A study on the differential and sub-differential of fuzzy mapping and its application problem

Volume 10, Issue 1, pp 1--17 http://dx.doi.org/10.22436/jnsa.010.01.01

Publication Date: January 26, 2017 Submission Date: July 04, 2016

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2915 Downloads
• 4578 Views

Authors

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

Abstract

In this paper, firstly, we gain some basic properties and characterization theorems of the differential and local sub-differential of the fuzzy mapping, obtain an important result that the local sub-differential of fuzzy mapping is an empty set or a convex set. Secondly, we generalize the concept of local differentiability of fuzzy mapping, and obtain some basic properties about the concept. At last, we study the relationships between sub-differential of fuzzy mapping and differential of convex fuzzy mappings. Moreover, a sufficient condition that a class of fuzzy mapping have convex extension is gained.

Share and Cite

Keywords

• Fuzzy number
• fuzzy mapping
• differential (sub-differential)
• convexification fuzzy mapping
• convex extension.

MSC

•  03E72
•  52A41
•  58C20
•  58C25

References

• [1] E. Ammar, J. Metz, On fuzzy convexity and parametric fuzzy optimization, Fuzzy Sets and Systems, 49 (1992), 135–141.
  • View Article
  • MathSciNet
  • Google Scholar


• [2] Y. E. Bao, C.-X. Wu, Convexity and strict convexity of fuzzy mappings, (Chinese) J. Harbin Inst. Tech., 39 (2007), 639–641.
  • MathSciNet


• [3] Y. E. Bao, C.-X. Wu, Semistrictly convex fuzzy mappings, J. Math. Res. Exposition, 30 (2010), 571–580.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [4] B. Bede, S. G. Gal, Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations, Fuzzy Sets and Systems, 151 (2005), 581–599.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


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


• [6] S. S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE Trans. Systems, Man, and Cybernet., SMC-, 2 (1972), 30–34.
  • View Article
  • MathSciNet
  • Google Scholar


• [7] D. Dubois, H. Prade, Towards fuzzy differential calculus, I, II, III, Fuzzy Sets and Systems, 8 (1982), 1–17, 105–116, 225–234.
  • MathSciNet
  • Google Scholar


• [8] Z. T. Gong, H. X. Li , Derivatives and gradients of fuzzy mappings and their applications, (Chinese) Appl. Math. J. Chinese Univ. Ser. A, 25 (2010), 229–238.
  • MathSciNet


• [9] S. Nanda, K. Kar , Convex fuzzy mappings, Fuzzy Sets and Systems, 48 (1992), 129–132.
  • View Article
  • Google Scholar


• [10] M. Panigrahi, G. Panda, S. Nanda , Convex fuzzy mapping with differentiability and its application in fuzzy optimization, European J. Oper. Res., 185 (2008), 47–62.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] M. L. Puri, D. A. Ralescu, Differentials of fuzzy functions, J. Math. Anal. Appl., 91 (1983), 552–558.
  • View Article
  • Google Scholar


• [12] R. T. Rockafellar , Convex analysis, Princeton Mathematical Series, Princeton University Press, Princeton, N.J. (1970)
  • Google Scholar


• [13] Y.-R. Syau, Differentiability and convexity of fuzzy mappings, Comput. Math. Appl., 41 (2001), 73–81.
  • View Article
  • Google Scholar


• [14] G.-X. Wang, C.-X. Wu, Directional derivatives and subdifferential of convex fuzzy mappings and application in convex fuzzy programming, Fuzzy Sets and Systems, 138 (2003), 559–591.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [15] C. X. Wu, C. Wu, The supremum and infimum of the [a] set of fuzzy numbers and its [their] application, J. Math. Anal. Appl., 210 (1997), 499–511.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] C. Zhang, X.-H. Yuan, E. S. Lee, Duality theory in fuzzy mathematical programming problems with fuzzy coefficients, Comput. Math. Appl., 49 (2005), 1709–1730.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [17] C. Zhang, X.-H. Yuan, E. S. Lee, Convex fuzzy mapping and operations of convex fuzzy mappings, Comput. Math. Appl., 51 (2006), 143–152.
  • View Article
  • MathSciNet
  • Google Scholar