On Denjoy-McShane-Stieltjes Integral of Fuzzy-Number-Valued Functions

Abstract

In this paper, we introduce the concepts of the McShane-Stieltjes integral and the Denjoy-McShane-Stieltjes integral for fuzzy-number-valued functions and give a characterization of the McShane-Stieltjes integrability and investigate some properties of the Denjoy-McShane-Stieltjes integral.

1 Introduction

The integrals of fuzzy-valued functions have been discussed in recent papers. It is well known that the notion of the Stieltjes integral of fuzzy-number-valued functions was originally proposed by Nanda [3] in 1989. In 1998, Wu [4] proposed the concept of the fuzzy Riemann-Stieltjes integral by means of the representation theorem of fuzzy-number-valued functions, whose membership function could be obtained by solving a nonlinear programming problem, but it is difficult to calculate and extend to the higher dimensional space. In 2006, Ren et al. introduced the concept of two kinds of fuzzy Riemann-Stieltjes integral for fuzzy-number-valued functions [5] and showed that a continuous fuzzy-number-valued function was fuzzy fuzzy Riemann-Stieltjes integrable with respect to a real-valued increasing function. However, we note that if a fuzzy-number-valued function has some kind of discontinuity, the existing methods have been restricted. In real analysis, The Henstock integral is designed to integrate highly oscillatory functions which the Lebesgue integral fails to do. It is known as nonabsolute integration and is a powerful tool. It is well-known that the Henstock integral includes the Riemann, improper Riemann, Lebesgue and Newton integrals [6–8]. Though such an integral was defined by Denjoy in 1912 and also by Perron in 1914, it was difficult to handle using their definitions. But with the Riemann-type definition introduced more recently by Henstock in 1963 and also independently by Kurzweil, the definition is now simple and furthermore the proof involving the integral also turns out to be easy. Wu and Gong [9–11] have combined the fuzzy set theory and nonabsolute theory and discussed the fuzzy Henstock integrals of fuzzy-number-valued functions which is extended Kaleva integration.

In this paper, we introduce the concept of the fuzzy McShane-Stieltjes integral and fuzzy Denjoy-McShane-Stieltjes integral. And we give a characterization of the fuzzy McShane-Stieltjes integrability and investigate some properties of fuzzy Denjoy-McShane-Stieltjes integral.

2 Preliminaries

Let $P_{k} (R^{n} )$ denote the family of all nonempty compact convex subset of $R^{n}$ and define the addition and scalar multiplication in $P_{k} (R^{n} )$ as usual. Let $A$ and $B$ be two nonempty bounded subset of $R^{n}$. The distance between $A$ and $B$ is defined by the Hausdorff metric [2]:

$$ d_{H} (A,B) = \hbox{max} \{ \mathop {\sup }\limits_{a \in A} \mathop {\inf }\limits_{b \in B} |a - b|,\mathop {\sup }\limits_{b \in B} \mathop {\inf }\limits_{a \in A} |b - a|\} . $$

Denote $E^{n} = \{ u:R^{n} \to [0,1],u$ satisfies (1)−(4) below $\}$ is a fuzzy number space. Where

• (1) $u$ is normal, i.e. there exists an $x_{0} \in R^{n}$ such that $u(x_{0} ) = 1$;

• (2) $u$ is fuzzy convex, i.e. $u(\lambda x + (1 - \lambda )y) \ge \hbox{min} \{ u(x),u(y)\}$ for any $x,y \in R^{n}$ and $0 \le \lambda \le 1,$

• (3) $u$ is upper semi-continuous;

• (4) $[u]^{0} = cl\{ x \in R^{n} |u(x) > 0\}$ is compact.

Define $D:E^{n} \times E^{n} \to (0, + \infty )$

$$ D(u,v) = \sup \{ d_{H} ([u]^{\alpha } ,[v]^{\alpha } ):\alpha \in [0,1]\} , $$

where $d_{H}$ is the Hausdorff metric defined in $P_{k} (R^{n} )$. Then it is easy see that $D$ is a metric in $E^{n}$. Using the results [3], we know that the metric space $(E^{n} ,D)$ has a linear structure, it can imbedded isomorphically as a cone in a Banach space of function $u^{ * } :I \times S^{n - 1}$, where $S^{n - 1}$ is the unit sphere in $E^{n}$, which an imbedding function $u^{ * } = j(u)$ defined by $u^{ * } (r,x) = \mathop {\sup }\limits_{{\alpha \in [u]^{\alpha } }} \left\langle {\alpha ,x} \right\rangle$.

