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On the eigenvalue problem for perturbed nonlinear maximal monotone operators in reflexive Banach spaces. (English) Zbl 1100.47050

Let \(X\) be a real reflexive Banach space with dual \(X^*\) and \(G\subset X\) be 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 \(C(0)\neq 0\). In the paper under review, 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 \[ T\,x+\lambda\,C\,x\ni 0. \] The “implicit” eigenvalue problem, with \(C(\lambda,\,x)\) in place of \(\lambda\,C\,x\), 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 are made in most of the results. The degree theories of Browder and Skrypnik are used, as well as the the degree theories of the authors, involving densely defined perturbations of maximal monotone operators. Applications to nonlinear partial differential equations are included.

MSC:

47H14 Perturbations of nonlinear operators
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H11 Degree theory for nonlinear operators
47J10 Nonlinear spectral theory, nonlinear eigenvalue problems
47N20 Applications of operator theory to differential and integral equations
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