PERIODIC SOLUTIONS OF A NONHOMOGENEOUS ITERATIVE FUNCTIONAL DIFFERENTIAL EQUATION

ZHAO Hou-yu

(School of Mathematics,Chongqing Normal University,Chongqing 401331,China)(Department of Pure Mathematics,University of Waterloo,Waterloo N2L 3G1,Canada)

1　Introduction

Recently,iterative functional differential equations of the form

appeared in several papers,here x[0](t)=t,x[1](t)=x(t),x[2](t)=x(x(t)),···,x[n](t)=x(xn?1(t)).In[1],Cooke pointed out that it is highly desirable to establish the existence and stability properties of periodic solutions for equations of the form

in which the lag h(t,x(t))implicitly involves x(t).Stephan[2]studied the existence of periodic solutions of equation

Eder[3]considered the iterative functional differential equation

and obtained that every solution either vanishes identically or is strictly monotonic.Feckan[4]studied the equation

and obtained an existence theorem for solutions satisfying x(0)=0.Later,Wang and Si[5]studied

and showed the existence theorem of analytic solutions.In particularly,Si and Cheng[6]discussed the smooth solutions of equation of

Some various properties of solutions for several iterative functional differential equations,we refer the interested reader to[7–10].

Since Burton[11]applied Krasnoselskii’s fixed theorem to prove the existence of periodic solutions,which was extensively used in proving stability,periodic of solutions and boundedness of solutions in functional differential(difference)equations.2005,Raffoul[12]used fixed point theorem to show a nonlinear neutral system

has a periodic solution.In[13],Guo and Yu discussed the existence and multiplicity of periodic of the second order difference equation.Some other works can also be found in[14–16].

In this paper,we consider the existence of periodic solutions of equation

where c1＞0.For convenience,we will make use of C(R,R)to denote the set of all real valued continuous functions map R into R.

For T＞0,we define

then PTis a Banach space with the norm

For P,L≥0,we define the set

which is a closed convex and bounded subset of PT,and we wish to find T-periodic functions x∈PT(P,L)satisfies(1.1).

2　Periodic Solutions of(1.1)

In this section,the existence of periodic solutions of equation(1.1)will be proved.Now let us state the Krasnoselskii’s fixed point theorem,it will be used to prove our main theorem.

Theorem 2.1(see[17])Let ? be a closed convex nonempty subset of a Banach space(B,‖ ·‖).Suppose that A and B map ? into B such that

(i)A is compact and continuous,

(ii)B is a contraction mapping,

(iii)x,y∈?,implies Ax+By∈?,

then there exists z∈? with z=Az+Bz.

We begin with the following lemma.

Lemma 2.2For any ?,ψ ∈ PT(P,L),

The result can be obtained by the definition of PT(P,L).

Lemma 2.3Suppose c1/=0.If x∈PT,then x(t)is a solution of equation(1.1)if and only if

where

ProofLet x(t)∈PT(P,L)be a solution of(1.1),multiply both sides of the resulting equation with e?c1tand integrate from t to t+T to obtain

Using the fact x(t+T)=x(t),the above expression can be put in the form

This completes the proof.

It is clear that G(t,s)=G(t+T,s+T)for all(t,s)∈R2,and for s∈[t,t+T],we have

Now we need to construct two mappings to satisfy Theorem 2.1.Set the map A,B:PT(P,L)→PTas the follwoing,

where F∈PT(P,L),G(t,s)defined as(2.3).

Lemma 2.4Operator A is continuous and compact on PT(P,L).

ProofTake ?,ψ ∈ PT(P,L),t∈ R,use(2.1)and(2.4),

This proves A is continuous.

Now we show that A is a compact map.It is easy to see that PT(P,L)is uniformly bounded and equicontinuous on R,thus by Arzela-Ascoli theorem,it is a compact set.Since A is continuous,it maps compact sets into compact sets,therefore A is compact.This completes the proof.

Lemma 2.5Operator B is a contraction mapping on PT(P,L).

ProofTake ?,ψ ∈ PT(P,L),

for any 0≤η＜1,hence B defines a contraction mapping.

Theorem 2.6Suppose F∈PT(P,L)is given,c1＞0 and the following inequalities are held

then eq.(1.1)has a periodic solution in PT(P,L).

ProofFor any ?,ψ ∈ PT(P,L),by(2.4)and(2.7),

Without loss of generality,we assume t2≥t1,by(2.7),

where t1≤ξ≤t2.

This shows that(A?)(t)+(Bψ)(t)∈ PT(P,L).By Lemma 2.4 and Lemma 2.5,we see that all the conditions of Krasnoselskii’s theorem are satisfied on the set PT(P,L).Thus there exists a fixed point x in PT(P,L)such that

Differential both sides of(2.10)and from Lemma 2.3,we can find(1.1)has a T-periodic solution.This completes the proof.

3　Uniqueness and Stability

In this section,uniqueness and stability of(1.1)will be proved.

Theorem 3.1In addition to the assumption of Theorem 2.6,suppose that

then(1.1)has a unique solution in PT(P,L).

ProofDefine an operator H from PT(P,L)into PT,

where G(t,s)defined as(2.3).Denote ?,ψ ∈ PT(P,L)are two different T-periodic solutions of(1.1),

where Γ=|c2|MT(1+L),thus

From(3.1),we know Γ ＜ 1 and the fixed point ? must be unique.

Theorem 3.2The unique solution obtained in Theorem 3.1 depends continuously on the given functions F and ci(i=1,2).

ProofUnder the assumptions of Theorem 3.1,for any two functions Fi(x)in PT(P,L)are given,λiand μi,i=1,2 are constants satisfy(2.7).Then there are two unique corresponding functions ?(t)and ψ(t)in PT(P,L)such that

and

where

We have

where

thus

From(3.1),

and

This completes the proof.

Example 1Now we will show that the conditions in Theorem2.6 do not self-contradict.Consider the following equation

where

A simple calculation yields 4.19＜and(1+|c2|)MT＜0.47＜1.Let P=1,L=8,2P(1+|c2|)=2.2＜8,then(2.7)is satisfied.By Theorem 2.6,equation(3.4)has aperiodic solution x such that‖x‖≤ 1,and

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