Boundary Value Problems for Systems of Second-Order Dynamic Equations on Time Scales with Δ-Carathéodory Functions

Abstract

We establish the existence of solutions to systems of second-order dynamic equations on time scales with the right member 𝑓, a Δ-Carathéodory function. First, we consider the case where the nonlinearity 𝑓 does not depend on the Δ-derivative, 𝑥^Δ(𝑡). 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 Δ-derivative and satisfies a linear growth condition with respect to 𝑥^Δ(𝑡). 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:𝑥^ΔΔ(𝑡)=𝑓(𝑡,𝑥(𝜎(𝑡))),Δ-a.e.𝑡∈𝕋^𝜅2₀,𝑥∈(BC),(1.1)𝑎₀𝑥^ΔΔ(𝑡)=𝑓𝑡,𝑥(𝜎(𝑡)),𝑥^Δ(𝑡),Δ-a.e.𝑡∈𝕋^𝜅2₀,𝑥(𝑎)−𝑥^Δ(𝑎)=𝑥₀,𝑎₁𝑥(𝑏)+𝛾₁𝑥^Δ(𝜌(𝑏))=𝑥₁.(1.2) Here, 𝕋 is a compact time scale where 𝑎=min𝕋, 𝑏=max𝕋, and 𝕋^𝜅2₀ is defined in (2.4). The map 𝑓∶𝕋^𝜅0×ℝ^𝑛→ℝ^𝑛 is Δ-Carathéodory (see Definition 2.9), and (BC) denotes one of the following boundary conditions:𝑥𝑥(𝑎)=𝑥(𝑏),_Δ(𝑎)=𝑥^Δ(𝜌(𝑏)),(1.3)𝑎₁𝑎₀𝑥(𝑎)−𝛾₀𝑥^Δ(𝑎)=𝑥₀,𝑥(𝑏)+𝛾₁𝑥^Δ(𝜌(𝑏))=𝑥₁,(1.4) where 𝑎₀,𝑎₁,𝛾₀,𝛾₁≥0, max{𝑎₀,𝛾₀}>0, and max{𝑎₁,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 Akın [1] for the Dirichlet boundary condition and by Stehlík [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:∃𝑅>0suchthat⟨𝑥,𝑓(𝑡,𝑥)⟩>0if‖𝑥‖=𝑅.(1.5)

The fact that the right member in the system (1.2) depends also on the Δ-derivative, 𝑥^Δ, 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(𝑡)<𝑡<𝜎(𝑡) or 𝕋=[𝑎,𝑏]). They established the existence of a solution of (1.2) under the following assumptions: (A1) there exists 𝑅>𝑚𝑎𝑥{‖𝑥₀‖,‖𝑥₁‖} such that 2⟨𝑥,𝑓(𝑡,𝑥,𝑦)⟩+‖𝑦‖²>0 if ‖𝑥‖=𝑅, −2⟨𝑥,𝑦⟩≤𝜇(𝑡)‖𝑦‖²,(A2) there exist 𝑐,𝑑≥0 such that 𝑑(𝜌(𝑏)−𝑎)<1 and ‖𝑓(𝑡,𝑥,𝑦)‖≤𝑐+𝑑‖𝑦‖ if ‖𝑥‖≤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)−𝑎)<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 Δ-Carathéodory 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 𝑊^Δ2,1(𝕋,ℝ^𝑛). 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 𝑎=min𝕋<𝑏=max𝕋. The forward jump operator 𝜎∶𝕋→𝕋 (resp., the backward jump operator 𝜌∶𝕋→𝕋) is defined by 𝜎(𝑡)=inf{𝑠∈𝕋∶𝑠>𝑡}𝑏if𝑡<𝑏,if𝑡=𝑏,resp.,𝜌(𝑡)=sup{𝑠∈𝕋∶𝑠<𝑡}𝑎if𝑡>𝑎,if𝑡=𝑎.(2.1) We say that 𝑡<𝑏 is right scattered (resp., 𝑡>𝑎 is left scattered) if 𝜎(𝑡)>𝑡 (resp., 𝜌(𝑡)<𝑡) otherwise, we say that 𝑡 is right dense (resp., left dense). The set of right-scattered points of 𝕋 is at most countable, see [14]. We denote it by 𝑅_𝕋∶={𝑡∈𝕋∶𝑡<𝜎(𝑡)}=𝑡_𝑖∶𝑖∈𝐼,(2.2) for some 𝐼⊂ℕ. The graininess function 𝜇∶𝕋→[0,∞) is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡. We denote 𝕋^𝜅=𝕋⧵(𝜌(𝑏),𝑏],𝕋₀=𝕋⧵{𝑏}.(2.3) So, 𝕋^𝜅=𝕋 if 𝑏 is left dense, otherwise 𝕋^𝜅=𝕋₀. Since 𝕋^𝜅 is also a time scale, we denote𝕋^𝜅2=(𝕋^𝜅)^𝜅,𝕋^𝜅2₀=𝕋^𝜅2⧵{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 𝑥^Δ(𝑡). This notion coincides with 𝑥′(𝑡) (resp., Δ𝑥(𝑡)) in the case where the time scale 𝕋 is an interval (resp., the discrete set {0,1,…,𝑁}).

Definition 2.1. A map 𝑓∶𝕋→ℝ^𝑛 is Δ-differentiable at 𝑡∈𝕋^𝜅 if there exists 𝑓^Δ(𝑡)∈ℝ^𝑛 (called the Δ-derivative of 𝑓 at 𝑡) such that for all 𝜀>0, there exists a neighborhood 𝑈 of 𝑡 such that ‖‖𝑓(𝜎(𝑡))−𝑓(𝑠)−𝑓^Δ(𝑡)(𝜎(𝑡)−𝑠)‖‖≤𝜀||𝜎(𝑡)−𝑠||∀𝑠∈𝑈.(2.5) We say that 𝑓 is Δ-differentiable if 𝑓^Δ(𝑡) exists for every 𝑡∈𝕋^𝜅.
If 𝑓 is Δ-differentiable and if 𝑓^Δ is Δ-differentiable at 𝑡∈𝕋^𝜅2, we call 𝑓^ΔΔ(𝑡)=(𝑓^Δ)^Δ(𝑡) the second Δ-derivative of 𝑓 at 𝑡.

Proposition 2.2. Let 𝑓∶𝕋→ℝ^𝑛 and 𝑡∈𝕋^𝜅. (i)If 𝑓 is Δ-differentiable at 𝑡, then 𝑓 is continuous at 𝑡.(ii)If 𝑓 is continuous at 𝑡∈𝑅_𝕋, then 𝑓^Δ(𝑡)=𝑓(𝜎(𝑡))−𝑓(𝑡)𝜇(𝑡).(2.6)(iii)The map 𝑓 is Δ-differentiable at 𝑡∈𝕋^𝜅⧵𝑅_𝕋 if and only if 𝑓^Δ(𝑡)=lim_𝑠→𝑡𝑓(𝑡)−𝑓(𝑠)𝑡−𝑠.(2.7)

Proposition 2.3. If 𝑓∶𝕋→ℝ^𝑛 and 𝑔∶𝕋→ℝ^𝑚 are Δ-differentiable at 𝑡∈𝕋^𝜅, then (i)if 𝑛=𝑚, (𝛼𝑓+𝑔)^Δ(𝑡)=𝛼𝑓^Δ(𝑡)+𝑔^Δ(𝑡) for every 𝛼∈ℝ,(ii)if 𝑚=1, (𝑓𝑔)^Δ(𝑡)=𝑔(𝑡)𝑓^Δ(𝑡)+𝑓(𝜎(𝑡))𝑔^Δ(𝑡)=𝑓(𝑡)𝑔^Δ(𝑡)+𝑔(𝜎(𝑡))𝑓^Δ(𝑡),(iii)if 𝑚=1 and 𝑔(𝑡)𝑔(𝜎(𝑡))≠0, then 𝑓𝑔^Δ(𝑡)=𝑔(𝑡)𝑓^Δ(𝑡)−𝑓(𝑡)𝑔^Δ(𝑡)𝑔(𝑡)𝑔(𝜎(𝑡)),(2.8)(iv)if 𝑊⊂ℝ^𝑛 is open and ℎ∶𝑊→ℝ is differentiable at 𝑓(𝑡)∈𝑊 and 𝑡∉𝑅_𝕋, then (ℎ∘𝑓)^Δ(𝑡)=⟨ℎ′(𝑓(𝑡)),𝑓^Δ(𝑡)⟩.

We denote 𝐶(𝕋,ℝ^𝑛) the space of continuous maps on 𝕋, and we denote 𝐶¹(𝕋,ℝ^𝑛) the space of continuous maps on 𝕋 with continuous Δ-derivative on 𝕋^𝜅. With the norm ‖𝑥‖₀=max{‖𝑥(𝑡)‖∶𝑡∈𝕋} (resp., ‖𝑥‖₁=max{‖𝑥‖₀,max{‖𝑥^Δ(𝑡)‖∶𝑡∈𝕋^𝜅}}), 𝐶(𝕋,ℝ^𝑛) (resp., 𝐶¹(𝕋,ℝ^𝑛)) is a Banach space.

We study the second Δ-derivative of the norm of a map.

