-CarathΓ©odory function. Assume that there exists a constant 𝑅>0 such that ⟨π‘₯,𝑓(𝑑,π‘₯)⟩β‰₯0Ξ”-a.e.π‘‘βˆˆπ•‹πœ…20,βˆ€π‘₯suchthatβ€–π‘₯β€–=𝑅.(3.27) Moreover, if (𝐡𝐢) denotes (3.3), assume that β€–π‘₯0β€–β‰€π‘Ž0𝑅 and β€–π‘₯1β€–β‰€π‘Ž1𝑅, then the system (3.1) has a solution π‘₯βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛) such that β€–π‘₯(𝑑)‖≀𝑅 for every π‘‘βˆˆπ•‹.

Here is an example in which one cannot find a solution-tube of the form (0,𝑅).

Example 3.7. Consider the system π‘₯ΔΔ(𝑑)=π‘˜(𝑑)(β€–π‘₯(𝜎(𝑑))βˆ’πœŽ(𝑑)π‘β€–βˆ’1)Ξ”-a.e.π‘‘βˆˆπ•‹πœ…20,π‘₯(π‘Ž)=0,π‘₯(𝑏)=0,(3.28) where π‘˜βˆˆπΏ1Ξ”(𝕋0,ℝ𝑛⧡{0}) and π‘βˆˆβ„π‘›β§΅{0} is such that ‖𝑐‖max{|π‘Ž|,|𝑏|}≀1. One can check that (𝑣,𝑀) is a solution-tube of (3.28) with 𝑣(𝑑)=𝑑𝑐, 𝑀(𝑑)≑1. By Theorem 3.2, this problem has at least one solution π‘₯ such that β€–π‘₯(𝑑)βˆ’π‘‘π‘β€–β‰€1. Observe that there is no 𝑅>0 such that (3.27) is satisfied.

4. Nonlinearity Depending on π‘₯Ξ”

In this section, we study more general systems of second-order dynamic equations on time scales. Indeed, we allow the nonlinearity 𝑓 to depend also on π‘₯Ξ”. We consider the problem π‘₯ΔΔ(𝑑)=𝑓𝑑,π‘₯(𝜎(𝑑)),π‘₯Ξ”ξ€Έ(𝑑),Ξ”-a.e.π‘‘βˆˆπ•‹πœ…20,π‘Ž0π‘₯(π‘Ž)βˆ’π‘₯Ξ”(π‘Ž)=π‘₯0,π‘Ž1π‘₯(𝑏)+𝛾1π‘₯Ξ”(𝜌(𝑏))=π‘₯1,(4.1) where π‘Ž0,π‘Ž1,𝛾1β‰₯0 and max{π‘Ž1,𝛾1}>0.

We also introduce a notion of solution-tube for this problem.

Definition 4.1. Let (𝑣,𝑀)βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)Γ—π‘ŠΞ”2,1(𝕋,(0,∞)). We say that (𝑣,𝑀) is a solution-tube of (4.1) if (i)for Ξ”-almost every π‘‘βˆˆ{π‘‘βˆˆπ•‹πœ…20βˆΆπ‘‘=𝜎(𝑑)}, π‘₯βˆ’π‘£(𝑑),𝑓(𝑑,π‘₯,𝑦)βˆ’π‘£Ξ”Ξ”ξ¬(𝑑)+β€–π‘¦βˆ’π‘£Ξ”(𝑑)β€–2β‰₯𝑀(𝑑)𝑀ΔΔ𝑀(𝑑)+Ξ”ξ€Έ(𝑑)2,(4.2) for every (π‘₯,𝑦)βˆˆβ„2𝑛 such that β€–π‘₯βˆ’π‘£(𝑑)β€–=𝑀(𝑑) and ⟨π‘₯βˆ’π‘£(𝑑),π‘¦βˆ’π‘£Ξ”(𝑑)⟩=𝑀(𝑑)𝑀Δ(𝑑),(ii)for every π‘‘βˆˆ{π‘‘βˆˆπ•‹πœ…20βˆΆπ‘‘<𝜎(𝑑)}, π‘₯βˆ’π‘£(𝜎(𝑑)),𝑓(𝑑,π‘₯,𝑦)βˆ’π‘£Ξ”Ξ”ξ¬(𝑑)β‰₯𝑀(𝜎(𝑑))𝑀ΔΔ(𝑑),(4.3) for every (π‘₯,𝑦)βˆˆβ„2𝑛 such that β€–π‘₯βˆ’π‘£(𝜎(𝑑))β€–=𝑀(𝜎(𝑑)),(iii)β€–π‘₯0βˆ’(π‘Ž0𝑣(π‘Ž)βˆ’π‘£Ξ”(π‘Ž))β€–β‰€π‘Ž0𝑀(π‘Ž)βˆ’π‘€Ξ”(π‘Ž), β€–π‘₯1βˆ’(π‘Ž1𝑣(𝑏)+𝛾1𝑣Δ(𝜌(𝑏)))β€–β‰€π‘Ž1𝑀(𝑏)+𝛾1𝑀Δ(𝜌(𝑏)).

