Existence Theorems for First-Order Equations on Time Scales with Δ -Carath ´eodory Functions

Advances in Difference Equations Volume 2010, Article ID 650827,20pages doi:10.1155/2010/650827

Research Article

Existence Theorems for First-Order Equations on Time Scales with Δ -Carath ´eodory Functions

Hugues Gilbert

D´epartement de Math´ematiques, Coll`ege ´Edouard-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

Copyrightq2010 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-Carath´eodory and, hence, not necessarily continuous.

1. Introduction

In this paper, we establish existence results for the following systems:

x^Δt ft, xσt, Δ-a.e. t∈^Ì0, x∈BC; 1.1 x^Δt ft, xt, Δ-a.e. t∈^Ì0, xa xb. 1.2

Here,^Ìis an arbitrary compact time scale where we notea min^Ì,b max^Ì, and^Ì0

Ì\ {b}. Moreover,f :^Ì₀×^Ên → ^Ên is aΔ-Carath´eodory function andBCdenotes one of the following boundary conditions:

xa x0, 1.3

xa xb. 1.4

In the literature, this kind of problem was mainly treated forn 1. The existence of extremal solutions was established in1,2. Moreover, some existence results in the particular

case where the time scale is a discrete setdifference equationwere 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 system1.1 resp.,1.2will be useful to get a new existence result for1.1 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 in5. 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 of3for equation1.2.

Some papers treat the existence of solutions to systems of first-order equations on time scales. Existence results are obtained in6,7under hypothesis different from ours. However, some particular cases obtained in7are corollaries of our existence result for problem1.1.

Also, our existence results treat the case where the right members in1.1and1.2areΔ- Carath´eodory functions which are more general than continuous functions used for systems studied in6,7. Let us mention that existence of extremal solutions for infinite systems of first-order equations of time scale withΔ-Carath´eodory functions is established in8.

This paper is organized as follows. The third section presents an existence result for the problem1.1, 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^Ìbe a time scale, which is a closed nonempty subset of^Ê. Fort ∈ ^Ì, we define the forward jump operatorσ:^Ì → ^Ìresp., the backward jump operatorρ :^Ì → ^Ìbyσt inf{s ∈ ^Ì : s > t}resp., by ρt sup{s ∈ ^Ì : s < t}. We suppose that σt tif tis the maximum of^Ìand thatρt tiftis the minimum of^Ì. We say thattis right-scattered resp., left-scatteredifσt> tresp., ifρt< t. We say thattis isolated if it is right-scattered and left-scattered. Also, ift < sup^Ìandσt t, we say thattis right-dense. Ift >inf^Ìand ρt t, we say thattis left dense. Points that are right dense and left dense are called dense.

The graininess functionμ:^Ì → 0,∞is defined byμt σt−t.

If ^Ì has a left-scattered maximum, then ^{Ìκ Ì}\ {sup^Ì}. Otherwise, ^{Ìκ Ì}. In summary,

Ì

κ

⎧⎨

⎩

Ì\ ρ

sup^Ì ,sup^Ì

if sup^Ì<∞,

Ì if sup^Ì∞. 2.1

If^Ìis bounded, then^Ì0⊂^Ìκwhere^Ì0:^Ì\ {max^Ì}.

Definition 2.1. Assumef :^Ì → ^Ênis a function and lett∈^Ìκ. We say thatfisΔ-differentiable attif there exists a vectorf^Δt∈^Ên such that for all >0, there exists a neighborhoodUof t, where

fσt−fs−f^Δtσt−s≤|σt−s| 2.2

for everys ∈ U. We callf^ΔttheΔ-derivative off at t. Iff isΔ-differentiable attfor every t∈^Ìκ, thenf^Δ:^Ìκ → ^Ênis called theΔ-derivative offon^Ìκ.

Theorem 2.2. Assumef :^Ì → ^Ên is a function and lett∈^Ìκ. Then, we have the following.

iIffisΔ-differentiable att, thenfis continuous att.

iiIffis continuous attand iftis right-scattered, thenfisΔ-differentiable attand

f^Δt fσt−ft

μt . 2.3

iiiIftis right dense, thenfisΔ-differentiable attif and only if lim_s→tft−fs/t−s exists in^Ên. In this case,f^Δt lim_s→tft−fs/t−s.

ivIffisΔ-differentiable att, thenfσt ft μtf^Δt.

