EXISTENCE OF SOLUTIONS OF BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS

In this paper, using a simple and classical application of the Leray-Schauder 
degree theory, we study the existence of solutions of the following boundary value 
problem for functional differential equations 
x″(t)


MAIN RESULTS
Befor stating our main Pesults we Pefe some lemmas which simplify the poof of he theorem bellow.LEMMA 2. i. [, pp 187] Le X be. a Banach space, A X-X be a completely eon.t.inuous mappJnK such that I-A is one to one, and let be a bounded set such that 0 e Then the completely continuous mapping S :-X has a fixed point in if fop any le(O,1), the equation x ISx+(I-A)Ax has no solution on the boundaPy of .
Obviously, the operator S is a compact operator defined on B and taking values in B.
Since |,e B.V.P. (1.1)-(1.2) is equivalent to (2.10s) and (2.108), the purpose of he following proof is to show that the mapping S has a fixed point.

It is clear hat
is open and bounded in B and A is a completely continuous operator First we prove that the operator I-A is one to one.Let (I-A)x =(I-A)y.If z(t) :x(t)-y() then (I-A)z =0 and z(O)+sz'(O)=0, z(T)+Bz'(T)=0.Hence, z is a solution of the B.V.P.
By l.,mn 2.3 tl,e last pPoblem has the unique solution z O, and consequently I-A is one to one.
ACKNOWLEDGEMENT.The aghoms would like to express their deep thanks to the mefeee for his useful comments and helpful suggestions.