Boundary Value Problems for Functional Differential Equations

AbstractThis paper is concerned with the existence of solutions of a boundary value problem for second-order functional differential equations. Specifically the following problem is considered: x''(t) = f(t,x,x'(t)), t \in [0,T], \alpha_0 x_0 - \alpha_1 x'(0) = \varphi \in C_r \beta_0 x(T) + \beta_1 x'(T) = A \in {\Bbb R}^n. The results are based on a nonlinear alternative of Granas and the use of a priori bounds on solutions. Some examples are also discussed to illustrate how these results may be used to yield the existence of solutions of specific boundary value problems