Lemma 2.4. Let 𝑥∶𝕋→ℝ^𝑛 be Δ-differentiable. (1)On {𝑡∈𝕋^𝜅2∶‖𝑥(𝜎(𝑡))‖>0 and 𝑥^ΔΔ(𝑡) exists}, ‖𝑥(𝑡)‖^ΔΔ≥⟨𝑥(𝜎(𝑡)),𝑥^ΔΔ(𝑡)⟩‖𝑥(𝜎(𝑡))‖.(2.9)(2) On {𝑡∈𝕋^𝜅2⧵𝑅_𝕋∶‖𝑥(𝜎(𝑡))‖>0 and 𝑥^ΔΔ(𝑡) exists}, ‖𝑥(𝑡)‖^ΔΔ=⟨𝑥(𝑡),𝑥^ΔΔ(𝑡)⟩+‖𝑥^Δ(𝑡)‖²‖𝑥(𝑡)‖−⟨𝑥(𝑡),𝑥^Δ(𝑡)⟩²‖𝑥(𝑡)‖³.(2.10)

Proof. Denote 𝐴={𝑡∈𝕋^𝜅2∶‖𝑥(𝜎(𝑡))‖>0 and 𝑥^ΔΔ(𝑡) exists}. By Proposition 2.3, on the set 𝐴⧵𝑅_𝕋, we have ‖𝑥(𝑡)‖^Δ=⟨𝑥(𝑡),𝑥^Δ(𝑡)⟩‖𝑥(𝑡)‖_ΔΔ‖𝑥(𝑡)‖,=⟨𝑥(𝑡),𝑥^ΔΔ(𝑡)⟩+‖𝑥^Δ(𝑡)‖²‖𝑥(𝑡)‖−⟨𝑥(𝑡),𝑥^Δ(𝑡)⟩²‖𝑥(𝑡)‖³≥⟨𝑥(𝜎(𝑡)),𝑥^ΔΔ(𝑡)⟩‖𝑥(𝜎(𝑡))‖.(2.11) If 𝑡∈𝐴 is such that 𝑡<𝜎(𝑡)=𝜎²(𝑡), then by Propositions 2.2 and 2.3, we have ‖𝑥(𝑡)‖^ΔΔ=‖𝑥(𝜎(𝑡))‖^Δ−‖𝑥(𝑡)‖^Δ=⟨𝑥(𝜎(𝑡)),𝜇𝑥(_Δ𝑡)(𝜎(𝑡))⟩𝜇(𝑡)‖𝑥(𝜎(𝑡))‖−‖𝑥(𝜎(𝑡)𝜇)(‖𝑡)−²‖𝑥(𝑡)‖=⟨𝑥(𝜎(𝑡)),𝑥^Δ(𝑡)+𝜇(𝑡)𝑥^ΔΔ(𝑡)⟩𝜇(𝑡)‖𝑥(𝜎(𝑡))‖−⟨𝑥(𝜎(𝑡)),𝑥(𝑡)+𝜇(𝑡)𝑥^Δ(𝑡)⟩𝜇(𝑡)²‖𝑥(𝜎(𝑡))‖+‖𝜇𝑥((𝑡𝑡))²‖=⟨𝑥(𝜎(𝑡)),𝑥^ΔΔ(𝑡)⟩‖𝑥(𝜎(𝑡))‖−𝜇⟨𝑥(𝑡()𝜎²‖(𝑥𝑡)()𝜎,𝑥(𝑡()𝑡))‖⟩+‖𝜇𝑥((𝑡𝑡))²‖≥⟨𝑥(𝜎(𝑡)),𝑥^ΔΔ(𝑡)⟩‖𝑥(𝜎(𝑡))‖.(2.12) If 𝑡∈𝐴 is such that 𝑡<𝜎(𝑡)<𝜎²(𝑡), then ‖𝑥(𝑡)‖^ΔΔ=‖𝑥(𝜎(𝑡))‖^Δ−‖𝑥(𝑡)‖^Δ=‖𝑥𝜎₂𝜇(𝑡)(𝑡)‖−‖𝑥(𝜎(𝑡))‖𝜇(𝜎(𝑡))𝜇(𝑡)−‖𝑥(𝜎(𝑡)𝜇)(‖𝑡)−²‖𝑥(𝑡)‖≥⟨𝑥(𝜎(𝑡)),𝑥𝜎²(𝑡)−𝑥(𝜎(𝑡))⟩𝜇(𝜎(𝑡))𝜇(𝑡)‖𝑥(𝜎(𝑡))‖−‖𝑥(𝜎(𝑡)𝜇)(‖𝑡)−²‖𝑥(𝑡)‖=⟨𝑥(𝜎(𝑡)),𝑥^Δ(𝜎(𝑡))⟩𝜇(𝑡)‖𝑥(𝜎(𝑡))‖−‖𝑥(𝜎(𝑡)𝜇)(‖𝑡)−²‖𝑥(𝑡)‖,(2.13) and we conclude as in the previous case.

Let 𝜖>0. The exponential function 𝑒_𝜖(⋅,𝑡₀) is defined by𝑒_𝜖𝑡,𝑡₀=exp_{[𝑡0,𝑡)∩𝕋}𝜉_𝜖(𝜇(𝑠))Δ𝑠,(2.14) where 𝜉_𝜖(ℎ)=⎧⎪⎨⎪⎩𝜖,log(1+ℎ𝜖)ifℎ=0,ℎ,ifℎ>0.(2.15) It is the unique solution to the initial value problem 𝑥^Δ(𝑡)=𝜖𝑥(𝑡),𝑥𝑡₀=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, 𝜖>0, and 𝑦∈𝐶(𝕋,ℝ). If 𝑦(𝑡)≤𝛼+_{[𝑎,𝑡)∩𝕋}𝜖𝑦(𝑠)Δ𝑠forevery𝑡∈𝕋,(2.17) then 𝑦(𝑡)≤𝛼𝑒_𝜖(𝑡,𝑎)forevery𝑡∈𝕋.(2.18)

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

Definition 2.6. A set 𝐴⊂𝕋 is said to be Δ-measurable if for every set 𝐸⊂𝕋, 𝑚^∗1(𝐸)=𝑚^∗1(𝐸∩𝐴)+𝑚^∗1(𝐸∩(𝕋⧵𝐴)),(2.19) where 𝑚^∗1(𝐸)=⎧⎪⎨⎪⎩inf^𝑚_𝑘=1𝑑_𝑘−𝑐_𝑘∶𝐸⊂^𝑚_𝑘=1𝑐_𝑘∞,𝑑_𝑘with𝑐_𝑘,𝑑_𝑘∈𝕋if𝑏∉𝐸,if𝑏∈𝐸.(2.20) The Δ-measure on ℳ(𝑚^∗1)∶={𝐴⊂𝕋∶𝐴isΔ-measurable}, denoted by 𝜇_Δ, is the restriction of 𝑚^∗1 to ℳ(𝑚^∗1). So, (𝕋,ℳ(𝑚^∗1),𝜇_Δ) is a complete measurable space.

Proposition 2.7 (see [14]). Let 𝐴⊂𝕋, then 𝐴 is Δ-measurable if and only if 𝐴 is Lebesgue measurable. Moreover, if 𝑏∉𝐴, 𝜇_Δ(𝐴)=𝑚(𝐴)+_{𝑡𝑖∈𝐴∩𝑅𝕋}𝜎𝑡_𝑖−𝑡_𝑖,(2.21) where 𝑚 is the Lebesgue measure.

The notions of Δ-measurable and Δ-integrable functions 𝑓∶𝕋→ℝ^𝑛 can be defined similarly to the general theory of Lebesgue integral.

Let 𝐸⊂𝕋 be a Δ-measurable set and 𝑓∶𝕋→ℝ^𝑛 a Δ-measurable function. We say that 𝑓∈𝐿^1Δ(𝐸,ℝ^𝑛) provided _𝐸‖𝑓(𝑠)‖Δ𝑠<∞.(2.22) The set 𝐿^1Δ(𝕋₀,ℝ^𝑛) is a Banach space endowed with the norm ‖𝑓‖_𝐿^1Δ∶=_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 {𝑓_𝑘}_𝑘∈ℕ be a sequence of functions in 𝐿^1Δ(𝕋₀,ℝ^𝑛). Assume that there exists a function 𝑓∶𝕋₀→ℝ^𝑛 such that 𝑓_𝑘(𝑡)→𝑓(𝑡)Δ-a.e. 𝑡∈𝕋₀, and there exists a function 𝑔∈𝐿^1Δ(𝕋₀) such that ‖𝑓_𝑘(𝑡)‖≤𝑔(𝑡)Δ-a.e. 𝑡∈𝕋₀ and for every 𝑘∈ℕ, then 𝑓_𝑘→𝑓 in 𝐿^1Δ(𝕋₀,ℝ^𝑛).

In our existence results, we will consider Δ-Carathéodory functions.

Definition 2.9. A function 𝑓∶𝕋₀×ℝ^𝑚→ℝ^𝑛 is Δ-Carathéodory if the following conditions hold: (i)𝑡↦𝑓(𝑡,𝑥) is Δ-measurable for every 𝑥∈ℝ^𝑚,(ii)𝑥↦𝑓(𝑡,𝑥) is continuous for Δ-almost every 𝑡∈𝕋₀,(iii)for every 𝑟>0, there exists ℎ_𝑟∈𝐿^1Δ(𝕋₀,[0,∞)) such that ‖𝑓(𝑡,𝑥)‖≤ℎ_𝑟(𝑡) for Δ-almost every 𝑡∈𝕋₀ and for every 𝑥∈ℝ^𝑚 such that ‖𝑥‖≤𝑟.

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

Definition 2.10. A function 𝑓∶𝕋→ℝ^𝑛 is said to be absolutely continuous on 𝕋 if for every 𝜀>0, there exists a 𝛿>0 such that if {[𝑎_𝑘,𝑏_𝑘)}^𝑚𝑘=1 with 𝑎_𝑘,𝑏_𝑘∈𝕋 is a finite pairwise disjoint family of subintervals satisfying ^𝑚_𝑘=1𝑏_𝑘−𝑎_𝑘<𝛿,then^𝑚_𝑘=1‖𝑓𝑏_𝑘−𝑓𝑎_𝑘‖<𝜀.(2.24)

Proposition 2.11 (see [17]). If 𝑓∶𝕋→ℝ^𝑛 is an absolutely continuous function, then the Δ-measure of the set {𝑡∈𝕋₀⧵𝑅_𝕋∶𝑓(𝑡)=0and𝑓^Δ(𝑡)≠0} is zero.

