Abstract
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.
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1 Introduction
In this paper, we investigate oscillation and nonoscillation of second order neutral functional dynamic equations with positive and negative coefficients of the form
where , , , ,
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 ;
(B6) are eventually positive, and satisfy
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 [13], 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. [14] 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
In order to prove our main results, we establish some fundamental results in this section. Now we introduce the auxiliary functions
where , , . For the convenience of discussion, let
Let . Furthermore, we need the following additional hypotheses:
or
First of all, we give the following estimates.
Lemma 2.1 Assume that conditions (B1)-(B6) and one of (2.2)-(2.3) hold. Let be an eventually positive solution of equation (1.1). Then there exists some such that
-
(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 .
Next, we show that is eventually positive. Otherwise there exists a sequence with such that and for some . Without loss of generality, we may assume that , i.e. . For and , let
We rewrite as follows:
By (B6), we see that and then clearly . Similarly, we also have and .
Then, for and , integrating (1.1) from to t, by (B1)-(B6) and the geometric sense of a definite integral, we obtain
It follows that
which contradicts .
Since is a solution of (1.1) satisfying and , we see that
Now, we claim that . By (2.2) or (2.3), and for , we have
(ii) It is clear that or is nonincreasing since for . For and , we have
Dividing both sides of above inequality by , we obtain
Likewise, we also have
and
Hence, (2.4) and (2.5) imply
This gives the desired result
For , , we also get
Therefore, we obtain
which yields the desired result
(iii) It can be proved similar to [[7], Lemma 2.1] and hence its proof is omitted here. □
In addition to the above lemmas, we need a method of studying separated boundary value problems (SBVP) to prove our main results. Namely, we will define functions called upper and lower solutions that not only imply the existence of a solution of a SBVP, but that also provide bounds on the location of the solution. Consider the SBVP
where the functions , with and on . We define the set
where the Banach space is equipped with the norm defined by
A function y is called a solution of the equation , on if and the equation holds for all . Next, we define for any the sector by
Definition 2.1 [[3], Definition 6.1]
We call a lower solution of the SBVP (2.6)-(2.7) on provided
Similarly, is called an upper solution of the SBVP (2.6)-(2.7) on provided
The following theorem is an extension of [[3], Theorem 6.5] to .
Theorem 2.1 [[10], Theorem 1.5]
Assume that there exist a lower solution α and an upper solution β of (2.6) with for all . Then
has a solution y with and on .
We end this section with time scale version of the Arzelà-Ascoli theorem (see [[7], Lemma 2.2]) and Kranoselskii’s fixed point theorem (see [7]). These will be used in the proof of Theorem 3.3.
For , let and . Further, let denote all continuous functions mapping into ℝ,
Endowed on with the norm , is a Banach space. Let , we say X is uniformly Cauchy if for any given , there exists a such that for any ,
X is said to be equi-continuous on if for any given , there exists a such that for any and with ,
Lemma 2.2 [[7], 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)
Suppose that X is a Banach space and Ω is a bounded, convex and closed subset of X. Suppose further that there exist two operators such that
-
(i)
for all ;
-
(ii)
U is a contraction mapping;
-
(iii)
S is completely continuous.
Then has a fixed point in Ω.
3 Main results
In this section, we establish our main results.
Theorem 3.1 Assume conditions (B1)-(B6) and one of (2.2)-(2.3) hold. Then any bounded solution of (1.1) is oscillatory in the case
for all and some sufficiently large , where , is given in (2.1).
Proof Assume this not to be the case and let be a bounded nonoscillatory solution of (1.1) which we may assume satisfies
For convenience, let . By Lemma 2.1, we have
Define the function as follows:
On one hand, by (2.2), Lemma 2.1, (B4)-(B6), the monotonicity of , , and , for , we have
On the other hand, we can also obtain (3.3) if (2.2) is replaced by (2.3).
Applying Theorem 2.1 with , the equation
has a solution with on .
It follows from Lemma 2.1 that is nonincreasing and exists (being finite). Integration for implies
Letting , we obtain
It follows that
Integrating again for and by change of integration order [[11], Lemma 1]
we obtain
Consequently, for , we obtain
Since is bounded and is an increasing function of t, it follows that
Let ; by (3.3) and being nondecreasing, we have
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.
