Solvability and iterative algorithms for a higher order nonlinear neutral delay differential equation

Applied Mathematics and Computation 215 (2009) 2534–2543

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

Solvability and iterative algorithms for a higher order nonlinearneutral delay differential equation

Zeqing Liu a, Lili Wang a, Shin Min Kang b,*, Jeong Sheok Ume c

a Department of Mathematics, Liaoning Normal University Dalian, Liaoning 116029, People’s Republic of Chinab Department of Mathematics and the Research Institute of Natural Science, Gyeongsang National University, Jinju 660-701, Republic of Koreac Department of Applied Mathematics, Changwon National University, Changwon 641-773, Republic of Korea

a r t i c l e i n f o

Keywords:Higher order nonlinear neutral delaydifferential equationUncountably many nonoscillatory solutionsContraction mappingMann iterative sequence with mixed errors

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.08.054

* Corresponding author.E-mail addresses: [email protected] (Z. Liu), lili_w

a b s t r a c t

The paper is concerned with the higher order nonlinear neutral delay differential equation

. All righ

ang@yah

rðtÞðxðtÞþ pðtÞxðt� sÞÞðmÞh iðn�mÞ

þ ð�1Þn�mþ1f t;x r1ðtÞð Þ; . . . ;xðrlðtÞÞð Þ ¼ gðtÞ; t P t0:

Using the Banach fixed-point theorem, we establish five existence results of uncountablymany bounded nonoscillatory solutions for the above equation, construct five Mann itera-tive sequences with mixed errors for approximating these nonoscillatory solutions and dis-cuss five error estimates between the approximate solutions and these nonoscillatorysolutions. To dwell the importance and applications of our results, five nontrivial examplesare constructed.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries

Recently, many researchers [1,2,4–10] and others investigated the existence of oscillatory and nonoscillatory solutions forvarious kinds of neutral delay differential equations. Especially, Parhi and Rath [5] obtained oscillation criteria for the forcedfirst order neutral differential equation with variable coefficients

yðtÞ � pðtÞyðt � sÞ½ �0 þ QðtÞG yðt � sÞð Þ ¼ f ðtÞ; t P t0: ð1:1Þ

Zhang et al. [8] studied the nonoscillatory behavior of solutions for the first order neutral delay differential equation

xðtÞ þ pðtÞxðt � sÞ½ �0 þ Q 1ðtÞxðt � r1Þ � Q 2ðtÞxðt � r2Þ ¼ 0; t P t0: ð1:2Þ

By employing a renormalization procedure and the Banach’s fixed point theorem, Wahlen [6] proved a nonoscillation resultfor the following second order nonlinear ordinary differential equation:

y00ðtÞ þ Fðt; yðtÞÞ ¼ 0; t P t0: ð1:3Þ

Kulenovic and Hadziomerspahic [1] investigated the neutral delay differential equation with positive and negative coeffi-cients below

xðtÞ þ pxðt � sÞ½ �00 þ Q 1ðtÞxðt � r1Þ � Q 2ðtÞxðt � r2Þ ¼ 0; t P t0; ð1:4Þ

ts reserved.

oo.cn (L. Wang), [email protected] (S.M. Kang), [email protected] (J.S. Ume).

http://dx.doi.org/10.1016/j.amc.2009.08.054

mailto:[email protected]




http://www.sciencedirect.com/science/journal/00963003

http://www.elsevier.com/locate/amc

Z. Liu et al. / Applied Mathematics and Computation 215 (2009) 2534–2543 2535

and gave some sufficient conditions for the existence of nonoscillatory solutions of Eq. (1.4) by using the Banach contractionmapping principle. Lin [2] suggested a few sufficient conditions for oscillation and nonoscillation of the second ordernonlinear neutral differential equation

xðtÞ þ pðtÞxðt � sÞ½ �00 þ qðtÞf xðt � sÞð Þ ¼ 0; t P t0: ð1:5Þ

Zhou [9] investigated the following second order nonlinear neutral differential equation:

rðtÞðxðtÞ þ pðtÞxðt � sÞÞ0� �0 þXm

i¼1

QiðtÞfi xðt � riÞð Þ ¼ 0; t P t0; ð1:6Þ

and discussed the existence of nonoscillatory solutions for Eq. (1.6) by applying the Krasnoselskii’s fixed point theorem. Zhouand Zhang [10] considered the existence of nonoscillatory solutions for the higher order neutral differential equation

xðtÞ þ cxðt � sÞ½ �ðnÞ þ ð�1Þnþ1 PðtÞxðt � rÞ � QðtÞxðt � dÞ½ � ¼ 0; t P t0: ð1:7Þ

Liu et al. [4] considered the nth-order neutral delay differential equation

xðtÞ þ cxðt � sÞ½ �ðnÞ þ ð�1Þnþ1f t; xðt � r1Þ; . . . ; xðt � rkÞð Þ ¼ gðtÞ; t P t0; ð1:8Þ

and studied the existence and iterative approximations of nonoscillatory solutions of Eq. (1.8).However, all papers mentioned above only dealt with the existence or iterative approximations of oscillatory or nonos-

cillatory solutions of Eqs. (1.1)–(1.7) and (1.8) and did not establish the existence of uncountably many nonoscillatory solu-tions for Eqs. (1.1)–(1.7) and (1.8).

