Research Article Existence for Eventually Positive Solutions of High-Order Nonlinear Neutral Differential Equations with Distributed Delay

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Volume 2012, Article ID 618058,13pages doi:10.1155/2012/618058

Research Article

Existence for Eventually Positive Solutions of High-Order Nonlinear Neutral

Differential Equations with Distributed Delay

Huanhuan Zhao,

¹

Youjun Liu,

¹

and Jurang Yan

²

1College of Mathematics and Computer Sciences, Shanxi Datong University, Datong, Shanxi 037009, China

2School of Mathematical Sciences, Shanxi University, Taiyuan, Shanxi 030006, China

Correspondence should be addressed to Huanhuan Zhao,[email protected] Received 14 February 2012; Accepted 12 March 2012

Academic Editor: Mingshu Peng

Copyrightq 2012 Huanhuan Zhao 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.

We consider the existence for eventually positive solutions of high-order nonlinear neutral diﬀerential equations with distributed delay. We use Lebesgue’s dominated convergence theorem to obtain new necessary and suﬃcient condition for the existence of eventually positive solutions.

1. Introduction and Preliminary

In this paper, we consider the high-order nonlinear neutral diﬀerential equation:

rtxt −

_b

a

pt, τxt − τdτ

_n

_d

c

qt, σfxt − σdσ 0, t ≥ t0, 1.1

and the associated inequality:

rtxt −

_b

a

_n

_d

c

qt, σfxt − σdσ ≤ 0, t ≥ t0, 1.2

1 where n is a positive integer, 0 < a < b, 0 < c < d,

2 p ∈ Ct0, ∞ × a, b , R,

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3 r ∈ Ct0, ∞, R, rt > 0, q ∈ Ct0, ∞ × c, d , R,

4 fu are continuously nondecreasing real function with respect to u defined on R such that ufu > 0, for u > 0, for n is positive odd integer; fu are continuously decreasing real function with respect to u defined on R such that ufu > 0, for u > 0, for n is positive even integer.

Recently, there has been a lot of activities concerning the existence of eventually positive solutions for nonlinear neutral diﬀerential equations. See 1–8 . In 1 , Liu et al. have studied the even-order neutral diﬀerential equation:

atxt −^m

i1

bitxt − τi

_n

−^l

j1

pjtfj

t, x t − σj

 0, t ≥ 0, 1.3

and the associated diﬀerential inequality:

atxt −^m

i1

bitxt − τi

_n

−^l

j1

pjtfj

t, x t − σj

≥ 0, t ≥ 0. 1.4

They have obtained that the existences of eventually positive solutions of1.3 and 1.4 are equivalent. In2 , Ouyang et al. has studied the odd-order neutral diﬀerential equation:

atxt −^m

i1

bitxt − τi

_n

^l

j1

pjtfj

x t − σj

 0, t ≥ 0, 1.5

and the associated diﬀerential inequality:

atxt −^m

i1

bitxt − τi

_n

^l

j1

pjtfj

x t − σj

≤ 0, t ≥ 0. 1.6

He has obtained that the existences of eventually positive solutions of 1.5 and 1.6 are equivalent.

As usual, a solution of1.1 is a continuous function xt defined on −μ, ∞ such that yt : rtxt −_b

apt, τxt − τdτ is n times diﬀerentiable and 1.1 holds for all n ≥ 0. Such a solution xt is called an eventually positive solution if there is T ≥ t0, such that xt > 0, for t ≥ T. Here, μ max{b, d}.

Lemma 1.1. Assume sup_t≥t₀rt < ∞, and_b

apt, τdτ/rt ≤ M, let xt is an eventually bounded positive solution of inequality1.2, and set

yt rtxt −

_b

a

pt, τxt − τdτ. 1.7

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If n is a positive odd integer, then eventually,

yⁿt ≤ 0, −1^k1y^kt < 0, k 0, 1, . . . , n − 1,

t → ∞limy^kt 0, k 1, 2, . . . , n − 1. 1.8

If n is a positive even integer, then eventually,

yⁿt ≤ 0, −1^ky^kt < 0, k 0, 1, . . . , n − 1,

t → ∞limy^kt 0, k 1, 2, . . . , n − 1. 1.9

Proof. We have the following cases.

