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A variational approach to a class of nonlinear eigenvalue problems.


Author: Peter Hess
Journal: Proc. Amer. Math. Soc. 29 (1971), 272-276
MSC: Primary 49.10; Secondary 47.00
DOI: https://doi.org/10.1090/S0002-9939-1971-0284890-3
MathSciNet review: 0284890
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Abstract: Let f be a real-valued differentiable function defined on the real reflexive Banach space X. The problem of minimizing f over a subset of X is investigated under the following mild monotonicity assumption on the derivative $ f'$ of f: if $ \{ {u_n}\} $ is a sequence in X converging weakly to some $ u \in X$, then $ \lim \sup (f'{u_n},{u_n} - u) \geqq 0$ . The eigenvalue problem $ f'u = \lambda g'u$ for some $ \lambda \in {R^1}$, with $ g'$ being the derivative of a further function g, is then reduced to that first question.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1971-0284890-3
Keywords: Variational problem, differentiable function, reflexive Banach space, operator of monotone type, nonlinear eigenvalue problem, multiple integral functional
Article copyright: © Copyright 1971 American Mathematical Society

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