In this paper, existence of solutions of third-order differential equation y′″（t）=f（t,y（t）,y′（t）,y″（t））with nonlinear three-point boundary condition{g（y（a）,y′（a）,y″（a））=0, h（y（b）,y′（b））=0, I（y（c）,y′（c）,y″（c））=0is obtained by embedding Leray-Schauder degree theory in upper and lower solutions method,where a, b, c∈ R,a〈 b〈 c; f ： [a,c]×R^3→R,g：R^3→R,h：R^2→R and I：R^3→R are continuous functions. The existence result is obtained by defining the suitable upper and lower solutions and introducing an appropriate auxiliary boundary value problem. As an application, an example with an explicit solution is given to demonstrate the validity of the results in this paper.
By two successive linear transformations,a singularly perturbed differential system with two parameters is quasi-diagonalized. The method of variation of constants and the principle of contraction map are used to prove the existence of the transformations.
Zheyan Zhou,Jinxia Yao(College of Math. and Computer Science,Fujian Normal University,Fuzhou 350007)
In this paper, we first obtain the existence of solution to some n-point boundary value problem for third-order differential equations using upper and lower solutions method. Based on the results, we explore singular perturbation of another n-point boundary value problem for third-order differential equations with a small positive parameter. Finally, a uniformly valid asymptotic solution is constructed and the error estimation is given.
