Nonresonance conditions for generalised φ-Laplacian problems with jumping nonlinearities

We consider the boundary value problem(0.1)- ? (x, u (x), u^' (x))^' = f (x, u (x), u^' (x)), a.e. x ? (0, 1),(0.2)c₀₀ u (0) = c₀₁ u^' (0), c₁₀ u (1) = c₁₁ u^' (1), where | c_{j 0} | + | c_{j 1} | > 0, for each j = 0, 1, and ?, f : [0, 1] × R² ? R are Carathéodory 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 f{symbol}-Laplacian (which corresponds to ? (x, s, t) = f{symbol} (t), with f{symbol} 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 Fucík spectrum or half-eigenvalues (for jumping f) of the p-Laplacian. In this paper, under suitable growth conditions on ? and f, we extend these conditions to the general problem (0.1), (0.2). © 2009 Elsevier Inc. All rights reserved.