Since Hausdorff metric is a kind of stronger metric, much problems could not be characterized. It is well known, the supremum (infimum) is a main concept in analysis, and how to characterized the supremum (infimum) of fuzzy number is an important problem in fuzzy analysis. Refer to (see [2, 10]), if $\{ u_{n} \}$ is a bounded fuzzy number sequence, then it has supremum and infimum and if $u$ is supremum (infimum) of $\{ u_{n} \}$, $D(u_{n} ,u) \to 0$ is not correct generally. The integral metric between two fuzzy numbers by using support functions of fuzzy numbers is defined by Gong (see [12]),

$$ D^{ * } (u,v) = \sqrt {{\text{sup}}_{{x \in S^{n - 1} ,\left\| x \right\| = 1}} \int_{0}^{1} {(u^{ * } (r,x) - u^{ * } (r,x))^{2} dr} } $$

We easy see that $(E^{n} ,D^{ * } )$ is a metric space, and for each fuzzy number sequence $\{ u_{n} \} \subset E^{n}$ and fuzzy number $u \in E^{n}$, if $D(u_{n} ,u) \to 0$, then $D^{ * } (u_{n} ,u) \to 0$. The converse result does not hold.

Definition 2.1 [14].

A fuzzy number valued function $f$ has H-difference property, i.e., for any $t_{1} ,t_{2} \in [a,b]$ satisfying $t_{1} < t_{2}$ there exist a fuzzy number $u \in E^{n}$ such that $f(t_{2} ) = f(t_{1} ) + u$, $u$ is called H-difference of $f(t_{1} )$ and $f(t_{2} )$, we denote $f(t_{2} ) -_{H} f(t_{1} ) = u$.

Definition 2.2 [12].

A McShane partition of $\left[ {a,b} \right]$ is a finite collection $P = \{ ([c_{i} ,d_{i} ] ,t_{i} ):1 \le i \le n\}$ such that $\{ [c_{i} ,d_{i} ]:1 \le i \le 2\}$ is a non-overlapping family of subintervals of $\left[ {a,b} \right]$ covering $\left[ {a,b} \right]$ and $t_{i} \in \left[ {a,b} \right]$ for each $i \le n$. A gauge on $\left[ {a,b} \right]$ is a positive real valued function $\delta :\left[ {a,b} \right] \to \left( {0,\infty } \right)$. A McShane partition $P = \{ ([c_{i} ,d_{i} ] ,t_{i} ):1 \le i \le n\}$ is subordinate to a gauge $\delta$ if $\left[ {c_{i} ,d_{i} } \right] \subset \left( {t_{i} - \delta \left( {t_{i} } \right),t_{i} + \delta \left( {t_{i} } \right)} \right)$ for every $i \le n$. If $f:\left[ {a,b} \right] \to {\text{E}}$ and if $P = \{ ([c_{i} ,d_{i} ] ,t_{i} ):1 \le i \le n\}$ is a McShane partition of $\left[ {a,b} \right]$, we will denote $f\left( P \right)$ for $\sum\nolimits_{i = 1}^{n} {f(\mathop t\nolimits_{i} )} (\mathop d\nolimits_{i} - \mathop c\nolimits_{i} )$.

Definition 2.3 [13].

A fuzzy number valued function $f:\left[ {a,b} \right] \to {\text{E}}$ is McShane integrable on $\left[ {a,b} \right]$, with a fuzzy number $A$, if for each $\varepsilon > 0$ there exists a gauge $\delta :\left[ {a,b} \right] \to \left( {0,\infty } \right)$ such that $D\left( {f\left( P \right),A} \right) < \varepsilon$ whenever $P = \{ ([c_{i} ,d_{i} ] ,t_{i} ):1 \le i \le n\}$ is a McShane partition of $\left[ {a,b} \right]$ subordinate to $\delta$.

