On the Asymptotic Integration of Nonlinear Dynamic Equations

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.


Introduction
This work is devoted to the study of the existence and asymptotic behavior of solutions to the nonlinear dynamic equation where the function f : T × R → R is continuous and T is a time scale i.e., a nonempty closed subset of the real numbers; see 1, 2 and Section 2 below that has a minimal element t 0 > 0 and is unbounded above, that is, lim n→∞ t n ∞ for some set t n : n ∈ N ⊂ T.

1.2
In this paper, we offer conditions that ensure that for given a, b ∈ R, there exists a solution u of 1.1 satisfying the asymptotic behavior where u σ u • σ, is proved to have the asymptotic representation 1.3 whenever ∞ t 0 σ s |h s |Δs < ∞.

1.5
The study is extended in 4 to the investigation of oscillatory solutions for the more general dynamic equation where the coefficients h and g satisfy some integral conditions. The existence of solutions converging to zero is considered in 5 for a linear nonhomogeneous dynamic equation in a selfadjoint form. In 6 , the authors considered the nonlinear dynamic equation 1.1 for the time scales T R and T kZ. They assumed the existence of some positive rd-continuous function h and a positive nondecreasing function g with g 0 0 and g u > 0 for u > 0 such that We mention that the dynamic equation 1.1 contains as special cases both differential T R and difference T Z equations of the form u f t, u 0, t ∈ R, Δ 2 u f t, u 0, t ∈ Z. where m ∈ N, α : N → R, and f : R → R, is studied in 8 . 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 equations m 2 , a solution u n is shown to have the asymptotic representation see 8, Theorem 2, page 4692 The problem of the existence and extendability of solutions for nonlinear ordinary differential equations has been widely investigated during the last couple of years see, e.g., 20-22 . Regarding the general theory of asymptotic integration of ODEs, more details and recent developments may be found in the works 23-30 and the references therein. Note also that 1.3 is referred to as Property L for the continuous case in 29 , and it seems that this notion was introduced first in 31 . Inspired and motivated by the results obtained both for difference and differential equations, 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 scale unbounded above and we will be interested in the asymptotic behavior 1.3 of a solution u of 1.1 . Here a and b are real numbers. Considered in the spirit of the linear asymptotic conditions 1.9 and 1.12 , the asymptotic development 1.3 will be used throughout this work. Indeed, 1.1 may be seen as a perturbation of the homogeneous equation u ΔΔ 0, the solutions of which are the straight lines u t at b. Taking into account the restrictions 1.5 , 1.8 , 1.9 , 1.12 , 1.13 , and 1.15 , our results will also depend heavily on the growth of the nonlinear function f with respect to the unknown u.
The setup of this paper is as follows. Section 2 contains some preliminary definitions and results from the theory of time scales. In Section 3, we only state the main theorems. These are four distinct results, each of which guarantees the existence of asymptotically linear solutions according to 1.3 . For the first two results, Lipschitz-like hypotheses are assumed on the nonlinearity, the third one is concerned with the sublinear growth case, while the fourth and last one generalizes a result from 6 . 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 4 features some examples that illustrate the applicability of the main results. The proofs of the main results are presented in Section 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 in Section 6.

Preliminaries
In this section, we gather some standard definitions, properties, and notations from the time scales calculus see 1, 2 . Definition 2.2. For t ∈ T and a function g : T → R, define the delta derivative g Δ t to be the number if it exists with the property that given ε > 0, there is a neighborhood U of t such that Define also the second delta derivative by g ΔΔ g Δ Δ .
Moreover, improper integrals are defined by ∞ a g t Δt lim T →∞ T a g t Δt.
and in the case T Z, we have Remark 2.6. In the theory of orthogonal polynomials and quantum calculus, an appropriate time scale is T q Z {q k : k ∈ Z} ∪ {0}, where q > 1, and thus we have We conclude this section with an auxiliary result that will be needed frequently for the proofs of the main theorems in Section 5. Then, Proof. For fixed T ∈ T and t ∈ t 0 , T , 1, Theorem 1.117 ii can be used to show that