In this paper we investigate the following higher order nonlinear neutral delay differential equation:

rðtÞðxðtÞ þ pðtÞxðt � sÞÞðmÞh iðn�mÞ

þ ð�1Þn�mþ1f t; x r1ðtÞð Þ; . . . ; xðrlðtÞÞð Þ ¼ gðtÞ; t P t0; ð1:9Þ

where l; n and m are positive integers with mþ 1 6 n; s > 0; R ¼ ð�1þ1Þ; Rþ ¼ ½0;þ1Þ; p; g 2 Cð½t0; þ1Þ; RÞ;r 2 Cð½t0; þ1Þ; R n f0gÞ; ri 2 Cð½t0; þ1Þ; RÞ with limt!þ1riðtÞ ¼ þ1 for i 2 f1; 2; . . . ; lg and f 2 Cð½t0; þ1Þ � Rl; RÞsatisfy the following assumptions:

(A1) there exist a constant M > 0 and functions w; q 2 Cð½t0; þ1Þ; RþÞ satisfying

jf ðt;u1;u2; . . . ;ulÞ � f ðt; �u1; �u2; . . . ; �ulÞj 6 wðtÞmax jui � �uij : 1 6 i 6 lf g

and

jf ðt;u1;u2; . . . ;ulÞj 6 qðtÞ

for any t 2 ½t0; þ1Þ; ui; �ui 2 ½0; M� and i 2 f1; 2; . . . ; lg;

(A2)Rþ1

t0

Rþ1u

um�1sn�m�1

jrðuÞj maxfwðsÞ; qðsÞ; jgðsÞjgdsdu < þ1.

Employing the Banach contraction mapping principle, we establish the existence of uncountably many bounded nonos-cillatory solutions for Eq. (1.9), construct Mann iterative schemes with mixed errors for these nonoscillatory solutions anddiscuss several error estimates between the approximate solutions and the nonoscillatory solutions. The results presented inthis paper extend and improve Theorem in [1], Theorems 2.1 and 2.2 in [4], Theorems 2.2–2.10 in [5], Theorems A and B in[7], Theorems 1-3 in [8] and Theorem 1 in [9]. To illustrate our results, five examples are also included.

Throughout this paper, we assume that Cð½t0; þ1Þ; RÞ denotes the Banach space of all continuous functions with thenorm kxk ¼ suptPt0

jxðtÞj < þ1. By a solution of Eq. (1.9), we mean a function x 2 Cð½t1 � s; þ1Þ; RÞ for some t1 P t0 þ s,such that xðtÞ þ pðtÞxðt � sÞ and rðtÞðxðtÞ þ pðtÞxðt � sÞÞðmÞ are m and n�m times continuously differentiable on ½t1; þ1Þ,respectively, and such that Eq. (1.9) is satisfied for t P t1. As a customary, a solution of Eq. (1.9) is said to be oscillatory ifit has arbitrary large zeros and nonoscillatory if it is eventually positive or eventually negative.

Lemma 1.1. ([3]) Let fangnP0; fbngnP0 and fcngnP0 be nonnegative sequences satisfying the inequality

anþ1 6 ð1� tnÞan þ tnbn þ cn; 8n P 0;

where ftngnP0 � ½0; 1�;P1

n¼0tn ¼ þ1; limn!1bn ¼ 0 andP1

n¼0cn < þ1. Then limn!1an ¼ 0.

2. Uncountably many nonoscillatory solutions and iterative approximations

In this section, sufficient conditions to ensure the existence of uncountably many bounded nonoscillatory solutions forEq. (1.9) are studied, Mann iterative schemes with mixed errors for approximating these nonoscillatory solutions areconstructed, and several error estimates between the approximate solutions and the nonoscillatory solutions are alsodiscussed under certain conditions. For notational purposes, let

2536 Z. Liu et al. / Applied Mathematics and Computation 215 (2009) 2534–2543

H ¼ 1ðm� 1Þ!ðn�m� 1Þ! :

Theorem 2.1. Let ðA1Þ and ðA2Þ hold and N be a constant with M > N > 0. Assume that fakgkP0 and fbkgkP0 are any sequencesin ½0; 1� and Cð½t0; þ1Þ; RÞ, respectively, satisfying

X1k¼0

ak ¼ þ1; ð2:1Þ

bkðtÞ ¼ b0kðtÞ þ b00kðtÞ;X1k¼0

kb0kk < þ1 and

kb00kk ¼ nkak for t P t0 and k P 0 with limk!1

nk ¼ 0: ð2:2Þ

If there exists a constant c1 2 0; M�N2M

� �satisfying

jpðtÞj 6 c1 for large t; ð2:3Þ

then

(a) for each L 2 ðN þ c1M; ð1� c1ÞMÞ, there exist h 2 ð0; 1Þ and T > t0 þ s such that for each x0 2 X, where

X ¼ x 2 Cð½t0;þ1Þ;RÞ : N 6 xðtÞ 6 M; 8t P t0f g;

the Mann iterative sequence with mixed errors fxkgkP0 generated by the scheme below

xkþ1ðtÞ ¼

ð1� akÞxkðtÞ þ ak L� pðtÞxkðt � sÞ þ HRþ1

t

R þ1u

ðu�tÞm�1ðs�uÞn�m�1

rðuÞ

n� ð�1Þmf ðs; xkðr1ðsÞÞ; . . . ; xkðrlðsÞÞÞ þ ð�1ÞngðsÞ� �

dsdu�

þbkðtÞ; t P T; k P 0;xkþ1ðTÞ; t0 6 t < T; k P 0

8>>>><>>>>:

ð2:4Þ

converges to a bounded nonoscillatory solution x 2 X of Eq. (1.9) and has the following error estimate

kxkþ1 � xk 6 1� ð1� hÞakð Þkxk � xk þ kbkk; k P 0; ð2:5Þ

(b) Eq. (1:9) possesses uncountably many bounded nonoscillatory solutions in X.

