Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 328956, 16 pages
doi:10.1155/2011/328956
Research Article
Positive Solutions for a Higher-Order Nonlinear
Neutral Delay Differential Equation
Zeqing Liu,1 Ming Jia,1 Jeong Sheok Ume,2 and Shin Min Kang3
1
Department of Mathematics, Liaoning Normal University, Liaoning, Dalian 116029, China
Department of Applied Mathematics, Changwon National University,
Changwon 641-773, Republic of Korea
3
Department of Mathematics and the Research Institute of Natural Science,
Gyeongsang National University, Jinju 660-701, Republic of Korea
2
Correspondence should be addressed to Jeong Sheok Ume, jsume@changwon.ac.kr
Received 23 May 2011; Accepted 31 July 2011
Academic Editor: Sergey V. Zelik
Copyright q 2011 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.
This paper deals
nonlinear neutral delay differential equation
m with the higher-order
n−1
dn /dtn xt /dtn−1 ft, xα1 t, . . . , xαk t ht, xβ1 t, . . . ,
i1 pi txTi t d
xβk t gt, t ≥ to , where n, m, k ∈ N, pi , τi , βj , g ∈ Cto , ∞, R, αj ∈ Cn−1 to , ∞, R, f ∈
Cn−1 to , ∞ × Rk , R, h ∈ Cto , ∞ × Rk , R, and limt → ∞ τi t limt → ∞ αj t limt → ∞ βj t ∞, i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , k}. By making use of the Leray-Schauder nonlinear alterative
theorem, we establish the existence of uncountably many bounded positive solutions for the
above equation. Our results improve and generalize some corresponding results in the field. Three
examples are given which illustrate the advantages of the results presented in this paper.
1. Introduction and Preliminaries
This paper is concerned with the higher-order nonlinear neutral delay differential equation:
m
dn−1
dn
xt
p
ft, xα1 t, . . . , xαk t
txτ
t
i
i
dtn
dtn−1
i1
h t, x β1 t , . . . , x βk t gt, t ≥ t0 ,
1.1
where n, m, k ∈ N, pi , τi , βj , g ∈ Ct0 , ∞, R, αj ∈ Cn−1 t0 , ∞, R, f ∈ Cn−1 t0 , ∞ ×
Rk , R, h ∈ Ct0 , ∞ × Rk , R, and
lim τi t lim αj t lim βj t ∞, i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , k}.
t−→∞
t−→∞
t−→∞
1.2
2
Abstract and Applied Analysis
Theory of neutral delay differential equations has undergone a rapid development in
the last over thirty years. We refer the readers to 1–8 and the references therein for a wealth
of reference materials on the subject. The authors 1–8 and others discussed the oscillation,
nonoscillation, and existence of a nonoscillatoy solution for some special cases of 1.1 under
various conditions. By using the Banach fixed point theorem, Zhang et al. 4 and Kulenović
and Hadžiomerspahić 1 studied, respectively, the existence of a nonoscillatory solution for
the first-order neutral delay differential equation:
d
xt ptxt − τ P txt − σ − Qtxt − δ 0,
dt
t ≥ t0 ,
1.3
where τ > 0, σ, δ ∈ R , P, Q ∈ Ct0 , ∞, R , and p ∈ Ct0 , ∞, R, and the second-order
neutral delay differential equation with positive and negative coefficients:
d2 xt pxt − τ P txt − σ − Qtxt − δ 0,
2
dt
t ≥ t0 ,
1.4
where p ∈ R \ {±1}, σ, δ ∈ R and P, Q ∈ Ct0 , ∞, R . Zhang et al. 6 considered the
second-order nonlinear neutral differential equation with positive and negative terms:
d2 xt − pxτt f1 t, xσ1 t − f2 t, xσ2 t 0,
2
dt
t ≥ t0
1.5
