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

Authors: M. Frigon; H. Gilbert
Journal: Abstract and Applied Analysis
 (2010)

---

## Abstract

We establish the existence of solutions to systems of second-order
dynamic equations on time scales with the right memberf, a Δ-Carathéodory
function. First, we consider the case where the nonlinearity f does not depend
on the Δ-derivative, xΔ(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 xΔ(t). Our existence results rely on notions
of solution-tube that are introduced in this paper.


---

## Body

## 1. Introduction

In this paper, we establish existence results for the following systems of second-order dynamic equations on time scales:(1.1)xΔΔ(t)=f(t,x(σ(t))),Δ-a.e.t∈T0κ2,x∈(BC),(1.2)xΔΔ(t)=f(t,x(σ(t)),xΔ(t)),Δ-a.e.t∈T0κ2,a0x(a)-xΔ(a)=x0,a1x(b)+γ1xΔ(ρ(b))=x1.
Here, 𝕋 is a compact time scale where a=min𝕋, b=max𝕋, and 𝕋0κ2 is defined in (2.4). The map f:𝕋0κ×ℝn→ℝn is Δ-Carathéodory (see Definition 2.9), and (BC) denotes one of the following boundary conditions:(1.3)x(a)=x(b),xΔ(a)=xΔ(ρ(b)),(1.4)a0x(a)-γ0xΔ(a)=x0,a1x(b)+γ1xΔ(ρ(b))=x1,
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:(1.5)∃R>0suchthat〈x,f(t,x)〉>0if∥x∥=R.The fact that the right member in the system (1.2) depends also on the Δ-derivative, xΔ, 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 existsR>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 existc,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 WΔ2,1(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.Let𝕋 be a compact time scale with a=min𝕋<b=max𝕋. The forward jump operatorσ:𝕋→𝕋 (resp.,  the backward jump operatorρ:𝕋→𝕋) is defined by (2.1)σ(t)={inf⁡{s∈T:s>t}ift<b,bift=b,({inf⁡{s∈T:s>t}ift<b,bift=b,resp.,ρ(t)={sup⁡{s∈T:s<t}ift>a,aift=a{inf⁡{s∈T:s>t}ift<b,bift=b,).
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 (2.2)RT:={t∈T:t<σ(t)}={ti:i∈I},
for some I⊂ℕ. The graininess functionμ:𝕋→[0,∞) is defined by μ(t)=σ(t)-t. We denote (2.3)Tκ=T∖(ρ(b),b],T0=T∖{b}.
So, 𝕋κ=𝕋 if b is left  dense, otherwise 𝕋κ=𝕋0. Since 𝕋κ is also a time scale, we denote(2.4)Tκ2=(Tκ)κ,T0κ2=Tκ2∖{b}.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 xΔ(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 mapf:𝕋→ℝn is Δ-differentiable at t∈𝕋κ if there exists fΔ(t)∈ℝn (called the Δ-derivative of f at t) such that for all ϵ>0, there exists a neighborhood U of t such that
(2.5)‖(f(σ(t))-f(s)-fΔ(t)(σ(t)-s))‖≤ϵ|σ(t)-s|∀s∈U.
We say that f is Δ-differentiable if fΔ(t) exists for every t∈𝕋κ.
Iff is Δ-differentiable and if fΔ is Δ-differentiable at t∈𝕋κ2, we call fΔΔ(t)=(fΔ)Δ(t) the secondΔ-derivative offatt.Proposition 2.2.
Letf:𝕋→ℝn and t∈𝕋κ.  (i)
Iff is Δ-differentiable at t, then f is continuous at t.(ii)
Iff is continuous at t∈R𝕋, then
(2.6)fΔ(t)=f(σ(t))-f(t)μ(t).(iii)
The mapf is Δ-differentiable at t∈𝕋κ∖R𝕋 if and only if(2.7)fΔ(t)=lim⁡s→tf(t)-f(s)t-s.Proposition 2.3.
Iff:𝕋→ℝn and g:𝕋→ℝm are Δ-differentiable at t∈𝕋κ, then  (i)
ifn=m, (αf+g)Δ(t)=αfΔ(t)+gΔ(t) for every α∈ℝ,(ii)
ifm=1, (fg)Δ(t)=g(t)fΔ(t)+f(σ(t))gΔ(t)=f(t)gΔ(t)+g(σ(t))fΔ(t),(iii)
ifm=1 and g(t)g(σ(t))≠0, then
(2.8)(fg)Δ(t)=g(t)fΔ(t)-f(t)gΔ(t)g(t)g(σ(t)),(iv)
ifW⊂ℝn is open and h:W→ℝ is differentiable at f(t)∈W and t∉R𝕋, then (h∘f)Δ(t)=〈h′(f(t)),fΔ(t)〉.We denoteC(𝕋,ℝn) the space of continuous maps on 𝕋, and we denote C1(𝕋,ℝn) the space of continuous maps on 𝕋 with continuous Δ-derivative on 𝕋κ. With the norm ∥x∥0=max{∥x(t)∥:t∈𝕋} (resp., ∥x∥1=max{∥x∥0,max{∥xΔ(t)∥:t∈𝕋κ}}), C(𝕋,ℝn) (resp., C1(𝕋,ℝn)) is a Banach space.We study the secondΔ-derivative of the norm of a map.Lemma 2.4.
Letx:𝕋→ℝn be Δ-differentiable.  (1)
On{t∈𝕋κ2:∥x(σ(t))∥>0 and  xΔΔ(t) exists},
(2.9)∥x(t)∥ΔΔ≥〈x(σ(t)),xΔΔ(t)〉∥x(σ(t))∥.(2)
On{t∈𝕋κ2∖R𝕋:∥x(σ(t))∥>0 and  xΔΔ(t) exists},(2.10)∥x(t)∥ΔΔ=〈x(t),xΔΔ(t)〉+∥xΔ(t)∥2∥x(t)∥-〈x(t),xΔ(t)〉2∥x(t)∥3.Proof.