Proposition 2.12 (see [17]). If 𝑔∈𝐿^1Δ(𝕋₀,ℝ^𝑛) and 𝑓∶𝕋→ℝ^𝑛 is the function defined by 𝑓(𝑡)∶=_{[𝑎,𝑡)∩𝕋}𝑔(𝑠)Δ𝑠,(2.25) then 𝑓 is absolutely continuous and 𝑓^Δ(𝑡)=𝑔(𝑡)Δ-almost everywhere on 𝕋₀.

Proposition 2.13 (see [16]). A function 𝑓∶𝕋→ℝ is absolutely continuous on 𝕋 if and only if 𝑓 is Δ-differentiable Δ-almost everywhere on 𝕋₀, 𝑓^Δ∈𝐿^1Δ(𝕋₀) and _{[𝑎,𝑡)∩𝕋}𝑓^Δ(𝑠)Δ𝑠=𝑓(𝑡)−𝑓(𝑎),forevery𝑡∈𝕋.(2.26)

We also recall a notion of Sobolev space, see [18],𝑊^Δ1,1(𝕋,ℝ^𝑛)=𝑥∈𝐿^1Δ𝕋₀,ℝ^𝑛∶∃𝑔∈𝐿^1Δ𝕋₀,ℝ^𝑛suchthat_𝕋0=−_𝕋0𝑥(𝑠)𝜑^Δ(𝑠)Δ𝑠𝑔(𝑠)𝜑(𝜎(𝑠))Δ𝑠forevery𝜑∈𝐶^10,rd(𝕋),(2.27) where 𝐶^10,rd(𝕋)={𝜑∶𝕋→ℝ∶𝜑(𝑎)=0=𝜑(𝑏),𝜑isΔ-differentiableand𝜑i^Δtsilsefcto-snitdineduoluimsiattsreixgihstt-daetnleseft-pdoeinnstesopfoi𝕋ntsanodf𝕋}.(2.28) A function 𝑥∈𝑊^Δ1,1(𝕋,ℝ^𝑛) can be identified to an absolutely continuous map.

Proposition 2.14 (see [18]). Suppose that 𝑥∈𝑊^Δ1,1(𝕋,ℝ) with some 𝑔∈𝐿^1Δ(𝕋₀,ℝ) satisfying (2.27), then there exists 𝑦∶𝕋→ℝ absolutely continuous such that 𝑦=𝑥,𝑦^Δ=𝑔Δ-a.e.on𝕋₀.(2.29) Moreover, if 𝑔 is 𝐶_rd(𝕋^𝜅,ℝ), then there exists 𝑦∈𝐶^1rd(𝕋,ℝ) such that 𝑦=𝑥Δ-a.e.on𝕋₀,𝑦^Δ=𝑔Δ-a.e.on𝕋^𝜅.(2.30)

Sobolev spaces of higher order can be defined inductively as follows: 𝑊^Δ2,1(𝕋,ℝ^𝑛)=𝑥∈𝑊^Δ1,1(𝕋,ℝ^𝑛)∶𝑥^Δ∈𝑊^Δ1,1(𝕋^𝜅,ℝ^𝑛).(2.31) With the norm ‖𝑥‖_𝑊^Δ1,1∶=‖𝑥‖_𝐿^1Δ+‖𝑥^Δ‖_𝐿^1Δ (resp., ‖𝑥‖_𝑊^Δ2,1∶=‖𝑥‖_𝐿^1Δ+‖𝑥^Δ‖_𝐿^1Δ+‖𝑥^ΔΔ‖_𝐿^1Δ), 𝑊^Δ1,1(𝕋,ℝ^𝑛) (resp., 𝑊^Δ2,1(𝕋,ℝ^𝑛)) is a Banach space.

Remark 2.15. By Proposition 2.7, we know that 𝜇_Δ({𝑡})>0 for every 𝑡∈𝑅_𝕋. 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 𝑊^Δ2,1(𝕋,ℝ^𝑛) and to consider Δ-Carathéodory maps 𝑓 in the case where the time scale is such that 𝜇_Δ(𝕋₀⧵𝑅_𝕋)=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 𝑗₁∶𝑊^Δ2,1(𝕋,ℝ^𝑛)→𝐶¹(𝕋,ℝ^𝑛) is continuous.

Proposition 2.17. The inclusion 𝑗₀∶𝑊^Δ2,1(𝕋,ℝ^𝑛)↪𝐶(𝕋,ℝ^𝑛) is a continuous, compact, linear operator.

Proof. Arguing as in the proof of the Arzelà-Ascoli Theorem, we can show that the inclusion 𝑖∶𝐶¹(𝕋,ℝ^𝑛)↪𝐶(𝕋,ℝ^𝑛) is linear, continuous, and compact. The conclusion follows from the previous proposition since 𝑗₀=𝑖∘𝑗₁.

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

Lemma 2.18 (see [19]). Let 𝑓∶𝕋→ℝ be a function with a local maximum at 𝑡₀∈(𝑎,𝑏)∩𝕋. If 𝑓^ΔΔ(𝜌(𝑡₀)) exists, then 𝑓^ΔΔ(𝜌(𝑡₀))≤0 provided 𝑡₀ is not at the same time left dense and right scattered.

Theorem 2.19. Let 𝑟∈𝑊^Δ2,1(𝕋) be a function such that 𝑟^ΔΔ(𝑡)>0Δ-almost everywhere on {𝑡∈𝕋^𝜅2₀∶𝑟(𝜎(𝑡))>0}. If one of the following conditions holds: (i)𝑎₀𝑟(𝑎)−𝛾₀𝑟^Δ(𝑎)≤0 and 𝑎₁𝑟(𝑏)+𝛾₁𝑟^Δ(𝜌(𝑏))≤0 (where 𝑎₀, 𝑎₁, 𝛾₀, and 𝛾₁ are defined as in (1.4)),(ii)𝑟(𝑎)=𝑟(𝑏) and 𝑟^Δ(𝑎)≥𝑟^Δ(𝜌(𝑏)), then 𝑟(𝑡)≤0, for every 𝑡∈𝕋.

Proof. If the conclusion is false, there exists 𝑡₀∈𝕋 such that 𝑟𝑡₀=max_𝑡∈𝕋𝑟(𝑡)>0.(2.32)
In the case where 𝑎≤𝜌(𝑡₀)<𝑡₀<𝑏, then 𝑟^ΔΔ(𝜌(𝑡₀)) exists since 𝜇_Δ({𝜌(𝑡₀)})=𝑡₀−𝜌(𝑡₀)>0 and 𝑟∈𝑊^Δ2,1(𝕋). By Lemma 2.18, 𝑟^ΔΔ(𝜌(𝑡₀))≤0, which is a contradiction since 𝑟(𝜎(𝜌(𝑡₀)))=𝑟(𝑡₀)>0.
If 𝑎<𝜌(𝑡₀)=𝑡₀=𝜎(𝑡₀)<𝑏, there exists 𝑡₁>𝑡₀ such that 𝑟(𝜎(𝑡))>0 for every 𝑡∈(𝑡₀,𝑡₁)∩𝕋. On the other hand, since 𝑟(𝑡₀) is a maximum, 𝑟^Δ(𝑡₀)=0, and there exists 𝑠∈(𝑡₀,𝑡₁) such that 𝑟^Δ(𝑠)≤0, thus, 0≥𝑟^Δ(𝑠)−𝑟^Δ𝑡₀=_{[𝑡0,𝑠)∩𝕋}𝑟^ΔΔ(𝜏)Δ𝜏>0,(2.33) by hypothesis and Proposition 2.13, which is a contradiction. Observe that the same argument applies if 𝑎=𝜌(𝑡₀)=𝑡₀=𝜎(𝑡₀) and 𝑟^Δ(𝑡₀)=0. Notice also that if 𝜌(𝑡₀)=𝑡₀=𝜎(𝑡₀)=𝑏 and 𝑟^Δ(𝑡₀)=0, we get a contradiction with an analogous argument for a suitable 𝑡₁<𝑡₀.
If 𝑎≤𝜌(𝑡₀)=𝑡₀<𝜎(𝑡₀)<𝑏 and 𝑟(𝑡₀)=𝑟(𝜎(𝑡₀)), we argue as in the first case replacing 𝑡₀ by 𝜎(𝑡₀) to obtain a contradiction.
In the case where 𝑎<𝜌(𝑡₀)=𝑡₀<𝜎(𝑡₀)≤𝑏 and 𝑟(𝑡₀)>𝑟(𝜎(𝑡₀)), then 𝑟^Δ(𝑡₀)<0. Since 𝑡↦𝑟^Δ(𝑡) is continuous, there exists 𝛿>0 such that 𝑟^Δ(𝑡)<0 on an interval (𝑡₀−𝛿,𝑡₀), which contradicts the maximality of 𝑟(𝑡₀).
Observe that 𝑎=𝑡₀ and 𝑟^Δ(𝑎)<0 could happen if 𝑎=𝑡₀=𝜎(𝑡₀) or if 𝑎=𝑡₀<𝜎(𝑡₀) and 𝑟(𝑎)>𝑟(𝜎(𝑎)). In this case, we get a contradiction if 𝑟 satisfies condition (i). On the other hand, if 𝑟 satisfies condition (ii), then 𝑟(𝑏)=𝑟(𝑎)>0 and 𝑟^Δ(𝜌(𝑏))<0. So 𝜌(𝑏)=𝑏 since 𝑟(𝑏)≥𝑟(𝜌(𝑏)). This contradicts the maximality of 𝑟(𝑏).
On the other hand, the case where 𝑡₀=𝑏 and 𝑟 satisfies condition (i) can be treated similarly to the previous case.

