In 2015, Ali et al. [3] has defined gauge spaces in the local of \(b_{s}\)-pseudo metrics, called b-gauge spaces. In order to introduce quasi-b-gauge spaces, we start the introduction of the notion of a quasi-pseudo-b metric.
Definition 3.1
Let U be a nonempty set and \(s \geq 1\). The map \(q: U\times U\rightarrow [0,\infty )\) is called to be a quasi-pseudo-b metric if it satisfies the following for all \(e,f,g\in U\):
-
(a)
\(q(e,e)=0\);
-
(b)
\(q(e,g)\leq s\{q(e,f)+q(f,g)\}\).
The pair \((U,q )\) is called a quasi-pseudo-b metric space. A Hausdorff quasi-pseudo-b metric space \((U,q )\) satisfies
$$\begin{aligned} e\neq f\quad \Rightarrow\quad q(e,f)>0 \vee q(f,e)>0 \end{aligned}$$
(3.1)
for all \(e,f\in U\).
Example 3.2
Let \(U=l_{p}= \{\{x_{n}\}_{n\geq 1}\subset \mathbb{R}, \sum^{\infty }_{n=1}|x_{n}|^{p}< \infty \}\), where \(1\leq p<\infty \). Define \(q: U\times U\rightarrow [0,\infty )\) for all \(x,y\in U\) by
$$\begin{aligned} q(x,y) = \textstyle\begin{cases} 0 & \text{if }x\leq y, \\ (\sum^{\infty }_{n=1} \vert x_{n} \vert ^{p})^{\frac{1}{p}} & \text{if }x>y. \end{cases}\displaystyle \end{aligned}$$
(3.2)
Then q is a quasi-pseudo-b-metric on U with \(s=p\geq 1\). Since symmetry property does not hold, q is not a pseudo-b-metric, and hence it is not a b-metric.
Example 3.3
Suppose \(U=[0,6]\). Define \(q: U\times U\rightarrow [0,\infty )\) for all \(e,f\in U\) by
$$\begin{aligned} q(e,f) = \textstyle\begin{cases} 0 & \text{if }e\geq f, \\ (e-f)^{2} & \text{if }e< f. \end{cases}\displaystyle \end{aligned}$$
(3.3)
Then q is a quasi-pseudo-b-metric on U. Indeed, \(q(e,e)=0\) for all \(e\in U\). Further, \(q(e,g)\leq 2\{q(e,f)+q(f,g)\}\) holds for all \(e,f,g\in U\) and for \(s=2\). Also, \((U,q )\) is a Hausdorff quasi-pseudo-b-metric space.
Definition 3.4
Each family \(\textsl{Q}_{s ; \Omega }=\{q_{\beta }: \beta \in \Omega \}\) of quasi-pseudo-b metrics \(q_{\beta }: U\times U\rightarrow [0,\infty )\) for \(\beta \in \Omega \) is said to be a quasi-b-gauge on U.
Definition 3.5
The family \(\textsl{Q}_{s ; \Omega }=\{q_{\beta }: \beta \in \Omega \}\) is called to be separating if for every pair \((e,f)\), where \(e\neq f\), there exists \(q_{\beta }\in \textsl{Q}_{s ; \Omega }\) such that either \(q_{\beta }(e,f)> 0\) or \(q_{\beta }(f,e)>0\).
Definition 3.6
Let the family \(\textsl{Q}_{s ; \Omega }=\{q_{\beta }: \beta \in \Omega \}\) be a quasi-b-gauge on U. The topology \(\mathcal{T}(\textsl{Q}_{s ; \Omega })\) whose subbase is defined by the family \(\mathcal{B}(\textsl{Q}_{s ; \Omega })=\{B(e,\epsilon _{\beta }):e\in U, \epsilon _{\beta }>0, \beta \in \Omega \}\) of all balls \(B(e,\varepsilon _{\beta })=\{f\in U: q_{\beta }(e,f)<\epsilon _{\beta } \}\) and is called the topology induced by \(\textsl{Q}_{s ; \Omega }\) on U.
Definition 3.7
Suppose \((U,\mathcal{T})\) is a topological space and \(\textsl{Q}_{s ; \Omega }\) is a quasi-b-gauge on U such that \(\mathcal{T}=\mathcal{T}(\textsl{Q}_{s ; \Omega })\). Then the topological space \((U,\textsl{Q}_{s ; \Omega })\) is called to be a quasi-b-gauge space. We note that \((U,\textsl{Q}_{s ; \Omega })\) is the Hausdorff if \(\textsl{Q}_{s ; \Omega }\) is separating.
Remark 3.8
-
(a)
Each topological space and quasi-uniform space is a quasi-gauge space [2]. Also, each quasi-gauge space is a quasi-b-gauge space (for \(s= 1\)). Therefore, in the asymmetric structure, we can term a quasi-b-gauge space as the largest general space.
-
(b)
We observe that if \(s=1\), the above definitions turn down to agree with the definitions in quasi-gauge spaces.
We now establish the notion of left (right) \(\mathcal{J}_{s ; \Omega }\)-families of generalized quasi-pseudo-b-distances on U [left (right) \(\mathcal{J}_{s ; \Omega }\)-families are the generalizations of quasi-b-gauges].