and its corresponding equation with forced term:
d2 xt − pxτt f1 t, xσ1 t − f2 t, xσ2 t gt,
2
dt
t ≥ t0 ,
1.6
where t ≥ t0 , p, τ, σi ∈ Ct0 , ∞, R, fi ∈ Ct0 , ∞ × R, R, and limt → ∞ τt limt → ∞ σi t ∞ for i ∈ {1, 2}. Lin 2 investigated sufficient conditions of oscillation and nonoscillation
for the second-order nonlinear neutral differential equation:
d2 xt − ptxt − τ qtfxt − σ 0,
2
dt
t ≥ 0,
1.7
0. Liu and Huang
where τ > 0, σ > 0, p, q ∈ CR , R , f ∈ CR, R with xfx > 0 for all x /
3 used the coincidence degree theory to establish the existence and uniqueness of T -periodic
solutions for the second-order neutral functional differential equation of the form
d2
dxt
gxt − τt pt,
xt Bxt − δ C
2
dt
dt
t ≥ 0,
1.8
Abstract and Applied Analysis
3
where τ, p, g : R → R are continuous functions, B, δ, C are constants, τ and p are T -periodic,
C/
0, |B| /
1, and T > 0. Zhou and Zhang 8 extended the results in 1 to the higher-order
neutral functional differential equation with positive and negative coefficients:
dn xt pxt − τ −1n1 P txt − σ − Qtxt − δ 0,
n
dt
t ≥ t0 ,
1.9
where p ∈ R \ {±1}, τ, σ, δ ∈ R and P, Q ∈ Ct0 , ∞, R . Zhou et al. 7 used the
Krasnoselskii fixed point theorem and the Schauder fixed point theorem to prove the existence results of a nonoscillatory solution for the forced higher-order nonlinear neutral
functional differential equation:
m
dn xt ptxt − τ qi tfxt − σi gt,
n
dt
i1
t ≥ t0 ,
1.10
where τ, σi ∈ R , p, qi , g ∈ Ct0 , ∞, R for i ∈ {1, 2, . . . , m} and f ∈ CR, R. Zhang et al.
5 obtained some sufficient conditions for the oscillation of all solutions of the even order
nonlinear neutral differential equations with variable coefficients:
dn xt ptxτt qtfxσt 0,
n
dt
t ≥ t0 ,
1.11
where n is an even number, p, q, σ, τ ∈ Ct0 , ∞, R with 0 ≤ pt < 1, for all t ≥ t0 ,
limt → ∞ τt limt → ∞ , σi t ∞ and f ∈ Ct0 , ∞, R.
The purpose of this paper is to investigate the solvability of 1.1. By constructing
appropriate mappings and using the Laray-Schauder nonlinear alternative theorem, we
establish a few sufficient conditions which ensure the existence of uncountably many
bounded positive solutions for 1.1. Our results improve and generalize some corresponding
results in 1, 2, 4, 6–8. Three examples are given to illustrate the advantages of the results
presented in this paper.
Throughout this paper, we assume that R, R , and N denote the sets of all real numbers,
nonnegative numbers, and positive integers, respectively, and
ν inf τi t, αj t, βj t : t ∈ t0 , ∞,
i ∈ {1, 2, . . . , m}, j ∈ {1, 2, . . . , k} .
1.12
Let CBν, ∞, R stand for the Banach space of all continuous and bounded functions in
ν, ∞ with norm x supt≥ν |xt| for all x ∈ CBν, ∞, R and
EN {x ∈ CBν, ∞ , R : xt ≥ N for t ≥ ν},
UM {x ∈ EN : x < M},
1.13
where M, N ∈ R with M > N > 0. Clearly, EN is a nonempty closed convex subset of
CBν, ∞, R and UM is an open subset of EN.
By a solution of 1.1, we mean a function x ∈ Cν, ∞, R with some T ≥ t0 |ν| such that xt m
i1 pi txτi t is n times continuously differentiable in T, ∞ and
4
Abstract and Applied Analysis
ft, xα1 t, . . . , xαk t is n − 1 times continuously differentiable in T, ∞ and 1.1 holds
for t ≥ T .