Finally, we define the Δ-differential operator associated to the problems that we will consider 𝐿∶𝑊^2,1Δ,𝐵(𝕋,ℝ^𝑛)⟶𝐿^1Δ𝕋^𝜅0,ℝ^𝑛definedby𝐿(𝑥)(𝑡)∶=𝑥^ΔΔ(𝑡)−𝑥(𝜎(𝑡)),(2.34) where 𝑊^2,1Δ,𝐵(𝕋,ℝ^𝑛)∶={𝑥∈𝑊^Δ2,1(𝕋,ℝ^𝑛)∶𝑥∈(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 𝐿⁻¹ is affine and continuous.

Proof. If (BC) denotes the Sturm-Liouville boundary condition (1.4), consider the associated homogeneous boundary condition 𝑎𝑎₀𝑥(𝑎)−𝛾₀𝑥^Δ(𝑎)=0,₁𝑥(𝑏)+𝛾₁𝑥^Δ(𝜌(𝑏))=0.(2.35) Denote 𝑊^2,1Δ,𝐵₀(𝕋,ℝ^𝑛)=𝑥∈𝑊^Δ2,1𝑥∈𝑊_Δ2,1(𝕋,ℝ^𝑛)∶𝑥satisfies(1.3)if(BC)denotes(1.3),(𝕋,ℝ^𝑛)∶𝑥satisfies(2.35)if(BC)denotes(1.4).(2.36) Notice that 𝑊^2,1Δ,𝐵₀(𝕋,ℝ^𝑛) is a Banach space. Define 𝐿₀∶𝑊^2,1Δ,𝐵₀(𝕋,ℝ^𝑛)⟶𝐿^1Δ𝕋^𝜅0,ℝ^𝑛by𝐿₀(𝑥)(𝑡)∶=𝑥^ΔΔ(𝑡)−𝑥(𝜎(𝑡)).(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(𝑡,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 ℎ∈𝐿^1Δ(𝕋^𝜅0,ℝ^𝑛), 𝑥(𝑡)=_{[𝑎,𝜌(𝑏))∩𝕋}𝐺(𝑡,𝑠)ℎ(𝑠)Δ𝑠(2.38) is a solution of 𝐿₀(𝑥)=ℎ. So, 𝐿₀ is bijective and, hence, invertible with 𝐿⁰⁻¹ 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 𝑦^ΔΔ(𝑡)−𝑦(𝜎(𝑡))=0on𝕋^𝜅2,𝑎₀𝑦(𝑎)−𝛾₀𝑦^Δ(𝑎)=𝑥₀,𝑎₁𝑦(𝑏)+𝛾₁𝑦^Δ(𝜌(𝑏))=𝑥₁,(2.39) then 𝐿⁻¹=𝑦+𝐿⁰⁻¹.

Remark 2.21. We could have considered the operator 𝐿∶𝑊^2,1Δ,𝐵(𝕋,ℝ^𝑛)⟶𝐿^1Δ𝕋^𝜅0,ℝ^𝑛definedby𝐿(𝑥)(𝑡)∶=𝑥^ΔΔ(𝑡),(2.40) when the boundary condition is (1.4) with suitable constants 𝑎₀, 𝑎₁, 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 𝑥^ΔΔ(𝑡)=𝑓(𝑡,𝑥(𝜎(𝑡))),Δ-a.e.𝑡∈𝕋^𝜅2₀,𝑥∈(BC),(3.1) where (BC) denotes the periodic boundary condition 𝑥𝑥(𝑎)=𝑥(𝑏),_Δ(𝑎)=𝑥^Δ(𝜌(𝑏)),(3.2) or the Sturm-Liouville boundary condition 𝑎₁𝑎₀𝑥(𝑎)−𝛾₀𝑥^Δ(𝑎)=𝑥₀,𝑥(𝑏)+𝛾₁𝑥^Δ(𝜌(𝑏))=𝑥₁,(3.3) where 𝑎₀,𝑎₁,𝛾₀,𝛾₁≥0, max{𝑎₀,𝛾₀}>0, and max{𝑎₁,𝛾₁}>0. We look for solutions in 𝑊^Δ2,1(𝕋,ℝ^𝑛).

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

Definition 3.1. Let (𝑣,𝑀)∈𝑊^Δ2,1(𝕋,ℝ^𝑛)×𝑊^Δ2,1(𝕋,[0,∞)). We say that (𝑣,𝑀) is a solution-tube of (3.1) if (i)⟨𝑥−𝑣(𝜎(𝑡)),𝑓(𝑡,𝑥)−𝑣^ΔΔ(𝑡)⟩≥𝑀(𝜎(𝑡))𝑀^ΔΔ(𝑡) for Δ-almost every 𝑡∈𝕋^𝜅2₀ and for every 𝑥∈ℝ^𝑛 such that ‖𝑥−𝑣(𝜎(𝑡))‖=𝑀(𝜎(𝑡)),(ii)𝑣^ΔΔ(𝑡)=𝑓(𝑡,𝑣(𝜎(𝑡))) and 𝑀^ΔΔ(𝑡)≤0 for Δ-almost every 𝑡∈𝕋^𝜅2₀ such that 𝑀(𝜎(𝑡))=0,(iii)(a)if (BC) denotes (3.2), then 𝑣(𝑎)=𝑣(𝑏), 𝑀(𝑎)=𝑀(𝑏), and ‖𝑣^Δ(𝜌(𝑏))−𝑣^Δ(𝑎)‖≤𝑀^Δ(𝜌(𝑏))−𝑀^Δ(𝑎), (b)if (BC) denotes (3.3), ‖𝑥₀−(𝑎₀𝑣(𝑎)−𝛾₀𝑣^Δ(𝑎))‖≤𝑎₀𝑀(𝑎)−𝛾₀𝑀^Δ(𝑎), ‖𝑥₁−(𝑎₁𝑣(𝑏)+𝛾₁𝑣^Δ(𝜌(𝑏)))‖≤𝑎₁𝑀(𝑏)+𝛾₁𝑀^Δ(𝜌(𝑏)). We denote 𝑇(𝑣,𝑀)=𝑥∈𝑊^Δ2,1(𝕋,ℝ^𝑛)∶‖𝑥(𝑡)−𝑣(𝑡)‖≤𝑀(𝑡)∀𝑡∈𝕋.(3.4)

We state the main theorem of this section.

Theorem 3.2. Let 𝑓∶𝕋^𝜅0×ℝ^𝑛→ℝ^𝑛 be a Δ-Carathéodory function. If (𝑣,𝑀)∈𝑊^Δ2,1(𝕋,ℝ^𝑛)×𝑊^Δ2,1(𝕋,[0,∞)) is a solution-tube of (3.1), then the system (3.1) has a solution 𝑥∈𝑊^Δ2,1(𝕋,ℝ^𝑛)∩𝑇(𝑣,𝑀).

In order to prove this result, we consider the following modified problem:𝑥^ΔΔ(𝑡)−𝑥(𝜎(𝑡))=𝑓𝑡,𝑥𝑥(𝜎∈(𝑡())−𝑥(𝜎(𝑡)),Δ-a.e.𝑡∈𝕋^𝜅2₀,BC),(3.5) where𝑥(𝑠)=⎧⎪⎨⎪⎩‖𝑥𝑀−𝑣(𝑠()𝑠)‖(𝑥−𝑣(𝑠))+𝑣(𝑠)𝑥if‖𝑥−𝑣(𝑠)‖>𝑀(𝑠),otherwise.(3.6)

We define the operator 𝐹∶𝐶(𝕋,ℝ^𝑛)→𝐿^1Δ(𝕋^𝜅0,ℝ^𝑛) by 𝐹(𝑥)(𝑡)∶=𝑓𝑡,𝑥(𝜎(𝑡))−𝑥(𝜎(𝑡)).(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(𝐶(𝕋,ℝ^𝑛)) is bounded. Fix 𝑅>‖𝑣‖₀+‖𝑀‖₀. Let ℎ_𝑅∈𝐿^1Δ(𝕋^𝜅0,[0,∞)) be given by Definition 2.9(iii). Thus, for every 𝑥∈𝐶(𝕋,ℝ^𝑛), ‖𝐹(𝑥)(𝑠)‖≤ℎ_𝑅(𝑠)+𝑅=∶ℎ(𝑠)Δ-a.e.𝑠∈𝕋^𝜅0.(3.8)
To prove the continuity of 𝐹, we consider {𝑥_𝑘}_𝑘∈ℕ a sequence of 𝐶(𝕋,ℝ^𝑛) converging to 𝑥∈𝐶(𝕋,ℝ^𝑛). We already know that for every 𝑘∈ℕ, ‖𝐹𝑥_𝑘(𝑠)‖≤ℎ(𝑠)Δ-a.e.𝑠∈𝕋^𝜅0.(3.9) One can easily check that 𝑥_𝑛(𝑡)→𝑥(𝑡) for all 𝑡∈𝕋. It follows from Definition 2.9(ii) that 𝐹𝑥_𝑘(𝑠)⟶𝐹(𝑥)(𝑠)Δ-a.e.𝑠∈𝕋^𝜅0.(3.10) Theorem 2.8 implies that 𝐹(𝑥_𝑘)→𝐹(𝑥) in 𝐿^1Δ(𝕋^𝜅0,ℝ^𝑛).