Proof. Let L 2 ðN þ c1M; ð1� c1ÞMÞ. With regard to (A2) and (2.3), we conclude that there exist h 2 ð0; 1Þ and T > t0 þ ssufficiently large such that

jpðtÞj 6 c1; 8 t P T; h ¼ c1 þ HZ þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞdsdu ð2:6Þ

and

HZ þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj qðsÞ þ jgðsÞjð Þdsdu 6minfL� N � c1M; ð1� c1ÞM � Lg: ð2:7Þ

It is easy to see that X is a closed subset of Cð½t0; þ1Þ; RÞ. Define a mapping SL : X ! Cð½t0; þ1Þ; RÞ by

SLxðtÞ ¼

L� pðtÞxðt � sÞ þ HRþ1

t

Rþ1u

ðu�tÞm�1ðs�uÞn�m�1

rðuÞ

� ð�1Þmf ðs; xðr1ðsÞÞ; . . . ; xðrlðsÞÞÞ þ ð�1ÞngðsÞ� �

dsdu;t P T; x 2 X;

SLxðTÞ; t0 6 t < T; x 2 X:

8>>>><>>>>:

ð2:8Þ

In view of (2.6), (2.7) and (2.8) and (A1), we derive that for every x 2 X and t P T ,

SLxðtÞ 6 Lþ jpðtÞjxðt � sÞ þ HZ þ1

t

Z þ1

u

ðu� tÞm�1ðs� uÞn�m�1

jrðuÞj jf ðs; xðr1ðsÞÞ; . . . ; xðrlðsÞÞÞj þ jgðsÞj½ �dsdu

6 Lþ c1M þ HZ þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj qðsÞ þ jgðsÞjð Þdsdu 6 Lþ c1M þM � c1M � L ¼ M


and

SLxðtÞP L� jpðtÞjxðt � sÞ � HZ þ1

t

Z þ1

u



P L� c1M � HZ þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj qðsÞ þ jgðsÞjð Þdsdu P L� c1M � Lþ N þ c1M ¼ N;

which imply that SLX # X.Now we show that SL is a contraction mapping in X. In fact, by (2.6) and (2.8) and (A1), we get that for each x; y 2 X and

t P T ,

jSLxðtÞ � SLyðtÞj 6 jpðtÞjjxðt � sÞ � yðt � sÞj þ HZ þ1

t

Z þ1

u


jrðuÞj jf ðs; xðr1ðsÞÞ; . . . ; xðrlðsÞÞÞ

� f ðs; yðr1ðsÞÞ; . . . ; yðrlðsÞÞÞjdsdu

6 c1kx� yk þ Hkx� ykZ þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞdsdu ¼ hkx� yk;

which means that

kSLx� SLyk 6 hkx� yk; 8x; y 2 X: ð2:9Þ

It follows that SL is a contraction mapping and it has a unique fixed point x in X, which is a bounded nonoscillatory solutionof Eq. (1.9).

In terms of (2.4) and (2.9), we deduce that for each k P 0 and t P T ,

jxkþ1ðtÞ � xðtÞj ¼ ð1� akÞxkðtÞ þ ak L� pðtÞxkðt � sÞ þ HZ þ1

t

Z þ1

u


rðuÞ ð�1Þmf ðs; xkðr1ðsÞÞ; . . . ; xkðrlðsÞÞÞ�(��

þ �1ÞngðsÞ� �

dsdu

)þ bkðtÞ � xðtÞ

�� 6 ð1� akÞjxkðtÞ � xðtÞj þ akjSLxkðtÞ � SLxðtÞj þ jbkðtÞ��

6 ð1� ð1� hÞakÞkxk � xk þ kbkk;

which yields that (2.5) holds. Thus (2.1), (2.2) and (2.5) and Lemma 1.1 ensure that limk!1xk ¼ x.Let L1; L2 2 ðN þ c1M; ð1� c1ÞMÞ and L1 – L2. For each j 2 f1; 2g, we choose constants hj 2 ð0; 1Þ; Tj P t0 þ s and a

mapping SLjsatisfying (2.6), (2.7) and (2.8), where h; L and T are replaced by hj; Lj and Tj, respectively, and

c1 þ HZ þ1

T3

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞdsdu <jL1 � L2j

2Mð2:10Þ

for some T3 > maxfT1; T2g. Obviously, the contraction mappings SL1 and SL2 have unique fixed points x and y in X, respec-tively, which are bounded nonoscillatory solutions of Eq. (1.9) in X. By (2.8) we obtain that

xðtÞ ¼ L1 � pðtÞxðt � sÞ þ HZ þ1

t

Z þ1

u


rðuÞ ð�1Þmf ðs; xðr1ðsÞÞ; . . . ; xðrlðsÞÞÞdsduþ ð�1ÞngðsÞ� �

dsdu; 8t P T1

ð2:11Þ

and

yðtÞ ¼ L2 � pðtÞyðt � sÞ þ HZ þ1

t

Z þ1

u


rðuÞ ð�1Þmf ðs; yðr1ðsÞÞ; . . . ; yðrlðsÞÞÞ þ ð�1ÞngðsÞ� �

ds du; 8t P T2:

