doi:10.1155/2008/739602
Research Article
On the Asymptotic Integration of Nonlinear Dynamic Equations
Elvan Akın-Bohner,1Martin Bohner,1Sma¨ıl Djebali,2and Toufik Moussaoui2
1Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA
2D´epartment de Math´ematiques, ´Ecole Normale Sup´erieure, P.O. Box 92, 16050 Kouba, Algiers, Algeria
Correspondence should be addressed to Martin Bohner,[email protected] Received 25 June 2007; Revised 12 November 2007; Accepted 29 January 2008 Recommended by Ondˇrej Doˇsl ´y
The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class of second-order nonlinear dynamic equations on unbounded time scales. Four different results are obtained by using the Banach fixed point theorem, the Boyd and Wong fixed point theorem, the Leray-Schauder nonlinear alternative, and the Schauder fixed point theorem. For each theorem, an illustrative example is presented. The results provide unification and some extensions in the time scale setup of the theory of asymptotic integration of nonlinear equations both in the continuous and discrete cases.
Copyrightq2008 Elvan Akın-Bohner et al. 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.
1. Introduction
This work is devoted to the study of the existence and asymptotic behavior of solutions to the nonlinear dynamic equation
uΔΔft, u 0, t∈T, 1.1
where the functionf :T×R→Ris continuous andTis a time scalei.e., a nonempty closed subset of the real numbers; see1,2 andSection 2belowthat has a minimal elementt0 >0 and is unbounded above, that is,
n→∞limtn∞for some set
tn: n∈N
⊂T. 1.2
In this paper, we offer conditions that ensure that for givena, b∈R, there exists a solutionuof 1.1satisfying the asymptotic behavior
ut atbo1 ast−→ ∞. 1.3
In3 , the general solution of the linear dynamic equation
uΔΔhuσ 0, t≥t0, 1.4
whereuσu◦σ, is proved to have the asymptotic representation1.3whenever ∞
t0
σs|hs|Δs <∞. 1.5
The study is extended in4 to the investigation of oscillatory solutions for the more general dynamic equation
uΔΔhtuΔσgt f◦uσ
0, 1.6 where the coefficientshandgsatisfy some integral conditions. The existence of solutions con- verging to zero is considered in5 for a linear nonhomogeneous dynamic equation in a self- adjoint form. In6 , the authors considered the nonlinear dynamic equation1.1for the time scalesTRandTkZ. They assumed the existence of some positiverd-continuous function hand a positive nondecreasing functiongwithg0 0 andgu>0 foru >0 such that
|ft, u| ≤htg |u|
t
1.7
with
∞
htΔt <∞. 1.8
Then, they obtained the linear behavior of solutionssee6, Theorem 4.1
t→∞lim ut
t a, a /0. 1.9
We mention that the dynamic equation1.1contains as special cases both differential TRand differenceTZequations of the form
u ft, u 0, t∈R, Δ2uft, u 0, t∈Z. 1.10 7, Chapter 8 is entirely devoted to the asymptotic behavior of linear difference equations and contains some classical and fundamental results. Themth order nonlinear difference equation Δmuαnfu 0, n∈N, 1.11 wherem∈N,α:N→R, andf:R→R, is studied in8 . Sufficient conditions which guarantee existence of solutions converging to some limit or having certain types of asymptotic behavior are given. In the particular case of second-order difference equationsm2, a solutionunis shown to have the asymptotic representationsee8, Theorem 2, page 4692
unanbo1, n≥n0, 1.12
provided that
n≥1
n|αn|<∞ 1.13
and both boundedness and uniform continuity offare assumed. For further related results in the discrete case, we refer the reader to7–15 . In the continuous case, that is,TR, the study of the linear differential equation
u htu0, 1.14
where
∞
t|ht|dt <∞, 1.15
goes back to at least the works of B ˆocher 16 and Dini 17 published at the beginning of the twentieth century, and it was also adapted by Bellman 18 in 1947, where the limit limt→∞ut ais obtained yielding by L’Hospital’s rule the limiting behavior limt→∞ut/t a. The nonlinear differential equation
u htgu 0 1.16
has been initially studied by Bihari in 1957 under1.15with further growth assumptions upon the nonlinearityg see19 . The problem of the existence and extendability of solutions for nonlinear ordinary differential equations has been widely investigated during the last couple of yearssee, e.g.,20–22 . Regarding the general theory of asymptotic integration of ODEs, more details and recent developments may be found in the works23–30 and the references therein. Note also that1.3is referred to as PropertyLfor the continuous case in29 , and it seems that this notion was introduced first in31 .
