On the existence of a global diffeomorphism between Fréchet spaces

Eftekharinasab, Kaveh

doi:10.31392/mfat-npu26_1.2020.05

Cited by 3 publications

(7 citation statements)

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“…We shall apply this theorem to generalize to Fréchet manifolds global diffeomorphism theorems for Fréchet spaces (see [1,Theorem 3.1] and [2,Theorem 4.1]). The proof is almost identical to the case of Fréchet spaces.…”

Section: Linking Results and Corollariesmentioning

confidence: 99%

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Some critical point results for Fréchet manifolds

Eftekharinasab¹

2022

Preprint

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We prove a so-called linking theorem and some of its corollaries, namely a mountain pass theorem and a three critical points theorem for Keller C 1 -functional on C 1 -Fréchet manifolds. Our approach relies on a deformation result which is not implemented by considering the negative pseudo-gradient flows. Furthermore, for mappings between Fréchet manifolds we provide a set of sufficient conditions in terms of the Palais-Smale condition that indicates when a local diffeomorphism is a global one.

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Section: Linking Results and Corollariesmentioning

confidence: 99%

“…If ϕ : Ñ R at x is of class C 1 , the derivative of ϕ at x, ϕ 1 pxq, is an element of the dual space 1 . The directional derivative of ϕ at x toward h P is given by Dϕpxqh " xϕ 1 pxq, hy, where x¨, ¨y is duality pairing.…”

Section: Preliminariesmentioning

confidence: 99%

Some critical point results for Fréchet manifolds

Eftekharinasab¹

2022

Preprint

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“…But it has not been the subject of study for more general Fréchet spaces. In [6] we found sufficient conditions that indicate when smooth tame maps are global diffeomorphisms. The purpose of this paper is to find weakened conditions for C 1 c -maps.…”

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confidence: 89%

A Global Diffeomorphism Theorem for Fréchet Spaces

Eftekharinasab

2020

J Math Sci

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We give sufficient conditions for a C 1 c -local diffeomorphism between Fréchet spaces to be a global one. We extend the Clarke's theory of generalized gradients to the more general setting of Fréchet spaces. As a consequence, we define the Chang Palais-Smale condition for Lipschitz functions and show that a function which is bounded below and satisfies the Chang Palais-Smale condition at all levels is coercive. We prove a version of the mountain pass theorem for Lipschitz maps in the Fréchet setting and show that along with the Chang Palais-Smale condition we can obtain a global diffeomorphism theorem.2010 Mathematics Subject Classification. 57R50;46A04;46A50 .

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“…Motivated by these results and recent developments in critical point theory in Fréchet spaces ( [2,3,4]), this paper aims to derive global implicit function theorems, which involve no loss of derivative, applicable to arbitrary Fréchet space for mappings which are only continuously differentiable by employing methods of critical point theory.…”

Section: Introductionmentioning

confidence: 99%

“…Interest in the broader context of Fréchet spaces has only recently started to gain traction. In [4,5], global inversion theorems, which are closely related to global implicit function theorems, have been obtained in these spaces. In [4], the result is closely linked to the Nash-Moser implicit function theorem, necessitating mappings to be at least twice continuously differentiable, and Fréchet spaces to be tame.…”

Section: Introductionmentioning

confidence: 99%