Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2010, Article ID 650827, 20 pages
doi:10.1155/2010/650827
Research Article
Existence Theorems for First-Order Equations on
Time Scales with Δ-Carathéodory Functions
Hugues Gilbert
Département de Mathématiques, Collège Édouard-Montpetit, 945 Chemin de Chambly, Longueuil,
QC, Canada J4H 3M6
Correspondence should be addressed to Hugues Gilbert, hugues.gilbert@college-em.qc.ca
Received 14 June 2010; Accepted 3 December 2010
Academic Editor: Kanishka Perera
Copyright q 2010 Hugues 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.
This paper concerns the existence of solutions for two kinds of systems of first-order equations on
time scales. Existence results for these problems are obtained with new notions of solution tube
adapted to these systems. We consider the general case where the right member of the system is
Δ-Carathéodory and, hence, not necessarily continuous.
1. Introduction
In this paper, we establish existence results for the following systems:
Δ-a.e. t ∈ Ì0 , x ∈ BC;
1.1
Δ-a.e. t ∈ Ì0 , xa xb.
1.2
xΔ t ft, xσt,
xΔ t ft, xt,
Here, Ì is an arbitrary compact time scale where we note a min Ì, b max Ì, and Ì0 Ì \ {b}. Moreover, f : Ì0 × Ên → Ên is a Δ-Carathéodory function and BC denotes one of
the following boundary conditions:
xa x0 ,
1.3
xa xb.
1.4
In the literature, this kind of problem was mainly treated for n 1. The existence of
extremal solutions was established in 1, 2. Moreover, some existence results in the particular
2
Advances in Difference Equations
case where the time scale is a discrete set difference equation were obtained with the lower
and upper solution method as in 3, 4. In this paper, we introduce a new notion which
generalizes to systems of first-order equations on time scales the notions of lower and upper
solutions. This notion called solution tube for system 1.1 resp., 1.2 will be useful to get
a new existence result for 1.1 resp., 1.2. Our notion of solution tube is in the spirit of the
notion of solution tube for systems of first-order differential equations introduced in 5. Our
notion is new even in the case of systems of first-order difference equations. In this case, we
generalize to the systems a result of 3 for equation 1.2.
Some papers treat the existence of solutions to systems of first-order equations on time
scales. Existence results are obtained in 6, 7 under hypothesis different from ours. However,
some particular cases obtained in 7 are corollaries of our existence result for problem 1.1.
Also, our existence results treat the case where the right members in 1.1 and 1.2 are ΔCarathéodory functions which are more general than continuous functions used for systems
studied in 6, 7. Let us mention that existence of extremal solutions for infinite systems of
first-order equations of time scale with Δ-Carathéodory functions is established in 8.
This paper is organized as follows. The third section presents an existence result for
the problem 1.1, and in the last section, we obtain an existence theorem for the problem
1.2. We start with some notations, definitions, and results on time scales equations which
are used throughout this paper.
2. Preliminaries and Notations
In this section, we establish notations, definitions, and results on equations on time scales
which are used throughout this paper. The reader may consult 9–11 and the references
therein to find the proofs and to get a complete introduction to this subject.
Let Ì be a time scale, which is a closed nonempty subset of Ê. For t ∈ Ì, we define
the forward jump operator σ : Ì → Ì resp., the backward jump operator ρ : Ì → Ì by σt inf{s ∈ Ì : s > t} resp., by ρt sup{s ∈ Ì : s < t}. We suppose that σt t if t is
the maximum of Ì and that ρt t if t is the minimum of Ì. We say that t is right-scattered
resp., left-scattered if σt > t resp., if ρt < t. We say that t is isolated if it is right-scattered
and left-scattered. Also, if t < sup Ì and σt t, we say that t is right-dense. If t > inf Ì and
ρt t, we say that t is left dense. Points that are right dense and left dense are called dense.
The graininess function μ : Ì → 0, ∞ is defined by μt σt − t.
If Ì has a left-scattered maximum, then Ìκ Ì \ {sup Ì}. Otherwise, Ìκ Ì. In
summary,
⎧
⎨Ì \ ρ sup Ì , sup Ì if sup Ì < ∞,
Ìκ ⎩
Ì
if sup Ì ∞.
2.1
If Ì is bounded, then Ì0 ⊂ Ìκ where Ì0 : Ì \ {max Ì}.
Definition 2.1. Assume f : Ì → Ên is a function and let t ∈ Ìκ . We say that f is Δ-differentiable
at t if there exists a vector f Δ t ∈ Ên such that for all > 0, there exists a neighborhood U of
t, where
fσt − fs − f Δ tσt − s ≤ |σt − s|
2.2
Advances in Difference Equations
3
for every s ∈ U. We call f Δ t the Δ-derivative of f at t. If f is Δ-differentiable at t for every
t ∈ Ìκ , then f Δ : Ìκ → Ên is called the Δ-derivative of f on Ìκ .
Theorem 2.2. Assume f : Ì →
Ên is a function and let t ∈ Ìκ . Then, we have the following.
i If f is Δ-differentiable at t, then f is continuous at t.
ii If f is continuous at t and if t is right-scattered, then f is Δ-differentiable at t and
f Δ t fσt − ft
.
μt
2.3
iii If t is right dense, then f is Δ-differentiable at t if and only if lims → t ft − fs/t − s
exists in Ên . In this case, f Δ t lims → t ft − fs/t − s.
iv If f is Δ-differentiable at t, then fσt ft μtf Δ t.
