Solution Of Boundary Value Problems

This paper is devoted into two sections.Firstly,we will study the existence and uniqueness of solutions of second-order multi-point boundary value problem. By using monotone iterative method, we obtains the suf-ficient conditions for the existence and uniqueness of solutions. The difference from the others, lower and upper solutions are in the reversed order. Moreover, this paper gives the iterative sequence for solving a solution and its error estimate formula under the condition of unique solution.Secondly,we will deal with the existence of positive solutions of third-order differential equation in ordered Banach spaces (φ(-x"(t)))’=f(t,x(t)), t∈J,subject to the following integral boundary conditions: x(0)=θ, x"(0)=θ, x(1)=/g(t)x(t)dt,where θ is the zero element of E, and g∈L[0,1] is nonnegative, φ:Râ†'R is the increasing homeomorphism and positive homomorphism and φ(0)=θ. The arguments are based upon the fixed-point principle in cone for strict set contraction operators. Meanwhile, as an application, we also give an example to illustrate our results.