Skip to main content

Generalized fuzzy GV-Hausdorff distance in GFGV-fractal spaces with application in integral equation


We propose a method for constructing a generalized fuzzy Hausdorff distance on the set of the nonempty compact subsets of a given generalized fuzzy metric space in the sence of George–Veeramni and Tian–Ha–Tian. Next, we define the generalized fuzzy fractal spaces. Morever, we obtain a fixed point theorem of a class of generalized fuzzy contractions and present an application in integranl equation.


George and Veeramani [1, 2] have rectified in absorbing manner the structure of Menger space and introduced a first countable Hausdorff topology on it, which is very popular in contemporary research.

We present and discuss an appropriate concept for the generalized fuzzy Hausdorff distance of a given generalized fuzzy GV-metric space on the set of its nonempty compact subsets. As an application, we use the concept of contraction on a generalized fuzzy GV-metric space to define a new concept of the generalized fuzzy fractal spaces and prove an interesting fixed point theorem in these spaces. In this paper, we present some results to extend and get uncertain models of recent results discused in [35] and [6].

Basic notions and preliminaries

In this paper, we denote \({\mathbb{I}}=[0,1]\), \({\mathbb{I}}^{\circ }=(0,1)\), \({\mathbb{J}}=[0,\infty )\), and \({\mathbb{J}}^{\circ }=(0,\infty )\).

Definition 2.1


A mapping \(\ast :{\mathbb{I}}^{2}\rightarrow {\mathbb{I}}\) is called a continuous t-norm (CTN) if

(i) is associative and commutative;

(ii) is continuous;

(iii) \(\jmath \ast 1=\jmath \) for all \(\jmath \in {\mathbb{I}}\);

(iv) \(\jmath \ast \imath \leq \jmath ' \ast \imath '\) whenever \(\jmath \leq \jmath '\) and \(\imath \leq \imath '\), \(\imath ,\jmath ,\imath ',\jmath ' \in {\mathbb{I}}\).

Some examples of CTNs are \(\imath {\ast _{P}}\jmath = \imath \cdot \jmath \) and \(\imath {\ast _{M}}\jmath = \min \{ {\imath ,\jmath } \} \). Note that \(\imath \ast \jmath \le \imath {\ast _{M}}\jmath \) for \(\imath ,\jmath \in {\mathbb{I}}\).

Assume that for every \(\text{\ss} \in {\mathbb{I}}^{\circ }\), there exists \(l \in {\mathbb{I}}^{\circ }\) (which does not depend on ) such that the following inequality holds:

$$\begin{aligned} \overbrace{(1-l )\ast \cdots \ast (1-l )}^{\ell }> 1-\text{\ss} \quad \text{for each } \ell \in \{2,3,\ldots \}. \end{aligned}$$

In this case, we say the CTN has the (D) property (CTND); for example, \(\ast _{M}\) is CTND. Now we consider a generalized fuzzy metric space in the George–Veeramani sense of (GFGVM-space); for details and results on fuzzy metric spaces introduced by George and Veeramani, we refer to [2, 1013], and [14].

Definition 2.2


The triple \((S,G,\ast )\) is a GFGVM-space if \(S\neq \emptyset \), is a CTN, and G is a fuzzy set on \(S \times S \times S \times {\mathbb{J}}^{\circ }\) such that for all \(w,x,y,z \in S\) and σ, δ in \({\mathbb{J}}^{\circ }\), we have:

(GGVFM-1) \(G_{x,y,z}(\sigma )\in {\mathbb{J}}^{\circ }\);

(GGVFM-2) \(G_{x,y,z}(\sigma )=1\) iff \(x=y=z\);

(GGVFM-3) \(G_{x,y,y}(\sigma )\ge G_{x,y,z}(\sigma ) \) for \(x \ne y\);

(GGVFM-4) \(G_{x,y,z}(\sigma )=G_{x,z,y}(\sigma )=G_{y,x,z}(\sigma )=\cdots \) ;

(GGVFM-5) \(G_{x,y,z}(\cdot):{\mathbb{J}}^{\circ }\to {\mathbb{I}}^{\circ }\) is continuous;

(GGVFM-6) \(G_{x,y,w}(\sigma +\delta )\ge G_{x,z,z}(\sigma ) \ast G_{z,y,w}( \delta )\).

Tian et al. [15] proved that in a GFGVM-space \((S,G,\ast )\), we have that \(G_{x,y,y}(\cdot )\) is increasing for all \(x,y \in S\). For \(\varepsilon \in {\mathbb{I}}^{\circ }\), \(\alpha \in {\mathbb{J}}^{\circ }\), and \(x_{0} \in S\), the set \(B_{G}(x_{0},\varepsilon ,\alpha )=\{y\in S:G_{x_{0},y,y}(\alpha )>1- \varepsilon ,G_{x_{0},x_{0},y}(\alpha )>1-\varepsilon \}\) is called the neighborhood with center \(x_{0}\) and radius ε. Consider

$$ \tau = \bigl\{ X \subset S: \forall x \in X, \exists \alpha \in { \mathbb{J}}^{\circ }, \varepsilon \in {\mathbb{I}}^{\circ }\text{ such that } {B_{G}} ( {x,\varepsilon ,\alpha } ) \subset X \bigr\} . $$

Then τ is the topology induced by GFGVM G on S. Moreover, the local base \(\{ B_{G}(u,\frac{1}{n},\frac{1}{n}):n=1,2,\ldots \}\) at u leads to the first countability of τ. Also, they proved that every GFGVM-space is Hausdorff.

Now we consider Cauchy sequences and completeness in GFGVM-spaces \((S,G,\ast )\) (see [15]).

(1) A sequence \(\{a_{n}\}\) in S converges to a point \(a \in S\) if for any \(\alpha \in {\mathbb{J}}^{\circ }\) and \(\varepsilon \in {\mathbb{I}}^{\circ }\), there is an integer \(N_{\alpha ,\varepsilon }\in {\mathbb{J}}^{\circ }\) such that \(a_{n} \in B_{G}(a,\varepsilon ,\alpha )\) whenever \(n>N_{\alpha ,\varepsilon }\).

(2) A sequence \(\{a_{n}\}\) in S is called a GFGV-Cauchy sequence (GFGVCS) if for any \(\alpha \in {\mathbb{J}}^{\circ }\) and \(\varepsilon \in {\mathbb{I}}^{\circ }\), there is an integer \(N_{\alpha ,\varepsilon }>0\) such that \({G_{{a_{n}},{a_{m}},{a_{p}}}} ( \alpha ) > 1 - \varepsilon \) whenever \(m,n,p>N_{\alpha ,\varepsilon }\).

(3) A GFGVM-space is said to be GFGV-complete if every GFGVCS is convergent in it.

Let \(\{ {{a_{n}}} \} \) be a sequence in S. Then the following statements are equivalent:

(i) The sequence \(\{ {{a_{n}}} \} \) in S converges to a point \(a \in S\);

(ii) As \(n \to \infty \), \({G_{{a_{n}},{a_{n}},a}} ( \alpha ) \to 1\) for each \(\alpha \in {\mathbb{J}}^{\circ }\);

(iii) As \(n \to \infty \), \({G_{{a_{n}},a,a}} ( \alpha ) \to 1\) for each \(\alpha \in {\mathbb{J}}^{\circ }\).

Also, the following statements are equivalent:

(i) A sequence \(\{ {{a_{n}}} \} \) in S is a GFGVCS;

(ii) For any \(\alpha \in {\mathbb{J}}^{\circ }\) and \(\varepsilon \in {\mathbb{I}}^{\circ }\), there is an integer \({N'_{\alpha ,\varepsilon }}>0\) such that \({G_{{a_{n}},{a_{m}},{a_{m}}}} ( \alpha ) > 1 - \varepsilon \) whenever \(n,m>{N'_{\alpha ,\varepsilon }}\).

Proposition 2.3


Consider the GFGVM-space \((S,G,\ast )\). Then the function G is continuous on \(S \times S \times S \times {\mathbb{J}}^{\circ }\).

