Fractional calculus of linear correlated fuzzy-valued functions related to Fréchet differentiability

Abstract

In this paper, we will introduce some types of Fréchet fractional derivative defined on the class of linear correlated fuzzy-valued functions. Firstly, we study Fréchet derivative and R-derivative of integer order and investigate their relationship with the well-known generalized Hukuhara derivatives in fuzzy metric space. Secondly, the Riemann-Liouville fractional integral of linear correlated fuzzy-valued functions is well-defined via an isomorphism between ${\mathbb{R}}^2$ and subspace of fuzzy numbers space ${\mathbb{R}}_{\mathcal{F}}$. That allows us to introduce three types of Fréchet fractional derivatives, which are Fréchet Caputo derivative, Fréchet Riemann-Liouville derivative and Fréchet Caputo-Fabrizio derivative. Moreover, some common properties of fuzzy Laplace transform for linear correlated fuzzy-valued function are investigated. Finally, some applications to fuzzy fractional differential equations are presented to demonstrate the usefulness of theoretical results.

Introduction

Fuzzy fractional differential equations (FFDEs) were first introduced by Agarwal et al. [2] ignoring the role of fuzzy derivative. In other words, they incorporated fuzzy uncertainty into dynamical systems to introduce a new class of FFDEs. Then, they presented the fuzzy solution in one special case of nonlinearity but no fuzzy derivative notion was introduced. However, the development of FFDEs has strongly been associated with the appearance of different types of fuzzy derivatives. FFDEs based on Riemann-Liouville Hukuhara differentiability was investigated by Allahviranloo et al. in [1]. Which is a direct generalization of the fractional Riemann-Liouville derivative using the Hukuhara difference. Mazandarani and Kamyad established the Caputo type fuzzy fractional derivative for FFDEs in [21]. However, fuzzy derivatives based on Hukuhara difference have some drawbacks due to the increasing of diameter of uncertainty in time. That is why the generalizations of Hukuhara derivative are recognized as the active research direction and it forms a significant development. Follow up the generalizations of Bede and Gal [6] on the differentiability of fuzzy number valued functions, the generalized Hukuhara derivatives of interval valued functions and fuzzy-valued functions were introduced in the references [8], [32], [33]. The generalized Caputo fractional derivative, Riemann-Liouville fractional derivative and Caputo-Katugampola fractional derivative were introduced with various applications in [14], [16], [21]. Recently, new notion of granular fuzzy fractional derivatives with application to FFDEs was established by Najariyan and Zhao [23]. The granular fuzzy fractional derivatives based on horizontal membership functions and granular difference have several advantages over previous derivatives since it can be calculated directly via inverse transformation of level sets, see [22], [23]. An extension application of granular fuzzy fractional derivatives to linear quadratic regulator problem governed by neutrosophic fractional partial differential equations was elaborated in [28], [29], [30], [31]. [9], [10] investigated the possibility of interactivity between fuzzy phenomena via joint possibility distribution. For additional applications of interactive fuzzy processes in fuzzy differential equations of integer order, we refer the readers to [5], [24], [34].

Recently, Esmi et al. [12] has introduced the concept of Fréchet derivative for linear correlated fuzzy-valued functions. They also presented a practical method to calculate the Fréchet derivative via appropriate standard functions when the fuzzy processes are autocorrected. Pedro et al. [25] employed Fréchet derivative and Riemann integral to study the integration and derivative theory for interactive fuzzy processes. They established some fundamental calculus for further studying fuzzy differential equations. Let us note that, [12], [25] considered Fréchet derivative to the integer order. This motivates us to extensive study Fréchet derivative to fractional calculus and investigate the use of this concept on some classes of FFDEs.

In this paper, we present some new concepts of Fréchet derivative in both integer order and fractional order cases. For the integer order case, we define the classical R-differentiability in Definition 2.4. The relationship of Fréchet derivative, R-derivative with generalized Hukuhara derivatives is distinguished in Lemma 2.2, Lemma 2.3 and Remark 2.2. For the fractional derivatives of linear correlated fuzzy-valued functions, we introduce the definition of Fréchet Caputo derivative, Fréchet Riemann-Liouville derivative in Definition 3.3 and Fréchet Caputo-Fabrizio derivative in Definition 3.4 with some illustrated examples. The proposed Fréchet fractional derivatives will be the foundation for many applications of fractional calculus operating on uncertainty environment. To demonstrate the use of theoretical concepts, we act on two fuzzy fractional dynamical systems under the Fréchet fractional differentiability in Section 4. In which, the contraction principle in complete metric space and the upper, lower solution method in partially ordered set are all employed to prove the existence of the unique fuzzy solution.