On the partial delta differentiability of fuzzy-valued functions via the generalized Hukuhara difference

Abstract

In this paper, we propose the concept of partial delta derivatives of binary fuzzy-valued functions on time scales via the generalized Hukuhara difference. The first focus is on the definition and essential properties of the partial delta derivatives that are naturally investigated based on the limit of fuzzy-valued functions on time scales. Then, we consider the fuzzy transport equation on time scales as an application of the proposed derivatives. Some examples are provided to illustrate their solutions.

Appendix

This section provides the definition and fundamental properties of the generalized exponential function on time scales.

Definition 20

( see Bohner and Peterson 2001.) A function $f : {\mathbb {T}}\rightarrow {\mathbb {R}}$ is said to be right-dense continuous (rd-continuous), if the following statements hold:

i):

$\displaystyle \lim _{t\dashrightarrow t_o}^\varDelta f(t)=f(t_0)$, for any right dense point $t_0\in {\mathbb {T}}$,

ii):

$\displaystyle \lim _{t\dashrightarrow t_1^-}^\varDelta f(t)$ exists, for any left dense point $t_1\in {\mathbb {T}}$.

Definition 21

(see Bohner and Peterson 2001.) A function $f: {\mathbb {T}}\rightarrow {\mathbb {R}}$ is said to be regressive, if $1 + (\sigma (t)-t) f(t) \ne 0$, for all $t\in {\mathbb {T}}$.

The generalized exponential function is defined as solutions of initial valued problems on time scales.

Definition 22

(see Bohner and Peterson 2001.) Let $p:{\mathbb {T}}\rightarrow \mathbb {R}$ be a rd-continuous and regressive function, and $t\in {\mathbb {T}}$. The unique solution, say $u:{\mathbb {T}}\rightarrow \mathbb {R}$, of the problem,

$$\begin{aligned} {\left\{ \begin{array}{ll} u^{\varDelta }(s)= p(s) u(s), s \in {\mathbb {T}}, \\ u(t) =1. \end{array}\right. } \end{aligned}$$

(18)

is defined as a generalized exponential function. It is denoted by $e_p(s,t)$.

In Table 1, one can see the generalized exponential function in particular time scales.

Table 1 Exponential functions in particular time scales

Moreover, let us note that the generalized exponential function $e_p(s,t)$, defined in Definition 22, is a unary function with variable $s\in {\mathbb {T}}$. It depends on the given function p and a constant $t\in {\mathbb {T}}$. However, when dealing with t as a variable in ${\mathbb {T}}$, one can consider $e_p(s,t)$ as a binary function on ${\mathbb {T}}\times {\mathbb {T}}$.

In what follows, we recall fundamental properties of the generalized exponential functions that are considered as binary functions on time scales:

Lemma 29

(see Bohner and Peterson 2001.) Let $e_p(s,t)$ be the generalized exponential function on time scale ${\mathbb {T}}\times {\mathbb {T}}$. The following statements hold:

i):

$e_0(s,t) =1$ and $e_p(s,s) = 1$.

ii):

$e_p(\sigma (s),t) = (1 + \mu (s)p(s)) e_p(s,t).$

iii):

$e_p(s,t) =\frac{1}{e_p(t,s)}.$

iv):

$(e_p(s,t))^{\varDelta }=-p(t)e_p (s,\sigma (t))$.

The further results on exponential functions can be found in Bohner and Georgiev (2016), Bohner and Peterson (2001) and the references there in.