Let \((S,g)\) be a G-metric space (for more detail, we refer to [1619] and [20]). Let \(G^{g}\) be the function defined on \(S \times S \times S \times {\mathbb{J}}^{\circ }\) by

$$ G^{g}_{x,y,z}(\alpha )=\frac{\alpha }{\alpha +g(x,y,z)} $$

for all \(x,y,z \in S\) and \(\alpha \in {\mathbb{J}}^{\circ }\). Then both \((S,G^{g},\ast _{P})\) and \((S,G^{g},\ast _{M})\) are GFGVM-spaces (standard GFGVM-spaces). Consider the GFGVM-space \((S,G,\ast )\). We denote by \(\mathfrak{T}_{0}(S)\), \(\mathfrak{F}_{0}(S)\), and \(\mathfrak{R}_{0}(S)\) the sets of nonempty subsets, of nonempty finite subsets, and of nonempty compact subsets of \((S,\tau _{G})\), respectively.

Let X and Y be two (nonempty) subsets of a GFGVM-space \((S,G,\ast )\). For \(s\in S\) and \(\alpha >0\), we define \(G_{s,X,Y}(\alpha ):=\sup \{G_{s,x,y}(\alpha ): x\in X, y\in Y\}\).

Lemma 2.4

Let \((S,G,\ast )\) be a GFGVM-space. Then, for all \(s\in S\), \(X,Y\in \mathfrak{R}_{0}(S)\), and \(\alpha \in {\mathbb{J}}^{\circ }\), there are \(x_{0}\in X\) and \(y_{0}\in Y\) such that

$$\begin{aligned} G_{s,X,Y}(\alpha )=G_{s,x_{0},y_{0}}(\alpha ). \end{aligned}$$


Let \(s\in S\), \(X,Y\in \mathfrak{R}_{0}(S)\), and \(\alpha >0\). By Proposition 2.3 the functions \(x\mapsto G_{s,x,y}(\alpha )\) and \(y\mapsto G_{s,x,y}(\alpha )\) are continuous. By the compactness of X and Y, there exist \(x_{0}\in X\) and \(y_{0}\in Y\) such that

$$ \sup_{x\in X,y \in Y}G_{s,x,y}(\alpha )=G_{s,x_{0},y_{0}}(\alpha ), $$

and thus

$$ G_{s,X,Y}(\alpha )=G_{s,x_{0},y_{0}}(\alpha ). $$


Lemma 2.5

Let \((S,G,\ast )\) be a GFGVM-space. Then, for all \(s\in S\) and \(X,Y\in \mathfrak{R}_{0}(S)\), the function \(\alpha \mapsto G_{s,X,Y}(\alpha )\) is continuous on \({\mathbb{J}}^{\circ }\).


From the equality

$$ G_{s,X,Y}(\alpha )=\sup_{x\in X,y \in Y}G_{s,x,y}(\alpha ) $$

and the continuity of the function \(\alpha \mapsto G_{s,x,y}(\alpha )\) for all \(x\in X\) and \(y\in Y\) on \({ \mathbb{J}}^{\circ }\), we get the lower semicontinuity of \(\alpha \mapsto G_{s,X,Y}(\alpha )\) on \({\mathbb{J}}^{\circ }\).

It suffices to show that \(\alpha \mapsto G_{s,X,Y}(\alpha )\) is upper semicontinuous on \({\mathbb{J}}^{\circ }\). Consider \(\alpha \in {\mathbb{J}}^{\circ }\) and a sequence \(\{\alpha _{n}\}_{n}\) in \({\mathbb{J}}^{\circ }\) converging to α. Lemma 2.4 implies that or every \(n\in \mathbb{N,}\) we can find \(x_{n}\in X\) and \(y_{n}\in Y\) such that \(G_{s,X,Y}(\alpha )=G_{s,x_{n},y_{n}}(\alpha _{n})\). Since \(X,Y\in \mathfrak{R}_{0}(S)\), we can find subsequences \(\{x_{n_{k}}\}_{k}\) of \(\{x_{n}\}_{n}\) and \(\{y_{n_{k}}\}_{k}\) of \(\{y_{n}\}_{n} \) and two points \(x_{0}\in X\) and \(y_{0}\in Y\) such that \(x_{n_{k}}\rightarrow x_{0}\) and \(y_{n_{k}}\rightarrow y_{0}\) in \((S,G,\ast )\). Hence \(\lim_{k}G_{s,x_{n_{k}},y_{n_{k}}}(\alpha _{n_{k}})=G_{s,x_{0},y_{0}}( \alpha )\). Using Proposition 2.3, we get

$$\begin{aligned} \lim_{k} G_{s,X,Y}(\alpha _{n_{k}})&= G_{s,x_{0},y_{0}}(\alpha ) \\ &\leq G_{s,X,Y}(\alpha ). \end{aligned}$$

Consequently, \(\alpha \mapsto G_{s,X,Y}(\alpha )\) is upper semicontinuous on \({\mathbb{J}}^{\circ }\). □

Lemma 2.6

Consider the GFGVM-space \((S,G,\ast )\). Then for all \(X\in \mathfrak{R}_{0}(S)\), \(Y,Z\in \mathfrak{T}_{0}(S)\), and \(\alpha \in {\mathbb{J}}^{\circ }\), we can find \(x_{0}\in X\) such that

$$\begin{aligned} \inf_{x\in X}G_{x,Y,Z}(\alpha )=G_{x_{0},Y,Z}(\alpha ). \end{aligned}$$


Put \(\delta =\inf_{x\in X}G_{x,Y,Z}(\alpha )\). Then we can find a sequence \(\{x_{n}\}_{n}\) in X such that \(\delta +\frac{1}{n}>G_{x_{n},Y,Z}(\alpha )\) for all \(n\in \mathbb{N}\). Since \(X\in \mathfrak{R}_{0}(S)\), we can find a subsequence \(\{x_{n_{k}}\}_{k}\) of \(\{x_{n}\}_{n}\) and \(x_{0}\in X\) such that \(x_{n_{k}}\rightarrow x_{0}\) in \((S,G,\ast )\).

Let \(y\in Y\) and \(z\in Z\). By Proposition 2.3, \(\lim_{k}G_{x_{n_{k}},y,z}(\alpha )=G_{x_{0},y,z}(\alpha )\). Since \(\delta +\frac{1}{n_{k}}>G_{x_{n_{k}},y,z}(\alpha )\) for each \(k\in \mathbb{N}\), we get \(\delta \geq G_{x_{0},y,z}(\alpha )\). Hence \(\delta =G_{x_{0},Y,Z}(\alpha )\). □

Now Lemmas 2.5 and 2.6 imply that the following result.

Corollary 2.7

Consider the GFGVM-space \((S,G,\ast )\). Let \(X,Y,Z\in \mathfrak{R}_{0}(S)\) and \(\alpha \in {\mathbb{J}}^{\circ }\). Then we can find \(x_{0}\in X\), \(y_{0}\in Y\), and \(z_{0}\in Z\) such that

$$\begin{aligned} \inf_{x\in X}G_{x,Y,Z}(\alpha )=G_{x_{0},y_{0},z_{0}}(\alpha ). \end{aligned}$$

Proposition 2.8

Consider the GFGVM-space \((S,G,\ast )\). Then for all \(X,Y,Z\in \mathfrak{R}_{0}(S)\), the function \(\alpha \mapsto \inf_{x\in X}G_{x,Y,Z}\) is continuous on \({\mathbb{J}}^{\circ }\).


The continuity of \(\alpha \mapsto G_{x,Y,Z}(\alpha )\) on \({\mathbb{J}}^{\circ }\) follows from Lemma 2.5, which implies the upper semicontinuity of \(\alpha \mapsto \inf_{x\in X}G_{x,Y,Z}(\alpha )\) on \({\mathbb{J}}^{\circ }\).

