A unified generalization for Hukuhara types differences and derivatives: Solid analysis and comparisons

Babak Shiri; Babak Shiri

doi:10.3934/math.2023112

AIMS Mathematics

2023, Volume 8, Issue 1: 2168-2190. doi: 10.3934/math.2023112

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Research article

A unified generalization for Hukuhara types differences and derivatives: Solid analysis and comparisons

Babak Shiri ^,

Data Recovery Key Laboratory of Sichuan Province, College of Mathematics and Information Science, Neijiang Normal University, Neijiang 641100, China

Received: 05 September 2022 Revised: 08 October 2022 Accepted: 11 October 2022 Published: 28 October 2022
MSC : 26E50, 54A40

Uncertain numbers, in a parallel definition of fuzzy numbers, are introduced. Model uncertainty and measurement uncertainty are our motivations for this study. A class of scalar multiplication and differences is proposed. Related algebra is investigated. A necessary and sufficient condition of the existence of the introduced differences is obtained. Then, the existing result for the derivative is studied. Many interestingly important results are obtained. For example, the Hukuhara derivative does not exist for any fuzzy function with the new viewpoint. Constructive conditions for the existence of the generalized Hukuhara derivative are introduced. Four possible categories for derivatives fall into two forms of the fuzzy derivative for the generalized Hukuhara derivative. Importantly, this bifurcation in the definition of the new generalized Hukuhara derivative does not happen. Finally, all definitions related to differences and derivatives of uncertain numbers are unified in one concrete form with concrete analysis. Some examples and counterexamples are provided to illustrate theories and theorems in detail.
- Hukuhara derivative,
- generalized Hukuhara derivative,
- fuzzy numbers,
- bifurcation in fuzzy derivative,
- measurement uncertainty,
- interval-valued functions
Citation: Babak Shiri. A unified generalization for Hukuhara types differences and derivatives: Solid analysis and comparisons[J]. AIMS Mathematics, 2023, 8(1): 2168-2190. doi: 10.3934/math.2023112

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Abstract

Uncertain numbers, in a parallel definition of fuzzy numbers, are introduced. Model uncertainty and measurement uncertainty are our motivations for this study. A class of scalar multiplication and differences is proposed. Related algebra is investigated. A necessary and sufficient condition of the existence of the introduced differences is obtained. Then, the existing result for the derivative is studied. Many interestingly important results are obtained. For example, the Hukuhara derivative does not exist for any fuzzy function with the new viewpoint. Constructive conditions for the existence of the generalized Hukuhara derivative are introduced. Four possible categories for derivatives fall into two forms of the fuzzy derivative for the generalized Hukuhara derivative. Importantly, this bifurcation in the definition of the new generalized Hukuhara derivative does not happen. Finally, all definitions related to differences and derivatives of uncertain numbers are unified in one concrete form with concrete analysis. Some examples and counterexamples are provided to illustrate theories and theorems in detail.

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