Lemma 1.1 the Leray-Schauder nonlinear alterative theorem 9. Let E be a closed convex
subset of a Banach space X and let U be an open subset of E with p ∈ U. Also, G : U → E is a
continuous, condensing mapping with GU bounded, where U denotes the closure of U Then,
A1 G has a fixed point in U, or
A2 there are x ∈ ∂U and λ ∈ 0, 1 with x 1 − λp λGx.
2. Main Results
Now, we apply the Leray-Schauder nonlinear alterative theorem to investigate the existence
of uncountably many bounded positive solutions of 1.1 under certain conditions.
Theorem 2.1. Assume that there exist constants M, N, p0 , t1 and functions F, H ∈ Ct0 , ∞, R satisfying
ft, u1 , . . . , uk ≤ Ft,
∀t, u1 , . . . , uk ∈ t0 , ∞ × N, Mk ,
2.1
|ht, v1 , . . . , vk | ≤ Ht,
∀t, v1 , . . . , vk ∈ t0 , ∞ × N, Mk ,
2.2
max gs, Hs ds < ∞,
2.3
∞
max
∞
Fs ds,
t0
n−1
s
t0
0 < N < 1 − 2p0 M,
m pi t ≤ p0 < 1 ,
2
i1
∀t ≥ t1 ≥ t0 .
2.4
Then, 1.1 has uncountably many bounded positive solutions in UM.
Proof. Let L ∈ p0 M N, 1 − p0 M. It follows from 2.3 and 2.4 that there exists a constant
T > 1 |t0 | |t1 | |ν| satisfying
∞
Fs ds ∞
T
T
M−N
sn−1 gs Hs ds < min L − p0 M − N, 1 − p0 M − L,
.
2
2.5
Choose 0 ∈ 0, min{L − p0 M − N, 1 − p0 M − L, M − N/2} with
∞
T
Fs ds ∞
T
M−N
sn−1 gs Hs ds < min L − p0 M − N, 1 − p0 M − L,
− 0 .
2
2.6
Abstract and Applied Analysis
5
Put p M − 0 . Clearly, p ∈ UM. Define two mappings AL , BL : UM → CBν, ∞, R
by
⎧
n ∞
m
⎪
⎨L − pi txτi t −1
s − tn−1 gs ds, t ≥ T,
−
1!
n
AL xt t
i1
⎪
⎩
ν ≤ t < T,
AL xT ,
⎧ ∞
⎪
⎪
fs, xα1 s, . . . , xαk s ds
⎪
⎪
⎪
⎨ t
−1n−1 ∞
BL xt s − tn−1 h s, x β1 s , . . . , x βk s ds, t ≥ T,
⎪
⎪
⎪
n − 1! t
⎪
⎪
⎩
ν≤t<T
BL xT ,
2.7
2.8
for all x ∈ UM. It is clear to see that AL x and BL x are continuous for each x ∈ UM. Let
DL AL BL . In view of 2.1, 2.2, and 2.4–2.8, we get that
AL xt BL xt
m
−1n ∞
L − pi txτi t s − tn−1 gs ds
−
1!
n
t
i1
∞
fs, xα1 s, . . . , xαk s ds
t
−1n−1
n − 1!
∞
≥ L − p0 M −
s − tn−1 h s, x β1 s , . . . , x βk s ds
t
∞
Fs ds −
∞
T
2.9
sn−1 gs Hs ds
T
M−N
≥ L − p0 M − min L − p0 M − N, 1 − p0 M − L,
ε0
2
> N,
∀t, x ∈ T, ∞ × UM,
which gives that DL : UM → EN.
Now, we show that BL : UM → CBν, ∞, R is continuous and compact. Let
{xm }m∈N ⊆ UM be an arbitrary sequence and x ∈ Cν, ∞, R with
xm − x −→
as m −→ ∞.
2.10
Since UM is closed, it follows that x ∈ UM. For any s, m ∈ T, ∞ × N, put
Fm s fs, xm α1 s, . . . , xm αk s − fs, xα1 s, . . . , xαk s,
Hm s h s, xm β1 s , . . . , xm βk s − h s, x β1 s , . . . , x βk s .
