Nonresonance conditions for generalised -Laplacian problems with jumping nonlinearities

We consider the boundary value problem where cj0+cj1>0, for each j=0,1, and 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 -Laplacian (which corresponds to ψ(x,s,t)= (t), with 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) and (0.2), have been obtained in terms of either eigenvalues (for non-jumping f) or the Fučík 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 (0.1) and (0.2).