Inspired and motivated by the results obtained both for difference and differential equa- tions, our aim in this paper is to extend some of these results to nonlinear dynamic equations on time scales. In order to obtain existence of global solutions and their asymptotic behavior at positive infinity, we consider an arbitrary time scaleunbounded aboveand we will be inter- ested in the asymptotic behavior1.3of a solutionuof1.1. Hereaandbare real numbers.
Considered in the spirit of the linear asymptotic conditions1.9and1.12, the asymptotic de- velopment1.3will be used throughout this work. Indeed,1.1may be seen as a perturbation of the homogeneous equationuΔΔ0, the solutions of which are the straight linesut atb.
Taking into account the restrictions1.5,1.8,1.9,1.12,1.13, and1.15, our results will also depend heavily on the growth of the nonlinear functionfwith respect to the unknownu.
The setup of this paper is as follows.Section 2contains some preliminary definitions and results from the theory of time scales. InSection 3, we only state the main theorems. These are four distinct results, each of which guarantees the existence of asymptotically linear solutions according to1.3. For the first two results, Lipschitz-like hypotheses are assumed on the non- linearity, the third one is concerned with the sublinear growth case, while the fourth and last one generalizes a result from6 . In the first three theorems, existence of solutions asymptotic
to any prescribed line is proved while in the last one, we describe linear behavior of some solution.Section 4features some examples that illustrate the applicability of the main results.
The proofs of the main results are presented inSection 5. They are based on the fixed point theorems of Banach, Boyd and Wong, the Leray-Schauder nonlinear alternative, and Schauder, respectively. We end this paper with some concluding remarks inSection 6.
2. Preliminaries
In this section, we gather some standard definitions, properties, and notations from the time scales calculussee1,2 .
Definition 2.1. Define the forward and backward jump operatorsσ :T→Tandρ:T→Tby σt:inf
s > t: s∈T
, ρt:sup
s < t : s∈T
, 2.1
respectively. A functiong :T→Ris calledrd-continuous if it is continuous at pointst∈Twith σt tand if it has finite left-sided limits at pointst∈Twithρt t.
Definition 2.2. Fort ∈ Tand a functiong : T → R, define the delta derivativegΔtto be the numberif it existswith the property that givenε >0, there is a neighborhoodUoftsuch that
|gσt−gs −gΔtσt−s |< ε|σt−s|, ∀s∈U. 2.2 Define also the second delta derivative bygΔΔ gΔΔ.
Definition 2.3. IfGis an antiderivative ofg : T → R, that is,GΔ g, then the integral ofg is defined by
b
a
gtΔtGb−Ga.z 2.3
Moreover, improper integrals are defined by ∞
a
gtΔt lim
T→∞
T
a
gtΔt. 2.4
Remark 2.4. A well-known existence theorem1, Theorem 1.74 says thatrd-continuous func- tions possess antiderivatives.
Remark 2.5. Note that in the caseTR, we have
σt ρt t, fΔt ft, fΔΔt f t,
b
a
ftΔt
b
a
ftdt, 2.5
and in the caseTZ, we have
σt t1, ρt t−1, fΔt Δft:ft1−ft, fΔΔt ft2−2ft1 ft,
b
a
ftΔtb−1
ta
ftifa < b. 2.6
Remark 2.6. In the theory of orthogonal polynomials and quantum calculus, an appropriate time scale isTqZ{qk : k∈Z} ∪ {0}, whereq >1, and thus we have
σt qt, ρt q−1t, fΔt fqt−ft q−1t , fΔΔt f
q2t
−2fqt ft q−12t2 ,
t
1
fsΔs q−1
logqt−1 i1
qifqiift >1,
2.7
see32, Lemma 2ii . In this case,1.1is called aq-difference equation.
We conclude this section with an auxiliary result that will be needed frequently for the proofs of the main theorems inSection 5.
