Directional differentiability of interval-valued functions

Volume 16, Issue 4, pp 507-515 http://dx.doi.org/10.22436/jmcs.016.04.04

Publication Date: December 15, 2016 Submission Date: July 30, 2016

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Authors

Yu-e Bao - College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia 028043, P. R. China. Bo Zao - College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia 028043, P. R. China. Eer-dun Bai - College of Computer Science and Technology, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia 028043, P. R. China.

Abstract

In this paper, by using the generalized Hukuhara difference (gH-difference) of interval numbers, we introduce and study the directional differentiability problem of interval-valued function. Firstly, we put forward the concept of directional differentiability of interval-valued function, discuss the characterizations of directional differentiability of interval-valued function, and propose the sufficient condition of the directional differentiability of interval-valued function. Secondly, we discuss the relations among the directional derivative, derivative and partial derivative of interval-valued function, and prove that the derivative and partial derivative are both special directional derivatives.

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Keywords

• Interval numbers
• interval-valued function
• Hausdorff metric
• generalized H-difference
• directional derivative.

MSC

•  46G05
•  58C25

References

• [1] B. Bede, L. Stefanini , Generalized differentiability of fuzzy-valued functions, Fuzzy Sets and Systems, 230 (2013), 119-141.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] Y. Chalco-Cano, H. Roman-Flores, M. D. Jimenez-Gamero, Generalized derivative and \(\pi\)-derivative for set-valued functions, Inform. Sci., 181 (2011), 2177-2188.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [3] Y. Chalco-Cano, A. Rufián-Lizan, H. Román-Flores, M. D. Jiménez-Gamero, Calculus for interval- valued functions using generalized Hukuhara derivative and applications, Fuzzy Sets and Systems, 219 (2013), 49-67.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] V. Lupulescu, Fractional calculus for interval-valued functions , Fuzzy Sets and Systems, 265 (2015), 63-85.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] R. E. Moore, Interval Analysis,Prentice-Hall , Englewood Cliffs, New Jersey (1966)


• [6] R. Osuna-Gómez, Y. Chalco-Cano, B. Hernández -Jiménez, G. Ruiz-Garzn, Optimality conditions for generalized differentiable interval-valued functions, Inform Sci., 321 (2015), 136-146.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71 (2009), 1311-1328.
  • View Article
  • MathSciNet
  • Google Scholar


• [8] H.-C. Wu, The KarushKuhnTucker optimality conditions in multiobjective programming problems with interval-valued objective functions , European J. Oper. Res., 196 (2009), 49-60.
  • View Article
  • Google Scholar