Research

# Oscillatory behavior of second-order nonlinear neutral differential equations with distributed deviating arguments

Tongxing Li1*, Blanka Baculíková2 and Jozef Džurina2

Author Affiliations

1 Qingdao Technological University, Feixian, Shandong, 273400, P.R. China

2 Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, Košice, 042 00, Slovakia

For all author emails, please log on.

Boundary Value Problems 2014, 2014:68  doi:10.1186/1687-2770-2014-68

 Received: 16 January 2014 Accepted: 6 March 2014 Published: 24 March 2014

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

### Abstract

We study oscillatory properties of a class of second-order nonlinear neutral functional differential equations with distributed deviating arguments. On the basis of less restrictive assumptions imposed on the neutral coefficient, some new criteria are presented. Three examples are provided to illustrate these results.

MSC: 34C10, 34K11.

##### Keywords:
oscillation; neutral differential equation; second-order equation; distributed deviating argument

### 1 Introduction

This paper is concerned with oscillation of the second-order nonlinear functional differential equation

(1.1)

where , is a constant, and . Throughout, we assume that the following hypotheses hold:

(H1) , , , and ;

(H2) and is not eventually zero on any , ;

(H3) , , and for ;

(H4) , , , and ;

(H5) is nondecreasing and the integral of (1.1) is taken in the sense of Riemann-Stieltijes.

By a solution of (1.1), we mean a function for some , which has the properties that , , and satisfies (1.1) on . We restrict our attention to those solutions x of (1.1) which exist on and satisfy for any . A solution x of (1.1) is termed oscillatory if it is neither eventually positive nor eventually negative; otherwise, it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions oscillate.

As is well known, neutral differential equations have a great number of applications in electric networks. For instance, they are frequently used in the study of distributed networks containing lossless transmission lines, which rise in high speed computers, where the lossless transmission lines are used to interconnect switching circuits; see [1]. Hence, there has been much research activity concerning oscillatory and nonoscillatory behavior of solutions to different classes of neutral differential equations, we refer the reader to [2-30] and the references cited therein.

In the following, we present some background details that motivate our research. Recently, Baculíková and Lacková [6], Džurina and Hudáková [12], Li et al.[15,18], and Sun et al.[22] established some oscillation criteria for the second-order half-linear neutral differential equation

where ,

Baculíková and Džurina [4,5] and Li et al.[17] investigated oscillatory behavior of a second-order neutral differential equation

where

(1.2)

Ye and Xu [26] and Yu and Fu [27] considered oscillation of the second-order differential equation

Assuming , Thandapani and Piramanantham [23], Wang [24], Xu and Weng [25], and Zhao and Meng [30] studied oscillation of an equation

As yet, there are few results regarding the study of oscillatory properties of (1.1) under the conditions or . Thereinto, Li and Thandapani [19] obtained several oscillation results for (1.1) in the case where (1.2) holds, , and

(1.3)

In the subsequent sections, we shall utilize the Riccati substitution technique and some inequalities to establish several new oscillation criteria for (1.1) assuming that (1.3) holds or

(1.4)

All functional inequalities are assumed to hold eventually, that is, they are satisfied for all t large enough.

### 2 Main results

In what follows, we use the following notation for the convenience of the reader:

where h, ρ, and η will be specified later.

Theorem 2.1Assume (H1)-(H5), (1.3), and let, , , andfor. Suppose further that there exists a real-valued functionsuch thatforand. If there exists a real-valued functionsuch that

(2.1)

then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a such that , , and for all and . Then . Applying (1.1), one has, for all sufficiently large t,

Using the inequality (see [[5], Lemma 1])

the definition of z, , and , we conclude that

(2.2)

By virtue of (1.1), we get

(2.3)

Thus, is nonincreasing. Now we have two possible cases for the sign of : (i) eventually, or (ii) eventually.

(i) Assume that for . Then we have by (2.3)

which yields

Then we obtain due to (1.3), which is a contradiction.