ð2:12Þ

Combining (2.10), (2.11) and (2.12) and (A1), we conclude that for any t P T3

jxðtÞ � yðtÞjP jL1 � L2j � jpðtÞjjxðt � sÞ � yðt � sÞj � HZ þ1

t

Z þ1

u


jrðuÞj jf ðs; xðr1ðsÞÞ; . . . ; xðrlðsÞÞÞ

� f ðs; yðr1ðsÞÞ; . . . ; yðrlðsÞÞÞjdsdu

P jL1 � L2j � c1kx� yk � kx� ykHZ þ1

T3

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞdsdu

P jL1 � L2j � 2M c1 þ HZ þ1

T3

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞdsdu�

> 0;

that is, x – y. Hence, Eq. (1.9) has uncountably many bounded nonoscillatory solutions in X. This completes the proof. h


Theorem 2.2. Let ðA1Þ and ðA2Þ hold and N be a constant with M > N > 0. Assume that fakgkP0 and fbkgkP0 are the same as inTheorem 2.1. If there exists a constant c2 >

MM�N such that

pðtÞP c2 for large t; ð2:13Þ
then
(a) for each L 2 ðN þ Mc2; MÞ, there exist h 2 ð0; 1Þ and T > t0 þ s such that for each x0 2 XT , where

XT ¼ x 2 Cð½t0;þ1Þ;RÞ :N

pðt þ sÞ 6 xðtÞ 6 Mpðt þ sÞ ; 8t P T;

NpðT þ sÞ 6 xðtÞ 6 M

pðT þ sÞ ; 8t 2 ½t0; TÞ �

; ð2:14Þ

the Mann iterative sequence with mixed errors fxkgkP0 generated by the following scheme:

xkþ1ðtÞ ¼ ð1� akÞxkðtÞ þak

pðt þ sÞ L� xkðt þ sÞ þ HZ þ1

tþs

Z þ1

u

ðu� t � sÞm�1ðs� uÞn�m�1

rðuÞ

((

� ð�1Þmf ðs; xkðr1ðsÞÞ; . . . ; xkðrlðsÞÞÞ þ ð�1ÞngðsÞ� �

dsdu�

þ bkðtÞ; t P T; k P 0; xkþ1ðTÞ; t0 6 t < T; k P 0; ð2:15Þ

converges to a bounded nonoscillatory solution x 2 XT of Eq. (1.9) and has the error estimate (2.5);

(b) Eq. (1:9) possesses uncountably many bounded nonoscillatory solutions in Cð½t0; þ1Þ; RÞ.

Proof. Set L 2 ðN þ Mc2; MÞ. It follows from (A2) and (2.13) that there exist h 2 ð0; 1Þ and T > t0 þ s sufficiently large such

that

pðtÞP c2; 8t P T; h ¼ 1c2þ H

c2

Z þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞdsdu;

HZ þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj qðsÞ þ jgðsÞjð Þdsdu 6 min L� N �Mc2;M � L

�:

ð2:16Þ

Clearly, XT is a closed subset of Cð½t0; þ1Þ; RÞ. Define a mapping SL : XT ! Cð½t0; þ1Þ; RÞ as follows:

SLxðtÞ ¼

1pðtþsÞ L� xðt þ sÞ þ H

Rþ1tþs

R þ1u

ðu�t�sÞm�1ðs�uÞn�m�1

rðuÞ ð�1Þmf ðs; xðr1ðsÞÞ; . . . ; xðrlðsÞÞÞ þ ð�1ÞngðsÞ� �

dsdun o

;

t P T; x 2 XT ;

SLxðTÞ; t0 6 t < T; x 2 XT :

8>><>>:

ð2:17Þ
By virtue of (2.16) and (2.17) and (A1), we arrive at for any x 2 XT and t P T ,
SLxðtÞ 6 1pðt þ sÞ L� N

pðt þ sÞ þ HZ þ1

tþs

Z þ1

u



( )

61

pðt þ sÞ L� Npðt þ sÞ þ H

Z þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj qðsÞ þ jgðsÞjð Þdsdu�

61

pðt þ sÞ L� Npðt þ sÞ þM � L

� 6

Mpðt þ sÞ

and

SLxðtÞP 1pðt þ sÞ L� M

pðt þ sÞ � HZ þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj qðsÞ þ jgðsÞjð Þdsdu �

P1

pðt þ sÞ L� Mpðt þ sÞ � LþM

c2þ N

� P

Npðt þ sÞ ;

which show that SLXT # XT . From (2.16) and (2.17) and (A1), we deduce that for any x; y 2 XT and t P T ,

jSLxðtÞ � SLyðtÞj 6 1pðt þ sÞ jxðt þ sÞ � yðt þ sÞj þ H

pðt þ sÞ

Z þ1

tþs

Z þ1

u


jrðuÞj� jf ðs; xðr1ðsÞÞ; . . . ; xðrlðsÞÞÞ � f ðs; yðr1ðsÞÞ; . . . ; yðrlðsÞÞÞjdsdu

61c2kx� yk þ kx� yk

c2HZ þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞdsdu ¼ hkx� yk;

which leads to (2.9) holds. Hence, SL is a contraction mapping and possesses a unique fixed point x in XT , which is a boundednonoscillatory solution of Eq. (1.9).


