Abstract
Based on a primal regularity criterion we provide lower bounds for the regularity modulus of a nonlinear single-valued mapping F from a Banach space X into another Banach space Y . We focus on the case when F is defined on a proper (closed convex) subset of X only rather than on the whole of X. Three possible ways of approximating F around the reference point are considered. First, we use a tangential approximation by set-valued mappings associated with the Bouligand’s tangent cone to the graph of F. Then we move on to approximations by positively homogeneous set-valued mappings whose graphs contain the graph of F, for example, by the strict prederivative. Finally, we use an approximation by bunches of continuous linear operators. In the first two cases finding approximating objects is relatively easy while in the third case the approximating object is very convenient to work with. On examples, we illustrate that these approaches are different and neither of them implies the other, unless the spaces in question are finite dimensional.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. R. Akhmerov, M. I. Kamenskii, A. S. Potapova, A. E. Rodkina and B. N. Sadovskii, Measure of Noncompactness and Condensing Operators. Birkhüser, Basel, 1992.
J.-P. Aubin, Contingent derivatives of set-valued maps and existence of solutions to nonlinear inclusions and differential inclusions. In: MathematicalAnalysis and Applications, Part A, Adv. in Math. Suppl. Stud. 7, Academic Press, New York, 1981, 159–229.
J. P. Aubin and H. Frankowska, Set-Valued Analysis. Systems & Control: Foundations & Applications 2, Birkhäuser Boston, Boston, MA, 1990.
Azé D.: A unified theory for metric regularity of multifunctions. J. Convex Anal. 13, 225–252 (2006)
J. M. Borwein and D. M. Zhuang, Verifiable necessary and sufficient conditions for openness and regularity of set-valued and single-valued maps. J. Math. Anal. Appl. 134 (1988), 441–459.
R. Cibulka and M. Fabian, A note on Robinson-Ursescu and Lyusternik-Graves theorem. Math. Program. 139 (2013), 89–101.
E. De Giorgi, A. Marino and M. Tosques, Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 68 (1980), 180–187.
R. Deville, G. Godefroy and V. Zizler, Smoothness and Renormings in Banach Spaces. Pitman Monographs and Surveys in Pure and Applied Mathematics 64, Longman Scientific & Technical, Harlow, 1993.
P. H. Dien, Some results on locally Lipschitzian mappings. Acta Math. Vietnam. 6 (1981), 97–105.
A. V. Dmitruk, A. A. Milyutin and N. P. Osmolovskii, Lyusternik’s theorem and the theory of extrema. Russian Math. Surveys 35 (1980), 11–51.
A. L. Dontchev, M. Quincampoix and N. Zlateva, Aubin criterion for metric regularity. J. Convex Anal. 13 (2006), 281–297.
A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings. Springer Monographs in Mathematics, Springer, Dordrecht, 2009.
N. Dunford and J. T. Schwartz, Linear Operators. I. General Theory. Interscience Publishers, New York, 1958.
M. Fabian, P. Habala, P. Hájek, V. Montesinos and V. Zizler, Banach Space Theory: The Basis for Linear and Nonlinear Analysis. CMS Books Math., Springer, New York, 2011.
M. Fabian and D. Preiss, A generalization of the interior mapping theorem of Clarke and Pourciau. Comment. Math. Univ. Carolin. 28 (1987), 311–324.
I. L. Glicksberg, A further generalization of the Kakutani fixed point theorem, with application to Nash equilibrium points. Proc. Amer. Math. Soc. 3 (1952), 170–174.
A. Granas and J. Dugunji, Fixed Point Theory. Springer, New York, 2003.
Graves L. M.: Some mapping theorems. Duke Math. J. 17, 111–114 (1950)
A. D. Ioffe, Nonsmooth Analysis: Differential calculus of nondifferentiable mappings. Trans. Amer. Math. Soc. 266 (1981), 1–56.
A. D. Ioffe, On the local surjection property. Nonlinear Anal. 11 (1987), 565–592.
A. D. Ioffe, Metric regularity and subdifferential calculus. Russian Math. Surveys 55 (2000), 501–558.
A. D. Ioffe, Regularity on a fixed set. SIAM J. Optim. 21 (2011), 1345–1370.
A. D. Ioffe, Separable reduction of metric regularity properties. In: Constructive Nonsmooth Analysis and Related Topics, V. F. Demyanov, P. M. Pardalos and M. Batsin, eds., Springer Optim. Appl. 87, Springer, New York, 2013, 25–37.
A. D. Ioffe, Metric regularity, fixed points and some associated problems of variational analysis. J. Fixed Point Theory Appl. 15 (2014), 67–99.
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation. I. Grundlehren Math. Wiss. 300, Springer, Berlin, 2006.
Z. Páles, Linear selections for set-valued functions and extensions of bilinear forms. Arch. Math. (Basel) 62 (1994), 427–432.
Z. Páles, Inverse and implicit function theorems for nonsmooth maps in Banach spaces. J. Math. Anal. Appl. 209 (1997), 202–220.
Z. Páles and V. Zeidan, Infinite dimensional Clarke generalized Jacobian. J. Convex Anal. 14 (2007), 433–454.
Penot J.-P.: Metric regularity, openness and Lipschitzian behavior of multifunctions. Nonlinear Anal. 13, 629–643 (1989)
J.-P. Penot, Calculus without Derivatives. Grad. Texts in Math. 266, Springer, New York, 2013.
V. Pták, A quantitative refinement of the closed graph theorem. Czechoslovak Math. J. 24 (1974), 503–506.
K. Sh. Tsiskaridze, Extremal problems in Banach spaces. In: Nekotorye Voprosy Matematicheskoy Theorii Optimalnogo Upravleniya (Some Problems of the Mathematical Theory of Optimal Control), Inst. Appl. Math., Tbilisi State Univ. 1975 (in Russian).
Author information
Authors and Affiliations
Corresponding author
Additional information
To Andrzej Granas en tout amitié et estime
Rights and permissions
About this article
Cite this article
Cibulka, R., Fabian, M. & Ioffe, A.D. On primal regularity estimates for single-valued mappings. J. Fixed Point Theory Appl. 17, 187–208 (2015). https://doi.org/10.1007/s11784-015-0240-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11784-015-0240-5