Nonresonance Conditions for Generalised Φ-laplacian Problems with Jumping Nonlinearities

We consider the boundary value problem −ψ(x, u(x), u′(x))′ = f(x, u(x), u′(x)), a.e. x ∈ (0, 1), (1) c00u(0) = c01u ′(0), c10u(1) = c11u ′(1), (2) where |cj0| + |cj1| > 0, for each j = 0, 1, and ψ, f : [0, 1] × R2 → R are Carathéodory functions, with suitable additional properties. The differential operator generated by the left-hand side of (1), together with the boundary conditions (2), is a generalisation of the usual p-Laplacian, and also of the so called φ-Laplacian (which corresponds to ψ(x, s, t) = φ(t), with φ 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 (1), (2), have been obtained in terms of either eigenvalues (for non-jumping f) or the Fuč́ık spectrum or half-eigenvalues (for jumping f) of the p-Laplacian. In this paper, under suitable growth conditions on ψ and f , we extend these conditions to the general problem (1), (2).