Nonresonance conditions for generalised ϕ-Laplacian problems with jumping nonlinearities

We consider the boundary value problemequation(0.1)−ψ(x,u(x),u′(x))′=f(x,u(x),u′(x)),a.e.x∈(0,1),equation(0.2)c00u(0)=c01u′(0),c10u(1)=c11u′(1), where |cj0|+|cj1|>0|cj0|+|cj1|>0, for each j=0,1j=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 (0.1), together with the boundary conditions (0.2), 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)ψ(x,s,t)=ϕ(t), with ϕ an odd, increasing homeomorphism). For the p  -Laplacian problem (and more particularly, the semilinear case p=2p=2), ‘nonresonance conditions’ which ensure the solvability of the problem  and , 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  and .