An m-point boundary value problem of Neumann type for a p-Laplacian like operator

Let phi and theta be two odd increasing homeomorphism from R onto R with phi(0) = 0, theta(0) = 0, and let f : [0, 1] x R x R --> R be a function satisfying Caratheodory's conditions. Let a(i)is an element ofR, zeta(i) is an element of(0, 1), i = 1 ,2, . . . ,m - 2, 0 < zeta(1) < xi(2) < ... <xi(m-2) < 1 be given. We are interested in the problem of existence of solutions for the m-point boundary value problem: (phi(x'))'=f(t'x'x')' t is an element of (0,1), x'(0) = 0, theta(x'.(1)) = Sigma(i=1)(m-2) a(i)theta(x'(zeta(i))). We note that this non-linear m-point boundary value problem is always at resonance since the associated m-point boundary value problem (phi(x'))' = 0, t is an element of (0, 1), x',(0) = 0, theta(x'(1)) = Sigma(i=1)(m-2)a(i)theta(x'(xi(i))) has non-trivial solutions x(t) = rho, to rho is an element of R (an arbitrary constant). Our results are obtained by a suitable homotopy, Leray-Schauder degree properties, and a priori bounds. (C) 2003 Elsevier Ltd. All rights reserved.