Theorem 2.3. If f, g : Ì →
Ê are Δ-differentiable at t ∈ Ìκ, then
i f g is Δ-differentiable at t and f gΔ t f Δ t g Δ t,
ii αf is Δ-differentiable at t for every α ∈ Ê and αfΔ t αf Δ t,
iii fg is Δ-differentiable at t and fgΔ t f Δ tgt fσtg Δ t ftg Δ t f Δ tgσt,
iv If gtgσt /
0, then f/g is Δ-differentiable at t and
Δ
f
f Δ tgt − ftg Δ t
.
t g
gtgσt
2.4
The next result is an adaptation of Theorem 1.87 in 10.
Theorem 2.4. Let W be an open set of Ên and t ∈ Ì a right-dense point. If g : Ì → Ên is Δdifferentiable at t and if f : W → Ê is differentiable at gt ∈ W, then f ◦ g is Δ-differentiable at t
and f ◦ gΔ t f gt, g Δ t
.
Example 2.5. Assume x : Ì → Ên is Δ-differentiable at t ∈ Ì. We know that · :
Ên \ {0} → 0, ∞ is differentiable. If t σt, by the previous theorem, we have xtΔ xt, xΔ t
/xt.
Definition 2.6. A function f : Ì → Ên is called rd-continuous provided it is continuous at
right-dense points in Ì and its left-sided limits exist finite at left-dense points in Ì. The
set of rd-continuous functions f : Ì → Ên is denoted by Crd Ì, Ên . The set of functions
f : Ì → Ên that are Δ-differentiable and whose Δ-derivative is rd-continuous is denoted by
1
Ì, Ên .
Crd
It is possible to define a theory of measure and integration for an arbitrary bounded
time scale Ì where a min Ì < max Ì b. We recall the notion of Δ-measure as introduced in
chapter 5 of 9. Define F1 the family of intervals of Ì of the form
c, d {t ∈ Ì : c ≤ t < d},
2.5
4
Advances in Difference Equations
where c, d ∈ Ì and c ≤ d. The interval c, c is understood as the empty set. An outer measure
m∗1 on PÌ is defined as follows: for E ⊂ Ì,
⎧ m
m
⎪
⎨inf
dk − ck : E ⊂ ck , dk with ck , dk ∈ F1
m∗1 E k1
k1
⎪
⎩
∞
if b /
∈ E,
2.6
if b ∈ E.
Definition 2.7. 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.
2.7
M m∗1 : {A ⊂ Ì : A is Δ-measurable}.
2.8
Now, denote
The Lebesgue Δ-measure on Mm∗1 , denoted by μΔ , is the restriction of m∗1 to Mm∗1 . We get a
complete measurable space with Ì, Mm∗1 , μΔ .
With this definition of complete measurable space for a bounded time scale Ì, we can
define the notions of Δ-measurability and Δ-integrability for functions f : Ì → Ê following
the same ideas of the theory of Lebesgue integral. We omit here these definitions that an
interested reader can find in 12. We only present definitions and results which will be useful
for this paper.
Definition 2.8. Let E ⊂ Ì be a Δ-measurable set and f :
We say that f ∈ L1Δ E provided
Ì
→
Ê be a Δ-measurable function.
fsΔs < ∞.
2.9
E
We say that a Δ-measurable function f : Ì →
Ên is in the set L1Δ E, Ên provided
fi sΔs < ∞
2.10
E
for each of its components fi : Ì →
Ê.
Proposition 2.9. Assume f ∈ L1Δ E, Ên . Then,
fsΔs ≤ fsΔs.
E
2.11
E
Many results of integration theory are established for measurable functions f : X → Ê
where X, τ, μ is a complete measurable space. These results are in particular true for the
measurable space Ì, Mm∗1 , μΔ . We recall two results of the theory of integration adapted
to our situation.
Advances in Difference Equations
5
Theorem 2.10 Lebesgue-dominated convergence Theorem. Let {fn }n∈Æ be a sequence of
functions in L1Δ Ì0 . If there exists a function f : Ì0 → Ê such that fn t → ftΔ-a.e. t ∈ Ì0 and
if there exists a function g ∈ L1Δ Ì0 such that fn t ≤ gtΔ-a.e. t ∈ Ì0 and for every n ∈ Æ , then
fn → f in L1Δ Ì0 .
Theorem 2.11. The set L1Δ Ì0 is a Banach space endowed with the norm fL1 : Ì0 |fs|Δs.
Δ
The following results were obtained in 12 where a useful relation between the Δmeasure on Ì resp., Δ-integral on Ì and the Lebesgue measure μL on Ê resp., Lebesgue
integral on Ê is established. To establish these results, the authors of 12 prove that the set
of right-scattered points of Ì is at most countable. Then, there are a set of index I ⊂ Æ and a
set {ti }i∈I ⊂ Ì such that RÌ : {t ∈ Ì : t < σt} {ti }i∈I .
Proposition 2.12. Let A ⊂ Ì. Then, A is Δ-measurable if and only if A is Lebesgue measurable. In
such a case, the following properties hold for every Δ-measurable set A.
i If b /
∈ A, then
μΔ A Σi∈IA σti − ti μL A.
2.12
∈ A and A has no right-scattered points. Here, IA {i ∈
ii μΔ A μL A if and only if b /
I : ti ∈ RÌ ∩ A}.