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

Proof. Since 𝑥∈𝑊^Δ2,1(𝕋,ℝ^𝑛) (resp., 𝑣∈𝑊^Δ2,1(𝕋,ℝ^𝑛), 𝑀∈𝑊^Δ2,1(𝕋,ℝ)), 𝑥^ΔΔ(𝑡) (resp., 𝑣^ΔΔ(𝑡), 𝑀^ΔΔ(𝑡)), there exists Δ-almost everywhere on 𝕋^𝜅2. Denote 𝐴=𝑡∈𝕋^𝜅2₀∶‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖>𝑀(𝜎(𝑡)).(3.11) By Lemma 2.4(1), (‖𝑥(𝑡)−𝑣(𝑡)‖−𝑀(𝑡))^ΔΔ≥⟨𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡)),𝑥^ΔΔ(𝑡)−𝑣^ΔΔ(𝑡)⟩‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖−𝑀^ΔΔ(𝑡)Δ-a.e.on𝐴.(3.12) We claim that (‖𝑥(𝑡)−𝑣(𝑡)‖−𝑀(𝑡))^ΔΔ>0Δ-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 Δ-almost everywhere on {𝑡∈𝐴∶𝑀(𝜎(𝑡))>0}, (‖𝑥(𝑡)−𝑣(𝑡)‖−𝑀(𝑡))^ΔΔ≥⟨𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡)),𝑓𝑡,𝑥(𝜎(𝑡))−𝑥(𝜎(𝑡))+𝑥(𝜎(𝑡))−𝑣^ΔΔ(𝑡)⟩‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖−𝑀^ΔΔ=⟨(𝑡)𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡)),𝑓𝑡,𝑥(𝜎(𝑡))−𝑣^ΔΔ(𝑡)⟩𝑀(𝜎(𝑡))+‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖−𝑀(𝜎(𝑡))−𝑀^ΔΔ>𝑀(𝜎(𝑡))𝑀_ΔΔ(𝑡)(𝑡)𝑀(𝜎(𝑡))−𝑀^ΔΔ(𝑡)=0,(3.14) and Δ-almost everywhere on {𝑡∈𝐴∶𝑀(𝜎(𝑡))=0}, (‖𝑥(𝑡)−𝑣(𝑡)‖−𝑀(𝑡))^ΔΔ≥⟨𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡)),𝑓𝑡,𝑥(𝜎(𝑡))−𝑥(𝜎(𝑡))+𝑥(𝜎(𝑡))−𝑣^ΔΔ(𝑡)⟩‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖−𝑀^ΔΔ=⟨𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡)),𝑓(𝑡,𝑣(𝜎(𝑡)))−𝑣_ΔΔ(𝑡)(𝑡)⟩‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖+‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖−𝑀^ΔΔ>−𝑀_ΔΔ(𝑡)(𝑡)≥0.(3.15)
Observe that if ‖𝑥(𝑎)−𝑣(𝑎)‖−𝑀(𝑎)>0, ‖𝑥(𝑎)−𝑣(𝑎)‖^Δ≥⟨𝑥(𝑎)−𝑣(𝑎),𝑥^Δ(𝑎)−𝑣^Δ(𝑎)⟩‖𝑥(𝑎)−𝑣(𝑎)‖.(3.16) Indeed, this follows from Proposition 2.3 when 𝑎=𝜎(𝑎). For 𝑎<𝜎(𝑎), ‖𝑥(𝑎)−𝑣(𝑎)‖^Δ=‖𝑥(𝜎(𝑎))−𝑣(𝜎(𝑎))‖−‖𝑥(𝑎)−𝑣(𝑎)‖≥⟨𝑥(𝑎)−𝑣(𝑎),𝑥(𝜎𝜇((𝑎𝑎)))−𝑣(𝜎(𝑎))−(𝑥(𝑎)−𝑣(𝑎))⟩=⟨𝑥(𝑎)−𝑣(𝑎),𝑥𝜇_Δ(𝑎)‖𝑥(𝑎)−𝑣(𝑎)‖(𝑎)−𝑣^Δ(𝑎)⟩‖𝑥(𝑎)−𝑣(𝑎)‖.(3.17) Similarly, if ‖𝑥(𝑏)−𝑣(𝑏)‖−𝑀(𝑏)>0, ‖𝑥(𝜌(𝑏))−𝑣(𝜌(𝑏))‖^Δ≤⟨𝑥(𝑏)−𝑣(𝑏),𝑥^Δ(𝜌(𝑏))−𝑣^Δ(𝜌(𝑏))⟩‖𝑥(𝑏)−𝑣(𝑏)‖.(3.18)
If (BC) denotes (3.2), ‖𝑥(𝑎)−𝑣(𝑎)‖−𝑀(𝑎)=‖𝑥(𝑏)−𝑣(𝑏)‖−𝑀(𝑏).(3.19) We deduce from (3.16), (3.18), and Definition 3.1 that ‖𝑥(𝑎)−𝑣(𝑎)‖−𝑀(𝑎)=‖𝑥(𝑏)−𝑣(𝑏)‖−𝑀(𝑏)≤0,(3.20) or ‖𝑥(≤𝜌(⟨𝑏𝑥))_Δ−𝑣(𝜌(𝑏))‖^Δ−𝑀^Δ(𝜌(𝑏))−‖𝑥(𝑎)−𝑣(𝑎)‖^Δ−𝑀^Δ(𝑎)(𝜌(𝑏))−𝑣^Δ(𝜌(𝑏)),𝑥(𝑏)−𝑣(𝑏)⟩−⟨𝑥_Δ‖𝑥(𝑏)−𝑣(𝑏)‖(𝑎)−𝑣^Δ(𝑎),𝑥(𝑎)−𝑣(𝑎)⟩‖𝑥(𝑎)−𝑣(𝑎)‖−𝑀^Δ=⟨𝑣_Δ(𝜌(𝑏))−𝑀^Δ(𝑎)(𝑎)−𝑣^Δ(𝜌(𝑏)),𝑥(𝑎)−𝑣(𝑎)⟩‖𝑥(𝑎)−𝑣(𝑎)‖−𝑀^Δ≤‖𝑣_Δ(𝜌(𝑏))−𝑀^Δ(𝑎)(𝑎)−𝑣^Δ(𝜌(𝑏))‖−𝑀^Δ(𝜌(𝑏))−𝑀^Δ(𝑎)≤0.(3.21)
If (BC) denotes (3.3), we deduce from (3.16), (3.18), and Definition 3.1 that ‖𝑥(𝑎)−𝑣(𝑎)‖−𝑀(𝑎)≤0,(3.22) or 𝑎₀(‖𝑥≤(𝑎⟨)𝑥−(𝑎𝑣)(−𝑎)𝑣‖(𝑎−),𝑀𝑎₀(𝑎))−𝛾₀‖𝑥(𝑎)−𝑣(𝑎)‖^Δ−𝑀^Δ(𝑎)(𝑥(𝑎)−𝑣(𝑎))−𝛾₀𝑥^Δ(𝑎)−𝑣^Δ(𝑎)⟩−𝑎₀‖𝑥(𝑎)−𝑣(𝑎)‖≤‖‖𝑥𝑀(𝑎)+𝛾₀𝑀^Δ(𝑎)₀≤0,−𝑎₀𝑣(𝑎)−𝛾₀𝑣^Δ(𝑎)‖‖−𝑎₀𝑀(𝑎)+𝛾₀𝑀^Δ(𝑎)(3.23) and similarly ‖𝑥(𝑏)−𝑣(𝑏)‖−𝑀(𝑏)≤0,(3.24) or 𝑎₁(‖𝑥(𝑏)−𝑣(𝑏)‖−𝑀(𝑏))+𝛾₁‖𝑥(𝜌(𝑏))−𝑣(𝜌(𝑏))‖^Δ−𝑀^Δ(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 𝑟(𝑡)=‖𝑥(𝑡)−𝑣(𝑡)‖−𝑀(𝑡) that ‖𝑥(𝑡)−𝑣(𝑡)‖≤𝑀(𝑡) for every 𝑡∈𝕋.

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 𝑇∶𝐶(𝕋,ℝ^𝑛)⟶𝐶(𝕋,ℝ^𝑛)definedby𝑇∶=𝑗₀∘𝐿⁻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 𝑥. By Lemma 3.4, this solution is in 𝑇(𝑣,𝑀). Thus, 𝑥 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 Akın [1] and Stehlík [2] for the Dirichlet and the periodic boundary conditions, respectively.

Corollary 3.5. Let 𝑓∶𝕋^𝜅0×ℝ→ℝ be a Δ-Carathéodory function. Assume that there exists 𝛼,𝛽∈𝑊^Δ2,1(𝕋,ℝ) such that (i)𝛼(𝑡)≤𝛽(𝑡) for Δ-almost every 𝑡∈𝕋,(ii)𝛼^ΔΔ(𝑡)≥𝑓(𝑡,𝛼(𝜎(𝑡))) and 𝛽^ΔΔ(𝑡)≤𝑓(𝑡,𝛽(𝜎(𝑡))) for Δ-almost every 𝑡∈𝕋^𝜅2,(iii)(a)if (BC) denotes (3.2), then 𝛼(𝑎)=𝛼(𝑏), 𝛼^Δ(𝑎)≥𝛼^Δ(𝜌(𝑏)), 𝛽(𝑎)=𝛽(𝑏), and 𝛽^Δ(𝑎)≤𝛽^Δ(𝜌(𝑏)),(b)if (BC) denotes (3.3), then 𝑎₀𝛼(𝑎)−𝛾₀𝛼^Δ(𝑎)≤𝑥₀≤𝑎₀𝛽(𝑎)−𝛾₀𝛽^Δ(𝑎) and 𝑎₁𝛼(𝑏)+𝛾₁𝛼^Δ(𝜌(𝑏))≤𝑥₁≤𝑎₁𝛽(𝑏)+𝛾₁𝛽^Δ(𝜌(𝑏)), then (3.1) has a solution 𝑥∈𝑊^Δ2,1(𝕋,ℝ) such that 𝛼(𝑡)≤𝑥(𝑡)≤𝛽(𝑡) for every 𝑡∈𝕋.

Proof. Observe that ((𝛼+𝛽)/2,(𝛽−𝛼)/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 𝑓∶𝕋^𝜅0×ℝ^𝑛→ℝ^𝑛 be a Δ-Carathéodory function. Assume that there exists a constant 𝑅>0 such that ⟨𝑥,𝑓(𝑡,𝑥)⟩≥0Δ-a.e.𝑡∈𝕋^𝜅2₀,∀𝑥suchthat‖𝑥‖=𝑅.(3.27) Moreover, if (𝐵𝐶) denotes (3.3), assume that ‖𝑥₀‖≤𝑎₀𝑅 and ‖𝑥₁‖≤𝑎₁𝑅, then the system (3.1) has a solution 𝑥∈𝑊^Δ2,1(𝕋,ℝ^𝑛) such that ‖𝑥(𝑡)‖≤𝑅 for every 𝑡∈𝕋.