(ii) Assume that for . It follows from (2.2) and that

(2.4)

We define a Riccati substitution

(2.5)

Then . From (2.3) and , we have

(2.6)

Differentiating (2.5), we get

(2.7)

Therefore, by (2.5), (2.6), and (2.7), we see that

(2.8)

Similarly, we introduce another Riccati transformation:

(2.9)

Then . From (2.3) and , we obtain

(2.10)

Differentiating (2.9), we have

(2.11)

Therefore, by (2.9), (2.10), and (2.11), we find

(2.12)

Combining (2.8) and (2.12), we get

It follows from (2.4) that

Integrating the latter inequality from to t, we obtain

(2.13)

Define

Using the inequality

(2.14)

we get

On the other hand, define

Then we have by (2.14)

Thus, from (2.13), we get

which contradicts (2.1). This completes the proof. □

Assuming (1.2), where and are constants, we obtain the following result.

Theorem 2.2Suppose (H1)-(H5), (1.2), (1.3), and let, , , andfor. If there exists a real-valued functionsuch that

(2.15)

then (1.1) is oscillatory.

Proof As above, let x be an eventually positive solution of (1.1). Proceeding as in the proof of Theorem 2.1, we have , (2.3), and (2.4) for all sufficiently large t. Using (1.2), (2.3), and (2.4), we obtain

(2.16)

The remainder of the proof is similar to that of Theorem 2.1, and hence it is omitted. □

Theorem 2.3Suppose we have (H1)-(H5), (1.3), and letandfor. Assume also that there exists a real-valued functionsuch thatforand. If there exists a real-valued functionsuch that

(2.17)

then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a such that , , and for all and . As in the proof of Theorem 2.1, we obtain (2.3) and (2.4). In view of (2.3), is nonincreasing. Now we have two possible cases for the sign of : (i) eventually, or (ii) eventually.

(i) Suppose that for . Then, with a proof similar to the proof of case (i) in Theorem 2.1, we obtain a contradiction.

(ii) Suppose that for . We define a Riccati substitution

(2.18)

Then . From (2.3) and , we have

(2.19)

Differentiating (2.18), we obtain

(2.20)

Therefore, by (2.18), (2.19), and (2.20), we see that

(2.21)

Similarly, we introduce another Riccati substitution:

(2.22)

Then . Differentiating (2.22), we have

(2.23)

Therefore, by (2.22) and (2.23), we get

(2.24)

Combining (2.21) and (2.24), we have

It follows from (2.4) and that

Integrating the latter inequality from to t, we obtain

(2.25)

Define

Using inequality (2.14), we have

On the other hand, define

Then, by (2.14), we obtain

Thus, from (2.25), we get

which contradicts (2.17). This completes the proof. □

Assuming we have (1.2), where and are constants, we get the following result.

Theorem 2.4Suppose we have (H1)-(H5), (1.2), (1.3), and letandfor. If there exists a real-valued functionsuch that

(2.26)

then (1.1) is oscillatory.

Proof Assume again that x is an eventually positive solution of (1.1). As in the proof of Theorem 2.1, we have , (2.3), and (2.4) for all sufficiently large t. By virtue of (1.2), (2.3), and (2.4), we have (2.16) for all sufficiently large t. The rest of the proof is similar to that of Theorem 2.3, and so it is omitted. □

In the following, we present some oscillation criteria for (1.1) in the case where (1.4) holds.

Theorem 2.5Suppose we have (H1)-(H5), (1.2), (1.4), and let, , for, andfor. Assume further that there exists a real-valued functionsuch that (2.15) is satisfied. If there exists a real-valued functionsuch that, , for, and

(2.27)

then (1.1) is oscillatory.

Proof Let x be a nonoscillatory solution of (1.1). Without loss of generality, we assume that there exists a such that , , and for all and . Then . As in the proof of Theorem 2.1, we get (2.2). By virtue of (1.1), we have (2.3). Thus, is nonincreasing. Now we have two possible cases for the sign of : (i) eventually, or (ii) eventually.

(i) Suppose that for . Then, by the proof of Theorem 2.2, we obtain a contradiction to (2.15).

