Abstract and Applied Analysis

Volume 2014 (2014), Article ID 105719, 22 pages

http://dx.doi.org/10.1155/2014/105719

Research Article

Positive Solutions and Mann Iterations of a Fourth Order Nonlinear Neutral Delay Differential Equation

Zeqing Liu,¹ Jingjing Zhu,² Jeong Sheok Ume,³ and Shin Min Kang⁴

¹Department of Mathematics, Liaoning Normal University, Dalian, Liaoning 116029, China

²Dalian No. 39 Middle School Dalian, Liaoning 116001, China

³Department of Mathematics, Changwon National University, Changwon 641-773, Republic of Korea

⁴Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 16 December 2013; Accepted 20 February 2014; Published 29 May 2014

Academic Editor: Allan Peterson

Copyright © 2014 Zeqing Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

This paper deals with a fourth order nonlinear neutral delay differential equation. By using the Banach fixed point theorem, we establish the existence of uncountably many bounded positive solutions for the equation, construct several Mann iterative sequences with mixed errors for approximating these positive solutions, and discuss some error estimates between the approximate solutions and these positive solutions. Seven nontrivial examples are given.

1. Introduction and Preliminaries

The oscillation, nonoscillation, and existence of solutions for various kinds of second order and third order neutral delay differential equations have been extensively studied over the last decades; for example, see [1–12]. Elbert [2] and Huang [3] established a few oscillation and nonoscillation criteria for the second order linear differential equation

where . Tang and Liu [9] studied the existence of bounded oscillation for the second order linear delay differential equation of unstable type

where , , and on any interval of length . Using the Banach fixed point theorem, Kulenović and Hadžiomerspahić [4] deduced the existence of a nonoscillatory solution for the second order linear neutral delay differential equation with positive and negative coefficients

where , , ,  , . Lin [5] suggested a few sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral differential equation

where , , , , , is nondecreasing, and , . Qin et al. [8] and Yang et al. [11] developed several oscillation criteria for the second order differential equation

where and are nonnegative constants, , and . By utilizing Krasnoselskii’s fixed point theorem, Zhou [12] discussed the existence of nonoscillatory solutions of the second order nonlinear neutral differential equation

where is an integer, , , , and for . Yu and Wang [10] studied the existence of a nonoscillatory solution for the second order nonlinear neutral delay differential equations with positive and negative coefficients

where , , , . Liu and Kang [7] investigated the existence of nonoscillatory solutions of the second order nonlinear neutral delay differential equation

where , with for , , , , and with

Kang et al. [13] discussed the existence of nonoscillatory solutions of the third order nonlinear neutral delay differential equation

where is an integer, , , , and .

Motivated by the papers mentioned above, in this paper, we investigate the following fourth order nonlinear neutral delay differential equation:

where , , , , , , , and with

Utilizing the contraction mapping principle, we show the existence of uncountably many bounded positive solutions for (11), construct a few Mann type iterative schemes with mixed errors for these positive solutions, and discuss error estimates between the approximate solutions and the bounded positive solutions. Seven nontrivial examples are considered to illustrate our results.

Throughout this paper, we assume that , denotes the set of positive integers, , and

By a solution of (11), we mean a function for some , such that is continuously differentiable in and (11) is satisfied for .

Let denote the Banach space of all continuous and bounded functions on with the norm for each and

It is easy to see that is a bounded closed and convex subset of .

The following lemma plays an important role in this paper.

Lemma 1 (see [6]). Let ,  ,  , and be four nonnegative real sequences satisfying the inequality

where , , , and . Then .

2. Uncountably Many Bounded Positive Solutions and Iterative Approximations

Now we study the solvability of (11).

Theorem 2. Assume that there exist constants , and and functions , and satisfying

Then,

(a) for any , there exist and such that for each , the Mann iterative sequence with mixed errors generated by the scheme

 converges to a bounded positive solution of (11) and has the following error estimate:

 where is an arbitrary sequence in and and are any sequences in such that

(b) equation (11) possesses uncountably many bounded positive solutions in .