Definition 3.9
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi-b-gauge space. The family \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) where \(\textsl{J}_{\beta } :U\times U\rightarrow [0,\infty )\), \(\beta \in \Omega \) is called the left (right) \(\mathcal{J}_{s ; \Omega }\)-family of generalized quasi-pseudo-b-distances on U (for short, left (right) \(\mathcal{J}_{s ; \Omega }\)-family on U) if the following statements hold for all \(\beta \in \Omega \) and for all \(x,y,z\in U\):
- (\(\mathcal{J}1\)):
-
\(\textsl{J}_{\beta }(x,z)\leq s_{\beta }\{\textsl{J}_{\beta }(x,y)+ \textsl{J}_{\beta }(y,z)\}\);
- (\(\mathcal{J}2\)):
-
for sequences \((u_{m} :m\in \mathbb{N} )\) and \((v_{m} :m\in N )\) in U fulfilling
$$\begin{aligned}& \lim_{m\rightarrow \infty }\sup_{n>m} \textsl{J}_{\beta }(u_{m},u_{n})=0, \end{aligned}$$
(3.4)
$$\begin{aligned}& \Bigl(\lim_{m\rightarrow \infty }\sup _{n>m}\textsl{J}_{\beta }(u_{n},u_{m})=0 \Bigr), \end{aligned}$$
(3.5)
and
$$\begin{aligned}& \lim_{m\rightarrow \infty } \textsl{J}_{\beta }(v_{m},u_{m})=0, \end{aligned}$$
(3.6)
$$\begin{aligned}& \Bigl(\lim_{m\rightarrow \infty } \textsl{J}_{\beta }(u_{m},v_{m})=0 \Bigr), \end{aligned}$$
(3.7)
the following hold:
$$\begin{aligned} \lim_{m\rightarrow \infty } q_{\beta }(v_{m},u_{m})=0, \end{aligned}$$
(3.8)
and
$$\begin{aligned} \Bigl(\lim_{m\rightarrow \infty } q_{\beta }(u_{m},v_{m})=0 \Bigr) ). \end{aligned}$$
(3.9)
We denote
$$ \mathbb{J}^{L}_{(U,\textsl{Q}_{s ; \Omega })}= \bigl\{ \mathcal{J}_{s ; \Omega }: \mathcal{J}_{s ; \Omega }= \{{{\textsl{J}}_{\beta }}: \beta \in \Omega \} \text{ is a left }\mathcal{J}_{s ; \Omega }\text{-family on }U \bigr\} $$
and
$$ \mathbb{J}^{R}_{(U,\textsl{Q}_{s ; \Omega })}= \bigl\{ \mathcal{J}_{s ; \Omega }: \mathcal{J}_{s ; \Omega }= \{{{\textsl{J}}_{\beta }}: \beta \in \Omega \} \bigr\} \text{ is a right }\mathcal{J}_{s ; \Omega }\text{-family on }U \}. $$
Now, we mention some trivial properties of left (right) \(\mathcal{J}_{s ; \Omega }\)-families.
Remark 3.10
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi-b-gauge space.
-
(a)
\(\textsl{Q}_{s ; \Omega }\in \mathbb{J}^{L}_{(U,\textsl{Q}_{s ; \Omega })} \cap \mathbb{J}^{R}_{(U,\textsl{Q}_{s ; \Omega })}\).
-
(b)
Let \(\mathcal{J}_{s ; \Omega } \in \mathbb{J}^{L}_{(U,\textsl{Q}_{s ; \Omega })}\) or \(\mathcal{J}_{s ; \Omega } \in \mathbb{J}^{R}_{(U,\textsl{Q}_{s ; \Omega })}\). If \(\textsl{J}_{\beta }(v,v)=0\) for all \(\beta \in \Omega \) and for all \(v\in U\), then for each \(\beta \in \Omega \), \(\textsl{J}_{\beta }\) is a quasi-pseudo-b metric.
-
(c)
There is an example of \(\mathcal{J}_{s ; \Omega } \in \mathbb{J}^{L}_{(U,\textsl{Q}_{s ; \Omega })}\) and \(\mathcal{J}_{s ; \Omega } \in \mathbb{J}^{R}_{(U,\textsl{Q}_{s ; \Omega })}\), which shows that the maps \(\textsl{J}_{\beta }\), \(\beta \in \Omega \) are not quasi-pseudo-b metric (see Example 3.12 below).
-
(d)
We note that if \(s =1\), the above definition reduces to the corresponding definition in quasi-gauge spaces.
Proposition 3.11
Let \((U,\textsl{Q}_{s ; \Omega })\) be the Hausdorff quasi-b-gauge space. Take the family \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) to be the left (right) \(\mathcal{J}_{s ; \Omega }\)-family of generalized quasi-pseudo-b-distances on U. Then there is \(\beta \in \Omega \) such that
$$ e\neq f \quad \Rightarrow\quad \textsl{J}_{\beta }(e,f)>0\vee \textsl{J}_{\beta }(f,e)>0 $$
for all \(e,f\in U\).
Proof
By incorporating the definition of the left (right) \(\mathcal{J}_{s ; \Omega }\)-family of generalized quasi-pseudo-b-distances on U in the proof of Proposition 3.11 of [15], the proof of our result can easily be obtained. □
Example 3.12
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi-b-gauge space, where U contains at least two distinct points and \(\textsl{Q}_{s ; \Omega }=\{q_{\beta }: \beta \in \Omega \}\) is the family of quasi-pseudo-b metrics \(q_{\beta }: U\times U\rightarrow [0,\infty )\), \(\beta \in \Omega \).
Let the set \(F\subset U\) contain at least two distinct, arbitrary and fixed points, and let \(d_{\beta }\in (0,\infty )\), \(\beta \in \Omega \) satisfy \(\delta _{\beta }(F)< d_{\beta }\) for all \(\beta \in \Omega \), where \(\delta _{\beta }(F)=\sup \{q_{\beta }(u,v): u,v \in F\}\) for all \(\beta \in \Omega \). Let the family \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) of maps \(\textsl{J}_{\beta } :U\times U\rightarrow [0,\infty )\), \(\beta \in \Omega \) be defined for all \(e,f\in U\) as:
$$\begin{aligned} \textsl{J}_{\beta }(e,f)= \textstyle\begin{cases} q_{\beta }(e,f) & \text{if }F\cap \{e,f\} = \{e,f\}, \\ d_{\beta } & \text{if }F\cap \{e,f\} \neq \{e,f\}. \end{cases}\displaystyle \end{aligned}$$
(3.10)
Then \(\mathcal{J}_{\varphi ; \Omega }\in \mathbb{J}^{L}_{(U,\textsl{Q})} \cap \mathbb{J}^{R}_{(U,\textsl{Q})}\).