2.11
6
Abstract and Applied Analysis
It follows from 2.1, 2.2, and 2.11 that
|Fm s| ≤ 2Fs,
|Hm s| ≤ 2Hs,
∀s, m ∈ T, ∞ × N,
2.12
which together with 2.8–2.11, the continuity of f, h, αj , βj for j ∈ {1, 2, . . . , k}, and the
Lebesgue dominated convergence theorem yields that
|BL xm t − BL xt|
∞
fs, xm α1 s, . . . , xm αk s − fs, xα1 s, . . . , xαk sds
≤
t
1
n − 1!
∞
×
s − tn−1 h s, xm β1 s , . . . , xm βk s − h s, x β1 s , . . . , x βk s ds
t
≤
∞
T
Fm sds 1
n − 1!
∞
sn−1 Hm sds,
T
lim supBL xm − BL x ≤ lim sup
m→∞
∞
m→∞
T
∀t ≥ T,
1
Fm sds n − 1!
∞
sn−1 Hm sds 0,
T
2.13
which means that BL is continuous in UM. It follow from 2.1, 2.2, 2.6, and 2.8 that
BL x sup|BL xt|
t≥ν
∞
∞
1
sn−1 Hsds
n − 1! T
T
M−N
− 0
< min L − p0 M − N, 1 − p0 M − L,
2
≤
< M,
Fsds 2.14
∀x ∈ UM,
which yields that BL UM is uniformly bounded in ν, ∞.
Let ε be an arbitrary positive number. Equation 2.3 ensures that there exists T ∗ > T
satisfying
∞
∞
ε
2.15
Fsds sn−1 Hsds < .
2
T∗
T∗
Set
δ
ε
,
1 4 Q M QT ∗ − T n−1
Q max{Ft, Ht : t ∈ T, T ∗ }.
2.16
For any x ∈ UM and t1 , t2 ∈ ν, ∞ with |t1 − t2 | < δ, we consider the following three cases.
Abstract and Applied Analysis
7
Case 1 T ∗ ≤ t1 < t2 . In view of 2.1, 2.2, 2.8, and 2.15, we deduce that
|BL xt2 − BL xt1 |
∞
fs, xα1 s, . . . , xαk sds
t2
−1n−1
n − 1!
−
∞
∞
s − t2 n−1 h s, x β1 s , . . . , x βk s ds
t2
fs, xα1 s, . . . , xαk sds
2.17
t1
−1n−1
−
n − 1!
≤2
∞
T∗
∞
s − t1 n−1
t1
2
Fsds n − 1!
h s, x β1 s , . . . , x βk s ds
∞
T∗
sn−1 Hsds
< ε.
Case 2 T ≤ t1 < t2 ≤ T ∗ . Suppose that n 1. It follows from 2.1, 2.2, 2.6, and 2.8 that
|BL xt2 − BL xt1 |
∞
∞
fs, xα1 s, . . . , xαk sds h s, x β1 s , . . . , x βk s ds
t2
t2
∞
−
fs, xα1 s, . . . , xαk sds −
t1
≤
t2
∞
t1
h s, x β1 s , . . . , x βk s ds
2.18
Fs Hsds
t1
≤ 2Q|t1 − t2 |
< ε.
Suppose that n ∈ N \ {1}. It follows from the mean value theorem that, for each s ∈ t2 , ∞,
there exists ζ ∈ s − t2 , s − t1 satisfying
s − t2 n−1 − s − t1 n−1 n − 1ζn−2 |t1 − t2 | ≤ n − 1sn−1 |t1 − t2 |,
2.19
8
Abstract and Applied Analysis
which together with 2.1, 2.2, 2.6, and 2.8 yields that
|BL xt2 − BL xt1 |
∞
fs, xα1 s, . . . , xαk sds
t2
−1n−1 ∞
s − t2 n−1 h s, x β1 s , . . . , x βk s ds
n − 1! t2
∞
fs, xα1 s, . . . , xαk sds
−
t1
−1n−1
−
n − 1!
