EXISTENCE OF SOLUTIONS FOR MULTIPOINT BOUNDARY VALUE PROBLEMS FOR N TH ORDER DIFFERENTIAL EQUATIONS J. M. DAVIS, J. HENDERSON, B. KARNA, Q. SHENG, AND C. C. TISDELL Abstract. Shooting methods are employed to obtain solutions of multipoint boundary value problem for the nth order equation, y (n) = f (x, y, y ′ ,...,y (n−1) ), satisfying boundary conditions for which solutions are unique, under a right disfocality assumption. 1. Introduction In this paper, we are concerned with the existence of solutions of multipoint boundary value problems for the nth order ordinary differ- ential equation, y (n) = f (x, y, y ′ ,...,y (n−1) ), a < x < b. (1) Namely, given y i ∈ R,1 ≤ i ≤ n, and points a<x 1 ≤ x 2 ≤···≤ x n < b, so that for the natural numbers, 1 ≤ k 1 < ··· <k ℓ ≤ n − 1, there are points x k i <η i <x k i+1 , 1 ≤ i ≤ ℓ, we shall consider, in some sense, a uniqueness implies existence result for solutions of (1) satisfying the (n + ℓ)-boundary conditions, y (j −1) (x j )= y j , 1 ≤ j ≤ n, j = k 1 +1,...,k ℓ +1, y (k i −1) (x k i +1 ) − y (k i −1) (η i )= y k i +1 , 1 ≤ i ≤ ℓ. (2) We assume throughout the following: (A) f :(a, b) × R n → R is continuous. (B) Solutions of initial value problems for (1) are unique and exist on all of (a, b). Multipoint boundary value problems, for which the number of bound- ary points is greater than the order of the ordinary differential equation, have received considerable interest. For a small sample of such works, we refer the reader to works by Bai and Fang [1], Gupta [5], Gupta and Trofimchuk [6] and Ma [17, 18, 19]. 1991 Mathematics Subject Classification. Primary: 34B15; Secondary: 34B10. Key words and phrases. boundary value problem, multipoint, shooting method. 1