Perturbation of error bounds
- Authors: Kruger, Alexander , López, Marco , Théra, Michel
- Date: 2018
- Type: Text , Journal article
- Relation: Mathematical Programming Vol. 168, no. 1-2 (2018), p. 533-554
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: Our aim in the current article is to extend the developments in Kruger et al. (SIAM J Optim 20(6):3280–3296, 2010. doi:10.1137/100782206) and, more precisely, to characterize, in the Banach space setting, the stability of the local and global error bound property of inequalities determined by lower semicontinuous functions under data perturbations. We propose new concepts of (arbitrary, convex and linear) perturbations of the given function defining the system under consideration, which turn out to be a useful tool in our analysis. The characterizations of error bounds for families of perturbations can be interpreted as estimates of the ‘radius of error bounds’. The definitions and characterizations are illustrated by examples. © 2017, Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society.
Metric regularity relative to a cone
- Authors: Van Ngai, Huynh , Tron, Nguyen , Théra, Michel
- Date: 2019
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics Vol. 47, no. 3 (2019), p. 733-756
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The purpose of this paper is to discuss some of the highlights of the theory of metric regularity relative to a cone. For example, we establish a slope and some coderivative characterizations of this concept, as well as some stability results with respect to a Lipschitz perturbation.
A new regularity criterion of weak solutions to the 3D micropolar fluid flows in terms of the pressure
- Authors: Gala, Sadek , Ragusa, Maria , Théra, Michel
- Date: 2021
- Type: Text , Journal article
- Relation: Bolletino dell Unione Matematica Italiana Vol. 14, no. 2 (2021), p. 331-337
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: In this study, we establish a new regularity criterion of weak solutions to the three-dimensional micropolar fluid flows by imposing a critical growth condition on the pressure field. © 2020, Unione Matematica Italiana.
Set-valued orthogonality and nearness
- Authors: Barbagallo, Annamaria , Ernst, Octavian , Théra, Michel
- Date: 2020
- Type: Text , Journal article
- Relation: AAPP Atti della Accademia Peloritana dei Pericolanti, Classe di Scienze Fisiche, Matematiche e Naturali Vol. 98, no. (2020), p.
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The theory of set-valued mappings has grown with the development of modern variational analysis. It is a key in convex and non-smooth analysis, in game theory, in mathematical economics and in control theory. The concepts of nearness and orthogonality have been known for functions since the pioneering works of Campanato, Birkhoff and James. In a recent paper Barbagallo et al. [J. Math. Anal. Appl., 484 (1), (2020)] a connection between these two concepts has been made. This note is mainly devoted to introduce nearness and orthogonality between set-valued mappings with the goal to study the solvability of generalized equations involving set-valued mappings. © 2020 Accademia Peloritana dei Pericolanti. All rights reserved.
Ekeland's inverse function theorem in graded Fréchet spaces revisited for multifunctions
- Authors: Huynh, Van Ngai , Théra, Michel
- Date: 2018
- Type: Text , Journal article
- Relation: Journal of Mathematical Analysis and Applications Vol. 457, no. 2 (2018), p. 1403-1421
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: In this paper, we present some inverse function theorems and implicit function theorems for set-valued mappings between Fréchet spaces. The proof relies on Lebesgue's Dominated Convergence Theorem and on Ekeland's variational principle. An application to the existence of solutions of differential equations in Fréchet spaces with non-smooth data is given.
Enlargements of the moreau–rockafellar subdifferential
- Authors: Abbasi, Malek , Kruger, Alexander , Théra, Michel
- Date: 2021
- Type: Text , Journal article
- Relation: Set-Valued and Variational Analysis Vol. 29, no. 3 (2021), p. 701-719
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: This paper proposes three enlargements of the conventional Moreau–Rockafellar subdifferential: the sup-, sup
Gateaux differentiability revisited
- Authors: Abbasi, Malek , Kruger, Alexander , Théra, Michel
- Date: 2021
- Type: Text , Journal article
- Relation: Applied Mathematics and Optimization Vol. 84, no. 3 (2021), p. 3499-3516
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: We revisit some basic concepts and ideas of the classical differential calculus and convex analysis extending them to a broader frame. We reformulate and generalize the notion of Gateaux differentiability and propose new notions of generalized derivative and generalized subdifferential in an arbitrary topological vector space. Meaningful examples preserving the key properties of the original notion of derivative are provided. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC part of Springer Nature.
On the regularity of weak solutions of the boussinesq equations in besov spaces
- Authors: Barbagallo, Annamaria , Gala, Sadek , Ragusa, Maria , Théra, Michel
- Date: 2021
- Type: Text , Journal article
- Relation: Vietnam Journal of Mathematics Vol. 49, no. 3 (2021), p. 637-649
- Relation: http://purl.org/au-research/grants/arc/DP160100854
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- Description: The main issue addressed in this paper concerns an extension of a result by Z. Zhang who proved, in the context of the homogeneous Besov space Ḃ