It suffices to show that \(\alpha \mapsto \inf_{x\in X}G_{x,Y,Z}(\alpha )\) is lower semicontinuous on \({\mathbb{J}}^{\circ }\). Consider \(\alpha \in {\mathbb{J}}^{\circ }\) and a sequence \(\{\alpha _{n}\}_{n}\) in \({\mathbb{J}}^{\circ }\) converging to α. Lemma 2.6 implies that we can find \(x_{n}\in X\) such that \(G_{x_{n},Y,Z}(\alpha _{n})=\inf_{x\in X}G_{x,Y,Z}(\alpha _{n})\). Since \(X\in \mathfrak{R}_{0}(S)\), we can find a subsequence \(\{x_{n_{k}}\}_{k}\) of \(\{x_{n}\}_{n}\) and \(x_{0}\in X\) such that \(x_{n_{k}}\rightarrow x_{0}\) in \((S,G,\ast )\). Then Lemma 2.4 implies that we can find \(y_{0}\in Y\) and \(z_{0}\in Z\) such that \(G_{x_{0},y_{0},z_{0}}(\alpha )=G_{x_{0},Y,Z}(\alpha )\), and then \(\lim_{k}G_{x_{n_{k}},y_{0},z_{0}}(\alpha _{n_{k}})=G_{x_{0},y_{0},z_{0}}( \alpha )\) by Proposition 2.3. Then for \(\varepsilon \in {\mathbb{I}}^{\circ }\), we can find \(k_{0}\in \mathbb{N}\) such that \(G_{x_{0},y_{0},z_{0}}(\alpha )<\varepsilon +G_{x_{n_{k}},y_{0},z_{0}}( \alpha )\) for every \(k\geq k_{0}\). So

$$\begin{aligned} \inf_{x\in X}G_{x,Y,Z}(\alpha ) &\leq G_{x_{0},y_{0},z_{0}}( \alpha ) \\ &< \varepsilon +G_{x_{n_{k}},Y,Z}(\alpha _{n_{k}}) \\ &=\varepsilon +\inf_{x\in X}G_{x,Y,Z}(\alpha _{n_{k}}) \end{aligned}$$

for every \(k\geq k_{0}\). Hence \(\alpha \mapsto \inf_{x\in X}G_{x,Y,Z}(\alpha )\) is lower semicontinuous on \({\mathbb{J}}^{\circ }\). □

Remark 2.9

Proposition 2.8 implies the continuity of \(\alpha \mapsto \inf_{y\in Y}G_{X,y,Z}(\alpha )\) and \(\alpha \mapsto \inf_{z\in Z}G_{X,Y,z}(\alpha )\) on \({\mathbb{J}}^{\circ }\) for \(X,Y,Z\in \mathfrak{R}_{0}(S)\).

GFGV-Hausdorff distance on \(\mathfrak{R}_{0}(S)\)

Consider the GFGVM-space \((S,G,\ast )\). We define the function \(H_{G}\) on \(\mathfrak{R}_{0}(S)\times \mathfrak{R}_{0}(S)\times \mathfrak{R}_{0}(S) \times {\mathbb{J}}^{\circ }\) by

$$\begin{aligned} H_{G}(X,Y,Z,\alpha )=\min \Bigl\{ \inf_{x\in X}G_{x,Y,Z}( \alpha ), \inf_{y\in Y}G_{X,y,Z}(\alpha ),\inf _{z\in Z}G_{X,Y,z}(\alpha ) \Bigr\} \end{aligned}$$

for \(X,Y,Z\in \mathfrak{R}_{0}(S)\) and \(\alpha \in {\mathbb{J}}^{\circ }\).

Lemma 3.1

Consider the GFGVM-space \((S,G,\ast )\). Let \(x\in S\), \(Y,Z\in \mathfrak{R}_{0}(S)\), \(W\in \mathfrak{T}_{0}(S)\), and \(\alpha ,\beta \in {\mathbb{J}}^{\circ }\). Then

$$\begin{aligned} G_{x,Y,W}(\alpha +\beta )\geq G_{x,Z,Z}(\alpha )\ast G_{z_{x},Y,W}( \beta ), \end{aligned}$$

where \(z_{x}\in Z\) satisfies \(G_{x,Z,Z}(\alpha )=G_{x,z_{x},z_{x}}(\alpha )\).


By Lemma 2.4, for \(z_{x}\in Z\), we have \(G_{x,Z,Z}(\alpha )=G_{x,z_{x},z_{x}}(\alpha )\). Now, for all \(y\in Y\) and \(w\in W\), we have

$$\begin{aligned} G_{x,Y,W}(\alpha +\beta )&\geq G_{x,y,w}(\alpha +\beta ) \\ &\geq G_{x,z_{x},z_{x}}(\alpha )\ast G_{z_{x},y,w}(\beta ). \end{aligned}$$

Then by the continuity of CTN we get

$$ G_{x,Y,W}(\alpha +\beta )\geq G_{x,Z,Z}(\alpha )\ast G_{z_{x},Y,W}( \beta ). $$


Theorem 3.2

Consider the GFGVM-space \((S,G,\ast )\). Then \((\mathfrak{R}_{0}(S),H_{G},\ast )\) is a GFGVM-space.


Let \(X,Y,Z,W\in \mathfrak{R}_{0}(S)\) and \(\alpha ,\beta \in {\mathbb{J}}^{\circ }\). By Lemma 2.6 there exist \(x_{0}\in X\), \(y_{0}\in Y\), and \(z_{0}\in Z\) such that

$$\begin{aligned}& \inf_{x\in X}G_{x,Y,Z}=G_{x_{0},Y,Z}, \\& \inf_{y\in Y}G_{X,y,Z}=G_{X,y_{0},Z}, \end{aligned}$$


$$ \inf_{z\in Z}G_{X,Y,z}=G_{X,Y,z_{0}}. $$

Then \(H_{G}(X,Y,Z,\alpha )>0\). Furthermore, it is obvious that \(X=Y=Z\) if and only if \(H_{G}(X,Y,Z,\alpha )=1\), and so we get \(H_{G}(X,Y,Z,\alpha )=H_{G}(X,Z,Y,\alpha )=H_{G}(Y,X,Z,\alpha )= \cdots \) .

Now by Lemma 3.1 and the continuity of CTN we have

$$\begin{aligned} \inf_{x\in X}G_{x,Y,W}(\alpha +\beta )\geq \inf _{x\in X}G_{x,Z,Z}( \alpha )\ast \inf_{x\in X}G_{z_{x},Y,W}( \beta ). \end{aligned}$$

Since \(\{z_{x}:x\in X\}\subseteq Z\), we have \(\inf_{x\in X}G_{z_{x},Y,W}(\beta )\geq \inf_{z\in Z}G_{z,Y,W}( \beta )\). Then

$$\begin{aligned} \inf_{x\in X}G_{x,Y,W}(\alpha +\beta )\geq \inf _{x\in X}G_{x,Z,Z}( \alpha )\ast \inf_{z\in Z }G_{z,Y,W}( \beta ). \end{aligned}$$

In the same way, we obtain

$$\begin{aligned} \inf_{y\in Y}G_{x,y,W}(\alpha +\beta )\geq \inf _{y\in Y}G_{y,Z,Z}( \alpha )\ast \inf_{z\in Z}G_{z,X,W}( \beta ) \end{aligned}$$


$$\begin{aligned} \inf_{w\in W}G_{x,Y,w}(\alpha +\beta )\geq \inf _{w\in W}G_{w,Z,Z}( \alpha )\ast \inf_{z\in Z}G_{z,X,W}( \beta ). \end{aligned}$$

Then it easily follows that

$$\begin{aligned} H_{G}(X,Y,W,\alpha +\beta )\geq H_{G}(X,Z,Z,\alpha )\ast H_{G}(Z,Y,W, \beta ). \end{aligned}$$

By Proposition 2.8 and Remark 2.9 we conclude that \(\alpha \mapsto H_{G}(X,Y,Z,\alpha )\), is continuous on \({\mathbb{J}}^{\circ }\). Then \((\mathfrak{R}_{0}(S),H_{G},\ast )\) is a GFGVM-space. □

We call the GFGVM \((H_{G},\ast )\) the GFGV-Hausdorff distance on \(\mathfrak{R}_{0}(S)\).

