On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces

Abstract:

Let $X$ be a real reflexive Banach space with dual $X^{*}$ and $G\subset X$ open and bounded and such that $0\in G.$ Let $T:X\supset D(T)\to 2^{X^{*}}$ be maximal monotone with $0\in D(T)$ and $0\in T(0),$ and $C:X\supset D(C)\to X^{*}$ with $0\in D(C)$ and $C(0)\neq 0.$ A general and more unified eigenvalue theory is developed for the pair of operators $(T,C).$ Further conditions are given for the existence of a pair $(\lambda ,x) \in (0,\infty )\times (D(T+C)\cap \partial G)$ such that \[ (**)\quad \qquad \qquad \qquad \qquad \qquad \qquad Tx+\lambda Cx\owns 0.\quad \qquad \qquad \qquad \qquad \qquad \qquad \] The “implicit" eigenvalue problem, with $C(\lambda ,x)$ in place of $\lambda Cx,$ is also considered. The existence of continuous branches of eigenvectors of infinite length is investigated, and a Fredholm alternative in the spirit of Necas is given for a pair of homogeneous operators $T,~C.$ No compactness assumptions have been made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the degree theories of the authors involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included.