We notice that \(\textsl{J}_{\beta }(e,g)\leq \frac{d_{\beta }}{\delta _{\beta }(F)}\{ \textsl{J}_{\beta }(e,f)+\textsl{J}_{\beta }(f,g)\}\) for all \(\beta \in \Omega \) and for all \(e,f,g\in U\), where \(\frac{d_{\beta }}{\delta _{\beta }(F)}=s_{\beta }>1\), \(\beta \in \Omega \). Thus, the condition \((\mathcal{J}_{1})\) holds. Indeed, the condition \((\mathcal{J}_{1})\) does not hold only if there are some \(\beta \in \Omega \) and \(e,f,g\in U\) such that \(\textsl{J}_{\beta }(e,g)=d_{\beta }\), \(\textsl{J}_{\beta }(e,f)=q_{\beta }(e,f)\), \(\textsl{J}_{\beta }(f,g)=q_{\beta }(f,g)\) and \(s_{\beta }\{q_{\beta }(e,f)+q_{\beta }(f,g)\}\leq d_{\beta }\). However, this implies that there exists \(h\in \{e,g\}\) such that \(h\notin F\) and, on the other hand, \(e,f,g\in F\), which is impossible.
Now, suppose that the sequences \(\{u_{m}\}\) and \(\{v_{m}\}\) in U are satisfying (3.4) and (3.6). Then in particular (3.6) yields that there exists \(m_{1}=m_{1}(\beta )\in \mathbb{N}\) such that for all \(m\geq m_{1}\), for all \(\beta \in \Omega \), and for all \(0<\epsilon < d_{\beta }\), we have
$$\begin{aligned} \textsl{J}_{\beta }(v_{m},u_{m})< \epsilon . \end{aligned}$$
(3.11)
By (3.11) and (3.10), denoting \(m'=\min \{m_{1}(\beta ):\beta \in \Omega \}\), we have for all \(m\geq m'\)
$$\begin{aligned} F\cap \{v_{m},u_{m}\}= \{v_{m},u_{m}\}. \end{aligned}$$
(3.12)
Let there exist \(m'\in \mathbb{N}\) such that for all \(m\geq m'\), for all \(\beta \in \Omega \), and for all \(0<\epsilon < d_{\beta }\), we have
$$\begin{aligned} q_{\beta }(v_{m},u_{m})= \textsl{J}_{\beta }(v_{m},u_{m})< \epsilon . \end{aligned}$$
(3.13)
Hence, the sequences \(\{u_{m}\}\) and \(\{v_{m}\}\) satisfy (3.8). Therefore, \(\mathcal{J}_{s ; \Omega }\) is a left \(\mathcal{J}_{s ; \Omega }\)-family.
Similarly, we show that if \(\{u_{m}\}\) and \(\{v_{m}\}\) in U satisfy (3.5) and (3.7), then (3.9) holds, and thus \(\mathcal{J}_{s ; \Omega }\) is a right \(\mathcal{J}_{s ; \Omega }\)-family.
Now, using the left (right) \(\mathcal{J}_{s ; \Omega }\)-family on U, we define the left (right) \(\mathcal{J}_{s ; \Omega }\)-completeness in the quasi-b-gauge space \((U,\textsl{Q}_{s ; \Omega })\).
Definition 3.13
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi-b-gauge space, and let \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) be a left (right) \(\mathcal{J}_{s ; \Omega }\)-family on U.
-
(A)
A sequence \(\{u_{m}\}_{m\in \mathbb{N}}\) is said to be a left (right) \(\mathcal{J}_{s ; \Omega }\)-Cauchy sequence in U if for all \(\beta \in \Omega \), we have
$$ \lim_{m\rightarrow \infty }\sup_{n>m}\textsl{J}_{\beta }(u_{m},u_{n})=0\quad \Bigl(\lim_{m\rightarrow \infty }\sup_{n>m} \textsl{J}_{\beta }(u_{n},u_{m})=0 \Bigr). $$
-
(B)
A sequence \(\{u_{m}\}_{m\in \mathbb{N}}\) is said to be the left (right) \(\mathcal{J}_{s ; \Omega }\)-convergent to \(u\in U\) if \(\lim_{m\rightarrow \infty }^{L-\mathcal{J}_{s ; \Omega }}u_{m} = u\) \((\lim_{m\rightarrow \infty }^{R-\mathcal{J}_{s ; \Omega }}u_{m} = u)\), where \(\lim_{m\rightarrow \infty }^{L-\mathcal{J}_{s ; \Omega }}u_{m} = u \Leftrightarrow \forall _{\beta \in \Omega }\{\lim_{m\rightarrow \infty }\textsl{J}_{\beta }(u,u_{m})=0\}\) (\(\lim_{m\rightarrow \infty }^{R-\mathcal{J}_{s ; \Omega }}u_{m} = u \Leftrightarrow \forall _{\beta \in \Omega }\{\lim_{m\rightarrow \infty }\textsl{J}_{\beta }(u_{m},u)=0\} \)).
-
(C)
If \(S_{({u_{m}: m\in \mathbb{N}})}^{L-\mathcal{J}_{s ; \Omega }}\neq \emptyset \) \((S_{({u_{m}: m\in \mathbb{N}})}^{R-\mathcal{J}_{s ; \Omega }}\neq \emptyset )\), where \(S_{({u_{m}: m\in \mathbb{N}})}^{L-\mathcal{J}_{s ; \Omega }}= \{u \in U : \lim^{L-\mathcal{J}_{s ; \Omega }}_{m\rightarrow \infty }u_{m} = u\}\) (\(S_{({u_{m}: m\in \mathbb{N}})}^{R-\mathcal{J}_{s ; \Omega }}= \{u \in U : \lim^{R-\mathcal{J}_{s ; \Omega }}_{m\rightarrow \infty }u_{m} = u\} \)). Then the sequence \(\{u_{m}\}_{m\in \mathbb{N}}\) in U is the left (right) \(\mathcal{J}_{s ; \Omega }\)-convergent in U.
-
(D)
The space \((U,\textsl{Q}_{s ; \Omega })\) is a left (right) \(\mathcal{J}_{s ; \Omega }\)-sequentially complete quasi-b-gauge space if each left (right) \(\mathcal{J}_{s ; \Omega }\)-Cauchy sequence in U is left (right) \(\mathcal{J}_{s ; \Omega }\)-convergent in U.
