EXISTENCE OF PERIODIC SOLUTION FOR A KIND OF(m,n)-ORDER GENERALIZED NEUTRAL DIFFERENTIAL EQUATION??

Shaowen Yao

(School of Math.and Information Science,Henan Polytechnic University,Jiaozuo 454000,Henan,PR China)

Abstract In this paper,we consider the following high-order p-Laplacian generalized neutral differential equation with variable parameter(φp(x(t)?c(t)x(t?σ))(n))(m)+g(t,x(t),x(t?τ(t)),x′(t),···,x(m)(t))=e(t).By the coincidence degree theory and some analysis skills,sufficient conditions for the existence of periodic solutions are established.

Keywords periodic solution;p-Laplacian;high-order;neutral operator;variable parameter

1 Introduction

In this paper,we consider the following high-order p-Laplacian neutral differential equation with variable parameter

where p>1,φp(x)=|x|p?2x for x0 and φp(0)=0;g:Rn+2→ R is a continuous periodic function with g(t+T,·,···,·) ≡ g(t,·,···,·),and g(t,a,a,0,···,0)?e(t)0 for all a∈R.e:R→R is a continuous periodic function with e(t+T)≡e(t)ande(t)d t=0;c∈Cn(R,R)and c(t+T)≡c(t),T is a positive constant;n and m are positive integers.

Complicated behavior of technical applications is often modeled by nonlinear high-order differential equations[1],for example,the Lorenz model of a simplified hydrodynamic flow,the dynamo model of erratic inversion of the earth’s magnetic field,etc.Oftentimes high-order equations are a result of combinations of lower order equations.Due to its obvious complexity,studies on high-order differential equation are rather few,especially on high-order neutral differential equation.In recent years,there has been many perfect results on periodic solutions for high-order differential equations(see[2-11]and the references cited therein).For example,in[8],Pan studied the n th-order differential equation

and obtained the existence of periodic solutions for(1.2).In[7],Liand Lu considered the following high-order p-Laplacian differential equation

and using the theory of Fourier series,Bernoulli number theory and continuation theorem of coincidence degree theory,they studied the existence of periodic solutions for(1.3).Afterwards,Wang and Lu[9]investigated the existence of periodic solution for the high-order neutral functional differential equation with distributed delay

Recently,in[10]and[11],Ren,Cheng and Cheung observed a high-order p-Laplacian neutral differential equation

and presented sufficient conditions for the existence of periodic solutions for(1.5)in the critical case(that is,|c|=1)and in the general case(that is,|c|=1),respectively.

Inspired by these results,we consider a generalized high-order neutral differential equation with variable parameter(1.1).Here A=x(t)? c(t)x(t? σ)is a natural generalization of the operator A1=x(t)? cx(t? σ),which typically possesses a more complicated nonlinearity than A1.For example,A1is homogeneous in the following sense(A1x)′(t)=(A1x′)(t),whereas A in general is inhomogeneous.As a consequence many new results of differential equations with the neutral operator A will not be direct generalizations of known theorems of neutral differential equations.

The paper is organized as follows:In Section 2,we first give qualitative properties of the neutral operator A which can be helpful to study differential equations with operator;in Section 3,based on Mawhin’s continuation theory and some new inequalities,we obtain sufficient conditions for the existence of periodic solutions for(1.1),also an example is also given to illustrate our results.

2 Lemmas

Lemma 2.1[12]If|c(t)|1,then the operator A has a continuous inverse A?1on CT,for any f∈CTsatisfying

Let X and Y be real Banach spaces and L:D(L)?X→Y be a Fredholm operator with index zero,here D(L)denotes the domain of L.This means that Im L is closed in Y and dim Ker L=dim(Y/Im L)<+∞.Consider supplementary subspaces X1,Y1of X,Y respectively,such that X=Ker L⊕X1,Y=Im L⊕Y1.Let P:X→Ker L and Q:Y→Y1denote the natural projections.Clearly,Ker L ∩ (D(L)∩ X1)={0}and so the restriction LP:=L|D(L)∩X1is invertible.Let K denote the inverse of LP.

Let ? be an open bounded subset of X with D(L)∩ ??.A map N:→Y is said to be L-compact inif QN()is bounded and the operator K(I?Q)N:→X is compact.

Lemma 2.2[13]Suppose that X and Y are two Banach spaces,and L:D(L)?X→Y is a Fredholm operator with index zero.Let??X be an open bounded set and N:→Y be L-compact on.Assume that the following conditions hold:

In order to apply Mawhin’s continuation degree theorem,we rewrite(1.1)in the form:

Now,Set X={x=(x1(t),x2(t))∈C(R,R2):x(t+T)≡x(t)}with the norm|x|∞=max{|x1|∞,|x2|∞};Y={x=(x1(t),x2(t))∈ C1(R,R2):x(t+T)≡ x(t)}with the norm||x||=max{|x|∞,|x′|∞}.Clearly,X and Y are both Banach spaces.Meanwhile,define

by

and N:X→Y by

Then(2.1)can be converted into the abstract equation Lx=N x.

we have

where a0,···,an?1,b0,···,bm?1∈ R are constants.From x1(t)? c(t)x1(t? σ) ∈CT,x2(t) ∈ CT,we have a1= ···=an?1=0,b1=b2= ···=bm?1=0.Let ?(t)0 be a solution for x(t)?c(t)x(t?σ)=1,then KerFrom the definition of L,one can easily see that