If 𝕋 is the real interval [π‘Ž,𝑏], condition (ii) of the previous definition becomes useless, and we get the notion of solution-tube introduced by the first author in [10] for a system of second-order differential equations.

Here is the main result of this section.

Theorem 4.2. Let π‘“βˆΆπ•‹πœ…0×ℝ2𝑛→ℝ𝑛 be a Ξ”-CarathΓ©odory function. Assume that (H1) there exists (𝑣,𝑀)βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)Γ—π‘ŠΞ”2,1(𝕋,(0,∞)) a solution-tube of (4.1),(H2)there exist constants 𝑐,𝑑>0 such that ‖𝑓(𝑑,π‘₯,𝑦)‖≀𝑐+𝑑‖𝑦‖ for Ξ”-almost every π‘‘βˆˆπ•‹πœ…0 and for every (π‘₯,𝑦)βˆˆβ„2𝑛 such that β€–π‘₯βˆ’π‘£(𝑑)‖≀𝑀(𝑑), then (4.1) has a solution π‘₯βˆˆπ‘ŠΞ”2,1(𝕋,ℝ𝑛)βˆ©π‘‡(𝑣,𝑀).

To prove this existence result, we consider the following modified problem:π‘₯ΔΔ(𝑑)βˆ’π‘₯(𝜎(𝑑))=𝑔𝑑,π‘₯(𝜎(𝑑)),π‘₯Ξ”ξ€Έ(𝑑),Ξ”-a.e.π‘‘βˆˆπ•‹πœ…20,π‘Ž0π‘₯(π‘Ž)βˆ’π‘₯Ξ”(π‘Ž)=π‘₯0,π‘Ž1π‘₯(𝑏)+𝛾1π‘₯Ξ”(𝜌(𝑏))=π‘₯1,(4.4) where π‘”βˆΆπ•‹πœ…0×ℝ2𝑛→ℝ𝑛 is defined by ⎧βŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽ©π‘”(𝑑,π‘₯,𝑦)=𝑀(𝜎(𝑑))𝑓‖π‘₯βˆ’π‘£(𝜎(𝑑))‖𝑑,ξ€Έβˆ’π‘₯(𝜎(𝑑)),̃𝑦(𝑑)+ξ‚΅π‘₯(𝜎(𝑑))1βˆ’π‘€(𝜎(𝑑))𝑣‖π‘₯βˆ’π‘£(𝜎(𝑑))‖ΔΔ𝑀(𝑑)+ΔΔ(𝑑)ξ‚Άβ€–π‘₯βˆ’π‘£(𝜎(𝑑))β€–(π‘₯βˆ’π‘£(𝜎(𝑑)))if𝑓‖π‘₯βˆ’π‘£(𝜎(𝑑))β€–>𝑀(𝜎(𝑑)),𝑑,π‘₯ξ€Έβˆ’(𝜎(𝑑)),̃𝑦(𝑑)π‘₯(𝜎(𝑑)),otherwise,(4.5) where π‘₯(𝜎(𝑑)) is defined as in (3.6), ⎧βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺ⎨βŽͺβŽͺβŽͺβŽͺβŽͺβŽͺβŽ©ξ‚΅π‘€Μƒπ‘¦(𝑑)=̂𝑦(𝑑)+Ξ”(𝑑)βˆ’βŸ¨π‘₯βˆ’π‘£(𝜎(𝑑)),̂𝑦(𝑑)βˆ’π‘£Ξ”(𝑑)βŸ©β€–π‘₯βˆ’π‘£(𝜎(𝑑))β€–ξ‚Άξ‚΅π‘₯βˆ’π‘£(𝜎(𝑑))ξ‚Άβ€–π‘₯βˆ’π‘£(𝜎(𝑑))β€–if𝑀𝑑=𝜎(𝑑),β€–π‘₯βˆ’π‘£(𝜎(𝑑))β€–>𝑀(𝜎(𝑑)),̂𝑦(𝑑)+Ξ”(𝑑)𝐾𝑀(𝜎(𝑑))1βˆ’β€–π‘¦βˆ’π‘£Ξ”ξ‚Ά(𝑑)β€–(π‘₯βˆ’π‘£(𝜎(𝑑)))if𝑑=𝜎(𝑑),β€–π‘₯βˆ’π‘£(𝜎(𝑑))‖≀𝑀(𝜎(𝑑)),β€–π‘¦βˆ’π‘£Ξ”(𝑑)β€–>𝐾,̂𝑦(𝑑)if𝑦𝑑<𝜎(𝑑),otherwise,𝐾̂𝑦(𝑑)=β€–π‘¦βˆ’π‘£Ξ”(𝑑)β€–π‘¦βˆ’π‘£Ξ”ξ€Έ(𝑑)+𝑣Δ(𝑑)ifβ€–π‘¦βˆ’π‘£Ξ”(𝑑)β€–>𝐾,𝑦,otherwise,(4.6) with 𝐾>0 a constant which will be fixed later.