Definition 3.14
Suppose \((U,\textsl{Q}_{s ; \Omega })\) is a quasi-b-gauge space, and let \(T:U \rightarrow 2^{U}\) be a set valued map. The map \(T^{[k]}\) (for \(k\in \mathbb{N}\)) is called a left (right) \(\textsl{Q}_{s ; \Omega }\)-quasi-closed map on U if for each sequence \(\{z_{m}\}_{m\in \mathbb{N}}\) within \(T^{[k]}(U)\), which is left (right) \(\textsl{Q}_{s ; \Omega }\)-convergent in U, thus \(S_{({z_{m}: m\in \mathbb{N}})}^{L-\textsl{Q}_{s ; \Omega }}\neq \emptyset \) (\(S_{({z_{m}: m\in \mathbb{N}})}^{R-\textsl{Q}_{s ; \Omega }}\neq \emptyset \)), having \(\{x_{m}\}_{m\in \mathbb{N}}\) and \(\{y_{m}\}_{m\in \mathbb{N}}\) as its subsequences satisfying \(y_{m}\in T^{[k]}(x_{m})\) for all \(m\in \mathbb{N}\), has the property that there exists \(z\in S^{L-\textsl{Q}_{s ; \Omega }}_{(z_{m}:m\in \mathbb{N})}(z\in S^{R- \textsl{Q}_{s ; \Omega }}_{(z_{m}:m\in \mathbb{N})})\) such that \(z= T^{[k]}(z)\) (\(z= T^{[k]}(z)\)).
Remark 3.15
Suppose \((U,\textsl{Q}_{s ; \Omega })\) is a quasi-b-gauge space.
-
(a)
If \(\{u_{m}:m\in \mathbb{N}\}\) is a left (right) \(\mathcal{J}_{s ; \Omega }\)-convergent sequence in U, then for every subsequence \(\{v_{m}\}_{m\in \mathbb{N}}\), we have
$$ S_{({u_{m}: m\in \mathbb{N}})}^{L-\mathcal{J}_{s ; \Omega }}\subset S_{({v_{m}: m\in \mathbb{N}})}^{L-\mathcal{J}_{s ; \Omega }} \quad \bigl(S_{({u_{m}: m \in \mathbb{N}})}^{R-\mathcal{J}_{s ; \Omega }}\subset S_{({v_{m}: m \in \mathbb{N}})}^{R-\mathcal{J}_{s ; \Omega }} \bigr). $$
-
(b)
We observe that if \(s=1\) for all \(\beta \in \Omega \), the above definitions turn down to agree with the definitions in quasi-gauge spaces.
Definition 3.16
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi-b-gauge space. Let \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) be a left (right) \(\mathcal{J}_{s ; \Omega }\)-family on U. The map \(T: U\rightarrow 2^{U}\) is the left (right) \(\mathcal{J}_{s ; \Omega }\)-admissible at a point \(z^{0}\in U\) if for any sequence \(\{z^{m}: m\in \{0\}\cup \mathbb{N}\}\) satisfying \(z^{m+1}\in T(z^{m})\) for all \(m \in \{0\}\cup \mathbb{N}\) and \(\lim_{m\rightarrow \infty }\sup_{n>m}\textsl{J}_{\beta }(z^{m},z^{n})=0\) \((\lim_{m\rightarrow \infty }\sup_{n>m}\textsl{J}_{\beta }(z^{n},z^{m})=0 )\) for all \(\beta \in \Omega \), there exists \(z\in U\) such that for all \(\beta \in \Omega \), \(\lim_{m\rightarrow \infty }\textsl{J}_{\beta }(z,z^{m})=0\) (\(\lim_{m\rightarrow \infty }\textsl{J}_{\beta }(z^{m},z)=0 \)).
The set-valued map \(T: U\rightarrow 2^{U}\) is the left (right) \(\mathcal{J}_{s ; \Omega }\)-admissible in U if \(T: U\rightarrow 2^{U}\) is the left (right) \(\mathcal{J}_{s ; \Omega }\)-admissible at each point \(z^{0}\in U\).
Remark 3.17
Suppose \((U,\textsl{Q}_{s ; \Omega })\) is a quasi-b-gauge space, and let \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) be a left (right) \(\mathcal{J}_{s ; \Omega }\)-family on U.
-
(a)
If \((U,\textsl{Q})\) is left (right) \(\mathcal{J}_{s ; \Omega }\)-sequentially complete, then \(T: U\rightarrow 2^{U}\) is the left (right) \(\mathcal{J}_{s ; \Omega }\)-admissible on U.
-
(b)
If \(s =1\), the above definitions reduce to the corresponding definition in quasi-gauge spaces.
Definition 3.18
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi b-gauge space, and let \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) be the a left (right) \(\mathcal{J}_{s ; \Omega }\)-family on U. A set \(W\in 2^{U}\) is the left (right) \(\mathcal{J}_{s ; \Omega }\)-closed in U if \(W = cl^{L-\mathcal{J}_{s ; \Omega }}_{U}(W)\) \((W = cl^{R-\mathcal{J}_{s ; \Omega }}_{U}(W))\), where \(cl^{L-\mathcal{J}_{s ; \Omega }}_{U}(W)\) \((cl^{R-\mathcal{J}_{s ; \Omega }}_{U}(W))\), is the left (right) \(\mathcal{J}_{s ; \Omega }\)-closure in U and is defined by \(cl^{L-\mathcal{J}_{s ; \Omega }}_{U}(W)=\{z\in U : \lim_{m \rightarrow \infty }^{L-\mathcal{J}_{s ; \Omega }}z_{m} = z\}\)
\((cl^{R-\mathcal{J}_{s ; \Omega }}_{U}(W)=\{z\in U : \lim_{m \rightarrow \infty }^{R-\mathcal{J}_{s ; \Omega }}z_{m} = z\})\).
