Oscillation and nonoscillation for second order neutral dynamic equations with positive and negative coefficients on time scales
© Deng et al.; licensee Springer. 2014
Received: 11 February 2014
Accepted: 22 April 2014
Published: 6 May 2014
We investigate oscillation and nonoscillation of certain second order neutral dynamic equations with positive and negative coefficients. We apply the results from the theory of lower and upper solutions for related dynamic equations along with some additional estimates on positive solutions and use different techniques to obtain some oscillatory theorems. Also, we apply Kranoselskii’s fixed point theorem to obtain nonoscillatory results and then give two sufficient and necessary conditions for the equations to be oscillatory. Some interesting examples are given to illustrate the versatility of our results.
Keywordsneutral dynamic equations positive and negative coefficients oscillation and nonoscillation lower and upper solutions Kranoselskii’s fixed point theorem
Throughout this paper, we shall assume that is a time scale satisfying and , and
(B1) satisfies ;
(B2) there exists a constant , such that ;
(B3) , , ;
(B4) are injective, , and ;
(B5) are injective, , and for sufficiently large , there exists such that and ;
A solution of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory. The equation itself is called oscillatory if all its solutions are oscillatory.
In recent years, there has been an increasing interest in studying the oscillation and nonoscillation of solutions of dynamic equations on time scales since Hilger introduced the theory of time scale which was excepted to unify continuous and discrete calculus. We refer the readers to the monographs [1–4], the papers [3, 5–17] and the references cited therein.
The results on oscillation of dynamic equations with positive and negative coefficients are mainly concentrated on differential equations or difference equations. To the best of our knowledge, there are few researches on dynamic equations with positive and negative coefficients on time scales. In , Özbekler and Zafer gave new oscillation criteria for superlinear and sublinear forced dynamic equations with positive and negative coefficients by means of nonprinciple solutions. Also, Özbekler et al.  made use of the concept of nonprinciple solutions to establish new oscillation criteria. However, in general, it is difficult for us to find a nonprinciple solution of second order dynamic equations. As a result, their approach may be difficult to apply to second order dynamic equations. Under the convergence of double integral of negative coefficients, sufficient conditions for oscillation was given in [12, 16–18]. The results on oscillation of difference equations with positive and negative coefficients can be found in [8, 15] and references therein.
In this paper, to obtain oscillatory theorems, we shall apply results from the theory of lower and upper solutions for related dynamic equations along with some additional estimates on positive solutions and use some different techniques. Also, we apply Kranoselskii’s fixed point theorem to obtain nonoscillatory results and then give two sufficient and necessary conditions for (1.1) being oscillatory. Our results cannot only be applied to differential equations and difference equations, but they can also be applied to other dynamic equations with positive and negative coefficients.
In Section 2, we present some preliminaries and important estimates, especially the estimate and the function if the solution of (1.1). In Section 3, we give several oscillatory and nonoscillatory results. In Section 4, we illustrate the versatility of our results by three examples.
2 Some preliminaries
First of all, we give the following estimates.
- (i)for all , , , , and
- (ii)for each , and for , we have
- (iii)if is nondecreasing, each , and for , we have
Proof (i) Suppose that is an eventually positive solution of (1.1). In view of conditions (B2)-(B6), there exists such that , , , , , , and for all . It is immediate to obtain by .
By (B6), we see that and then clearly . Similarly, we also have and .
which contradicts .
(iii) It can be proved similar to [, Lemma 2.1] and hence its proof is omitted here. □
Definition 2.1 [, Definition 6.1]
The following theorem is an extension of [, Theorem 6.5] to .
Theorem 2.1 [, Theorem 1.5]
has a solution y with and on .
Lemma 2.2 [, Lemma 2.2]
Suppose that is bounded and uniformly Cauchy. Further, suppose that X is equi-continuous on for any . Then X is relatively compact.
Lemma 2.3 (Kranoselskii’s fixed point theorem)
for all ;
U is a contraction mapping;
S is completely continuous.
Then has a fixed point in Ω.
3 Main results
In this section, we establish our main results.
for all and some sufficiently large , where , is given in (2.1).
On the other hand, we can also obtain (3.3) if (2.2) is replaced by (2.3).
has a solution with on .