Case 1 If n is a positive odd integer. Because xt is bounded, and sup_t≥t₀rt <

∞, _b

apt, τdτ/rt ≤ M, thus yt is bounded. From 1.2, we have yⁿt ≤

−_d

c qt, σfxt − σdσ ≤ 0. Assume yⁿ⁻¹t ≤ 0, since yⁿt ≤ 0, yⁿ⁻¹t decreases, set lim_{t → ∞}yⁿ⁻¹t L, then −∞ ≤ L < 0.

Thus,

yⁿ⁻²t − yⁿ⁻²t0

_t

t0

yⁿ⁻¹tdt ≤ L εt − t0 −→ −∞, t −→ ∞, 1.10

that is, limt → ∞yⁿ⁻²t −∞. Simile, limt → ∞y^kt −∞, k 0, 1, . . . , n − 2, this is a contradiction and yt is bounded, therefore, yⁿ⁻¹t > 0.

Again, since it decreases, set lim_{t → ∞}yⁿ⁻¹t L, thus L ≥ 0. Next, we proof L 0. Assume L > 0, then,

yⁿ⁻²t − yⁿ⁻²t0

_t

t0

yⁿ⁻¹tdt ≥ L − εt − t0 −→ ∞, t −→ ∞, 1.11

then, limt → ∞yⁿ⁻²t ∞. Simile, limt → ∞y^kt ∞, k 0, 1, . . . , n − 2, this is a contradiction and yt is bounded, therefore, L 0. That is, limt → ∞yⁿ⁻¹t 0. Similarly, we obtain

−1^k1y^kt < 0, k 0, 1, . . . , n − 1, lim

t → ∞y^kt 0, k 1, 2, . . . , n − 1. 1.12

Case 2If n is a positive even integer. The proof of Case2is similar to that of part Case1, therefore, it is omitted. We obtain

−1^ky^kt < 0, k 0, 1, . . . , n − 1, lim

t → ∞y^kt 0, k 1, 2, . . . , n − 1. 1.13

The proof is complete.

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Lemma 1.2 see 3, page 21 . Let η ∈ −∞, 0, τ ∈ 0, ∞, t0 ∈ R, and suppose that a function xt ∈ Ct0− τ, ∞, R satisfies the inequality:

xt ≤ η max

t−τ≤s≤txs, t ≥ t0. 1.14

Then, xt cannot be a nonnegative function.

Lemma 1.3. Suppose that sup_t≥t₀rt < ∞, and_b

apt, τdτ/at ≤ 1. Let xt be an eventually positive solution of1.2 and set

yt rtxt −

_b

a

pt, τxt − τdτ, 1.15

then eventually

yt > 0. 1.16

Proof. From1.2 and 1.7, eventually, we have

yⁿt ≤ −

_d

c

qt, σfxt − σdσ ≤ 0, 1.17

and the hypotheses on qt, σ, fx, and xt yield that yⁿ is not eventually zero. Thus, yⁱt i 0, l, . . . , n − 1 is eventually nonzero. Hence, if yt > 0 does not hold, then eventually yt < 0, yt < 0, or yt < 0, yt > 0.

Case 1. yt < 0, yt < 0, then there exists t1 > 0 such that yt ≤ yt1 < 0 for t > t1. Then, α : −yt1/sup_t≥t₀rt > 0. In view of 1.3 and_b

apt, τdτ/at ≤ 1, we obtain

xt  yt

rt 1 rt

_b

a

≤ −α  1 rt

_b

a

pt, τdτ max

t−b≤s≤txs

≤ −α max

t−b≤s≤txs.

1.18

According toLemma 1.2, we obtain xt < 0. This is a contradiction and so yt is eventually positive.

Case 2. yt < 0, yt > 0, one argues that yt < 0 and repeats the argument to obtain that yⁿt > 0 which contradicts the oﬀset after yⁿt < 0.

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2. Comparison Theory of Existence for Eventually Positive Solution

Theorem 2.1. Assume all conditions ofLemma 1.1hold, n is a positive odd integer. And

H1 pt, τ qt, σ > 0, for suﬃciently large t.

Then1.1 has an eventually bounded positive solution if and only if inequality 1.2 has an eventually bounded positive solution.