(ii) Suppose that for . It follows from (2.2), (2.3), and that

(2.28)

We define the function u by

(2.29)

Then . Noting that is nondecreasing, we get

Integrating this inequality from to l, we obtain

Letting , we have

That is,

Thus, we get by (2.29)

(2.30)

Similarly, we define another function v by

(2.31)

Then . Noting that is nondecreasing and , we get

Thus, . Hence, by (2.30), we see that

(2.32)

Differentiating (2.29), we obtain

By (2.3) and , we have , and so

(2.33)

Similarly, we see that

(2.34)

Combining (2.33) and (2.34), we get

(2.35)

Using (2.28), (2.35), and , we obtain

(2.36)

Multiplying (2.36) by and integrating the resulting inequality from to t, we have

Set

Using inequality (2.14), we get

Similarly, we set

Then we have by (2.14)

Thus, from (2.30) and (2.32), we find

which contradicts (2.27). This completes the proof. □

With a proof similar to the proof of Theorems 2.4 and 2.5, we obtain the following result.

Theorem 2.6Suppose we have (H1)-(H5), (1.2), (1.4), and let, for, andfor. Assume also that there exists a real-valued functionsuch that (2.26) is satisfied. If there exists a real-valued functionsuch that, , for, and (2.27) holds, then (1.1) is oscillatory.

### 3 Applications and discussion

In this section, we provide three examples to illustrate the main results.

Example 3.1 Consider the second-order neutral functional differential equation

(3.1)

Let , , , , , , , , , and . Then , , for , and . Moreover, letting , then

Hence, by Theorem 2.2, (3.1) is oscillatory. As a matter of fact, one such solution is .

Example 3.2 Consider the second-order neutral functional differential equation

(3.2)

where is a constant. Let , , , , , , , , , and . Then , for , , and for . Further, setting ,

and

Therefore, we have

Hence, (3.2) is oscillatory due to Theorem 2.3.

Example 3.3 Consider the second-order neutral functional differential equation

(3.3)

where , and β are positive constants. Let , , , , , , , , , and . Then , , for , , for , and . Further,

and

Hence, by Theorem 2.6, (3.3) is oscillatory when .

Remark 3.1 In this paper, we establish some new oscillation theorems for (1.1) in the case where p is finite or infinite on . The criteria obtained extend the results in [22] and improve those reported in [19]. Similar results can be presented under the assumption that . In this case, using [[5], Lemma 2], one has to replace with and proceed as above. It would be interesting to find another method to investigate (1.1) in the case where .

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally to this work. They all read and approved the final version of the manuscript.

### Acknowledgements

The authors express their sincere gratitude to the anonymous referees for the careful reading of the original manuscript and useful comments that helped to improve the presentation of the results and accentuate important details.

### References

1. Hale, JK: Theory of Functional Differential Equations, Springer, New York (1977)

2. Agarwal, RP, Bohner, M, Li, W-T: Nonoscillation and Oscillation Theory for Functional Differential Equations, Dekker, New York (2004)

3. Agarwal, RP, Grace, SR, O’Regan, D: Oscillation Theory for Difference and Functional Differential Equations, Kluwer Academic, Dordrecht (2000)

4. Baculíková, B, Džurina, J: Oscillation theorems for second order neutral differential equations. Comput. Math. Appl.. 61, 94–99 (2011). Publisher Full Text

5. Baculíková, B, Džurina, J: Oscillation theorems for second order nonlinear neutral differential equations. Comput. Math. Appl.. 62, 4472–4478 (2011). Publisher Full Text

6. Baculíková, B, Lacková, D: Oscillation criteria for second order retarded differential equations. Stud. Univ. Zilina Math. Ser.. 20, 11–18 (2006)

7. Baculíková, B, Li, T, Džurina, J: Oscillation theorems for second-order superlinear neutral differential equations. Math. Slovaca. 63, 123–134 (2013). Publisher Full Text

8. Candan, T: The existence of nonoscillatory solutions of higher order nonlinear neutral equations. Appl. Math. Lett.. 25, 412–416 (2012). Publisher Full Text

9. Candan, T, Dahiya, RS: Existence of nonoscillatory solutions of higher order neutral differential equations with distributed deviating arguments. Math. Slovaca. 63, 183–190 (2013). Publisher Full Text

