Existence-Uniqueness Theorems for Three-Point Boundary Value Problems for nth-Order Nonlinear Differential Equations

where 01, ,l3, y and the x’s are real. It will be assumed throughout this paper that f (t, Ul , u, ,.-., un) is continuous on [CX, r] x R”. The approach taken here is similar to that of Barr and Sherman [I] and is based on the use of a solutionmatching technique that assumes existence and/or uniqueness of solutions to corresponding problems for the subintervals [cu, ,Kj and [p, y]. We refer to the problems (l.l), (1.2) as three-point problems. This terminology differs from Jackson’s definition [3] of a “k-point problem” where the postulation of the value of the ith derivative of y(t) at a point presupposes a knowledge of the values of all derivatives of order less than i at that point. The result in [l] for the third-order problem refers only to the situation when i = j = k = 0 in (1.2) and will thus represent a particular case of our results. Even their extension to the nth order relates to the “three-point problem” as defined in 131. The results of this paper cover a wider class of problems. Section 2 deals with results obtained by relying on uniqueness of solutions to certain three-point problems. This assumption is replaced in Section 3 by suitable monotonicity conditions imposed onf(t, zli , us ,..., u,?) to yield improved