Remark 4.3. (1) Remark that β€–π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))‖≀𝑀(𝜎(𝑑)),‖̃𝑦(𝑑)‖≀2𝐾+‖𝑣Δ||𝑀(𝑑)β€–+Ξ”||.(𝑑)(4.7)
(2) If β€–π‘₯βˆ’π‘£(𝜎(𝑑))β€–>𝑀(𝜎(𝑑)), ‖π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–=𝑀(𝜎(𝑑)),ξ‚‹π‘₯π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑)),Ξ”(𝑑)βˆ’π‘£Ξ”ξ‚­(𝑑)=𝑀(𝜎(𝑑))𝑀Δ(𝑑),for𝑑=𝜎(𝑑).(4.8)
(3) If β€–π‘₯βˆ’π‘£(𝜎(𝑑))β€–>𝑀(𝜎(𝑑)) and 𝑑=𝜎(𝑑), β€–ξ‚‹π‘₯Ξ”(𝑑)βˆ’π‘£Ξ”(𝑑)β€–2ξ‚Šπ‘₯=β€–Ξ”(𝑑)βˆ’π‘£Ξ”(𝑑)β€–2+𝑀Δ(𝑑)2βˆ’ξ‚¬ξ‚Šπ‘₯π‘₯(𝑑)βˆ’π‘£(𝑑),Ξ”(𝑑)βˆ’π‘£Ξ”ξ‚­(𝑑)2β€–π‘₯(𝑑)βˆ’π‘£(𝑑)β€–2.(4.9)
(4) Since 𝑓 is Ξ”-CarathΓ©odory, by (1), there exists β„ŽβˆˆπΏ1Ξ”(π•‹πœ…0,ℝ) such that for every π‘₯,π‘¦βˆˆβ„π‘›, ‖𝑔(𝑑,π‘₯,𝑦)β€–β‰€β„Ž(𝑑)Ξ”-a.e.π‘‘βˆˆπ•‹πœ…0.(4.10)
We associate to 𝑔 the operator 𝐺∢𝐢1(𝕋,ℝ𝑛)→𝐿1Ξ”(π•‹πœ…0,ℝ𝑛) defined by 𝐺(π‘₯)(𝑑)∢=𝑔𝑑,π‘₯(𝜎(𝑑)),π‘₯Ξ”ξ€Έ.(𝑑)(4.11)

Proposition 4.4. Let π‘“βˆΆπ•‹πœ…0×ℝ2𝑛→ℝ𝑛 be a Ξ”-CarathΓ©odory function. Assume that (H1) is satisfied, then 𝐺 is continuous.