Theorem 2.3. Iff, g:^Ì → ^ÊareΔ-differentiable att∈^Ìκ, then if gisΔ-differentiable attandf g^Δt f^Δt g^Δt, iiαf isΔ-differentiable attfor everyα∈^Êandαf^Δt αf^Δt,

iiifg is Δ-differentiable at t and fg^Δt f^Δtgt fσtg^Δt ftg^Δt f^Δtgσt,

ivIfgtgσt/0, thenf/gisΔ-differentiable attand f

g Δ

t f^Δtgt−ftg^Δt

gtgσt . 2.4

The next result is an adaptation of Theorem 1.87 in10.

Theorem 2.4. LetW be an open set of^Ên and t ∈ ^Ìa right-dense point. Ifg : ^Ì → ^Ên isΔ- differentiable attand iff :W → ^Êis differentiable atgt∈W, thenf◦gisΔ-differentiable att andf◦g^Δt fgt, g^Δt.

Example 2.5. Assumex : ^Ì → ^Ên isΔ-differentiable att ∈ ^Ì. We know that · :

Ê

n \ {0} → 0,∞is differentiable. Iftσt, by the previous theorem, we havext^Δ xt, x^Δt/xt.

Definition 2.6. A functionf : ^Ì → ^Ên is called rd-continuous provided it is continuous at right-dense points in^Ì and its left-sided limits existfinite at left-dense points in^Ì. The set of rd-continuous functionsf : ^Ì → ^Ên is denoted by Crd^Ì,^Ên. The set of functions f :^Ì → ^Ênthat areΔ-differentiable and whoseΔ-derivative is rd-continuous is denoted by C¹_rd^Ì,^Ên.

It is possible to define a theory of measure and integration for an arbitrary bounded time scale^Ìwhereamin^Ì<max^Ìb. We recall the notion ofΔ-measure as introduced in chapter 5 of9. DefineF1the family of intervals of^Ìof the form

c, d {t∈^Ì:c≤t < d}, 2.5

wherec, d∈^Ìandc≤d. The intervalc, cis understood as the empty set. An outer measure m^∗₁onP^Ìis defined as follows: forE⊂^Ì,

m^∗₁E

⎧⎪

⎨

⎪⎩ inf

_m

k1

dk−ck:E⊂^m

k1

ck, dkwithck, dk∈ F1

ifb /∈E,

∞ ifb∈E.

2.6

Definition 2.7. A setA⊂^Ìis said to beΔ-measurable if for every setE⊂^Ì,

m^∗₁E m^∗₁E∩A m^∗₁E∩^Ì\A. 2.7

Now, denote

M m^∗₁

:{A⊂^Ì:AisΔ-measurable}. 2.8

The LebesgueΔ-measure onMm^∗₁, denoted byμ_Δ, is the restriction ofm^∗₁toMm^∗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 ofΔ-measurability andΔ-integrability for functionsf :^Ì → ^Êfollowing the same ideas of the theory of Lebesgue integral. We omit here these definitions that an interested reader can find in12. We only present definitions and results which will be useful for this paper.

Definition 2.8. LetE⊂^Ìbe aΔ-measurable set andf :^Ì → ^Êbe aΔ-measurable function.

We say thatf∈L¹_ΔEprovided

E

fsΔs <∞. 2.9

We say that aΔ-measurable functionf:^Ì → ^Ên is in the setL¹_ΔE,^Ênprovided

E

fisΔs <∞ 2.10

for each of its componentsf_i:^Ì → ^Ê. Proposition 2.9. Assumef ∈L¹_ΔE,^Ên. Then,

E

fsΔs ≤

E

fsΔs. 2.11 Many results of integration theory are established for measurable functionsf :X → ^Ê whereX, τ, μis 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.

Theorem 2.10 Lebesgue-dominated convergence Theorem. Let {fn}_n∈^Æ be a sequence of functions inL¹_Δ^Ì0. If there exists a functionf:^Ì0 → ^Êsuch thatfnt → ftΔ-a.e.t∈^Ì0and if there exists a functiong ∈L¹_Δ^Ì0such thatfnt ≤gtΔ-a.e.t∈^Ì0and for everyn∈^Æ, then f_n → finL¹_Δ^Ì0.

Theorem 2.11. The setL¹_Δ^Ì0is a Banach space endowed with the normf_L1 Δ :

Ì0|fs|Δs.

The following results were obtained in 12 where a useful relation between theΔ- measure on^Ìresp.,Δ-integral on^Ìand the Lebesgue measureμLon^Êresp., Lebesgue integral on^Êis established. To establish these results, the authors of12prove that the set of right-scattered points of^Ìis at most countable. Then, there are a set of indexI ⊂^Æ and a set{ti}_i∈I ⊂^Ìsuch thatR^Ì:{t∈^Ì:t < σt}{ti}_i∈I.

