Abstract and Applied Analysis

Volume 2013 (2013), Article ID 165382, 28 pages

http://dx.doi.org/10.1155/2013/165382

Research Article

Existence and Iterative Algorithms of Positive Solutions for a Higher Order Nonlinear Neutral Delay Differential Equation

Zeqing Liu,¹ Ling Guan,¹ Sunhong Lee,² and Shin Min Kang²

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

²Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Republic of Korea

Received 23 October 2012; Accepted 10 December 2012

Academic Editor: Jifeng Chu

Copyright © 2013 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 is concerned with the higher order nonlinear neutral delay differential equation for all . Using the Banach fixed point theorem, we establish the existence results of uncountably many positive solutions for the equation, construct Mann iterative sequences for approximating these positive solutions, and discuss error estimates between the approximate solutions and the positive solutions. Nine examples are included to dwell upon the importance and advantages of our results.

1. Introduction and Preliminaries

In recent years, the existence problems of nonoscillatory solutions for neutral delay differential equations of first, second, third, and higher order have been studied intensively by using fixed point theorems; see, for example, [1–12] and the references therein.

Using the Banach, Schauder, and Krasnoselskii fixed point theorems, Zhang et al. [9] and Liu et al. [7] considered the existence of nonoscillatory solutions for the following first order neutral delay differential equations:

where and . Making use of the Banach and Krasnoselskii fixed point theorems, Kulenović and Hadžiomerspahić [2] and Zhou [10] studied the existence of a nonoscillatory solution for the following second order neutral differential equations:

where and . Zhou and Zhang [11], Zhou et al. [12], and Liu et al. [4], respectively, investigated the existence of nonoscillatory solutions for the following higher order neutral delay differential equations:

where , and . Candan [1] proved the existence of a bounded nonoscillatory solution for the higher order nonlinear neutral differential equation:

where .

Motivated by the results in [1–12], in this paper we consider the following higher order nonlinear neutral delay differential equation:

where and with , , , , and with

It is clear that (5) includes (1)–(4) as special cases. Utilizing the Banach fixed point theorem, we prove several existence results of uncountably many positive solutions for (5), construct a few Mann iterative schemes, and discuss error estimates between the sequences generated by the Mann iterative schemes and the positive solutions. Nine examples are given to show that the results presented in this paper extend substantially the existing ones in [1, 2, 4, 5, 8, 9, 11].

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

stands for the Banach space of all continuous and bounded functions in with norm , and for any

It is easy to check that , and are closed subsets of .

By a solution of (5), we mean a function for some , such that are times continuously differentiable in and such that (5) is satisfied for .

Lemma 1. Let , and . Then

(a); (b); (c)if , then

(d)if , then

Proof. Let denote the largest integral number not exceeding . Note that

Clearly (12) means that

Thus (a) follows from (11) and (13).

Assume that . As in the proof of (a), we infer that

that is, (c) holds.

Similar to the proofs of (a) and (c), we conclude that (b) and (d) hold. This completes the proof.

2. Existence of Uncountably Many Positive Solutions and Mann Iterative Schemes

Now we show the existence of uncountably many positive solutions for (5) and discuss the convergence of the Mann iterative sequences to these positive solutions.

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

Then

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

converges to a positive solution of (5) and has the following error estimate:

where is an arbitrary sequence in such that

Equation (5) has uncountably many positive solutions in .

Proof. Firstly, we prove that (a) holds. Set . From (15) and (18), we know that there exist and satisfying

Define a mapping by

It is obvious that is continuous for each . By means of (16), (22), (23), and (25), we deduce that for any and

which yields that

On the basis of (17), (22), (24), and (25), we acquire that for any and

which guarantee that . Consequently, (27) gives that is a contraction mapping in and it has a unique fixed point . It is easy to see that is a positive solution of (5).

It follows from (19), (25), and (27) that

which yields that

That is, (20) holds. Thus (20) and (21) ensure that .