Proof. It is clear that an eventually bounded positive solution of1.1 is also an eventually bounded positive solution of 1.2. So, it suﬃces to prove that if 1.2 has an eventually bounded positive solution xt, for t > t0, then so does1.1. Set

yt rtxt −

_b

a

pt, τxt − τdτ. 2.1

It follows fromLemma 1.1and1.2 that eventually,

yⁿt ≤ 0, −1^k1y^kt > 0, k 0, 1, . . . , n − 1,

t → ∞limy^kt 0, k 1, 2, . . . , n − 1. 2.2

By using1.8 and integrating 1.2 from t to ∞, we obtain

yⁿ⁻¹∞ − yⁿ⁻¹t ≤ −

_∞

t

_d

c

qt, σfxt − σdσ

ds,

yⁿ⁻¹t ≥

_∞

t

_d

c

qs, σfxs − σdσ

ds.

2.3

By repeating the same procedure n times and by using 1.8, we are led to the inequality:

yt ≥

_∞

t

dtn

_∞

tn

dtn−1

_∞

tn−1

dtn−2· · ·

_∞

t2

_d

c

ds, t ≥ t0. 2.4

Using Tonellis theorem, we reverse the order of integration and obtain

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds, t ≥ t0. 2.5

That is,

xt ≥ 1 rt

_b

a

pt, τxt − τdτ  1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds, t ≥ t0.

2.6

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Let T ≥ t0be such that2.6 hold, and xt−μ > 0, t ≥ T. Now, we consider the set of functions Ω

z ∈ C

T − μ, ∞ , R

: 0≤ zt ≤ 1, t ≥ T − μ

2.7

and define an operator S on Ω as follows:

Szt 

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

t − T μ

μ SzT 

1−t − T μ μ

, T − μ ≤ t < T,

1 xt

1 rt

_b

a

pt, τzt − τxt − τdτ

1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

qs, σfzs − σxs − σdσ

ds

, t ≥ T.

2.8

Then, it follows from Lebesgues dominated convergence theorem that S is continuous. By using2.6, it is easy to see that S maps Ω into itself, and for any z ∈ Ω, we have Szt > 0, for T − μ ≤ t < T. Next, we define the sequence zkt ∈ Ω

z0t ≡ 1, t ≥ T − μ,

zk1t Szkt, t ≥ T − μ, k 0, 1, . . . . 2.9

Then, by using2.6 and a simple induction, we can easily see that

0≤ zk1t ≤ zkt ≤ 1, t ≥ T − μ. 2.10

Set limk → ∞zkt zt, t ≥ T − μ then zt satisfies

zt 

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

t − T μ

μ SzT 

1−t − T μ μ

, T − μ ≤ t < T,

1 xt

1 rt

_b

a

1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds

, t ≥ T.

2.11

Again, set ωt ztxt, then ωt satisfies ωt > 0, T − μ ≤ t ≤ T, and

ωt  1 rt

_b

a

pt, τωt − τdτ  1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

qs, σfωs − σdσ

ds, t ≥ T.

2.12

Thus, ωt is a positive solution of 1.1 for t ≥ T.

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The following:

ωt > 0, t ≥ T − μ. 2.13

Assume that there exists t^∗≥ T − μ, such that ωt > 0, for T − μ ≤ t ≤ t^∗, and ωt^∗ 0. Then,

0 ωt^∗ 1 rt^∗

_b

a

pt^∗, τωt^∗− τdτ

1

rt^∗

_∞

t^∗

s − t^∗ⁿ⁻¹

n − 1!

_d

c

ds, t^∗≥ T,

2.14

which implies

pt^∗, τ 0,

qs, σfωs − σ ≡ 0, 2.15

which contradictsH1. Thus, ωt is an eventually bounded positive solution of 1.1.

Theorem 2.2. Assume all conditions ofLemma 1.3hold, n is a positive odd integer. And

Then,1.1 has an eventually positive solution if and only if inequality 1.2 has an eventually positive solution.

Proof. It is clear that an eventually positive solution of1.1 is also an eventually positive solution of1.2. So, it suﬃces to prove that if 1.2 has an eventually positive solution xt, for t > t0, then so does1.1. Set

yt rtxt −

_b

a

pt, τxt − τdτ. 2.16

It follows from Lemma 1.3 and 1.2 that eventually, yⁿt ≤ 0, yt > 0, which implies that there exists a nonnegative even integer n^∗≤ n − 1, such that eventually

y^kt > 0, k 0, 1, . . . , n^∗,

−1^ky^kt > 0, k n^∗, . . . , n − 1. 2.17

We consider the following possible cases.