Proposition 3.3

The GFGV-Hausdorff distance \((H_{G^{g}},\ast _{P})\) of the standard GFGVM \((G^{g},\ast _{P})\) coincides with the standard GFGVM \((G^{h_{g}},\ast _{P})\) of the Hausdorff distance

$$ h_{g}(X,Y,Z):=\max \Bigl\{ \sup_{x\in X}g(x,Y,Z),\sup _{y\in Y}g(X,y,Z), \sup_{z\in Z}g(X,Y,z) \Bigr\} $$

on \(\mathfrak{R}_{0}(S)\).


Let \(X,Y,Z\in \mathfrak{R}_{0}(S)\) and \(\alpha \in {\mathbb{J}}^{\circ }\), and let

$$\begin{aligned} G^{g}_{x,Y,Z}(\alpha )= \frac{\alpha }{\alpha +g(x,Y,Z)} \end{aligned}$$

for \(x\in X\). Now we have

$$\begin{aligned} \inf_{x\in X}G^{g}_{x,Y,Z}(\alpha )=\inf _{x\in X} \biggl( { \frac{\alpha }{{\alpha + g ( {x,Y,Z} )}}} \biggr)= \frac{\alpha }{\alpha +\sup_{x\in X}g(x,Y,Z)}. \end{aligned}$$

Similarly, we obtain

$$\begin{aligned}& \inf_{y\in Y}G^{g}_{X,y,Z}(\alpha )=\inf _{y\in Y} \biggl( { \frac{\alpha }{{\alpha + g ( {X,y,Z} )}}} \biggr)= \frac{\alpha }{\alpha +\sup_{y\in Y}g(X,y,Z)}, \\& \inf_{z\in Z}G^{g}_{X,Y,z}(\alpha )=\inf _{z\in Z} \biggl( { \frac{\alpha }{{\alpha + g ( {X,Y,z} )}}} \biggr)= \frac{\alpha }{\alpha +\sup_{z\in Z}g(X,Y,z)}. \end{aligned}$$

Therefore \(H_{G^{g}}(X,Y,Z,\alpha )=G^{h_{g}}_{X,Y,Z}(\alpha )\). □

Now we present some examples to support our idea.

Example 3.4

Consider the discrete G-metric g on S (see [18]) with \(\vert S \vert \geq 3\). Let X, Y, and Z be nonempty finite subsets of S such that \(X\neq Y\)(\(Y\neq Z\) or \(X\neq Z\)). Then \(h_{g}(X,Y,Z)=1\), and so, by Proposition 3.3, we get \(H_{G_{g}}(X,Y,Z,\alpha )=\frac{\alpha }{\alpha +1}\) for all \(\alpha \in {\mathbb{J}}^{\circ }\).

Example 3.5

Let g be the Euclidean G-metric on \(\mathbb{R}\) (see [18]), and let \(X = [ {{x_{1}},{x_{2}}} ]\), \(Y = [ {{y_{1}},{y_{2}}} ]\), and \(Z = [ {{z_{1}},{z_{2}}} ]\) be compact intervals. Then

$$ {h_{g}} ( {X,Y,Z} ) = \max \bigl\{ { \vert {{x_{1}} - {y_{1}}} \vert , \vert {{x_{2}} - {y_{2}}} \vert , \vert {{x_{1}} - {z_{1}}} \vert , \vert {{x_{2}} - {z_{2}}} \vert , \vert {{y_{1}} - {z_{1}}} \vert , \vert {{y_{2}} - {z_{2}}} \vert } \bigr\} . $$

Now by Proposition 3.3 we get

$$ {H_{{G^{g}}}} ( {X,Y,Z,\alpha } ) = \frac{\alpha }{{\alpha + {h_{g}} ( {X,Y,Z} )}} $$

for all \(\alpha \in {\mathbb{J}}^{\circ }\).


Now we assume that all are CTND.

Lemma 4.1

Consider the GFGVM-space \((S,G,\ast )\). For each \(\mu \in {\mathbb{I}}^{\circ }\), define the function

$$ E_{\mu ,G}(x,y,z)=\inf \bigl\{ \alpha >0,G_{x,y,z}(\alpha )>1-\mu \bigr\} $$

for \(x,y,z\in S\). Then:

(i) For any \(\lambda \in {\mathbb{I}}^{\circ }\), we can find \(\mu \in {\mathbb{I}}^{\circ }\) such that

$$\begin{aligned}& E_{\lambda ,G}(s_{0},s_{m},s_{m})\leq \sum _{i=0}^{m-1}E_{\mu ,G}(s_{i},s_{i+1},s_{i+1}), \\& E_{\lambda ,G}(s_{0},s_{0},s_{m})\leq \sum _{i=0}^{m-1}E_{\mu ,G}(s_{i},s_{i},s_{i+1}) \quad \textit{for all} \ s_{0},s_{1},\ldots ,s_{m} \in S. \end{aligned}$$

(ii) Let \(\{s_{n}\}_{n}\) be a convergent sequence in \((S,G,\ast )\). Then we have \(E_{\lambda ,G}(s,s_{n},s_{n})\to 0\) and \(E_{\lambda ,G}(s_{n},s,s)\to 0\) and vice versa.

Also, if \(\{s_{n}\}\) is a GFGVCS in \((S,G,\ast )\), then it is a GFGVCS with \(E_{\lambda ,G}\) and vice versa.


(i) For every \(\lambda \in {\mathbb{I}}^{\circ }\), we can find \(\mu \in {\mathbb{I}}^{\circ }\) such that

$$ \overbrace{(1-\mu )\ast ,\ldots,\ast (1-\mu )}^{m}>1-\lambda . $$

For any given \(m\in Z^{+}\), we put

$$ E_{\mu ,G}(s_{i},s_{i+1},s_{i+1})=\alpha _{i},\quad i=0,1,2, \ldots ,m-1. $$

It is obvious that for every \(\varepsilon >0\),

$$ E_{\mu ,G}(s_{i},s_{i+1},s_{i+1})< \alpha _{i}+\frac{\varepsilon }{m}. $$

For \(i=0,1,\ldots,m-1\), we have \(G_{s_{i},s_{i+1},s_{i+1}}(\alpha _{i}+\frac{\varepsilon }{m})>1-\mu \), and so

$$\begin{aligned}& G_{s_{0},s_{m},s_{m}}(\alpha _{0}+\alpha _{1}+\cdots+\alpha _{m-1}+ \varepsilon ) \\& \quad \geq \overbrace{ G_{s_{0},s_{1},s_{1}}\biggl(\alpha _{0}+ \frac{\varepsilon }{m}\biggr)\ast ...\ast G_{s_{m-1},s_{m},s_{m}}\biggl(\alpha _{m-1} +\frac{\varepsilon }{m}\biggr)}^{m} \\& \quad \geq \overbrace{(1-\mu )\ast ,\ldots,\ast (1-\mu )}^{m} \\& \quad >1-\lambda . \end{aligned}$$


$$ E_{\lambda ,G}(s_{0},s_{m},s_{m})\leq \alpha _{0}+\alpha _{1}+\cdots+ \alpha _{m-1}+\varepsilon , $$

and so

$$\begin{aligned} E_{\lambda ,G}(s_{0},s_{m},s_{m}) \leq \sum_{i=0}^{m-1}E_{\mu ,G}(s_{i},s_{i},s_{i})+ \varepsilon . \end{aligned}$$

Taking the limit in (4.1) as \(\varepsilon \downarrow 0\), we get

$$\begin{aligned} E_{\lambda ,G}(s_{0},s_{m},s_{m}) \leq \sum_{i=0}^{m-1}E_{\mu ,G}(s_{i},s_{i},s_{i}) \end{aligned}$$

for all \(s_{0},s_{1},\ldots,s_{m}\in S\).

Similarly, we get

$$ E_{\lambda ,G}(s_{0},s_{0},s_{m})\leq \sum _{i=0}^{m-1}E_{\mu ,G}(s_{i},s_{i},s_{i+1}) $$

for all \(s_{0},s_{1},\ldots,s_{m}\in S\).

