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 [112]. 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).