To establish the relation between Δ-integration on Ì and Lebesgue integration on a
real compact interval, the function f : Ì → Ê is extended to a, b in the following way.
ft :
⎧
⎨ft,
if t ∈ Ì,
⎩ft ,
if t ∈ ti , σti , for an i ∈ IÌ.
i
2.13
∈ E and let E E ∪ i∈IE ti , σti .
Theorem 2.13. Let E ⊂ Ì be a Δ-measurable set such that b /
Let f : Ì → Ê be a Δ-measurable function and f : a, b → Ê its extension on a, b. Then, f is
In such a case, one has
Δ-integrable on E if and only if f is Lebesgue integrable on E.
fsΔs E
Also, the function f :
a, b → Ê by
ft :
Ì
→
E
fsds.
2.14
Ê can be extended on a, b in another way. Define ft :
⎧
⎪
⎨ft,
if t ∈ Ì,
fσti − fti ⎪
⎩fti t − ti ,
μti if t ∈ ti , σti , for an i ∈ IÌ.
2.15
Definition 2.14. A function f : Ì → Ê is said to be absolutely continuous on Ì if for every > 0,
there exists a δ > 0 such that if {ak , bk }nk1 with ak , bk ∈ Ì is a finite pairwise disjoint family
of subintervals of Ì satisfying Σnk1 bk − ak < δ, then Σnk1 |fbk − fak | < .
6
Advances in Difference Equations
The three following results were obtained in 13.
Lemma 2.15. If f is differentiable at t ∈ a, b ∩ Ì, then f is Δ-differentiable at t and f Δ t f t.
Theorem 2.16. Consider a function f : Ì → Ê and its extension f : a, b →
absolutely continuous on Ì if and only if f is absolutely continuous on a, b.
Theorem 2.17. A function f : Ì → Ê is absolutely continuous on
differentiable Δ-almost everywhere on Ì0 , f Δ ∈ L1Δ Ì0 and
f Δ sΔs ft − fa,
a,t∩Ì
Ê.
Then, f is
Ì
if and only if f is Δ-
for every t ∈ Ì.
2.16
We also recall the Banach Lemma.
Lemma 2.18. Let E be a Banach space and u : a, b → E an absolutely continuous function, then
the measure of the set {t ∈ a, b : ut 0 and u t /
0} is zero.
Using the previous results, we now prove two propositions that will be used later.
Proposition 2.19. Let g ∈ L1Δ Ì0 and G : Ì →
Ê the function defined by
Gt :
gsΔs.
2.17
for every t ∈ Ì.
2.18
a,t∩Ì
Then, GΔ t gtΔ-almost everywhere on Ì0 .
Proof. By Theorem 2.13, remark that
Gt a,t
g sds
We can also check that for ti right scattered,
Gt a,ti g sds ti ,t
g sds if t ∈ ti , σti .
2.19
for every t ∈ a, b.
2.20
Obviously,
Gt a,t
g sds
Advances in Difference Equations
7
It is well known that G t g t almost everywhere on a, b. By Lemma 2.15, we have that
GΔ t G t g t gt except on a set A ⊂ Ì0 such that μL A 0. Since G is continuous,
for ti ∈ RÌ ∩ Ì0 right scattered,
Δ
G ti Gσti − Gti gti μti 2.21
by Theorem 2.2ii. Then, G is Δ-differentiable at t ∈ Ì0 \A∪RÌ∩ Ì0 . By Proposition 2.12ii,
μΔ A \ RÌ ∩ Ì0 0 and, then, the proposition is proved.
Proposition 2.20. Let u : Ì → Ê be an absolutely continuous function, then the Δ-measure of the
0} is zero.
set {t ∈ Ì0 \ RÌ0 : ut 0 and uΔ t /
Proof. It suffices to consider the extension u of u on a, b and successively apply
Theorem 2.16, Lemmas 2.18, 2.15, and the Proposition 2.12ii.
We recall a notion of Sobolev’s space of functions defined on a bounded time scale Ì
where a min Ì < max Ì b. The definition and the result are from 14.
Definition 2.21. We say that a function u : Ì → Ê belongs to WΔ1,1 Ì if and only if u ∈ L1Δ Ì0 and there exists a function g : Ìκ → Ê such that g ∈ L1Δ Ì0 and
Ì0
usφΔ sΔs −
Ì0
gsφσsΔs for every φ ∈ C10,rd Ì
2.22
with
1
1
C0,rd
Ì : f : Ì −→ Ê : f ∈ Crd
Ì, fa 0 fb .
We say that a function f : Ì →
WΔ1,1 Ì.
2.23
Ên is in the set WΔ1,1 Ì, Ên if each of its components fi are in
Theorem 2.22. Suppose that u ∈ WΔ1,1 Ì and that 2.22 holds for a function g ∈ L1Δ Ì0 . Then,
there exists a unique function x : Ì → Ê absolutely continuous such that Δ-almost everywhere on
Ì0 , one has x u and xΔ g. Moreover, if g is rd-continuous on Ì0, then there exists a unique
1
function x ∈ Crd
Ì such that x u Δ-almost everywhere on Ì0 and such that xΔ g on Ì0 .
By the previous theorem, we can conclude that u ∈ WΔ1,1 Ì is also continuous.
Remark 2.23. If x ∈ WΔ1,1 Ì, Ên , then its components xi are in WΔ1,1 Ì. By Theorems
2.22 and 2.17, x is Δ-differentiable Δ-a.e. on Ì. From Example 2.5, we obtain xtΔ xt, xΔ t
/xt Δ-a.e. on {t ∈ Ì : t σt}.