DenoteA={t∈𝕋κ2:∥x(σ(t))∥>0 and xΔΔ(t) exists}. By Proposition 2.3, on the set A∖R𝕋, we have
(2.11)∥x(t)∥Δ=〈x(t),xΔ(t)〉∥x(t)∥,∥x(t)∥ΔΔ=〈x(t),xΔΔ(t)〉+∥xΔ(t)∥2∥x(t)∥-〈x(t),xΔ(t)〉2∥x(t)∥3≥〈x(σ(t)),xΔΔ(t)〉∥x(σ(t))∥.
If t∈A is such that    t<σ(t)=σ2(t), then by Propositions 2.2 and 2.3, we have
(2.12)∥x(t)∥ΔΔ=∥x(σ(t))∥Δ-∥x(t)∥Δμ(t)=〈x(σ(t)),xΔ(σ(t))〉μ(t)∥x(σ(t))∥-∥x(σ(t))∥-∥x(t)∥μ(t)2=〈x(σ(t)),xΔ(t)+μ(t)xΔΔ(t)〉μ(t)∥x(σ(t))∥-〈x(σ(t)),x(t)+μ(t)xΔ(t)〉μ(t)2∥x(σ(t))∥+∥x(t)∥μ(t)2=〈x(σ(t)),xΔΔ(t)〉∥x(σ(t))∥-〈x(σ(t)),x(t)〉μ(t)2∥x(σ(t))∥+∥x(t)∥μ(t)2≥〈x(σ(t)),xΔΔ(t)〉∥x(σ(t))∥.
If t∈A is such that t<σ(t)<σ2(t), then
(2.13)∥x(t)∥ΔΔ=∥x(σ(t))∥Δ-∥x(t)∥Δμ(t)=∥x(σ2(t))∥-∥x(σ(t))∥μ(σ(t))μ(t)-∥x(σ(t))∥-∥x(t)∥μ(t)2≥〈x(σ(t)),x(σ2(t))-x(σ(t))〉μ(σ(t))μ(t)∥x(σ(t))∥-∥x(σ(t))∥-∥x(t)∥μ(t)2=〈x(σ(t)),xΔ(σ(t))〉μ(t)∥x(σ(t))∥-∥x(σ(t))∥-∥x(t)∥μ(t)2,
and we conclude as in the previous case.Letɛ>0. The exponential function eɛ(·,t0) is defined by(2.14)eɛ(t,t0)=exp⁡(∫[t0,t)∩Tξɛ(μ(s))Δs),
where (2.15)ξɛ(h)={ɛ,ifh=0,log⁡(1+hɛ)h,ifh>0.
It is the unique solution to the initial value problem  (2.16)xΔ(t)=ɛx(t),x(t0)=1.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.
Letα>0, ɛ>0, and y∈C(𝕋,ℝ). If
(2.17)y(t)≤α+∫[a,t)∩Tɛy(s)Δsforeveryt∈T,
then
(2.18)y(t)≤αeɛ(t,a)foreveryt∈T.We recall some notions and results related to the theory ofΔ-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 setA⊂𝕋 is said to be Δ-measurable if for every set E⊂𝕋,
(2.19)m1*(E)=m1*(E∩A)+m1*(E∩(T∖A)),
where
(2.20)m1*(E)={inf⁡{∑k=1m(dk-ck):E⊂⋃k=1m[ck,dk)withck,dk∈T}ifb∉E,∞ifb∈E.
The Δ-measure on ℳ(m1*):={A⊂𝕋:AisΔ-measurable}, denoted by μΔ, is the restriction of m1* to ℳ(m1*). So, (𝕋,ℳ(m1*),μΔ) is a complete measurable space.Proposition 2.7 (see [14]).
LetA⊂𝕋, then A is Δ-measurable if and only if A is Lebesgue measurable. Moreover, if b∉A,
(2.21)μΔ(A)=m(A)+∑ti∈A∩RT(σ(ti)-ti),
where m is the Lebesgue measure.The notions ofΔ-measurable and Δ-integrable functions f:𝕋→ℝn can be defined similarly to the general theory of Lebesgue integral.LetE⊂𝕋 be a Δ-measurable set and f:𝕋→ℝn a Δ-measurable function. We say that f∈LΔ1(E,ℝn) provided (2.22)∫E∥f(s)∥Δs<∞.
The set LΔ1(𝕋0,ℝn) is a Banach space endowed with the norm (2.23)∥f∥LΔ1:=∫T0∥f(s)∥Δs.
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{fk}k∈ℕ be a sequence of functions in LΔ1(𝕋0,ℝn). Assume that  there exists a function f:𝕋0→ℝn such that fk(t)→f(t)Δ-a.e. t∈𝕋0, and there exists a function g∈LΔ1(𝕋0) such that ∥fk(t)∥≤g(t)Δ-a.e. t∈𝕋0 and for every k∈ℕ, then fk→f in LΔ1(𝕋0,ℝn).In our existence results, we will considerΔ-Carathéodory functions.Definition 2.9.
A functionf:𝕋0×ℝm→ℝn is Δ-Carathéodory if the following conditions hold:  (i)
t↦f(t,x) is Δ-measurable for every x∈ℝm,(ii)
x↦f(t,x) is continuous for Δ-almost every t∈𝕋0,(iii)
for everyr>0, there exists hr∈LΔ1(𝕋0,[0,∞)) such that ∥f(t,x)∥≤hr(t)  for Δ-almost every  t∈𝕋0 and for every x∈ℝm such that ∥x∥≤r.In this context, there is also a notion of absolute continuity introduced in [16].Definition 2.10.
A functionf:𝕋→ℝn is said to be absolutely continuous on𝕋 if for every ϵ>0, there exists a δ>0 such that if {[ak,bk)}k=1m with ak,bk∈𝕋 is a finite pairwise disjoint family of subintervals satisfying
(2.24)∑k=1m(bk-ak)<δ,then∑k=1m∥f(bk)-f(ak)∥<ϵ.Proposition 2.11 (see [17]).
