RT Journal Article
T1 Metric regularity, pseudo-jacobians and global inversion theorems on Finsler manifols
A1 Gutú, Olivia
A1 Jaramillo Aguado, Jesús Ángel
A1 Madiedo Castro, Óscar Reynaldo
AB 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.
YR 2022
FD 2022-03-01
LK https://hdl.handle.net/20.500.14352/71789
UL https://hdl.handle.net/20.500.14352/71789
LA eng
NO [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.
NO Ministerio de Ciencia e Innovación (MICINN)
DS Docta Complutense
RD 6 may 2024