The metric derivative of set-valued functions
-
Mohamad Muslikh
,
Adem Kilicman
,
Siti Hasana bt Sapar
Abstract
In this article, we introduce the notion of “metrically differentiable” for set-valued functions. By using this notion, it is shown that each Lipschitz set-valued function is differentiable almost everywhere. Its relationship to the differentiation in the sense of Hukuhara and its generalizations are also discussed.
Acknowledgements
The authors are very grateful to the referees for their useful comments that improve the manuscript.
References
[1] L. Ambrosio and P. Tilli, Topics on Analysis in Metric Spaces, Oxford Lecture Ser. Math. Appl. 25, Oxford University Press, Oxford, 2004. Search in Google Scholar
[2] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966. Search in Google Scholar
[3] M. Hukuhara, Intégration des applications mesurables dont la valeur est un compact convexe, Funkcial. Ekvac. 10 (1967), 205–223. Search in Google Scholar
[4] B. Kirchheim, Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. Amer. Math. Soc. 121 (1994), no. 1, 113–123. 10.1090/S0002-9939-1994-1189747-7Search in Google Scholar
[5] M. Kisielewicz, Differential Inclusions and Optimal Control, Math. Appl. (East European Ser.) 44, Kluwer Academic, Dordrecht, 1991. Search in Google Scholar
[6] V. Lakshmikantham, T. G. Bhaskar and J. Vasundhara Devi, Theory of Set Differential Equations in Metric Spaces, Cambridge Scientific, Cambridge, 2006. Search in Google Scholar
[7] R. E. Moore, R. B. Kearfott and M. J. Cloud, Introduction to Interval Analysis, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 2009. 10.1137/1.9780898717716Search in Google Scholar
[8] R. Moriello, Partial differentiation on a metric space, Pi Mu Epsilon J. 5 (1974), 514–519. Search in Google Scholar
[9] M. Muslikh, A. Kılıçman, S. H. Sapar and N. Bachok, Absolute differentiation of set-valued functions on metric spaces, preprint. Search in Google Scholar
[10] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, 1966. Search in Google Scholar
[11] K. Skaland, Differentiation on metric spaces, Proc. S. D. Acad. Sci. 54 (1975), 75–77. Search in Google Scholar
[12] L. Stefanini, A generalization of Hukuhara difference and division for interval and fuzzy arithmetic, Fuzzy Sets and Systems 161 (2010), no. 11, 1564–1584. 10.1016/j.fss.2009.06.009Search in Google Scholar
[13] L. Stefanini and B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal. 71 (2009), no. 3–4, 1311–1328. 10.1016/j.na.2008.12.005Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Meromorphic function sharing a small function with a homogeneous differential polynomial
- On generalized quasi-Einstein manifolds
- On Green’s function of the Robin problem for the Poisson equation
- Moment functions on hypergroup joins
- Sparse reconstruction with multiple Walsh matrices
- On certain equations in semiprime rings and standard operator algebras
- Derivatives of meromorphic functions sharing two sets with least cardinalities
- The metric derivative of set-valued functions
- The relativistic Enskog equation near the vacuum in the Robertson–Walker space-time
Articles in the same Issue
- Frontmatter
- Meromorphic function sharing a small function with a homogeneous differential polynomial
- On generalized quasi-Einstein manifolds
- On Green’s function of the Robin problem for the Poisson equation
- Moment functions on hypergroup joins
- Sparse reconstruction with multiple Walsh matrices
- On certain equations in semiprime rings and standard operator algebras
- Derivatives of meromorphic functions sharing two sets with least cardinalities
- The metric derivative of set-valued functions
- The relativistic Enskog equation near the vacuum in the Robertson–Walker space-time