(ii) We have

$$ G_{s,s_{n},s_{n}}(\eta )>1-\lambda \quad \Longleftrightarrow\quad E_{\lambda ,G}(s,s_{n},s_{n})< \eta $$

for every \(\eta >0\).


$$ G_{s_{n},s,s}(\eta )>1-\lambda \quad \Longleftrightarrow\quad E_{\lambda ,G}(s_{n},s,s)< \eta $$

for every \(\eta \in {\mathbb{J}}^{\circ }\). □

Lemma 4.2

Consider the GFGVM-space \((S,G,\ast )\). If

$$\begin{aligned} G_{x,y,z}(\alpha )=C \end{aligned}$$

for all \(x,y,z\in S\) and \(\alpha \in {\mathbb{J}}^{\circ }\), then \(C=1\).


Taking \(x=y=z\) in (4.3), we get \(C=1\). □

Consider the class of mappings \(\phi :{\mathbb{J}}^{\circ }\to {\mathbb{J}}^{\circ }\) that are onto, strictly increasing, and \(\phi (\alpha )<\alpha \) for all \(\alpha \in {\mathbb{J}}^{\circ }\).

Lemma 4.3

Consider the GFGVM-space \((S,G,\ast )\). Then

$$ \inf \bigl\{ \phi ^{n}(\alpha )>0:G_{x,y,z}(\alpha )>1-\lambda \bigr\} \leq \phi ^{n}\bigl( \inf \bigl\{ \alpha >0:G_{x,y,z}( \alpha )>1-\lambda \bigr\} \bigr) $$

for all \(x,y,z\in S\), \(\lambda \in {\mathbb{I}}^{\circ }\), and \(n\in \mathbb{N}\).


Fix \(\alpha \in {\mathbb{J}}^{\circ }\) with \(G_{x,y,z}(\alpha )>1-\lambda \). Then \(\phi ^{n}(\alpha )\in {\mathbb{J}}^{\circ }\). Also,

$$ \phi ^{n}(\alpha )>\inf \bigl\{ \phi ^{n}(\beta )>0:G_{x,y,z}(\beta )>1- \lambda \bigr\} , $$

and so we have

$$ \alpha \geq \bigl(\phi ^{n}\bigr)^{-1}\bigl(\inf \bigl\{ \phi ^{n}(\beta )>0:G_{x,y,z}( \beta )>1-\lambda \bigr\} \bigr). $$


$$ \inf \bigl\{ \alpha >0:G_{x,y,z}(\alpha )>1-\lambda \bigr\} \geq \bigl(\phi ^{n}\bigr)^{-1}\bigl( \inf \bigl\{ \phi ^{n}(\beta )>0:G_{x,y,z}(\beta )>1-\lambda \bigr\} \bigr), $$

and we conclude that

$$ \inf \bigl\{ \phi ^{n}(\alpha )>0:G_{x,y,z}(\alpha )>1-\lambda \bigr\} \leq \phi ^{n}\bigl( \inf \bigl\{ \alpha >0:G_{x,y,z}( \alpha )>1-\lambda \bigr\} \bigr). $$


Lemma 4.4

Consider the GFGVM-space \((S,G,\ast )\). Suppose that \(\{ {{s_{n}}} \} \subseteq S\) satisfies

$$ G_{s_{n},s_{n+1},s_{n+1}}\bigl(\phi ^{n}(\alpha )\bigr)\ge G_{s_{0},s_{1},s_{1}}( \alpha )\quad \textit{for all} \ \alpha \in {\mathbb{J}}^{\circ }. $$

Then \(\{ {{s_{n}}} \} \) is a GFGVCS.


Using Lemma 4.3, we get

$$\begin{aligned}& {{\mathrm{{E}}}_{\mu ,G}} ( {{s_{n}},{s_{n+1}},{s_{n + 1}}} ) \\& \quad = \inf \bigl\{ {{\phi ^{n}} ( \alpha ) > 0:{G_{{s_{n}},{s_{n + 1}},{s_{n + 1}}}} \bigl( {{\phi ^{n}} ( \alpha )} \bigr) > 1 - \mu } \bigr\} \\& \quad \le \inf \bigl\{ {{\phi ^{n}} ( \alpha ) > 0:{G_{{s_{0}},{s_{1}},{s_{1}}}} ( \alpha ) > 1 - \mu } \bigr\} \\& \quad \le {\phi ^{n}} \bigl( {\inf \bigl\{ {\alpha > 0:{G_{{s_{0}},{s_{1}},{s_{1}}}} ( \alpha ) > 1 - \mu } \bigr\} } \bigr) \\& \quad = {\phi ^{n}} \bigl( {{{\mathrm{{E}}}_{\mu ,G}} ( {{s_{0}},{s_{1}},{s_{1}}} )} \bigr) \end{aligned}$$

for every \(\mu \in {\mathbb{I}}^{\circ }\).

For every \(\lambda \in {\mathbb{I}}^{\circ }\), there exists \(\theta \in {\mathbb{I}}^{\circ }\) such that

$$\begin{aligned}& {{\mathrm{{E}}}_{\lambda ,G}} ( {{s_{n}},{s_{m}},{s_{m}}} ) \\& \quad \le {{\mathrm{{E}}}_{\theta ,G}} ( {{s_{m - 1}},{s_{m}},{s_{m}}} ) + {{\mathrm{{E}}}_{\theta ,G}} ( {{s_{m - 2}},{s_{m - 1}},{s_{m - 1}}} ) + \cdots \\& \qquad {}+ {{\mathrm{{E}}}_{\theta ,G}} ( {{s_{n}},{s_{n + 1}},{s_{n + 1}}} ) \\& \quad \le \sum_{i = n}^{m - 1} {{\phi ^{i}} \bigl( {{E_{ \theta , G}} ( {{s_{0}},{s_{1}},{s_{1}}} )} \bigr)} \\& \quad \to 0 \end{aligned}$$

as \(m,n\to \infty \). By Lemma 4.1, \(\{ {{s_{n}}} \} \) is a GFGVCS. □

GFGV-fractal spaces

Hutchinson [21] considered the concept of fractal theory by studying the iterated function system (IFS). This subject was generalized by Barnsley [22], Bisht [6], Imdad [23], and Ri [5].

Definition 5.1

Consider the GFGVM-space \((S,G,\ast )\). A mapping \(\Omega :S \to S\) is said to be a GFGV-ϕ-contractive mapping if

$$ {G_{\Omega ( x ),\Omega ( y ),\Omega ( z )}} \bigl( {\phi ( \alpha )} \bigr) \ge {G_{x,y,z}} ( \alpha ) $$

for all \(x,y,z \in S\) and \(\alpha \in {\mathbb{J}}^{\circ }\).

Definition 5.2

A GFGV iterated function system (GFGVIFS) is a finite set of GFGV-ϕ-contractions \(\{ {{\Omega _{1}},{\Omega _{2}}, \ldots ,{\Omega _{m}}} \} \ ( {m \ge 2} )\) defined on a complete GFGVM-space \((S,G,\ast )\).

Consider the given GFGVIFS, if there is a unique nonempty compact set Γ of the complete GFGVM-space \((S,G,\ast )\) such that \(\Gamma = \bigcup_{i = 1}^{m} {\Omega _{i} (\Gamma )}\) in which Γ is a fractal set called the attractor of the respective GFGVIFS. The related attractor GFGVIFS is called a GFGV-fractal space.

Lemma 5.3

Consider the GFGVM-space \((S,G,\ast )\). Assume that \(\Omega :S \to S\) is a mapping such that

$$\begin{aligned} {G_{\Omega ( x ),\Omega ( y ),\Omega ( z )}} \bigl( {\phi ( \alpha )} \bigr) \ge {G_{x,y,z}} ( \alpha ) \end{aligned}$$

for all \(x,y,z \in S\) and \(\alpha \in {\mathbb{J}}^{\circ }\). Then the sequence \(\{ {{\Omega ^{n}} ( x )} \} _{n = 1}^{ + \infty }\) is GFGVCS.