Define \(Cl^{L-\mathcal{J}_{s ; \Omega }}(U)=\{W\in 2^{U}: W = cl^{L- \mathcal{J}_{s ; \Omega }}_{U}(W)\}\) \((Cl^{R-\mathcal{J}_{s ; \Omega }}(U)=\{W\in 2^{U}: W = cl^{R- \mathcal{J}_{s ; \Omega }}_{U}(W)\})\). Thus, \(Cl^{L-\mathcal{J}_{s ; \Omega }}(U)\) \((Cl^{R-\mathcal{J}_{s ; \Omega }}(U))\) symbolizes the class of all non-empty left (right) \(\mathcal{J}_{s ; \Omega }\)-closed subsets of U.
Remark 3.19
We note that if \(s =1\), the above definition reduces to the corresponding definition in quasi-gauge spaces.
In a quasi-b-gauge space, we describe the left (right) Hausdorff type quasi-b-distances and Nadler type left (right) contractions in the following way.
Definition 3.20
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi-b-gauge space, and let \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) be a left (right) \(\mathcal{J}_{s ; \Omega }\)-family on U. Let \(\zeta \in \{1,2,3\}\) and suppose that for all \(\beta \in \Omega \), for all \(u\in U\), and for all \(V \in 2^{U}\),
$$\begin{aligned}& \textsl{J}_{\beta }(u,V)=\inf \bigl\{ \textsl{J}_{\beta }(u,w): w\in V \bigr\} \\& \quad {}\wedge \textsl{J}_{\beta }(V,u)=\inf \bigl\{ \textsl{J}_{\beta }(w,u): w\in V \bigr\} . \end{aligned}$$
(3.14)
-
(a)
Define on \(Cl^{L-\mathcal{J}_{s ; \Omega }}(U)\) \((Cl^{R-\mathcal{J}_{s ; \Omega }}(U))\), the left (right) quasi-b-distance \(\mathcal{D}^{L-\mathcal{J}_{s ; \Omega }}_{\zeta }=\{D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }, \beta \in \Omega \}(\mathcal{D}^{R- \mathcal{J}_{s ; \Omega }}_{\zeta }=\{D^{R-\mathcal{J}_{s ; \Omega }}_{ \zeta ;\beta }, \beta \in \Omega \})\) of the Hausdorff type, where \(D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }:Cl^{L-\mathcal{J}_{s ; \Omega }}(U)\times Cl^{L-\mathcal{J}_{s ; \Omega }}(U)\rightarrow [0, \infty ]\) \((D^{R-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }: Cl^{R-\mathcal{J}_{s ; \Omega }}(U)\times Cl^{R-\mathcal{J}_{s ; \Omega }}(U)\rightarrow [0, \infty ], \beta \in \Omega )\) for all \(\beta \in \Omega \) and for all \(U,V \in Cl^{\mathcal{J}_{s ; \Omega }}(U)\) as:
-
(a.1)
$$\begin{aligned}& D^{L-\mathcal{J}_{s ; \Omega }}_{1;\beta }(U,V)= \max \Bigl\{ \sup_{u \in U} \textsl{J}_{\beta }(u,V), \sup_{v\in V}\textsl{J}_{\beta }(U,v) \Bigr\} ,\\& D^{L-\mathcal{J}_{s ; \Omega }}_{2;\beta }(U,V)= \max \Bigl\{ \sup_{u \in U} \textsl{J}_{\beta }(u,V), \sup_{v\in V}\textsl{J}_{\beta }(v,U) \Bigr\} \quad \text{and}\\& D^{L-\mathcal{J}_{s ; \Omega }}_{3;\beta }(U,V)= \sup_{u\in U} \textsl{J}_{\beta }(u,V), \quad \text{if } \mathcal{J}_{s ; \Omega }\in \mathbb{J}^{L}_{(U,\textsl{Q})}; \end{aligned}$$
-
(a.2)
$$\begin{aligned}& D^{R-\mathcal{J}_{s ; \Omega }}_{1;\beta }(U,V)= \max \Bigl\{ \sup_{u \in U} \textsl{J}_{\beta }(u,V), \sup_{v\in V}\textsl{J}_{\beta } \Bigr\} (U,v),\\& D^{R-\mathcal{J}_{s ; \Omega }}_{2;\beta }(U,V)= \max \Bigl\{ \sup_{u \in U} \textsl{J}_{\beta }(u,V), \sup_{v\in V}\textsl{J}_{\beta }(v,U) \Bigr\} \quad \text{and}\\& D^{R-\mathcal{J}_{s ; \Omega }}_{3;\beta }(U,V)= \sup_{u\in U} \textsl{J}_{\beta }(u,V),\quad \text{if } \mathcal{J}_{s ; \Omega } \in \mathbb{J}^{R}_{(U,\textsl{Q})}. \end{aligned}$$
-
(b)
Let \(\mu =\{\mu _{\beta }\}_{\beta \in \Omega }\in [0,1)^{\Omega }\). The set-valued map \(T:U\rightarrow Cl^{L-\mathcal{J}_{s ; \Omega }}(U)\) \((T:U\rightarrow Cl^{R-\mathcal{J}_{s ; \Omega }}(U))\) is a left (right) \((\mathcal{D}^{L-\mathcal{J}_{s ; \Omega }}_{\zeta },\mu )\)-contraction on U (\((\mathcal{D}^{R-\mathcal{J}_{s ; \Omega }}_{\zeta },\mu )\)-contraction on U) if for all \(\beta \in \Omega \) and for all \(x,y \in U\):
-
(b.1)
\(D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }(T(x),T(y))\leq \mu _{ \beta }\textsl{J}_{\beta }(x,y)\), if \(\mathcal{J}_{s ; \Omega } \in \mathbb{J}^{L}_{(U,\textsl{Q})}\);
-
(b.2)
\(D^{R-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }(T(x),T(y))\leq \mu _{ \beta }\textsl{J}_{\beta }(x,y)\), if \(\mathcal{J}_{s ; \Omega } \in \mathbb{J}^{R}_{(U,\textsl{Q})}\).
Remark 3.21
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi-b-gauge space, and let \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) of maps \(\textsl{J}_{\beta } :U\times U\rightarrow [0,\infty )\), \(\beta \in \Omega \) be a left (right) \(\mathcal{J}_{s ; \Omega }\)-family on U.
