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Springerplus. 2016; 5: 459.
Published online 2016 April 14. doi:  10.1186/s40064-016-2111-y
PMCID: PMC4831960

Oscillation of certain higher-order neutral partial functional differential equations

Abstract

In this paper, we study the oscillation of certain higher-order neutral partial functional differential equations with the Robin boundary conditions. Some oscillation criteria are established. Two examples are given to illustrate the main results in the end of this paper.

Keywords: Oscillation, Partial functional differential equation, Robin boundary condition

Background

It is well known that the theory of partial functional differential equations can be applied to many fields, such as population dynamics, cellular biology, meteorology, viscoelasticity, engineering, control theory, physics and chemistry (Wu 1996). In the monograph, Wu (1996) provided some fundamental theories and applications of partial functional differential equations.

The oscillation theory as a part of the qualitative theory of partial functional differential equations has been developed in the past few years. Many researchers have established some oscillation results for partial functional differential equations. For example, see the monograph (Yoshida 2008) and the papers (Bainov et al. 1996; Fu and Zhuang 1995; Li and Cui 1999; Li 2000; Li and Cui 2001; Ouyang et al. 2005; Gao and Luo 2008; Li and Han 2006; Wang et al. 2010). We especially note that the monograph (Yoshida 2008) contained large material on oscillation theory for partial differential equations.

Li and Cui (2001) studied the oscillation of even order partial functional differential equations

ntn[u(x,t)+μ(t)u(x,t-ρ)]=a(t)Δu(x,t)+k=1sak(t)Δu(x,ρk(t))-q(x,t)u(x,t)-abp(x,t,ξ)u(x,g(t,ξ))dσ(ξ),(x,t)Ω×[0,)G,
E1

where n ≥ 2 is an even integer, with the two kinds of boundary conditions:

u(x,t)N+ν(x,t)u(x,t)=0,(x,t)Ω×[0,),
B1

and

u(xt) = 0,  (xt) ∈ Ω × [0, ).
B2

Ouyang et al. (2005) established the oscillation of odd order partial functional differential equations

nu(x,t)tn-a(t)Δu(x,t)-k=1spk(x,t)u(x,t-σk)-j=1mqj(x,t)u(x,t-τj)+h(t)f(u(x,t-r1),,u(x,t-r))=0,(x,t)Ω×[0,)G,
E2

where n is an odd integer and s ≤ m, with the boundary conditions (B1), (B2) and

u(x,t)N=0,(x,t)Ω×[0,).
B3

In this paper, we investigate the oscillation of the following higher-order neutral partial functional differential equations

ntn[u(x,t)+μ(t)u(x,t-τ)]=a(t)Δu(x,t)+k=1sak(t)Δu(x,ρk(t))-abp(t,ξ)u(x,g(t,ξ))dσ(ξ),(x,t)Ω×[0,)G,
1

with the Robin boundary condition

α(x)u(x,t)N+β(x)u(x,t)=0,(x,t)Ω×[0,),
2

where n ≥ 2 is an even integer, Ω is a bounded domain in M with a piecewise smooth boundary Ω, and Δ is the Laplacian in the Euclidean M-space M, αβ ∈ C(Ω, [0, )), α2(x) + β2(x) ≠ 0, and N is the unite exterior normal vector to Ω.

Throughout this paper, we always suppose that the following conditions hold:

  1. μ ∈ Cn([0, );[0, )), 0 ≤ μ(t) ≤ 1, τ const. > 0;
  2. aak ∈ C([0, );[0, )), ρk ∈ C([0, );[0, )), ρk(t) ≤ tlimt→+ρk(t) =  + k ∈ Is = {1, 2, …, s
  3. p ∈ C([0, ) × [ab];[0, )), g ∈ C([0, ) × [ab];[0, )), g(tξ) ≤ tξ ∈ [ab], g(tξ) is a nondecreasing function with respect to t and ξ, respectively, and limt→+infξ∈[a,b]{g(tξ)} =  + ;
  4. σ ∈ ([ab];ℝ) and σ(ξ) is nondecreasing in ξ, the integral in (1) is Stieltjes integral.