Lemma 2.7. Letg:T→0,∞berd-continuous and assume
G∗: ∞
t0
σs−t0
gsΔs <∞. 2.8
Then, i∞
t σs−tgsΔs≤G∗, for allt≥t0, ii∞
t σs−tgsΔs→0 ast→ ∞, iii∞
t0gsΔs <∞.
Proof. For fixedT ∈Tandt∈t0, T,1, Theorem 1.117ii can be used to show that Gt:
T
t
σs−tgsΔsimpliesGΔt − T
t
gsΔs≤0 2.9
so thatGis nonincreasing and henceGt≤Gt0≤G∗. Now, letT→ ∞to obtaini. Next, 0≤
∞
t
σs−tgsΔs≤ ∞
t
σs−t0gsΔs−→0 ast−→ ∞ 2.10
so thatiiholds. Finally, for sufficiently largeT ∈T, lett∗∈t01, T ∩Tso that T
t0
gsΔs
T
t∗
gsΔs
t∗
t0
gsΔs
≤ T
t∗
σs−t0gsΔs t∗
t0
gsΔs
≤G∗ t∗
t0
gsΔs.
2.11
LettingT → ∞, we deriveiii.
3. Main results
Throughout this paper, for a given nonnegativerd-continuous functionh:T→R, we consider when they existthe constants
H∗: ∞
t0
σs−t0hsΔs, H∗∗: ∞
t0
σs−t0shsΔs. 3.1 We are now in position to state the four main results of this paper.
Theorem 3.1. Assume
∃L >0 with ∞
t0
σs−t0|fs,0|Δs≤L, 3.2
|ft, u1−ft, u2| ≤ht|u1−u2|, ∀t≥t0, u1, u2∈R, 3.3
H∗<1, H∗∗<∞. 3.4
Then, for alla, b∈R,1.1has a solutionuont0,∞satisfying1.3.
Theorem 3.2. AssumeH∗<∞,3.2, and
ft, u1−ft, u2≤ht u1−u2
H∗u1−u2, ∀t≥t0, u1, u2∈R. 3.5 Then, the conclusion ofTheorem 3.1holds true.
It is clear that3.5is stronger than3.3ofTheorem 3.1. However, the assumptionH∗<
∞inTheorem 3.2 is weaker than the restrictionH∗ < 1 in3.4and no further restriction is made on the second integralH∗∗.
Theorem 3.3. Assume3.4and
|ft, u| ≤ht|u|, ∀t, u∈T×R. 3.6 Then, the conclusion ofTheorem 3.1holds true.
In the last existence result, we are rather concerned with existence of at least one solution asymptotic to a specified line.
Theorem 3.4. AssumeH∗<∞and suppose there exists a nondecreasing functiong:0,∞→0,∞ such that
|ft, u| ≤htg |u|
t
, ∀t, u∈T×R. 3.7
Suppose also that there exista, b∈R,K >0, andt∗≥t0such that H∗sup
gs: 0≤s≤K t∗ |b|
t∗ |a|
≤K. 3.8
Then,1.1has a solutionuont∗,∞satisfying1.3.
4. Examples
In this section, we illustrate each of the four theorems given inSection 3by means of an exam- ple.
Example 4.1application ofTheorem 3.1. Letγ :R→Rsatisfy γ0 1, γ
u1
−γ
u2≤u1−u2, ∀u1, u2∈R, 4.1 for example,γu arctanu 1. ByTheorem 3.1, for anya, b∈R, the difference equation
Δ2uγu
4t22t 0, t∈TN, 4.2
has at least one solutionuonNsatisfying1.3. Indeed, let
ht 1
4t22t, ft, u htγu, L1
4. 4.3
According to4.1, clearly3.3is satisfied as well as3.2:
∞
1
σs−1|fs,0|Δs1 4
∞ n1
1 n
1 2
n
≤ 1 4
∞ n1
1 2
n
1
4 L. 4.4
Moreover,
0< H∗≤H∗∗
∞
1
σs−1shsΔs ∞
n1
n2 1 4n2
1 2
n
1
4 <1 4.5
so that3.4also holds.