≤
t2
t1
∞
s − t1 t1
1
Fsds n − 1!
n−1
h s, x β1 s , . . . , x βk s ds
∞
t2
n−1
n−1 n−1
s − t1 Hsds
s − t2 − s − t1 Hsds t2
t1
∞
n − 1|t1 − t2 |
sn−1 Hsds T ∗ − T n−1 Q|t1 − t2 |
n − 1!
t2
≤ Q M T ∗ − T n−1 Q |t1 − t2 |
≤ Q|t1 − t2 | < ε.
2.20
Case 3 ν ≤ t1 < t2 ≤ T . Equation 2.8 gives that
|BL xt2 − BL xt1 | |BL xT − BL xT | 0.
2.21
Thus, BL UM is equicontinuous in ν, ∞. Hence, BL UM is a relatively compact
subset of Cν, ∞, R. That is, BL is a compact mapping.
Note that for any x, y ∈ UM and t ≥ T
AL xt − AL y t
m
−1n ∞
L − pi txτi t s − tn−1 gsds
−
1!
n
t
i1
m
−1n ∞
n−1
−L pi tyτi t −
s − t gsds
−
1!
n
t
i1
≤
m pi tx − y
i1
≤ p0 x − y,
2.22
Abstract and Applied Analysis
9
which implies that
AL x − AL y ≤ p0 x − y,
∀x, y ∈ UM,
2.23
which together with 2.4 gives that AL is a contraction mapping. It follows that DL :
UM → EN is a continuous and condensing mapping. Let x0 t N for all t ∈ ν, ∞.
Notice that x0 ∈ UM. Thus, 2.4–2.7, 2.14, and 2.23 yield that
DL x ≤ AL x BL x
≤ AL x − AL x0 AL x0 M
m
−1n ∞
n−1
≤ p0 x − x0 M supL − pi tx0 τi t s − t gsds
−
1!
n
t
t≥T
i1
∞
≤ p0 M N M L p0 N sn−1 gsds
2.24
T
≤ 2 p0 M 2p0 N L,
∀x ∈ UM,
that is, DL UM is uniformly bounded in γ, ∞.
Put
S1 {x ∈ CBν, ∞, R : N ≤ xt ≤ M,
∀t ≥ ν, x M},
S2 {x ∈ CBν, ∞, R : N ≤ xt ≤ M,
∀t ≥ ν and there exists t∗ ≥ ν satisfying xt∗ N .
2.25
It is easy to verify that ∂UM S1 ∪ S2 .
Next, we show that A2 in Lemma 1.1 does not hold. Otherwise, there exist x ∈
∂UM and λ ∈ 0, 1 satisfying x 1 − λp λDL x. We have to discuss the following
possible cases.
Case 1. Let x ∈ S1 . By means of 2.1, 2.2, and 2.4–2.8, we get that, for t ≥ T ,
xt 1 − λp λAL xt BL xt
≤ 1 − λM − 0 m λ L pi txτi t i1
∞
t
1
n − 1!
∞
sn−1 gsds
t
fs, xα1 s, . . . , xαk sds
10
Abstract and Applied Analysis
∞
sn−1 h s, x β1 s , . . . , x βk s ds
t
∞
∞
1
≤ 1 − λM − 0 λ L p0 M Fsds sn−1 gs Hs ds
n − 1! t
t
M−N
< 1 − λM − 0 λ L p0 M min L − p0 M − N, 1 − p0 M − L,
− 0
2
≤ M − 0 ,
1
n − 1!
2.26
which implies that
M x sup |xt| ≤ M − 0 < M,
2.27
t≥ν
which is a contradiction.
Case 2. Let x ∈ S2 . It follows from 2.1, 2.2, and 2.4–2.8 that
N xt∗ 1 − λp λAL xt∗ BL xt∗ 1 − λM − 0 λAL xmax{t∗ , T } BL xmax{t∗ , T }
m ≥ 1 − λM − 0 λ L − pi max{t∗ , T }xτi max{t∗ , T }
1
−
n − 1!
