On the Krein–Milman–Ky Fan theorem for convex compact metrizable sets

AbstractThe improvements of Ky Fan theorem are given for tensors. First, based on Brauer-type eigenvalue inclusion sets, we obtain some new Ky Fan-type theorems for tensors. Second, by characterizing the ratio of the smallest and largest values of a Perron vector, we improve the existing results. Third, some new eigenvalue localization sets for tensors are given and proved to be tighter than those presented by Li and Ng (Numer Math 130(2):315–335, 2015) and Wang et al. (Linear Multilinear Algebra 68(9):1817–1834, 2020). Finally, numerical examples are given to validate the efficiency of our new bounds.

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Inequalities for certain Classes of Convex Functions

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500021969 ◽

1959 ◽

Vol 11 (4) ◽

pp. 231-235 ◽

Cited By ~ 9

Author(s):

L. Mirsky

Keyword(s):

Simple Proof ◽

Convex Functions ◽

Stochastic Matrices ◽

Doubly Stochastic ◽

Fan Theorem ◽

Special Cases ◽

Gauge Functions ◽

Ky Fan ◽

Ky Fan Theorem ◽

Symmetric Gauge

Making use of properties of doubly-stochastic matrices, I recently gave a simple proof (4) of a theorem of Ky Fan (Theorem 2b below) on symmetric gauge functions. I now propose to show that the same idea can be employed to derive a whole series of results on convex functions ; in particular, certain well-known inequalities of Hardy-Littlewood-Pólya and of Pólya will emerge as specìal cases.

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On solvability of the impulsive Cauchy problem for integro-differential inclusions with non-densely defined operators

Philosophical Transactions of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rsta.2019.0384 ◽

2021 ◽

Vol 379 (2191) ◽

pp. 20190384

Author(s):

Irene Benedetti ◽

Valeri Obukhovskii ◽

Valentina Taddei

Keyword(s):

Cauchy Problem ◽

Fixed Point ◽

Compact Convex ◽

Linear Part ◽

Main Tool ◽

Densely Defined Operator ◽

Fan Theorem ◽

Locally Convex ◽

Ky Fan ◽

Densely Defined

We prove the existence of at least one integrated solution to an impulsive Cauchy problem for an integro-differential inclusion in a Banach space with a non-densely defined operator. Since we look for integrated solution we do not need to assume that A is a Hille Yosida operator. We exploit a technique based on the measure of weak non-compactness which allows us to avoid any hypotheses of compactness both on the semigroup generated by the linear part and on the nonlinear term. As the main tool in the proof of our existence result, we are using the Glicksberg–Ky Fan theorem on a fixed point for a multivalued map on a compact convex subset of a locally convex topological vector space. This article is part of the theme issue ‘Topological degree and fixed point theories in differential and difference equations’.

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