Proof. Firstly, we show that (a) holds. Let . It follows from (18) and (19) that there exist and satisfying

Define a mapping by

Obviously, is continuous for each . Combining (16), (17), (19), (20), and (25)–(27), we derive that for and

which mean that

That is, is a contraction mapping in and has a unique fixed point , which is a bounded positive solution of (11). By virtue of (21), (27), and (29), we get that for any and

which yielded that

That is, (22) holds. Thus Lemma 1, (23), and (24) ensure that .

Next we show that (b) holds. Let with . As in the proof of (a), we conclude that for each there exist ,  , and satisfying (25)–(27), where are replaced by , and , respectively, and the contraction mapping has a unique fixed point , which is also a bounded positive solution of (11). In order to prove (b), we need only to show that . Put . Note that for and

which together with (16) and (25) implies that for

which yields that

That is, . This completes the proof.

Theorem 3. Assume that there exist constants , and and functions ,  and satisfying (16)–(18) and

Then

(a) for any , there exist and such that, for each , the Mann iterative sequence generated by the scheme (21) with (23) and (24) converges to a bounded positive solution of (11) and has the error estimate (22); (b) equation (11) has uncountably many bounded positive solutions in .

Proof. Let . It follows from (18) and (35) that there exist and satisfying

Define a mapping by (27). Clearly is continuous for each . On account of (16), (17), (27), (35), (36), and (37), we infer that for and

which imply that (29) holds. The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 4. Assume that there exist constants and and functions and satisfying (17), (18), and

Then

(a) for any , there exist and such that, for each , the Mann iterative sequence generated by the scheme (21) with (23) and (24) converges to a bounded positive solution of (11) and has the error estimate (22); (b) equation (11) has uncountably many bounded positive solutions in .

Proof. Let . Equations (18) and (39) guarantee that there exist and satisfying (36) and

Let the mapping be defined by (27). Using (17), (27), (39), and (40), we deduce that for any and

which mean that (29) holds. That is, is a contraction mapping and possesses a unique fixed point , which is a bounded positive solution of (11). The rest of the proof is similar to that of Theorem 2 and is omitted. This completes the proof.

Theorem 5. Assume that there exist constants , and and functions ,  and satisfying (16)–(18) and

Then

(a) for any , there exist and such that, for each , the Mann iterative sequence generated by the scheme

 converges to a bounded positive solution of (11) and has the error estimate (22), where is an arbitrary sequence in and and are any sequences in satisfying (23) and (24); (b) equation (11) possesses uncountably many bounded positive solutions in .

Proof. First of all, we show that (a) holds. Let . It follows from (18) that there exist and satisfying

Define a mapping by

Clearly, is continuous for each . Notice that (16), (17), (42), and (44)–(46) ensure that for and

which imply that (29) holds. That is, is a contraction mapping and has a unique fixed point . It follows that

Adding (48) and (49), we infer that

which yields that is a bounded positive solution of (11). By means of (29), (43), and (46), we know that for any and

which gives (22). Thus Lemma 1, (23), and (24) ensure that .

Now we show that (b) holds. Let with . As in the proof of (a), for each , we infer that there exists ,  , and satisfying (44)–(46), where are replaced by , respectively, and the contraction mapping possesses a unique fixed point , and is a bounded positive solution of (11); that is,

Put . Using (16), (44), and (52), we conclude that for

which yields that

that is, . This completes the proof.

Theorem 6. Assume that there exist constants and and functions and satisfying (16), (17), and

Then

(a) for any , there exist and such that, for each , the Mann iterative sequence generated by the scheme

 converges to a bounded positive solution of (11) and has the error estimate (22), where is an arbitrary sequence in , and and are any sequences in satisfying (23) and (24);(b) equation (11) possesses uncountably many bounded positive solutions in .

Proof. Firstly, we show that (a) holds. Let . It follows from (56) that there exist and satisfying

Define a mapping by

Clearly, is continuous for each . In light of (16), (17), and (58)–(60), we conclude that for and

which yield that (29) holds. That is, is a contraction mapping in and has a unique fixed point ; that is,

By virtue of (62) and (63), we get that

which yields that

which ensures that

which gives that is a bounded positive solution of (11). By means of (29), (57), (58), and (60), we deduce that, for any and ,

which implies (22). Thus Lemma 1, (23), and (24) ensure that .