Proposition 2.12. LetA⊂^Ì. Then,AisΔ-measurable if and only ifAis Lebesgue measurable. In such a case, the following properties hold for everyΔ-measurable set A.

iIfb /∈A, then

μ_ΔA Σi∈IAσti−ti μLA. 2.12

iiμ_ΔA μ_LAif and only ifb /∈Aand A has no right-scattered points. Here,I_A {i∈ I :t_i ∈R^Ì∩A}.

To establish the relation betweenΔ-integration on^Ìand Lebesgue integration on a real compact interval, the functionf :^Ì → ^Êis extended toa, bin the following way.

ft:

⎧⎨

⎩

ft, ift∈^Ì,

fti, ift∈ti, σti, for ani∈I^Ì. 2.13 Theorem 2.13. LetE ⊂ ^Ìbe aΔ-measurable set such thatb /∈Eand letE E∪

i∈IEti, σti. Letf :^Ì → ^Êbe aΔ-measurable function andf:a, b → ^Êits extension ona, b. Then,f is Δ-integrable onEif and only iffis Lebesgue integrable onE. In such a case, one has

E

fsΔs

E

fsds. 2.14

Also, the functionf :^Ì → ^Ê can be extended ona, bin another way. Defineft: a, b → ^Êby

ft:

⎧⎪

⎨

⎪⎩

ft, if t∈^Ì,

fti fσti−fti

μti t−ti, if t∈ti, σti, for ani∈I^Ì. 2.15 Definition 2.14. A functionf :^Ì → ^Êis said to be absolutely continuous on^Ìif for every >0, there exists aδ >0 such that if{ak, bk}ⁿ_k1withak, bk∈^Ìis a finite pairwise disjoint family of subintervals of^ÌsatisfyingΣⁿ_k1bk−ak< δ, thenΣⁿ_k1|fbk−fak|< .

The three following results were obtained in13.

Lemma 2.15. Iffis differentiable att∈a, b∩^Ì, thenfisΔ-differentiable attandf^Δt ft.

Theorem 2.16. Consider a functionf : ^Ì → ^Ê and its extensionf : a, b → ^Ê. Then,f is absolutely continuous on^Ìif and only iffis absolutely continuous ona, b.

Theorem 2.17. A function f : ^Ì → ^Ê is absolutely continuous on ^Ì if and only if f is Δ- differentiableΔ-almost everywhere on^Ì0,f^Δ∈L¹_Δ^Ì0and

a,t∩^Ìf^ΔsΔsft−fa, for everyt∈^Ì. 2.16

We also recall the Banach Lemma.

Lemma 2.18. LetEbe a Banach space andu:a, b → Ean absolutely continuous function, then the measure of the set{t∈a, b:ut 0 andut/0}is zero.

Using the previous results, we now prove two propositions that will be used later.

Proposition 2.19. Letg∈L¹_Δ^Ì0andG:^Ì → ^Êthe function defined by

Gt:

a,t∩^ÌgsΔs. 2.17

Then,G^Δt gtΔ-almost everywhere on^Ì0. Proof. ByTheorem 2.13, remark that

Gt

a,tgsds for everyt∈^Ì. 2.18

We can also check that fort_iright scattered,

Gt

a,tigsds

ti,tgsds ift∈ti, σti. 2.19

Obviously,

Gt

a,tgsds for everyt∈a, b. 2.20

It is well known thatGt gt almost everywhere ona, b. ByLemma 2.15, we have that G^Δt Gt gt gtexcept on a setA⊂^Ì0such thatμLA 0. SinceGis continuous, fort_i∈R^Ì∩^Ì0right scattered,

G^Δti Gσti−Gti

μti gti 2.21

byTheorem 2.2ii. Then,GisΔ-differentiable att∈^Ì0\A∪R^Ì∩^Ì0. ByProposition 2.12ii, μ_ΔA\R^Ì∩^Ì0 0 and, then, the proposition is proved.

Proposition 2.20. Letu:^Ì → ^Êbe an absolutely continuous function, then theΔ-measure of the set{t∈^Ì0\R^Ì₀:ut 0 andu^Δt/0}is zero.

Proof. It suffices to consider the extension u of u on a, b and successively apply Theorem 2.16, Lemmas2.18,2.15, and theProposition 2.12ii.

We recall a notion of Sobolev’s space of functions defined on a bounded time scale^Ì whereamin^Ì<max^Ìb. The definition and the result are from14.

Definition 2.21. We say that a functionu:^Ì → ^Êbelongs toW_Δ^1,1Ìif and only ifu∈L¹_Δ^Ì0 and there exists a functiong:^Ìκ → ^Êsuch thatg∈L¹_Δ^Ì0and

Ì0

usφ^ΔsΔs−

Ì0

gsφσsΔs for everyφ∈C¹_0,rd^Ì 2.22

with

C¹_0,rd^Ì:

f:^Ì−→^Ê:f∈C¹_rd^Ì, fa 0fb

. 2.23

We say that a functionf:^Ì → ^Ênis in the setW_Δ^1,1Ì,^Ênif each of its componentsf_iare in W_Δ^1,1Ì.