We use induction. In (5.1), taking \(y=z=\Omega (x)\), we get

$$ {G_{\Omega (x),\Omega ^{2}(x),\Omega ^{2}(x)}}\bigl( \phi (\alpha )\bigr) \ge {G_{x, \Omega (x),\Omega (x)}} ( \alpha ). $$

Let \({G_{{\Omega ^{n}} ( x ),{\Omega ^{n + 1}} ( x ),{\Omega ^{n + 1}} ( x )}} ( {{\phi ^{n}} ( \alpha )} ) \ge {G_{x,\Omega ( x ), \Omega ( x )}} ( \alpha )\). Then

$$\begin{aligned}& {G_{{\Omega ^{n + 1}} ( x ),{\Omega ^{n + 2}} ( x ),{\Omega ^{n + 2}} ( x )}} \bigl( {{\phi ^{n + 1}} ( \alpha )} \bigr) \\& \quad = {G_{\Omega ( {{\Omega ^{n}} ( x )} ), \Omega ( {{\Omega ^{n + 1}} ( x )} ),\Omega ( {{\Omega ^{n + 1}} ( x )} )}} \bigl( { \phi \bigl( {{\phi ^{n}} ( \alpha )} \bigr)} \bigr) \\& \quad \ge {G_{{\Omega ^{n}} ( x ),{\Omega ^{n + 1}} ( x ),{\Omega ^{n + 1}} ( x )}} \bigl( {{\phi ^{n}} ( \alpha )} \bigr) \\& \quad \ge {G_{x,\Omega ( x ),\Omega ( x )}} ( \alpha ). \end{aligned}$$

Put \(\{ s_{n}\}_{n = 1}^{ + \infty } = \{\Omega ^{n} ( x )\}_{n = 1}^{ + \infty }\). Then \(\{ {{s_{n}}} \} \) is a sequence that satisfies the conditions of Lemma 4.4. Therefore

$$ {G_{{s_{n}},{s_{n + 1}},{s_{n + 1}}}} \bigl( {{\phi ^{n}} ( \alpha )} \bigr) \ge {G_{{s_{0}},{s_{1}},{s_{1}}}} ( \alpha ), $$

and hence \(\{ s_{n}\}_{n = 1}^{ + \infty } = \{\Omega ^{n} ( x )\}_{n = 1}^{ + \infty }\) is a GFGVCS. □

Lemma 5.4

Consider the GFGVM-space \((S,G,\ast )\) and GFGVF-ϕ-contractive map Ω such that

$$ {G_{\Omega ( x ),\Omega ( y ),\Omega ( z )}} \bigl( {\phi ( \alpha )} \bigr) \ge {G_{x,y,z}} ( \alpha ) $$

for all \(x,y,z \in S\) and \(\alpha \in {\mathbb{J}}^{\circ }\). Then Ω has a unique fixed point δ in S.


Lemma 5.3 and (5.2) imply that the sequence \(\{ {{\Omega ^{n}} ( x )} \} _{n = 1}^{ + \infty }\) is GFGVCS for each \(x \in S\) and \(\lim_{n \to + \infty } {\Omega ^{n}} ( x ) = \delta \in S\).

Letting \({x_{0}} = x\) and \({x_{n}} = {\Omega ^{n}} ( x )\) for each \(n \ge 1\), since \(\lim_{n \to + \infty } {\Omega ^{n}} ( x ) = \delta \), we have \(\lim_{n \to + \infty } {G_{{x_{n}},\delta , \delta }} ( \alpha ) = 1\) for each \(\alpha \in {\mathbb{J}}^{\circ }\).

On the other hand, we have

$$ {G_{\Omega ( \delta ),{x_{n + 1}},{x_{n + 1}}}} \bigl( { \phi ( \alpha )} \bigr) \ge {G_{\delta ,{x_{n}},{x_{n}}}} ( \alpha ) $$

for each \(n \in \mathbb{N}\) and each \(\alpha >0\). Then

$$ {G_{\Omega ( \delta ),\delta ,\delta }} \bigl( {\phi ( \alpha )} \bigr) = \lim_{n \to + \infty } {G_{\Omega ( \delta ),{x_{n + 1}},{x_{n + 1}}}} \bigl( {\phi ( \alpha )} \bigr) \ge \lim_{n \to + \infty } {G_{\delta ,{x_{n}},{x_{n}}}} ( \alpha ) = 1 $$

for each \(\alpha >0\). Therefore \(\delta =\Omega ( \delta )\), that is, δ is a fixed point of Ω.

Now we prove that δ is the unique fixed point of Ω. If σ is another fixed point of Ω, then for any \(\alpha \in {\mathbb{J}}^{\circ }\),

$$ {G_{\delta ,\delta ,\sigma }} ( \alpha ) = {G_{\Omega ( \delta ),\Omega ( \delta ),\Omega ( \sigma )}} ( \alpha ) \le {G_{\Omega ( \delta ),\Omega ( \delta ),\Omega ( \sigma )}} \bigl( {\phi ( \alpha )} \bigr). $$

On the other hand, since \({G_{x,y,y}} ( \alpha )\) is nondecreasing and \(\phi ( \alpha ) < \alpha \), we have

$$ {G_{\Omega ( \delta ),\Omega ( \delta ), \Omega ( \sigma )}} \bigl( {\phi ( \alpha )} \bigr) \le {G_{\Omega ( \delta ),\Omega ( \delta ),\Omega ( \sigma )}} ( \alpha ) = {G_{\delta ,\delta ,\sigma }} ( \alpha ). $$

Hence \({G_{\delta ,\delta ,\sigma }} ( \alpha ) = C\) for all \(\alpha \in {\mathbb{J}}^{\circ }\). From Lemma 4.2 we get \(C=1\). Therefore \(\delta = \sigma \), that is, δ is a unique fixed point of Ω. □

Now we present an example illustrating our results; for more applications, we refer to [11, 17, 2429].

Example 5.5

Let \(S=C({\mathbb{I}})\) be the set of all continuous functions defined on \({ \mathbb{I}}\). Define G on \(S\times S\times S\times \mathbb{J}^{0}\) by

$$\begin{aligned} G_{u,v,w}(\alpha )=\inf_{\delta \in \mathbb{I}} \biggl( \frac{\alpha }{\alpha + \vert u(\delta )-v(\delta ) \vert + \vert v(\delta )-w(\delta ) \vert + \vert w(\delta )-u(\delta ) \vert } \biggr) \end{aligned}$$

for \(u,v,w\in S\) and \(\alpha \in \mathbb{J}^{0}\). Then

$$\begin{aligned} G_{u,v,w}(\alpha )= \frac{\alpha }{\alpha +\sup_{\delta \in \mathbb{I}} \vert u(\delta )-v(\delta ) \vert + \sup_{\delta \in \mathbb{I}} \vert v(\delta )-w(\delta ) \vert + \sup_{\delta \in \mathbb{I}} \vert w(\delta )-u(\delta ) \vert }. \end{aligned}$$

We denote

$$ g(u,v,w)=\sup_{\delta \in \mathbb{I}} \bigl\vert u(\delta )-v( \delta ) \bigr\vert + \sup_{\delta \in \mathbb{I}} \bigl\vert v( \delta )-w(\delta ) \bigr\vert + \sup_{\delta \in \mathbb{I}} \bigl\vert w(\delta )-u(\delta ) \bigr\vert . $$

It is obvious that \((S,g)\) is a complete G-metric space [16, 27]. Then \((S,G,\ast _{M})\) is a GFGVM-space.

Let \(\phi (\alpha ):\mathbb{J}\rightarrow \mathbb{J}\) be defined as \(\phi (\alpha )=\frac{\alpha }{\alpha +1}\).

Consider the following integral equation:

$$\begin{aligned} \Omega \bigl(u(\delta )\bigr)= \int _{0}^{1}p(\delta ,\sigma )f\bigl(\sigma ,u( \sigma )\bigr)\,d\sigma , \quad \sigma \in \mathbb{I}. \end{aligned}$$

Suppose that the following conditions are satisfied:

(i) \(p:{\mathbb{I}}\times {\mathbb{I}}\rightarrow \mathbb{R}^{+}\) is continuous.

