Let f:[0,1]×ℝ3→ℝ be a function satisfying Caratheodory'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,
u(η)=0, 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.