Theorem 2.22. Suppose thatu ∈W_Δ^1,1Ìand that2.22holds for a functiong ∈L¹_Δ^Ì0. Then, there exists a unique functionx :^Ì → ^Ê absolutely continuous such thatΔ-almost everywhere on

Ì0, one hasx uandx^Δ g. Moreover, if g isrd-continuous on^Ì0, then there exists a unique functionx∈C_rd^{1 Ì}such thatxuΔ-almost everywhere on^Ì0and such thatx^Δgon^Ì₀.

By the previous theorem, we can conclude thatu∈W_Δ^1,1Ìis also continuous.

Remark 2.23. Ifx ∈ W_Δ^1,1Ì,^Ên, then its componentsxi are inW_Δ^1,1Ì. By Theorems 2.22 and 2.17, x isΔ-differentiableΔ-a.e. on^Ì. From Example 2.5, we obtainxt^Δ xt, x^Δt/xt Δ-a.e. on {t∈^Ì:tσt}.

We prove two maximum principles that will be useful to get a priori bounds for solutions of systems considered in this paper.

Lemma 2.24. Letr ∈W_Δ^1,1Ìsuch thatr^Δt< 0Δ-a.e.t∈ {t∈^Ì0 :rσt>0}. If one of the following conditions holds,

ira≤0, iira≤rb, thenrt≤0, for everyt∈^Ì.

Proof. Suppose the conclusion is false. Then, there existst₀∈^Ìsuch thatrt0 max_t∈^Ìrt>

0, sinceris continuous on^Ì. Ift0> ρt0, thenr^Δρt0exists, sinceμ_Δ{ρt0} t0−ρt0>0 and becauser∈W_Δ^1,1Ì. Then,

r^Δ ρt0

rt0−r ρt0

t0−ρt0 ≥0, 2.24

which is a contradiction sincert0 rσρt0 > 0. Ift0 ρt0 > a, then there exists an intervalt1, ρt0such thatrσt>0 for allt∈t1, ρt0∩^Ì. Thus,

0≤rt0−rt1 r ρt0

−rt1

t1,ρt0∩^Ìr^ΔsΔs <0 2.25 by hypothesis and by Theorem 2.17. Hence, we get a contradiction. The case t₀ a is impossible if hypothesisiholds and ifra ≤ rb, we must havera rb. If we take t0b, by using previous steps of this proof, one can check thatrb≤0 and, then, the lemma is proved.

Lemma 2.25. Letr ∈W_Δ^1,1Ìbe a function such thatr^Δt > 0 Δ-a.e. on{t ∈ ^Ìk :rt > 0}if ra≥rb, thenrt≤0, for everyt∈^Ì.

Proof. If there exists t ∈ ^Ì such that rt > 0, then there exists at0 ∈ ^Ìsuch that rt0

max_t∈^Ìrt> 0. Ift₀ < bandt₀ < σt0, thenμ_Δ{t0} σt0−t₀ >0. Sincer ∈ W_Δ^1,1Ì,r^Δ existsΔ-almost everywhere. Then, we must have

r^Δt0 rσt0−rt0

σt0−t₀ ≤0, 2.26

which contradicts the hypothesis of the lemma. Ift0< bandt0σt0, there exists an interval t0, t₁such thatrt>0 for everyt∈t0, t₁∩^Ì. Then,

0<

t0,t1∩^Ìr^ΔsΔsrt1−rt0, 2.27 by Theorem 2.17, which contradicts the fact that rt0 is a maximum. If t₀ b, then by hypothesis, we must havera rb. Thus, we can taket0 a, and by using the previous steps of this proof, one can check thatra≤0. Then, the lemma is proved.

