Journal of Applied Mathematics and Simulation 
Volume 2 (1989), Issue 1, Pages 33-51
doi:10.1155/S1048953389000031

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

Chaitan P. Gupta1,2

 1Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL 60439-4801, USA
2Department of Mathematical Sciences, Northern Illinois University, DeKalb, IL 60115, USA
Received 1 January 1989; Revised 1 March 1989

Abstract

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.