So L is a Fredholm operator with index zero.Let P:X→Ker L and Q:Y→Im Q?R2be defined by

then Im P=Ker L,Ker Q=Im L.Set LP=L|D(L)∩KerPand:Im L → D(L)denotes the inverse of LP,then

where(Ax1)(i)(0),i=1,2,···,n ? 1 are defined by E1Z=B,with

Z=((Ax1)(n?1)(0),···,(Ax1)′(0),(Ax1)′(0))?,B=(b1,b2,···,bn?1)?,bi=and,j=1,2,···,n ? 2.And1,2,···,m ? 1 are decided by the equation E2W=F,where

with W=((x2)(m?1)(0),···,(x2)′(0),(x2)′(0))?,F=(d1,d2,···,dn?1)?,di=and,j=1,2,···,m ? 2.

From(2.2)and(2.3),it is clear that QN and K(I?Q)N are continuous,QN()is bounded and then K(I?Q)Nis compact for any open bounded ? ?X which means N is L-compact on.

3 Existence of Periodic Solutions for(1.1)

For convenience,we list the following assumptions which will be used repeatedly in the sequel:

(H1)There exists a constant D>0 such that

for any(t,v1,v2,v3,···,vn+2)∈ [0,T]× Rn+2with|v1|>D;(H2)there exists a constant D1>0 such that

for any(t,v1,v2,v3,···,vn+2)∈ [0,T]× Rn+2with|v1|>D1;

(H3)there exist non-negative constants α1,α2,α3,···,αn+2,m such that

|g(t,v1,v2,v3,···,vn+2)|≤ α1|v1|p?1+α2|v2|p?1+α3|v3|p?1+···+αn+2|vn+2|p?1+m,for any(t,v1,v2,v3,···,vn+2)∈ [0,T]× Rn+1.

Theorem 3.1Assume(H1)and(H3)hold.Suppose the following one of conditions is satisfied:

(i)|c|∞<1 and

where ci=|c(i)(t)|,i=1,2,···,n.Then(1.1)has at least one non-constant T-periodic solution.

ProofConsider the equation

Set ?1={x:Lx= λ N x,λ ∈ (0,1)}.If x(t)=(x1(t),x2(t))?∈ ?1,then

Substituting x2(t)= λ1?pφp[(Ax1)(n)(t)]into the second equation of(3.1),we obtain

Integrating both sides of(3.2)from 0 to T,we have

From(3.3),there exists a point ξ∈[0,T]such that

In view of(H1),we obtain

Then,we have

and

Combing the above two inequalities,we obtain

From(3.4)and Wirtinger inequality(see[14],Lemma 2.4),we get

Since(Ax1)(t)=x1(t)?c(t)x1(t? σ),we have

(Ax1(t))(n)=(x1(t)?c(t)x1(t?σ))(n)

So,we get

Case(I)If|c(t)|≤ |c|∞<1,by applying Lemma 2.1,(3.5)and(3.6),we have

so we have

For a given constant δ>0,which only depends on k>0,we have

From(3.9),we have

Substituting(3.6),(3.7)into(3.10),we have

Combination of(3.8)and(3.11)implies

So,we have

Since

there exists a positive constant M1such that

Case(ii)If|c(t)|≥|c|L>1,by applying Lemma 2.1,we have

where ci=|c(i)(t)|.We know that

so we have

Similarly,we can get|x2|∞≤M1.

On the other hand,from(3.7),we have

From(3.6),we have

Hence,from(3.4),we have

From(3.6)and(3.9),we have

From(3.8),we can get

Let M=max{M1,M2,M3,M4}+1,?2={x∈ Ker L:QN x=0}.We shall prove that ?2is a bounded set.For any x∈?2,x2=0,x1=a0?(t),a0∈ R,then we have

From assumption(H1),we have|a0?(t)|≤ D.So?2is a bounded set.

Let ? ={x ∈ (x1,x2)?:||x||≤ M},then ?1∪ ?2? ?,for any(x,λ)∈? ? × (0,1).From the above proof,Lxλ N x is satisfied,so conditions(1)and(2)of Lemma 2.2 are both satisfied.Define an isomorphism J:Im Q→Ker L as follows:

Let H(μ,x)= ?μx+(1?μ)J QN x,(μ,x)∈ [0,1]×?,then for any(μ,x)∈ (0,1)×(?? ∩ Ker L),

for any(μ,x)∈ (0,1)× (?? ∩ Ker L).

From(H1),it is obvious that x?H(μ,x)<0,for any(μ,x)∈ (0,1)×(??∩Ker L).Hence

So condition(3)of Lemma 2.2 is satisfied.By applying Lemma 2.2,we conclude that equation Lx=N x has a solution x=(x1,x2)?on∩D(L),that is,(2.1)has a T-periodic solution x1(t).

Similarly,we can get the following result:

Theorem 3.2Assume(H2)and(H3)hold,Suppose the following one of conditions is satisfied

(i)|c|∞<1 and

(ii)|c|L>1 and

Then(1.1)has at least one non-constant T-periodic solution.

We illustrate our results with an example.

Example 3.1Consider the following neutral functional differential

which means(H3)holds with α1=,α2=0,α3=0,α4=0,α5=0,m=1.Obviously

So by Theorem 3,(3.16)has at least one non-constantperiodic solution.