Example 4.2application ofTheorem 3.2. LetTbe any time scale which is unbounded above such that its graininess is bounded above. Suppose also 1∈T. Letp >1. By2, Example 5.72 ,
M: ∞
1
Δs
sp <∞. 4.6
Letγ :R→Rbe such that γ0 0, γ
u1
−γ
u2≤ u1−u2
Mu1−u2, ∀u1, u2∈R, 4.7 for example,γu |u|/M|u|:
|γu1−γu2| Mu1−u2 Mu1Mu2
≤ Mu1−u2 M2Mu1u2
≤ u1−u2 Mu1−u2 γ
u1−u2
.
4.8
ByTheorem 3.2, for anya, b∈R, the dynamic equation uΔΔ γu
tpσt0, t≥1, t∈T, 4.9
has at least one solutionuonTsatisfying1.3. To prove this, let
ht 1
tpσt, ft, u htγu, L1. 4.10
Now note that H∗
∞
1
σs−1hsΔs≤ ∞
1
σshsΔs
∞
1
Δs
sp M <∞. 4.11 Thus, together with4.7, we clearly have3.5and3.2.
Example 4.3application ofTheorem 3.3. For anya, b∈R, the dynamic equation
uΔΔhtln1|u| 0, t≥t0, t∈T, 4.12 has a solution satisfying the asymptotic representation1.3provided3.4is fulfilled, for ex- ample,ht 1/4t22t. This follows directly fromTheorem 3.3.
Example 4.4application ofTheorem 3.4. Letq >1. ByTheorem 3.4, theq-difference equation uΔΔ e|u|/t
tlogqt2 0, t∈qN, 4.13 has at least one solutionuonqNwhich behaves asut twhent→ ∞. In fact, setting
ht 1
tlogqt2, gs es, ft, u htg |u|
t
, 4.14
we can first seerefer toRemark 2.6that the integral H∗
∞
q
σs−qhsΔs≤q ∞
q
Δs
slogqs2Δsqq−1∞
n1
1
n2 4.15
converges. Moreover,3.7is clearly satisfied. Letb0 andKH∗e2|a|. SinceTis unbounded above, there existsα≥1 such thatt∗αK∈T. Then,
H∗sup
gs: 0≤s≤K t∗ |b|
t∗ |a|
H∗sup
es: 0≤s≤1|a|1 α
H∗e1|a|1/α≤H∗e2|a|K
4.16
so that3.8holds true as well.
5. Proofs
For anya, b∈R, consider the transformationvt ut−at−b. Then,uis a solution of1.1 if and only ifvis a solution of
vΔΔt ft, vt atb 0, t≥t0. 5.1
Consider the space
C0:C0 t0,∞
T
v∈C
t0,∞
T,R : lim
t→∞vt 0
, 5.2
wheret0,∞T : t0,∞∩T. Endowed with the supnormv sup{|vt| : t ≥ t0},C0is a Banach space. Define the mappingAforv∈C0if the improper integral existsby
Avt ∞
t
t−σs f
s, vs asb
Δs. 5.3
It is clear that fixed points of the operatorAare solutions of 5.1. Observe also thatAv is continuous ifv is continuous, since thennote thatf is assumed to be continuous and that σ isrd-continuousanrd-continuous function is integratednote that this is possible due to Remark 2.4which yields a delta differentiable and hence a continuous function; see1, Theo- rem 1.16i . Now, we are ready to prove the four results giving not only asymptotic behavior but also existence of global solutions.
5.1. Result based on the Banach fixed point theorem
The proof ofTheorem 3.1relies on the Banach fixed point theorem, which we recall here for completeness.
Theorem 5.1the Banach fixed point theorem. LetXbe a Banach space and letA:X →X be a contraction. Then,Ahas a unique fixed point inX.
Proof ofTheorem 3.1. Letv∈C0. First, we use3.3to find f
s, vs asb
−fs,0≤hsvs asb≤hs
v|a|s|b|
5.4 so that
Avt≤ ∞
t
σs−tfs, vs asb−fs,0Δs ∞
t
σs−tfs,0Δs
≤
v|b|∞
t
σs−t
hsΔs|a|
∞
t
σs−t
shsΔs ∞
t
σs−tfs,0Δs, 5.5
which tends to 0 ast → ∞when applying3.2and3.4together withLemma 2.7iithree times. Hence,Av∈C0and thereforeA:C0 →C0. Moreover, passing to the supremum above and usingLemma 2.7ithree times, we also find thatAis indeed well defined and that
Av ≤
v|b|
H∗|a|H∗∗L. 5.6
Next, letv1, v2∈C0. With3.3, we get f
s, v1s asb
−f
s, v2s asb≤hsv1−v2 5.7 so that
|Av1t−Av2t| ≤v1−v2∞
t
σs−t
hsΔs≤H∗v1−v2, 5.8 where we used the first part of3.4together withLemma 2.7i. Passing to the supremum, we get
Av1−Av2≤H∗v1−v2, 5.9 and due to the first part of3.4,Ais a contraction. According toTheorem 5.1,Ahas a fixed point inC0.