Next we show that (b) holds. Let with . As in the proof of (a), for each , we infer that there exists , and satisfying (58)–(60), where are replaced by , respectively, and the contraction mapping possesses a unique fixed point , and is a bounded positive solution of (11); that is,

Put . Using (16), (58), and (68), we conclude that for

which yields that

that is, . This completes the proof.

Theorem 7. Assume that there exist constants and and functions and satisfying (16)–(18) and

Then

(a) for any , there exist and such that, for each , the Mann iterative sequence with mixed errors generated by the scheme

 with (23) and (24) converges to a bounded positive solution of (11) and has the error estimate (22); (b) equation (11) possesses uncountably many bounded positive solutions in .

Proof. In the first place, we prove that (a) holds. Let . It follows from (18) and (71) that there exist and satisfying

Define a mapping by

Obviously, is continuous for each . In view of (16), (17), (71), (72), and (74)–(76), we conclude that for and

which imply that (29) holds. That is, is a contraction mapping in and has a unique fixed point , which is a bounded positive solution of (11). By means of (29), (73), and (76), we obtain that for any and

which gives (22). Thus Lemma 1, (23), and (24) mean that .

Next we show that (b) holds. Let with . As in the proof of (a), we conclude that, for each , there exist and satisfying (74)–(76), where are replaced by , and , respectively, and the contraction mapping has a unique fixed point , which is also a bounded positive solution of (11). In order to prove (b), we need only to show that . Put . Note that for and

which together with (16) and (74) implies that for

which yields that

that is, . This completes the proof.

Theorem 8. Assume that there exist constants and and functions and satisfying (16)–(18) and

Then

(a) for any , there exist and such that, for each , the Mann iterative sequence with mixed errors generated by the scheme

 with (23) and (24) converges to a bounded positive solution of (11) and has the error estimate (22);(b) equation (11) possesses uncountably many bounded positive solutions in .

Proof. First of all, we prove that (a) holds. Let . It follows from (18) and (82) that there exist and satisfying (74) and

Define a mapping by

Obviously is continuous for each . On account of (17), (82), (84), and (85), we get that for any and

which imply (29). That is, is a contraction mapping in and has a unique fixed point , which is a bounded positive solution of (11). The rest of the proof is similar to that of Theorem 7 and is omitted. This completes the proof.

3. Examples

Now we construct seven examples as applications of the results presented in Section 2.

Example 9. Consider the following fourth order nonlinear neutral delay differential equation:

where , , , , , , , , and

It is easy to verify that (16)–(20) are satisfied. It follows from Theorem 2 that (11) possesses uncountably many bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with mixed errors generated by (21) converges to a bounded positive solution of (11) and has the error estimate (22), where is an arbitrary sequence in , and   and are any sequences in satisfying (23) and (24).

Example 10. Consider the following fourth order nonlinear neutral delay differential equation:

where , , , , , , , , and

It is easy to verify that (16)–(18) and (35) are satisfied. It follows from Theorem 3 that (11) possesses uncountably many bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with mixed errors generated by (21) converges to a bounded positive solution of (11) and has the error estimate (22), where is an arbitrary sequence in , and and are any sequences in satisfying (23) and (24).

Example 11. Consider the following fourth order nonlinear neutral delay differential equation:

where , , , , , , , , and

It is easy to verify that (17), (18), and (39) are satisfied. It follows from Theorem 4 that (11) possesses uncountably many bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with mixed errors generated by (21) converges to a bounded positive solution of (11) and has the error estimate (22), where is an arbitrary sequence in , and and are any sequences in satisfying (23) and (24).

Example 12. Consider the following fourth order nonlinear neutral delay differential equation:

where , , , , , , and

It is easy to verify that (16)–(18) and (42) are satisfied. It follows from Theorem 5 that (11) possesses uncountably many bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with mixed errors generated by (43) converges to a bounded positive solution of (11) and has the error estimate (22), where is an arbitrary sequence in , and and are any sequences in satisfying (23) and (24).

