A remark on a third-order three-point boundary value problem

Salvatore A. Marano
1994 Bulletin of the Australian Mathematical Society  
Let / be a real function denned on [0, l]xR s and let r\ e (0, 1). Very recently, C.P. Gupta and V. Lakshimikantham, making use of the Leray-Schauder continuation theorem and Wirtinger-type inequalities, established an existence result for the problem r x" = f (t, x, x 1 , x") (Theorem 1 and Remark 4 of [Nonlinear Anal. 16 (1991), 949-957]). The aim of the present paper is simply to point out how, by means of a completely different approach, it is possible to improve that result not only by
more » ... iring much more general conditions on / , but also by giving a precise pointwise estimate on x . Let / be a real function denned on [0, 1] X R s ; •q £ (0, 1); Jb £ [1, +00); I*([0, 1]) the space of all (equivalence classes of) measurable functions ij): [0, 1] -» R such that IMIL*([O,I]) = (So W*)l* d *) < + °°; W S> *([0, 1]) the space of all u £ C 2 ([0, 1]) such that u is absolutely continuous in [0, 1] and u £ L k ([0, 1]). Consider the problem Our interest in problem (P) originated reading [3] . In that paper the authors implicitly established the following existence theorem (see [3, Theorem 1 and Remark 4])-
doi:10.1017/s0004972700016014 fatcat:imndmzf4g5bejdsxk3lyum4jai