# Study on differentiability problems of interval-valued functions

Volume 10, Issue 11, pp 5677--5689 Publication Date: November 10, 2017       Article History
• 534 Views

### Authors

Yu-E Bao - College of Mathematics, Inner Mongolia University for Nationalities, Tongliao, Inner Mongolia 028043, P. R. China. Jin-Jun Li - 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 Universities, Tongliao, Inner Mongolia 028043, P. R. China.

### Abstract

In this paper, we give the concepts of $H$-directional differentiability and $D$-directional differentiability of interval-valued functions. Then we discuss the properties of $H$-directional differentiable interval-valued functions and $D$-directional differentiable interval-valued functions. The necessary and sufficient conditions for the $H$-directional differentiability are given together with the sufficient conditions and the necessary and sufficient conditions for $D$-directional differentiability of interval-valued functions. Then we discuss the relationship between the two directional differentiability and prove these directional differentiability can be equivalent under a certain conditions.

### Keywords

• Hukuhara difference
• Hausdorff distance
• interval-valued function
• $H$-directional differentiability
• $D$-directional differentiability

•  46T20
•  46G05

### References

• [1] Y. Chalco-Cano, H. Román-Flores, M. D. Jiménez-Gamero, Generalized derivative and $\pi$-derivative for set-valued functions, Inform. Sci., 181 (2011), 2177–2188.

• [2] 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.

• [3] V. Lupulescu , Fractional calculus for interval-valued functions, Fuzzy Sets and Systems, 265 (2015), 63–85.

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

• [5] R. Osuna-Gómez, Y. Chalco-Cano, B. Hernández-Jiménez, G. Ruiz-Garzón, Optimality conditions for generalized differentiable interval-valued functions, Inform. Sci., 321 (2015), 136–146.

• [6] L. Stefanini, B. Bede , Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Anal., 71 (2009), 1311–1328.

• [7] H.-C. Wu, The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective functions, European J. Oper. Res., 176 (2007), 46–59.

• [8] H.-C. Wu , The Karush-Kuhn-Tucker optimality conditions in multiobjective programming problems with interval-valued objective functions, European J. Oper. Res., 196 (2009), 49–60.