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

Example 3.7. Consider the system 𝑥^ΔΔ(𝑡)=𝑘(𝑡)(‖𝑥𝑥((𝑎𝜎)(=𝑡))0−,𝜎(𝑡)𝑥𝑐(‖𝑏)−=1)0,Δ-a.e.𝑡∈𝕋^𝜅2₀,(3.28) where 𝑘∈𝐿^1Δ(𝕋₀,ℝ^𝑛⧵{0}) and 𝑐∈ℝ^𝑛⧵{0} is such that ‖𝑐‖max{|𝑎|,|𝑏|}≤1. One can check that (𝑣,𝑀) is a solution-tube of (3.28) with 𝑣(𝑡)=𝑡𝑐, 𝑀(𝑡)≡1. By Theorem 3.2, this problem has at least one solution 𝑥 such that ‖𝑥(𝑡)−𝑡𝑐‖≤1. Observe that there is no 𝑅>0 such that (3.27) is satisfied.

4. Nonlinearity Depending on 𝑥^Δ

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 𝑥^Δ. We consider the problem 𝑎₀𝑥^ΔΔ(𝑡)=𝑓𝑡,𝑥(𝜎(𝑡)),𝑥^Δ(𝑡),Δ-a.e.𝑡∈𝕋^𝜅2₀,𝑥(𝑎)−𝑥^Δ(𝑎)=𝑥₀,𝑎₁𝑥(𝑏)+𝛾₁𝑥^Δ(𝜌(𝑏))=𝑥₁,(4.1) where 𝑎₀,𝑎₁,𝛾₁≥0 and max{𝑎₁,𝛾₁}>0.

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

Definition 4.1. Let (𝑣,𝑀)∈𝑊^Δ2,1(𝕋,ℝ^𝑛)×𝑊^Δ2,1(𝕋,(0,∞)). We say that (𝑣,𝑀) is a solution-tube of (4.1) if (i)for Δ-almost every 𝑡∈{𝑡∈𝕋^𝜅2₀∶𝑡=𝜎(𝑡)}, 𝑥−𝑣(𝑡),𝑓(𝑡,𝑥,𝑦)−𝑣^ΔΔ(𝑡)+‖𝑦−𝑣^Δ(𝑡)‖²≥𝑀(𝑡)𝑀^ΔΔ(𝑡)+𝑀^Δ(𝑡)²,(4.2) for every (𝑥,𝑦)∈ℝ^2𝑛 such that ‖𝑥−𝑣(𝑡)‖=𝑀(𝑡) and ⟨𝑥−𝑣(𝑡),𝑦−𝑣^Δ(𝑡)⟩=𝑀(𝑡)𝑀^Δ(𝑡),(ii)for every 𝑡∈{𝑡∈𝕋^𝜅2₀∶𝑡<𝜎(𝑡)}, 𝑥−𝑣(𝜎(𝑡)),𝑓(𝑡,𝑥,𝑦)−𝑣^ΔΔ(𝑡)≥𝑀(𝜎(𝑡))𝑀^ΔΔ(𝑡),(4.3) for every (𝑥,𝑦)∈ℝ^2𝑛 such that ‖𝑥−𝑣(𝜎(𝑡))‖=𝑀(𝜎(𝑡)),(iii)‖𝑥₀−(𝑎₀𝑣(𝑎)−𝑣^Δ(𝑎))‖≤𝑎₀𝑀(𝑎)−𝑀^Δ(𝑎), ‖𝑥₁−(𝑎₁𝑣(𝑏)+𝛾₁𝑣^Δ(𝜌(𝑏)))‖≤𝑎₁𝑀(𝑏)+𝛾₁𝑀^Δ(𝜌(𝑏)).

If 𝕋 is the real interval [𝑎,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 𝑓∶𝕋^𝜅0×ℝ^2𝑛→ℝ^𝑛 be a Δ-Carathéodory function. Assume that (H1) there exists (𝑣,𝑀)∈𝑊^Δ2,1(𝕋,ℝ^𝑛)×𝑊^Δ2,1(𝕋,(0,∞)) a solution-tube of (4.1),(H2)there exist constants 𝑐,𝑑>0 such that ‖𝑓(𝑡,𝑥,𝑦)‖≤𝑐+𝑑‖𝑦‖ for Δ-almost every 𝑡∈𝕋^𝜅0 and for every (𝑥,𝑦)∈ℝ^2𝑛 such that ‖𝑥−𝑣(𝑡)‖≤𝑀(𝑡), then (4.1) has a solution 𝑥∈𝑊^Δ2,1(𝕋,ℝ^𝑛)∩𝑇(𝑣,𝑀).

To prove this existence result, we consider the following modified problem:𝑥^ΔΔ𝑎₀(𝑡)−𝑥(𝜎(𝑡))=𝑔𝑡,𝑥(𝜎(𝑡)),𝑥^Δ(𝑡),Δ-a.e.𝑡∈𝕋^𝜅2₀,𝑥(𝑎)−𝑥^Δ(𝑎)=𝑥₀,𝑎₁𝑥(𝑏)+𝛾₁𝑥^Δ(𝜌(𝑏))=𝑥₁,(4.4) where 𝑔∶𝕋^𝜅0×ℝ^2𝑛→ℝ^𝑛 is defined by 𝑔(𝑡,𝑥,𝑦)=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝑀(𝜎(𝑡))‖𝑥−𝑣(𝜎(𝑡))‖𝑓𝑡,+1−𝑀(𝜎(𝑡𝑥))(𝜎(𝑡)),𝑦̃(𝑡)−𝑥(𝜎(𝑡))‖𝑥−𝑣(𝜎(𝑡))‖𝑣^ΔΔ(𝑡)+𝑀^ΔΔ(𝑡)‖𝑥−𝑣(𝜎(𝑡))‖(𝑥−𝑣(𝜎(𝑡)))𝑓𝑡,if‖𝑥−𝑣(𝜎(𝑡))‖>𝑀(𝜎(𝑡)),𝑥(𝜎(𝑡)),𝑦̃(𝑡)−𝑥(𝜎(𝑡)),otherwise,(4.5) where 𝑥(𝜎(𝑡)) is defined as in (3.6), 𝑦̃(𝑡)=⎧⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎩𝑦̂(𝑡)+𝑀^Δ(𝑡)−⟨𝑥−𝑣(𝜎(𝑡)),𝑦̂(𝑡)−𝑣^Δ(𝑡)⟩‖𝑥−𝑣(𝜎(𝑡))‖‖𝑥𝑥−−𝑣𝑣((𝜎𝜎((𝑡𝑡))))‖𝑦̂(𝑡)+𝑀_Δif𝑡=𝜎(𝑡),‖𝑥−𝑣(𝜎(𝑡))‖>𝑀(𝜎(𝑡)),(𝑡)𝑀(𝜎(𝑡))1−‖𝑦−𝐾𝑣^Δ(𝑡)‖(𝑥−𝑣(𝜎(𝑡)))𝑦̂(𝑡)if𝑡=𝜎(𝑡),‖𝑥−𝑣(𝜎(𝑡))‖≤𝑀(𝜎(𝑡)),‖𝑦−𝑣^Δ(𝑡)‖>𝐾,𝑦if𝑡<𝜎(𝑡),𝑦̂(𝑡)=𝐾otherwise,‖𝑦−𝑣^Δ(𝑡)‖𝑦−𝑣^Δ𝑦,(𝑡)+𝑣^Δ(𝑡)if‖𝑦−𝑣^Δ(𝑡)‖>𝐾,otherwise,(4.6) with 𝐾>0 a constant which will be fixed later.

Remark 4.3. (1) Remark that ‖‖𝑦𝑥̃(𝑡()𝜎‖(𝑡≤))2−𝐾𝑣+(𝜎‖(𝑣𝑡_Δ))‖≤𝑀(𝜎(𝑡)),(𝑡)‖+||𝑀^Δ(𝑡)||.(4.7)
(2) If ‖𝑥−𝑣(𝜎(𝑡))‖>𝑀(𝜎(𝑡)), ‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖=𝑀(𝜎(𝑡)),𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡)),𝑥^Δ(𝑡)−𝑣^Δ(𝑡)=𝑀(𝜎(𝑡))𝑀^Δ(𝑡),for𝑡=𝜎(𝑡).(4.8)
(3) If ‖𝑥−𝑣(𝜎(𝑡))‖>𝑀(𝜎(𝑡)) and 𝑡=𝜎(𝑡), ‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖²=‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖²+𝑀^Δ(𝑡)²−𝑥(𝑡)−𝑣(𝑡),𝑥^Δ(𝑡)−𝑣^Δ(𝑡)²‖𝑥(𝑡)−𝑣(𝑡)‖².(4.9)
(4) Since 𝑓 is Δ-Carathéodory, by (1), there exists ℎ∈𝐿^1Δ(𝕋^𝜅0,ℝ) such that for every 𝑥,𝑦∈ℝ^𝑛, ‖𝑔(𝑡,𝑥,𝑦)‖≤ℎ(𝑡)Δ-a.e.𝑡∈𝕋^𝜅0.(4.10)
We associate to 𝑔 the operator 𝐺∶𝐶¹(𝕋,ℝ^𝑛)→𝐿^1Δ(𝕋^𝜅0,ℝ^𝑛) defined by 𝐺(𝑥)(𝑡)∶=𝑔𝑡,𝑥(𝜎(𝑡)),𝑥^Δ(𝑡).(4.11)

Proposition 4.4. Let 𝑓∶𝕋^𝜅0×ℝ^2𝑛→ℝ^𝑛 be a Δ-Carathéodory function. Assume that (H1) is satisfied, then 𝐺 is continuous.