-
(a)
In general, \(D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta } (D^{R-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta })\) are not symmetric, thus \(D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }(U,V)=D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }(V,U) (D^{R-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }(U,V)=D^{R-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }(V,U))\) does not necessarily hold. Also, \(D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }(U,U)=0 (D^{R- \mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }(U,U)=0)\) does not necessarily hold (See Remark 3.27 (b) and (c) for details).
-
(b)
Each \((\mathcal{D}^{L-\mathcal{J}_{s ; \Omega }}_{\zeta },\mu )\)-contraction on U (\((\mathcal{D}^{R-\mathcal{J}_{s ; \Omega }}_{\zeta },\mu )\)-contraction on U), \(\zeta \in \{1,2,3\}\) is \((\mathcal{D}^{L-\mathcal{J}_{s ; \Omega }}_{3},\mu )\)-contraction on U (\((\mathcal{D}^{R-\mathcal{J}_{s ; \Omega }}_{3},\mu )\)-contraction on U), but the converse is not generally true.
Our main result for set-valued mappings is given below.
Theorem 3.22
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi-b-gauge space. Let \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) be a left (right) \(\mathcal{J}_{s ; \Omega }\)-family on U and let \(\zeta \in \{1,2,3\}\). Assume, moreover, that \(\mu =\{\mu _{\beta }\}_{\beta \in \Omega }\in [0,1)\), and the set-valued map \(T:U\rightarrow Cl^{L-\mathcal{J}_{s ; \Omega }}(U)\) \((T:U\rightarrow Cl^{R-\mathcal{J}_{s ; \Omega }}(U))\) satisfies:
-
(i)
T is a \((\mathcal{D}^{L-\mathcal{J}_{s ; \Omega }}_{\zeta },\mu )\)-contraction on U (\((\mathcal{D}^{R-\mathcal{J}_{s ; \Omega }}_{\zeta },\mu )\)-contraction on U);
-
(ii)
for any \(u\in U\) and any \(\gamma =\{\gamma _{\beta }\}_{\beta \in \Omega }\in (0,\infty )\), there exists \(v\in T(u)\) such that for all \(\beta \in \Omega \),
$$\begin{aligned}& \textsl{J}_{\beta }(u,v)< \textsl{J}_{\beta } \bigl(u,T(u) \bigr)+\gamma _{\beta }, \end{aligned}$$
(3.15)
$$\begin{aligned}& \bigl(\textsl{J}_{\beta }(v,u)< \textsl{J}_{\beta } \bigl(T(u),u \bigr)+\gamma _{ \beta } \bigr). \end{aligned}$$
(3.16)
We have the following:
-
(I)
If \((U,T)\) at a point \(z^{0}\in U\) is the left (right) \(\mathcal{J}_{s ; \Omega }\)-admissible, then there exists a sequence \(\{z^{m}:m\in \{0\}\cup \mathbb{N}\}\) starting at \(z^{0}\in U\) such that \(z^{m}\in T(z^{m-1})\) for all \(m\in \mathbb{N}\), a point \(z\in U\) and \(r=\{r_{\beta }\}_{\beta \in \Omega }\in (0,\infty )\) such that \(z^{m}\in B^{L-\mathcal{J}_{s ; \Omega }}(z^{0},r)\) (\(z^{m}\in B^{R-\mathcal{J}_{s ; \Omega }}(z^{0},r)\}\)) for all \(m\in \mathbb{N}\) and \(\lim_{m\rightarrow \infty }^{L-\mathcal{J}_{s ; \Omega }}z_{m} = z\) (\(\lim_{m\rightarrow \infty }^{R-\mathcal{J}_{s ; \Omega }}z_{m} = z \)).
-
(II)
If \((U,T)\) at a point \(z^{0}\in U\) is the left (right) \(\mathcal{J}_{s ; \Omega }\)-admissible and if \(T^{[k]}\), for some \(k\in \mathbb{N}\), is a left (right) \(\textsl{Q}_{s ; \Omega }\)-quasi-closed map on U, then \(\mathrm{Fix} (T^{[k]})\) is non-empty and there exists a sequence \(\{z^{m}:m\in \{0\}\cup \mathbb{N}\}\) starting at \(z^{0}\in U\) such that \(z^{m}\in T(z^{m-1})\) for all \(m\in \mathbb{N}\), a point \(z\in \operatorname{Fix}(T^{[k]})\) and \(r=\{r_{\beta }\}_{\beta \in \Omega }\in (0,\infty )\) such that \(z^{m}\in B^{L-\mathcal{J}_{s ; \Omega }}(z^{0},r)\) (\(z^{m}\in B^{R- \mathcal{J}_{s ; \Omega }}(z^{0},r)\)) for all \(m\in \mathbb{N}\) and \(\lim_{m\rightarrow \infty }^{L-\mathcal{J}_{s ; \Omega }}z_{m} = z\) (\(\lim_{m\rightarrow \infty }^{R-\mathcal{J}_{s ; \Omega }}z_{m} = z \)).
Proof
(I) Suppose that \((U,T)\) is the left (right) \(\mathcal{J}_{s ; \Omega }\)-admissible at a point \(z^{0}\in U\).
