Nonoscillatory Solutions for Super-linear Emden–fowler Type Dynamic Equations on Time Scales

In this paper, we consider the following Emden–Fowler type dynamic equations on time scales a(t)|x ∆ (t)| α sgn x ∆ (t) ∆ + b(t)|x(t)| β sgn x(t) = 0, when α < β. The classification of the nonoscillatory solutions are investigated and some necessary and sufficient conditions of the existence of oscillatory and nonoscilla-tory solutions are given by using the Schauder–Tychonoff fixed point theorem. Three possibilities of two classes of double integrals which are not only related to the coefficients of the equation but also linked with the classification of the nonoscillatory solutions and oscillation of solutions are put forward. Moreover, an important property of the intermediate solutions on time scales is indicated. At last, an example is given to illustrate our main results.


Introduction
Emden-Fowler dynamic equations originated in the early 20th century and they were established in the early research of gas dynamics in astrophysics [8].They also occur in the study of fluid mechanics, relativity, nuclear physics and chemical reaction systems, one can see the survey article by Wong [15] for detailed background of the generalized Emden-Fowler equation.With the development of science and technology, the super-linear Emden-Fowler type dynamic equations on time scales have played an important and extensive role in physics and engineering technology.We refer the reader to [16] and the references cited therein.The basic theorems and applications can be found in Agarwal et al. [1].In the recent years, there have been lots of results for Emden-Fowler type equations in [2,4,7,9].
In 2011, Erbe et al. [7] considered the asymptotic behavior of solutions for Emden-Fowler equations on time scales where p ∈ C rd ([t 0 , ∞) T , R), α is the quotient of odd positive integers, and T denotes a time scale which is unbounded from above.This article proposed an important property about the solution of (1.1) under the condition Zhou and Lan gave a classification of nonoscillatory solutions for the second-order neutral delay dynamic equations on time scales and some existence results of each kind of nonoscillatory solutions were also established in [17].
In 2014, Došlá and Marini [4] studied the nonoscillatory solutions for second order Emden-Fowler type differential equation This article has an important and far-reaching influence because it solved the open problem on the possible coexistence of three types of nonoscillatory solutions for super-linear Emden-Fowler differential equations.However, to the best of our knowledge, the coexistence of nonoscillatory solutions for dynamic equations on time scales has been scarcely investigated.
Motivated by [4], we consider the second order super-linear dynamic equations on time scales where 0 < α < β are constants and a(t) > 0, b(t) ≥ 0 are rd-continuous functions on [0, ∞) T , and If α = β, then the prototype of (1.3) is the Emden-Fowler equation Moreover, the half-linear case of (1.2) is the following form We will consider only the eventually positive solutions of (1.2) in the following section and denote for convenience.
The main work of this article can be listed as follows.Firstly, we improve the result in [6].We will show that the case when the solution is a constant as x [1] (t) tends to a constant is impossible.Secondly, we investigate the necessary and sufficient conditions for the existence of oscillatory and nonoscillatory solutions by methods different from [6].Thirdly, we present an important property about intermediate solutions on time scales which generalize the related contributions to the subject in [4].The research about the second order super-linear dynamic equations on time scales unifies the cases of differential equations and difference equations.
The paper is organized as follows.In Section 2, we introduce some definitions and a lemma about oscillatory and nonoscillatory solutions and the Schauder-Tychonoff fixed point theorem.In Section 3, we investigate the classification of the nonoscillatory solutions.Then we give some necessary and sufficient conditions for the existence of some oscillatory and nonoscillatory solutions by the Schauder-Tychonoff fixed point theorem.We propose three possibilities of two classes of double integrals which is related to the coefficients of the equation and an important property of the intermediate solutions.Moreover, an example is given to illustrate our main results.

Preliminaries
In this section, we collect some definitions and a lemma about dynamic equations on time scales.

Definition 2.1 ([14]
).We say that a nontrivial solution x of (1.2) has a generalized zero at t, if x(t)x(σ(t)) ≤ 0. If x(t) = 0 we say that solution x has a common zero at t.

Definition 2.2 ([14]
).We say that a solution x of equation (1.2) is nonoscillatory on T, if there exists τ ∈ T such that there does not exist any generalized zero at t for t ∈ [τ, ∞) T .
A nontrivial solution x of equation (1.2) is called oscillatory on T, if for every τ ∈ T has x a generalized zero on [τ, ∞) T .

Definition 2.3 ([6]
).We say that equation (1.2) is super-linear, if there exists a constant γ > 0 such that |v −γ ||b(s)v β | is nondecreasing in |v| for each fixed s and (2.1) Lemma 2.4 (Schauder-Tychonoff fixed point theorem [12]).Let X be a locally convex space, K ⊂ X be nonempty and convex, S ⊂ K, S be compact.Given a continuous map F : K → S, then there exists x ∈ S such that F( x) = x.