Main results
Throughout this paper, for a given nonnegative rd-continuous function h : T → R, we consider when they exist the constants We are now in position to state the four main results of this paper.
Then, for all a, b ∈ R, 1.1 has a solution u on t 0 , ∞ satisfying 1.3 .
Then, the conclusion of Theorem 3.1 holds true.
It is clear that 3.5 is stronger than 3.3 of Theorem 3.1. However, the assumption H * < ∞ in Theorem 3.2 is weaker than the restriction H * < 1 in 3.4 and no further restriction is made on the second integral H * * .
Then, the conclusion of Theorem 3.1 holds true.
In the last existence result, we are rather concerned with existence of at least one solution asymptotic to a specified line.
Suppose also that there exist a, b ∈ R, K > 0, and t * ≥ t 0 such that Then, 1.1 has a solution u on t * , ∞ satisfying 1.3 .

Examples
In this section, we illustrate each of the four theorems given in Section 3 by means of an example.
for example, γ u arctan u 1. By Theorem 3.1, for any a, b ∈ R, the difference equation According to 4.1 , clearly 3.3 is satisfied as well as 3.2 :  Let γ : R → R be such that for example, γ u |u|/ M |u| :

Proofs
For any a, b ∈ R, consider the transformation v t u t − at − b. Then, u is a solution of 1.
Consider the space It is clear that fixed points of the operator A are solutions of 5.1 . Observe also that Av is continuous if v is continuous, since then note that f is assumed to be continuous and that σ is rd-continuous an rd-continuous function is integrated note that this is possible due to Remark 2.4 which yields a delta differentiable and hence a continuous function; see 1, Theorem 1.16 i . Now, we are ready to prove the four results giving not only asymptotic behavior but also existence of global solutions.

Result based on the Banach fixed point theorem
The proof of Theorem 3.1 relies on the Banach fixed point theorem, which we recall here for completeness.    which tends to 0 as t → ∞ when applying 3.2 and 3.4 together with Lemma 2.7 ii three times. Hence, Av ∈ C 0 and therefore A : C 0 → C 0 . Moreover, passing to the supremum above and using Lemma 2.7 i three times, we also find that A is indeed well defined and that Av ≤ v |b| H * |a|H * * L.

5.6
Next, let v 1 , v 2 ∈ C 0 . With 3.3 , we get so that where we used the first part of 3.4 together with Lemma 2.7 i . Passing to the supremum, we get and due to the first part of 3.4 , A is a contraction. According to Theorem 5.1, A has a fixed point in C 0 .

Result based on the Boyd and Wong fixed point theorem
To prove Theorem 3.2, we employ the Boyd and Wong fixed point theorem from 33 , which extends Theorem 5.1 and is recalled here together with a pertinent definition for completeness.   which tends to 0 as t → ∞ when applying 3.2 and H * < ∞ together with Lemma 2.7 ii twice. Hence, Av ∈ C 0 and therefore A : C 0 → C 0 . Furthermore, using Lemma 2.7 i twice, we find that A is well defined and that Av ≤ H * L.

5.13
We introduce a continuous nondecreasing function ψ : 0, ∞ → 0, ∞ satisfying ψ 0 0 and ψ x < x, for all x > 0 by Let v 1 , v 2 ∈ C 0 . Assumption 3.5 yields that so that where we used 0 < H * < ∞ together with Lemma 2.7 i . Passing to the supremum, we get and by Definition 5.2, A is a nonlinear contraction. According to Theorem 5.3, A has a fixed point in C 0 .

Result based on the Leray-Schauder nonlinear alternative
The celebrated Leray-Schauder nonlinear alternative see, e.g., 34 is fundamental in the proof of Theorem 3.3. Recall that an operator is said to be completely continuous if it is continuous and maps bounded sets into relatively compact sets. We need the time scales version of the compactness criterion for subsets of C 0 which is due to Avramescu for the case T R see 20, 35 .