Iff:𝕋→ℝn is an absolutely continuous function, then the Δ-measure of the set {t∈𝕋0∖R𝕋:f(t)=0andfΔ(t)≠0} is zero.Proposition 2.12 (see [17]).
Ifg∈LΔ1(𝕋0,ℝn) and f:𝕋→ℝn is the function defined by
(2.25)f(t):=∫[a,t)∩Tg(s)Δs,
then f is absolutely continuous and fΔ(t)=g(t)Δ-almost everywhere on 𝕋0.Proposition 2.13 (see [16]).
A functionf:𝕋→ℝ is absolutely continuous on 𝕋 if and only if f is Δ-differentiable Δ-almost everywhere on 𝕋0, fΔ∈LΔ1(𝕋0) and
(2.26)∫[a,t)∩TfΔ(s)Δs=f(t)-f(a),foreveryt∈T.We also recall a notion of Sobolev space, see [18],(2.27)WΔ1,1(T,Rn)={∫T0x∈LΔ1(T0,Rn):∃g∈LΔ1(T0,Rn)suchthat∫T0x(s)ϕΔ(s)Δs=-∫T0g(s)ϕ(σ(s))Δsforeveryϕ∈C0,rd⁡1(T)∫T0},
where (2.28)C0,rd⁡1(T)={ϕ:T→R:ϕ(a)=0=ϕ(b),ϕisΔ-differentiableandϕΔiscontinuousatright-densepointsofTanditsleft-sidedlimitsexistatleft-densepointsofT}.
A function x∈WΔ1,1(𝕋,ℝn) can be identified to an absolutely continuous map.Proposition 2.14 (see [18]).
Suppose thatx∈WΔ1,1(𝕋,ℝ) with some g∈LΔ1(𝕋0,ℝ) satisfying (2.27), then there exists y:𝕋→ℝ absolutely continuous such that
(2.29)y=x,yΔ=gΔ-a.e.onT0.
Moreover, if g is Crd(𝕋κ,ℝ), then there exists y∈Crd1(𝕋,ℝ) such that
(2.30)y=xΔ-a.e.onT0,yΔ=gΔ-a.e.onTκ.Sobolev spaces of higher order can be defined inductively as follows:(2.31)WΔ2,1(T,Rn)={x∈WΔ1,1(T,Rn):xΔ∈WΔ1,1(Tκ,Rn)}.
With the norm ∥x∥WΔ1,1:=∥x∥LΔ1+∥xΔ∥LΔ1 (resp., ∥x∥WΔ2,1:=∥x∥LΔ1+∥xΔ∥LΔ1+∥xΔΔ∥LΔ1), WΔ1,1(𝕋,ℝn) (resp.,  WΔ2,1(𝕋,ℝn)) is a Banach space.Remark 2.15.
By Proposition2.7, we know that μΔ({t})>0 for every t∈R𝕋. 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 WΔ2,1(𝕋,ℝn)  and to consider Δ-Carathéodory maps f in the case where the time scale is such that μΔ(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 inclusionj1:WΔ2,1(𝕋,ℝn)→C1(𝕋,ℝn) is continuous.Proposition 2.17.
The inclusionj0:WΔ2,1(𝕋,ℝn)↪C(𝕋,ℝn) is a continuous, compact, linear operator.Proof.
Arguing as in the proof of the Arzelà-Ascoli Theorem, we can show that the inclusioni:C1(𝕋,ℝn)↪C(𝕋,ℝn) 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]).
Letf:𝕋→ℝ be a function with a local maximum at t0∈(a,b)∩𝕋. If fΔΔ(ρ(t0)) exists, then fΔΔ(ρ(t0))≤0 provided t0 is not at the same time left dense and right scattered.Theorem 2.19.
Letr∈WΔ2,1(𝕋) be a function such that  rΔΔ(t)>0Δ-almost everywhere on {t∈𝕋0κ2:r(σ(t))>0}. If one of the following conditions holds: (i)
a0r(a)-γ0rΔ(a)≤0 and a1r(b)+γ1rΔ(ρ(b))≤0 (where a0, a1, γ0, and γ1 are defined as in (1.4)),(ii)
r(a)=r(b) and rΔ(a)≥rΔ(ρ(b)), then r(t)≤0, for every t∈𝕋.Proof.
If the conclusion is false, there existst0∈𝕋 such that
(2.32)r(t0)=max⁡t∈Tr(t)>0.
In the case wherea≤ρ(t0)<t0<b, then rΔΔ(ρ(t0)) exists since μΔ({ρ(t0)})=t0-ρ(t0)>0 and r∈WΔ2,1(𝕋). By Lemma 2.18, rΔΔ(ρ(t0))≤0, which is a contradiction since r(σ(ρ(t0)))=r(t0)>0.
Ifa<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, rΔ(t0)=0, and there exists s∈(t0,t1) such that rΔ(s)≤0, thus,
(2.33)0≥rΔ(s)-rΔ(t0)=∫[t0,s)∩TrΔΔ(τ)Δτ>0,
by hypothesis and Proposition 2.13, which is a contradiction. Observe that the same argument applies if a=ρ(t0)=t0=σ(t0) and rΔ(t0)=0. Notice also that if ρ(t0)=t0=σ(t0)=b and rΔ(t0)=0, we get a contradiction with an analogous argument for a suitable t1<t0.
Ifa≤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 wherea<ρ(t0)=t0<σ(t0)≤b and r(t0)>r(σ(t0)), then rΔ(t0)<0. Since t↦rΔ(t) is continuous, there exists δ>0 such that rΔ(t)<0 on an interval (t0-δ,t0), which contradicts the maximality of r(t0).
Observe thata=t0 and rΔ(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 rΔ(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 wheret0=b and r satisfies condition (i) can be treated similarly to the previous case.Finally, we define theΔ-differential operator associated to the problems that we will consider (2.34)L:WΔ,B2,1(T,Rn)⟶LΔ1(T0κ,Rn)definedbyL(x)(t):=xΔΔ(t)-x(σ(t)),
where WΔ,B2,1(𝕋,ℝn):={x∈WΔ2,1(𝕋,ℝn):x∈(BC)} with (BC) denoting the periodic boundary condition (1.3) or the Sturm-Liouville boundary condition (1.4).Proposition 2.20.