Case 1n^∗  0. Since n is a positive integer, n − 1 is an even integer, we can easily see that there exists a T> 0, such yⁿ⁻¹t > 0, and yⁿt ≤ 0, t ≥ T, yⁿ⁻¹∞ ≥ 0.

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yⁿ⁻¹∞ − yⁿ⁻¹t ≤ −

_∞

t

d c

ds,

yⁿ⁻¹t ≥

_∞

t

d c

ds.

2.18

yt ≥

_∞

t

dtn

_∞

tn

dtn−1

_∞

tn−1

dtn−2· · ·

_∞

t2

_d

c

ds, t ≥ t0. 2.19

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds, t ≥ t0. 2.20

That is,

xt ≥ 1 rt

_b

a

pt, τxt − τdτ  1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

d c

qt, σfxt − σdσ

ds, t ≥ t0.

2.21

Let T ≥ t0 be such that2.21 hold, and xt − μ > 0, t ≥ T. Now, we consider the set of functions

Ω z ∈ C

T − μ, ∞ , R

: 0≤ zt ≤ 1, t ≥ T − μ

, 2.22

Szt 

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

t − T μ

μ SzT 

1−t − T μ μ

, T − μ ≤ t < T,

1 xt

1 rt

_b

a

1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds

, t ≥ T.

2.23

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Then, it follows from Lebesgues dominated convergence theorem that S is continuous. By using2.21, it is easy to see that S maps Ω into itself, and for any z ∈ Ω, we have Szt > 0, for T − μ ≤ t < T. Next, we define the sequence zkt ∈ Ω

z0t ≡ 1, t ≥ T − μ,

zk1t Szkt, t ≥ T − μ, k 0, 1, . . . . 2.24

0≤ zk1t ≤ zkt ≤ 1, t ≥ T − μ. 2.25

Set limk → ∞zkt zt, t ≥ T − μ, then zt satisfies:

zt 

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

t − T μ

μ SzT 

1−t − T μ μ

, T − μ ≤ t < T,

1 xt

1 rt

_b

a

1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds

, t ≥ T.

2.26

Again, set ωt ztxt, then ωt satisfies ωt > 0, T − μ ≤ t ≤ T, and

ωt  1 rt

_b

a

pt, τωt − τdτ  1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds, t ≥ T.

2.27

Consider the following:

ωt > 0, t ≥ T − μ. 2.28

0 ωt^∗ 1 rt^∗

_b

a

1

rt^∗

_∞

t^∗

s − t^∗ⁿ⁻¹

n − 1!

d c

ds, t^∗≥ T,

2.29

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which implies

pt^∗, τ 0,

qs, σfωs − σ ≡ 0, 2.30

which contradictsH1. Thus, ωt is an eventually positive solution of equation.

Case 22 ≤ n^∗≤ n − 1. By using 2.17 and integrating 1.2 from t to ∞, we obtain

yⁿ^∗t ≥

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds, t ≥ t0. 2.31

Let T ≥ 0 be such that 2.17 and H1 hold. Integrating 2.31 from T to t and using 2.17, we have

xt ≥ 1 rt

_b

a

1 rt

_t

T

t − sⁿ^∗⁻¹

n^∗− 1!

_∞

t

u − sⁿ⁻ⁿ^∗⁻¹

n − n^∗− 1!

d c

du ds, t ≥ t0.

2.32

Using a method similar to the proof of Case1yields that1.1 also has an eventually positive solution.

Theorem 2.3. Assume all conditions ofLemma 1.1hold, n is a positive even integer, and

Then,1.1 has an eventually positive solution if and only if inequality 1.2 has an eventually positive solution.

Proof. It is clear that an eventually bounded positive solution of1.1 is also an eventually bounded positive solution of 1.2. So, it suﬃces to prove that if 1.2 has an eventually bounded positive solution xt, for t > t0, then so does1.1. Set

yt rtxt −

_b

a

pt, τxt − τdτ. 2.33

It follows fromLemma 1.1and1.2 that eventually,

yⁿt ≤ 0, −1^ky^kt < 0, k 0, 1, . . . , n − 1,

t → ∞limy^kt 0, k 1, 2, . . . , n − 1. 2.34

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yⁿ⁻¹∞ − yⁿ⁻¹t ≤ −

_∞

t

_d

c

ds,

yⁿ⁻¹t ≥

_∞

t

_d

c

ds.

