Existence result for a third-order ODE with nonlinear boundary conditions in presence of a sign-type Nagumo control

In this work we provide an existence and location result for the third-order nonlinear differential equation u'"(t) = f (t, u(t), ul(t), u'(t), u"(t)), where f : [a, b] x R-3 -> R is a continuous function, and two types of boundary conditions: u(a) = A, phi(u'(b), u"(b)) 0, u"(a) = B, or u(a) = A, psi(u'(a), u" (a)) 0, u"(b) = C, with phi, psi : R-2 -> R continuous functions, monotonous in the second variable and A, B, C E R. We assume that f satisfies a sign-type Nagumo condition which allows an asymmetric unbounded behaviour on the nonlinearity. The arguments used concern Leray-Schauder degree theory and lower and upper solutions technique. (c) 2005 Elsevier Inc. All rights reserved.