PERIODIC SOLUTIONS OF NEUTRAL DUFFING EQUATIONS

We consider the following neutral delay Dung equation ax 00 (t) + bx 0 (t) + cx(t) + g(x(t 1); x 0 (t 2); x 00 (t 3)) = p(t) = p(t + 2 ); where a, b and c are constants, i, i = 1; 2; 3, are nonnegative constants, g : R R R ! R is continuous, and p(t) is a continuous 2 -periodic function. In this paper, combining the Brouwer degree theory with a continuation theorem based on Mawhin's coincidence degree, we obtain a sucien t condition for the existence of 2 -periodic solution of above equation.


Introduction
On the existence problem of periodic solutions for the Duffing equations so far there has been a wide literature since the interest in studying Eq.(1.1) comes from different sources.Under the conditions which exclude the resonance cases, The Project Supported by NNSF of China (No:19971026, 19831030) Typeset by A M S-T E X EJQTDE, 2000 No. 5, p.1 many results have been obtained [1,2,3,4] .At resonance, many authors have paid much attention to the problem in recent years.[5] and [6] resolved the existence problem of 2π-periodic solutions of Eq.(1.1) under some different conditions, respectively.
On the other hand, a few papers have appeared [7,8,9,10,11,12] which dealt with the existence problem of periodic solutions to the delay Duffing equations such as Under some conditions which exclude the resonance cases, some results have been obtained [13,14,15] .
Next, [17] discussed the Duffing equations of the form where m is a positive integer, and proved the existence of 2π-periodic solutions of Eq.( 1.3) under some conditions.
Jack Hale [21] and [22] put forward the Euler's equations which are of the form where r is a positive constant.
Motivated by above papers, in the present paper, we consider the neutral Duffing equations of the form where a, b, c are constants, τ 1 , τ To the best of our knowledge, in this direction, few papers can be found in the literature.In this paper, combining the Brouwer degree theory with a continuation theorem based on Mawhin's coincidence degree [16] , we obtain a sufficient condition for the existence of 2π-periodic solution of Eq.(1.4).

Existence of a Periodic Solution
In order to obtain the existence of a periodic solution of Eq. (1.4), we first make the following preparations.
Let X and Z be two Banach spaces.Consider an operator equation where L: Dom L ∩ X → Z is a linear operator and λ ∈ [0, 1] a parameter.Let P and Q denote two projectors such that In the sequel, we will use the following result of Mawhin [16] .
LEMMA 2.1.Let X and Z be two Banach spaces and L a Fredholm mapping Then Lx = N x has at least one solution in Ω.
Recall that a linear mapping L: Dom L ⊂ X → Z with Ker L = L −1 (0) and Im L = L(DomL), will be called a Fredholm mapping if the following two conditions hold: (i).Ker L has a finite dimension; (ii).Im L is closed and has a finite codimension.
Recalled also that the codimension of Im L is the dimension of Z/Im L, i.e., the dimension of the cokernel coker L of L.
When L is a Fredholm mapping, its (Fredholm) index is the integer We shall say that a mapping N is L-compact on Ω if the mapping QN : , it is continuous and K P (I − Q)N ( Ω) is relatively compact, where K P : Im L →Dom L∩Ker P is a inverse of the restriction L P of L to Dom L∩Ker P , so that LK P = I and K P L = I − P .
from which, together with (2.4), it implies that .
(2.10) EJQTDE, 2000 No. 5, p.9 In view of (2.8) and (2.10), we can obtain A}.We now will show that N is L-compact on Ω.For any x ∈ Ω, EJQTDE, 2000 No. 5, p.10 where is continuous with respect to z, and For ∀x ∈ Ω, we have Thus, the set {K P (I − Q)N x|x ∈ Ω} is equicontinuous and uniformly bounded.
Consequently, N is L-compact.This satisfies condition (a) in Lemma 2.1.