We prove two maximum principles that will be useful to get a priori bounds for
solutions of systems considered in this paper.
8
Advances in Difference Equations
Lemma 2.24. Let r ∈ WΔ1,1 Ì such that r Δ t < 0 Δ-a.e. t ∈ {t ∈
following conditions holds,
Ì0 : rσt > 0}. If one of the
i ra ≤ 0,
ii ra ≤ rb,
then rt ≤ 0, for every t ∈ Ì.
Proof. Suppose the conclusion is false. Then, there exists t0 ∈ Ì such that rt0 maxt∈Ìrt >
0, since r is continuous on Ì. If t0 > ρt0 , then r Δ ρt0 exists, since μΔ {ρt0 } t0 −ρt0 > 0
and because r ∈ WΔ1,1 Ì. Then,
r
rt0 − r ρt0 ≥ 0,
ρt0 t0 − ρt0 Δ
2.24
which is a contradiction since rt0 rσρt0 > 0. If t0 ρt0 > a, then there exists an
interval t1 , ρt0 such that rσt > 0 for all t ∈ t1 , ρt0 ∩ Ì. Thus,
0 ≤ rt0 − rt1 r ρt0 − rt1 t1 ,ρt0 ∩Ì
r Δ sΔs < 0
2.25
by hypothesis and by Theorem 2.17. Hence, we get a contradiction. The case t0 a is
impossible if hypothesis i holds and if ra ≤ rb, we must have ra rb. If we take
t0 b, by using previous steps of this proof, one can check that rb ≤ 0 and, then, the lemma
is proved.
Lemma 2.25. Let r ∈ WΔ1,1 Ì be a function such that r Δ t > 0 Δ-a.e. on {t ∈
ra ≥ rb, then rt ≤ 0, for every t ∈ Ì.
Ìk
: rt > 0} if
Proof. If there exists t ∈ Ì such that rt > 0, then there exists a t0 ∈ Ì such that rt0 maxt∈Ìrt > 0. If t0 < b and t0 < σt0 , then μΔ {t0 } σt0 − t0 > 0. Since r ∈ WΔ1,1 Ì, r Δ
exists Δ-almost everywhere. Then, we must have
r Δ t0 rσt0 − rt0 ≤ 0,
σt0 − t0
2.26
which contradicts the hypothesis of the lemma. If t0 < b and t0 σt0 , there exists an interval
t0 , t1 such that rt > 0 for every t ∈ t0 , t1 ∩ Ì. Then,
0<
t0 ,t1 ∩Ì
r Δ sΔs rt1 − rt0 ,
2.27
by Theorem 2.17, which contradicts the fact that rt0 is a maximum. If t0 b, then by
hypothesis, we must have ra rb. Thus, we can take t0 a, and by using the previous
steps of this proof, one can check that ra ≤ 0. Then, the lemma is proved.
Advances in Difference Equations
9
Definition 2.26. For > 0, the exponential function e ·, t0 :
unique solution of the initial value problem
xΔ t xt,
Ì
→
Ê
may be defined as the
xt0 1.
2.28
More explicitly, the exponential function e ·, t0 is given by the formula
t
e t, t0 exp
ξ μs Δs ,
2.29
t0
where for h ≥ 0, we define ξ h as
ξ h ⎧
⎪
⎨,
if h 0,
log1 h
⎪
⎩
,
h
otherwise.
2.30
As direct consequences of Proposition 2.19 and Theorem 2.3, we get the following
results.
Proposition 2.27. If g ∈ L1Δ Ì0 , Ên , the function x : Ì →
Ên defined by
xt e1 a, t x0 a,t∩Ì
2.31
e1 s, agsΔs
is a solution of the problem
xΔ t xσt gt,
Δ-a.e. t ∈ Ì0 ,
2.32
xa x0 .
Proposition 2.28. If g ∈ L1Δ Ì0 , Ên , then the function x : Ì →
1
xt e1 t, a
1
e1 b, a − 1
Ên defined by
a,b∩Ì
gse1 s, aΔs a,t∩Ì
gse1 s, aΔs
2.33
is a solution of the problem
xΔ t xσt gt,
xa xb.
Δ-a.e. t ∈ Ì0 ,
2.34
10
Advances in Difference Equations
Proposition 2.29. If g ∈ L1Δ Ì0 , Ên , then the function x : Ì →
xt e1 t, a
e1 b, a
1 − e1 b, a
gs
Δs e
a
σs,
a,b∩Ì 1
Ên defined by
gs
Δs
e
a
σs,
a,t∩Ì 1
2.35
is a solution of the problem
xΔ t − xt gt,
Δ-a.e. t ∈ Ì0 ,
xa xb.
2.36
We now define a notion of Carathéodory functions on a compact time scale.
Definition 2.30. A function f :
following conditions hold.
Ì0 × Ên
→
Ên
is called a Δ-Carathéodory function if the three
C-i The map t → ft, x is Δ-measurable for every x ∈ Ên .
C-ii The map x → ft, x is continuous Δ-a.e. t ∈ Ì0 .
C-iii For every r > 0, there exists a function hr ∈ L1Δ Ì0 , 0, ∞ such that ft, x ≤
hr t Δ-a.e. t ∈ Ì0 and for every x ∈ Ên such that x ≤ R.