The operatorL 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
(2.35)a0x(a)-γ0xΔ(a)=0,a1x(b)+γ1xΔ(ρ(b))=0.
Denote
(2.36)WΔ,B02,1(T,Rn)={{x∈WΔ2,1(T,Rn):xsatisfies(1.3)}if(BC)denotes(1.3),{x∈WΔ2,1(T,Rn):xsatisfies(2.35)}if(BC)denotes(1.4).
Notice that WΔ,B02,1(𝕋,ℝn) is a Banach space. Define
(2.37)L0:WΔ,B02,1(T,Rn)⟶LΔ1(T0κ,Rn)byL0(x)(t):=xΔΔ(t)-x(σ(t)).
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∈LΔ1(𝕋0κ,ℝn),
(2.38)x(t)=∫[a,ρ(b))∩TG(t,s)h(s)Δs
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
(2.39)yΔΔ(t)-y(σ(t))=0onTκ2,a0y(a)-γ0yΔ(a)=x0,a1y(b)+γ1yΔ(ρ(b))=x1,
then L-1=y+L0-1.Remark 2.21.
We could have considered the operator(2.40)L̃:WΔ,B2,1(T,Rn)⟶LΔ1(T0κ,Rn)definedbyL̃(x)(t):=xΔΔ(t),
when the boundary condition is (1.4) with suitable constants a0, a1, γ0, γ1 such that L̃ 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(3.1)xΔΔ(t)=f(t,x(σ(t))),Δ-a.e.t∈T0κ2,x∈(BC),
where (BC) denotes the periodic boundary condition  (3.2)x(a)=x(b),xΔ(a)=xΔ(ρ(b)),
or the Sturm-Liouville boundary condition  (3.3)a0x(a)-γ0xΔ(a)=x0,a1x(b)+γ1xΔ(ρ(b))=x1,
where a0,a1,γ0,γ1≥0, max{a0,γ0}>0, and max{a1,γ1}>0. We look for solutions in WΔ2,1(𝕋,ℝn).We introduce the notion of solution-tube for the problem (3.1).Definition 3.1.
Let(v,M)∈WΔ2,1(𝕋,ℝn)×WΔ2,1(𝕋,[0,∞)). We say that (v,M) is a solution-tube of (3.1) if  (i)
〈x-v(σ(t)),f(t,x)-vΔΔ(t)〉≥M(σ(t))MΔΔ(t) for Δ-almost every  t∈𝕋0κ2 and for every x∈ℝn such that ∥x-v(σ(t))∥=M(σ(t)),(ii)
vΔΔ(t)=f(t,v(σ(t))) and MΔΔ(t)≤0 for Δ-almost every t∈𝕋0κ2 such that M(σ(t))=0,(iii)
(a)
if(BC) denotes (3.2), then v(a)=v(b), M(a)=M(b), and ∥vΔ(ρ(b))-vΔ(a)∥≤MΔ(ρ(b))-MΔ(a),(b)
if(BC) denotes (3.3), ∥x0-(a0v(a)-γ0vΔ(a))∥≤a0M(a)-γ0MΔ(a), ∥x1-(a1v(b)+γ1vΔ(ρ(b)))∥≤a1M(b)+γ1MΔ(ρ(b)). We denote
(3.4)T(v,M)={x∈WΔ2,1(T,Rn):∥x(t)-v(t)∥≤M(t)∀t∈T}.We state the main theorem of this section.Theorem 3.2.
Letf:𝕋0κ×ℝn→ℝn be a Δ-Carathéodory function. If (v,M)∈WΔ2,1(𝕋,ℝn)×WΔ2,1(𝕋,[0,∞)) is a solution-tube of (3.1), then the system (3.1) has a solution x∈WΔ2,1(𝕋,ℝn)∩T(v,M).In order to prove this result, we consider the following modified problem:(3.5)xΔΔ(t)-x(σ(t))=f(t,x¯(σ(t)))-x¯(σ(t)),Δ-a.e.t∈T0κ2,x∈(BC),
where(3.6)x¯(s)={M(s)∥x-v(s)∥(x-v(s))+v(s)if∥x-v(s)∥>M(s),xotherwise.We define the operatorF:C(𝕋,ℝn)→LΔ1(𝕋0κ,ℝn) by (3.7)F(x)(t):=f(t,x¯(σ(t)))-x¯(σ(t)).Proposition 3.3.
Under the assumptions of Theorem3.2, the operator F defined above is continuous and bounded.Proof.
First of all, we show that the setF(C(𝕋,ℝn)) is bounded. Fix R>∥v∥0+∥M∥0. Let hR∈LΔ1(𝕋0κ,[0,∞)) be given by Definition 2.9(iii). Thus, for every x∈C(𝕋,ℝn),
(3.8)∥F(x)(s)∥≤hR(s)+R=:h(s)Δ-a.e.s∈T0κ.
To prove the continuity ofF, we consider {xk}k∈ℕ a sequence of C(𝕋,ℝn) converging to x∈C(𝕋,ℝn). We already know that for every k∈ℕ,
(3.9)∥F(xk)(s)∥≤h(s)Δ-a.e.s∈T0κ.
One can easily check that x¯n(t)→x¯(t) for all t∈𝕋. It follows from Definition 2.9(ii) that
(3.10)F(xk)(s)⟶F(x)(s)Δ-a.e.s∈T0κ.
Theorem 2.8 implies that F(xk)→F(x) in LΔ1(𝕋0κ,ℝn).Lemma 3.4.
Under the assumptions of Theorem3.2, every solution x of (3.5) is in T(v,M).Proof.
Sincex∈WΔ2,1(𝕋,ℝn) (resp., v∈WΔ2,1(𝕋,ℝn), M∈WΔ2,1(𝕋,ℝ)), xΔΔ(t) (resp., vΔΔ(t), MΔΔ(t)), there exists Δ-almost everywhere on 𝕋κ2. Denote
(3.11)A={t∈T0κ2:∥x(σ(t))-v(σ(t))∥>M(σ(t))}.