In terms of (2.9) and (2.15), we deduce that for each k P 0 and t P T ,

jxkþ1ðtÞ � xðtÞj ¼ jð1� akÞxkðtÞ þak

pðt þ sÞ L� xkðt þ sÞ þ HZ þ1

tþs

Z þ1

u


rðuÞ

(

� ð�1Þmf ðs; xkðr1ðsÞÞ; . . . ; xkðrlðsÞÞÞ þ ð�1ÞngðsÞ� �

dsdu

)þ bkðtÞ � xðtÞj

6 ð1� akÞjxkðtÞ � xðtÞj þ akjSLxkðtÞ � SLxðtÞj þ jbkðtÞj 6 ð1� ð1� hÞakÞkxk � xk þ kbkk;
which implies that (2.5) holds. Thus (2.1), (2.2) and (2.5) and Lemma 1.1 ensure that limk!1xk ¼ x.
Let L1 and L2 be in ðN þ Mc2;MÞ with L1 – L2. Obviously there exist constants hj 2 ð0; 1Þ; Tj P t0 þ r and a mapping

SLj: XTj

! Cð½t0; þ1Þ; RÞ satisfying (2.14), (2.16) and (2.17) for each j 2 f1; 2g, where h; L; T and XT are replaced byhj; Lj; Tj and XTj

, respectively, and

HZ þ1

T3

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞdsdu <c2jL1 � L2j

2Mð2:18Þ

for some T3 > maxfT1; T2g. Notice that for every j 2 f1; 2g the contraction mapping SLjhas a unique fixed point zj 2 XTj

,which is a bounded nonoscillatory solution of Eq. (1.9) in XTj

. It follows from (2.17) that:Z Z( )
zjðtÞ ¼
1pðt þ sÞ L� zjðt � sÞ þ H

þ1

t

þ1

u


rðuÞ ð�1Þmf ðs; zjðr1ðsÞÞ; . . . ; zjðrlðsÞÞÞ þ ð�1ÞngðsÞ� �

dsdu ;

8t P Tj; j 2 f1;2g: ð2:19Þ

In view of (2.18) and (2.19) and (A1), we get that for any t P T3,"
z1ðtÞ � z2ðtÞ þ
z1ðt þ sÞ � z2ðt þ sÞpðt þ sÞ

�� P 1

pðt þ sÞ jL1 � L2j � HZ þ1

t

Z þ1

u


jrðuÞj jf ðs; z1ðr1ðsÞÞ; . . . ; z1ðrlðsÞÞÞ

� f ðs; z2ðr1ðsÞÞ; . . . ; z2ðrlðsÞÞÞjdsdu

#

P1

pðt þ sÞ jL1 � L2j � kz1 � z2kHZ þ1

T3

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞdsdu�

P1

pðt þ sÞ jL1 � L2j �2MH

c2

Z þ1

T3

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞÞdsdu�

> 0;

that is, z1 – z2. Hence, Eq. (1.9) has uncountably many bounded nonoscillatory solutions in Cð½t0; þ1Þ; RÞ. This completes theproof. h

Theorem 2.3. Let ðA1Þ and ðA2Þ hold and N be a constant with M > N > 0. Assume that fakgkP0 and fbkgkP0 are the same as inTheorem 2.1. If there exists a constant c2 >

MM�N satisfying

pðtÞ 6 �c2 for large t; ð2:20Þ
then
(a) for each L 2 N; Mð1� 1c2Þ

� �, there exist h 2 ð0; 1Þ and T > t0 þ s such that for each x0 2 XT , where

XT ¼ x 2 Cð½t0;þ1Þ;RÞ :�N

pðt þ sÞ 6 xðtÞ 6 �Mpðt þ sÞ ; 8t P T;

�NpðT þ sÞ 6 xðtÞ 6 �M

pðT þ sÞ ; 8t 2 ½t0; TÞ �

;

the Mann iterative sequence with mixed errors fxkgkP0 generated by the following scheme:

xkþ1ðtÞ ¼ð1� akÞxkðtÞ þ ak

pðtþsÞ �L� xkðt þ sÞ þ HRþ1

tþsR þ1

uðu�t�sÞm�1ðs�uÞn�m�1

rðuÞ

n� ð�1Þmf ðs; xkðr1ðsÞÞ; . . . ; xkðrlðsÞÞÞ þ ð�1ÞngðsÞ� �

dsdu�þ bkðtÞ; t P T; k P 0;

xkþ1ðTÞ; t0 6 t < T; k P 0;

8>><>>: ð2:21Þ

converges to a nonoscillatory solution x 2 XT of Eq. (1.9) and has the error estimate (2.5);

(b) Eq. (1:9) possesses uncountably many nonoscillatory solutions in Cð½t0; þ1Þ; RÞ.