1
−
n − 1!
i1
∞
max{t∗ ,T }
sn−1 gsds −
∞
max{t∗ ,T }
∞
max{t∗ ,T }
fs, xα1 s, . . . , xαk sds
h s, x β1 s , . . . , x βk s ds
n−1 s
≥ 1 − λM − 0 ∞
∞
1
n−1 gs Hs ds
λ L − p0 M −
Fsds −
s
n − 1! max{t∗ ,T }
max{t∗ ,T }
M−N
≥ 1 − λM − 0 λ L − p0 M − min L − p0 M − N, 1 − p0 M − L,
0
2
≥ 1 − λM − 0 λN 0 ≥ min{M − 0 , N 0 }
N 0 ,
2.28
which is absurd.
Abstract and Applied Analysis
11
Thus, Lemma 1.1 ensures that DL has a fixed point x ∈ UM; that is,
m
−1n ∞
xt L − pi txτi t s − tn−1 gsds
−
1!
n
t
i1
∞
fs, xα1 s, . . . , xαk sds
t
−1n−1
n − 1!
∞
s − tn−1 h s, x β1 s , . . . , x βk s ds,
2.29
∀t ≥ T,
t
which yields that
m
dn
dn−1
xt pi txτi t n−1 ft, xα1 t, . . . , xαk t
n
dt
dt
i1
h t, x β1 t , . . . , x βk t gt, ∀t ≥ T,
2.30
which means that x ∈ UM is a bounded positive solution of 1.1.
L2 . Similarly, we can prove that, for each
Let L1 , L2 ∈ p0 M N, 1 − p0 M with L1 /
r ∈ {1, 2}, there exist a constant Tr > 1 |t0 | |t1 | |ν| and two mappings ALr , BLr : UM →
CBν, ∞, R satisfying 2.6–2.8, where T, L, AL , and BL are replaced by Tr , Lr , ALr , and
BLr , respectively, and ALr BLr has a fixed point zr ∈ UM, which is a bounded positive
solution of 1.1 in UM. In order to prove that 1.1 possesses uncountably many bounded
positive solutions in UM, we need only to prove that z1 /
z2 . By means of 2.1–2.3, we
know that there exists T3 > max{T1 , T2 } satisfying
∞
∞
|L1 − L2 |
.
2.31
Fsds sn−1 Hsds <
4
T3
T3
It follows from 2.1, 2.2, 2.4, 2.7, 2.8, and 2.31 that for t ≥ T3
|z1 t − z2 t|
m
m
L1 − L2 − pi tz1 τi t pi tz2 τi t
i1
i1
∞
∞
fs, z1 α1 s, . . . , z1 αk sds −
fs, z2 α1 s, . . . , z2 αk sds
t
t
∞
−1
s − tn−1 h s, z1 β1 s , . . . , z1 βk s ds
n − 1! t
−1n−1 ∞
n−1 −
s − t h s, z2 β1 s , . . . , z2 βk s ds
n − 1! t
∞
∞
≥ |L1 − L2 | − p0 z1 − z2 − 2
Fsds − 2
sn−1 Hsds
n−1
T3
≥ |L1 − L2 | − p0 z1 − z2 −
|L1 − L2 |
,
2
T3
2.32
12
Abstract and Applied Analysis
which implies that
|L1 − L2 |
> 0,
z1 − z2 ≥ 2 1 p0
2.33
that is, z1 / z2 . This completes the proof.
Theorem 2.2. Assume that there exist constants M, N, p0 , t1 and functions F, H ∈ Ct0 , ∞, R satisfying 2.1–2.3 and
N < 1 p0 M, max pi t : 1 ≤ i ≤ m ≤ 0,
m
pi t ≥ p0 > −1,
∀t ≥ t1 ≥ t0 .
2.34
i1
Then, 1.1 has uncountably many bounded positive solutions in UM.
