A Unified Approach to Some Two-Point, Three-Point, and Four-Point Boundary Value Problems with Carath´eodory Functions

This paper deals with boundary value problems y g u9..9sf t , u, u9., a-t-b, u a. yj u a . sA, u b. yhu b . sB and y g u9.. sf t , u, u9., a-t-b, u a. yj u b. sA, u9 b. yhu9 a. sB, where a-a Fb -b, j G0, h G0, A and B are given real numbers, g is a continuous, strictly increasing function with g R.sR, and f is a Carath´eodory function satisfying a Nagumo condition. The problems have solutions if and only if there exist a lower solution x and an upper solution y and x t.Fy t. for all aFtFb.