5.2. Result based on the Boyd and Wong fixed point theorem
To proveTheorem 3.2, we employ the Boyd and Wong fixed point theorem from33 , which extendsTheorem 5.1and is recalled heretogether with a pertinent definitionfor complete- ness.
Definition 5.2. LetX be a Banach space and letA : X → X be a mapping. Ais said to be a nonlinear contraction if there exists a continuous nondecreasing functionψ : 0,∞ → 0,∞ such thatψ0 0 andψx< xfor allx >0 with the property
Au−Av ≤ψ
u−v
, ∀u, v∈X. 5.10
Theorem 5.3the Boyd and Wong fixed point theorem. LetXbe a Banach space and letA:X→ Xbe a nonlinear contraction. Then,Ahas a unique fixed point inX.
Proof ofTheorem 3.2. Letv∈C0. From3.5, we infer the estimates
|f
s, vs asb
−fs,0≤hs |vs asb|
H∗|vs asb| ≤hs 5.11 so that
|Avt| ≤ ∞
t
σs−tf
s, vs asb
−fs,0Δs ∞
t
σs−tfs,0Δs
≤ ∞
t
σs−t
hsΔs
∞
t
σs−tfs,0Δs,
5.12
which tends to 0 ast→ ∞when applying3.2andH∗<∞together withLemma 2.7iitwice.
Hence,Av ∈C0and thereforeA:C0 →C0. Furthermore, usingLemma 2.7itwice, we find thatAis well defined and that
Av ≤H∗L. 5.13
We introduce a continuous nondecreasing functionψ:0,∞→0,∞satisfyingψ0 0 and ψx< x,for allx >0 by
ψx H∗x
H∗x, ∀x≥0. 5.14
Letv1, v2∈C0. Assumption3.5yields that f
s, v1s asb
−f
s, v2s asb≤hs H∗ψ
v1−v2
5.15
so that
Av1t−Av2t≤ψ
v1−v2∞
t
σs−ths
H∗ Δs≤ψ
v1−v2
, 5.16
where we used 0< H∗<∞together withLemma 2.7i. Passing to the supremum, we get Av1−Av2 ≤ψ
v1−v2
, 5.17
and byDefinition 5.2,Ais a nonlinear contraction. According toTheorem 5.3,Ahas a fixed point inC0.
5.3. Result based on the Leray-Schauder nonlinear alternative
The celebrated Leray-Schauder nonlinear alternative see, e.g., 34 is fundamental in the proof ofTheorem 3.3. Recall that an operator is said to be completely continuous if it is con- tinuous and maps bounded sets into relatively compact sets.
Theorem 5.4the Leray-Schauder nonlinear alternative. LetXbe a Banach space,Ω⊂Xbounded and open, 0∈Ω, andA :Ω→X a completely continuous operator. Then, either there existu∈∂Ω andλ >1 such thatAuλuorAhas a fixed point inΩ.
We need the time scales version of the compactness criterion for subsets ofC0which is due to Avramescu for the caseTRsee20,35 .
Proposition 5.5. Assume that the subsetB⊂C0has the following properties:
iBis uniformly bounded, that is, there exists a constantN >0 with
|ut| ≤N, ∀t≥t0, u∈B, 5.18
iiBis equicontinuous, that is, for everyε >0 there existsδε>0 with
ut1−ut2< ε, ∀t1, t2≥t0, |t1−t2|< δε, u∈B, 5.19
iiiBis equiconvergent, that is, for everyε >0 there existst∗ε> t0with
|ut|< ε, ∀t≥t∗ε, u∈B. 5.20
Then,Bis relatively compact.