Definition 2.26. For > 0, the exponential functione·, t0 : ^Ì → ^Ê may be defined as the unique solution of the initial value problem

x^Δt xt, xt0 1. 2.28

More explicitly, the exponential functione·, t0is given by the formula

et, t0 exp _t

t0

ξ

μs

Δs

, 2.29

where forh≥0, we defineξhas

ξh

⎧⎪

⎨

⎪⎩

, ifh0,

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. Ifg∈L¹_Δ^Ì0,^Ên, the functionx:^Ì → ^Ên defined by

xt e1a, t

x0

a,t∩^Ìe1s, agsΔs

2.31

is a solution of the problem

x^Δt xσt gt, Δ-a.e. t∈^Ì0,

xa x0. 2.32

Proposition 2.28. Ifg∈L¹_Δ^Ì0,^Ên, then the functionx:^Ì → ^Êndefined by

xt 1

e₁t, a

1 e₁b, a−1

a,b∩^Ìgse1s, aΔs

a,t∩^Ìgse1s, aΔs

2.33

is a solution of the problem

x^Δt xσt gt, Δ-a.e. t∈^Ì0,

xa xb. 2.34

Proposition 2.29. Ifg∈L¹_Δ^Ì0,^Ên, then the functionx:^Ì → ^Êndefined by

xt e1t, a

e1b, a 1−e₁b, a

a,b∩^Ì

gs e₁σs, aΔs

a,t∩^Ì

gs e₁σs, aΔs

2.35

is a solution of the problem

x^Δt−xt gt, Δ-a.e. t∈^Ì0,

xa xb. 2.36

We now define a notion of Carath´eodory functions on a compact time scale.

Definition 2.30. A functionf :^Ì0×^Ên → ^Ên is called aΔ-Carath´eodory function if the three following conditions hold.

C-iThe mapt→ft, xisΔ-measurable for everyx∈^Ên. C-iiThe mapx→ft, xis continuousΔ-a.e.t∈^Ì0.

C-iiiFor everyr > 0, there exists a function hr ∈ L¹_Δ^Ì0,0,∞such thatft, x ≤ h_rt Δ-a.e.t∈^Ì0and for everyx∈^Ên such thatx ≤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 functionx ∈ W_Δ^1,1Ì,^Ênsatisfying1.1. Let us recall that^Ìis compact andamin^Ì<max^Ìb. We introduce the notion of solution tube for the problem1.1.

Definition 3.1. Letv, M∈W_Δ^1,1Ì,^Ên×W_Δ^1,1Ì,0,∞. We say thatv, Mis a solution tube of1.1if

ix−vσt, ft, x−v^Δt ≤MσtM^ΔtΔ-a.e.t∈^Ì0and for everyx∈^Ênsuch

thatx−vσtMσt,

iiv^Δt ft, vσtΔ-a.e.t∈^Ì0such thatMσt 0, iiiMt 0 for everyt∈^Ì0such thatMσt 0,

ivIfBCdenotes1.3,x0−va ≤Ma; ifBCdenotes1.4, thenvb−va ≤

Ma−Mb.

We denote

Tv, M

x∈W_Δ^1,1Ì,^Ên:xt−vt ≤Mtfor everyt∈^Ì

. 3.1

If^Ìis a real intervala, b, our definition of solution tube is equivalent to the notion of solution tube introduced in5.

We consider the following problem.

x^Δt xσt ft,xσt xσt, Δ-a.e. t∈^Ì0,

x∈BC, 3.2

where

xs

⎧⎪

⎨

⎪⎩

Ms

x−vsx−vs vs ifx−vs> Ms,

x otherwise.

3.3

Let us define the operatorT_I :C^Ì,^Ên → C^Ì,^Ênby

T_Ixt e₁a, t

x₀

a,t∩^Ìe₁s, a

fs,xσs xσs Δs

. 3.4

Proposition 3.2. Ifv, M∈W_Δ^1,1Ì,^Ên×W_Δ^1,1Ì,0,∞is a solution tube of 1.1,1.3, then T_I :C^Ì,^Ên → C^Ì,^Ênis compact.

Proof. We first observe that from Definitions 2.30 and 3.1, there exists a function h ∈ L¹_Δ^Ì0,0,∞such thatft,xσt xσt ≤ ht Δ-a.e.t∈^Ì0for everyx∈C^Ì,^Ên.

Let{xn}_n∈^Æbe a sequence ofC^Ì,^Ênconverging tox∈C^Ì,^Ên. ByProposition 2.9, TIxnt−T_Ixt

a,t∩^Ìe₁s, t

fs,x_nσs x_nσs

−

fs,xσs xσs Δs

≤K

a,b∩^Ì

fs,x_nσs x_nσs−

fs,xσs xσs Δs

,

3.5

whereK:maxt,t1∈^Ì|e1t1, t|.

Then, we must show that the sequence{gn}_n∈Ædefined by

gns:fs,xnσs xnσs 3.6

converges to the functionginL¹_Δ^Ì0,^Ênwhere

gs fs,xσs xσs. 3.7

We can easily check thatx_nt → xt for everyt∈^Ì0and, then, byC-iiofDefinition 2.30, gns → gs Δ-a.e.s∈^Ì0. Using also the fact thatgns ≤hsΔ-a.e.s ∈^Ì0, we deduce thatgn → ginL¹_Δ^Ì0,^ÊnbyTheorem 2.10. This prove the continuity ofTI.