From using (3.14) and the fact that \(\textsl{J}_{\beta } :U\times U\rightarrow [0,\infty )\), \(\beta \in \Omega \), we choose
$$\begin{aligned}& r=\{r_{\beta }\}_{\beta \in \Omega }\in (0,\infty ), \end{aligned}$$
(3.17)
$$\begin{aligned}& s=\{s_{\beta }\}_{\beta \in \Omega }\in [1,\infty ) \end{aligned}$$
(3.18)
such that for all \(\beta \in \Omega \)
$$\begin{aligned} \textsl{J}_{\beta } \bigl(z^{0},T \bigl(z^{0} \bigr) \bigr)< \frac{(1-\mu _{\beta })r_{\beta }}{s_{\beta }}. \end{aligned}$$
(3.19)
Put
$$\begin{aligned} \gamma ^{(0)}_{\beta }= \frac{(1-\mu _{\beta })r_{\beta }}{s_{\beta }}- \textsl{J}_{\beta } \bigl(z^{0},T \bigl(z^{0} \bigr) \bigr) \quad \textrm{for all } \beta \in \Omega . \end{aligned}$$
(3.20)
From (3.17), (3.18), and (3.19), we have \(\gamma ^{(0)}=\{\gamma ^{(0)}_{\beta }\}_{\beta \in \Omega }\in (0, \infty )\). Applying (3.15), we get \(z^{1}\in T(z^{(0)})\) such that
$$\begin{aligned} \textsl{J}_{\beta } \bigl(z^{0},z^{1} \bigr)< \textsl{J}_{\beta } \bigl(z^{0},T \bigl(z^{0} \bigr) \bigr)+ \gamma ^{(0)}_{\beta } \quad \textrm{for all } \beta \in \Omega . \end{aligned}$$
(3.21)
We see from (3.20) and (3.21) that
$$\begin{aligned} \textsl{J}_{\beta } \bigl(z^{0},z^{1} \bigr)< \frac{(1-\mu _{\beta })r_{\beta }}{s_{\beta }} \quad \textrm{for all } \beta \in \Omega . \end{aligned}$$
(3.22)
Observe that (3.22) implies \(z^{1}\in B^{L-\mathcal{J}_{s ; \Omega }}(z^{0},r)\).
Put now
$$\begin{aligned} \gamma ^{(1)}_{\beta }=\mu _{\beta } \biggl[ \frac{(1-\mu _{\beta })r_{\beta }}{s^{2}_{\beta }}-\textsl{J}_{\beta } \bigl(z^{0},z^{1} \bigr) \biggr] \quad \textrm{for all } \beta \in \Omega . \end{aligned}$$
(3.23)
From (3.22), we have \(\gamma ^{(1)}=\{\gamma ^{(1)}_{\beta }\}_{\beta \in \Omega }\in (0, \infty )\), and we apply (3.15) to find \(z^{2}\in T(z^{(1)})\) such that
$$\begin{aligned} \textsl{J}_{\beta } \bigl(z^{1},z^{2} \bigr)< \textsl{J}_{\beta } \bigl(z^{1},T \bigl(z^{1} \bigr) \bigr)+ \gamma ^{(1)}_{\beta } \quad \textrm{for all } \beta \in \Omega . \end{aligned}$$
(3.24)
Also, note that
$$\begin{aligned} \textsl{J}_{\beta } \bigl(z^{1},z^{2} \bigr)< \frac{\mu _{\beta }(1-\mu _{\beta })r_{\beta }}{s^{2}_{\beta }} \quad \textrm{for all } \beta \in \Omega . \end{aligned}$$
(3.25)
Indeed, from (3.24), (3.14), Definition 3.20, and (3.23), we get for all \(\beta \in \Omega \),
$$\begin{aligned} \textsl{J}_{\beta } \bigl(z^{1},z^{2} \bigr)&< \textsl{J}_{\beta } \bigl(z^{1},T \bigl(z^{1} \bigr) \bigr)+ \gamma ^{(1)}_{\beta } \\ & \leq \sup_{u\in T(z^{0})} \textsl{J}_{\beta } \bigl(u,T \bigl(z^{1} \bigr) \bigr)+ \gamma ^{(1)}_{\beta } \\ & \leq D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta } \bigl(T \bigl(z^{0} \bigr),T \bigl(z^{1} \bigr) \bigr)+ \gamma ^{(1)}_{\beta } \\ &\leq \mu _{\beta }\textsl{J}_{\beta } \bigl(z^{0},z^{1} \bigr)+ \gamma ^{(1)}_{\beta } = \frac{\mu _{\beta }(1-\mu _{\beta })r_{\beta }}{s^{2}_{\beta }}, \quad \zeta \in \{1,2,3\}. \end{aligned}$$
Thus, (3.25) holds. Further, by \((\mathcal{J}_{1})\) there exists \(s=\{s_{\beta }\}_{\beta \in \Omega }\in [1,\infty )\). Using (3.22) and (3.25), we have for all \(\beta \in \Omega \),
$$\begin{aligned} \textsl{J}_{\beta } \bigl(z^{0},z^{2} \bigr)&\leq s_{\beta } \bigl\{ \textsl{J}_{\beta } \bigl(z^{0},z^{1} \bigr)+ \textsl{J}_{\beta } \bigl(z^{1},z^{2} \bigr) \bigr\} \\ &< s_{\beta } \biggl\{ \frac{(1-\mu _{\beta })r_{\beta }}{s_{\beta }} + \frac{\mu _{\beta }(1-\mu _{\beta })r_{\beta }}{s^{2}_{\beta }} \biggr\} \\ &\leq (1-\mu _{\beta })r_{\beta } \biggl(1+\frac{\mu _{\beta }}{s_{\beta }} \biggr) \leq (1-\mu _{\beta })r_{\beta }(1+\mu _{\beta }) \\ & \leq (1-\mu _{\beta })r_{\beta }\sum^{\infty }_{k=0} \mu ^{k}_{\beta }=r_{ \beta }. \end{aligned}$$
Thus, \(z^{2}\in B^{L-\mathcal{J}_{s ; \Omega }}(z^{0},r)\). Repeating the above process, using Definition 3.20 and property (3.15), we find a sequence \(\{z^{m}\}_{m\in \mathbb{N}}\) in U satisfying
