Numerical ranges for pairs of operators, duality mappings with gauge function, and spectra of nonlinear operators

We define and study numerical ranges for pairs of nonlinear operators F and J which act between some Banach space X and its dual X*, with respect to some increasing gauge function (p. Connections with spectra for certain classes of nonlinear operators introduced recently in the literature are also established. As a sample example, we consider the case when F is the duality map of the Lebesgue space L-p(Omega), J is the duality map of the corresponding Sobolev space W-0(1,P)(Omega), and phi(t) = t(P-1) (1 < p < infinity). This leads to existence, uniqueness, and perturbation results for a homogeneous eigenvalue problem involving the p-Laplace operator.