Boundary Value Problems for Systems of Second-Order Dynamic

Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2010, Article ID 234015, 26 pages doi:10.1155/2010/234015

Research Article Boundary Value Problems for Systems of Second-Order Dynamic Equations on Time Scales with Δ-Caratheodory ´ Functions M. Frigon1 and H. Gilbert2 1

D´epartement de Math´ematiques et de Statistique, Universit´e de Montr´eal, CP 6128, Succursale Centre-Ville, Montr´eal, QC, Canada H3C 3J7 2 ´ D´epartement de Math´ematiques, Coll`ege Edouard-Montpetit, 945 Chemin de Chambly, Longueuil, QC, Canada J4H 3M6 Correspondence should be addressed to M. Frigon, [email protected] Received 31 August 2010; Accepted 30 November 2010 Academic Editor: J. Mawhin Copyright q 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. We establish the existence of solutions to systems of second-order dynamic equations on time scales with the right member f , a Δ-Carath´eodory 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.

1. Introduction In this paper, we establish existence results for the following systems of second-order dynamic equations on time scales:

xΔΔ t  ft, xσt,

Δ-a.e. t ∈ Ìκ0 , x ∈ BC, 2

  xΔΔ t  f t, xσt, xΔ t , Δ

a0 xa − x a  x0 ,

Δ-a.e. t ∈ Ìκ0 ,

a1 xb  γ1 x

1.1

2

Δ



ρb  x1 .

1.2

2

Abstract and Applied Analysis

Here, Ì is a compact time scale where a  min Ì, b  max Ì, and Ìκ0 is defined in 2.4. The map f : Ìκ0 × Ên → Ên is Δ-Carath´eodory see Definition 2.9, and BC denotes one of the following boundary conditions: 2

xa  xb,   xΔ a  xΔ ρb ,

1.3

a0 xa − γ0 xΔ a  x0 ,   a1 xb  γ1 xΔ ρb  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 Stehl´ık 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 > 0

  such that x, ft, x > 0 if x  R.

1.5

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 Ì a regular time scale i.e., ρt < t < σt or Ì  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 2x, ft, x, y  y2 > 0 if x  R, −2x, y ≤ μty2 , A2 there exist c, d ≥ 0 such that dρb − a < 1 and ft, x, y ≤ c  dy 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 solutiontube 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.

Abstract and Applied Analysis

3

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 Ì does not need to be regular, and we do not require the restriction dρb − 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-Carath´eodory function. This condition is interesting in the case where the points of Ì are not all right scattered. We obtain the existence of solutions to 1.1 and to 1.2 in the Sobolev space WΔ2,1 Ì, Ên . 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 Ì be a compact time scale with a  min Ì < b  max Ì. The forward jump operator σ : Ì → Ì resp., the backward jump operator ρ : Ì → Ì is defined by σt  ⎛ ⎝resp., ρt 

⎧ ⎨inf{s ∈ Ì : s > t} if t < b, ⎩b

if t  b,

⎧ ⎨sup{s ∈ Ì : s < t}

if t > a,

⎩a

if t  a

⎞

2.1

⎠.

We say that t < b is right scattered resp., t > a is left scattered if σt > t resp., ρt < t otherwise, we say that t is right dense resp., left dense. The set of right-scattered points of Ì is at most countable, see 14. We denote it by RÌ : {t ∈ Ì : t < σt}  {ti : i ∈ I},

2.2

for some I ⊂ Æ . The graininess function μ : Ì → 0, ∞ is defined by μt  σt − t. We denote

Ìκ  Ì \

  ρb, b ,

Ì0  Ì \ {b}.

2.3

So, Ìκ  Ì if b is left dense, otherwise Ìκ  Ì0 . Since Ìκ is also a time scale, we denote

Ìκ

2

 Ìκ κ ,

Ìκ0

2

 Ìκ \ {b}. 2

2.4

4

Abstract and Applied Analysis

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., Δxt in the case where the time scale Ì is an interval resp., the discrete set {0, 1, . . . , N}. Definition 2.1. A map f : Ì → Ên is Δ-differentiable at t ∈ Ìκ if there exists f Δ t ∈ Ên called the Δ-derivative of f at t such that for all  > 0, there exists a neighborhood U of t such that      fσt − fs − f Δ tσt − s  ≤ |σt − s|

∀s ∈ U.

2.5

We say that f is Δ-differentiable if f Δ t exists for every t ∈ Ìκ . 2 If f is Δ-differentiable and if f Δ is Δ-differentiable at t ∈ Ìκ , we call f ΔΔ t  f Δ Δ t the second Δ-derivative of f at t. Proposition 2.2. Let f : Ì →

Ên and t ∈ Ìκ.

i If f is Δ-differentiable at t, then f is continuous at t. ii If f is continuous at t ∈ RÌ, then f Δ t 

fσt − ft . μt

2.6

iii The map f is Δ-differentiable at t ∈ Ìκ \ RÌ if and only if ft − fs . s→t t−s

f Δ t  lim Proposition 2.3. If f : Ì →

Ên and g : Ì

→

2.7

Êm are Δ-differentiable at t ∈ Ìκ , then

i if n  m, αf  gΔ t  αf Δ t  g Δ t for every α ∈ Ê, ii if m  1, fgΔ t  gtf Δ t  fσtg Δ t  ftg Δ t  gσtf Δ t, iii if m  1 and gtgσt /  0, then  Δ f gtf Δ t − ftg Δ t , t  g gtgσt iv if W ⊂ Ên is open and h : W → h ◦ fΔ t  h ft, f Δ t.

