Abstract
We study the solvability of a nonlinear eigenvalue problem for maximal monotone operators under a normalization observation. The investigation is based on degree theories for appropriate classes of operators, and a regularization method by the duality operator is used. Let X be a real reflexive Banach space with its dual and Ω be a bounded open set in X. Suppose that is a maximal monotone operator and is a bounded demicontinuous operator satisfying condition ( ). Applying the Browder degree theory, we solve a nonlinear eigenvalue problem of the form . In the case where , an eigenvalue result for generalized pseudomonotone densely defined perturbations is obtained by the Kartsatos-Skrypnik degree theory. MSC:47J10, 47H05, 47H14, 47H11.
Highlights
1 Introduction and preliminaries Eigenvalue theory is closely related to the problem of solving nonlinear equations which was initiated by Krasnosel’skii [ ] for compact operators
It has been extensively investigated by many researchers in various aspects, with applications to evolution equations and differential equations; see, e.g., [ – ]
The study was mostly based on degree theories for appropriate classes of operators and the usual method of regularization by means of the duality operator; see [ – ]
Summary
Jxn + Txn ≤ ( – ε)Jxn + C( n, xn) ≤ M( n) for sufficiently large n. Since the sequence { C( n, xn) – C( n, xn)} is bounded in the reflexive Banach space X*, we may suppose that C( n, xn) – C( n, xn) converges weakly to some h ∈ X*. Using (c ), it is clear that h =.
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