Proof. Let L ∈ N, 1 p0 M. It follows from 2.3 and 2.34 that there exists a constant
T > 1 |t0 | |t1 | |ν| satisfying
∞
Fsds ∞
T
T
M−N
sn−1 gs Hs ds < min L − N, 1 p0 M − L,
.
2
2.35
Take 0 ∈ 0, min{L − N, 1 p0 M − L, M − N/2} such that
∞
Fsds T
∞
n−1
s
T
gs Hs ds ≤ min L − N, 1 p0 M − L, M − N − 0 .
2
2.36
The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the
proof.
Theorem 2.3. Assume that there exist constants M, N, p0 , t1 and functions F, H ∈ Ct0 , ∞, R satisfying 2.1–2.3 and
N < 1 − p0 M, min pi t : 1 ≤ i ≤ m ≥ 0,
m
pi t ≤ p0 < 1,
∀t ≥ t1 ≥ t0 .
2.37
i1
Then, 1.1 has uncountably many bounded positive solutions in UM.
Proof. Let L ∈ p0 M N, M. It follows from 2.3 and 2.37 that there exists a constant
T > 1 |t0 | |t1 | |ν| satisfying
∞
T
Fsds ∞
n−1
s
T
gs Hs ds < min L − p0 M − N, M − L, M − N .
2
2.38
Abstract and Applied Analysis
13
Choose 0 ∈ 0, min{L − p0 M − N, M − L, M − N/2} such that
∞
Fsds ∞
T
T
M−N
− 0 . 2.39
sn−1 gs Hs ds ≤ min L − p0 M − N, M − L,
2
The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the
proof.
Remark 2.4. Theorems 2.1–2.3 extend, improve, and unify the theorem in 1, Theorem 2.2 in
2, Theorem 1 in 4, Theorems 2.1 and 2.3 in 6, Theorems 1 and 3 in 7, and Theorems 1
and 3 in 8.
3. Examples and Applications
Now, we construct three nontrivial examples to show the superiority and applications of
Theorems 2.1–2.3, respectively.
Example 3.1. Consider the higher-order nonlinear neutral delay differential equation:
−1n t sin 1 − t3 3
t2 cos t
dn
xt −
x t −t
xt 2 dtn
1 4t
1 6t2
tx2 t2 2 sin2 t − t3 x2 t − 2
dn−1 1 t2 x4 t − 1 − tx2 t2 2
n−1
1 t5
1 x2 t − 2 t3
dt
ln 1 x2 2t t2 sin x t2
t3 1/t
3
2
cos x t ln t 1 tn3 t2
1 tn4
1 − t3
,
tn4 ln1 t3 3.1
t ≥ 3.
Let t0 t1 3, p0 5/12, m 2, k 3, M 36, N 3, ν 1,
−1n t sin 1 − t3
,
p2 t 1 4t
t2 cos t
,
p1 t −
1 6t2
α1 t t − 1,
α2 t t2 2,
τ1 t t 2,
τ2 t t3 − t,
β2 t 2t ,
β1 t t2 ln t,
2
1 t2 u4 − tv2 tv2 sin t − t3 w2
2
ft, u, v, w ,
β3 t t ,
1 t5
1 w 2 t3
ln 1 v2 t2 sin w
t3 1/t
3
cos u ,
ht, u, v, w 1 tn3 t2
1 tn4
1 t2 M4 tM2
tM2
t3 1/t ln 1 M2 t2
Ft ,
Ht ,
1 t5
1 N 2 t3
1 tn3 t2
1 tn4
gt α3 t t − 2,
1 − t3
,
tn4 ln1 t3 ∀t, u, v, w ∈ t0 , ∞ × R3 .
3.2
14
Abstract and Applied Analysis
It is clear that 2.1–2.4 hold. Consequently, Theorem 2.1 ensures that 3.1 has uncountably
many bounded positive solutions in UM. But Theorem in 1, Theorems 2.1 and 2.3 in 6,
Theorems 1 and 3 in 7, and Theorems 1 and 3 in 8 are null for 3.1.
