A multi-point boundary value problem with two critical conditions

We apply a coincidence degree theorem due to Mawhin to show the existence of at least one solution of the nonlinear second order multi-point boundary value problem u"(t) = f(t,u(t),u'(t)), t is an element of(0,1), u'(0) = u'(eta), (n)(i=l)Sigma alpha(i)u(eta(i))=u(1), where 0 < eta <= 1, 0 < eta(i) < 1, i = 1,..., n, Sigma(n)(i=l)alpha(i) = Sigma(n)(i=l)alpha(i)eta(i) = 1,n >= 2, and f: [0,1] x R-2 -> R satisfying the Caratheodory conditions. In our setting both boundary conditions are responsible for resonance. (c) 2005 Elsevier Ltd. All rights reserved.