As it is customary, the solution u(x,t)Cn(G)C1(G¯) of the problem (1), (2) is said to be oscillatory in the domain G ≡ Ω × [0, ) if for any positive number μ there exists a point (x0t0) ∈ Ω × [μ) such that the equality u(x0t0) = 0 holds.

To the best of our knowledge, no result is known regarding the oscillatory behavior of higher-order partial functional differential equations with the Robin boundary condition (2) up to now.

The paper is organized as follows. In “Main results” section, we establish some results for the oscillation of the problem (1), (2). In “Examples” section, we construct two examples to illustrate our main results.

Main results

In this section, we establish the oscillation criteria of the problem (1), (2). First, we introduce the following lemma which is very useful for establishing our main results.

Lemma 1

Ye and Li (1990). Suppose thatλ0is the smallest eigenvalue of the problem

Δφ(x)+λφ(x)=0,inΩ,α(x)φ(x)N+β(x)φ(x)=0,onΩ
3

andφ(x)is the corresponding eigenfunction ofλ0. Thenλ0 = 0, φ(x) = 1asβ(x) = 0(x ∈ Ω)andλ0 > 0, φ(x) > 0(x ∈ Ω)asβ(x) ≢ 0(x ∈ Ω).

Next, we give our main results.

Theorem 2

Ifβ(x) ≢ 0forx ∈ Ω, then the necessary and sufficient condition for all solutions of the problem (1), (2) to oscillate is that all solutions of the differential equation

[y(t)+μ(t)y(t-τ)](n)+λ0a(t)y(t)+λ0k=1sak(t)y(ρk(t))+abp(t,ξ)y(g(t,ξ))dσ(ξ)=0,t0,
4

to oscillate, whereλ0is the smallest eigenvalue of (3).

Proof

(i) Sufficiency. Suppose to the contrary that there is a non-oscillatory solution u(xt) of the problem (1), (2) which has no zero in Ω × [t0) for some t0 ≥ 0. Without loss of generality we assume that u(xt) > 0,  u(xtτ) > 0,  u(xρk(t)) > 0, u(xg(tξ)) > 0, (xt) ∈ Ω × [t1), k ∈ Is.

Multiplying both sides of (1) by φ(x) and integrating with respect to x over the domain Ω, we have

dndtn[Ωu(x,t)φ(x)dx+μ(t)Ωu(x,t-τ)φ(x)dx]=a(t)ΩΔu(x,t)φ(x)dx+k=1sak(t)ΩΔu(x,ρk(t))φ(x)dx-Ωabp(t,ξ)u(x,g(t,ξ))φ(x)dσ(ξ)dx,tt1.
5

From Green’s formula and boundary condition (2), it follows that

ΩΔu(x,t)φ(x)dx=Ω(φ(x)u(x,t)N-u(x,t)φ(x)N)dS+Ωu(x,t)Δφ(x)dx=Ω(φ(x)u(x,t)N-u(x,t)φ(x)N)dS-λ0Ωu(x,t)φ(x)dx,tt1,

where dS is the surface element on Ω.

If α(x) ≡ 0, x ∈ Ω,  then from (2) we have

β(x) ≢ 0,  u(xt) = 0,  (xt) ∈ Ω × [0, ).

Hence, we obtain

Ω(φ(x)u(x,t)N-u(x,t)φ(x)N)dS0,tt1.