Ê

2.8

is differentiable at ft ∈ W and t / ∈ RÌ, then

We denote CÌ, Ên  the space of continuous maps on Ì, and we denote C1 Ì, Ên  the space of continuous maps on Ì with continuous Δ-derivative on Ìκ . With the norm x0  max{xt : t ∈ Ì} resp., x1  max{x0 , max{xΔ t : t ∈ Ìκ }}, CÌ, Ên  resp., C1 Ì, Ên  is a Banach space.

Abstract and Applied Analysis

5

We study the second Δ-derivative of the norm of a map. Lemma 2.4. Let x : Ì →

Ên be Δ-differentiable.

1 On {t ∈ Ìκ : xσt > 0 and xΔΔ t exists}, 2

xtΔΔ ≥

xσt, xΔΔ t . xσt

2.9

2 On {t ∈ Ìκ \ RÌ : xσt > 0 and xΔΔ t exists}, 2

xtΔΔ 

Proof. Denote A  {t ∈ A \ RÌ, we have

Ìκ

2

xt, xΔΔ t  xΔ t2 xt, xΔ t2 . − xt xt3

: xσt > 0 and xΔΔ t exists}. By Proposition 2.3, on the set

xtΔ  xtΔΔ

2.10

xt, xΔ t , xt

xt, xΔΔ t  xΔ t2 xt, xΔ t2 xσt, xΔΔ t  ≥ − . xt xσt xt3

2.11

If t ∈ A is such that t < σt  σ 2 t, then by Propositions 2.2 and 2.3, we have

xtΔΔ 

xσtΔ − xtΔ μt



xσt, xΔ σt xσt − xt − μtxσt μt2



xσt, xΔ t  μtxΔΔ t xσt, xt  μtxΔ t xt −  μtxσt μt2 xσt μt2



xσt, xΔΔ t xσt, xt xt −  xσt μt2 xσt μt2

≥

xσt, xΔΔ t . xσt

2.12

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Abstract and Applied Analysis

If t ∈ A is such that t < σt < σ 2 t, then xσtΔ − xtΔ μt  2  x σ t  − xσt xσt − xt  − μσtμt μt2   xσt, x σ 2 t − xσt xσt − xt ≥ − μσtμtxσt μt2

xtΔΔ 



2.13

xσt, xΔ σt xσt − xt − , μtxσt μt2

and we conclude as in the previous case. Let ε > 0. The exponential function eε ·, t0  is defined by  eε t, t0   exp

Ì

t0 ,t∩

   ξε μs Δs ,

2.14

where

ξε h 

⎧ ⎪ ⎨ε,

if h  0,

⎪ ⎩ log1  hε , if h > 0. h

2.15

It is the unique solution to the initial value problem xΔ t  εxt,

xt0   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 ∈ CÌ, Ê. If  yt ≤ α 

Ì

a,t∩

εysΔs for every t ∈ Ì,

2.17

then yt ≤ αeε t, a for every t ∈ Ì.

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

Abstract and Applied Analysis

7

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 ⊂ Ì is said to be Δ-measurable if for every set E ⊂ Ì, m∗1 E  m∗1 E ∩ A  m∗1 E ∩ Ì \ A,

2.19

where  ⎧ m m   ⎪ ⎨inf if b / ∈ E, dk − ck  : E ⊂ ck , dk  with ck , dk ∈ Ì m∗1 E  k1 k1 ⎪ ⎩ ∞ if b ∈ E.

2.20

The Δ-measure on Mm∗1  : {A ⊂ Ì : A is Δ-measurable}, denoted by μΔ , is the restriction of m∗1 to Mm∗1 . So, Ì, Mm∗1 , μΔ  is a complete measurable space. Proposition 2.7 see 14. Let A ⊂ measurable. Moreover, if b / ∈ A,

Ì,

then A is Δ-measurable if and only if A is Lebesgue 

μΔ A  mA 

ti ∈A∩R

σti  − ti ,

Ì

2.21

where m is the Lebesgue measure. The notions of Δ-measurable and Δ-integrable functions f : Ì → Ên can be defined similarly to the general theory of Lebesgue integral. Let E ⊂ Ì be a Δ-measurable set and f : Ì → Ên a Δ-measurable function. We say that f ∈ L1Δ E, Ên  provided  fsΔs < ∞.

2.22

E

The set L1Δ Ì0 , Ên  is a Banach space endowed with the norm  fL1Δ :

Ì0

fsΔs.

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 {fk }k∈Æ be a sequence of functions in L1Δ Ì0 , Ên . Assume that there exists a function f : Ì0 → Ên such that fk t → ft Δ-a.e. t ∈ Ì0 , and there exists a function g ∈ L1Δ Ì0  such that fk t ≤ gt Δ-a.e. t ∈ Ì0 and for every k ∈ Æ , then fk → f in L1Δ Ì0 , Ên . In our existence results, we will consider Δ-Carath´eodory functions.

8

Abstract and Applied Analysis

Definition 2.9. A function f : hold:

Ì0 × Êm

→

Ên

is Δ-Carath´eodory if the following conditions

i t → ft, x is Δ-measurable for every x ∈ Êm , ii x → ft, x is continuous for Δ-almost every t ∈ Ì0 , iii for every r > 0, there exists hr ∈ L1Δ Ì0 , 0, ∞ such that ft, x ≤ hr t for Δ-almost every t ∈ Ì0 and for every x ∈ Êm such that x ≤ r. In this context, there is also a notion of absolute continuity introduced in 16. Definition 2.10. A function f : Ì → Ên is said to be absolutely continuous on Ì if for every  > 0, there exists a δ > 0 such that if {ak , bk }m k1 with ak , bk ∈ Ì is a finite pairwise disjoint family of subintervals satisfying m  bk − ak  < δ,

then

k1

m 

fbk  − fak  < .