For the second part of the proof, we have to show that the setTIC^Ì,^Ênis relatively compact. LetyTIx∈TIC^Ì,^Ên. Therefore,

TIxt ≤K

x0

a,b∩^Ì

fs,xσs xσs Δs

≤K

x0 h_L¹_Δ^Ì₀

.

3.8

So,TIC^Ì,^Ênis uniformly bounded. This set is also equicontinuous since for every t₁, t₂∈^Ì,

TIxt2−TIxt1 ≤K

t1,t2∩^ÌhsΔs. 3.9

By an analogous version of the Arzel`a-Ascoli Theorem adapted to our context, TIC^Ì,^Ênis relatively compact. Hence,TIis compact.

We now define the operatorTP :C^Ì,^Ên → C^Ì,^Ênby

TPxt 1

e1t, a

1 e1b, a−1

a,b∩^Ì

fs,xσs xσs

e1s, aΔs

a,t∩^Ì

fs,xσs xσs

e1s, aΔs

.

3.10

The following result can be proved as the previous one.

Proposition 3.3. Ifv, M∈W_Δ^1,1Ì,^Ên×W_Δ^1,1Ì,0,∞is a solution tube of 1.1,1.4, then the operatorTP :C^Ì,^Ên → C^Ì,^Ênis compact.

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

Theorem 3.4. Ifv, M∈W_Δ^1,1Ì,^Ên×W_Δ^1,1Ì,0,∞is a solution tube of1.1, then the problem 1.1has a solutionx∈W_Δ^1,1Ì,^Ên∩Tv, M.

Proof. ByProposition 3.2resp.,Proposition 3.3,TIresp.,TPis compact. It has a fixed point by the Schauder fixed-point Theorem.Proposition 2.27resp.,Proposition 2.28implies that this fixed point is a solution for the problem3.2. Then, it suffices to show that for every solutionxof3.2,x∈Tv, M.

Consider the setA{t∈^Ì0 :xσt−vσt> Mσt}. ByRemark 2.23,Δ-a.e.

on the setA{t∈A:tσt}, we have

xt−vt −Mt^Δ

xt−vt, x^Δt−v^Δt

xt−vt −M^Δt

xσt−vσt, x^Δt−v^Δt

xσt−vσt −M^Δt.

3.11

Ift∈Ais right scattered, thenμ_Δ{t}>0 and

xt−vt −Mt^Δ

xσt−vσt − xt−vt

μt −M^Δt

xσt−vσt²− xσt−vσtxt−vt

μtxσt−vσt −M^Δt

≤ xσt−vσt, xσt−vσt−xt−vt

μtxσt−vσt −M^Δt

xσt−vσt, x^Δt−v^Δt

xσt−vσt −M^Δt.

3.12

Therefore, sincev, Mis a solution tube of1.1, we haveΔ-a.e. on{t∈A:Mσt>

0}that

xt−vt −Mt^Δ

≤

xσt−vσt, ft,xσt xσt −xσt−v^Δt

xσt−vσt −M^Δt

xσt −vσt, ft,xσt −v^Δt Mσt

Mσt− xσt−vσt−M^Δt

< MσtM^Δt

Mσt −M^Δt 0.

3.13

On the other hand, we haveΔ-a.e. on{t∈A:Mσt 0}that xt−vt −Mt^Δ

≤

xσt−vσt, ft,xσt xσt −xσt−v^Δt

xσt−vσt −M^Δt

xσt−vσt, ft, vσt−v^Δt xσt−vσt

−xσt−vσt−M^Δt

<−M^Δt 0.

3.14

This last equality follows fromDefinition 3.1iiiandProposition 2.20.

If we setrt : xt−vt −Mt, thenr^Δt < 0Δ-almost everywhere onA {t ∈^Ì0 :rσt> 0}. Moreover, sincev, Mis a solution tube of1.1andxsatisfies1.3 resp.,xsatisfies1.4, thenra≤0resp.,ra−rb≤ va−vb −Ma−Mb≤0.

Lemma 2.24implies thatA∅. Therefore,x∈Tv, Mand, hence, the theorem is proved.

Existence theorems are obtained in7for the problem1.1,1.3whenf :^Ì×^Ên →

Ê

nis continuous by using a hypothesis different of ours. Whenfis 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 wheref is unbounded, Theorems 4.7 and 4.8 of7become corollaries of our existence theorem.

Corollary 3.5. Letf : ^Ìκ×^Ên → ^Ên be an unbounded continuous function. If there exist non- negative constantsLandNsuch that

f

t, p≤ −2L p, f

t, p

N 3.15

for everyt∈^Ìκand everyp∈^Ên, then the problem1.1,1.3has at least one solution.

