Positive solutions to a second order three-point boundary value problem

The existence, nonexistence, and multiplicity of nonnegative solutions are established for the three-point boundary value problem −u (t ) = p(t)f (u(t)), 0 < t <1, u(0) = 0, u(1) −αu(β) = λ, where β ∈ (0, 1), α ∈ (0, 1/β), and λ is a nonnegative parameter, under appropriate hypotheses. The key idea is that the problem of finding a nonnegative solution is transformed into the problem of finding a fixed point of a completely continuous operator. The arguments involve the Schauder fixed point, the method of upper and lower solutions for three-point boundary value problems and the Leray–Schauder degree theory.  2003 Elsevier Inc. All rights reserved.