Secondly, we show that (b) holds. Let with . In light of (15) and (18), we know that for each , there exist , and with and satisfying (22)–(24) and

where and are replaced by and , respectively. Let the mapping be defined by (25) with and replaced by and , respectively. As in the proof of (a), we deduce easily that the mapping possesses a unique fixed point , that is, is a positive solution of (5) in . In order to prove (b), we need only to show that . In fact, (25) means that for each and

It follows from (16), (22), (31), and (32) that for each

which implies that

that is, . This completes the proof.

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

Then

(a)for any , there exist and such that for each , the Mann iterative sequence generated by (19) converges to a positive solution of (5) and has the error estimate (20), where is an arbitrary sequence in satisfying (21);(b)Equation (5) has uncountably many positive solutions in .

Proof. Let . Equations (18) and (36) ensure that there exist and satisfying (23),

Define a mapping by (25). Obviously, is continuous for every . Using (16), (23), (25), and (36), we conclude that for any and

which implies that (27) holds. In light of (17), (25), (36), and (37), we know that for any and

which mean that . Equation (27) guarantees that is a contraction mapping in and it possesses a unique fixed point . As in the proof of Theorem 2, we infer that is a positive solution of (5). 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 three constants , , and and four functions satisfying (16)–(18) and

Then

(a)for any , there exist and such that for each , the Mann iterative sequence generated by (19) converges to a positive solution of (5) and has the error estimate (20), where is an arbitrary sequence in satisfying (21);(b)Equation (5) has uncountably many positive solutions in .

Proof. Set . It follows from (18) and (40) that there exist and satisfying (23),

Define a mapping by (25). Distinctly, is continuous for each . In terms of (16), (23), (25), and (41), we reason that for any and

which means that (27) holds. Owing to (17), (25), (41), and (42), we earn that for any and

which yield that . Thus (27) ensures that is a contraction mapping in and it owns a unique fixed point . As in the proof of Theorem 2, we infer that is a positive solution of (5). The rest of the proof is parallel to that of Theorem 2, and hence is elided. This completes the proof.

Theorem 5. Assume that there exist three constants , , and and four functions satisfying (18) and

Then

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

converges to a positive solution of (5) and has the error estimate (20), where is an arbitrary sequence in with (21);

Equation (5) has uncountably many positive solutions in .

Proof. First of all, we show that (a) holds. Set . It follows from (18) and (45) that there exist and such that

Define a mapping by

In light of (46), (49), (50), and (52), we conclude that for and

which yields that

In view of (47), (49), (51), and (52), we obtain that for any and

which imply that . It follows from (50) and (54) that is a contraction mapping in and it has a unique fixed point . It is clear that is a positive solution of (5).

Note that (48), (52), and (54) undertake that

which indicates that (20) holds. Thus (20) and (21) assure that .

Next we prove that (b) holds. Let with . As in the proof of (a) we infer that for each there exist , and satisfying (49)–(52), where , , , and are replaced by , , , and , respectively, and has a unique fixed point , which is a positive solution of (5) in . It follows that for each and

On behalf of proving (b), we need only to show that . Notice that (18) guarantees that there exits satisfying

Due to (46), (51), (57), and (58), we conclude that for each

which yields that . This completes the proof.

Theorem 6. Assume that there exist three constants , , and and four functions satisfying (18), (46), (47), and

Then

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

converges to a positive solution of (5) and has the error estimate (20), where is an arbitrary sequence in satisfying (21);

Equation (5) has uncountably many positive solutions in .

Proof. Put . It follows from (18) and (60) that there exist and satisfying (50) and

Define a mapping by

By virtue of (47), (62), and (63), we know that for any and

which imply that . The rest of the proof is identical with the proof of Theorem 5 and hence is omitted. This completes the proof.

Theorem 7. Let . Assume that there exist two constants , with and four functions satisfying (16)–(18) and

Then

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

converges to a positive solution of (5) and has the error estimate (20), where is an arbitrary sequence in with (21);

Equation (5) has uncountably many positive solutions in .