Proof. Observe thatL >0 sincefis unbounded. By hypothesis, there exists a constantK : N/2Lsuch thatp, ft, p ≤K. Let us defineM:^Ì → 0,∞by

Mt:x0 1

a,t∩^ÌKΔs. 3.16

Then,M^Δt Kfor everyt∈^Ìand, thus, p, f

t, p

≤K≤M^ΔtMσt 3.17

for everyt∈^Ìand everyp∈^Ên. Then, if we takev≡0, we get a solution tubev, Mfor our problem and byTheorem 3.4, the problem has a solutionxsuch thatxt ≤ x0 1 Kt−a for everyt∈^Ì.

Corollary 3.6. Let f : ^Ìκ ×^Ên → ^Ên be an unbounded continuous function. If there exists a nonnegative constantKsuch that

p, f t, p

≤K 3.18

for everyt∈^Ìκand everyp∈^Ên, then the problem1.1,1.3has 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 functionx∈W_Δ^1,1Ì,^Ênfor which1.2is satisfied. As before,^Ìis compact anda min^Ì <max^Ìb. We introduce the notion of solution tube for the problem1.2.

Conditions of this definition are slightly different than conditions inDefinition 3.1.

Definition 4.1. Letv, M∈W_Δ^1,1Ì,^Ên×W_Δ^1,1Ì,0,∞. We say thatv, Mis a solution tube of1.2if

ix−vt, ft, x−v^Δt ≥MtM^Δt Δ-a.e.t∈^Ì0and for everyx∈^Ênsuch that x−vtMt,

iiv^Δt ft, vtandM^Δt 0,Δ-a.e.t∈^Ì0such thatMt 0, iiivb−va ≤Mb−Ma.

We consider the following problem.

x^Δt−xt ft,xt −xt, Δ-a.e. t∈^Ì0,

xa xb. 4.1

wherext is defined in3.3.

Let us define the operatorT_P∗:C^Ì,^Ên → C^Ì,^Ênby

T_P∗xt e₁t, a

e₁b, a 1−e1b, a

a,b∩^Ì

fs,xs −xs e1σs, a Δs

a,t∩^Ì

fs,xs −xs e1σs, a Δs

.

4.2

The following result can be proved asProposition 3.2.

Proposition 4.2. Ifv, M ∈ W_Δ^1,1Ì,^Ên×W_Δ^1,1Ì,0,∞is a solution tube of 1.2, then the operatorT_P∗:C^Ì,^Ên → C^Ì,^Ênis compact.

Here is the main existence theorem for problem1.2.

Theorem 4.3. Ifv, M∈W_Δ^1,1Ì,^Ên×W_Δ^1,1Ì,0,∞is a solution tube of1.2, then the problem 1.2has a solutionx∈W_Δ^1,1Ì,^Ên∩Tv, M.

Proof. ByProposition 4.2,T_P∗is compact. Then, by the Schauder fixed-point Theorem,T_P∗has a fixed point which is a solution of4.1byProposition 2.29. It suffices to show that for every solutionxof4.1,x∈Tv, M.

Let us consider the setA{t∈^Ì0 :xt−vt> Mt}. ByRemark 2.23,Δ-a.e. on the setA{t∈A:tσt}we have

xt−vt −Mt^Δ

xt−vt, x^Δt−v^Δt

xt−vt −M^Δt. 4.3

Ift∈Ais right scattered, thenμ_Δ{t}>0 and

xt−vt −Mt^Δ

xσt−vσt − xt−vt

μt −M^Δt

xσt−vσtxt−vt − xt−vt²

μtxt−vt −M^Δt

≥ xt−vt, xσt−vσt−xt−vt

μtxt−vt −M^Δt

xt−vt, x^Δt−v^Δt

xt−vt −M^Δt.

4.4

Sincev, Mis a solution tube of1.2, we haveΔ-a.e. on{t∈A:Mt>0}that

xt−vt −Mt^Δ

≥

xt−vt, ft,xt xt −xt −v^Δt

xt−vt −M^Δt

xt −vt, ft,xt −v^Δt Mt

−Mt− xt−vt−M^Δt

> MtM^Δt

Mt −M^Δt 0.

4.5

On the other hand, we haveΔ-a.e. on{t∈A:Mt 0}that xt−vt −Mt^Δ

≥

xt−vt, ft,xt xt −xt −v^Δt

xt−vt −M^Δt

xt−vt, ft, vt−v^Δt xt−vt

xt−vt−M^Δt.