Definition 2.4.

Let $F:\left[ {a,b} \right] \to E$ be a fuzzy-number-valued function and let $t \in (a,b)$. A fuzzy number $A$ in $E$ is the approximate derivative of $F$ at $t$ if there exists a measurable set ${\text{S}} \subset \left[ {a,b} \right]$ that has $t$ as a point of density such that $\mathop {\lim }\limits_{\begin{subarray}{l} s \to t \\ s \in E \end{subarray} } \frac{F(s) - F(t)}{s - t} = z$.we will write $\mathop F\nolimits_{ap}^{'} \left( t \right) = A$

Definition 2.5.

A fuzzy number valued function $f:\left[ {a,b} \right] \to E$ is fuzzy Denjoy integrable on $\left[ {a,b} \right]$ if there exists an $ACG$ function $F:\left[ {a,b} \right] \to E$ such that $\mathop F\nolimits_{ap}^{'} = f$ almost everywhere on $\left[ {a,b} \right]$.

Definition 2.6.

Let a fuzzy number valued function $F:\left[ {a,b} \right] \to E$ and let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function and let $E \subset \left[ {a,b} \right]$.

• (a) The fuzzy-number-valued function $F$ is $BV$ with respect to $\alpha$ on $E$ if $V\left( {F,\alpha ,E} \right) = \sup \left\{ {\sum\limits_{i = 1}^{n} {D(F(d_{i} ),F\left( {c_{i} } \right))} \frac{{\alpha \left( {d_{i} } \right) - \alpha \left( {c_{i} } \right)}}{{d_{i} - c_{i} }}} \right\}$ is finite where the supremum is taken over all finite collections $\{ [c_{i} ,d_{i} ]:1 \le i \le n\}$ of non-overlapping intervals that have endpoints in $E$.

• (b) The fuzzy valued function $F$ is $AC$ with respect to $\alpha$ on $X$ if for each $\varepsilon > 0$ there exists $\delta > 0$ such that $\sum\limits_{i = 1}^{n} {D(F\left( {d_{i} } \right)} ,F\left( {c_{i} } \right)) < \varepsilon$ whenever $\{ [c_{i} ,d_{i} ]:1 \le i \le n\}$ is a finite collection of non-overlapping intervals that have endpoints in $E$ and satisfy $\sum\limits_{i = 1}^{n} {\left[ {\alpha \left( {d_{i} } \right) - \alpha \left( {c_{i} } \right)} \right]} < \delta$.

• (c) The fuzzy-number-valued function $F$ is $BVG$ with respect to $\alpha$ on $E$ if $E$ can be expressed as a countable union of sets on each of which $F$ is $BVG$ with respect to $\alpha$.

• (d) The fuzzy-number-valued function $F$ is $ACG$ with respect to α on $E$ if $F$ is continuous on $E$ and if $E$ can be expressed as a countable union of sets on each of which $F$ is $AC$ with respect to $\alpha$.

Definition 2.7.

Let $F:\left[ {a,b} \right] \to E,_{{}} t \in (a,b)$ and let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$. A fuzzy number $A \in E$ is the approximate derivative of $F$ with respect to $\alpha$ at $t$ if there exists a measurable set $E \subset \left[ {a,b} \right]$ that has $t$ as a point of density such that $\mathop {\lim }\limits_{\begin{subarray}{l} s \to t \\ s \in E \end{subarray} } \frac{F\left( s \right) - F\left( t \right)}{\alpha \left( s \right) - \alpha \left( t \right)} = z .$ we will write $F^{\prime}_{\alpha ,\alpha \rho } \left( t \right) = z$

Definition 2.8.

A fuzzy-number-valued function $f:\left[ {a,b} \right] \to E$ is Denjoy-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$ if there exists an $ACG$ function $F:\left[ {a,b} \right] \to E$ with respect to $\alpha$ such that $F^{\prime}_{a,ap} = f$ almost everywhere on $\left[ {a,b} \right]$.

