ON A BOUNDARY VALUE PROBLEM FOR NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS ROBERT HAKL Received 21 August 2004 and in revised form 1 March 2005 We consider the problem u (t) = H(u)(t) + Q(u)(t), u(a) = h(u), where H,Q : C([a,b];R) αβ → L([a,b];R) are, in general, nonlinear continuous operators, H ∈ ab (g0 ,g1 , p0 , p1 ), and h : C([a,b];R) → R is a continuous functional. Efficient conditions sufficient for the solvability and unique solvability of the problem considered are established. 1. Notation The following notation is used throughout the paper: N is the set of all natural numbers | ON A BOUNDARY VALUE PROBLEM FOR NONLINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS ROBERT HAKL Received 21 August 2004 and in revised form 1 March 2005 We consider the problemu t H u t Q u t u a h u where H Q C a b R aS L a b R are in general nonlinear continuous operators H e ab go g1 p0 p1 and h C a b R R is a continuous functional. Efficient conditions sufficient for the solvability and unique solvability of the problem considered are established. 1. Notation The following notation is used throughout the paper N is the set of all natural numbers. R is the set of all real numbers R 0 TO x 1 2 x x x _ 1 2 x -x . C a b R is the Banach space of continuous functions u a b R with the norm u c max u t t e a b . C a b R is the set of absolutely continuous functions u a b R. L a b R is the Banach space of Lebesgue integrable functions p a b R with the norm II p L b p s ds. L a b R p e L a b R p t 0 for t e a b . Mab is the set of measurable functions T a b a b . 3ỉab is the set of continuous operators F C a b R L a b R satisfying the Caratheodory condition that is for each r 0 there exists qr e L a b R such that F v t I qr t for t e a b v e C a b R v c r. K a b X A B where A c R2 B Q R is the set of functions f a b X A B satisfying the Caratheodory conditions that is f x a b B is a measurable function for all x e A f t A B is a continuous function for almost all t e a b and for each r 0 there exists qr e L a b R such that I f t x I qr t for t e a b x e A x r. Copyright 2006 Hindawi Publishing Corporation BoundaryValue Problems 2005 3 2005 263-288 DOI 264 On a BVP for nonlinear FDE 2. Statement of the problem We consider the equation u t H u t Q u t aft where H e ab g0 g1 p0 pl see Definition and Q e 3 ab. By a solution of we understand a function u e C a b R satisfying the equality almost everywhere in a b . Definition . We will say that an operator H belongs to the set X b g0 g1 p0 p1 where g0 g1 p0 p1 e L a b R and a ft e 0 1 .
