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Existence of multiple principal eigenvalues for some indefinite linear eigenvalue problems

J. Fleckinger;  J. Hernández;  F. Thélin — 2004

Bollettino dell'Unione Matematica Italiana

We study the existence of principal eigenvalues for differential operators of second order which are not necessarily in divergence form. We obtain results concerning multiplicity of principal eigenvalues in both the variational and the general case. Our approach uses systematically the Krein-Rutman theorem and fixed point arguments for the spectral radius of some associated problems. We also use a variational characterization for both the self-adjoint and the general case.

Characterization of Globally Lipschitz Nemytskiĭ Operators Between Spaces of Set-Valued Functions of Bounded φ-Variation in the Sense of Riesz

N. Merentes;  J. L. Sánchez Hernández — 2004

Bulletin of the Polish Academy of Sciences. Mathematics

Let (X,∥·∥) and (Y,∥·∥) be two normed spaces and K be a convex cone in X. Let CC(Y) be the family of all non-empty convex compact subsets of Y. We consider the Nemytskiĭ operators, i.e. the composition operators defined by (Nu)(t) = H(t,u(t)), where H is a given set-valued function. It is shown that if the operator N maps the space R V φ ₁ ( [ a , b ] ; K ) into R W φ ₂ ( [ a , b ] ; C C ( Y ) ) (both are spaces of functions of bounded φ-variation in the sense of Riesz), and if it is globally Lipschitz, then it has to be of the form H(t,u(t)) = A(t)u(t)...

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