CRITICAL CASE FOR PERIODIC SOLUTIONS OF A CLASS OF NEUTRAL EQUATIONS WITH A SMALL PARAMETER

Introduction In this palier we study a class of neutral functio nal differential equations which arises from a coupled sys tem of differential-difference and ordinary diff erence equa tions that occur in various applications, as electrical cir cuits with lossless transmision lines, [11 . In [101, [151 is studied the stability of such system and in [131 are given conditions for the existence of periodic solutions. In this paper, we consider a nonlinear system with a small parame-ter and we study the existente of periodic solutions when the corresponding linear system can have periodic solutions (critical case) . This is done by applying the method of Hale [5,61 and its extension to delay diferential equations (see [141) to the neutral equations obtained from the coupled system after taking into account the results of [101 .


Introduction
In this palier we study a class of neutral functio nal differential equations which arises from a coupled sys tem of differential-difference and ordinary diff erence equa tions that occur in various applications, as electrical cir cuits with lossless transmision lines, [11 . In [101,[151 is studied the stability of such system and in [131 are givenconditions for the existence of periodic solutions.In this paper, we consider a nonlinear system with a small parame-ter and we study the existente of periodic solutions whenthe corresponding linear system can have periodic solutions (critical case) .This is done by applying the method of Hale [5,61 and its extension to delay diferential equations -(see [141) to the neutral equations obtained from the coupled system after taking into account the results of [101 .

l . Notation and summary of known results
Let En be a complex n-dimensional linear vector space with norm 1 .1 and let r be a fixed positive number .-r 0 L ( 1 ) = f -r [d 1 (9)]f (8) where li is an n x n matrix, det li =~0, rt , ~are n x n matrix Functions of.bounded variation on [-r, 0] with M nonatomic at zero .We assume N has no singular part . If.
x is a continuous function mapping [a'-r, ) into En, then for any tE[0,w) we define xt in C by -xt (9) = x (t + A ) ,9E [-r, 0] .An autonomous linear homoge-- fferentiable on [0, A) and equation (1 .1) is sati,sfied onthis interval .It is proved in [2,4] that there is a unique for t>r, f E C, where it is always understood that the inte grals in (1 .6)are actually an integrals in En.
Formula (1 .6)suggert the change of variables Definition 1 .1 .The operator D is said to be stable if there is a -4 > 0 such that all roots of the equations det D (e x .I) = 0 stisfy ReÁ<-a .
If D(+) = H j(0) -M«-r), then D is stable if the roots of the polynomial equation det (H -PM) = 0 sátisfy -An important property of equation (1 .1)when D is stable is the following (see [3]) : If D is stable, thenthere is a constant aD < 0 such that for any a > aD, thereare only a finite nimber of roots of det (j(ñ) = 0 with --Reñ) a .
Let D be stable .If A = t ~: det á (X) = 0, Re X > .A solution through f at t = T of (1 .3) is defi- ned as before and is known to exist on [o'-r, co) .
The variation of constants formula for (1 .3)(see [9]) states that the solution of (1 .3)through (r, f) is given by t t+ for t>l T, where X is the fundamental matrix solution given by (1 .2) .Equation (1 .4)can be written as for t>,(' .Now, let P C be the space of functions taking -[-r,0] into E n which are uniformly continuous on E-r.0) and may be discontinuous at zero .With the matrix X as defined 0 before, it is clear that PC= C + ( X0 ), where (X0 ) is the -- where the superscrits P and Q designate the projections ofthe correspondine functions onto the subspaces P and Q, respectively, and they can be determined by means of adjoint differential equation to (1 .1),see ( 2 .The li near problem In this section, we consider the system where x, y are n-vector and all matrices are constants .For any aEEn ,--tEC, one can define a solution of (2 .1)with - , where is a basis for P and T(t) ~_ eBt, - the spectrum of B is 11., then u satisf¡es the equation where IP is a basis for the initial values of those solu-tions of (2 .10) of the form p(t)e-~t , p a polynomial,IEA.where u" (t) and zt 9 are the unique solutions of (2 .13)and (2 .12a) in 9.