Advances in Difference Equations
ii B is equicontinuous, that is, for every ε > 0 there exists δ ε > 0 with iii B is equiconvergent, that is, for every ε > 0 there exists t * ε > t 0 with

5.20
Then, B is relatively compact.
Proof. Following 36, proof of Proposition 2.2 , consider an interval α, β T α, β ∩ T, α < β, and C C α, β T , R . The spaces C 0 and C are isomorphic by the mapping Φ defined by where ϕ : α, β T → T is a continuous, strictly nondecreasing function with lim t→β − ϕ t ∞. From ii and iii , B is equicontinuous in C 0 . Then, Φ B is equicontinuous and uniformly bounded in C. By the Arzelà-Ascoli theorem for time scales 37, Lemma 2.6 , we conclude that Φ B is relatively compact in C, which completes the proof.

Proof of Theorem 3.3. Define
β : |a|H * * |b|H * , m: We also introduce which tends to 0 as t → ∞ when applying 3.4 together with Lemma 2.7 ii twice. This means that A Ω is equiconvergent observe Proposition 5.5 iii and that Av ∈ C 0 and therefore A : Ω → C 0 . By Lemma 2.7 i , A is well defined, and passing to the supremum in 5.25 , we get Av ≤ m |b| H * |a|H * * mH * β.

5.26
We conclude that A Ω is uniformly bounded observe Proposition 5.5 i . Now use 5.24 again to deduce Using 3.4 and Lemma 2.7 iii twice, we find that the right-hand side above is equal to a finite constant, say R. Thus, and so A Ω is equicontinuous observe Proposition 5.5 ii . Altogether, by Proposition 5.5, A Ω is relatively compact. It remains to prove that A is continuous. Let v ∈ Ω and let v n ⊂ Ω be a sequence converging strongly to the limit v, that is, v n − v → 0 as n → ∞.

5.30
Then, Av n → Av pointwise as n → ∞. In addition, A Ω is relatively compact. Then, there exists a subsequence Av n k of Av n converging strongly to a certain w ∈ C 0 . As the strong convergence implies the pointwise convergence keeping the limit function, we find that w Av. Now, Av n converges strongly to Av as n → ∞ and thus the mapping A is continuous. Altogether, A : Ω → C 0 is completely continuous. Let v ∈ ∂Ω and λ > 1 be such that Av λv. Then, using 5.26 , a contradiction. Hence, by Theorem 5.4, A has a fixed point in Ω.
14 Advances in Difference Equations

Result based on the Schauder fixed point theorem
To prove Theorem 3.4, we appeal to the Schauder fixed point theorem see, e.g., 34 .

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 difference and differential equations, for example to q-difference equations see Remark 2.6 and to other cases of arbitrary time scales. Not only did our work extend the continuous and the discrete, 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 nonlinearity f f t, u, u Δ may be treated in an analogous manner yielding the asymptotic behavior 1.3 . For this purpose, additional restrictions on the growth of f with respect to the derivative u Δ need to be assumed. The space C 0 introduced in Section 5 is then extended to a space involving also the limit at infinity of the delta derivative; accordingly, a new compactness criterion is required. Also notice that the most informative condition is 3.7 which shows how the nonlinearity grows in terms of the ratio u/t. Apart from Theorem 3.4, Theorems 3.1, 3.2, and 3.3 are concerned with what is usually called the inverse problem of seeking a solution asymptotic to a given line see 28,29 . We point out that further to the asymptotic behavior, these theorems also provide existence of solutions to initial value problems for the dynamic equation 1.1 . Moreover, the existence of solutions with behavior described by 1.3 does not mean that all solutions behave in the same manner as shown in the nonlinear ordinary differential equation u 3/t 5 u 2 , t ≥ 1 see also 28, Section 5 . Indeed, this equation admits by Theorem 3.1 a solution having Property L while the solution u t 2t 3 has not. Finally, we mention that similar Bihari-type existence results of solutions which can be expanded asymptotically as u t P t o t near positive infinity may also be obtained for the nonhomogeneous dynamic equation u ΔΔ f t, u p t , t ∈ T, 6.1 where P ΔΔ p and p is a polynomial. For the case T R, we refer to 24, Section 8, Theorem 18 see also 30 .