Publication: Metric regularity, pseudo-jacobians and global inversion theorems on Finsler manifols
Loading...
Official URL
Full text at PDC
Publication Date
2022-03-01
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
Citations
Exportar
Abstract
Our aim in this paper is to study the global invertibility of a locally Lipschitz map f : X → Y between (possibly infinite-dimensional) Finsler manifolds, stressing the connections with covering properties and metric regularity of f. To this end, we introduce a natural notion of pseudo-Jacobian Jf in this setting, as is a kind of set-valued differential object associated to f. By means of a suitable index, we study the relations between properties of pseudo-Jacobian Jf and local metric properties of the map f, which lead to conditions for f to be a covering map, and for f to be globally invertible. In particular, we obtain a version of Hadamard integral condition in this context.
Description
UCM subjects
Unesco subjects
Keywords
Citation
[1] F. H. Clarke, Optimization and Nonsmooth Analysis, Classics in Applied Mathematics 5, SIAM, Philadelphia, 1990.
[2] M. Durea, V. H. Huynh, H. T. Nguyen, and R. Strugariu. Metric regularity of composition set-valued mappings: metric setting and coderivative conditions. J. Math. Anal. Appl. 412 (2014), no. 1, 41–62.
[3] M. Fabian, P. Habala, P. Hájek, V. Montesinos and V. Zizler, Banach Space Theory - The basis for Linear and Nonlinear Analysis, Springer, New York, 2011.
[4] I. Garrido, O. Gutú and J. A. Jaramillo, Global inversion and covering maps on length spaces, Nonlinear Anal. 73 (2010), no. 5, 1364–1374.
[5] E. Ghahraei, S. Hosseini and M. R. Pouryayevali, Pseudo-Jacobian and global inversion of nonsmooth mappings on Riemannian manifolds, Nonlinear Anal. 130 (2016), 229–240.
[6] O. Gutú, On global inverse theorems, Topol. Methods in Nonlinear Anal. 49 (2017), no. 2, 401–444.
[7] O. Gutú, Chang palais-smale condition and global inversion, Bull. Math. Soc. Sci. Math. Roumanie, 109 (2018), no. 3, 293–303.
[8] O. Gutú, Global inversion for metrically regular mappings between Banach spaces, Rev. Mat. Complut. 35 (2022), no. 1, 25–51.
[9] O. Gutú and J. A. Jaramillo, Global homeomorphisms and covering projections on metric spaces, Math. Ann. 338 (2007), no. 5, 75–95.
[10] O. Gutú and J. A. Jaramillo, Surjection and inversion for locally Lipschitz maps between Banach spaces, J. Math. Anal. Appl. 478 (2019), no. 2, 578–594.
[11] J. Hadamard, Sur les transformations ponctuelles, Bull. Soc. Math. France 34 (1906), 71–84.
[12] A. D. Ioffe, Metric regularity: a survey Part 1. Theory, J. Aust. Math. Soc. 101 (2016), no. 2, 188–243.
[13] A. D. Ioffe, Nonsmooth analysis: differential calculus of nondifferentiable mappings, Trans. Amer. Math. Soc. 266 (1981), 1–56.
[14] J. A. Jaramillo, O. Madiedo and L. Sánchez-González, Global inversion of nonsmooth mappings on Finsler manifolds, J. Convex Anal. 20 (2013), no. 4, 1127–1146.
[15] J. A. Jaramillo, O. Madiedo and L. Sánchez-González, Global inversion of nonsmooth mappings using pseudo-Jacobian matrices Nonlinear Anal. 108 (2014), 57–65.
[16] J. A. Jaramillo, S. Lajara and O. Madiedo, Inversion of nonsmooth maps between Banach spaces, SetValued Var. Anal. 27 (2019), no. 4, 921–947.
[17] V. Jeyakumar and D.T. Luc, Approximate Jacobian matrices for nonsmooth continuous maps and C1- optimization, SIAM J. Control Optim. 36 (1998), no. 5, 1815–1832.
[18] V. Jeyakumar and D.T. Luc, Nonsmooth Vector Functions and Continuous Optimization, Optimization and its Applications 10, Springer, New York, 2008.
[19] F. John, On quasi-isometric maps I, Comm. Pure Appl. Math. 21 (1968), 77–110.
[20] G. Katriel, Mountain pass theorems and global homeomorphism theorems, Ann. de l’I.H.P., 11 (1994), no. 2, 189–209.
[21] M. Montenegro and A. E. Presoto, Invertibility of nonsmooth mappings, Ark. Mat. 55 (2017), no. 1, 217–228.
[22] R. S. Palais. Lusternik-Schnirelman theory on Banach manifolds. Topology. 5 (1966), 115–132.
[23] Z. P´ales and V. Zeidan, Generalized Jacobian for Functions with Infinite Dimensional Range and Domain, Set-Valued Anal. 15 (2007), no. 4, 331–375.
[24] R.R. Phelps, Convex functions, monotone operators and differentiability, Lecture Notes in Mathematics 1363, Springer-Verlag, Berling-Heidelberg, 1993.
[25] R. Plastock, Homeomorphisms between Banach spaces, Trans. Amer. Math. Soc. 200 (1974), 169–183.
[26] P. J. Rabier, Ehresmann fibrations and Palais-Smale conditions for morphisms of Finsler manifolds, Ann. of Math. 146 (1997), no. 3, 647–691.
[27] M. Schechter The use of Cerami sequences in critical point theory, Abstract and Applied Analysis, (2007), Art. ID 58948, 28 pp.
[28] A. Shapiro On concepts of directional differentiability, J. Optim. Theory Appl. 66 (1990), no. 3, 477–487.
[29] L. Thibault, On generalized differentials and subdifferentials of Lipschitz vector-valued functions, Nonlinear Anal. 10 (1982), no. 10, 1037–1053.