3 McShane-Stieltjes Integral of Fuzzy-Number-Valued Functions

In this section we introduce the concept of the fuzzy McShane-Stieltjes integral and give a characterization of the fuzzy McShane-Stieltjes integrability.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be an increasing function. If a fuzzy number valued function $f:\left[ {a,b} \right] \to E$ and if $P = \{ ([c_{i} ,d_{i} ] ,t_{i} ):1 \le i \le n\}$ is a McShane partition of $\left[ {a,b} \right]$, we will denote $f_{\alpha } \left( P \right)$ for $\sum\limits_{i = 1}^{n} {f(\mathop t\nolimits_{i} )\left[ {\alpha (\mathop d\nolimits_{i} ) - \alpha (\mathop c\nolimits_{i} )} \right]}$.

Definition 3.1.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be an increasing function. A fuzzy-number -valued function $f:\left[ {a,b} \right] \to E$ is fuzzy McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$, with fuzzy McShane-Stieltjes integral $A$, if for each $\varepsilon > 0$ there exists a gauge $\delta :\left[ {a,b} \right] \to \left( {0,\infty } \right)$ such that $D(f_{\alpha } (p),A) < \varepsilon$ whenever $P = \{ ([c_{i} ,d_{i} ] ,t_{i} ):1 \le i \le n\}$ is a McShane partition of $\left[ {a,b} \right]$ subordinate to $\delta$.

Theorem 3.2.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$ and let $f:\left[ {a,b} \right] \to E$ be a bounded function. Then $f$ is fuzzy McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$ if and only if $\alpha 'f$ is McShane integrable on $\left[ {a,b} \right]$.

Proof.

Since fuzzy number valued function $f:\left[ {a,b} \right] \to E$ is a bounded function, there exists $M > 0$ such that $D(f(x),\tilde{0}) \le M$ for all $x \in \left[ {a,b} \right]$. Continuity of $\alpha '$ on $\left[ {a,b} \right]$ implies uniform continuity on $\left[ {a,b} \right]$. Hence for each $\varepsilon > 0$ there exists $\eta > 0$ such that for all $x,y \in \left[ {a,b} \right]$ and $\left| {x - y} \right| < \eta$, we have $\left| {\alpha '\left( x \right) - \alpha '\left( y \right)} \right| < \frac{\varepsilon }{{3M\left( {b - a} \right)}}$.

Choose a gauge $\mathop \delta \nolimits_{1}$ on $\left[ {a,b} \right]$ with $\mathop \delta \nolimits_{1} (x) < \eta$ for all $x \in \left[ {a,b} \right]$ Let $P = \{ ([\mathop c\nolimits_{i} ,\mathop d\nolimits_{i} ],\mathop t\nolimits_{i} ):1 \le i \le n\}$ be a McShane partition of $\left[ {a,b} \right]$ subordinate to $\mathop \delta \nolimits_{1}$. Then by the Mean Value Theorem, there exists $\mathop x\nolimits_{i} \in (\mathop c\nolimits_{i} ,\mathop d\nolimits_{i} )$ such that $\alpha \left( {d_{i} } \right) - \alpha \left( {c_{i} } \right) =$ $\alpha '(\mathop x\nolimits_{i} )(\mathop d\nolimits_{i} - \mathop c\nolimits_{i} )$ for $1 \le i \le n$ since $|\mathop t\nolimits_{i} - \mathop x\nolimits_{i} | < \mathop \delta \nolimits_{i} (\mathop t\nolimits_{i} ) < \eta$ for all $1 \le i \le n$,$|\alpha '(\mathop t\nolimits_{i} ) - \alpha '(\mathop x\nolimits_{i} )| <$ $\frac{\varepsilon }{{3M\left( {b - a} \right)}}$ for $1 \le i \le n$. Hence we have

$$ \begin{aligned} & D(f_{\alpha } \left( P \right),\left( {\alpha 'f} \right)\left( P \right)) \\ & = D(\sum\limits_{i = 1}^{n} {f\left( {t_{i} } \right)\left[ {\alpha \left( {d_{i} } \right) - \alpha \left( {c_{i} } \right)} \right],\sum\limits_{i = 1}^{n} {\alpha '\left( {t_{i} } \right)f\left( {t_{i} } \right)\left( {d_{i} - c_{i} } \right)} } ) \\ & = D(\sum\limits_{i = 1}^{n} {f\left( {t_{i} } \right)\left[ {\alpha '\left( {x_{i} } \right) - \alpha '\left( {t_{i} } \right)} \right]\left( {d_{i} - c_{i} } \right)} ,\tilde{0}) \\ & = \frac{\varepsilon }{3} \\ \end{aligned} $$

Whenever $P = \{ ([\mathop c\nolimits_{i} ,\mathop d\nolimits_{i} ],\mathop t\nolimits_{i} ):1 \le i \le n\}$ is a McShane partition of $\left[ {a,b} \right]$ subordinate to $\mathop \delta \nolimits_{1}$.