If α(x) ≢ 0, x ∈ Ω. Noting that Ω is piecewise smooth, αβ ∈ C(Ω, [0, )), α2(x) + β2(x) ≠ 0, without loss of generality, we can assume that α(x) > 0,  x ∈ Ω. Then by (2) and (3) we have

Ω(φ(x)u(x,t)N-u(x,t)φ(x)N)dS=Ω(-φ(x)β(x)α(x)u(x,t)+u(x,t)β(x)α(x)φ(x))dS=0,tt1,

Therefore, using Lemma 1, we obtain

ΩΔu(xt)φ(x)dx =  - λ0Ωu(xt)φ(x)dxt ≥ t1.
6

Similarly, we have

ΩΔu(xρk(t))φ(x)dx =  - λ0Ωu(xρk(t))φ(x)dxt ≥ t1k ∈ Is.
7

It is easy to see that

Ωabp(t,ξ)u(x,g(t,ξ))φ(x)dσ(ξ)dx=abp(t,ξ)Ωu(x,g(t,ξ))φ(x)dxdσ(ξ),tt1.
8

Set

V(t) = ∫Ωu(xt)φ(x)dxt ≥ t1.

Combining (5)–(8) we have

[V(t)+μ(t)V(t-τ)](n)+λ0a(t)V(t)+λ0k=1sak(t)V(ρk(t))+abp(t,ξ)V(g(t,ξ))dσ(ξ)=0,tt1.
9

Obviously, it follows from (9) that V(t) is a positive solution of Eq. (4), which contradicts the fact that all solutions of Eq. (4) are oscillatory.

(ii) Necessity. Suppose that Eq. (4) has a non-oscillatory solution

An external file that holds a picture, illustration, etc.
Object name is 40064_2016_2111_Article_IEq57.gif
. Without loss of generality we assume
An external file that holds a picture, illustration, etc.
Object name is 40064_2016_2111_Article_IEq58.gif
for t ≥ t ≥ 0, where t is some large number. From (4), we have

[V~(t)+μ(t)V~(t-τ)](n)+λ0a(t)V~(t)+λ0k=1sak(t)V~(ρk(t))+abp(t,ξ)V~(g(t,ξ))dσ(ξ)=0,tt.
10

Multiplying both sides of (10) by φ(x), we obtain

ntn[V~(t)φ(x)+μ(t)V~(t-τ)φ(x)]+λ0a(t)V~(t)φ(x)+λ0k=1sak(t)V~(ρk(t))φ(x)+abp(t,ξ)V~(g(t,ξ))φ(x)dσ(ξ)=0,tt,xΩ.
11

Let

An external file that holds a picture, illustration, etc.
Object name is 40064_2016_2111_Article_IEq62.gif
. By Lemma 1, we have Δφ(x) =  - λ0φ(x),  x ∈ Ω. Then (11) implies

ntn[u~(x,t)+μ(t)u~(x,t-τ)]=a(t)Δu~(x,t)+k=1sak(t)Δu~(x,ρk(t))-abp(t,ξ)u~(x,g(t,ξ))dσ(ξ),tt,xΩ,
12

which shows that

An external file that holds a picture, illustration, etc.
Object name is 40064_2016_2111_Article_IEq64.gif
satisfies (1).

From Lemma 1, we get

α(x)φ(x)N+β(x)φ(x)=0,xΩ,

which implies

α(x)u~(x,t)N+β(x)u~(x,t)=0,(x,t)Ω×[0,).
13

Hence

An external file that holds a picture, illustration, etc.
Object name is 40064_2016_2111_Article_IEq65.gif
is a non-oscillatory solution of the problem (1), (2), which is a contradiction. The proof is complete.

Remark 3

Theorem 2 shows that the oscillation of problem (1), (2) is equivalent to the oscillation of the differential equation (4).

Theorem 4

Ifβ(x) ≢ 0forx ∈ Ω, and the neutral differential inequality

[y(t)+μ(t)y(t-τ)](n)+abp(t,ξ)y(g(t,ξ))dσ(ξ)0,t0,
14

has no eventually positive solutions, then every solution of the problem (1), (2) is oscillatory inG.