2.24

k1

Proposition 2.11 see 17. If f : Ì → Ên is an absolutely continuous function, then the Δ-measure of the set {t ∈ Ì0 \ RÌ : ft  0 and f Δ t /  0} is zero. Proposition 2.12 see 17. If g ∈ L1Δ Ì0 , Ên  and f : Ì →

Ên is the function defined by

 ft :

Ì

a,t∩

2.25

gsΔs,

then f is absolutely continuous and f Δ t  gt Δ-almost everywhere on Ì0 . Proposition 2.13 see 16. A function f : Ì → Ê is absolutely continuous on Ì if and only if f is Δ-differentiable Δ-almost everywhere on Ì0 , f Δ ∈ L1Δ Ì0  and 

Ì

a,t∩

f Δ sΔs  ft − fa,

for every t ∈ Ì.

2.26

We also recall a notion of Sobolev space, see 18, 

Ì, Ê

WΔ1,1 

n



x∈

Ì Ê

L1Δ  0 ,

 : ∃g ∈



Ì Ê

L1Δ  0 ,

n

 such that

Ì0

xsφΔ sΔs



 −

n

Ì0

1 gsφσsΔs for every φ ∈ C0,rd Ì ,

2.27

Abstract and Applied Analysis

9

where  1 C0,rd Ì  φ : Ì →

Ê : φa  0  φb, φ is Δ-differentiable and

φΔ is continuous at right-dense points of

Ì and

its left-sided limits exist at left-dense points of

2.28  .

Ì

A function x ∈ WΔ1,1 Ì, Ên  can be identified to an absolutely continuous map. Proposition 2.14 see 18. Suppose that x ∈ WΔ1,1 Ì, Ê with some g ∈ L1Δ Ì0 , Ê satisfying 2.27, then there exists y : Ì → Ê absolutely continuous such that y  x,

yΔ  g

Δ-a.e. on

Ì0 .

2.29

1 Ì, Ê such that Moreover, if g is Crd Ìκ , Ê, then there exists y ∈ Crd

yx

Δ-a.e. on

Ì0 ,

yΔ  g

Δ-a.e. on

Ìκ.

2.30

Sobolev spaces of higher order can be defined inductively as follows:   WΔ2,1 Ì, Ên   x ∈ WΔ1,1 Ì, Ên  : xΔ ∈ WΔ1,1 Ìκ , Ên  .

2.31

With the norm xW 1,1 : xL1Δ  xΔ L1Δ resp., xW 2,1 : xL1Δ  xΔ L1Δ  xΔΔ L1Δ , Δ

WΔ1,1 Ì, Ên  resp., WΔ2,1 Ì, Ên  is a Banach space.

Δ

Remark 2.15. By Proposition 2.7, we know that μΔ {t} > 0 for every t ∈ RÌ. 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 Ì, Ên  and to consider Δ-Carath´eodory maps f in the case where the time scale is such that μΔ Ì0 \ RÌ  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 : WΔ2,1 Ì, Ên  → C1 Ì, Ên  is continuous. Proposition 2.17. The inclusion j0 : WΔ2,1 Ì, Ên  → CÌ, Ên  is a continuous, compact, linear operator. Proof. Arguing as in the proof of the Arzel`a-Ascoli Theorem, we can show that the inclusion i : C1 Ì, Ên  → CÌ, Ên  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.

10

Abstract and Applied Analysis

Lemma 2.18 see 19. Let f : Ì → Ê be a function with a local maximum at t0 ∈ a, b ∩ Ì. If f ΔΔ ρt0  exists, then f ΔΔ ρt0  ≤ 0 provided t0 is not at the same time left dense and right scattered. Theorem 2.19. Let r ∈ WΔ2,1 Ì be a function such that r ΔΔ t > 0 Δ-almost everywhere on {t ∈

Ìκ0

2

: rσt > 0}. If one of the following conditions holds: i a0 ra − γ0 r Δ a ≤ 0 and a1 rb  γ1 r Δ ρb ≤ 0 (where a0 , a1 , γ0 , and γ1 are defined as in 1.4), ii ra  rb and r Δ a ≥ r Δ ρb,

then rt ≤ 0, for every t ∈ Ì. Proof. If the conclusion is false, there exists t0 ∈ Ì such that rt0   max rt > 0.

Ì

t∈

2.32

In the case where a ≤ ρt0  < t0 < b, then r ΔΔ ρt0  exists since μΔ {ρt0 }  t0 −

ρt0  > 0 and r ∈ WΔ2,1 Ì. By Lemma 2.18, r ΔΔ ρt0  ≤ 0, which is a contradiction since rσρt0   rt0  > 0. If a < ρt0   t0  σt0  < b, there exists t1 > t0 such that rσt > 0 for every t ∈ t0 , t1  ∩ Ì. On the other hand, since rt0  is a maximum, r Δ t0   0, and there exists s ∈ t0 , t1  such that r Δ s ≤ 0, thus, Δ

