Existence and uniqueness theorems for a third-order generalized boundary value problem

Let f:[0,1]×ℝ³⇒ℝ be a function satisfying Carathéodory’s conditions, e(x)∈L¹[0,1], η∈[0,1], h∈0, k∈0, h+k∈0. This paper studies existence and uniqueness questions for the third-order three-point generalized boundary value problem u∈+f(x,u,u',u∈)=e(x), 0∈x∈1, with u(η)=0 and u∈(0)-hu'(0)=u∈(1)+ku'(1)=0, and the associated special cases corresponding to one or both of h and k equal to infinity. The conditions on the nonlinearity f turn out to be related to the spectrum of the linear boundary value problem u∈=λu', u(η)=0, u∈(0)-hu'(0)=u∈(1)+ku'(1)=0, in a natural way.