By Lemma 2.4(1),
(3.12)(∥x(t)-v(t)∥-M(t))ΔΔ≥〈x(σ(t))-v(σ(t)),xΔΔ(t)-vΔΔ(t)〉∥x(σ(t))-v(σ(t))∥-MΔΔ(t)Δ-a.e.onA.
We claim that
(3.13)(∥x(t)-v(t)∥-M(t))ΔΔ>0Δ-a.e.onA.
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},
(3.14)(∥x(t)-v(t)∥-M(t))ΔΔ≥〈x(σ(t))-v(σ(t)),f(t,x¯(σ(t)))-x¯(σ(t))+x(σ(t))-vΔΔ(t)〉∥x(σ(t))-v(σ(t))∥-MΔΔ(t)=〈x¯(σ(t))-v(σ(t)),f(t,x¯(σ(t)))-vΔΔ(t)〉M(σ(t))+∥x(σ(t))-v(σ(t))∥-M(σ(t))-MΔΔ(t)>M(σ(t))MΔΔ(t)M(σ(t))-MΔΔ(t)=0,
and Δ-almost everywhere on {t∈A:M(σ(t))=0},
(3.15)(∥x(t)-v(t)∥-M(t))ΔΔ≥〈x(σ(t))-v(σ(t)),f(t,x¯(σ(t)))-x¯(σ(t))+x(σ(t))-vΔΔ(t)〉∥x(σ(t))-v(σ(t))∥-MΔΔ(t)=〈x(σ(t))-v(σ(t)),f(t,v(σ(t)))-vΔΔ(t)〉∥x(σ(t))-v(σ(t))∥+∥x(σ(t))-v(σ(t))∥-MΔΔ(t)>-MΔΔ(t)≥0.
Observe that if∥x(a)-v(a)∥-M(a)>0,
(3.16)∥x(a)-v(a)∥Δ≥〈x(a)-v(a),xΔ(a)-vΔ(a)〉∥x(a)-v(a)∥.
Indeed, this follows from Proposition 2.3 when a=σ(a). For a<σ(a),
(3.17)∥x(a)-v(a)∥Δ=∥x(σ(a))-v(σ(a))∥-∥x(a)-v(a)∥μ(a)≥〈x(a)-v(a),x(σ(a))-v(σ(a))-(x(a)-v(a))〉μ(a)∥x(a)-v(a)∥=〈x(a)-v(a),xΔ(a)-vΔ(a)〉∥x(a)-v(a)∥.
Similarly, if ∥x(b)-v(b)∥-M(b)>0,
(3.18)∥x(ρ(b))-v(ρ(b))∥Δ≤〈x(b)-v(b),xΔ(ρ(b))-vΔ(ρ(b))〉∥x(b)-v(b)∥.
If(BC) denotes (3.2),
(3.19)∥x(a)-v(a)∥-M(a)=∥x(b)-v(b)∥-M(b).
We deduce from (3.16), (3.18), and Definition 3.1 that
(3.20)∥x(a)-v(a)∥-M(a)=∥x(b)-v(b)∥-M(b)≤0,
or
(3.21)(∥x(ρ(b))-v(ρ(b))∥Δ-MΔ(ρ(b)))-(∥x(a)-v(a)∥Δ-MΔ(a))≤〈xΔ(ρ(b))-vΔ(ρ(b)),x(b)-v(b)〉∥x(b)-v(b)∥-〈xΔ(a)-vΔ(a),x(a)-v(a)〉∥x(a)-v(a)∥-(MΔ(ρ(b))-MΔ(a))=〈vΔ(a)-vΔ(ρ(b)),x(a)-v(a)〉∥x(a)-v(a)∥-(MΔ(ρ(b))-MΔ(a))≤∥vΔ(a)-vΔ(ρ(b))∥-(MΔ(ρ(b))-MΔ(a))≤0.
If(BC) denotes (3.3), we deduce from (3.16), (3.18), and Definition 3.1 that
(3.22)∥x(a)-v(a)∥-M(a)≤0,
or
(3.23)a0(∥x(a)-v(a)∥-M(a))-γ0(∥x(a)-v(a)∥Δ-MΔ(a))≤〈x(a)-v(a),a0(x(a)-v(a))-γ0(xΔ(a)-vΔ(a))〉∥x(a)-v(a)∥-a0M(a)+γ0MΔ(a)≤‖x0-(a0v(a)-γ0vΔ(a))‖-a0M(a)+γ0MΔ(a)≤0,
and similarly
(3.24)∥x(b)-v(b)∥-M(b)≤0,
or
(3.25)a1(∥x(b)-v(b)∥-M(b))+γ1(∥x(ρ(b))-v(ρ(b))∥Δ-MΔ(ρ(b)))≤0.
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 Theorem3.2.
A solution of (3.5) is a fixed point of the operator
(3.26)T:C(T,Rn)⟶C(T,Rn)definedbyT:=j0∘L-1∘F,
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 wheren=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.
Letf:𝕋0κ×ℝ→ℝ be a Δ-Carathéodory function. Assume that  there exists α,β∈WΔ2,1(𝕋,ℝ) such that  (i)
α(t)≤β(t) for Δ-almost every t∈𝕋,(ii)
αΔΔ(t)≥f(t,α(σ(t))) and βΔΔ(t)≤f(t,β(σ(t))) for Δ-almost every t∈𝕋κ2,(iii)
(a)
if(BC) denotes (3.2), then α(a)=α(b), αΔ(a)≥αΔ(ρ(b)), β(a)=β(b), and βΔ(a)≤βΔ(ρ(b)),(b)
if(BC) denotes (3.3), then a0α(a)-γ0αΔ(a)≤x0≤a0β(a)-γ0βΔ(a)  and a1α(b)+γ1αΔ(ρ(b))≤x1≤a1β(b)+γ1βΔ(ρ(b)), then    (3.1) has a solution x∈WΔ2,1(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.Theorem3.2 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.