(ii) \(f:{\mathbb{I}}\times \mathbb{R}\rightarrow \mathbb{R}^{+}\) is continuous.

(iii) There exists a constant \(\lambda >0\) such that

$$\begin{aligned} \bigl\vert f(\delta ,u)-f(\delta ,v) \bigr\vert \leq \lambda \vert u-v \vert \end{aligned}$$

for all \(\delta \in \mathbb{I}\) and \(u,v\in \mathbb{R}\).

(iv) \(\lambda \Vert p \Vert _{\infty }\leq \frac{1}{\alpha +1}\), where

$$\begin{aligned} \Vert p \Vert _{\infty }=\sup \bigl\{ p(\delta ,\sigma ):\delta ,\sigma \in {\mathbb{I}}\bigr\} . \end{aligned}$$

Then, under conditions (i)–(iv), integral (5.3) has a unique solution in \(C({\mathbb{I}})\).


First, consider \(\Omega :S\rightarrow S\). It is clear that Ω is well defined (i.e., for \(u\in S\), we have \(\Omega (u)\in S\)). Then we have

$$\begin{aligned} &G_{\Omega (u(\delta )),\Omega (v(\delta )),\Omega (w(\delta ))}\bigl( \phi (\alpha )\bigr) \\ &\quad = \inf_{\delta \in \mathbb{I}} \biggl( \frac{\phi (\alpha )}{\phi (\alpha )+ \vert \Omega (u(\delta ))-\Omega (v(\delta )) \vert + \vert \Omega (v(\delta ))-\Omega (w(\delta )) \vert + \vert \Omega (w(\delta ))-\Omega (u(\delta )) \vert } \biggr) \\ &\quad = \bigl(\phi (\alpha )\bigr)\Big/ \Bigl(\phi (\alpha )+\sup _{\delta \in {\mathbb{I}}} \bigl\vert \Omega \bigl(u(\delta )\bigr)-\Omega \bigl(v(\delta )\bigr) \bigr\vert \\ &\qquad {} +\sup_{\delta \in {\mathbb{I}}} \bigl\vert \Omega \bigl(v( \delta )\bigr)-\Omega \bigl(w(\delta )\bigr) \bigr\vert +\sup _{\delta \in {\mathbb{I}}}|\Omega \bigl(w(\delta )\bigr)-\Omega _{(}u( \delta )\Bigr)|) \\ &\quad = \bigl(\phi (\alpha )\bigr)\Big/\biggl(\phi (\alpha )+\biggl( \sup _{\delta \in \mathbb{I}} \biggl\vert \int _{0}^{1}p(\delta ,\sigma )f\bigl(\sigma ,u( \sigma )\bigr)\,d\sigma - \int _{0}^{1}p(\delta ,\sigma )f\bigl(\sigma ,v( \sigma )\bigr)\,d\sigma \biggr\vert \\ &\qquad {}+\sup_{\delta \in \mathbb{I}} \biggl\vert \int _{0}^{1}p(\delta ,\sigma )f\bigl(\sigma ,v( \sigma )\bigr)\,d\sigma - \int _{0}^{1}p(\delta ,\sigma )f\bigl(\sigma ,w( \sigma )\bigr)\,d\sigma \biggr\vert \\ &\qquad {}+\sup_{\delta \in \mathbb{I}} \biggl\vert \int _{0}^{1}p(\delta ,\sigma )f\bigl(\sigma ,w( \sigma )\bigr)\,d\sigma - \int _{0}^{1}p(\delta ,\sigma )f\bigl(\sigma ,u( \sigma )\bigr)\,d\sigma \biggr\vert \biggr)\biggr) \\ &\quad = \bigl(\phi (\alpha )\bigr)\Big/\biggl(\phi (\alpha )+\biggl( \sup _{\delta \in \mathbb{I}} \biggl\vert \int _{0}^{1}p(\delta ,\sigma ) \bigl(f\bigl(\sigma ,u(\sigma )\bigr)-f\bigl(\sigma ,v(\sigma )\bigr)\bigr)\,d\sigma \biggr\vert \\ &\qquad {}+\sup_{\delta \in \mathbb{I}} \biggl\vert \int _{0}^{1}p(\delta ,\sigma ) \bigl(f\bigl(\sigma ,v(\sigma )\bigr)-f\bigl(\sigma ,w(\sigma )\bigr)\bigr)\,d\sigma \biggr\vert \\ &\qquad {}+\sup_{\delta \in \mathbb{I}} \biggl\vert \int _{0}^{1}p(\delta ,\sigma ) \bigl(f\bigl(\sigma ,w(\sigma )\bigr)-f\bigl(\sigma ,u(\sigma )\bigr)\bigr)\,d\sigma \biggr\vert \biggr)\biggr) \\ &\quad \geq \bigl(\phi (\alpha )\bigr)\Big/\biggl(\phi (\alpha )+\biggl( \lambda \sup_{\delta \in \mathbb{I}} \int _{0}^{1}p(\delta ,\sigma ) \bigl\vert u( \sigma )-v(\sigma ) \bigr\vert \,d\sigma \\ &\qquad {}+\lambda \sup_{\delta \in \mathbb{I}} \int _{0}^{1}p(\delta ,\sigma ) \bigl\vert v( \sigma )-w(\sigma ) \bigr\vert \,d\sigma \\ &\qquad {}+\lambda \sup_{\delta \in \mathbb{I}} \int _{0}^{1}p(\delta ,\sigma ) \bigl\vert w( \sigma )-u(\sigma ) \bigr\vert \,d\sigma \biggr)\biggr). \end{aligned}$$

Using the Cauchy–Schwarz inequality, we have

$$\begin{aligned} \int _{0}^{1}p(\delta ,\sigma ) \bigl\vert u( \sigma )-v(\sigma ) \bigr\vert \,d \sigma &\leq \biggl( \int _{0}^{1}p^{2}(\delta ,\sigma )\,d \sigma \biggr)^{ \frac{1}{2}} ( \int _{0}^{1}\bigl( \bigl\vert u(\sigma )-v( \sigma ) \bigr\vert ^{2}\,d \sigma \bigr)^{\frac{1}{2}} \\ &\leq \Vert p \Vert _{\infty }\sup_{\delta \in {\mathbb{I}}} \bigl\vert u(\delta )-v(\delta ) \bigr\vert . \end{aligned}$$

In the same way, we have

$$\begin{aligned} & \int _{0}^{1}p(\delta ,\sigma ) \bigl\vert v( \sigma )-w(\sigma ) \bigr\vert \,d \sigma \leq \Vert p \Vert _{\infty } \sup_{\delta \in { \mathbb{I}}} \bigl\vert v(\delta )-w(\delta ) \bigr\vert \end{aligned}$$


$$\begin{aligned} & \int _{0}^{1}p(\delta ,\sigma ) \bigl\vert w( \sigma )-u(\sigma ) \bigr\vert \,d \sigma \leq \Vert p \Vert _{\infty } \sup_{\delta \in { \mathbb{I}}} \bigl\vert w(\delta )-u(\delta ) \bigr\vert . \end{aligned}$$