So we obtain a contradiction to (3.1). □
Under the assumptions of Theorem 3.1, noting that (3.3), it is easy to obtain the following corollary.
for some positive constant k, .
holds for all and some sufficiently large , where , is given in (2.1), then all solutions of (1.1) are oscillatory.
However, letting in above inequality, the left side is bounded whereas the right side is unbounded by assumptions (3.6), (3.7). This contradiction shows that (3.6) is sufficient for all solutions of (1.1) to be oscillatory. □
Next we will give two sufficient and necessary conditions for (1.1) being oscillatory under the case . However, we need a sufficient condition for (1.1) having a bounded nonoscillatory solution.
for some and some sufficiently large , then (1.1) has a bounded nonoscillatory solution.
According to (B3) and (B4), we see that there exists with such that and , , for .
- (i)We first prove that for any . Note that for any , . For any and , by (3.10)-(3.11), we haveand
Similarly, we can show that for any and .
It is not difficult to check that U is a contraction mapping.
We will prove that S is a completely continuous mapping. It is easy to check that S maps Ω into Ω.
Thus S is continuous.
Third, we show S Ω is relatively compact. According to Lemma 2.2, it suffices to show that S Ω is bounded, uniformly Cauchy, and equi-continuous. The boundedness is obvious.
So S Ω is uniformly Cauchy.
For , .
Therefore there exists such that if and . This means that S Ω is equi-continuous.
It follows from Lemma 2.2 that S Ω is relatively compact, and then S is completely continuous.
Let , we obtain the desired result. □
for some , then (1.1) has a bounded nonoscillatory solution.
Theorem 3.3 plays an important role in excluding (1.1) to have a unbounded nonoscillatory solution under that (3.8) holds.
holds for all and some sufficiently large , where is given in (2.1).
where . The rest of the proof is the same as Theorem 3.2. So we leave details to readers.
which gives (3.8). Therefore, by Theorem 3.3, equation (1.1) has a bounded nonoscillatory solution. This contradiction shows that (3.14) is necessary. □
We give another sufficient and necessary condition for (1.1) to be oscillatory.
holds for all and some sufficiently large , where is given in (2.1).
Since the proof of Theorem 3.5 is similar to that of Theorem 3.4, we leave the details to the readers.
for all , some sufficiently large , and , where is given in (2.1).
which contradicts . □
We will show that Theorem 3.6 is also true if in (3.20) for .
which contradicts (3.20) for .
So we have the following conclusion.
for all and some sufficiently large , where is given in (2.1).
Is Theorem 3.7 also true if is replaced by ? In general, it is not true. It is easy to see that (3.23) holds for , which does not imply (3.1) for all . Then, if (3.12)-(3.13) hold, by Theorem 3.3, all bounded solutions of (1.1) may be nonoscillatory. But it is true for unbounded solutions of (1.1).
for all and some sufficiently large , where is given in (2.1).
which contradicts (3.24). The proof is complete. □
We would like to illustrate the results by means of the following examples.
According to Corollary 3.1, any bounded solution of (4.1) is oscillatory.
for some positive constant k and . Condition (C1) is satisfied. Since , (3.12) holds. Equations (4.2) and (3.17) imply (3.18) holds, so we give a sufficient and necessary condition for all solutions of (4.1) being oscillatory.
Let , , then (3.19) holds. Hence, by Theorem 3.6, all solutions of (1.1) are oscillatory because (4.2) and (3.17) imply (3.20).
In particular, (4.1) becomes the classical difference equation if .
Remark 4.1 Let , be a constant; ; , where are the so-called harmonic numbers, , for . Then the conclusion of Example 4.1 is also true under the same assumptions.
According to Corollary 3.1, any bounded solution of (4.3) is oscillatory.
Similar to Example 4.1, for any , it is easy to check that condition (C1) () and (3.19)-(3.20) () hold, respectively. Then we see that (4.3) is oscillatory.
In particular, (4.3) becomes the classical differential equations if .
By Theorem 3.5, (4.4) is oscillatory.
The authors would like to express their great gratitude to the anonymous valuable suggestions and comments, which helped the authors to improve the previous version of this article. This work was supported by the NNSF of P.R. China (Grant No. 11271379).
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