10. Candan, T, Karpuz, B, Öcalan, Ö: Oscillation of neutral differential equations with distributed deviating arguments. Nonlinear Oscil.. 15, 65–76 (2012)

11. Dix, JG, Karpuz, B, Rath, R: Necessary and sufficient conditions for the oscillation of differential equations involving distributed arguments. Electron. J. Qual. Theory Differ. Equ.. 2011, 1–15 (2011)

12. Džurina, J, Hudáková, D: Oscillation of second order neutral delay differential equations. Math. Bohem.. 134, 31–38 (2009)

13. Hasanbulli, M, Rogovchenko, YuV: Oscillation criteria for second order nonlinear neutral differential equations. Appl. Math. Comput.. 215, 4392–4399 (2010). Publisher Full Text

14. Karpuz, B, Öcalan, Ö, Öztürk, S: Comparison theorems on the oscillation and asymptotic behaviour of higher-order neutral differential equations. Glasg. Math. J.. 52, 107–114 (2010). Publisher Full Text

15. Li, T, Agarwal, RP, Bohner, M: Some oscillation results for second-order neutral differential equations. J. Indian Math. Soc.. 79, 97–106 (2012)

16. Li, T, Han, Z, Zhang, C, Li, H: Oscillation criteria for second-order superlinear neutral differential equations. Abstr. Appl. Anal.. 2011, Article ID 367541 (2011)

17. Li, T, Rogovchenko, YuV, Zhang, C: Oscillation of second-order neutral differential equations. Funkc. Ekvacioj. 56, 111–120 (2013)

18. Li, T, Sun, S, Han, Z, Han, B, Sun, Y: Oscillation results for second-order quasi-linear neutral delay differential equations. Hacet. J. Math. Stat.. 42, 131–138 (2013)

19. Li, T, Thandapani, E: Oscillation of second-order quasi-linear neutral functional dynamic equations with distributed deviating arguments. J. Nonlinear Sci. Appl.. 4, 180–192 (2011)

20. Li, WN: Oscillation of higher order delay differential equations of neutral type. Georgian Math. J.. 7, 347–353 (2000)

21. Şenel, MT, Candan, T: Oscillation of second order nonlinear neutral differential equation. J. Comput. Anal. Appl.. 14, 1112–1117 (2012)

22. Sun, S, Li, T, Han, Z, Li, H: Oscillation theorems for second-order quasilinear neutral functional differential equations. Abstr. Appl. Anal.. 2012, Article ID 819342 (2012)

23. Thandapani, E, Piramanantham, V: Oscillation criteria of second order neutral delay dynamic equations with distributed deviating arguments. Electron. J. Qual. Theory Differ. Equ.. 2010, 1–15 (2010)

24. Wang, P: Oscillation criteria for second order neutral equations with distributed deviating arguments. Comput. Math. Appl.. 47, 1935–1946 (2004). Publisher Full Text

25. Xu, Z, Weng, P: Oscillation of second-order neutral equations with distributed deviating arguments. J. Comput. Appl. Math.. 202, 460–477 (2007). Publisher Full Text

26. Ye, L, Xu, Z: Interval oscillation of second order neutral equations with distributed deviating argument. Adv. Dyn. Syst. Appl.. 1, 219–233 (2006)

27. Yu, Y, Fu, X: Oscillation of second order neutral equation with continuous distributed deviating argument. Rad. Mat.. 7, 167–176 (1991)

28. Zhang, C, Agarwal, RP, Bohner, M, Li, T: Oscillation of second-order nonlinear neutral dynamic equations with noncanonical operators. Bull. Malays. Math. Sci. Soc. (2014, in press)

29. Zhang, C, Baculíková, B, Džurina, J, Li, T: Oscillation results for second-order mixed neutral differential equations with distributed deviating arguments. Math. Slovaca (2014, in press)

30. Zhao, J, Meng, F: Oscillation criteria for second-order neutral equations with distributed deviating argument. Appl. Math. Comput.. 206, 485–493 (2008). Publisher Full Text