Existence Of Solution For Third-Order m-Point Boundary Value Problem

In this paper, we consider the following nonlinear third-order m-point boundary value problem {u"'(t)'+ f(t, u(t), u'(t), u"(t)) - 0, t is an element of [0,1], u(0) = Sigma(m-2)(i=1) a(i)u(eta(i)), u'(1) = u"(0) = 0, where 0 < eta(1) < eta(2) < center dot center dot center dot < eta(m-2) < 1, a(i) >= 0 (i = 1,2, center dot center dot center dot , m - 2) and Sigma m(i=1)(-2) < 1. By imposing some conditions on the nonlinear term f, we construct a lower solution and an upper solution and prove the existence of solution to the above boundary value problem. Our main tools are upper and lower solution method and Schauder fixed point theorem.