Proof. Set L 2 ðN; Mð1� 1c2ÞÞ. In terms of (A2) and (2.20), we conclude that there exist h 2 ð0; 1Þ and T > t0 þ s sufficiently

large satisfying

pðtÞ 6 �c2; 8t P T; h ¼ 1c2þ H

c2

Z þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj wðsÞdsdu;

HZ þ1

T

Z þ1

u

um�1sn�m�1

jrðuÞj qðsÞ þ jgðsÞjð Þdsdu 6 min L� N;M 1� 1c2

� � L

�: ð2:22Þ


Obviously, XT is a closed subset of Cð½t0; þ1Þ; RÞ. Define a mapping SL : XT ! Cð½t0; þ1Þ; RÞ by

SLxðtÞ ¼

1pðtþsÞ �L� xðt þ sÞ þ H

Rþ1tþs

Rþ1u

ðu�t�sÞm�1ðs�uÞn�m�1

rðuÞ

n� ð�1Þmf ðs; xðr1ðsÞÞ; . . . ; xðrlðsÞÞÞ þ ð�1ÞngðsÞ� �

dsdu�; t P T; x 2 XT ;

SLxðTÞ; t0 6 t < T; x 2 XT :

8>><>>: ð2:23Þ

In light of (2.22) and (2.23) and (A1), we know that for any x 2 XT and t P T ,

SLxðtÞ 6 1pðt þ sÞ �Lþ M

pðt þ sÞ � HZ þ1

tþs

Z þ1

u

um�1sn�m�1


61

pðt þ sÞ �Lþ Mpðt þ sÞ �M 1� 1

c2

� þ L

� ¼ 1

pðt þ sÞ M1c2þ 1

pðt þ sÞ

� �M

� 6�M

pðt þ sÞ

and

SLxðtÞP 1pðt þ sÞ �Lþ N

pðt þ sÞ þ HZ þ1

T

Z þ1

u

um�1sn�m�1


P1

pðt þ sÞ �Lþ Npðt þ sÞ þ L� N

� P

�Npðt þ sÞ ;

which infer that SLXT # XT . The rest of the proof is identical with the proof of Theorem 2.2 and is omitted. This completesthe proof. h

Analogous to the proofs of Theorems 2.1, 2.2 and 2.3, we have the following results and omit their proofs.

Theorem 2.4. Let ðA1Þ and ðA2Þ hold and N be a constant with M > N > 0. Assume that X; fakgkP0 and fbkgkP0 are the same asin Theorem 2.1. If there exists a constant c3 2 ½0; M�N

M Þ satisfying

0 6 pðtÞ 6 c3 for large t;

then

(a) for each L 2 ðN þ c3M;MÞ, there exist h 2 ð0; 1Þ and T > t0 þ s such that for each x0 2 X, the Mann iterative sequencewith mixed errors fxkgkP0 generated by the scheme (2:4) converges to a nonoscillatory solution x 2 X of Eq. (1:9) and hasthe error estimate (2:5);


Theorem 2.5. Let ðA1Þ and ðA2Þ hold and N be a constant with M > N > 0. Assume that X; fakgkP0 and fbkgkP0 are the same as inTheorem 2.1. If there exists a constant c3 2 ½0; M�N

M Þ satisfying

�c3 6 pðtÞ 6 0 for any large t;

then

(a) for each L 2 ðN; ð1� c3ÞMÞ, there exist h 2 ð0; 1Þ and T > t0 þ s such that for each x0 2 X, the Mann iterative sequencewith mixed errors fxkgkP0 generated by the scheme (2:4) converges to a nonoscillatory solution x 2 X of Eq. (1:9) and hasthe error estimate (2:5);


Remark 2.1. Theorems 2.1–2.4 and 2.5 improve essentially Theorem in [1], Theorems 2.1 and 2.2 in [4], Theorems 2.2–2.10in [5], Theorems A and B in [7], Theorems 1–3 in [8] and Theorem 1 in [9].

3. Some examples

In this section, five examples are given to illustrate the advantage of the results presented in Section 2.

Example 3.1. Consider the following three order nonlinear neutral delay differential equation:

t2 xðtÞ þ t arctan t1þ 8t

xðt � sÞ� 00� 0

þ x3ðlnð1þ t2ÞÞ sin tt3 ¼ �t

t4 þ sin2 t; t P 1; ð3:1Þ

where l ¼ 1; m ¼ 2; n ¼ 3; s > 0 is a constant, t0 ¼ 1; M ¼ 10; N ¼ 5; c1 ¼ p16. Let fakgkP0 and fbkgkP0 be any sequences in

[0,1] and Cð½1; þ1Þ; RÞ, respectively, satisfying (2.1) and (2.2). Put


gðtÞ ¼ �t

t4 þ sin2 t; f ðt;uÞ ¼ u3 sin t

t3 ; rðtÞ ¼ t2; pðtÞ ¼ t arctan t1þ 8t

;

rðtÞ ¼ lnð1þ t2Þ; qðtÞ ¼ M3

t3 ; wðtÞ ¼ 3M2

t3 ; 8t P 1; u 2 ½0;10�:

Clearly, the conditions of Theorem 2.1 are satisfied. Thus Theorem 2.1 ensures that for any L 2 5þ 58 p;10� 5

8 p� �

, there existh 2 ð0; 1Þ and T > 1þ s such that for each x0 2 X, where

X ¼ x 2 Cð½1;þ1Þ;RÞ : 5 6 xðtÞ 6 10; 8t P 1f g;

the Mann iterative sequence with mixed errors fxkgkP0 generated by (2.4) converges to a nonoscillatory solution of Eqs. (3.1)and (3.1) has uncountably many bounded nonscillatory solutions in X. But Theorem in [1], Theorem 2.1 in [4], Theorems 2.2,2.3 and 2.6 in [5], Theorem A in [7], Theorems 1–3 in [8] and Theorem 1 in [9] are not valid for Eq. (3.1).