Δ



0 ≥ r s − r t0  

Ì

t0 ,s∩

r ΔΔ τΔτ > 0,

2.33

by hypothesis and Proposition 2.13, which is a contradiction. Observe that the same argument applies if a  ρt0   t0  σt0  and r Δ t0   0. Notice also that if ρt0   t0  σt0   b and r Δ t0   0, we get a contradiction with an analogous argument for a suitable t1 < t0 . If a ≤ ρt0   t0 < σt0  < b and rt0   rσt0 , we argue as in the first case replacing t0 by σt0  to obtain a contradiction. In the case where a < ρt0   t0 < σt0  ≤ b and rt0  > rσt0 , then r Δ t0  < 0. Since Δ t → r t is continuous, there exists δ > 0 such that r Δ t < 0 on an interval t0 − δ, t0 , which contradicts the maximality of rt0 . Observe that a  t0 and r Δ a < 0 could happen if a  t0  σt0  or if a  t0 < σt0  and ra > rσa. In this case, we get a contradiction if r satisfies condition i. On the other hand, if r satisfies condition ii, then rb  ra > 0 and r Δ ρb < 0. So ρb  b since rb ≥ rρb. This contradicts the maximality of rb. 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 2,1 L : WΔ,B Ì, Ên  −→ L1Δ



Ìκ0 , Ên



defined by Lxt : xΔΔ t − xσt,

2.34

Abstract and Applied Analysis

11

2,1 Ì, Ên  : {x ∈ WΔ2,1 Ì, Ên  : x ∈ BC} with BC denoting the periodic boundary where WΔ,B 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 a0 xa − γ0 xΔ a  0,

2.35

  a1 xb  γ1 xΔ ρb  0. Denote ⎧  2,1 ⎪ if BC denotes 1.3, ⎨ x ∈ WΔ Ì, Ên  : x satisfies 1.3 2,1 n Ì , Ê WΔ,B      0 ⎪ ⎩ x ∈ W 2,1 Ì, Ên  : x satisfies 2.35 if BC denotes 1.4. Δ

2.36

2,1 Notice that WΔ,B Ì, Ên  is a Banach space. Define 0 2,1 L0 : WΔ,B Ì, Ên  −→ L1Δ 0



Ìκ0 , Ên



by L0 xt : xΔΔ t − xσt.

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 Gt, 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 ∈ L1Δ Ìκ0 , Ên ,  xt 

Ì

a,ρb∩

2.38

Gt, shsΔs

is a solution of L0 x  h. So, L0 is bijective and, hence, invertible with L−1 0 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 t − yσt  0 on

Ìκ , 2

a0 ya − γ0 yΔ a  x0 ,

  a1 yb  γ1 yΔ ρb  x1 , 2.39

then L−1  y  L−1 0 . Remark 2.21. We could have considered the operator  : W 2,1 Ì, Ên  −→ L1 L Δ Δ,B



Ìκ0 , Ên



 defined by Lxt : xΔΔ t,

2.40

12

Abstract and Applied Analysis

 is when the boundary condition is 1.4 with suitable constants a0 , a1 , γ0 , γ1 such that L 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 xΔΔ t  ft, xσt,

Δ-a.e. t ∈ Ìκ0 , x ∈ BC, 2

3.1

where BC denotes the periodic boundary condition xa  xb,

3.2

  xΔ a  xΔ ρb , or the Sturm-Liouville boundary condition a0 xa − γ0 xΔ a  x0 ,

3.3

  a1 xb  γ1 xΔ ρb  x1 ,

where a0 , a1 , γ0 , γ1 ≥ 0, max{a0 , γ0 } > 0, and max{a1 , γ1 } > 0. We look for solutions in

WΔ2,1 Ì, Ên . We introduce the notion of solution-tube for the problem 3.1.

Definition 3.1. Let v, M ∈ WΔ2,1 Ì, Ên  × WΔ2,1 Ì, 0, ∞. We say that v, M is a solution-tube of 3.1 if i x − vσt, ft, x − vΔΔ t ≥ MσtMΔΔ t for Δ-almost every t ∈ Ìκ0 and for 2

every x ∈ Ên such that x − vσt  Mσt,

ii vΔΔ t  ft, vσt and MΔΔ t ≤ 0 for Δ-almost every t ∈ Mσt  0,

Ìκ0

2

such that

iii a if BC denotes 3.2, then va  vb, Ma  Mb, and vΔ ρb−vΔ a ≤ MΔ ρb − MΔ a, b if BC denotes 3.3, x0 − a0 va − γ0 vΔ a ≤ a0 Ma − γ0 MΔ a, x1 − a1 vb  γ1 vΔ ρb ≤ a1 Mb  γ1 MΔ ρb. We denote   Tv, M  x ∈ WΔ2,1 Ì, Ên  : xt − vt ≤ Mt ∀t ∈ Ì . We state the main theorem of this section.

3.4

Abstract and Applied Analysis

13

Theorem 3.2. Let f : Ìκ0 × Ên → Ên be a Δ-Carath´eodory function. If v, M ∈ WΔ2,1 Ì, Ên  × WΔ2,1 Ì, 0, ∞ is a solution-tube of 3.1, then the system 3.1 has a solution x ∈ WΔ2,1 Ì, Ên  ∩ Tv, M. In order to prove this result, we consider the following modified problem: xΔΔ t − xσt  ft, xσt − xσt,

Δ-a.e. t ∈ Ìκ0 , 2

x ∈ BC,

3.5

where

xs 

⎧ ⎪ ⎨

Ms x − vs  vs if x − vs > Ms, x − vs

⎪ ⎩x

3.6

otherwise.

We define the operator F : CÌ, Ên  → L1Δ Ìκ0 , Ên  by Fxt : ft, xσt − xσt.

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 FCÌ, Ên  is bounded. Fix R > v0  M0 . Let hR ∈ L1Δ Ìκ0 , 0, ∞ be given by Definition 2.9iii. Thus, for every x ∈ CÌ, Ên , Fxs ≤ hR s  R : hs Δ-a.e. s ∈ Ìκ0 .

3.8

To prove the continuity of F, we consider {xk }k∈Æ a sequence of CÌ, Ên  converging to x ∈ CÌ, Ên . We already know that for every k ∈ Æ , Fxk s ≤ hs Δ-a.e. s ∈ Ìκ0 .