Letf:𝕋0κ×ℝn→ℝn be a Δ-Carathéodory function. Assume that there exists a constant R>0 such that
(3.27)〈x,f(t,x)〉≥0Δ-a.e.t∈T0κ2,∀xsuchthat∥x∥=R.
Moreover, if (BC) denotes (3.3), assume that ∥x0∥≤a0R and ∥x1∥≤a1R, then  the system (3.1) has a solution x∈WΔ2,1(𝕋,ℝn) 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(3.28)xΔΔ(t)=k(t)(∥x(σ(t))-σ(t)c∥-1)Δ-a.e.t∈T0κ2,x(a)=0,x(b)=0,
where k∈LΔ1(𝕋0,ℝn∖{0}) and c∈ℝn∖{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 onxΔ

In this section, we study more general systems of second-order dynamic equations on time scales. Indeed, we allow the nonlinearityf to depend also on xΔ. We consider the problem (4.1)xΔΔ(t)=f(t,x(σ(t)),xΔ(t)),Δ-a.e.t∈T0κ2,a0x(a)-xΔ(a)=x0,a1x(b)+γ1xΔ(ρ(b))=x1,
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)∈WΔ2,1(𝕋,ℝn)×WΔ2,1(𝕋,(0,∞)). We  say that (v,M) is a solution-tube of (4.1) if  (i)
forΔ-almost every t∈{t∈𝕋0κ2:t=σ(t)},
(4.2)〈x-v(t),f(t,x,y)-vΔΔ(t)〉+∥y-vΔ(t)∥2≥M(t)MΔΔ(t)+(MΔ(t))2,
for every (x,y)∈ℝ2n such that ∥x-v(t)∥=M(t) and 〈x-v(t),y-vΔ(t)〉=M(t)MΔ(t),(ii)
for everyt∈{t∈𝕋0κ2:t<σ(t)},
(4.3)〈x-v(σ(t)),f(t,x,y)-vΔΔ(t)〉≥M(σ(t))MΔΔ(t),
for every (x,y)∈ℝ2n such that ∥x-v(σ(t))∥=M(σ(t)),(iii)
∥x0-(a0v(a)-vΔ(a))∥≤a0M(a)-MΔ(a), ∥x1-(a1v(b)+γ1vΔ(ρ(b)))∥≤a1M(b)+γ1MΔ(ρ(b)).If𝕋 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.
Letf:𝕋0κ×ℝ2n→ℝn be a Δ-Carathéodory function. Assume  that (H1)
there exists(v,M)∈WΔ2,1(𝕋,ℝn)×WΔ2,1(𝕋,(0,∞)) a solution-tube of (4.1),(H2)
there exist constantsc,d>0 such that  ∥f(t,x,y)∥≤c+d∥y∥  for Δ-almost every t∈𝕋0κ and for every (x,y)∈ℝ2n such that ∥x-v(t)∥≤M(t), then  (4.1) has a solution x∈WΔ2,1(𝕋,ℝn)∩T(v,M).To prove this existence result, we consider the following modified problem:(4.4)xΔΔ(t)-x(σ(t))=g(t,x(σ(t)),xΔ(t)),Δ-a.e.t∈T0κ2,a0x(a)-xΔ(a)=x0,a1x(b)+γ1xΔ(ρ(b))=x1,
where g:𝕋0κ×ℝ2n→ℝn is defined by (4.5)g(t,x,y)={M(σ(t))∥x-v(σ(t))∥f(t,x¯(σ(t)),ỹ(t))-x¯(σ(t))+(1-M(σ(t))∥x-v(σ(t))∥)(vΔΔ(t)+MΔΔ(t)∥x-v(σ(t))∥(x-v(σ(t))))if∥x-v(σ(t))∥>M(σ(t)),f(t,x¯(σ(t)),ỹ(t))-x¯(σ(t)),otherwise,
where x¯(σ(t)) is defined as in (3.6), (4.6)ỹ(t)={ŷ(t)+(MΔ(t)-〈x-v(σ(t)),ŷ(t)-vΔ(t)〉∥x-v(σ(t))∥)(x-v(σ(t))∥x-v(σ(t))∥)ift=σ(t),∥x-v(σ(t))∥>M(σ(t)),ŷ(t)+MΔ(t)M(σ(t))(1-K∥y-vΔ(t)∥)(x-v(σ(t)))ift=σ(t),∥x-v(σ(t))∥≤M(σ(t)),∥y-vΔ(t)∥>K,ŷ(t)ift<σ(t),yotherwise,ŷ(t)={K∥y-vΔ(t)∥(y-vΔ(t))+vΔ(t)if∥y-vΔ(t)∥>K,y,otherwise,
with K>0 a constant which will be fixed later.Remark 4.3.
(1) Remark that(4.7)∥x¯(σ(t))-v(σ(t))∥≤M(σ(t)),∥ỹ(t)∥≤2K+∥vΔ(t)∥+|MΔ(t)|.
(2) If∥x-v(σ(t))∥>M(σ(t)),
(4.8)∥x¯(σ(t))-v(σ(t))∥=M(σ(t)),〈x¯(σ(t))-v(σ(t)),xΔ̃(t)-vΔ(t)〉=M(σ(t))MΔ(t),fort=σ(t).
(3) If∥x-v(σ(t))∥>M(σ(t)) and t=σ(t),
(4.9)∥xΔ̃(t)-vΔ(t)∥2=∥xΔ̂(t)-vΔ(t)∥2+(MΔ(t))2-〈x(t)-v(t),xΔ̂(t)-vΔ(t)〉2∥x(t)-v(t)∥2.
(4) Sincef is Δ-Carathéodory, by (1), there exists h∈LΔ1(𝕋0κ,ℝ) such that for every x,y∈ℝn,
(4.10)∥g(t,x,y)∥≤h(t)Δ-a.e.t∈T0κ.
We associate tog the operator G:C1(𝕋,ℝn)→LΔ1(𝕋0κ,ℝn) defined by
(4.11)G(x)(t):=g(t,x(σ(t)),xΔ(t)).Proposition 4.4.
