Optimal existence criteria for symmetric positive solutions to a three-point boundary value problem

In this paper we shall consider the following second-order three-point boundary value problem: u"(t) + a(t) f(t, u (t)) = 0, 0 < t < 1, u(0) = u(1) = alpha u(eta), where alpha is an element of (0, 1), eta is an element of (0, (1)/(2)], a is an element of L-1(0, 1) is nonnegative and symmetric on (0, 1), f : [0, 1] x [0, infinity) -> [0, infinity) satisfies Caratheodory conditions and f ((.), u) is symmetric on [0, 1] for all u is an element of [0, infinity). By using fixed point index theorems, we get some optimal existence criteria for the existence of one or two symmetric solutions which involve the principal eigenvalue of a related linear operator. (c) 2006 Elsevier Ltd. All rights reserved.