If $f$ is fuzzy McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$, then there exists a gauge $\mathop \delta \nolimits_{2}$ on $\left[ {a,b} \right]$ such that

$$ D(f_{\alpha } \left( {P_{1} } \right),f_{\alpha } \left( {P_{2} } \right)) < {\varepsilon \mathord{\left/ {\vphantom {\varepsilon 3}} \right. \kern-0pt} 3} $$

Whenever $\mathop P\nolimits_{1}$ and $\mathop P\nolimits_{2}$ are McShane partitions of $\left[ {a,b} \right]$ subordinate to $\mathop \delta \nolimits_{2}$. Define $\delta$ on [a,b] by ${\delta (x) = min}\left\{ {\updelta_{1} (\text{x}),\,\updelta_{\text{2}} (\text{x})} \right\}$ for $x \in \left[ {a,b} \right]$. Then $\delta$ is a gauge on $\left[ {a,b} \right]$ and

$$ \begin{aligned} D(\left( {\alpha 'f} \right)\left( {P_{1} } \right), & \left( {\alpha 'f} \right)\left( {P_{2} } \right)) < D(\left( {\alpha 'f} \right)\left( {P_{1} } \right),f_{\alpha } \left( {P_{1} } \right)) \\ \, & \, + D(f_{\alpha } \left( {P_{1} } \right),f_{\alpha } \left( {P_{2} } \right)) \\ \, & + D(f_{\alpha } \left( {P_{2} } \right),\left( {\alpha 'f} \right)\left( {P_{2} } \right)) < \varepsilon \\ \, \\ \end{aligned} $$

Whenever $\mathop P\nolimits_{1}$ and $\mathop P\nolimits_{2}$ are McShane partitions of $\left[ {a,b} \right]$ subordinate to $\delta$. Hence $\alpha 'f$ is fuzzy McShane integrable on $\left[ {a,b} \right]$.

Conversely, if $\alpha 'f$ is fuzzy McShane integrable on $\left[ {a,b} \right]$, then for each $\varepsilon > 0$ there exists a gauge $\mathop \delta \nolimits_{3}$ on $\left[ {a,b} \right]$ such that $D((\alpha 'f)(\mathop P\nolimits_{1} ),(\alpha 'f)(\mathop P\nolimits_{2} ))$ $< {\varepsilon \mathord{\left/ {\vphantom {\varepsilon 3}} \right. \kern-0pt} 3}$ whenever $\mathop P\nolimits_{1}$ and $\mathop P\nolimits_{2}$ are McShane partitions of $\left[ {a,b} \right]$ subordinate to $\mathop \delta \nolimits_{3}$. Define $\delta$ on $\left[ {a,b} \right]$ by $\delta (x) = \hbox{min} \{ \mathop \delta \nolimits_{1} (x),\mathop \delta \nolimits_{3} (x)\}$ for $x \in \left[ {a,b} \right]$. Then $\delta$ is a gauge on $\left[ {a,b} \right]$ and

$$ \begin{aligned} D(f_{\alpha } \left( {P_{1} } \right) & ,f_{\alpha } \left( {P_{2} } \right)) \le D(f_{\alpha } \left( {P_{1} } \right),\left( {\alpha 'f} \right)\left( {P_{1} } \right)) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & + D(\left( {\alpha 'f} \right)\left( {P_{1} } \right),\left( {\alpha 'f} \right)\left( {P_{2} } \right)) \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, & + D(\left( {\alpha 'f} \right)\left( {P_{2} } \right),f_{\alpha } \left( {P_{2} } \right)) < \varepsilon \\ \end{aligned} $$

