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

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R E S E A R C H

Open Access

Solvability and Mann iterative

approximations for a higher order nonlinear

neutral delay differential equation

Guojing Jiang

1

_{, Young Chel Kwun}

2

_{and Shin Min Kang}

3,4* *_{Correspondence:}

[email protected]

3_{Department of Mathematics and}

RINS, Gyeongsang National University, Jinju, 52828, Korea

4_{Center for General Education,}

China Medical University, Taichung, 40402, Taiwan

Full list of author information is available at the end of the article

Abstract

The purpose of this paper is to study solvability of the higher order nonlinear neutral delay diﬀerential equation

dn

dtn

x(t) +c(t)x(t–

τ

)+ (–1)n+1f

(

t,x

(σ

1(t)

)

,x

(σ

2(t)

)

,. . .,x

(σ

k(t)

))

=g(t), t≥t0,

wherenandkare positive integers,

τ

> 0,t0∈R,f∈C([t0, +∞)×Rk,R),

c,g,

σ

i∈C([t0, +∞),R) and limt→+∞

σ

i(t) = +∞fori∈ {1, 2,. . .,k}. Under suitable

conditions, several existence results of uncountably many nonoscillatory solutions and convergence of Mann iterative approximations for the above equation are shown. Three nontrivial examples are given to demonstrate the advantage of our results over the existing ones in the literature.

MSC: 34K07; 34K11; 34C15

Keywords: higher order nonlinear neutral delay diﬀerential equation; uncountably many nonoscillatory solutions; contraction mapping; Mann iterative sequence; error estimate

1 Introduction and preliminaries

In the past twenty years or so, the existence of oscillatory and nonoscillatory solutions for a lot of neutral delay linear and nonlinear diﬀerential equations has been studied and discussed by many researchers, for example, see [–] and the references therein.

In , Chuanxi and Ladas [] investigated the ﬁrst order neutral delay diﬀerential equation

d dt

x(t) +p(t)x(t–τ)+Q(t)x(t–σ) = , t≥t. (.)

In ,  and , Kulenović and Hadžiomerspahić [, ] and Cheng and Annie [] investigated, respectively, the first and second order neutral delay differential equations with positive and negative coefficients:

d dt

x(t) +cx(t–τ)+Q(t)x(t–σ) –Q(t)x(t–σ) = , t≥t (.)

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and

d

dt

x(t) +cx(t–τ)+Q(t)x(t–σ) –Q(t)x(t–σ) = , t≥t. (.)

Underc=± and other conditions, they obtained suﬃcient conditions for the existence

of nonoscillatory solutions of Eqs. (.) and (.), respectively. In , Zhou and Zhang [] extended the result in [] to thenth order neutral functional diﬀerential equation with positive and negative coeﬃcients

dn

dtn

x(t) +cx(t–τ)+ (–)n+Q(t)x(t–σ) –Q(t)x(t–σ)

= , t≥t, (.)

wherec=±. In , Liu et al. [] extended and improved the results in [, , , ] to the

followingnth order neutral delay nonlinear diﬀerential equation:

dn

dtn

x(t) +cx(t–τ)+ (–)n+ft,x(t–σ),x(t–σ), . . . ,x(t–σk)

=g(t), t≥t, (.)

wherec= –. They gave suﬃcient conditions for the existence of nonoscillatory solutions

for Eq. (.), constructed the Mann iterative approximations for these nonoscillatory so-lutions and established the error estimates between these nonoscillatory soso-lutions and the Mann iterative approximations. At the same time, they also proved the existence of inﬁnitely many nonoscillatory solutions for Eq. (.).

