Abstract
We study the bifurcation of the time map of positive solutions of the nonlinear two-point boundary value problem u’' + f(u) = 0,-L < x < L, u(-L) = u(L) = 0 for f(u) =-(u-a)(u-b)(u-c) satisfying 0 ≤ a < b < c and c > 2b-a, Under one additional hypothesis on the cubic polynomial f, we are able to show that the time map not only has exactly one critical point, a minimum, but is also a strictly convex function by modifying a time map technique introduced by J. Smeller and A. Wasserman (1). Combined with some results of J. Smeller and A. Wasserman (1) or of S.-H. Wang (2), our result implies that for some cubic polynomials f(u) =-(u-a)(u-b)(u-c) with fixed numbers 0 < a < b, the time map has exactly one critical point, a minimum, for any number c > 2b-a. Our method can be generalized to general functions f with f’'' < 0. © 1994, Khayyam Publishing.