Proof. Let {π‘₯π‘˜} be a sequence of 𝐢1(𝕋,ℝ𝑛) converging to π‘₯∈𝐢1(𝕋,ℝ𝑛). It is clear that π‘₯π‘˜(𝑑)βŸΆξ‚Šπ‘₯π‘₯(𝑑),Ξ”π‘˜ξ‚Šπ‘₯(𝑑)βŸΆΞ”(𝑑).(4.12)
On {π‘‘βˆˆπ•‹βˆΆπ‘‘<𝜎(𝑑)}, we have 𝑔𝑑,π‘₯π‘˜(𝜎(𝑑)),π‘₯Ξ”π‘˜ξ€Έξ€·(𝑑)βŸΆπ‘”π‘‘,π‘₯(𝜎(𝑑)),π‘₯Ξ”ξ€Έ,(𝑑)(4.13) since 𝑓 is Ξ”-CarathΓ©odory.
Similarly, Ξ”-almost everywhere on {π‘‘βˆˆπ•‹πœ…0βˆΆπ‘‘=𝜎(𝑑),β€–π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))‖≠𝑀(𝜎(𝑑))}, 𝑔𝑑,π‘₯π‘˜(𝜎(𝑑)),π‘₯Ξ”π‘˜ξ€Έξ€·(𝑑)βŸΆπ‘”π‘‘,π‘₯(𝜎(𝑑)),π‘₯Ξ”ξ€Έ,(𝑑)(4.14) since ξ‚‹π‘₯Ξ”π‘˜ξ‚‹π‘₯(𝑑)β†’Ξ”(𝑑) and, for π‘˜ sufficiently large, β€–π‘₯π‘˜(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))‖≠𝑀(𝜎(𝑑)).
Denote π‘†βˆΆ={π‘‘βˆˆπ•‹πœ…0βˆΆπ‘‘=𝜎(𝑑)andβ€–π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–=𝑀(𝜎(𝑑))} and 𝐼𝑑={π‘˜βˆˆβ„•βˆΆβ€–π‘₯π‘˜(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))‖≀𝑀(𝜎(𝑑))}. As before, it is easy to check that Ξ”-almost everywhere on {π‘‘βˆˆπ‘†βˆΆcard𝐼𝑑=∞}, 𝑔𝑑,π‘₯π‘˜(𝜎(𝑑)),π‘₯Ξ”π‘˜ξ€Έξ€·(𝑑)βŸΆπ‘”π‘‘,π‘₯(𝜎(𝑑)),π‘₯Ξ”ξ€Έ(𝑑)asπ‘˜βˆˆπΌπ‘‘goestoinfinity.(4.15) On the other hand, Proposition 2.11 implies that π‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑)),π‘₯Ξ”(𝑑)βˆ’π‘£Ξ”ξ¬(𝑑)=𝑀(𝜎(𝑑))𝑀Δ(𝑑)Ξ”-a.e.π‘‘βˆˆπ‘†.(4.16) So, Ξ”-almost everywhere on {π‘‘βˆˆπ‘†βˆΆcard(ℕ⧡𝐼𝑑)=∞}, ξ‚‹π‘₯Ξ”π‘˜ξ‚Šπ‘₯(𝑑)=Ξ”π‘˜+βŽ›βŽœβŽœβŽπ‘€(𝑑)Δπ‘₯(𝑑)βˆ’π‘˜ξ‚Šπ‘₯(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑)),Ξ”π‘˜(𝑑)βˆ’π‘£Ξ”ξ‚­(𝑑)β€–π‘₯π‘˜(βŽžβŽŸβŽŸβŽ ξ‚΅π‘₯𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–π‘˜(𝜎(𝑑))βˆ’π‘£(𝜎(𝑑))β€–π‘₯π‘˜(ξ‚Ά