Corollary 3.1 Assume conditions (B1)-(B6) and one of (2.2)-(2.3) hold. Then any bounded solution of (1.1) is oscillatory in the case
or
In order to extend Theorem 3.1 to unbounded solutions, we introduce the class Φ of functions ϕ such that is a nondecreasing continuous function of u satisfying , with
We say that satisfies condition (C1) provided for some there exists such that for all , ,
for some positive constant k, .
Theorem 3.2 Suppose and . Assume (B1)-(B6), and one of (2.2)-(2.3) hold. Furthermore, suppose that the function h satisfies condition (C1). If
holds for all and some sufficiently large , where , is given in (2.1), then all solutions of (1.1) are oscillatory.
Proof Assume (3.6) holds for all and let be an eventually positive solution of (1.1) with
As the proof of Theorem 3.1, we have
where .
Let , then . Now multiplying above inequality by , we get
We next define the continuously differentiable real-valued function
Observe that . By the Pötzsche chain rule [[2], Theorem 1.90],
where .
We claim that
for . By change of integration order [[11], Lemma 1], we obtain
For sufficiently large , by condition (C1), it immediately follows that
where . Since , we have
by assumption. Therefore,
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.
Theorem 3.3 Assume conditions (B1)-(B6) hold, and (2.2) or (2.3) holds. If
for some and some sufficiently large , then (1.1) has a bounded nonoscillatory solution.
Proof Let
Assume that (3.8) holds. Without loss of generality, we may assume . Since and on , by change of integration order [[11], Lemma 1], we obtain
By (B2) and (3.9), we can choose large enough such that
According to (B3) and (B4), we see that there exists with such that and , , for .
Define the Banach space as in (2.8), and let
It is easy to verify that Ω is a bounded, convex, and closed subset of . For the sake of convenience, set
For any and , we have
Now we define two operators U and as follows:
and
Next, we will show that U and S satisfy the conditions in Lemma 2.3.
-
(i)
We first prove that for any . Note that for any , . For any and , by (3.10)-(3.11), we have
and
Similarly, we can show that for any and .
-
(ii)
It is not difficult to check that U is a contraction mapping.
-
(iii)
We will prove that S is a completely continuous mapping. It is easy to check that S maps Ω into Ω.
Again, for the sake of convenience, let
Next, we show that the continuity of S. Let and as , then and as . By the monotonicity and continuity of h, as , we have
For , we have
and for . Employing Lebesgue’s dominated convergence theorem [[3], Chapter 5], we get
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.
For any , by (3.10)-(3.11), we have
Then for any given , there exists large enough such that
Hence, for any and , we have
So S Ω is uniformly Cauchy.
Finally, we will prove that S Ω is equi-continuous. For , we have
For , we have
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.
By Lemma 2.3, there exists such that , which indicates that is a solution of (1.1). In particular, for , we have
Let , we obtain the desired result. □
Remark 3.1 Similar to the proof of Theorem 3.3, under the assumptions of Theorem 3.3, if
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.
Theorem 3.4 Suppose and . Assume (B1)-(B6) and one of (2.2)-(2.3) hold, and the function h satisfies condition (C1). Furthermore, suppose that there exists such that
and
Then a sufficient and necessary condition for (1.1) to be oscillatory is that
holds for all and some sufficiently large , where is given in (2.1).
Proof Assume that (3.14) holds for all and let be a nonoscillatory solution of (1.1). Similar to the proof of Theorem 3.2, , we obtain (3.7):
By (3.12)-(3.14), for sufficiently large , we have
where . The rest of the proof is the same as Theorem 3.2. So we leave details to readers.
Conversely, assume that (3.12)-(3.13) hold and (3.14) does not hold for some , then we have
Note that (3.12) implies that is bounded on , and (3.15) holds if and only if
According to (3.12), it follows that for any with , there exists such that for . It follows that for . Then by the monotonicity of h and the fact that for , we have
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. □
If (3.13) is false, (3.14) is not a necessary condition for (1.1) being oscillatory. Suppose that there exist constants , , such that
and
We give another sufficient and necessary condition for (1.1) to be oscillatory.