3.9

One can easily check that xn t → xt for all t ∈ Ì. It follows from Definition 2.9ii that Fxk s −→ Fxs Δ-a.e. s ∈ Ìκ0 .

3.10

Theorem 2.8 implies that Fxk  → Fx in L1Δ Ìκ0 , Ên . Lemma 3.4. Under the assumptions of Theorem 3.2, every solution x of 3.5 is in Tv, M. Proof. Since x ∈ WΔ2,1 Ì, Ên  resp., v ∈ WΔ2,1 Ì, Ên , M ∈ WΔ2,1 Ì, Ê, xΔΔ t resp., vΔΔ t, 2 MΔΔ t, there exists Δ-almost everywhere on Ìκ . Denote   2 A  t ∈ Ìκ0 : xσt − vσt > Mσt .

3.11

14

Abstract and Applied Analysis

By Lemma 2.41,

xt − vt − MtΔΔ ≥

xσt − vσt, xΔΔ t − vΔΔ t − MΔΔ t Δ-a.e. on A. xσt − vσt 3.12

We claim that xt − vt − MtΔΔ > 0

Δ-a.e. on A.

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 : Mσt > 0}, xt − vt − MtΔΔ ≥

xσt − vσt, ft, xσt − xσt  xσt − vΔΔ t − MΔΔ t xσt − vσt



xσt − vσt, ft, xσt − vΔΔ t  xσt − vσt − Mσt − MΔΔ t Mσt

>

MσtMΔΔ t − MΔΔ t  0, Mσt

3.14

and Δ-almost everywhere on {t ∈ A : Mσt  0}, xt − vt − MtΔΔ ≥

xσt − vσt, ft, xσt − xσt  xσt − vΔΔ t − MΔΔ t xσt − vσt

xσt − vσt, ft, vσt − vΔΔ t  xσt − vσt − MΔΔ t  xσt − vσt

3.15

> −MΔΔ t ≥ 0. Observe that if xa − va − Ma > 0,

xa − vaΔ ≥

xa − va, xΔ a − vΔ a . xa − va

3.16

Abstract and Applied Analysis

15

Indeed, this follows from Proposition 2.3 when a  σa. For a < σa, xa − vaΔ 

xσa − vσa − xa − va μa

≥

xa − va, xσa − vσa − xa − va μaxa − va



xa − va, xΔ a − vΔ a . xa − va

3.17

Similarly, if xb − vb − Mb > 0,        Δ xb − vb, xΔ ρb − vΔ ρb  x ρb − v ρb  ≤ . xb − vb 

3.18

If BC denotes 3.2, xa − va − Ma  xb − vb − Mb.

3.19

We deduce from 3.16, 3.18, and Definition 3.1 that xa − va − Ma  xb − vb − Mb ≤ 0,

3.20

or 

        x ρb − v ρb Δ − MΔ ρb − xa − vaΔ − MΔ a     xΔ ρb − vΔ ρb , xb − vb ≤ xb − vb   xΔ a − vΔ a, xa − va  Δ  − M ρb − MΔ a xa − va    vΔ a − vΔ ρb , xa − va  Δ   − M ρb − MΔ a  xa − va       ≤ vΔ a − vΔ ρb  − MΔ ρb − MΔ a ≤ 0. −

3.21

If BC denotes 3.3, we deduce from 3.16, 3.18, and Definition 3.1 that xa − va − Ma ≤ 0,

3.22

16

Abstract and Applied Analysis

or   a0 xa − va − Ma − γ0 xa − vaΔ − MΔ a   xa − va, a0 xa − va − γ0 xΔ a − vΔ a  ≤ xa − va 3.23

− a0 Ma  γ0 MΔ a      ≤ x0 − a0 va − γ0 vΔ a  − a0 Ma  γ0 MΔ a ≤ 0, and similarly xb − vb − Mb ≤ 0,

3.24

or        a1 xb − vb − Mb  γ1 x ρb − v ρb Δ − MΔ ρb ≤ 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 rt  xt − vt − Mt that xt − vt ≤ Mt for every t ∈ Ì. 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 : CÌ, Ên  −→ CÌ, Ên  defined by T : 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 Tv, 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 Stehl´ık 2 for the Dirichlet and the periodic boundary conditions, respectively. Corollary 3.5. Let f : WΔ2,1 Ì, Ê such that

Ìκ0 × Ê

→

Ê be a Δ-Carath´eodory function. Assume that there exists α, β ∈

i αt ≤ βt for Δ-almost every t ∈ Ì, ii αΔΔ t ≥ ft, ασt and βΔΔ t ≤ ft, βσt for Δ-almost every t ∈ Ìκ , 2

Abstract and Applied Analysis

17

iii a if BC denotes 3.2, then αa  αb, αΔ a ≥ αΔ ρb, βa  βb, and βΔ a ≤ βΔ ρb, b if BC denotes 3.3, then a0 αa − γ0 αΔ a ≤ x0 ≤ a0 βa − γ0 βΔ a and a1 αb  γ1 αΔ ρb ≤ x1 ≤ a1 βb  γ1 βΔ ρb, then 3.1 has a solution x ∈ WΔ2,1 Ì, Ê such that αt ≤ xt ≤ βt for every t ∈ Ì. 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 : Ìκ0 × Ên → constant R > 0 such that   x, ft, x ≥ 0