Proof. Let . It follows from (18) and (65) that there exist and satisfying

Define a mapping by

With a view to (16), (68), and (70), we derive that for any and

which gives (27). By virtue of (17), (69), and (70), we deduce that for any and

which mean that . Coupled with (27) and (68), we get that is a contraction mapping in and it possesses a unique fixed point . Clearly, is a positive solution of (5).

From (27), (66), and (70), we gain that

which yields (20). It follows from (20) and (21) that .

Now we prove that (b) holds. Let and . As in the proof of (a), we conclude that for each , there exist , and satisfying (69)–(77), where , , , and are replaced by , , , and , respectively, and has a unique fixed point , which is a positive solution of (5) in , that is,

For purpose of proving (b), we just need to show that . It follows from (16), (27), (68), and (74) that

which yields that

that is, . This completes the proof.

Theorem 8. Let . Assume that there exist two constants , with and four functions satisfying (16), (17), (65), and

Then

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

converges to a positive solution of (5) and has the error estimate (20), where is an arbitrary sequence in with (21);

Equation (5) has uncountably many positive solutions in .

Proof. Let . It follows from (65) and (77) that there exist and satisfying (67),

Define a mapping by

By virtue of (16), (79), and (81), we derive that for any and

which gives (27). It follows from (17), (80), and (81) that for any and

which mean that . Combined with (27) and (79), we know that is a contraction mapping in and it possesses a unique fixed point . Obviously, is a positive solution of (5).

In light of (27), (78), and (81), we gain that

which yields (20). It follows from (20) and (21) that .

Now we prove that (b) holds. Let and . As in the proof of (a), we conclude that for each , there exist , and satisfying (67) and (79)–(81), where , , , and are replaced by , , , and , respectively, and has a unique fixed point , which is a positive solution of (5) in , that is,

In order to prove (b), we just need to show that . In view of (16), (27), (79), and (85), we get that

which implies that

that is, . This completes the proof.

Theorem 9. Let . Assume that there exist two constants , with and four functions satisfying (16), (17),

Then

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

converges to a positive solution of (5) and has the error estimate (20), where is an arbitrary sequence in with (21);

Equation (5) has uncountably many positive solutions in .

Proof. Set . In view of (88) and (89), there exist and such that

Define a mapping by

By virtue of (16), (92), (94), and Lemma 1, we acquire that for any and

which yields that (27) holds. From (17), (94), (98), and Lemma 1, we obtain that for any and

which means that . It follows from (27) and (92) that is a contraction mapping and it has a unique fixed point . It is clear that is a positive solution of (5).

On the basis of (27), (90), and (94), we deduce that

which signifies that (20) holds. It follows from (20) and (21) and that .

Now we show that (b) holds. Let and . As in the proof of (a), we conclude that for each , there exist , and satisfying (91)–(94), where , , , and are replaced by , , , and , respectively, and has a unique fixed point , which is a positive solution of (5) in , that is,

In order to prove (b), it is sufficient to show that . Note that (16), (92), (98), and Lemma 1 lead to

which means that

that is, . This completes the proof.

Theorem 10. Let . Assume that there exist two constants , with and four functions satisfying (16), (17), (89), and

Then

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

converges to a positive solution of (5) and has the error estimate (20), where is an arbitrary sequence in with (21);

Equation (5) has uncountably many positive solutions in .

Proof. Set . Due to (101), there exist and satisfying (91),

Define a mapping by

In view of (16), (103), (105), and Lemma 1, we achieve that for any and

which means that (27) holds. It follows from (17), (104), (105), and Lemma 1 that for any and

which means that . Coupled with (27), we know that is a contraction mapping and it has a unique fixed point . It follows that is a positive solution of (5).

In view of (27), (102), and (105), we deduce that

which signifies that (20) holds. It follows from (20) and (21) that .

Now we show that (b) holds. Let and . As in the proof of (a), we conclude that for each , there exist , and satisfying (91) and (103)–(105), where , , , and are replaced by , , and , respectively, and has a unique fixed point , which is a positive solution of (5) in . It follows that for any and

In order to prove (b), we just need to show that . Notice that (16), (103), (109), and Lemma 1 ensure that

which yields that

that is, . This completes the proof.

