Affiliations: [a] Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran | [b] Department of Mathematics, Yadegar-e-Imam Khomeini (RAH) Shahr-e-Rey Branch, Islamic Azad University, Tehran, Iran
Correspondence:
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Corresponding author. M. R. Balooch Shahryari, Department of Mathematics, Kerman Branch, Islamic Azad University, Kerman, Iran. E-mail: [email protected].
Abstract: This article identifies and presents the generalized difference (g-difference) of fuzzy numbers, Fréchet and Gâteaux generalized differentiability (g-differentiability) for fuzzy multi-dimensional mapping which consists of a new concept, fuzzy g-(continuous linear) function; Moreover, the relationship between Fréchet and Gâteaux g-differentiability is studied and shown. The concepts of directional and partial g-differentiability are further framed and the relationship of which will the aforementioned concepts are also explored. Furthermore, characterization is pointed out for Fréchet and Gâteaux g-differentiability; based on level-set and through differentiability of endpoints real-valued functions a characterization is also offered and explored for directional and partial g-differentiability. The sufficient condition for Fréchet and Gâteaux g-differentiability, directional and partial g-differentiability based on level-set and through employing level-wise gH-differentiability (LgH-differentiability) is expressed. Finally, to illustrate the ability and reliability of the aforementioned concepts we have solved some application examples.
