JOURNAL OF DIFFERENTIAL EQUATIONS 67, 414440 (1987) Uniqueness, Existence, and Optimality for Fourth-Order Lipschitz Equations JOHNNY HENDERSON AND ROBERT W. MCGWIER, JR. Department of Mathematics, Auburn University, Auburn, Alabama 36849 Received April 18, 1986 1. INTRODUCTION In this paper we will be concerned with solutions of boundary value problems for the fourth-order differential equation Y (4)=f(t, y, Y’, Y”, Y”‘), (1) where we assume throughout that (A) f: (a, b) x R4 + R is continuous, and (B) f satisfies the Lipschitz condition for each (t, y,, y,, y,, y4), (t, zi, z2, z3, Z~)E (a, 6) x R4. In particular, we will characterize optimal length subintervals of (a, b), in terms of the Lipschitz coefficients k,, i= 1, 2, 3, 4, on which certain two, three, and four point boundary value problems for (1) have unique solutions. The techni- ques we employ here involve applications of the Pontryagin Maximum Principle [ 14, p. 3141 in conjunction with uniqueness implies existence results for solutions of boundary value problems for (1). These techniques are motivated by works of Melentsova [15], and Melentsova and Mil’shtein [ 16, 171, and most notably by the two papers by Jackson [ll, 121. Furthermore, the results contained herein can be con- sidered as extensions of a recent paper by Henderson [6] dealing with boundary value problems for third-order equations. In relating the results of this paper to previous works, we will formulate the boundary value problems in terms of the nth-order differential equation Y ‘“‘=f(t, y, y’,..., y’“-I’). (2) 414 0022-0396187 $3.00 Copyright 0 1987 by Academc Press, Inc. All rights ol reproductmn in any form reserved.