Survey of partial and directional derivatives of fuzzy multi-dimensional mappings under generalized differentiability

Subject Areas : Statistics

Mohsen Miri Karbasaki ¹ (Department of Mathematics Basis of Science Islamic Azad University of KermanIran)
Mohammadreza Balooch Shahryari ² (Department of mathematics‎ ‎Haj Qassem Soleimani Branch‎ ‎Islamic Azad University‎ Kerman‎ ‎Iran)
Omolbanin Sedaghatfar ³ (Department of mathematics Yadegare Emam Khomeini Branch Islamic Azad University Tehran Iran)

Received: 2021-02-05 Accepted : 2021-11-01 Published : 2022-10-23

Keywords: نگاشت های چند بعدی فازی, اعداد فازی n-بعدی, مشتق پذیری تعمیم یافته جهتی, مشتق پذیری تعمیم یافته کلی, مشتق پذیری تعمیم یافته جزئی,

Abstract :

Fuzzy optimization issues have been discussed by many authors in recent articles, inspired by the concepts of generalized Hukuhara difference and generalized Hukuhara differentiability for one-dimensional functions from R to E, and have made considerable progress. In this paper, the concept of total generalized differentiability is introduced by using the concept of generalized difference from R^n to E for multi-dimensional mappings. Furthermore, the total generalized differentiability of the above is also examined and then the concept of directional generalized differentiability and partial generalized differentiability for fuzzy multi-dimensional mappings is defined and discussed in detail. In addition, directional generalized differentiability and partial generalized differentiability are expressed in terms of level-wise generalized differentiability. Also, the properties and the relationship between them are discussed. Finally, the relationships between total generalized differentiability, directional generalized differentiability and partial generalized differentiability are shown to illustrate the power and ability of relationships between them by mentioning some examples.

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