3. Existence Theorem for the Problem 1.1
In this section, we establish an existence result for the problem 1.1. A solution of this
problem will be a function x ∈ WΔ1,1 Ì, Ên satisfying 1.1. Let us recall that Ì is compact
and a min Ì < max Ì b. We introduce the notion of solution tube for the problem 1.1.
Definition 3.1. Let v, M ∈ WΔ1,1 Ì, Ên × WΔ1,1 Ì, 0, ∞. We say that v, M is a solution tube
of 1.1 if
i x − vσt, ft, x − vΔ t
≤ MσtMΔ tΔ-a.e. t ∈ Ì0 and for every x ∈ Ên such
that x − vσt Mσt,
ii vΔ t ft, vσtΔ-a.e. t ∈ Ì0 such that Mσt 0,
iii Mt 0 for every t ∈ Ì0 such that Mσt 0,
iv If BC denotes 1.3, x0 − va ≤ Ma; if BC denotes 1.4, then vb − va ≤
Ma − Mb.
We denote
Tv, M x ∈ WΔ1,1 Ì, Ên : xt − vt ≤ Mt for every t ∈ Ì .
3.1
If Ì is a real interval a, b, our definition of solution tube is equivalent to the notion
of solution tube introduced in 5.
Advances in Difference Equations
11
We consider the following problem.
xΔ t xσt ft, xσt
xσt,
Δ-a.e. t ∈ Ì0 ,
x ∈ BC,
3.2
where
⎧
⎪
⎨
Ms
x − vs vs if x − vs > Ms,
− vs
x
xs
⎪
⎩x
otherwise.
3.3
Let us define the operator TI : CÌ, Ên → CÌ, Ên by
TI xt e1 a, t x0 a,t∩Ì
e1 s, a fs, xσs
xσs
Δs .
3.4
Proposition 3.2. If v, M ∈ WΔ1,1 Ì, Ên × WΔ1,1 Ì, 0, ∞ is a solution tube of 1.1, 1.3, then
TI : CÌ, Ên → CÌ, Ên is compact.
Proof. We first observe that from Definitions 2.30 and 3.1, there exists a function h ∈
L1Δ Ì0 , 0, ∞ such that ft, xσt
xσt
≤ ht Δ-a.e. t ∈ Ì0 for every x ∈ CÌ, Ên .
n
Let {xn }n∈Æ be a sequence of CÌ, Ê converging to x ∈ CÌ, Ên . By Proposition 2.9,
TI xn t − TI xt
e s, t fs, x
n σs x
n σs − fs, xσs
xσs
Δs
a,t∩Ì 1
fs, x
Δs ,
≤K
n σs x
n σs − fs, xσs
xσs
a,b∩Ì
3.5
where K : maxt,t1∈Ì|e1 t1 , t|.
Then, we must show that the sequence {gn }n∈Æ defined by
gn s : fs, x
n σs x
n σs
3.6
converges to the function g in L1Δ Ì0 , Ên where
gs fs, xσs
xσs.
3.7
We can easily check that xn t → xt
for every t ∈ Ì0 and, then, by C-ii of Definition 2.30,
gn s → gs Δ-a.e. s ∈ Ì0 . Using also the fact that gn s ≤ hsΔ-a.e. s ∈ Ì0 , we deduce
that gn → g in L1Δ Ì0 , Ên by Theorem 2.10. This prove the continuity of TI .
12
Advances in Difference Equations
For the second part of the proof, we have to show that the set TI CÌ, Ên is relatively
compact. Let y TI x ∈ TI CÌ, Ên . Therefore,
Δs
fs, xσs
xσs
TI xt ≤ K x0 a,b∩Ì
3.8
≤ K x0 hL1Δ Ì0 .
So, TI CÌ, Ên is uniformly bounded. This set is also equicontinuous since for every
t1 , t2 ∈ Ì,
TI xt2 − TI xt1 ≤ K
t1 ,t2 ∩Ì
hsΔs.
3.9
By an analogous version of the Arzelà-Ascoli Theorem adapted to our context,
TI CÌ, Ên is relatively compact. Hence, TI is compact.
We now define the operator TP : CÌ, Ên → CÌ, Ên by
1
TP xt e1 t, a
1
e1 b, a − 1
a,t∩Ì
a,b∩Ì
e1 s, aΔs
fs, xσs
xσs
e1 s, aΔs .
fs, xσs
xσs
3.10
The following result can be proved as the previous one.
Proposition 3.3. If v, M ∈ WΔ1,1 Ì, Ên × WΔ1,1 Ì, 0, ∞ is a solution tube of 1.1, 1.4, then
the operator TP : CÌ, Ên → CÌ, Ên is compact.
Now, we can obtain the main theorem of this section.
Theorem 3.4. If v, M ∈ WΔ1,1 Ì, Ên ×WΔ1,1 Ì, 0, ∞ is a solution tube of 1.1, then the problem
1.1 has a solution x ∈ WΔ1,1 Ì, Ên ∩ Tv, M.
Proof. By Proposition 3.2 resp., Proposition 3.3, TI resp., TP is compact. It has a fixed point
by the Schauder fixed-point Theorem. Proposition 2.27 resp., Proposition 2.28 implies that
this fixed point is a solution for the problem 3.2. Then, it suffices to show that for every
solution x of 3.2, x ∈ Tv, M.