Inspired and motivated by the results in [–], in this paper, we study the following higher order nonlinear neutral delay diﬀerential equation:

dn

dtn

x(t) +c(t)x(t–τ)+ (–)n+_f_t_,_x_σ (t)

,xσ(t)

, . . . ,xσk(t)

=g(t), t≥t, (.)

wherenandkare positive integers,τ> ,t∈R,c,g,σi∈C([t, +∞),R),limt→+∞σi(t) =

+∞fori∈ {, , . . . ,k}, andf ∈C([t, +∞)×Rk,R) satisﬁes the following condition:

(H) there exist constantsM>N> and functionsp,q∈C([t, +∞),R+)satisfying

f(t,u, . . . ,uk) –f(t,u¯, . . . ,u¯k)

≤p(t)max|ui–u¯i|: ≤i≤k

, t∈[t, +∞),ui,u¯i∈[N,M], ≤i≤k

and

f(t,u, . . . ,uk)≤q(t), t∈[t, +∞),ui∈[N,M], ≤i≤k,

whereR= (–∞, +∞),R+_{= [, +}_∞₎_.

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to discuss the error estimates between the approximate solutions and the nonoscillatory solutions. These results obtained in this paper extend, improve and unify the correspond-ing results in [, –].

Throughout this paper, we assume that

(H) _t+_∞snmax{p(s),q(s),|g(s)|}ds< +∞;

(H) _t+_∞sn–max{p(s),q(s),|g(s)|}ds< +∞;

(H) {λn}n≥is an arbitrary sequence in[, ]satisfying

∞

m=

λm= +∞.

By a solution of Eq. (.), we mean a functionx∈C([t–τ,∞),R) for somet≥tsuch thatx(t) +cx(t–τ) isn-times continuously diﬀerentiable in [t,∞) and such that Eq. (.) is satisﬁed fort≥t. As is customary, a solution of Eq. (.) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory otherwise.

LetXdenote the Banach space of all continuous and bounded functions on [t, +∞) with normx=sup_t_≥_t

|x(t)|, andA(N,M) ={x∈X:N≤x(t)≤M,t≥t}forM>N> . It is easy to see thatA(N,M) is a bounded closed and convex subset ofX.

2 Main results

Now we study those conditions under which Eq. (.) possesses uncountably many nonoscillatory solutions, and the Mann-type iterative sequences converge to these nonoscillatory solutions.

Theorem . Let(H), (H)and(H)be fulﬁlled and

c(t) = –, t≥t. (.)

Then

(a) for anyL∈(N,M),there existθ∈(, )andT>t+τ such that for any

x∈A(N,M),the Mann iterative sequence{xm}m≥generated by the following iterative scheme:

xm+(t) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

( –λm)xm(t) +λm[L– _∞

i= +∞

t+iτ

+∞

sn– · · · +∞

s f(s,xm(σ(s)), . . . ,xm(σk(s)))ds · · ·dsn–dsn–– (–)n

∞

i= +∞

t+iτ

+∞

sn– · · · +∞

s g(s)ds· · ·dsn–dsn–], t≥T,m≥,

xm+(T), t≤t<T,m≥

(.)

converges to a nonoscillatory solutionx∈A(N,M)of Eq. (.)and has the following error estimate:

xm+–x ≤e–(–θ)

m i=λi_x

–x, m≥; (.)

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–fs,y

σ(s)

, . . . ,yσk(s)

ds· · ·dsn–dsn–

≥ |L–L|– ∞

i=

+∞

t+iτ +∞

sn– · · ·

+∞

s

p(s)ds· · ·dsn–dsn–x–y

≥ |L–L|–  (n– )!

∞

i=

+∞

T+iτ

sn_–p(s)dsx–y

≥ |L–L|–θx–y, t≥T,

which yields that

x–y ≥ |L–L|–θx–y.

That is,

x–y ≥|L–L|

 +θ > .

Hencex=y. It follows that for any distinctLandL∈(N,M), the corresponding nonoscil-latory solutions x and y∈ A(N,M) of Eq. (.) are distinct. Consequently, the set of nonoscillatory solutions of Eq. (.) is uncountable. This completes the proof.

The proofs of Theorems .-. are similar to the proof of Theorem ., hence are omit-ted.

Theorem . Let(H), (H)and(H)hold.Assume that there exists C∈(, )such that ≤c(t)≤C for t≥tand M>_–_CN.Then

(a) for anyL∈(CM+N,M),there existθ∈(, )andT>t+τ such that for each

xm+(t) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

( –λm)xm(t) +λm{L–c(t)xm(t–τ)

+₍_n_–)! _t+∞(s–t)n–_f₍_s_,_x

m(σ(s)),

. . . ,xm(σk(s)))ds

+₍(–)_n_–)!n _t+∞(s–t)n–_g₍_s₎_ds_}, _t_≥_T_,_m_≥_,

xm+(T), t≤t<T,m≥

(.)

converges to a nonoscillatory solutionx∈A(N,M)of Eq. (.),and the error estimate

(.)holds;

(b) the set of nonoscillatory solutions of Eq. (.)is uncountable.