Main results
In this section, we investigate the classification of the nonoscillatory solutions.Then we give some necessary and sufficient conditions for the existence of some oscillatory and nonoscillatory solutions by Schauder-Tychonoff fixed point theorem.We also present three possibilities of two classes of double integrals which is related to the coefficients of the equation and an important property of the intermediate solutions.We extend some results of [4] to time scales.Theorem 3.1.The class P of all eventually positive solutions of (1.2) can be divided into three subclasses: [1] (∞) = 0, 0 < < ∞}.The superscript symbol "+" means that solutions are eventually positive increasing.We call solutions in M + ∞, , M + ∞,0 , M + ,0 dominant solutions, intermediate solutions and subdominant solutions.
Proof.Let x(t) be a positive solution of (1.2) for large t.Then there exists a t 0 > 0 such that x(t) > 0 as t ≥ t 0 .From (1.2) we have (x [1]  [1] (t) is non-increasing for t ≥ t 0 , which implies that x [1] (t) is eventually positive or negative.
Thus for any , 0 < < ∞, the possible cases of x [1] (t) and x(t) when t → ∞ are as follows: Now we prove the case (i) is impossible.If lim t→∞ x [1] (t) = , then there exists t 0 > 0 such that x [1] (t) ≥ 2 for t ≥ t 0 , i.e. x ∆ (t) ≥ 1/α (2a) 1/α .Integrating the last inequality and by the condition Now we give some sufficient and necessary conditions for the existence of oscillatory and nonoscillatory solutions.,0 is nonempty if and only if J < ∞.Moreover, for any , 0 < < ∞, there exists x ∈ M + ,0 such that lim t→∞ x(t) = .
So c 2 ≤ Tx ≤ c, which implies that Tx ∈ X.
(ii) T is continuous.Obviously Tx is compact.Let {x n } be a sequence of measurable functions of X converging to x ∈ X as n → ∞ in the topology of C rd [t 1 , ∞) T .
Since 0 ≤ x n (t) ≤ c, we get c 2 ≤ Tx n (t) ≤ c.The Lebesgue dominated convergence theorem shows that Tx n (t 1 ) → Tx(t 1 ), which implies Tx n (t) → Tx(t) uniformly on [t 1 , ∞), that is T is continuous.
Therefore, applying the Schauder-Tychonoff fixed point theorem, we see that there exists an element x ∈ X such that x = Tx, which shows that x(t) is a positive solution of equation (1.2) for t ≥ t 1 .Since x(t) is uniformly bounded, then from Theorem 3.1 we get that x(t) is an element in M + ,0 .i 2 ) (The "only if" part) Let x(t) be a nonoscillatory solution in M + ∞, , t > t 1 > 0, i.e. x(∞) = ∞, x [1] (∞) = .According to the definition of limit, we know Integrating from 0 to t, we get ∆s.
Substituting the above to equation (1.2) and integrating from 0 to ∞, we have By contrary, if K = ∞, then x [1] (∞) ≤ −∞, which is a contradiction with the condition (The "if" part) Suppose K < ∞ holds.We can choose proper > 0 such that Define the subset X of C rd [t 1 , ∞) T and the mapping A : and Now, in order to use the Schauder-Tychonoff fixed point theorem in Lemma 2.4 we separate the proof into the following three steps.
(i) A maps X into itself.For any x ∈ X, we have (2 ) which implies that Ax ∈ X.
(ii) AX is compact.Since A maps X into itself, we only need to illustrate X is compact.Let For any x n ∈ X, since a 1/α (s) ∆s is bounded, from the compactness theorem, we can know that there exists a convergent subsequence y n k .So for any x n ∈ X there exists a convergent subsequence x n k , which shows that X is compact.
(iii) A is continuous.Let {x n } be a sequence of measurable functions of X converging to The Lebesgue dominated convergence theorem shows that Therefore, applying the Schauder-Tychonoff fixed point theorem, we see that there exists an element x ∈ X such that x = Ax, which shows that x(t) is a positive solution of equation (1.2) for t ≥ t 1 .So x(t) is an element in M + ∞, .i 3 ) The "only if" part follows from i 1 ).To prove the "if" part.Assume for contradiction that (1.2) has a nonoscillatory solution x(t).We may assume without loss of generality that x(t) > 0 for t ≥ t 0 > 0. Integrating (1.2) from t to ∞ and noting that lim t→∞ a(t)(x ∆ (t)) α ≥ 0, we have Dividing the above by x γ/α (t), we obtain Since x(t) is an eventually positive solution, there exists a positive constant c 0 such that x(t) ≥ c 0 for t ≥ t 0 .We have b Integrating above over [t 0 , t] and by condition (2.1), we have The proof of i 4 ) is similar to that of i 3 ), so it is omitted here.The proof is completed.
Motivated by [5], now we give the following proof.
Proof.Let p = 1/m.Obviously p ≥ 1.The integrals J m , K m can be written as Put 1 a 1/α (t,s) = 0 for s < t and 1 a 1/α (t,s) = 1 a 1/α (t) for s ≥ t.Then we obtain The proof is completed.
Case (i): Let t 0 ≥ 0 be such that Case (ii): We have Integrating by parts we have Since 1 < α < β, by the Hölder inequality, we obtain where M is a finite positive constant.So we can choose proper M such that J ≤ M α K 1/β .Since J = ∞, this inequality yields the assertion.The proof is completed.
where X, Y are positive numbers, we obtain  We obtain the assertion from (3.6).The proof is completed.

Examples
In this section, we will present an example to illustrate our main results.From Theorem 3.2 we can get that (4.1) above has subdominant solutions for t ≥ 0.

Conclusion
At the end of this paper, let us suggest the further possible research in the theory of dynamic equations, concretely for Emden-Fowler dynamic equations.First, the coexistence of three classes of nonoscillatory solutions can be studied.Second, the sufficient and necessary conditions for the existence of intermediate solutions may be established.Third, the cases of sub-linear and half-linear about the corresponding conclusions can also be considered.