Example 3.2. Consider the following higher order nonlinear neutral delay differential equation:

ð1þ tÞn�m xðtÞ þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnðt þ 2Þ

qxðt � sÞ

� ðmÞ" #ðn�mÞ

þ ð�1Þn�mþ1 sin2ðtxðetÞÞð1þ t2Þnþ

m2þ2 �

ð1þ x2ðetÞÞ cos2ffiffitp

ð1þffiffiffiffiffiffiffiffiffiffiffi1þ tp

Þ2nþ3mþ1

" #

¼ � tm3

ð1þ t3nþmþ4Þ13; t P 0; ð3:2Þ

where l ¼ 1; n and m are two positive integers with mþ 1 6 n; s > 0 is a constant, t0 ¼ 0; M ¼ 4; N ¼ 1; c2 ¼ 32. Let fakgkP0

and fbkgkP0 be any sequences in [0,1] and Cð½0; þ1Þ; RÞ, respectively, satisfying (2.1) and (2.2). Put

gðtÞ ¼ � tm3

ð1þ t3nþmþ4Þ13; f ðt;uÞ ¼ sin2ðtuÞ

ð1þ t2Þnþm2þ2 �

ð1þ u2Þ cos2ffiffitp

ð1þffiffiffiffiffiffiffiffiffiffiffi1þ tp

Þ2nþ3mþ1 ;

rðtÞ ¼ ð1þ tÞn�m; pðtÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnðt þ 2Þ

q; rðtÞ ¼ et;

qðtÞ ¼ 1

ð1þ t2Þnþm2þ2 þ

3þM2

3ð1þffiffiffiffiffiffiffiffiffiffiffi1þ tp

Þ2nþ3mþ1 ;

wðtÞ ¼ 2t

ð1þ t2Þnþm2þ2 þ

2Mffiffiffi3pð1þ

ffiffiffiffiffiffiffiffiffiffiffi1þ tp

Þ2nþ3mþ1 ; 8t P 0; u 2 ½0;4�:

It is easy to show that the conditions of Theorem 2.2 are satisfied. Hence Theorem 2.2 ensures that for any L 2 ð113 ; 4Þ, there

exist h 2 ð0; 1Þ and T > s such that for each x0 2 XT , where

XT ¼ x 2 Cð½0;þ1Þ;RÞ :1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lnðt þ 2Þp 6 xðtÞ 6 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

lnðt þ 2Þp ; 8t P T;

1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnðT þ 2Þ

p 6 xðtÞ 6 4ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffilnðT þ 2Þ

p ; 8t 2 ½0; TÞ( )

;

the Mann iterative sequence with mixed errors fxkgkP0 generated by (2.15) converges to a nonoscillatory solution of Eqs.(3.2) and (3.2) possesses uncountably many nonscillatory solutions in Cð½t0; þ1Þ; RÞ. However, Theorem in [1], Theorem2.2 in [4], Theorems 2.4, 2.5 and 2.10 in [5] and Theorems 1–3 in [8] are not applicable for Eq. (3.2).

Example 3.3. Consider the following higher order nonlinear neutral delay differential equation:

1þ tnþm

lnð2þ tÞ xðtÞ � ðt þ 1Þ2xðt � sÞ� �ðmÞ� ðn�mÞ

þ ð�1Þn�mþ1 t sin2ðtx3ðtn arctan tÞÞðt þ 2Þnðt þ 5Þ2mþ5 ¼ t lnð1þ tÞ

1þ t2nþ3 ; t P 0; ð3:3Þ


and fbkgkP0 be any sequences in [0,1] and Cð½0; þ1Þ; RÞ, respectively, satisfying (2.1) and (2.2). Set

gðtÞ ¼ t lnð1þ tÞ1þ t2nþ3 ; f ðt;uÞ ¼ t sin2ðtu3Þ

ðt þ 2Þnðt þ 5Þ2mþ5 ;

rðtÞ ¼ 1þ tnþm

lnð2þ tÞ ; pðtÞ ¼ �ðt þ 1Þ2; rðtÞ ¼ tn arctan t;

qðtÞ ¼ t

ðt þ 2Þnðt þ 5Þ2mþ5 ; wðtÞ ¼ 3t2M2

8ðt þ 2Þnðt þ 5Þ2mþ5 ; 8t P 0; u 2 ½0;4�:

It is a simple matter to verify that the conditions of Theorem 2.3 are satisfied. Hence Theorem 2.3 ensures that for anyL 2 ð1; 4

3Þ, there exist h 2 ð0; 1Þ and T > s such that for each x0 2 XT , where


XT ¼ x 2 Cð½0;þ1Þ;RÞ :1

ðt þ 1Þ26 xðtÞ 6 4

ðt þ 1Þ2; 8t P T;

1

ðT þ 1Þ26 xðtÞ 6 4

ðT þ 1Þ2; 8t 2 ½0; TÞ

( );

the Mann iterative sequence with mixed errors fxkgkP0 generated by (2.21) converges to a nonoscillatory solution of Eqs.(3.3) and (3.3) possesses uncountably many nonscillatory solutions in Cð½0; þ1Þ; RÞ. But Theorem in [1], Theorem 2.2 in[4], Theorem 2.8 in [5] and Theorems 1–3 in [8] can not be applied to Eq. (3.3).