Whenever $\mathop P\nolimits_{1}$ and $\mathop P\nolimits_{2}$ are McShane partitions of $\left[ {a,b} \right]$ subordinate to $\delta$. Hence $f$ is fuzzy McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

Theorem 3.3.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be an increasing function. A fuzzy number valued function $f:\left[ {a,b} \right] \to E$ is fuzzy McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$, if and only if for each $\varepsilon > 0$ there exists a gauge $\delta :\left[ {a,b} \right] \to \left( {0,\infty } \right)$ such that $D(\mathop f\nolimits_{\alpha } (\mathop p\nolimits_{1} ),\mathop f\nolimits_{\alpha } (\mathop p\nolimits_{2} )) < \varepsilon$ whenever $\mathop P\nolimits_{1}$ and $\mathop P\nolimits_{2}$ are McShane partitions of $\left[ {a,b} \right]$ subordinate to $\delta$.

4 Denjoy-McShane-Stieltjes Integral of Fuzzy-Number-Valued Functions

In this section we introduce the concept of the fuzzy Denjoy-McShane-Stieltjes integral and investigate some properties of this integral.

Definition 4.1.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$. A fuzzy-number-valued function $f:\left[ {a,b} \right] \to E$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$ if there exists a continuous function $F:\left[ {a,b} \right] \to F$ such that $F$ is $ACG$ with respect to $\alpha$ on $\left[ {a,b} \right]$ and is approximately differentiable with respect to $\alpha$ almost everywhere on $\left[ {a,b} \right]$ and $(F)_{\alpha ,ap}^{'} = f$ almost everywhere on $\left[ {a,b} \right]$.

Theorem 4.2.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$. Then a fuzzy-number-valued function $f:\left[ {a,b} \right] \to E$ is Denjoy-McShane- Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$ if and only if $\alpha 'f$ is Denjoy- McShane integrable on $\left[ {a,b} \right]$.

The following three corollaries are obtained from Theorem 4.2.

Corollary 4.3.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$ and let $f:\left[ {a,b} \right] \to E$ be a fuzzy-number-valued function. If $\alpha 'f$ is fuzzy McShane integrable on $\left[ {a,b} \right]$, then $f$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

Corollary 4.4.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$ and let $f:\left[ {a,b} \right] \to E$ be a fuzzy-number-valued function. If $\alpha 'f$ is Denjoy-Bochner integrable on $\left[ {a,b} \right]$, then $f$ is Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

Corollary 4.5.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$ and let $f:\left[ {a,b} \right] \to E$ If $f$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$ then $\alpha 'f$ is fuzzy Denjoy-Pettis integrable on $\left[ {a,b} \right]$.

Theorem 4.6.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$. If $f:\left[ {a,b} \right] \to E$ is a bounded fuzzy McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$, then $f$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

Proof.

If $f:\left[ {a,b} \right] \to E$ is a bounded fuzzy McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$, then by Theorem 3.2 $\alpha 'f$ is fuzzy McShane integrable on $\left[ {a,b} \right]$. Then we have $\alpha 'f$ is fuzzy Denjoy-McShane integrable on $\left[ {a,b} \right]$. By Theorem 4.2, $f$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

Definition 4.7.

A fuzzy-number-valued function $f:\left[ {a,b} \right] \to E$ is Denjoy-Stieltjes-Bochner integrable with respect to $\alpha$ on $\left[ {a,b} \right]$ if there exists an $ACG$ function $F:\left[ {a,b} \right] \to E$ with respect to $\alpha$ such that $F$ is approximately differentiable with respect to $\alpha$ almost everywhere on $\left[ {a,b} \right]$ and $F^{\prime}_{a,ap} = f$ almost everywhere on $\left[ {a,b} \right]$.

Theorem 4.8.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$. If $f:\left[ {a,b} \right] \to E$ is fuzzy Denjoy-Stieltjes-Bochner integrable with respect to $\alpha$ on $\left[ {a,b} \right]$, then $f:\left[ {a,b} \right] \to E$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

Proof.