Letf:𝕋0κ×ℝ2n→ℝn be a Δ-Carathéodory function. Assume that   (H1) is satisfied, then G is continuous.Proof.
Let{xk} be a sequence of C1(𝕋,ℝn) converging to x∈C1(𝕋,ℝn). It is clear that
(4.12)x¯k(t)⟶x¯(t),xΔ̂k(t)⟶xΔ̂(t).
On{t∈𝕋:t<σ(t)}, we have
(4.13)g(t,xk(σ(t)),xkΔ(t))⟶g(t,x(σ(t)),xΔ(t)),
since f is Δ-Carathéodory.
Similarly,Δ-almost everywhere on {t∈𝕋0κ:t=σ(t),∥x(σ(t))-v(σ(t))∥≠M(σ(t))},
(4.14)g(t,xk(σ(t)),xkΔ(t))⟶g(t,x(σ(t)),xΔ(t)),
since xΔ̃k(t)→xΔ̃(t) and, for k sufficiently large, ∥xk(σ(t))-v(σ(t))∥≠M(σ(t)).
DenoteS:={t∈𝕋0κ:t=σ(t)and∥x(σ(t))-v(σ(t))∥=M(σ(t))} and It={k∈ℕ:∥xk(σ(t))-v(σ(t))∥≤M(σ(t))}. As before, it is easy to check that Δ-almost everywhere on {t∈S:cardIt=∞},
(4.15)g(t,xk(σ(t)),xkΔ(t))⟶g(t,x(σ(t)),xΔ(t))ask∈Itgoestoinfinity.
On the other hand, Proposition 2.11 implies that
(4.16)〈x(σ(t))-v(σ(t)),xΔ(t)-vΔ(t)〉=M(σ(t))MΔ(t)Δ-a.e.t∈S.
So, Δ-almost everywhere on {t∈S:card(ℕ∖It)=∞},
(4.17)xkΔ̃(t)=xkΔ̂(t)+(MΔ(t)-〈xk(σ(t))-v(σ(t)),xkΔ̂(t)-vΔ(t)〉∥xk(σ(t))-v(σ(t))∥)(xk(σ(t))-v(σ(t))∥xk(σ(t))-v(σ(t))∥)⟶xΔ̂(t)+(MΔ(t)-〈x(σ(t))-v(σ(t)),xΔ̂(t)-vΔ(t)〉∥x(σ(t))-v(σ(t))∥)(x(σ(t))-v(σ(t))∥x(σ(t))-v(σ(t))∥)={xΔ̂(t)+MΔ(t)M(σ(t))(1-K∥xΔ(t)-vΔ(t)∥)(x(σ(t))-v(σ(t)))if∥xΔ(t)-vΔ(t)∥>K,xΔ̂(t)if∥xΔ(t)-vΔ(t)∥≤K,=xΔ̃(t),
as k∈ℕ∖It goes to infinity. Thus, Δ-almost everywhere on {t∈S:card(ℕ∖It)=∞},
(4.18)g(t,xk(σ(t)),xkΔ(t))⟶g(t,x(σ(t)),xΔ(t))ask∈N∖Itgoestoinfinity.
By Remark4.3(4), we have
(4.19)∥g(t,xk(σ(t)),xkΔ(t))∥≤h(t)Δ-a.e.t∈T0κ.
Theorem 2.8 implies that
(4.20)G(xk)⟶G(x)inLΔ1(T0κ,Rn).Lemma 4.5.
Assume  (H1), thenx∈T(v,M) for every solution x of (4.4).Proof.
Observe thatxΔΔ(t) (resp., vΔΔ(t), MΔΔ(t)) exists Δ-almost everywhere on 𝕋κ2, since x∈WΔ2,1(𝕋,ℝn) (resp.,  v∈WΔ2,1(𝕋,ℝn), M∈WΔ2,1(𝕋,ℝ)). Denote
(4.21)A={t∈T0κ2:∥x(σ(t))-v(σ(t))∥>M(σ(t))}.
Observe that by (H1) and Remark 4.3(2), for Δ-almost every t∈A,
(4.22)〈x(σ(t))-v(σ(t)),xΔΔ(t)-vΔΔ(t)〉=〈x(σ(t))-v(σ(t)),g(t,x(σ(t)),xΔ(t))+x(σ(t))-vΔΔ(t)〉=〈x¯(σ(t))-v(σ(t)),f(t,x¯(σ(t)),xΔ̃(t))-vΔΔ(t)〉+MΔΔ(t)(∥x(σ(t))-v(σ(t))∥-M(σ(t)))+∥x(σ(t))-v(σ(t))∥-M(σ(t))>〈x¯(σ(t))-v(σ(t)),f(t,x¯(σ(t)),xΔ̃(t))-vΔΔ(t)〉+MΔΔ(t)(∥x(σ(t))-v(σ(t))∥-M(σ(t)))≥{MΔΔ(t)∥x(σ(t))-v(σ(t))∥ift<σ(t),MΔΔ(t)∥x(σ(t))-v(σ(t))∥+(MΔ(t))2-∥xΔ̃(t)-vΔ(t)∥2ift=σ(t).
This inequality  with Lemma2.4(1) imply that for Δ-almost every t∈{t∈A:t<σ(t)},
(4.23)(∥x(t)-v(t)∥-M(t))ΔΔ>0.
Also, (4.22), Lemma 2.4(2), and Remark 4.3(3) imply that for Δ-almost every t∈{t∈A:t=σ(t)},
(4.24)(∥x(t)-v(t)∥-M(t))ΔΔ>(MΔ(t))2-∥xΔ̃(t)-vΔ(t)∥2+∥xΔ(t)-vΔ(t)∥2∥x(t)-v(t)∥-〈x(t)-v(t),xΔ(t)-vΔ(t)〉2∥x(t)-v(t)∥3=∥xΔ(t)-vΔ(t)∥2-∥xΔ̂(t)-vΔ(t)∥2∥x(t)-v(t)∥+〈x(t)-v(t),xΔ̂(t)-vΔ(t)〉2-〈x(t)-v(t),xΔ(t)-vΔ(t)〉2∥x(t)-v(t)∥3={0if∥xΔ(t)-vΔ(t)∥≤K,(1-K2∥xΔ(t)-vΔ(t)∥2)(∥xΔ(t)-vΔ(t)∥2∥x(t)-v(t)∥-〈x(t)-v(t),xΔ(t)-vΔ(t)〉2∥x(t)-v(t)∥3)if∥xΔ(t)-vΔ(t)∥>K,≥0.
