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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

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 [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