On a global diffeomorphism between two Banach spaces and some application

Galewski, Marek; Koniorczyk, Marcin

doi:10.1556/sscmath.52.2015.1.1302

Cited by 7 publications

(17 citation statements)

References 11 publications

(32 reference statements)

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“…See also [18] for some recent related results along this line. The connection between Idczak's result and the above conditions can be easily made by means of the following fact: If f : X → Y is a Fredholm map of index 0 between Hilbert spaces then the Katriel condition implies that F y satisfies the PS-condition for all y ∈ Y ; see Lemma 17 in the appendix.…”

Section: 23mentioning

confidence: 92%

On global inverse theorems

Gutú¹

2016

TMNA

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Abstract. Since the Hadamard Theorem, several metric and topological conditions have emerged in the literature to date, yielding global inverse theorems for functions in different settings. Relevant examples are the mappings between infinite-dimensional Banach-Finsler manifolds, which are the focus of this work. Emphasis is given to the nonlinear Fredholm operators of nonnegative index between Banach spaces. The results are based on good local behavior of f at every x, namely, f is a local homeomorphism or f is locally equivalent to a projection. The general structure includes a condition that ensures a global property for the fibres of f , ideally expecting to conclude that f is a global diffeomorphism or equivalent to a global projection. A review of these results and some relationships between different criteria are shown. Also, a global version of Graves Theorem is obtained for a suitable submersion f with image in a Banach space: given r > 0 and x 0 in the domain of f we give a radius ̺(r) > 0, closely related to the hypothesis of the Hadamard Theorem, such that B̺(f (x 0 )) ⊂ f (Br (x 0 )).

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Section: 23mentioning

confidence: 92%

On global inverse theorems

Gutú¹

2016

TMNA

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“…In this section we give an example where the conditions for global invertibility introduced in the previous sections can be easily checked. We will be concerned with the following integro-differential equation, which has been considered, with several variants, in [16], [8] and [9]:…”

Section: Examplementioning

confidence: 99%

Surjection and inversion for locally Lipschitz maps between Banach spaces

Gutú

Jaramillo

2019

Journal of Mathematical Analysis and Applications

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We study the global invertibility of non-smooth, locally Lipschitz maps between infinite-dimensional Banach spaces, using a kind of Palais-Smale condition. To this end, we consider the Chang version of the weighted Palais-Smale condition for locally Lipschitz functionals in terms of the Clarke subdifferential, as well as the notion of pseudo-Jacobians in the infinite-dimensional setting, which are the analog of the pseudo-Jacobian matrices defined by Jeyakumar and Luc. Using these notions, we derive our results about existence and uniqueness of solution for nonlinear equations. In particular, we give a version of the classical Hadamard integral condition for global invertibility in this context.2000 Mathematics Subject Classification. 47J07, 58E05, 49J52.

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“…It should be emphasized that we do not make any assumption on the constantĀ in (A3). If we compare this with the corresponding assumption in papers [6,9] or [7] we see that it is usually assumed that the coefficient in the linearity should satisfy a specific additional condition connected with the diagonal of the square [0, 1] 2 . This condition is removed in our paper thanks to the application of the Bielecki norm.…”

Section: Lemma 22mentioning

confidence: 99%

“…The problem of the smooth dependence of solutions on the parameter of integral or integro-differential equations was discussed in papers [2,6,7,9,10].…”

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confidence: 99%

Control system defined by some integral operator

Majewski¹

2017

OpMath

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Abstract. In the paper we consider a nonlinear control system governed by the Volterra integral operator. Using a version of the global implicit function theorem we prove that the control system under consideration is well-posed and robust, i.e. for any admissible control u there exists a uniquely defined trajectory xu which continuously depends on control u and the operator u → xu is continuously differentiable. The novelty of this paper is, among others, the application of the Bielecki norm in the space of solutions which allows us to weaken standard assumptions.

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On a global diffeomorphism between two Banach spaces and some application

Cited by 7 publications

References 11 publications

On global inverse theorems

On global inverse theorems

Surjection and inversion for locally Lipschitz maps between Banach spaces

Control system defined by some integral operator

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