EXISTENCE OF PERIODIC SOLUTION FOR FIRST ORDER NONLINEAR NEUTRAL DELAY EQUATIONS GENQIANG WANG

In this paper by using the coincidence degree theory, sufficient conditions are given for the existence of periodic solutions of the first order nonlinear neutral delay differential equation.


Introduction
In [3], Kuang and Feldstein proposed to study the existence of a periodic solution for the first order periodic neutral delay equation.In particular, Gopalsamy, He and Wen [2] studied the existence of periodic solutions of the first order neutral delay logistic equation.In this paper, we discuss the following nonlinear neutral delay equation" [.(t) + c.(t )]' + (t..(t-)) p(t). (1) where v, cr and c are constants, and r>_O, cr>_O, [c] <1; geC(R2,R), g(t,x) is a function with period T( > O) for t and g(t,x)is nondecreasing for x in [0, /c); p E C(R,R), p(t,T) p(t) for t G'R and f ToP(t)dt O. Using coincidence degree theory developed by Mawhin [1], we establish a theorem of the existence of periodic solutions with period T of Equation (1).

Main Result
The following result provides sufficient conditions for the existence of periodic solution of Equation (1).
Theorem: Assume that there exist constants D > 0 and M > 0 such that xg(t,x) > O for t R and x > D, g(t,x) >_ M for t e R and x < D, and g(t,x) <_ g(t, Ix I) for (t,x) R2. (4) Then there exists a periodic solution with period T of Equation (1).
In order to prove the above theorem, we introduce the following preliminaries.Let X and Z be two Banach spaces.Consider an operator equation, Lx-Nx, where L: DomL gl XZ is a linear operator and A E [0, 1] is a parameter.
Q denote two projectors, Let P and P: DomL N X-KerL and Q: ZZ/ImL.
We will use the following result of Mawhin [1].
Lemma 1" Let X and Y be two Banach spaces and L be a Fredholm mapping with index null.Assume that f2 is open bounded in X and N:f--,Z is L-compact on Furthermore, suppose that (a) for each E (0, 1), x 0 DomL, Lx :/: $Nx; for each x Of fq KerL, QNz -Tt: O and deg{QN, f2 f3 KerL, 0} 5 0, then Lx-Nx has at least one solution in f2 fq DomL.
To prove Lemma 2, we make the following preparations.Set and define the norm on Xas IIxll e[O,T] Z: {z e C(R,n) z(t + T)-z(t)} and define the norm on Z as (Z, II" I I 0)are Banach spaces.
I I z I I o-max Iz(t) l.Then both (X, l[ II) and e [O,T] Define respectively the operators L and N as L" XZ, x(t)-x'(t), We know that Karl-R.Define, respectively, the projective operators P and Q as and Hence, we have ImP-Karl and ImL-KerQ.Consider the equation '(t) + ( 9) where E (0, 1) is a parameter.Lemma 2: Suppose that conditions (2)-(4) are satisfied.solution with period T of Equation (9), then there Dj(j O, 1) independent of A and such that If z(t) is any periodic exist positive constants and x'(t) <_ D, t [0,T].
(10) Proof: Suppose that x(t) is a periodic solution with period T of Equation (9).By integrating (9) from 0 to T, we find Since g C(R2, R) and g(t,x) is a function with period T for t, we know that sup g(t,x) max g(t,x) < c.
where K o is a positive constant.
follows from (9) that Since x'(t) is a periodic function with period T, it x(t-))ldt+T max [p(t)[.
Let D 2 max(Do, D,D} + 1, and   In view of (2), we see that T T 1/ og(t _D2)d/>0 and --/ g(t, D2)dt < O (25) Z 0 By (5)-( 7) and (8_), we know that L is the Fredholm operator with index null and N is L-compact on fl (see [1]).In terms of evaluation of a bound of periodic solutions in Lemma 2, we know that for any x E 0fl V DomL and A E (0, 1), Lx y ANx.Since for any xOflNKerL, x-D2(>D or x--D2, in view of (25) and f Top(t)dt O, we have T QNx / [-cx'(t r)-g(t, x(t )) + p(t)]dt By Lemma 1, there exists a periodic solution with period T of Equation (1).
proof is complete.
Example: Consider the equation The Ix(t) 1/2x(t r)]' + e sin tx(t r)e x(' r) -cos t sin t. (26)   It is easy to verify that for Equation 926), all the conditions of the theorem are satisfied with D > 0 and M-3.Thus Equation (26) has a periodic solution with period 2r.We see that x(t) sin is such a periodic solution of Equation (26).