Proof. Following36, proof of Proposition 2.2 , consider an intervalα, β T α, β ∩T,α < β, andCCα, β T,R. The spacesC0andCare isomorphic by the mappingΦdefined by
Φxt x
ϕt
, ift∈α, βT,
x∞, iftβ, 5.21
whereϕ :α, βT →Tis a continuous, strictly nondecreasing function with limt→β−ϕt ∞.
Fromii andiii,B is equicontinuous inC0. Then,ΦBis equicontinuous and uniformly bounded inC. By the Arzel`a-Ascoli theorem for time scales37, Lemma 2.6 , we conclude that ΦBis relatively compact inC, which completes the proof.
Proof ofTheorem 3.3. Define
β:|a|H∗∗|b|H∗, m:
β1−H∗−1, ifβ /0,
1, ifβ 0. 5.22
We also introduce
Ω:
v∈C0: v< m
5.23
and note thatΩ⊂C0is open and by3.4satisfies 0∈Ωsincem >0. Letv∈Ω. From3.6, we get
f
s, vs asb≤hsvs
asb≤hs
m|b||a|s
5.24 so that
Avt≤
m|b|∞
t
σs−thsΔs|a|
∞
t
σs−t
shsΔs, 5.25
which tends to 0 ast→ ∞when applying3.4together withLemma 2.7iitwice. This means thatAΩis equiconvergent observe Proposition 5.5iii and thatAv ∈ C0 and therefore A:Ω →C0. ByLemma 2.7i,Ais well defined, and passing to the supremum in5.25, we get
Av ≤ m|b|
H∗|a|H∗∗mH∗β. 5.26
We conclude that AΩis uniformly bounded observeProposition 5.5i. Now use5.24 again to deduce
AvΔt ∞t f
s, vs asb
Δs ≤
m|b|∞
t0hsΔs|a|∞
t0shsΔs. 5.27 Using3.4andLemma 2.7iiitwice, we find that the right-hand side above is equal to a finite constant, sayR. Thus,
Av t2
−Av
t1≤Rt2−t1−→0 ast2−→t1 5.28 and soAΩis equicontinuous observeProposition 5.5ii. Altogether, byProposition 5.5, AΩis relatively compact.
It remains to prove thatA is continuous. Letv ∈ Ω and letvn ⊂ Ω be a sequence converging strongly to the limitv, that is,vn −v → 0 asn → ∞. By 5.24, we have the estimate
f
s, vns asb≤
m|b||a|s
hs. 5.29
SinceH∗ <∞andH∗∗ <∞and because ofLemma 2.7i, we infer from the Lebesgue domi- nated convergence theoremsee38 and37, Theorem 10.1 that
n→∞lim ∞
t
σs−tfs, vns asbΔs ∞
t
σs−tfs, vs asbΔs. 5.30
Then,Avn → Avpointwise asn→ ∞. In addition,AΩis relatively compact. Then, there exists a subsequenceAvnkofAvnconverging strongly to a certainw ∈ C0. As the strong convergence implies the pointwise convergence keeping the limit function, we find thatw Av. Now,Avnconverges strongly toAvasn→ ∞and thus the mappingAis continuous.
Altogether,A : Ω → C0is completely continuous. Let v ∈ ∂Ωandλ > 1 be such that Avλv. Then, using5.26,
λmλvAv ≤H∗mβ 5.31
so that
1< λ≤H∗ β m
H∗, ifβ0 1, ifβ /0
≤1, 5.32
a contradiction. Hence, byTheorem 5.4,Ahas a fixed point inΩ.
5.4. Result based on the Schauder fixed point theorem
To proveTheorem 3.4, we appeal to the Schauder fixed point theoremsee, e.g.,34 .
Theorem 5.6the Schauder fixed point theorem. LetX be a Banach space and let B ⊂ X be nonempty, bounded, closed, and convex. LetA:B →Bbe a completely continuous operator. Then,A has a fixed point inB.