Ên

be a Δ-Carath´eodory function. Assume that there exists a

Δ-a.e. t ∈ Ìκ0 , ∀x such that x  R. 2

3.27

Moreover, if BC denotes 3.3, assume that x0  ≤ a0 R and x1  ≤ a1 R, then the system 3.1 has a solution x ∈ WΔ2,1 Ì, Ên  such that xt ≤ R for every t ∈ Ì. Here is an example in which one cannot find a solution-tube of the form 0, R. Example 3.7. Consider the system xΔΔ t  ktxσt − σtc − 1 Δ-a.e. t ∈ Ìκ0 , 2

xa  0,

xb  0,

3.28

where k ∈ L1Δ Ì0 , Ên \ {0} and c ∈ Ên \ {0} is such that c max{|a|, |b|} ≤ 1. One can check that v, M is a solution-tube of 3.28 with vt  tc, Mt ≡ 1. By Theorem 3.2, this problem has at least one solution x such that xt − tc ≤ 1. Observe that there is no R > 0 such that 3.27 is satisfied.

4. Nonlinearity Depending on xΔ 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 xΔ . We consider the problem   xΔΔ t  f t, xσt, xΔ t , a0 xa − xΔ a  x0 ,

Δ-a.e. t ∈ Ìκ0 , 2

  a1 xb  γ1 xΔ ρb  x1 ,

where a0 , a1 , γ1 ≥ 0 and max{a1 , γ1 } > 0. We also introduce a notion of solution-tube for this problem.

4.1

18

Abstract and Applied Analysis

Definition 4.1. Let v, M ∈ WΔ2,1 Ì, Ên  × WΔ2,1 Ì, 0, ∞. We say that v, M is a solution-tube of 4.1 if i for Δ-almost every t ∈ {t ∈ Ìκ0 : t  σt}, 2

! 2    x − vt, f t, x, y − vΔΔ t  y − vΔ t2 ≥ MtMΔΔ t  MΔ t , for every x, y ∈ MtMΔ t,

Ê2n

4.2

such that x − vt  Mt and x − vt, y − vΔ t 

ii for every t ∈ {t ∈ Ìκ0 : t < σt}, 2

!   x − vσt, f t, x, y − vΔΔ t ≥ MσtMΔΔ t,

4.3

for every x, y ∈ Ê2n such that x − vσt  Mσt, iii x0 − a0 va − vΔ a ≤ a0 Ma − MΔ a, x1 − a1 vb  γ1 vΔ ρb ≤ a1 Mb  γ1 MΔ ρb. If Ì 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 : Ìκ0 × Ê2n →

Ên be a Δ-Carath´eodory function. Assume that H1 there exists v, M ∈ WΔ2,1 Ì, Ên  × WΔ2,1 Ì, 0, ∞ a solution-tube of 4.1, H2 there exist constants c, d > 0 such that ft, x, y ≤ c  dy for Δ-almost every t ∈ and for every x, y ∈ Ê2n such that x − vt ≤ Mt,

Ìκ0

then 4.1 has a solution x ∈ WΔ2,1 Ì, Ên  ∩ Tv, M. To prove this existence result, we consider the following modified problem:   xΔΔ t − xσt  g t, xσt, xΔ t , a0 xa − xΔ a  x0 , where g : Ìκ0 × Ê2n →

Δ-a.e. t ∈ Ìκ0 , 2

  a1 xb  γ1 xΔ ρb  x1 ,

4.4

Ên is defined by

⎧   Mσt ⎪ ⎪ f t, xσt, yt  − xσt ⎪ ⎪ ⎪ x − vσt ⎪ ⎪   ⎪   ⎪ ΔΔ ⎪ Mσt M t ⎪ ΔΔ ⎪ v t  x − vσt ⎪  1− x − vσt x − vσt   ⎨ g t, x, y  ⎪ if x − vσt > Mσt, ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ − xσt,  f t, xσt, yt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ otherwise,

4.5

Abstract and Applied Analysis

19

where xσt is defined as in 3.6,   ⎧  x − vσt, yt " − vΔ t x − vσt ⎪ Δ ⎪ ⎪ yt "  M t − ⎪ ⎪ x − vσt x − vσt ⎪ ⎪ ⎪ ⎪ ⎪ if t  σt, x − vσt > Mσt, ⎪ ⎪ ⎪ ⎪   ⎪ ⎪ ⎪ K MΔ t ⎪ ⎪ 1 − yt "  x − vσt ⎪ ⎪ Mσt ⎪ y − vΔ t ⎪ ⎨ if t  σt, x − vσt ≤ Mσt, y − vΔ t > K, yt   ⎪ ⎪ ⎪ ⎪ ⎪ yt " ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ if t < σt, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y ⎪ ⎪ ⎪ ⎪ ⎩ otherwise, ⎧   K ⎪ ⎨ y − vΔ t  vΔ t if y − vΔ t > K, Δ t y − v yt "  ⎪ ⎩y, otherwise, 4.6 with K > 0 a constant which will be fixed later. Remark 4.3. 1 Remark that xσt − vσt ≤ Mσt,

4.7

# # yt  ≤ 2K  vΔ t  #MΔ t#. 2 If x − vσt > Mσt, xσt − vσt  Mσt, ! xσt − vσt, x$Δ t − vΔ t  MσtMΔ t,

for t  σt.

4.8

3 If x − vσt > Mσt and t  σt, !2 xt − vt, x%Δ t − vΔ t 2 . x$Δ t − vΔ t2  x%Δ t − vΔ t2  MΔ t − xt − vt2 

4.9

4 Since f is Δ-Carath´eodory, by 1, there exists h ∈ L1Δ Ìκ0 , Ê such that for every x, y ∈ Ên ,   g t, x, y  ≤ ht Δ-a.e. t ∈ Ìκ0 .

