Báo Cáo Báo cáo khoa học On a boundary value problem for nonlinear functional differential equations robert

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. R is the set of all real numbers, R+ = [0,+∞[,[x]+ = (1/2)(|x| + x), [x]ư = (1/2)(|x| ư 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 ∈ [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 b norm p L = a