Advances in Difference Equations
13
Consider the set A {t ∈ Ì0 : xσt − vσt > Mσt}. By Remark 2.23, Δ-a.e.
{t ∈ A : t σt}, we have
on the set A
xt − vt, xΔ t − vΔ t
− MΔ t
xt − vt
xσt − vσt, xΔ t − vΔ t
− MΔ t.
xσt − vσt
xt − vt − MtΔ 3.11
If t ∈ A is right scattered, then μΔ {t} > 0 and
xt − vt − MtΔ
xσt − vσt − xt − vt
− MΔ t
μt
xσt − vσt2 − xσt − vσtxt − vt
− MΔ t
μtxσt − vσt
3.12
xσt − vσt, xσt − vσt − xt − vt
− MΔ t
μtxσt − vσt
xσt − vσt, xΔ t − vΔ t
− MΔ t.
xσt − vσt
≤
Therefore, since v, M is a solution tube of 1.1, we have Δ-a.e. on {t ∈ A : Mσt >
0} that
xt − vt − MtΔ
xσt − vσt, ft, xσt
xσt
− xσt − vΔ t
≤
− MΔ t
xσt − vσt
xσt
− vσt, ft, xσt
− vΔ t
Mσt
Mσt − xσt − vσt − MΔ t
<
MσtMΔ t
− MΔ t 0.
Mσt
3.13
14
Advances in Difference Equations
On the other hand, we have Δ-a.e. on {t ∈ A : Mσt 0} that
xt − vt − MtΔ
xσt − vσt, ft, xσt
xσt
− xσt − vΔ t
≤
− MΔ t
xσt − vσt
xσt − vσt, ft, vσt − vΔ t
xσt − vσt
3.14
− xσt − vσt − MΔ t
< −MΔ t 0.
This last equality follows from Definition 3.1iii and Proposition 2.20.
If we set rt : xt − vt − Mt, then r Δ t < 0 Δ-almost everywhere on A {t ∈ Ì0 : rσt > 0}. Moreover, since v, M is a solution tube of 1.1 and x satisfies 1.3
resp., x satisfies 1.4, then ra ≤ 0 resp., ra − rb ≤ va − vb − Ma − Mb ≤ 0.
Lemma 2.24 implies that A ∅. Therefore, x ∈ Tv, M and, hence, the theorem is proved.
Existence theorems are obtained in 7 for the problem 1.1, 1.3 when f : Ì × Ên →
Ê is continuous by using a hypothesis different of ours. When f is bounded, we can directly
use the Schauder fixed-point Theorem to deduce the existence of a solution to 1.1, 1.3.
We now show that in the case where f is unbounded, Theorems 4.7 and 4.8 of 7 become
corollaries of our existence theorem.
n
Corollary 3.5. Let f : Ìκ × Ên →
negative constants L and N such that
Ên
be an unbounded continuous function. If there exist non-
f t, p ≤ −2L p, f t, p N
3.15
for every t ∈ Ìκ and every p ∈ Ên , then the problem 1.1, 1.3 has at least one solution.
Proof. Observe that L > 0 since f is unbounded. By hypothesis, there exists a constant K :
N/2L such that p, ft, p
≤ K. Let us define M : Ì → 0, ∞ by
Mt : x0 1 a,t∩Ì
KΔs.
3.16
Then, MΔ t K for every t ∈ Ì and, thus,
p, f t, p ≤ K ≤ MΔ tMσt
3.17
for every t ∈ Ì and every p ∈ Ên . Then, if we take v ≡ 0, we get a solution tube v, M for our
problem and by Theorem 3.4, the problem has a solution x such that xt ≤ x0 1Kt−a
for every t ∈ Ì.
Advances in Difference Equations
Corollary 3.6. Let f : Ìκ × Ên →
nonnegative constant K such that
15
Ên
be an unbounded continuous function. If there exists a
p, f t, p ≤ K
3.18
for every t ∈ Ìκ and every p ∈ Ên , then the problem 1.1, 1.3 has at least one solution.
4. Existence Theorem for the Problem 1.2
In this section, we establish an existence result for the problem 1.2. A solution of this
problem will be a function x ∈ WΔ1,1 Ì, Ên for which 1.2 is satisfied. As before, Ì is compact
and a min Ì < max Ì b. We introduce the notion of solution tube for the problem 1.2.
Conditions of this definition are slightly different than conditions in Definition 3.1.
Definition 4.1. Let v, M ∈ WΔ1,1 Ì, Ên × WΔ1,1 Ì, 0, ∞. We say that v, M is a solution tube
of 1.2 if
i x − vt, ft, x − vΔ t
≥ MtMΔ t Δ-a.e. t ∈ Ì0 and for every x ∈ Ên such that
x − vt Mt,
ii vΔ t ft, vt and MΔ t 0, Δ-a.e. t ∈ Ì0 such that Mt 0,
iii vb − va ≤ Mb − Ma.
We consider the following problem.
xΔ t − xt ft, xt
− xt,
Δ-a.e. t ∈ Ì0 ,
xa xb.
4.1
where xt
is defined in 3.3.
Let us define the operator TP ∗ : CÌ, Ên → CÌ, Ên by
TP ∗ xt e1 t, a
fs, xs
− xs
Δs
e1 σs, a
a,b∩Ì
fs, xs
− xs
Δs .
e1 σs, a
a,t∩Ì
e1 b, a
1 − e1 b, a
4.2
The following result can be proved as Proposition 3.2.
