Nonresonance conditions for generalised φ-Laplacian problems with jumping nonlinearities

We consider the boundary value problem -psi(x, u(x), u'(x))' = f(x, u(x), u'(x)), a.e.x is an element of (0, 1), (0.1) c(00)u(0) = c(01)u'(0), c(10)u(1) = c(11)u'(1) (0.2) where vertical bar c(j0)vertical bar + vertical bar c(j1)vertical bar > 0, for each j = 0, 1, and psi, f : inverted right perpendicular 0, 1 inverted left perpendicular x R(2) -> R are Caratheodory functions, with suitable additional properties. The differential operator generated by the left-hand side of (0.1), together with the boundary conditions (0.2), is a generalisation of the usual p-Laplacian, and also of the so-called phi-Laplacian (which corresponds to psi(x, s, t) = phi(t), with phi an odd, increasing homeomorphism). For the p-Laplacian problem (and more particularly, the semilinear case p = 2), 'nonresonance conditions' which ensure the solvability of the problem (0.1), (0.2), have been obtained in terms of either eigenvalues (for non-jumping f) or the Fucik spectrum or half-eigenvalues (for jumping f) of the p-Laplacian. In this paper, under suitable growth conditions on psi and f, we extend these conditions to the general problem (0.1), (0.2). (C) 2009 Elsevier Inc. All rights reserved.