Replacing (5.5), (5.6), and (5.7) in (5.4), we obtain that

$$\begin{aligned} &G_{\Omega (u),\Omega (v),\Omega (w)}\bigl(\phi (\alpha )\bigr) \\ &\quad \geq \bigl(\phi (\alpha )\bigr)\Big/ \biggl(\phi (\alpha )+\biggl( \lambda \sup_{\delta \in \mathbb{I}} \int _{0}^{1}p(\delta ,\sigma ) \bigl\vert u( \sigma )-v(\sigma ) \bigr\vert \,d\sigma \\ &\qquad {}+\lambda \sup_{\delta \in \mathbb{I}} \int _{0}^{1}p(\delta ,\sigma ) \bigl\vert v( \sigma )-w(\sigma ) \bigr\vert \,d\sigma \\ &\qquad {}+\lambda \sup_{\delta \in \mathbb{I}} \int _{0}^{1}p(\delta ,\sigma ) \bigl\vert w( \sigma )-u(\sigma ) \bigr\vert \,d\sigma \biggr)\biggr) \\ &\quad \geq \frac{\phi (\alpha )}{\phi (\alpha )+\lambda \Vert p \Vert _{\infty }(\sup_{\delta \in \mathbb{I}} \vert u(\delta )-v(\delta ) \vert +\sup_{\delta \in \mathbb{I}} \vert v(\delta )-w(\delta ) \vert +\sup_{\delta \in \mathbb{I}} \vert w(\delta )-u(\delta ) \vert )} \\ &\quad = \frac{\phi (\alpha )}{\phi (\alpha )+\lambda \Vert p \Vert _{\infty }g(u,v,w)} \\ &\quad \geq \frac{\frac{\alpha }{\alpha +1}}{\frac{\alpha }{\alpha +1}+\frac{1}{\alpha +1}g(u,v,w)} \\ &\quad =\frac{\alpha }{\alpha +g(u,v,w)} \\ &\quad =G_{u,v,w}(\alpha ). \end{aligned}$$

By Lemma 5.4 Ω has a unique fixed point, that is, \(\Omega (u)=u\), and so u is the unique solution of equation (5.3). □

Availability of data and materials

Not applicable.


  1. 1.

    George, A., Veeramani, P.: On some results in fuzzy metric spaces. Fuzzy Sets Syst. 64(3), 395–399 (1994)

    MathSciNet  Article  Google Scholar 

  2. 2.

    George, A., Veeramani, P.: On some results of analysis for fuzzy metric spaces. Fuzzy Sets Syst. 90(3), 365–368 (1997)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Rodriguez-Lopez, J., Romaguera, S.: The Hausdorff fuzzy metric on compact sets. Fuzzy Sets Syst. 147(2), 273–283 (2004)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Alihajimohammad, A., Saadati, R.: Generalized modular fractal spaces and fixed point theorems. Adv. Differ. Equ. 2021, Paper No. 383, 10 pp. (2021)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Ri, S.-I.: A new fixed point theorem in the fractal space. Indag. Math. 27(1), 85–93 (2016)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Bisht, R.K.: Comment on: A new fixed point theorem in the fractal space. Indag. Math. 29(2), 819–823 (2018)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Hadzic, O., Pap, E.: Fixed Point Theory in Probabilistic Metric Spaces. Mathematics and Its Applications, vol. 536. Kluwer Academic, Dordrecht (2001)

    Book  Google Scholar 

  8. 8.

    Schweizer, B., Sklar, A.: Probabilistic Metric Spaces. North-Holland Series in Probability and Applied Mathematics. North-Holland, New York (1983)

    MATH  Google Scholar 

  9. 9.

    Mihet, D.: A note on a paper of O. Hadzic and E. Pap: “New classes of probabilistic contractions and applications to random operators” [in Fixed point theory and applications, 97–119, Nova Sci. Publ., Hauppauge, NY, 2003; MR2043680]. In: Fixed Point Theory and Applications, vol. 7, pp. 127–133. Nova Science Publishers, New York (2007)

    MATH  Google Scholar 

  10. 10.

    Rakic, D., Mukheimer, A., Dosenovic, T., Mitrovic, Z.D., Radenovic, S.: On some new fixed point results in fuzzy b-metric spaces. J. Inequal. Appl. 2020, Paper No. 99, 14 pp. (2020)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Ahmed, M.A., Beg, I., Khafagy, S.A., Nafadi, H.A.: Fixed points for a sequence of L-fuzzy mappings in non-Archimedean ordered modified intuitionistic fuzzy metric spaces. J. Nonlinear Sci. Appl. 14(2), 97–108 (2021)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Gregori, V., Romaguera, S.: Some properties of fuzzy metric spaces. Fuzzy Sets Syst. 115(3), 485–489 (2000)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Pap, E., Park, C., Saadati, R.: Additive σ-random operator inequality and rhom-derivations in fuzzy Banach algebras. Sci. Bull. “Politeh.” Univ. Buchar., Ser. A, Appl. Math. Phys. 82(2), 3–14 (2020)

    MathSciNet  Google Scholar 

  14. 14.

    Gregori, V., Minana, J.-J.: On fuzzy ψ-contractive sequences and fixed point theorems. Fuzzy Sets Syst. 300, 93–101 (2016)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Tian, J.-F., Ha, M.-H., Tian, D.-Z.: Tripled fuzzy metric spaces and fixed point theorem. Inf. Sci. 518, 113–126 (2020)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Hashemi, E., Ghaemi, M.B.: Ekeland’s variational principle in complete quasi-G-metric spaces. J. Nonlinear Sci. Appl. 12(3), 184–191 (2019)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Kumar, M., Arora, S., Imdad, M., Alfaqih, W.M.: Coincidence and common fixed point results via simulation functions in G-metric spaces. J. Math. Comput. Sci. 19(4), 288–300 (2019)

    Article  Google Scholar 

  18. 18.

    Mustafa, Z., Sims, B.: A new approach to generalized metric spaces. J. Nonlinear Convex Anal. 7(2), 289–297 (2006)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Hussain, N., Roshan, J.R., Parvaneh, V., Latif, A.: A unification of G-metric, partial metric, and b-metric spaces. Abstr. Appl. Anal. 2014, Article ID 180698 (2014)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Mustafa, Z., Parvaneh, V., Abbas, M., Roshan, J.R.: Some coincidence point results for generalized \((\psi ,\phi )\)-weakly contractive mappings in ordered G-metric spaces. Fixed Point Theory Appl. 2013, 23 pp. 326 (2013)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Hutchinson, J.E.: Fractals and self-similarity. Indiana Univ. Math. J. 30(5), 713–747 (1981)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Barnsley, M.: Fractals Everywhere. Academic Press, Boston (1988)

    MATH  Google Scholar 

  23. 23.

    Imdad, M., Alfaqih, W.M., Khan, I.A.: Weak θ-contractions and some fixed point results with applications to fractal theory. Adv. Differ. Equ. 2018, Paper No. 439, 18 pp. (2018)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Gu, F., Yin, Y.: A new common coupled fixed point theorem in generalized metric space and applications to integral equations. Fixed Point Theory Appl. 2013, 266, 16 pp. (2013)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Nashine, H.K., Ibrahim, R.W., Rhoades, B.E., Pant, R.: Unified Feng–Liu type fixed point theorems solving control problems. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 115(1), 5 (2021)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Abu-Donia, H.M., Atia, H.A., Khater, O.M.A.: Common fixed point theorems in intuitionistic fuzzy metric spaces and intuitionistic \((\phi ,\psi )\)-contractive mappings. J. Nonlinear Sci. Appl. 13(6), 323–329 (2020)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Mustafa, Z., Jaradat, M.M.M.: Some remarks concerning \(D^{*}\)-metric spaces. J. Math. Comput. Sci. 22(2), 128–130 (2021)

    Article  Google Scholar 

  28. 28.

    Gu, F., Zhou, S.: Coupled common fixed point theorems for a pair of commuting mappings in partially ordered G-metric spaces. Fixed Point Theory Appl. 2013, 64, 18 pp. (2013)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Patriciu, A.-M., Popa, V.: A fixed point theorem in G-metric spaces for mappings using auxiliary functions. Bull. Transilv. Univ. Braşov Ser. III 12(61)(2), 419–428 (2019)

    MathSciNet  Google Scholar 

Download references


The authors are thankful to the area editor and referees for giving valuable comments and suggestions.


No funding.

Author information




Both authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Corresponding author

Correspondence to Reza Saadati.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Alihajimohammad, A., Saadati, R. Generalized fuzzy GV-Hausdorff distance in GFGV-fractal spaces with application in integral equation. J Inequal Appl 2021, 143 (2021).

Download citation


  • 54H25
  • 54A40
  • 28A80
  • 47H10


  • GFGV-metric spaces
  • GFGV-Hausdorff distance
  • Fixed points
  • Fuzzy fractal space