Proposition 4.2. If v, M ∈ WΔ1,1 Ì, Ên × WΔ1,1 Ì, 0, ∞ is a solution tube of 1.2, then the
operator TP ∗ : CÌ, Ên → CÌ, Ên is compact.
Here is the main existence theorem for problem 1.2.
Theorem 4.3. If v, M ∈ WΔ1,1 Ì, Ên ×WΔ1,1 Ì, 0, ∞ is a solution tube of 1.2, then the problem
1.2 has a solution x ∈ WΔ1,1 Ì, Ên ∩ Tv, M.
16
Advances in Difference Equations
Proof. By Proposition 4.2, TP ∗ is compact. Then, by the Schauder fixed-point Theorem, TP ∗ has
a fixed point which is a solution of 4.1 by Proposition 2.29. It suffices to show that for every
solution x of 4.1, x ∈ Tv, M.
Let us consider the set A {t ∈ Ì0 : xt − vt > Mt}. By Remark 2.23, Δ-a.e. on
{t ∈ A : t σt} we have
the set A
xt − vt, xΔ t − vΔ t
− MΔ t.
xt − vt − Mt xt − vt
Δ
4.3
If t ∈ A is right scattered, then μΔ {t} > 0 and
xt − vt − MtΔ
xσt − vσt − xt − vt
− MΔ t
μt
xσt − vσtxt − vt − xt − vt2
− MΔ t
μtxt − vt
4.4
xt − vt, xσt − vσt − xt − vt
− MΔ t
μtxt − vt
xt − vt, xΔ t − vΔ t
− MΔ t.
xt − vt
≥
Since v, M is a solution tube of 1.2, we have Δ-a.e. on {t ∈ A : Mt > 0} that
xt − vt − MtΔ
xt − vt, ft, xt
xt − xt
− vΔ t
≥
− MΔ t
xt − vt
xt
− vt, ft, xt
− vΔ t
Mt
− Mt − xt − vt − MΔ t
>
MtMΔ t
− MΔ t 0.
Mt
4.5
Advances in Difference Equations
17
On the other hand, we have Δ-a.e. on {t ∈ A : Mt 0} that
xt − vt − MtΔ
xt − vt, ft, xt
xt − xt
− vΔ t
≥
− MΔ t
xt − vt
xt − vt, ft, vt − vΔ t
xt − vt
4.6
xt − vt − MΔ t.
>0
If we set rt : xt − vt − Mt, then r Δ t > 0 Δ-almost everywhere on A {t ∈ Ì0 : rt > 0}. Moreover, since v, M is a solution tube of 1.2 and x satisfies 1.4,
rb − ra ≤ vb − va − Mb − Ma ≤ 0. Lemma 2.25 implies that A ∅. So,
x ∈ Tv, M and the theorem is proved.
Let us observe that the following results obtained in 6 and 15, respectively, are
different from ours.
Theorem 4.4. Let f : Ìκ × Ên → Ên be a continuous function with
exist nonnegative constants α and K such that
f t, p − p
ht
for every t, p ∈ Ìκ × Ên , where h : Ì →
Ì such that μt / 1. If there
≤ 2α p, f t, p K
4.7
Ê is defined by ht : exp aσt ξ−1 μsΔs with
⎧
⎪
⎪−1,
⎨
ξ−1 μs log1 − μs
⎪
⎪
,
⎩
μs
if μs 0,
4.8
otherwise,
then the problem 1.2 has a solution.
Theorem 4.5. Let N ∈ Æ , Ì {0, 1, . . . , N, N 1} and f : {0, 1, . . . , N} × Ên →
function. If there exist nonnegative constants α and K such that
f t, p − p
2t1
Ên a continuous
≤ 2α p, f t, p K
4.9
for every t, p ∈ {0, 1, . . . , N} × Ên , then the difference equation 1.2 has one solution.
Observe that Theorem 4.3 is valid for every arbitrary time scale Ì. Here is an example
where 4.7 and 4.9 are not satisfied, but where Theorem 4.3 can be applied to deduce the
existence of a solution.
18
Advances in Difference Equations
Example 4.6. Consider the system
xΔ t −a1 xt2 xt a2 xt − a3 φt,
t ∈ Ìκ ,
xa xb,
4.10
where a1 , a2 , a3 are real positive constants such that −a1 a2 − a3 0 and where φ : Ìκ → Ên
is a continuous function such that φt 1 for every t ∈ Ìκ .
We first show that this system do not satisfy 4.7. Suppose there exist non-negative
constants α and K such that
−a1 x2 x a2 x − a3 φt − x
ht
≤ 2α x, −a1 x2 x a2 x − a3 φt K
4.11
for every t, x ∈ Ìκ × Ên .
If we define k : maxt∈Ìht, then
2
−a1 x3 a2 x − a3 − x −a1 x x a2 x − a3 φt − x
≤
k
k
≤ 2α x, −a1 x2 x a2 x − a3 φt K
4.12
≤ −2a1 αx4 2a2 αx2 2a3 αx K
for every t, x ∈ Ìκ × Ên .
Then,
2a1 αx4 bx3 cx2 dx e ≤ K
4.13
for every x ∈ Ên , where b −a1 /k, c −2a2 α, d −2a3 α a2 /k − 1/k, and e −a3 /k.