Example 3.4. Consider the higher order nonlinear neutral delay differential equation below

ð1þ t2Þm2þ1 xðtÞ þ t

t2 þ 1xðt � sÞ

� ðmÞ" #ðn�mÞ

þ ð�1Þn�mþ1 ð1� tÞx2ðlnð1þ t4ÞÞ � ð1þ tÞ sin ttn�mþ3

�

¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ t2

p� sinð1þ 2t2Þ ln t

tnþmþ4 ; t P 1; ð3:4Þ



gðtÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ t2

p� sinð1þ 2t2Þ ln t

tnþmþ4 ; f ðt;uÞ ¼ ð1� tÞu2 � ð1þ tÞ sin ttn�mþ3 ;

rðtÞ ¼ ð1þ t2Þm2þ1

; pðtÞ ¼ tt2 þ 1

; rðtÞ ¼ lnð1þ t4Þ;

qðtÞ ¼ ð1þ tÞð1þM2Þtn�mþ3 ;wðtÞ ¼ 2Mðt � 1Þ

tn�mþ3 ; 8t P 1; u 2 ½0;7�:

It is clear that the conditions of Theorem 2.4 are satisfied. Thus Theorem 2.4 ensures that for any L 2 ð335 ; 7Þ, there exist

h 2 ð0; 1Þ and T > 1þ s such that for each x0 2 X, where


the Mann iterative sequence with mixed errors fxkgkP0 generated by (2.4) converges to a nonoscillatory solution of Eqs. (3.4)and (3.4) possesses uncountably many bounded nonscillatory solutions in X. But Theorem in [1], Theorem 2.1 in [4], Theo-rem 2.2 in [5], Theorem B in [7], Theorems 1–3 in [8] and Theorem 1 in [9] are unapplicable for Eq. (3.4).

Example 3.5. Consider the higher order nonlinear neutral delay differential equation below

tn xðtÞ � 45

� t

ðsin2 tÞxðt � sÞ !ðmÞ2

435ðn�mÞ

þ ð�1Þn�mþ1 �t2x2ðt3Þ þ xðt3Þx2ðt þffiffiffiffiffiffiffiffiffiffiffiffiffi1þ t3

pÞ

tnþ5 þ cos2 t

" #

¼ ð�1Þn�m

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ t2

ptn�mþ3 ; t P 1; ð3:5Þ



gðtÞ ¼ ð�1Þn�m

ffiffiffiffiffiffiffiffiffiffiffiffiffi1þ t2

ptn�mþ3 ; f ðt; u1; u2Þ ¼

�t2u21 þ u1u2

2

tnþ5 þ cos2 t;

rðtÞ ¼ tn; pðtÞ ¼ � 45

� t

sin2 t; r1ðtÞ ¼ t3; r2ðtÞ ¼ t þffiffiffiffiffiffiffiffiffiffiffiffiffi1þ t3

p;

qðtÞ ¼ t2M2 þM3

tnþ5 ; wðtÞ ¼ 2Mt2 þ 3M2

tnþ5 ; 8t P 1; u1;u2 2 ½0;7�:

It is obvious that the conditions of Theorem 2.5 are satisfied. Thus Theorem 2.5 ensures that for any L 2 ð1; 75Þ, there exist

h 2 ð0; 1Þ and T > 1þ s such that for each x0 2 X, where


the Mann iterative sequence with mixed errors fxkgkP0 generated by (2.4) converges to a nonoscillatory solution of Eqs. (3.5)and (3.5) possesses uncountably many bounded nonscillatory solutions in X. However, Theorem in [1], Theorem 2.1 in [4],Theorems 2.2 and 2.6 in [5], Theorem B in [7], Theorems 1–3 in [8] and Theorem 1 in [9] are not valid for Eq. (3.5).


Acknowledgements

This work was supported by the Science Research Foundation of Educational Department of Liaoning Province(2009A419) and the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promo-tion Fund) (KRF-2008-313-C00042).

References

[1] M.R.S. Kulenovic, S. Hadziomerspahic, Existence of nonoscillatory solution of second order linear neutral delay equation, J. Math. Anal. Appl. 228 (1998)436–448.

[2] X.Y. Lin, Oscillation of second order nonlinear neutral differential equations, J. Math. Anal. Appl. 309 (2005) 442–452.[3] L.S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (2005)

114–125.[4] Z. Liu, H.Y. Gao, S.M. Kang, S.H. Shim, Existence and Mann iterative approximations of nonoscillatory solutions of nth-order neutral delay differential

equations, J. Math. Anal. Appl. 329 (2007) 515–529.[5] N. Parhi, R.N. Rath, Oscillation criteria for forced first order neutral differential equations with variable coefficients, J. Math. Anal. Appl. 256 (2001) 525–

541.[6] E. Wahlen, Positive solutions of second order differential equations, Nonlinear Anal. 58 (2004) 359–366.[7] Y.H. Yu, H.Z. Wang, Nonoscillatory solutions of second-order nonlinear neutral delay equations, J. Math. Anal. Appl. 311 (2005) 445–456.[8] W.P. Zhang, W. Feng, J. Yan, J.S. Song, Existence of nonoscillatory solutions of first order linear neutral delay differential equations, Comput. Math. Appl

49 (2005) 1021–1027.[9] Y. Zhou, Existence for nonoscillatory solutions of second order nonlinear differential equations, J. Math. Anal. Appl. 331 (2007) 91–96.

[10] Y. Zhou, B.G. Zhang, Existence of nonoscillatory solutions of higher-order neutral differential equations with positive and negative coefficients, Appl.Math. Lett. 15 (2002) 867–874.

Documents

Solvability and iterative algorithms for a higher order nonlinear neutral delay differential equation