Theorem . Let(H), (H)and(H)hold.Assume that there exists C∈(, )such that –C≤c(t)≤for t≥tand M>_–_CN.Then

(a) for anyL∈(N, ( –C)M),there existθ∈(, )andT>t+τ such that for each

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Theorem . Let(H), (H)and(H)hold.Assume that there exists C> such that c(t)≥

C for t≥tand_CC_–N<M.Then

(a) for anyL∈(_CM+N,M),there existθ∈(, )andT>t+τ such that for each

xm+(t) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

( –λm)xm(t) +λm{L–_c₍_t₊τ)xm(t+τ)

+ 

c(t+τ)(n–)! +∞

t+τ (s–t–τ) n–

×f(s,xm(σ(s)), . . . ,xm(σk(s)))ds

+_c₍_t₊(–)_τ₎₍_nn_–)! _t+₊∞_τ (s–t–τ)n–_g₍_s₎_ds_}_, _t_≥_T_,_m_≥_,

xm+(T), t≤t<T,m≥

(.)

(.)holds;

Theorem . Let(H), (H)and(H)hold.Assume that there exists C> such that c(t)≤ –C for t≥tand_CC_–N<M.Then

(a) for anyL∈(N, ( – 

C)M),there existθ∈(, )andT>t+τ such that for each

x∈A(N,M),the Mann iterative sequence{xm}m≥generated by(.)converges to a nonoscillatory solutionx∈A(N,M)of Eq. (.),and the error estimate(.)holds; (b) the set of nonoscillatory solutions of Eq. (.)is uncountable.

Theorem . Let(H), (H)and(H)hold.Assume that there exists C∈(,_)such that |c(t)| ≤C for t≥tand N< ( – C)M.Then

(a) for anyL∈(CM+N, ( –C)M),there existθ∈(, )andT>t+τ such that for each

x∈A(N,M),the Mann iterative sequence{xm}m≥generated by(.)converges to a nonoscillatory solutionx∈A(N,M)of Eq. (.),and the error estimate(.)holds; (b) the set of nonoscillatory solutions of Eq. (.)is uncountable.

Theorem . Let n= , (H), (H)and(H)hold and c(t) = for t≥t.Then

(a) for anyL∈(N,M),there existθ∈(, )andT>t+τ such that for every

xm+(t) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

( –λm)xm(t) +λm{L+ +∞

j=

t+jτ t+(j–)τ

×f(s,xm(σ(s)), . . . ,xm(σk(s)))ds

–+_j_=∞ _tt_+(+j_jτ_–)_τg(s)ds}, t≥T,m≥,

xm+(T), t≤t<T,m≥

(.)

(.)holds;

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(a) for anyL∈(N,M),there existθ∈(, )andT>t+τ such that for arbitrary

xm+(t) =

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

( –λm)xm(t)

+λm{L+₍_n_–)! +∞

j=

t+jτ t+(j–)τ

+∞

u (s–u)n–

×f(s,xm(σ(s)), . . . ,xm(σk(s)))ds du

+₍(–)_n_–)!n _j+_=∞ _tt_+(+j_jτ_–)_τ _u+∞g(s)ds du}, t≥T,m≥,

xm+(T), t≤t<T,m≥

(.)

(.)holds;

Remark . Theorems .-. extend, improve and unify a few known results due to

Cheng and Annie [], Kulenović and Hadžiomerspahić [, ], Liu et al. [] and Zhou and Zhang [] and others.

3 Examples

In this section, in order to illustrate the advantage of our results, we consider the following three examples.