Theorem 3.5 Suppose and . Assume (B1)-(B6) and one of (2.2)-(2.3) hold, and the function h satisfies condition (C1). Furthermore, suppose that (3.12) and one of (3.16)-(3.17) hold. Then a sufficient and necessary condition for the second order nonlinear neutral dynamic equation (1.1) to be oscillatory is that
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 the case , Theorems 3.2, 3.4, and 3.5 do not hold. We introduce another class Ψ of functions ψ such that is a nondecreasing continuous function of u satisfying , with
Theorem 3.6 Assume , conditions (B1)-(B6) and one of (2.2)-(2.3) hold. Furthermore, suppose that there exists a function which satisfies (3.19). Then any solution of (1.1) is oscillatory in the case
for all , some sufficiently large , and , where is given in (2.1).
Proof Assume not and let be a bounded nonoscillatory solution of (1.1) which we may assume satisfies
Let
As the proof of Theorem 3.1, we also get (3.3), i.e.
By Lemma 2.1, for . Integrating it from to t, for , we have
Let
By (3.21) and (3.22), we obtain
Integrating the above inequality from t to , , we have
Then, similar to the proof of Theorem 3.2, we have
According to (3.19), we have
However, by (3.20) and (3.22), we have
which contradicts . □
We will show that Theorem 3.6 is also true if in (3.20) for .
Let with . Suppose that be a solution of (1.1). Integrating (1.1) from v to s and dividing the resulting equation by , we have
Again, integrating the above mentioned inequality from u to t, by Lemma 2.1 and (3.3), we have
Set and , using (3.22) and , we find
and hence
Thus, by [[9], Lemma 2.1], we get
which implies
Integrating it from to t and letting , we obtain
which contradicts (3.20) for .
So we have the following conclusion.
Theorem 3.7 Assume , conditions (B1)-(B6) and one of (2.2)-(2.3) hold. Then any solution of (1.1) is oscillatory in the case
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).
Theorem 3.8 Assume , conditions (B1)-(B6), and one of (2.2)-(2.3) hold. Then any unbounded solution of (1.1) is oscillatory in the case
for all and some sufficiently large , where is given in (2.1).
Proof Assume be an unbounded solution of (1.1). Integrating (1.1) from to t, using , and by (B1) and Lemma 2.1, we have
which implies that
By (3.3) and for , we get
Let , then
Both sides divide ; integrating it from to , , and noting that , , we have
Letting and , we obtain
which contradicts (3.24). The proof is complete. □
4 Examples
We would like to illustrate the results by means of the following examples.
Example 4.1 Let , , , , is a constant, time scale . Consider the dynamic equation
where , is a constant. , , , , , , , , . We assume that is nondecreasing and satisfies (B1). For sufficiently large , it is not difficult to check that conditions (B2)-(B6), (2.2), and (3.17) hold. By Lemma 2.1, for sufficiently large , we have
and
According to Corollary 3.1, any bounded solution of (4.1) is oscillatory.
Let and , we obtain
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.
Example 4.2 Let , the time scale , , are constants. Consider the dynamic equation
where . We assume that is rd-continuous on which satisfies (B2). For , and , , are defined as follows:
and
Then , , , , . It is easy to check that conditions (B1)-(B6) and (2.2) hold,
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 .
Example 4.3 Let and time scale . Consider the differential equation
where , , , , , , , , , , , . It is not difficult to check that conditions (B1)-(B6) hold. Since , it is easy to see that (2.3) and (3.17) hold. Similar to Example 4.1, we see that the function
satisfies condition (C1). Finally, we show that (3.12) holds. By Lemma 2.1(iii),
so (3.12) holds. For any ,
By Theorem 3.5, (4.4) is oscillatory.
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Acknowledgements
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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The first author discovered the topic, offered the main ideas for the proof of the paper, and carried out writing this article. The first and the second authors discussed the paper together, and all the authors interchanged ideas about this paper. The second and the third authors gave some helpful suggestions for writing the paper and checked the proof of the paper. All authors read and approved the final manuscript.
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Deng, XH., Wang, QR. & Agarwal, R.P. Oscillation and nonoscillation for second order neutral dynamic equations with positive and negative coefficients on time scales. Adv Differ Equ 2014, 115 (2014). https://doi.org/10.1186/1687-1847-2014-115
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DOI: https://doi.org/10.1186/1687-1847-2014-115