On the stability of two functional equations for $ (S, N) $-implications

Sizhao Li; Xinyu Han; Dapeng Lang; Songsong Dai; Sizhao Li; Xinyu Han; Dapeng Lang; Songsong Dai

doi:10.3934/math.2021110

AIMS Mathematics

2021, Volume 6, Issue 2: 1822-1832. doi: 10.3934/math.2021110

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On the stability of two functional equations for $ (S, N) $-implications

1.
College of Computer Science and Technology, Harbin Engineering University, Harbin 150000, China
2.
School of Electronics and Information Engineering, Taizhou University, Taizhou 318000, China

Received: 29 August 2020 Accepted: 29 November 2020 Published: 01 December 2020
MSC : 03E72, 39B82

The iterative functional equation $ \alpha\rightarrow(\alpha\rightarrow \beta) = \alpha\rightarrow \beta $ and the law of importation $ (\alpha\wedge \beta)\rightarrow \gamma = \alpha\rightarrow (\beta\rightarrow \gamma) $ are two tautologies in classical logic. In fuzzy logics, they are two important properties, and are respectively formulated as $ I(\alpha, \beta) = I(\alpha, I(\alpha, \beta)) $ and $ I(T(\alpha, \beta), \gamma) = I(\alpha, I(\beta, \gamma)) $ where $ I $ is a fuzzy implication and $ T $ is a $ t $-norm. Over the past several years, solutions to these two functional equations involving different classes of fuzzy implications have been studied. However, there are no results about stability study of fuzzy functional equations involving fuzzy implication. This paper discusses fuzzy implications that do not strictly satisfying these equations, but approximately satisfy these equations. Then we establish the Hyers-Ulam stability of the iterative functional equation involving the $ (S, N) $-implication, where the $ (S, N) $-implication is a common class of fuzzy implications generated by a continuous $ t $-conorm $ S $ and a continuous fuzzy negation $ N $. Furthermore, given a fixed $ t $-norm (the minimum $ t $-norm or the product $ t $-norm) the Hyers-Ulam stability of the law of importation involving the $ (S, N) $-implication is studied.
- fuzzy implications,
- functional equations,
- iterative Boolean-like law,
- law of importation,
- stability,
- $ (S, N) $-implication
Citation: Sizhao Li, Xinyu Han, Dapeng Lang, Songsong Dai. On the stability of two functional equations for $ (S, N) $-implications[J]. AIMS Mathematics, 2021, 6(2): 1822-1832. doi: 10.3934/math.2021110

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Abstract

The iterative functional equation $ \alpha\rightarrow(\alpha\rightarrow \beta) = \alpha\rightarrow \beta $ and the law of importation $ (\alpha\wedge \beta)\rightarrow \gamma = \alpha\rightarrow (\beta\rightarrow \gamma) $ are two tautologies in classical logic. In fuzzy logics, they are two important properties, and are respectively formulated as $ I(\alpha, \beta) = I(\alpha, I(\alpha, \beta)) $ and $ I(T(\alpha, \beta), \gamma) = I(\alpha, I(\beta, \gamma)) $ where $ I $ is a fuzzy implication and $ T $ is a $ t $-norm. Over the past several years, solutions to these two functional equations involving different classes of fuzzy implications have been studied. However, there are no results about stability study of fuzzy functional equations involving fuzzy implication. This paper discusses fuzzy implications that do not strictly satisfying these equations, but approximately satisfy these equations. Then we establish the Hyers-Ulam stability of the iterative functional equation involving the $ (S, N) $-implication, where the $ (S, N) $-implication is a common class of fuzzy implications generated by a continuous $ t $-conorm $ S $ and a continuous fuzzy negation $ N $. Furthermore, given a fixed $ t $-norm (the minimum $ t $-norm or the product $ t $-norm) the Hyers-Ulam stability of the law of importation involving the $ (S, N) $-implication is studied.

References

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