Proof

Suppose to the contrary that there is a non-oscillatory solution u(xt) of the problem (1), (2) which has no zero in Ω × [t0) for some t0 ≥ 0. Without loss of generality we assume that u(xt) > 0,  u(xtτ) > 0,  u(xρk(t)) > 0, u(xg(tξ)) > 0, (xt) ∈ Ω × [t1), k ∈ Is. As in the proof of Theorem 2, we obtain Eq. (9). By Lemma 1, from (9) we have

[V(t)+μ(t)V(t-τ)](n)+abp(t,ξ)V(g(t,ξ))dσ(ξ)=-λ0a(t)V(t)-λ0k=1sak(t)V(ρk(t))0,tt1,
15

which shows that V(t) > 0 is a solution of the inequality (14). This is a contradiction. The proof of Theorem 4 is complete.

Using Theorems 1 and 2 in Li and Cui (2001), we can obtain the following two conclusions, respectively.

Theorem 5

Assume thatβ(x) ≢ 0forx ∈ Ω. If fort0 > 0, 

t0+abp(s,ξ)[1-μ(g(s,ξ))]dσ(ξ)ds=+,
16

then every solution of the problem (1), (2) is oscillatory inG.

Theorem 6

Assume thatβ(x) ≢ 0forx ∈ Ω, μ(t) ≡ μis a positive constant,p(tξ)is periodic in t with periodρ. If fort0 > 0, 

g(t-c,ξ)=g(t,ξ)-cforanynumberc>0,
17
t0+abp(s,ξ)dσ(ξ)ds=+,
18

then every solution of the problem (1), (2) is oscillatory inG.

Examples

In this section, we give two examples to illustrate our main results.

Example 7

Consider the partial functional differential equation

6t6[u(x,t)+15u(x,t-π)]=3Δu(x,t)+115Δux,t-3π2--π-π/2115u(x,t+ξ)dξ,(x,t)(0,π)×[0,),
19

with the boundary condition

u(0, t) = u(πt) = 0,  t ≥ 0.
20

Here

An external file that holds a picture, illustration, etc.
Object name is 40064_2016_2111_Article_IEq85.gif
It is easy to see that for t0 > 0, 

t0+abp(s,ξ)[1-μ(g(s,ξ))]dσ(ξ)ds=t0+-π-π/21151-15dξds=+.

Then the conditions of Theorem 5 are fulfilled. Therefore every solution of the problem (19), (20) is oscillatory in (0, π) × [0, ). Indeed, u(xt) = sinxcost is such a solution.

Example 8

Consider the partial functional differential equation

4t4[u(x,t)+12u(x,t-π)]=13Δu(x,t)+16Δux,t-π2--π-π/216u(x,t+ξ)dξ,(x,t)(0,π)×[0,),
21

with the boundary condition

ux(0, t) + u(0, t) = ux(πt) + u(πt) = 0,  t ≥ 0.
22

Here

An external file that holds a picture, illustration, etc.
Object name is 40064_2016_2111_Article_IEq89.gif
It is easy to see that for t0 > 0, 

t0+abp(s,ξ)[1-μ(g(s,ξ))]dσ(ξ)ds=t0+-π-π/2161-12dξds=+,

which shows that the conditions of Theorem 5 are satisfied. By Theorem 5, we obtain that every solution of the problem (21), (22) is oscillatory in (0, π) × [0, ). In fact, u(xt) = e-xcost is such a solution.

Conclusions

This paper provides some oscillation criteria for solutions of higher-order neutral partial functional differential equations with Robin boundary conditions. Using Lemma 1, we obtain Theorems 2 and 4. We should note that Theorem 2 shows that the oscillation of the problem (1), (2) is equivalent to the oscillation of the functional differential equation (4). Using the results in Li and Cui (2001), two useful conclusions are established in Theorems 5 and 6.

Authors’ contributions

Both authors contributed equally to the writing of this paper. Both authors read and approved the final manuscript.

Acknowledgements

The authors are grateful to anonymous referees for their kind comments and suggestions on this paper. This work is supported by the National Natural Science Foundation of China (10971018).

Competing interests

Both authors declare that they have no competing interests.

Contributor Information

Wei Nian Li, ten.362@ilnw.

Weihong Sheng, moc.361@gnehs-hw.

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