Proof. Let {𝑥_𝑘} be a sequence of 𝐶¹(𝕋,ℝ^𝑛) converging to 𝑥∈𝐶¹(𝕋,ℝ^𝑛). It is clear that 𝑥_𝑘(𝑡)⟶𝑥(𝑡),𝑥^Δ𝑘(𝑡)⟶𝑥^Δ(𝑡).(4.12)
On {𝑡∈𝕋∶𝑡<𝜎(𝑡)}, we have 𝑔𝑡,𝑥_𝑘(𝜎(𝑡)),𝑥^Δ𝑘(𝑡)⟶𝑔𝑡,𝑥(𝜎(𝑡)),𝑥^Δ(𝑡),(4.13) since 𝑓 is Δ-Carathéodory.
Similarly, Δ-almost everywhere on {𝑡∈𝕋^𝜅0∶𝑡=𝜎(𝑡),‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖≠𝑀(𝜎(𝑡))}, 𝑔𝑡,𝑥_𝑘(𝜎(𝑡)),𝑥^Δ𝑘(𝑡)⟶𝑔𝑡,𝑥(𝜎(𝑡)),𝑥^Δ(𝑡),(4.14) since 𝑥_Δ𝑘(𝑡)→𝑥^Δ(𝑡) and, for 𝑘 sufficiently large, ‖𝑥_𝑘(𝜎(𝑡))−𝑣(𝜎(𝑡))‖≠𝑀(𝜎(𝑡)).
Denote 𝑆∶={𝑡∈𝕋^𝜅0∶𝑡=𝜎(𝑡)and‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖=𝑀(𝜎(𝑡))} and 𝐼_𝑡={𝑘∈ℕ∶‖𝑥_𝑘(𝜎(𝑡))−𝑣(𝜎(𝑡))‖≤𝑀(𝜎(𝑡))}. As before, it is easy to check that Δ-almost everywhere on {𝑡∈𝑆∶card𝐼_𝑡=∞}, 𝑔𝑡,𝑥_𝑘(𝜎(𝑡)),𝑥^Δ𝑘(𝑡)⟶𝑔𝑡,𝑥(𝜎(𝑡)),𝑥^Δ(𝑡)as𝑘∈𝐼_𝑡goestoinfinity.(4.15) On the other hand, Proposition 2.11 implies that 𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡)),𝑥^Δ(𝑡)−𝑣^Δ(𝑡)=𝑀(𝜎(𝑡))𝑀^Δ(𝑡)Δ-a.e.𝑡∈𝑆.(4.16) So, Δ-almost everywhere on {𝑡∈𝑆∶card(ℕ⧵𝐼_𝑡)=∞}, 𝑥_Δ𝑘(𝑡)=+𝑥^Δ𝑘⎛⎜⎜⎝𝑀(𝑡)_Δ(𝑡)−𝑥^𝑘(𝜎(𝑡))−𝑣(𝜎(𝑡)),𝑥^Δ𝑘(𝑡)−𝑣^Δ(𝑡)‖𝑥_𝑘(𝜎(𝑡))−𝑣(𝜎(𝑡))‖⎞⎟⎟⎠𝑥^𝑘(𝜎(𝑡))−𝑣(𝜎(𝑡))⟶𝑥‖𝑥_𝑘(𝜎(𝑡))−𝑣(𝜎(𝑡))‖_Δ+⎛⎜⎜⎝𝑀(𝑡)_Δ(𝑡)−𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡)),𝑥^Δ(𝑡)−𝑣^Δ(𝑡)=⎧⎪⎪⎪⎨⎪⎪⎪⎩𝑥‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖⎞⎟⎟⎠‖𝑥𝑥((𝜎𝜎((𝑡𝑡))))−−𝑣𝑣((𝜎𝜎((𝑡𝑡))))‖_Δ(𝑡)+𝑀^Δ(𝑡)𝑀(𝜎(𝑡))1−‖𝑥^Δ(𝑡)𝐾−𝑣^Δ(𝑡)‖(𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡)))𝑥if‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖>𝐾,_Δ(𝑡)=𝑥if‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖≤𝐾,_Δ(𝑡),(4.17) as 𝑘∈ℕ⧵𝐼_𝑡 goes to infinity. Thus, Δ-almost everywhere on {𝑡∈𝑆∶card(ℕ⧵𝐼_𝑡)=∞}, 𝑔𝑡,𝑥_𝑘(𝜎(𝑡)),𝑥^Δ𝑘(𝑡)⟶𝑔𝑡,𝑥(𝜎(𝑡)),𝑥^Δ(𝑡)as𝑘∈ℕ⧵𝐼_𝑡goestoinfinity.(4.18)
By Remark 4.3(4), we have ‖𝑔𝑡,𝑥_𝑘(𝜎(𝑡)),𝑥^Δ𝑘(𝑡)‖≤ℎ(𝑡)Δ-a.e.𝑡∈𝕋^𝜅0.(4.19) Theorem 2.8 implies that 𝐺𝑥_𝑘⟶𝐺(𝑥)in𝐿^1Δ𝕋^𝜅0,ℝ^𝑛.(4.20)

Lemma 4.5. Assume (H1), then 𝑥∈𝑇(𝑣,𝑀) for every solution 𝑥 of (4.4).

Proof. Observe that 𝑥^ΔΔ(𝑡) (resp., 𝑣^ΔΔ(𝑡), 𝑀^ΔΔ(𝑡)) exists Δ-almost everywhere on 𝕋^𝜅2, since 𝑥∈𝑊^Δ2,1(𝕋,ℝ^𝑛) (resp., 𝑣∈𝑊^Δ2,1(𝕋,ℝ^𝑛), 𝑀∈𝑊^Δ2,1(𝕋,ℝ)). Denote 𝐴=𝑡∈𝕋^𝜅2₀∶‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖>𝑀(𝜎(𝑡)).(4.21) Observe that by (H1) and Remark 4.3(2), for Δ-almost every 𝑡∈𝐴, ⟨𝑥(𝜎=(𝑡)⟨)𝑥−(𝜎𝑣((𝑡𝜎))(𝑡−))𝑣,𝑥(𝜎^ΔΔ(𝑡)()𝑡),−𝑔𝑣𝑡,^ΔΔ𝑥((𝑡𝜎)⟩(𝑡)),𝑥_Δ=⟨(𝑡)+𝑥(𝜎(𝑡))−𝑣^ΔΔ(𝑡)⟩+𝑥𝑀(𝜎(_ΔΔ𝑡))−𝑣(𝜎(𝑡)),𝑓𝑡,𝑥(𝜎(𝑡)),𝑥^Δ(𝑡)−𝑣^ΔΔ(𝑡)⟩>+⟨‖𝑥(𝜎((𝑡𝑡)))(‖−𝑥𝑣(𝜎(𝜎(𝑡()𝑡)))−‖𝑣−(𝜎𝑀(𝑡()𝜎)‖(𝑡−))𝑀(𝜎(𝑡)))+𝑥𝑀(𝜎(_ΔΔ𝑡))−𝑣(𝜎(𝑡)),𝑓𝑡,𝑥(𝜎(𝑡)),𝑥^Δ(𝑡)−𝑣^ΔΔ(𝑡)⟩≥𝑀(𝑡)(‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖−𝑀(𝜎(𝑡)))_ΔΔ𝑀(𝑡)‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖if𝑡<𝜎(𝑡),_ΔΔ(𝑡)‖𝑥(𝜎(𝑡))−𝑣(𝜎(𝑡))‖+𝑀^Δ(𝑡)²−‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖²if𝑡=𝜎(𝑡).(4.22)
This inequality with Lemma 2.4(1) imply that for Δ-almost every 𝑡∈{𝑡∈𝐴∶𝑡<𝜎(𝑡)}, (‖𝑥(𝑡)−𝑣(𝑡)‖−𝑀(𝑡))^ΔΔ>0.(4.23) Also, (4.22), Lemma 2.4(2), and Remark 4.3(3) imply that for Δ-almost every 𝑡∈{𝑡∈𝐴∶𝑡=𝜎(𝑡)}, (‖𝑥(𝑡)−𝑣(𝑡)‖−𝑀(𝑡))^ΔΔ>𝑀^Δ(𝑡)²−‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖²+‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖²−⟨𝑥(𝑡)−𝑣(𝑡),𝑥_Δ‖𝑥(𝑡)−𝑣(𝑡)‖(𝑡)−𝑣^Δ(𝑡)⟩²‖𝑥(𝑡)−𝑣(𝑡)‖³=‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖²−‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖²+⟨𝑥(𝑡)−𝑣(‖𝑡)𝑥,(𝑥𝑡_Δ)−𝑣(𝑡)‖(𝑡)−𝑣^Δ(𝑡)⟩²−⟨𝑥(𝑡)−𝑣(𝑡),𝑥^Δ(𝑡)−𝑣^Δ(𝑡)⟩²‖𝑥(𝑡)−𝑣(𝑡)‖³=⎧⎪⎪⎪⎨⎪⎪⎪⎩01−𝐾₂if‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖≤𝐾,‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖²‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖²‖𝑥(𝑡)−𝑣(𝑡)‖−⟨𝑥(𝑡)−𝑣(𝑡),𝑥^Δ(𝑡)−𝑣^Δ(𝑡)⟩²‖𝑥(𝑡)−𝑣(𝑡)‖³≥0.if‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖>𝐾,(4.24)
Let us denote 𝑟(𝑡)=‖𝑥(𝑡)−𝑣(𝑡)‖−𝑀(𝑡). Inequalities (4.23) and (4.24) imply that 𝑟^ΔΔ(𝑡)>0 for Δ-almost every 𝑡∈{𝑡∈𝕋^𝜅2₀∶𝑟(𝜎(𝑡))>0}. Arguing as in the proof of Lemma 3.4, we can show that 𝑟(𝑏𝑟)(𝑎≤)0≤0or𝑎₀𝑟(𝑎)−𝑟^Δ(𝑎)≤0,or𝑎₁𝑟(𝑏)+𝛾₁𝑟^Δ(𝜌(𝑏))≤0.(4.25) Theorem 2.19 implies that 𝑥∈𝑇(𝑣,𝑀).

