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We consider positive solutions of a class of semilinear problems
u'' + λa(x)f(u) = 0, −1 < x < 1, u(−1) = u(1) = 0,
with even and positive a(x), depending on a positive parameter λ. In case f(u) is convex, an exact multiplicity result was given in P. Korman, Y. Li and T. Ouyang ; see also P. Korman  for the details. It was observed by P. Korman and J. Shi  that convexity requirement can be relaxed for large u (see also ). We show that convexity requirement can also be relaxed for small u.