2.35

yt ≤ −

_∞

t

dtn

_∞

tn

dtn−1

_∞

tn−1

dtn−2· · ·

_∞

t2

_d

c

ds, t ≥ t0. 2.36

−

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds, t ≥ t0. 2.37

That is,

xt ≤ 1 rt

_b

a

pt, τωt − τdτ −

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds, t ≥ t0.

2.38

Let T ≥ 0 be such that 2.38 hold, and xt − μ > 0, t ≥ T. Now, we consider the set of functions:

Ω z ∈ C

T − μ, ∞ , R

: 0 < zt ≤ 1, t ≥ T − μ

, 2.39

Szt 

⎧⎪

⎪⎪

⎪⎨

⎪⎪

⎩

t − T μ

μ SzT 

1−t − T μ μ

, T − μ ≤ t < T,

xt

1 rt

_b

apt, τxt − τ

zt − τdτ − 1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c qs, σf

xs − σ

zs − σ

dσ

ds

,

t ≥ T.

2.40

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Then, it follows from Lebesgues dominated convergence theorem that S is continuous. By using2.38, it is easy to see that S maps Ω into itself and, for any z ∈ Ω, we have Szt < 0, for T − μ ≤ t < T. Next, we define the sequence zkt ∈ Ω:

z0t ≡ 1, t ≥ T − μ,

zk1t Szkt, t ≥ T − μ, k 0, 1, . . . . 2.41

0 < zk1t ≤ zkt ≤ 1, t ≥ T − μ. 2.42

Set lim_{k → ∞}zkt zt, t ≥ T − μ, then zt satisfies

zt 

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

t − T μ

μ SzT 

1−t − T μ μ

, T − μ ≤ t < T,

xt

1 rt

_b

apt, τxt − τ

zt − τdτ − 1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c qs, σf

xs − σ

zs − σ

dσ

ds

,

t ≥ T.

2.43

Again, set ωt xt/zt, then ωt satisfies ωt > 0, T − μ ≤ t ≤ T, and

ωt  1 rt

_b

a

pt, τωt − τdτ

− 1 rt

_∞

t

s − tⁿ⁻¹

n − 1!

_d

c

ds, t ≥ T.

2.44

Consider the following

ωt > 0, t ≥ T − μ. 2.45

0 ωt^∗ 1 rt^∗

_b

a

− 1

rt^∗

_∞

t^∗

s − t^∗ⁿ⁻¹

n − 1!

d c

ds, t^∗≥ T,

2.46

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which implies

pt^∗, τ 0,

qs, σfωs − σ ≡ 0, 2.47

which contradictsH1. Thus, ωt is an eventually positive solution of 1.1.

Example 2.4. Consider high-order neutral diﬀerential equation with distributed delay

xt −

₂

1

e^−txt − τdτ

_n

1

2e⁻¹− 3e⁻²

₂

1

σxt − σdσ 0. 2.48

Here, rt ≡ 1, pt − τ e^−t, qt, σ σ/2e⁻¹− 3e⁻², a c 1, b d 2. It is easy to see that

rt ≡ 1 < ∞,

₂

1e^−tdτ

1  e^−t< 1, t > 0. 2.49

From Theorem 2.2, we have that 1.1 has an eventually positive solution if and only if inequality1.2 has an eventually positive solution. In fact, xt e^t is a positive solution of1.1.

Acknowledgments

This research is supported by the Natural Sciences Foundation of Shanxi Province and Scientific Research Project of Shanxi Datong University. The authors are greatly indebted to the referees for their valuable suggestions and comments.

References

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2 Z. Ouyang, Y. Li, and M. Qing, “Eventually positive solutions of odd-order neutral diﬀerential equations,” Applied Mathematics Letters, vol. 17, no. 2, pp. 159–166, 2004.

3 I. Gy¨ori and G. Ladas, Oscillation Theory of Delay Diﬀerential Equations With Applications, Clarendon Presss, Oxford, UK, 1991.

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6 J. G. Dong, “Oscillation of solutions for first order neutral diﬀerential equations with distributed deviating arguments,” Computers and Mathematics with Applications, vol. 58, no. 4, pp. 784–790, 2009.

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Research Article Existence for Eventually Positive Solutions of High-Order Nonlinear Neutral Differential Equations with Distributed Delay