Example 13. Consider the following fourth order nonlinear neutral delay differential equation:

where , , , , , , and

It is easy to verify that (16), (17), (55), and (56) are satisfied. It follows from Theorem 6 that (11) possesses uncountably many bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with mixed errors generated by (57) converges to a bounded positive solution of (11) and has the error estimate (22), where is an arbitrary sequence in , and and are any sequences in satisfying (23) and (24).

Example 14. Consider the following fourth order nonlinear neutral delay differential equation:

where , , , , , , , , and

It is easy to verify that (16)–(18), (71), and (72) are satisfied. It follows from Theorem 7 that (11) possesses uncountably many bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with mixed errors generated by (73) converges to a bounded positive solution of (11) and has the error estimate (22), where is an arbitrary sequence in , and and are any sequences in satisfying (23) and (24).

Example 15. Consider the following fourth order nonlinear neutral delay differential equation:

where , , , , , , , , and

It is easy to verify that (16)–(18) and (82) are satisfied. It follows from Theorem 8 that (11) possesses uncountably many bounded positive solutions in . On the other hand, for any , there exist and such that the Mann iterative sequence with mixed errors generated by (83) converges to a bounded positive solution of (11) and has the error estimate (22), where is an arbitrary sequence in , and and are any sequences in satisfying (23) and (24).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The first author was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380). The third author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (2013R1A1A2057665).

References

R. P. Agarwal and S. R. Grace, “Oscillation theorems for certain neutral functional differential equations,” Computers and Mathematics with Applications, vol. 38, no. 11, pp. 1–11, 1999.

Á. Elbert, “Oscillation/Nonoscillation criteria for linear second order differential equations,” Journal of Mathematical Analysis and Applications, vol. 226, no. 1, pp. 207–219, 1998.

C. Huang, “Oscillation and nonoscillation for second order linear differential equations,” Journal of Mathematical Analysis and Applications, vol. 210, no. 2, pp. 712–723, 1997.

M. R. S. Kulenović and S. Hadžiomerspahić, “Existence of nonoscillatory solution of second order linear neutral delay equation,” Journal of Mathematical Analysis and Applications, vol. 228, no. 2, pp. 436–448, 1998.

X. Lin, “Oscillation of second-order nonlinear neutral differential equations,” Journal of Mathematical Analysis and Applications, vol. 309, no. 2, pp. 442–452, 2005.

L. S. Liu, “Ishikawa and mann iterative process with errors for nonlinear strongly accretive mappings in banach spaces,” Journal of Mathematical Analysis and Applications, vol. 194, no. 1, pp. 114–125, 1995.

Z. Liu and S. M. Kang, “Infinitely many nonoscillatory solutions for second order nonlinear neutral delay differential equations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 70, no. 12, pp. 4274–4293, 2009.

H. Z. Qin, N. N. Shang, and Y. M. Lu, “A note on oscillation criteria of second order nonlinear neutral delay differential equations,” Computers and Mathematics with Applications, vol. 56, no. 12, pp. 2987–2992, 2008.

X. H. Tang and Y. Liu, “Bounded oscillation for second-order delay differential equations with unstable type in a critical case,” Applied Mathematics Letters, vol. 16, no. 3, pp. 263–268, 2003.

Y. H. Yu and H. Z. Wang, “Nonoscillatory solutions of second-order nonlinear neutral delay equations,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 445–456, 2005.

Q. G. Yang, L. J. Yang, and S. M. Zhu, “Interval criteria for oscillation of second-order nonlinear neutral differential equations,” Computers and Mathematics with Applications, vol. 46, no. 5-6, pp. 903–918, 2003.

Y. Zhou, “Existence for nonoscillatory solutions of second-order nonlinear differential equations,” Journal of Mathematical Analysis and Applications, vol. 331, no. 1, pp. 91–96, 2007.

S. M. Kang, Z. Liu, L. Chen, and S. Y. Cho, “Existence of nonoscillatory solutions for a third-order nonlinear neutral delay differential equation,” Abstract and Applied Analysis, vol. 2011, Article ID 693890, 23 pages, 2011.