4.10

20

Abstract and Applied Analysis We associate to g the operator G : C1 Ì, Ên  → L1Δ Ìκ0 , Ên  defined by   Gxt : g t, xσt, xΔ t .

Proposition 4.4. Let f : Ìκ0 × satisfied, then G is continuous.

Ê2n

→

Ên

4.11

be a Δ-Carath´eodory function. Assume that (H1) is

Proof. Let {xk } be a sequence of C1 Ì, Ên  converging to x ∈ C1 Ì, Ên . It is clear that

xk t −→ xt,

x%Δ k t −→ x%Δ t.

4.12

On {t ∈ Ì : t < σt}, we have     g t, xk σt, xkΔ t −→ g t, xσt, xΔ t ,

4.13

since f is Δ-Carath´eodory. Similarly, Δ-almost everywhere on {t ∈ Ìκ0 : t  σt, xσt − vσt /  Mσt},     g t, xk σt, xkΔ t −→ g t, xσt, xΔ t ,

4.14

 Mσt. since x$Δ k t → x$Δ t and, for k sufficiently large, xk σt − vσt / Denote S : {t ∈ Ìκ0 : t  σt and xσt − vσt  Mσt} and It  {k ∈ Æ : xk σt − vσt ≤ Mσt}. As before, it is easy to check that Δ-almost everywhere on {t ∈ S : card It  ∞},     g t, xk σt, xkΔ t −→ g t, xσt, xΔ t

as k ∈ It goes to infinity.

4.15

On the other hand, Proposition 2.11 implies that ! xσt − vσt, xΔ t − vΔ t  MσtMΔ t

Δ-a.e. t ∈ S.

4.16

Abstract and Applied Analysis

21

So, Δ-almost everywhere on {t ∈ S : cardÆ \ It   ∞}, x$kΔ t  x%kΔ t ⎛

!⎞   xk σt − vσt, x%kΔ t − vΔ t ⎜ Δ ⎟ xk σt − vσt  ⎝M t − ⎠ xk σt − vσt xk σt − vσt

−→ x%Δ t ⎛

!⎞   xσt − vσt, x%Δ t − vΔ t ⎜ Δ ⎟ xσt − vσt  ⎝M t − ⎠ xσt − vσt xσt − vσt

4.17

⎧   ⎪ MΔ t K ⎪ % Δ t  ⎪ x 1 − xσt − vσt ⎪ ⎪ Mσt xΔ t − vΔ t ⎪ ⎪ ⎪ ⎨ if xΔ t − vΔ t > K,  ⎪ ⎪ ⎪ x%Δ t ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ if xΔ t − vΔ t ≤ K,  x$Δ t, as k ∈ Æ \ It goes to infinity. Thus, Δ-almost everywhere on {t ∈ S : cardÆ \ It   ∞},     g t, xk σt, xkΔ t −→ g t, xσt, xΔ t

as k ∈ Æ \ It goes to infinity.

4.18

By Remark 4.34, we have   g t, xk σt, xkΔ t  ≤ ht Δ-a.e. t ∈ Ìκ0 .

4.19

Theorem 2.8 implies that Gxk  −→ Gx

in L1Δ



Ìκ0 , Ên

 .

4.20

Lemma 4.5. Assume (H1), then x ∈ Tv, M for every solution x of 4.4. Proof. Observe that xΔΔ t resp., vΔΔ t, MΔΔ t exists Δ-almost everywhere on Ìκ , since 2

x ∈ WΔ2,1 Ì, Ên  resp., v ∈ WΔ2,1 Ì, Ên , M ∈ WΔ2,1 Ì, Ê. Denote

  2 A  t ∈ Ìκ0 : xσt − vσt > Mσt .

4.21

22

Abstract and Applied Analysis

Observe that by H1 and Remark 4.32, for Δ-almost every t ∈ A, xσt − vσt, xΔΔ t − vΔΔ t    xσt − vσt, g t, xσt, xΔ t  xσt − vΔΔ t    xσt − vσt, f t, xσt, x$Δ t − vΔΔ t  MΔΔ txσt − vσt − Mσt 4.22

 xσt − vσt − Mσt   > xσt − vσt, f t, xσt, x$Δ t − vΔΔ t  MΔΔ txσt − vσt − Mσt ⎧ ⎨MΔΔ txσt − vσt ≥ ⎩MΔΔ txσt − vσt  MΔ t2 − x$Δ t − vΔ t2

if t < σt, if t  σt.

This inequality with Lemma 2.41 imply that for Δ-almost every t ∈ {t ∈ A : t < σt}, xt − vt − MtΔΔ > 0.

4.23

Also, 4.22, Lemma 2.42, and Remark 4.33 imply that for Δ-almost every t ∈ {t ∈ A : t  σt}, xt − vt − MtΔΔ  Δ 2 M t − x$Δ t − vΔ t2  xΔ t − vΔ t2 > xt − vt − 

xt − vt, xΔ t − vΔ t2 xt − vt3

xΔ t − vΔ t2 − x%Δ t − vΔ t2 xt − vt 

xt − vt, x%Δ t − vΔ t2 − xt − vt, xΔ t − vΔ t2 xt − vt3

⎧ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ if xΔ t − vΔ t ≤ K, ⎪ ⎨     ⎪ xΔ t − vΔ t2 xt − vt, xΔ t − vΔ t2 K2 ⎪ ⎪ 1− Δ − ⎪ ⎪ xt − vt xt − vt3 x t − vΔ t2 ⎪ ⎪ ⎪ ⎩ Δ if x t − vΔ t > K, ≥ 0.