Taking the limit as x → ∞, we get a contradiction. Similarly, if Ì {0, 1, . . . , N, N1}, it can
be shown that 4.9 is not satisfied. On the other hand, it is easy to verify that v ≡ 0, M ≡ 1
is a solution tube of 4.10. By Theorem 4.3, this problem has a solution x such that xt ≤ 1
for every t ∈ Ì.
Definition 4.1 generalizes the notions of lower and upper solutions α and β introduced
in 3 in the particular case where the problem 1.2 is considered with n 1, Ì {0, 1, . . . , N, N 1} for some N ∈ Æ and with f depending only on xt. We recall these
definitions. Consider the problem
Δxt fxt, for every t ∈ {0, 1, . . . , N},
x0 xN 1,
where Δxt xt 1 − xt.
4.14
Advances in Difference Equations
19
Definition 4.7. A vector β β0, β1, . . . , βN 1 ∈ ÊN2 resp., α α0, α1, . . . , αN 1 ∈ ÊN2 is called an upper solution resp., a lower solution of 4.14 if
i for every t ∈ {0, 1, . . . , N, N 1}, fβt ≥ Δβt resp., fαt ≤ Δαt,
ii β0 βN 1 resp., α0 αN 1.
Remark that if α, β ∈ ÊN2 are, respectively, lower and upper solutions of 4.14 such
that αt ≤ βt for every t ∈ {0, 1, . . . , N, N 1}, then β α/2, β − α/2 is a solution tube
for this problem. Conversely, if v, M is a solution tube of 4.14, then v − M and v M are,
respectively, lower and upper solution for the same problem if, in addition, condition ii of
Definition 4.7 is satisfied. Then, Theorem 5 of 3 becomes a corollary of Theorem 4.3.
Corollary 4.8. If α, β ∈ ÊN2 are, respectively, lower and upper solutions of 4.14 such that αt ≤
βt for every t ∈ {0, 1, . . . , N, N 1}, then this equation has a solution x x0, x1, . . . , xN 1 ∈ ÊN2 such that αt ≤ xt ≤ βt for every t ∈ {0, 1, . . . , N, N 1}.
Acknowledgment
The author would like to thank Professor Marlene Frigon for useful discussion and comments
and the FQRNT for financial support.
References
1 V. Otero-Espinar and D. R. Vivero, “Existence of extremal solutions by approximation to a first-order
initial dynamic equation with Carathéodory’s conditions and discontinuous non-linearities,” Journal
of Difference Equations and Applications, vol. 12, no. 12, pp. 1225–1241, 2006.
2 V. Otero-Espinar and D. R. Vivero, “The existence and approximation of extremal solutions to several
first-order discontinuous dynamic equations with nonlinear boundary value conditions,” Nonlinear
Analysis: Theory, Methods & Applications, vol. 68, no. 7, pp. 2027–2037, 2008.
3 C. Bereanu and J. Mawhin, “Existence and multiplicity results for periodic solutions of nonlinear
difference equations,” Journal of Difference Equations and Applications, vol. 12, no. 7, pp. 677–695, 2006.
4 D. Franco, D. O’Regan, and J. Perán, “Upper and lower solution theory for first and second order
difference equations,” Dynamic Systems and Applications, vol. 13, no. 2, pp. 273–282, 2004.
5 B. Mirandette, Résultats d’Existence pour des Systèmes d’Équations Différentielles du Premier Ordre avec
Tube-Solution, Mémoire de Maı̂trise, Université de Montréal, Montréal, Canada, 1996.
6 Q. Dai and C. C. Tisdell, “Existence of solutions to first-order dynamic boundary value problems,”
International Journal of Difference Equations, vol. 1, no. 1, pp. 1–17, 2006.
7 C. C. Tisdell and A. Zaidi, “Basic qualitative and quantitative results for solutions to nonlinear,
dynamic equations on time scales with an application to economic modelling,” Nonlinear Analysis:
Theory, Methods & Applications, vol. 68, no. 11, pp. 3504–3524, 2008.
8 V. Otero-Espinar and D. R. Vivero, “Existence and approximation of extremal solutions to first-order
infinite systems of functional dynamic equations,” Journal of Mathematical Analysis and Applications,
vol. 339, no. 1, pp. 590–597, 2008.
9 M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass,
USA, 2003.
10 M. Bohner and A. Peterson, Dynamic Equations on Time Scales, Birkhäuser, Boston, Mass, USA, 2001.
11 S. Hilger, “Analysis on measure chains—a unified approach to continuous and discrete calculus,”
Results in Mathematics. Resultate der Mathematik, vol. 18, no. 1-2, pp. 18–56, 1990.
12 A. Cabada and D. R. Vivero, “Expression of the Lebesgue Δ-integral on time scales as a usual
Lebesgue integral: application to the calculus of Δ-antiderivatives,” Mathematical and Computer
Modelling, vol. 43, no. 1-2, pp. 194–207, 2006.
13 A. Cabada and D. R. Vivero, “Criterions for absolute continuity on time scales,” Journal of Difference
Equations and Applications, vol. 11, no. 11, pp. 1013–1028, 2005.
20
Advances in Difference Equations
14 R. P. Agarwal, V. Otero-Espinar, K. Perera, and D. R. Vivero, “Basic properties of Sobolev’s spaces on
time scales,” Advances in Difference Equations, vol. 2006, Article ID 38121, 14 pages, 2006.
15 C. C. Tisdell, “On first-order discrete boundary value problems,” Journal of Difference Equations and
Applications, vol. 12, no. 12, pp. 1213–1223, 2006.