Let us denoter(t)=∥x(t)-v(t)∥-M(t). Inequalities (4.23) and (4.24) imply that rΔΔ(t)>0 for Δ-almost every t∈{t∈𝕋0κ2:r(σ(t))>0}. Arguing as in the proof of Lemma 3.4, we can show that
(4.25)r(a)≤0ora0r(a)-rΔ(a)≤0,r(b)≤0ora1r(b)+γ1rΔ(ρ(b))≤0.
Theorem 2.19 implies that x∈T(v,M).We can now prove the existence theorem of this section.Proof of Theorem4.2.
We first show that for every solutionx of (4.4), there exists a constant K>0 such that
(4.26)∥xΔ(t)-vΔ(t)∥≤Kforeveryt∈Tκ.
By (H2), Proposition 2.13 and Lemma 4.5, for any x solution of (4.4), we have for Δ-almost every  t∈𝕋κ,
(4.27)∥xΔ(t)∥≤∥xΔ(a)∥+∫[a,t)∩T∥xΔΔ(s)∥Δs=∥x0-a0x(a)∥+∫[a,t)∩T‖f(s,x(σ(s)),xΔ̃(s))‖Δs≤c0+∫[a,t)∩Tc+d∥xΔ̃(s)∥Δs≤c0+∫[a,t)∩Tc+d(∥xΔ̂(s)-vΔ(s)∥+∥vΔ(s)∥+|MΔ(s)|)Δs≤c0+∫[a,t)∩Tc+d(∥xΔ(s)-vΔ(s)∥+∥vΔ(s)∥+|MΔ(s)|)Δs≤c1+∫[a,t)∩Td∥xΔ(s)∥Δs,
where
(4.28)c0:=∥x0∥+a0(M(a)+∥v(a)∥),c1:=c0+∫[a,b)∩Tc+d(2∥vΔ(s)∥+|MΔ(s)|)Δs.
Fix K≥∥vΔ∥0+c1∥ed(·,a)∥0. By Theorem 2.5,
(4.29)∥xΔ(t)-vΔ(t)∥≤∥xΔ(t)∥+∥vΔ(t)∥≤∥vΔ(t)∥+c1ed(t,a)≤K∀t∈Tκ.
Consider the operator(4.30)T=j1∘L-1∘G:C1(T,Rn)⟶C1(T,Rn),
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∈LΔ1(𝕋0κ,[0,∞)) such that for every z∈T(C1(𝕋,ℝn)), there exists x∈C1(𝕋,ℝn) such that z=T(x) and
(4.31)∥G(x)(s)∥≤h(s)Δ-a.e.t∈T0κ.
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
(4.32)∥z∥1≤k0.
Moreover, z∈WΔ2,1(𝕋,ℝn) and
(4.33)L(z)(s)=zΔΔ(s)-z(σ(s))=G(x)(s)Δ-a.e.s∈Tκ2.
So, for every t<τ in 𝕋κ,
(4.34)∥zΔ(t)-zΔ(τ)∥≤∫[t,τ)∩T∥z(σ(s))+G(x)(s)∥Δs≤∫[t,τ)∩Tk0+h(s)Δs.
Thus, T(C1(𝕋,ℝn)) is bounded and equicontinuous in C1(𝕋,ℝn). By an analog of the Arzelà-Ascoli Theorem for our context, T(C1(𝕋,ℝn)) is relatively compact in C1(𝕋,ℝn).
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.
Let𝕋=[0,1]∪{2}∪[3,4] and consider the system
(4.35)xΔΔ(t)=x(σ(t))(l(t)+∥xΔ(t)∥)+k(t)Δ-a.e.t∈T0,xΔ(0)=x0,xΔ(4)=x1,
where x0,x1∈ℝn are such that ∥x0∥≤1, ∥x1∥≤1, l∈LΔ1(𝕋0,[0,∞)),  and k∈LΔ1(𝕋0,ℝn) 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.
Takev(t)≡0 and
(4.36)M(t)={5-tift∈[0,2]∩T,tift∈[3,4]∩T.
So, v∈WΔ2,1(𝕋,ℝn), M∈WΔ2,1(𝕋,(0,∞)), and
(4.37)MΔ(t)={-1ift∈[0,1],0ift=2,1ift∈[3,4],MΔΔ(t)={0ift∈[0,1),1ift∈{1,2},0ift∈[3,4].
One has ∥x0∥≤-MΔ(0) and ∥x1∥≤MΔ(4). Observe that Δ-almost everywhere on [0,1)∪[3,4]  if ∥x∥=M(t) and 〈x,y〉=M(t)MΔ(t), one has ∥y∥≥1 and
(4.38)〈x,f(t,x,y)〉+∥y∥2=∥x∥2(l(t)+∥y∥)+〈x,k(t)〉+∥y∥2≥M(t)(M(t)-∥k(t)∥)+1≥1=M(t)MΔΔ(t)+(MΔ(t))2.
If t∈{1,2} and ∥x∥=M(σ(t))=3,
(4.39)〈x,f(t,x,y)〉=∥x∥2(l(t)+∥y∥)+〈x,k(t)〉≥9l(t)-3∥k(t)∥≥3=M(σ(t))MΔΔ(t).
So, (v,M) is a solution-tube of (4.35). Moreover,
(4.40)∥f(t,x,y)∥≤2+5r+5∥y∥Δ-a.e.t∈T0,all(x,y)suchthat∥x∥≤M(t).
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.



---

*Source: 234015-2011-02-06.xml*