4.24

Abstract and Applied Analysis

23

Let us denote rt  xt − vt − Mt. Inequalities 4.23 and 4.24 imply that

r t > 0 for Δ-almost every t ∈ {t ∈ Ìκ0 : rσt > 0}. Arguing as in the proof of Lemma 3.4, we can show that ΔΔ

2

a0 ra − r Δ a ≤ 0,

ra ≤ 0 or rb ≤ 0

or

4.25

  a1 rb  γ1 r Δ ρb ≤ 0.

Theorem 2.19 implies that x ∈ Tv, 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 xΔ t − vΔ t ≤ K

for every t ∈ Ìκ .

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 ∈ Ìκ , Δ



Δ

x t ≤ x a 

Ì

a,t∩



 x0 − a0 xa   ≤ c0   ≤ c0   ≤ c0   ≤ c1 

Ì

a,t∩

xΔΔ sΔs      f s, xσs, x$Δ s Δs

Ì

a,t∩

c  dx$Δ sΔs # # # # c  d x%Δ s − vΔ s  vΔ s  #MΔ s# Δs 

Ì

a,t∩

Ì

a,t∩

Ì

a,t∩

4.27

# #  # # c  d xΔ s − vΔ s  vΔ s  #MΔ s# Δs dxΔ sΔs,

where c0 : x0   a0 Ma  va,  # #  # # c  d 2vΔ s  #MΔ s# Δs. c1 : c0 

4.28

Ì

a,b∩

Fix K ≥ vΔ 0  c1 ed ·, a0 . By Theorem 2.5, xΔ t − vΔ t ≤ xΔ t  vΔ t ≤ vΔ t  c1 ed t, a ≤ K

∀t ∈ Ìκ .

4.29

24

Abstract and Applied Analysis Consider the operator T  j1 ◦ L−1 ◦ G : C1 Ì, Ên  −→ C1 Ì, Ên ,

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 4.34, there exists h ∈ L1Δ Ìκ0 , 0, ∞ such that for every z ∈ TC1 Ì, Ên , there exists x ∈ C1 Ì, Ên  such that z  Tx and Gxs ≤ hs Δ-a.e. t ∈ Ìκ0 .

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 z1 ≤ k0 .

4.32

Moreover, z ∈ WΔ2,1 Ì, Ên  and Lzs  zΔΔ s − zσs  Gxs Δ-a.e. s ∈ Ìκ . 2

4.33

So, for every t < τ in Ìκ , zΔ t − zΔ τ ≤





Ì

t,τ∩

zσs  GxsΔs ≤

Ì

t,τ∩

k0  hsΔs.

4.34

Thus, TC1 Ì, Ên  is bounded and equicontinuous in C1 Ì, Ên . By an analog of the Arzel`aAscoli Theorem for our context, TC1 Ì, Ên  is relatively compact in C1 Ì, Ên . By the Schauder fixed point theorem, T has a fixed point x which is a solution of 4.4. By Lemma 4.5, x ∈ Tv, 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 Ì  0, 1 ∪ {2} ∪ 3, 4 and consider the system   xΔΔ t  xσt lt  xΔ t  kt Δ-a.e. t ∈ Ì0 , xΔ 0  x0 ,

xΔ 4  x1 ,

4.35

where x0 , x1 ∈ Ên are such that x0  ≤ 1, x1  ≤ 1, l ∈ L1Δ Ì0 , 0, ∞, and k ∈ L1Δ Ì0 , Ên  such that l1 ≥ 1, l2 ≥ 1, kt ≤ 2, and lt ≤ r for Δ-almost every t ∈ Ì0 for some r > 0.

Abstract and Applied Analysis

25

Take vt ≡ 0 and

Mt 

⎧ ⎨5 − t

if t ∈ 0, 2 ∩ Ì,

⎩

if t ∈ 3, 4 ∩ Ì.

t

4.36

So, v ∈ WΔ2,1 Ì, Ên , M ∈ WΔ2,1 Ì, 0, ∞, and ⎧ ⎪ −1 if t ∈ 0, 1, ⎪ ⎪ ⎪ ⎨ if t  2, MΔ t  0 ⎪ ⎪ ⎪ ⎪ ⎩1 if t ∈ 3, 4, ⎧ ⎪ 0 if t ∈ 0, 1, ⎪ ⎪ ⎨ MΔΔ t  1 if t ∈ {1, 2}, ⎪ ⎪ ⎪ ⎩ 0 if t ∈ 3, 4.

4.37

One has x0  ≤ −MΔ 0 and x1  ≤ MΔ 4. Observe that Δ-almost everywhere on 0, 1 ∪ 3, 4 if x  Mt and x, y  MtMΔ t, one has y ≥ 1 and     x, f t, x, y   y2  x2 lt  y  x, kt  y2 ≥ MtMt − kt  1 ≥1

4.38

2   MtMΔΔ t  MΔ t . If t ∈ {1, 2} and x  Mσt  3,     x, f t, x, y   x2 lt  y  x, kt ≥ 9lt − 3kt ≥3

4.39

 MσtMΔΔ t. So, v, M is a solution-tube of 4.35. Moreover,   f t, x, y  ≤ 2  5r  5y

  Δ-a.e. t ∈ Ì0 , all x, y such that x ≤ Mt.

4.40

Theorem 4.2 implies that 4.35 has at least one solution x such that xt ≤ Mt. 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 4.1iii would not be satisfied. This explains why Henderson and

26

Abstract and Applied Analysis

Tisdell 9 did not consider the Neumann boundary condition. Notice also that the restriction dρb − a in A2 is not satisfied in this example.

Acknowledgments The authors would like to thank, respectively, CRSNG-Canada and FQRNT-Qu´ebec for their financial support.

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