We can now prove the existence theorem of this section.

Proof of Theorem 4.2. We first show that for every solution 𝑥 of (4.4), there exists a constant 𝐾>0 such that ‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖≤𝐾forevery𝑡∈𝕋^𝜅.(4.26) By (H2), Proposition 2.13 and Lemma 4.5, for any 𝑥 solution of (4.4), we have for Δ-almost every 𝑡∈𝕋^𝜅, ‖𝑥^Δ(𝑡)‖≤=‖‖𝑥𝑥^Δ₀(𝑎)‖+_{[𝑎,𝑡)∩𝕋}‖𝑥^ΔΔ(𝑠)‖Δ𝑠≤𝑐₀−𝑎₀𝑥(𝑎)‖+_{[𝑎,𝑡)∩𝕋}‖‖𝑓𝑠,𝑥(𝜎(𝑠)),𝑥_Δ(𝑠)‖‖Δ𝑠≤𝑐₀+_{[𝑎,𝑡)∩𝕋}𝑐+𝑑‖𝑥^Δ(𝑠)‖Δ𝑠≤𝑐₀+_{[𝑎,𝑡)∩𝕋}𝑐+𝑑‖𝑥^Δ(𝑠)−𝑣^Δ(𝑠)‖+‖𝑣^Δ(𝑠)‖+||𝑀^Δ(𝑠)||Δ𝑠≤𝑐₁+_{[𝑎,𝑡)∩𝕋}𝑐+𝑑‖𝑥^Δ(𝑠)−𝑣^Δ(𝑠)‖+‖𝑣^Δ(𝑠)‖+||𝑀^Δ(𝑠)||Δ𝑠+_{[𝑎,𝑡)∩𝕋}𝑑‖𝑥^Δ(𝑠)‖Δ𝑠,(4.27) where 𝑐𝑐₀∶=‖𝑥₀‖+𝑎₀(𝑀(𝑎)+‖𝑣(𝑎)‖),₁∶=𝑐₀+_{[𝑎,𝑏)∩𝕋}𝑐+𝑑2‖𝑣^Δ(𝑠)‖+||𝑀^Δ(𝑠)||Δ𝑠.(4.28) Fix 𝐾≥‖𝑣^Δ‖₀+𝑐₁‖𝑒_𝑑(⋅,𝑎)‖₀. By Theorem 2.5, ‖𝑥^Δ(𝑡)−𝑣^Δ(𝑡)‖≤‖𝑥^Δ(𝑡)‖+‖𝑣^Δ(𝑡)‖≤‖𝑣^Δ(𝑡)‖+𝑐₁𝑒_𝑑(𝑡,𝑎)≤𝐾∀𝑡∈𝕋^𝜅.(4.29)
Consider the operator 𝑇=𝑗₁∘𝐿⁻¹∘𝐺∶𝐶¹(𝕋,ℝ^𝑛)⟶𝐶¹(𝕋,ℝ^𝑛),(4.30) where 𝐿 and 𝑗₁ are defined, respectively, in (2.34) and Proposition 2.16. By Propositions 2.16, 2.20, and 4.4, 𝑇 is continuous. Moreover, 𝑇 is compact. Indeed, by Remark 4.3(4), there exists ℎ∈𝐿^1Δ(𝕋^𝜅0,[0,∞)) such that for every 𝑧∈𝑇(𝐶¹(𝕋,ℝ^𝑛)), there exists 𝑥∈𝐶¹(𝕋,ℝ^𝑛) such that 𝑧=𝑇(𝑥) and ‖𝐺(𝑥)(𝑠)‖≤ℎ(𝑠)Δ-a.e.𝑡∈𝕋^𝜅0.(4.31) Since 𝑗₁ and 𝐿⁻¹ are continuous and affine, they map bounded sets in bounded sets. Thus, there exists a constant 𝑘₀>0 such that ‖𝑧‖₁≤𝑘₀.(4.32) Moreover, 𝑧∈𝑊^Δ2,1(𝕋,ℝ^𝑛) and 𝐿(𝑧)(𝑠)=𝑧^ΔΔ(𝑠)−𝑧(𝜎(𝑠))=𝐺(𝑥)(𝑠)Δ-a.e.𝑠∈𝕋^𝜅2.(4.33) So, for every 𝑡<𝜏 in 𝕋^𝜅, ‖𝑧^Δ(𝑡)−𝑧^Δ(𝜏)‖≤_{[𝑡,𝜏)∩𝕋}‖𝑧(𝜎(𝑠))+𝐺(𝑥)(𝑠)‖Δ𝑠≤_{[𝑡,𝜏)∩𝕋}𝑘₀+ℎ(𝑠)Δ𝑠.(4.34) Thus, 𝑇(𝐶¹(𝕋,ℝ^𝑛)) is bounded and equicontinuous in 𝐶¹(𝕋,ℝ^𝑛). By an analog of the Arzelà-Ascoli Theorem for our context, 𝑇(𝐶¹(𝕋,ℝ^𝑛)) is relatively compact in 𝐶¹(𝕋,ℝ^𝑛).
By the Schauder fixed point theorem, 𝑇 has a fixed point 𝑥 which is a solution of (4.4). By Lemma 4.5, 𝑥∈𝑇(𝑣,𝑀). Also, 𝑥 satisfies (4.26). Hence, 𝑥 is also a solution of (4.1).
Here is an example in which one cannot find a solution-tube of the form (0,𝑅). Moreover, Assumption (A2) stated in the introduction and assumed in [9] is not satisfied.

Example 4.6. Let 𝕋=[0,1]∪{2}∪[3,4] and consider the system 𝑥^ΔΔ(𝑡)=𝑥(𝜎(𝑥𝑡)_Δ)𝑙(𝑡)+‖𝑥^Δ(𝑡)‖+𝑘(𝑡)Δ-a.e.𝑡∈𝕋₀,(0)=𝑥₀,𝑥^Δ(4)=𝑥₁,(4.35) where 𝑥₀,𝑥₁∈ℝ^𝑛 are such that ‖𝑥₀‖≤1, ‖𝑥₁‖≤1, 𝑙∈𝐿^1Δ(𝕋₀,[0,∞)), and 𝑘∈𝐿^1Δ(𝕋₀,ℝ^𝑛) such that 𝑙(1)≥1, 𝑙(2)≥1, ‖𝑘(𝑡)‖≤2, and 𝑙(𝑡)≤𝑟 for Δ-almost every 𝑡∈𝕋₀ for some 𝑟>0.
Take 𝑣(𝑡)≡0 and 𝑀(𝑡)=5−𝑡𝑡if𝑡∈[0,2]∩𝕋,if𝑡∈[3,4]∩𝕋.(4.36) So, 𝑣∈𝑊^Δ2,1(𝕋,ℝ^𝑛), 𝑀∈𝑊^Δ2,1(𝕋,(0,∞)), and 𝑀^Δ(𝑡)=⎧⎪⎨⎪⎩−10if𝑡∈[0,1],1if𝑡=2,𝑀if𝑡∈[3,4_ΔΔ],(𝑡)=⎧⎪⎨⎪⎩01if𝑡∈[0,1),0if𝑡∈{1,2},if𝑡∈[3,4].(4.37) One has ‖𝑥₀‖≤−𝑀^Δ(0) and ‖𝑥₁‖≤𝑀^Δ(4). Observe that Δ-almost everywhere on [0,1)∪[3,4] if ‖𝑥‖=𝑀(𝑡) and ⟨𝑥,𝑦⟩=𝑀(𝑡)𝑀^Δ(𝑡), one has ‖𝑦‖≥1 and ⟨𝑥,𝑓(𝑡,𝑥,𝑦)⟩+‖𝑦‖²=‖𝑥‖²(𝑙(𝑡)+‖𝑦‖)+⟨𝑥,𝑘(𝑡)⟩+‖𝑦‖²≥≥=𝑀1𝑀((𝑡𝑡))(𝑀𝑀(_ΔΔ𝑡)−‖𝑘(𝑡)‖)+1(𝑡)+𝑀^Δ(𝑡)².(4.38) If 𝑡∈{1,2} and ‖𝑥‖=𝑀(𝜎(𝑡))=3, ⟨𝑥,𝑓(𝑡,𝑥,𝑦)⟩=≥≥=‖93𝑀𝑙𝑥(‖(𝑡𝜎²)(−(𝑙𝑡)(3)𝑡‖)𝑀𝑘+(‖𝑡_ΔΔ)𝑦‖‖)+⟨𝑥,𝑘(𝑡)⟩(𝑡).(4.39) So, (𝑣,𝑀) is a solution-tube of (4.35). Moreover, ‖𝑓(𝑡,𝑥,𝑦)‖≤2+5𝑟+5‖𝑦‖Δ-a.e.𝑡∈𝕋₀,all(𝑥,𝑦)suchthat‖𝑥‖≤𝑀(𝑡).(4.40) Theorem 4.2 implies that (4.35) has at least one solution 𝑥 such that ‖𝑥(𝑡)‖≤𝑀(𝑡). Observe that if 𝑥₀≠0 or 𝑥₁≠0, this problem has no solution-tube of the form (0,𝑅) with 𝑅 a positive constant since Definition 4.1(iii) would not be satisfied. This explains why Henderson and Tisdell [9] did not consider the Neumann boundary condition. Notice also that the restriction 𝑑(𝜌(𝑏)−𝑎) in (A2) is not satisfied in this example.

Acknowledgments

The authors would like to thank, respectively, CRSNG-Canada and FQRNT-Québec for their financial support.