Example . Consider thenth order neutral delay diﬀerential equation

dn dtn

x(t) –x(t–τ)

+ (–)n+

e–tcos(√t–t) + sin √

t

 +tn+x

_t_{– }_τ_x₍√_t₎

=te–tsint– t+ , t≥, (.)

whereτis a positive number. Let{λn}n≥be an arbitrary sequence in [, ] satisfying (H),

MandNbe constants withM>N> . Putt= ,k= ,

σ(t) =t– τ, σ(t) = √

t,

f(t,u,u) =e–t 

cos(√t–t) + sin √

t

 +tn+u  u,

c(t) = –, g(t) =te–tsint– t+ ,

p(t) = M 

 +tn+, q(t) =e –t

+ M



 +tn+

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Example . Consider thenth order neutral delay diﬀerential equation

dn

dtn

x(t) +tx(t–τ)+ (–)n+

t_{– }

tn+ ln

 +x(√t– τ)

+ –t _cos_t

t₊_tn+ x

_(_t_{– ) –} x(t– √

t– )

tn+₊_x₍_t_{– )}

= √

t+tsin(t₎

tn+₊_ln_t , t≥, (.)

MandNbe constants with  <_N<M. Putt= ,k= ,C= ,

σ(t) = √

t– τ, σ(t) = t– ,

σ(t) =t– √

t– , σ(t) =t– ,

f(t,u,u,u,u) =

t_{– }

tn+ ln

 +|u|

+ –t _cos_t

t₊_tn+ u  –

u 

tn+₊_u  ,

c(t) =t, g(t) = √

t+tsin(t₎

tn+₊_ln_t ,

p(t) =t _{– }

tn+ +

M| –tcost| t₊_tn+ +

M(M+ tn+) (N₊_tn+₎ ,

q(t) =t _{– }

tn+ ln( +M) +

M| –tcost| t₊_tn+ +

M N₊_tn+

fort≥,ui∈Randi∈ {, , , }. Clearly, (H) and (H) hold. It follows from Theorem . that Eq. (.) possesses uncountably many nonoscillatory solutions, and for anyL∈

(_M+N,M), the Mann iterative sequence{xn}n≥generated by (.) converges to some nonoscillatory solution of Eq. (.), and the error estimate (.) holds. However, the results in [, –] are not applicable for Eq. (.).

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

dn

dtn

x(t) –t–sint t x(t–τ)

+ (–)n+

 – tn+

tn+ x ₍√_t₎

+ t

₍_t_{– )}

tn+₊_cos_tsin

_x_(_t₎₊  – t–t

tn+₊|_x_(_t)|

= +t

n_cos_tn

tn+ , t≥, (.)

MandNbe constants withM> N> . Putt= ,k= ,C= –_,

σ(t) = √

t, σ(t) = t, σ(t) = t,

f(t,u,u,u) =

 – tn+

tn+ u  +

t₍_t_{– )}

tn+₊_cos_tsin _u

+

 – t–t

tn+₊|_u |

,

c(t) = –t–sint

t , g(t) =

 +tn_cos_tn

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p(t) =M(t n+_{– )}

tn+ +

M_t_|_t_{– |}

tn+₊_cos_t +

M₍_t_{+ }_t_{– )} (tn+₊_N₎ ,

q(t) =M

_(_tn+_{– )}

tn+ +

t|t– |

tn+₊_cos_t+

t+ t– 

tn+₊_N

fort≥,ui∈Randi∈ {, , }. Obviously, (H) and (H) hold. It follows from Theo-rem . that Eq. (.) possesses uncountably many nonoscillatory solutions, and for any

L∈(N,_M), the Mann iterative sequence{xn}n≥generated by (.) converges to some nonoscillatory solution of Eq. (.), and the error estimate (.) holds. However, the results in [, –] are not applicable for Eq. (.).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{Basic Teaching Department, Dalian Vocational Technical College, Dalian, Liaoning 116035, People’s Republic of China.} 2_{Department of Mathematics, Dong-A University, Busan, 49315, Korea.}3_{Department of Mathematics and RINS,}

Gyeongsang National University, Jinju, 52828, Korea. 4_{Center for General Education, China Medical University, Taichung,}

40402, Taiwan.

Acknowledgements

This work was supported by the Dong-A University research fund.

Received: 29 November 2016 Accepted: 6 February 2017

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