Proof ofTheorem 3.4. Consider the closed ballB:{v∈C0:v ≤K}and define θ:sup
gu: 0≤u≤ K t∗ |b|
t∗ |a|
. 5.33
Letv∈B. By3.7, we find f
s, vs asb≤hsg
|vs asb|
s
≤θhs, s≥t∗, 5.34
since fors≥t∗we have
0≤ vs asb
s ≤ v|b|
s |a| ≤ K|b|
s |a| ≤K|b|
t∗ |a|. 5.35
By5.34, fort≥t∗, Avt≤
∞
t
σs−tf
s, vs asbΔs≤θ ∞
t
σs−t
hsΔs, 5.36
which tends to 0 ast→ ∞when applyingH∗ < ∞together withLemma 2.7ii. This means thatAB is equiconvergent observeProposition 5.5iiiand that Av ∈ C0 and therefore A : B → C0. Thanks toLemma 2.7i, we also get thatAis well defined and, passing to the supremum above, we have
Av ≤θH∗≤K, 5.37
where we used 3.8. We conclude thatA : B → B and that ABis uniformly bounded observeProposition 5.5i. Next, lett1, t2∈Tbe such thatt2≥t1≥t∗. Then,
Avt2−Avt1
∞
t2
t2−σs f
s, vs asb Δs−
∞
t1
t1−σs f
s, vs asb Δs
∞
t2
t2−σs
−
t1−σs f
s, vs asb Δs−
t2
t1
t1−σs f
s, vs asb Δs
t2−t1 ∞
t2
f
s, vs asb Δs
t2
t1
σs−t1 f
s, vs asb Δs.
5.38
Using again5.34, we find Av
t2
−Av t1≤
t2−t1 θ
∞
t0
hsΔsθ t2
t1
σshsΔs, 5.39
which tends to zero ast2→t1due toH∗<∞andLemma 2.7iii for the second integral, use thatσhisrd-continuous and hence has an antiderivativeQbyRemark 2.4, and thus this inte- gral equals toQt2−Qt1andQis continuous. Therefore,ABis equicontinuousobserve Proposition 5.5ii. Altogether, byProposition 5.5,ABis relatively compact. As in the proof ofTheorem 3.3, we may check thatAis continuous. Thus,A:B→Bis completely continuous.
According toTheorem 5.6,Ahas a fixed point inB.
6. Concluding remarks
In this work, specific results regarding the asymptotic behavior of the nonlinear dynamic equation 1.1 have been obtained, extending some known results in the theories of differ- ence and differential equations, for example toq-difference equationsseeRemark 2.6and to other cases of arbitrary time scales. Not only did our work extend the continuous and the dis- crete, but it also unified those two important cases and illuminated the common grounds of the corresponding differential and difference equations. As a fundamental contribution to the now well-established theory of time scales, it is hoped that our results will advance the area and stimulate future research on this and related topics. For example, the more general case of delta-derivative depending nonlinearityfft, u, uΔmay be treated in an analogous manner yielding the asymptotic behavior1.3. For this purpose, additional restrictions on the growth offwith respect to the derivativeuΔneed to be assumed. The spaceC0introduced inSection 5 is then extended to a space involving also the limit at infinity of the delta derivative; accord- ingly, a new compactness criterion is required. Also notice that the most informative condition is3.7which shows how the nonlinearity grows in terms of the ratiou/t.
Apart fromTheorem 3.4, Theorems3.1,3.2, and3.3are concerned with what is usually called the inverse problem of seeking a solution asymptotic to a given linesee28,29 . We point out that further to the asymptotic behavior, these theorems also provide existence of solutions to initial value problems for the dynamic equation1.1. Moreover, the existence of solutions with behavior described by1.3does not mean that all solutions behave in the same manner as shown in the nonlinear ordinary differential equationu 3/t5u2,t≥1see also 28, Section 5 . Indeed, this equation admits byTheorem 3.1a solution having PropertyL while the solutionut 2t3has not.
Finally, we mention that similar Bihari-type existence results of solutions which can be expanded asymptotically asut Pt otnear positive infinity may also be obtained for the nonhomogeneous dynamic equation
uΔΔft, u pt, t∈T, 6.1
wherePΔΔpandpis a polynomial. For the caseTR, we refer to24, Section 8, Theorem 18 see also30 .
Acknowledgments
S. Djebali would like to thank his laboratory, `Equations aux D`eriv`ees Partielles & Histoire des Math`ematiques, for supporting this work. M. Bohner acknowledges support by NSF Grant no.
0624127. The authors are grateful to two anonymous referees for their careful reading of the first version of this manuscript and their constructive comments. Their suggestions substan- tially influenced the final presentation of the authors’ results.
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