Example 3.2. Consider the higher-order nonlinear neutral delay differential equation:
t2
2t2
dn
2
2
xt −
x tt −
x t − 4t
dtn
1 3t2
1 7t2
t2 x2 t − 1/t
dn−1 2 t4 x2 t − 1/t
n−1
1 t6
dt
1 t4 1 tx2 2t2 − t
2
3.3
t3 − x t2 − t − t2 x3 t ln1 t
1 tn4 2 sin2 t3 xt2 − tx4 t ln1 t
√
1 t 1 − t t2 ln1 2|t| ,
1 tn4
t ≥ 1.
Let t0 t1 1, p0 −13/21, m 2, k 2, M 400, N 100, ν −4,
p1 t −
t2
,
1 3t2
1
α1 t t − ,
t
ft, u, v p2 t −
2t2
,
1 7t2
α2 t 2t2 − t,
t 2 u2
2 t 4 u2
,
6
4
1t
1 t 1 tv2 Ft τ1 t t t2 ,
β1 t t2 − t,
τ2 t t2 − 4t,
β2 t t ln1 t,
t3 − u2 − t2 v3
1 tn4 2 sin2 t3 uv4
ht, u, v 2 t4 M2
t2 M 2
,
1 t6
1 t4 1 tN 2 √
1 t gt 1 − t t2 ln1 2|t| ,
1 tn4
Ht 3.4
t3 M 2 t 2 M 3
,
2 2tn4
∀t, u, v ∈ t0 , ∞ × R2 .
It is easy to verify that 2.1–2.3 and 2.34 hold. Consequently, Theorem 2.2 guarantees
that 3.3 has uncountably many bounded positive solutions in UM. But Theorem in 1,
Theorem 1 in 4, Theorem 2.3 in 6, Theorem 3 in 7, and Theorem 3 in 8 are useless for
3.3.
Abstract and Applied Analysis
15
Example 3.3. Consider the higher-order nonlinear neutral delay differential equation:
2 ln 1 t2
dn
t4
2
4
xt x 2 ln t
x t 1 dtn
1 3 ln1 t2 1 t2 4t4
n
dn−1 t x3 t3 2t2 − −1 t2 sin x t2
n−1
dt
1 t5 t2 x4 t2 √
t − −1n ln 1 1 t2
t3 − x2 t − 2 t2 x5 t − 2
,
√
1 tn5 1 tx2 t2 ln1 t
tn5/2 1 cos2 t
3.5
t ≥ 2.
Let t0 t1 2, p0 11/12, m 2, k 2, M 24, N 1, ν 0,
2 ln 1 t2
,
p2 t 1 3 ln1 t2 t4
p1 t ,
1 t2 4t4
α1 t t3 2t2 ,
ft, u, v α2 t t2 ,
t u3 − −1n t2 sin v
,
1 t5 t2 v 4
τ1 t t4 1,
β1 t t − 2,
ht, u, v τ2 t 2 ln2 t,
β2 t t2 ln1 t,
t 3 − u2 t 2 u 5
,
1 tn5 1 tv2 3.6
t M 3 t2
t3 M 2 t 2 M 5
,
,
Ht
1 tn5 1 tN 2 1 t5 t2 N 4
√
t − −1n ln 1 1 t2
gt , ∀t, u, v ∈ t0 , ∞ × R2 .
√
tn5/2 1 cos2 t
Ft Obviously, 2.1–2.3 and 2.37 hold. It follows from Theorem 2.3 that 3.5 has uncountably
many bounded positive solutions in UM. But Theorem in 1, Theorem 2.2 in 2,
Theorem 2.1 in 4, Theorem 2.1 in 6, Theorem 1 in 7, and Theorem 1 in 8 are inapplicable for 3.5.
Acknowledgments
This work was supported by the Science Research Foundation of Educational Department of
Liaoning Province 2011 and Changwon National University in 2011.
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Abstract and Applied Analysis
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