On the Solvability of Nonlinear Boundary Value Problems for Functional Differential Equations

Sufficient conditions are established for the solvability of the boundary value problem dx(t) dt = p(x, x)(t) + q(x)(t), l(x, x) = c(x) , where p : C(I; Rn)× C(I; Rn) → L(I; Rn), q : C(I; Rn) → L(I; Rn), l : C(I, Rn)× C(I; Rn) → Rn, and cn : C(I, Rn) → Rn are continuous operators, and p(x, ·) and l(x, ·) are linear operators for any fixed x ∈ C(I; Rn). 1. Formulation of the Main Results 1.1. Formulation of the problem. Let n be a natural number, I = [a, b], −∞ < a < b + ∞ and p : C(I; Rn) × C(I; Rn) → L(I, Rn), q : C(I; Rn) → L(I; Rn), l : C(I;Rn) × C(I;Rn) → Rn and c : C(I; Rn) → Rn be continuous operators. We consider the vector functional differential equation dx(t) dt = p(x, x)(t) + q(x)(t) (1.1) with the boundary condition l(x, x) = c(x) . (1.2) By a solution of (1.1) we mean an absolutely continuous vector function x : I → Rn which satisfies it almost everywhere in I, and by a solution of problem (1.1), (1.2) a solution of (1.1) satisfying condition (1.2). 1991 Mathematics Subject Classification. 34K10.