New metric-based derivatives for fuzzy functions and some of their properties

Khastan, A.; Rodríguez‐López, Rosana; Shahidi, M.

doi:10.1016/j.fss.2021.09.007

Cited by 10 publications

(3 citation statements)

References 13 publications

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“…A similar derivative was later considered in [6,14,15,22,27,28,36]. The examples can be constructed where the Huygens derivative exists, but the Hukuhara derivative does not exist (see [6,36]).…”

Section: Lemma 1 ([38]mentioning

confidence: 99%

Existence of basic solutions of first order linear homogeneous set-valued differential equations

Plotnikov,

Komleva,

Skripnik

2024

Mat. Stud.

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The paper presents various derivatives of set-valued mappings,their main properties and how they are related to each other.Next, we consider Cauchy problems with linear homogeneousset-valued differential equations with different types ofderivatives (Hukuhara derivative, PS-derivative andBG-derivative). It is known that such initial value problems withPS-derivative and BG-derivative have infinitely many solutions.Two of these solutions are called basic. These are solutions suchthat the diameter function of the solution section is amonotonically increasing (the first basic solution) or monotonicallydecreasing (the second basic solution) function. However, the secondbasic solution does not always exist. We provideconditions for the existence of basic solutions of such initialvalue problems. It is shown that their existence depends on thetype of derivative, the matrix of coefficients on the right-handand the type of the initial set. Model examples are considered.

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Section: Lemma 1 ([38]mentioning

confidence: 99%

Existence of basic solutions of first order linear homogeneous set-valued differential equations

Plotnikov,

Komleva,

Skripnik

2024

Mat. Stud.

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“…This concept was further developed to solve the Cauchy problem for fuzzy as well as intuitionistic fuzzy differential equations [42,43] and set valued functions [44]. Khastan et al [45] introduced new metric-based derivatives and compared these with other existing metric-based derivatives. L r -norm-based derivatives were introduced by Caldern and Zygmund in 1961 [46] in order to solve elliptic partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

On Extended Lr-Norm-Based Derivatives to Intuitionistic Fuzzy Sets

Wungreiphi,

Mazarbhuiya,

Shenify

2023

Mathematics

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The study of differential equation theory has come a long way, with applications in various fields. In 1961, Zygmund and Calderón introduced the notion of derivatives to metric Lr, which proved to be better in applications than approximate derivatives. However, most of the studies available are on Fuzzy Set Theory. In view of this, intuitionistic fuzzy Lr-norm-based derivatives deserve study. In this study, the Lr-norm-based derivative for intuitionistic fuzzy number valued functions is introduced. Some of its basic properties are also discussed, along with numerical examples. The results obtained show that the proposed derivative is not dependent on the existence of the Hukuhara difference. Lastly, the Cauchy problem for the intuitionistic fuzzy differential equation is discussed.

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“…Khastan et. al [45] introduced a new metric -based derivatives and compared with other existing metric -based derivatives. 𝐿 𝑟 -norm -based derivatives were introduced by Calderón and Zygmund in 1961 [46] in order to solve elliptic partial differential equations.…”

Section: Introductionmentioning

confidence: 99%

On Extending Lr-Norm Based Derivatives to Intuitionistic Fuzzy Set

Wungreiphi,

Mazarbhuiya,

Shenify

2023

Preprint

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Study of differential equation theory has come a long way with applications in the various fields. In 1961, Zygmund and Calderón introduced the notion of derivatives on metric Lr which proved to be better in applications than approximate derivatives. But most of the studies involved are on fuzzy set theory, so it seems likely that intuitionistic fuzzy Lr - norm - based derivatives deserve study. In the study, the fuzzy derivative is extended to intuitionistic fuzzy derivatives with respect.to Lr - norm - based derivatives using intuitionistic fuzzy number valued functions. The efficacy of the proposed method is established by proving the associated theorems along with numerical examples. Further, the Cauchy problem is also studied with respect to intuitionistic fuzzy setting.

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New metric-based derivatives for fuzzy functions and some of their properties

Cited by 10 publications

References 13 publications

Existence of basic solutions of first order linear homogeneous set-valued differential equations

Existence of basic solutions of first order linear homogeneous set-valued differential equations

On Extended Lr-Norm-Based Derivatives to Intuitionistic Fuzzy Sets

On Extending Lr-Norm Based Derivatives to Intuitionistic Fuzzy Set

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