If $f:\left[ {a,b} \right] \to E$ is Denjoy-Stieltes-Bocner integrable with respect to $\alpha$ on $\left[ {a,b} \right]$, then there exists an $ACG$ function $F:\left[ {a,b} \right] \to E$ with respect to $\alpha$ such that $F$ is approximately differentiable with respect to $\alpha$ almost everywhere on $\left[ {a,b} \right]$ and $F^{\prime}_{a,ap} = f$ almost everywhere on $\left[ {a,b} \right]$. It is easy to show that for each $j \circ F$ is $ACG$ with respect to $\alpha$ on $\left[ {a,b} \right]$ and $j \circ F$ is approximately differentiable with respect to $\alpha$ almost everywhere on $\left[ {a,b} \right]$ and $\left( {j \circ F} \right)_{ap}^{'} = j \circ f$ almost everywhere on $\left[ {a,b} \right]$. Hence $f$ is Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

Theorem 4.9.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$. If $f:\left[ {a,b} \right] \to E$ is Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$, then $f:\left[ {a,b} \right] \to E$ is Denjoy-Stieltjes-Pettis integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

Proof.

Suppose that $f:\left[ {a,b} \right] \to E$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$. Let $F\left( t \right) = \left( {DMS} \right)\int_{a}^{t} {fd\alpha }$ Since $j \circ F$ is $ACG$ with respect to $\alpha$ on $\left[ {a,b} \right]$ and $\left( {j \circ F} \right)_{ap}^{'} = j \circ f$ almost everywhere on [a,b]. $j \circ f$ is Denjoy-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

For every interval $\left[ {c,d} \right]$ in $\left[ {a,b} \right]$ we have

$$ \begin{aligned} j \circ \left( {F\left( d \right) - F\left( c \right)} \right) & = j \circ F\left( d \right) - j \circ F\left( c \right) \\ & \, = \left( {DS} \right)\int_{a}^{b} {j \circ } fd\alpha - \left( {DS} \right)\int_{c}^{d} {j \circ } fd\alpha \\ & = \left( {DS} \right)\int_{c}^{d} {j \circ fd\alpha } \\ \end{aligned} $$

Since $F\left( d \right) - F\left( c \right) \in E$, $f$ is fuzzy Denjoy-Stieltjes-Pettis integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

Theorem 4.10.

Let $\alpha :\left[ {a,b} \right] \to \Re$ be a strictly increasing function such that $\alpha \in C^{1} \left( {\left[ {a,b} \right]} \right)$ If fuzzy number valued function $f:\left[ {a,b} \right] \to E$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$ and $T:E \to E$ is a fuzzy bounded linear operator, then $T \circ f:\left[ {a,b} \right] \to E$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.

Proof.

If fuzzy-number-valued function $f:\left[ {a,b} \right] \to E$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$, then there exists a continuous fuzzy valued function $F:\left[ {a,b} \right] \to E$ such that

• (i) for each $j \circ F$ is with respect to $\alpha$ on $\left[ {a,b} \right]$ and

• (ii) for each $j \circ F$ is approximately differentiable with respect to $\alpha$ almost everywhere on $\left[ {a,b} \right]$ and $\left( {j \circ F} \right)_{ap}^{'} = j \circ f$ almost everywhere on $\left[ {a,b} \right]$.

Let $G = T \circ F$. Then $G:\left[ {a,b} \right] \to Y$ is a continuous function such that

• (i) for each $j \circ G$ $= j \circ \left( {T \circ F} \right)$ $= \left( {j \circ T} \right)F$ is $ACG$ with respect to $\alpha$ on $\left[ {a,b} \right]$ and

• (ii) for each $j \circ G$ $= j \circ \left( {T \circ F} \right)$ $= \left( {j \circ T} \right)F$ is approximately differentiable which respect to $\alpha$ almost everywhere on $\left[ {a,b} \right]$ and

  $$ \left( {j \circ G} \right)_{\alpha ,ap}^{'} = \left( {\left( {j \circ T} \right)F} \right)_{\alpha 'ap}^{'} = \left( {j \circ T} \right)f = j \circ \left( {T \circ f} \right) . $$

Hence $T \circ f:\left[ {a,b} \right] \to E$ is fuzzy Denjoy-McShane-Stieltjes integrable with respect to $\alpha$ on $\left[ {a,b} \right]$.