>0

4.6

If we setrt : xt−vt −Mt, thenr^Δt > 0Δ-almost everywhere onA {t ∈ ^Ì0 : rt > 0}. Moreover, sincev, M is a solution tube of1.2andxsatisfies1.4, rb − ra ≤ vb − va − Mb − Ma ≤ 0. Lemma 2.25implies thatA ∅. So, x∈Tv, Mand the theorem is proved.

Let us observe that the following results obtained in 6 and 15, respectively, are different from ours.

Theorem 4.4. Letf :^Ìκ×^Ên → ^Ên be a continuous function with^Ìsuch thatμt/1. If there exist nonnegative constantsαandKsuch that

f t, p

−p

ht ≤2α

p, f t, p

K 4.7

for everyt, p∈^Ìκ×^Ên, whereh:^Ì → ^Êis defined byht:exp_σt

a ξ₋₁μsΔswith

ξ₋₁

μs

⎧⎪

⎪⎨

⎪⎪

⎩

−1, if μs 0, log

1−μs

μs , otherwise, 4.8

then the problem1.2has a solution.

Theorem 4.5. LetN∈^Æ,^Ì{0,1, . . . , N, N 1}andf:{0,1, . . . , N} ×^Ên → ^Êna continuous function. If there exist nonnegative constantsαandKsuch that

f t, p

−p 2^{t 1} ≤2α

p, f t, p

K 4.9

for everyt, p∈ {0,1, . . . , N} ×^Ên, then the difference equation1.2has one solution.

Observe thatTheorem 4.3is valid for every arbitrary time scale^Ì. Here is an example where4.7and4.9are not satisfied, but whereTheorem 4.3can be applied to deduce the existence of a solution.

Example 4.6. Consider the system

x^Δt −a1xt²xt a₂xt−a₃φt, t∈^Ìκ,

xa xb, 4.10

wherea₁, a₂, a₃are real positive constants such that−a1 a₂−a₃0 and whereφ:^Ìκ → ^Ên is a continuous function such thatφt1 for everyt∈^Ìκ.

We first show that this system do not satisfy4.7. Suppose there exist non-negative constantsαandKsuch that

−a1x²x a2x−a3φt−x

ht ≤2α

x,−a1x²x a₂x−a₃φt K 4.11

for everyt, x∈^Ìκ×^Ên.

If we definek:max_t∈^Ìht, then

−a1x³ a2x −a3− x

k ≤

−a1x²x a₂x−a₃φt−x k

≤2α

x,−a1x²x a2x−a3φt K

≤ −2a1αx⁴ 2a₂αx² 2a₃αx K

4.12

for everyt, x∈^Ìκ×^Ên. Then,

2a1αx⁴ bx³ cx² dx e≤K 4.13

for everyx∈^Ên, whereb−a1/k,c −2a2α,d −2a3α a2/k−1/k, ande −a3/k.

Taking the limit asx → ∞, we get a contradiction. Similarly, if^Ì{0,1, . . . , N, N 1}, it can be shown that4.9is not satisfied. On the other hand, it is easy to verify thatv≡0, M≡1 is a solution tube of4.10. ByTheorem 4.3, this problem has a solutionxsuch thatxt ≤1 for everyt∈^Ì.

Definition 4.1generalizes the notions of lower and upper solutionsαandβintroduced 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 everyt∈ {0,1, . . . , N},

x0 xN 1, 4.14

whereΔxt xt 1−xt.

Definition 4.7. A vectorβ β0, β1, . . . , βN 1∈^{ÊN 2} resp.,α α0, α1, . . . , αN 1∈^{ÊN 2}is called an upper solutionresp., a lower solutionof4.14if

ifor everyt∈ {0,1, . . . , N, N 1},fβt≥Δβt resp.,fαt≤Δαt,

iiβ0 βN 1 resp.,α0 αN 1.

Remark that ifα, β∈^{ÊN 2} are, respectively, lower and upper solutions of4.14such thatαt≤βtfor everyt∈ {0,1, . . . , N, N 1}, thenβ α/2,β−α/2is a solution tube for this problem. Conversely, ifv, Mis a solution tube of4.14, thenv−Mandv Mare, respectively, lower and upper solution for the same problem if, in addition, conditioniiof Definition 4.7is satisfied. Then, Theorem 5 of3becomes a corollary ofTheorem 4.3.

Corollary 4.8. Ifα, β∈^{ÊN 2} are, respectively, lower and upper solutions of4.14such thatαt≤ βtfor everyt∈ {0,1, . . . , N, N 1}, then this equation has a solutionx x0, x1, . . . , xN 1∈^{ÊN 2} such thatαt≤xt≤βtfor everyt∈ {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.

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