3. Remark and Examples

Remark 11. Theorems 2–10 extend, improve, and unifies Theorems 1–4 in [1], the theorem in [2], Theorems 2.1–2.4 in [4], Theorems 2.1–2.5 in [5, 8], Theorems 1–3 in [9], and Theorems 1–4 in [11], respectively. The examples below prove that Theorems 2–10 extend substantially the corresponding results in [1, 2, 4, 5, 8, 9, 11]. Note that none of the known results can be applied to these examples.

Example 12. Consider the higher order nonlinear neutral delay differential equation

where and . Let , , , , , and

It is easy to verify that the conditions of Theorem 2 are satisfied. Thus Theorem 2 ensures that (112) has uncountably many positive solutions in , and for any , there exist and such that the Mann iterative sequence generated by (19) and (21) converges to a positive solution of (112) and has the error estimate (20).

Example 13. Consider the higher order nonlinear neutral delay differential equation

where and . Let , , , , , and

It is easy to check that the conditions of Theorem 3 are satisfied. Therefore (114) has uncountably many positive solutions in , and for any , there exist and such that the Mann iterative sequence generated by (19) and (21) converges to a positive solution of (114) and has the error estimate (20).

Example 14. Consider the higher order nonlinear neutral delay differential equation

where and . Let , , , , , and

It is easy to prove that the conditions of Theorem 4 are satisfied. Hence (116) has uncountably many positive solutions in , and for any , there exist and such that the Mann iterative sequence generated by (19) and (21) converges to a positive solution of (116) and has the error estimate (20).

Example 15. Consider the higher order nonlinear neutral delay differential equation

where and . Let , , , , , and

It is easy to verify that the conditions of Theorem 5 are satisfied. Hence Theorem 5 ensures that (118) has uncountably many positive solutions in , and, for any , there exist and such that the Mann iterative sequence generated by (48) and (21) converges to a positive solution of (118) and has the error estimate (20).

Example 16. Consider the higher order nonlinear neutral delay differential equation

where , and . Let , , , , , and

It is easy to check that the conditions of Theorem 6 are satisfied. Thus Theorem 6 ensures that (120) has uncountably many positive solutions in , and, for any , there exist and such that the Mann iterative sequence generated by (61) and (21) converges to a positive solution of (120) and has the error estimate (20).

Example 17. Consider the higher order nonlinear neutral delay differential equation

where , and . Let , , , , and

It is easy to check that the conditions of Theorem 7 are satisfied. Thus (122) has uncountably many positive solutions in , and, for any , there exist and such that the Mann iterative sequence generated by (66) and (21) converges to a positive solution of (122) and has the error estimate (20).

Example 18. Consider the higher order nonlinear neutral delay differential equation

where , and . Let , , , , and

It is easy to check that the conditions of Theorem 8 are satisfied. Thus Theorem 8 ensures that (124) has uncountably many positive solutions in , and, for any , there exist and such that the Mann iterative sequence generated by (78) and (21) converges to a positive solution of (124) and has the error estimate (20).

Example 19. Consider the higher order nonlinear neutral delay differential equation

where , and . Let , , , , and

It is easy to check that the conditions of Theorem 9 are satisfied. Thus Theorem 9 ensures that (126) has uncountably many positive solutions in , and, for any , there exist and such that the Mann iterative sequence generated by (90) and (21) converges to a positive solution of (126) and has the error estimate (20).

Example 20. Consider the higher order nonlinear neutral delay differential equation

where , and . Let , , , , and

It is easy to check that the conditions of Theorem 10 are satisfied. Thus Theorem 10 ensures that (128) has uncountably many positive solutions in , and, for any , there exist and such that the Mann iterative sequence generated by (102) and (21) converges to a positive solution of (128) and has the error estimate (20).

Acknowledgments

The authors would like to thank the referees for useful comments and suggestions. This paper was supported by the Science Research Foundation of Educational Department of Liaoning province (L2012380).

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