$$\begin{aligned} z^{m+1}\in T \bigl(z^{m} \bigr) \quad \textrm{for all } m\in \{0\}\cup \mathbb{N}. \end{aligned}$$
(3.26)
Letting \(\gamma ^{(m)}=\{\gamma ^{(m)}_{\beta }\}_{\beta \in \Omega }\) for all \(m\in \mathbb{N}\), where
$$ \gamma ^{(m)}_{\beta }= \mu _{\beta } \biggl[ \frac{\mu ^{m-1}_{\beta }(1-\mu _{\beta })r_{\beta }}{s^{m+1}_{\beta }}- \textsl{J}_{\beta } \bigl(z^{m-1},z^{m} \bigr) \biggr]. $$
We also notice that \(\{\gamma ^{(m)}\in (0,\infty ): m\in \mathbb{N}\}\) and for all \(\beta \in \Omega \) and for all \(m\in \{0\}\cup \mathbb{N}\), we have
$$\begin{aligned}& \textsl{J}_{\beta } \bigl(z^{m},z^{m+1} \bigr)< \textsl{J}_{\beta } \bigl(z^{m},T \bigl(z^{m} \bigr) \bigr)+ \gamma ^{(m)}, \\& \textsl{J}_{\beta } \bigl(z^{m},z^{m+1} \bigr)< \frac{\mu ^{m}_{\beta }(1-\mu _{\beta })r_{\beta }}{s^{m+1}_{\beta }}. \end{aligned}$$
(3.27)
For all \(\beta \in \Omega \) and for all \(m\in \{0\}\cup \mathbb{N}\), we can write
$$\begin{aligned} \textsl{J}_{\beta } \bigl(z^{0},z^{m+1} \bigr) \leq& s_{\beta } \textsl{J}_{\beta } \bigl(z^{0},z^{1} \bigr)+s^{2}_{\beta }\textsl{J}_{\beta } \bigl(z^{1},z^{2} \bigr)+s^{3}_{ \beta } \textsl{J}_{\beta } \bigl(z^{2},z^{3} \bigr) \\ &{} +\cdots+s^{m}_{\beta }\textsl{J}_{\beta } \bigl(z^{m-1},z^{m} \bigr)+s^{m}_{ \beta } \textsl{J}_{\beta } \bigl(z^{m},z^{m+1} \bigr) \\ < &s_{\beta } \frac{(1-\mu _{\beta })r_{\beta }}{s_{\beta }}+s^{2}_{\beta } \frac{\mu _{\beta }(1-\mu _{\beta })r_{\beta }}{s^{2}_{\beta }} +s^{3}_{ \beta }\frac{\mu ^{2}_{\beta }(1-\mu _{\beta })r_{\beta }}{s^{3}_{\beta }} \\ &{} +\cdots+s^{m}_{\beta } \frac{\mu ^{m-1}_{\beta }(1-\mu _{\beta })r_{\beta }}{s^{m}_{\beta }} +s^{m}_{ \beta } \frac{\mu ^{m}_{\beta }(1-\mu _{\beta })r_{\beta }}{s^{m+1}_{\beta }} \\ =&(1-\mu _{\beta })r_{\beta } \biggl\{ 1+\mu _{\beta }+\mu ^{2}_{\beta }\cdots \mu ^{m-1}_{\beta }+ \frac{\mu ^{m}_{\beta }}{s_{\beta }} \biggr\} \\ \leq& (1-\mu _{\beta })r_{\beta } \bigl\{ 1+\mu _{\beta }+\mu ^{2}_{\beta }\cdots \mu ^{m-1}_{\beta }+\mu ^{m}_{\beta } \bigr\} \\ =&(1-\mu _{\beta })r_{\beta }\sum^{m}_{k=0} \mu ^{k}_{\beta }. \\ < & (1-\mu _{\beta })r_{\beta }\sum^{\infty }_{k=0} \mu ^{k}_{\beta }=r_{ \beta }. \end{aligned}$$
Hence, this implies that \(z^{m}\in B^{L-\mathcal{J}_{s ; \Omega }}(z^{0},r)\) for all \(m\in \mathbb{N}\).
Using \((\mathcal{J}1)\) and (3.27), for all \(m,n\in \mathbb{N}\) such that \(n>m\), we have
$$\begin{aligned} \lim_{m \rightarrow \infty }\sup_{n>m} \textsl{J}_{\beta } \bigl(z^{m},z^{n} \bigr) \leq& \lim_{m\rightarrow \infty }\sup_{n>m} \bigl\{ s_{\beta } \textsl{J}_{ \beta } \bigl(z^{m},z^{m+1} \bigr)+s^{2}_{\beta }\textsl{J}_{\beta } \bigl(z^{m+1},z^{m+2} \bigr) \\ &{} +\cdots+s^{n-m-1}_{\beta }\textsl{J}_{\beta } \bigl(z^{n-2},z^{n-1} \bigr)+s^{n-m-1}_{ \beta } \textsl{J}_{\beta } \bigl(z^{n-1},z^{n} \bigr) \bigr\} \\ \leq& \lim_{m\rightarrow \infty }\sup_{n>m} \biggl\{ s_{\beta } \frac{\mu ^{m}_{\beta }(1-\mu _{\beta })r_{\beta }}{s^{m+1}_{\beta }}+s^{2}_{ \beta } \frac{\mu ^{m+1}_{\beta }(1-\mu _{\beta })r_{\beta }}{s^{m+2}_{\beta }} \\ &{}+\cdots+s^{n-m-1}_{\beta } \frac{\mu ^{n-2}_{\beta }(1-\mu _{\beta })r_{\beta }}{s^{n-1}_{\beta }}+s^{n-m-1}_{ \beta } \frac{\mu ^{n-1}_{\beta }(1-\mu _{\beta })r_{\beta }}{s^{n-1+1}_{\beta }} \biggr\} \\ \leq& \lim_{m\rightarrow \infty }\sup_{n>m}(1-\mu _{\beta })r_{\beta } \biggl\{ \frac{\mu ^{m}_{\beta }}{s^{m}_{\beta }}+ \frac{\mu ^{m+1}_{\beta }}{s^{m}_{\beta }}+ \cdots +\frac{\mu ^{n-2}_{\beta }}{s^{m}_{\beta }}+ \frac{\mu ^{n-1}_{\beta }}{s^{m+1}_{\beta }} \biggr\} \\ \leq& \lim_{m\rightarrow \infty }\sup_{n>m}(1-\mu _{\beta })r_{\beta } \bigl\{ \mu ^{m}_{\beta }+\mu ^{m+1}_{\beta }+\cdots+\mu ^{n-2}_{\beta }+\mu ^{n-1}_{ \beta } \bigr\} \\ =& (1-\mu _{\beta })r_{\beta } \lim_{m \rightarrow \infty }\sup _{n>m} \sum^{n-1}_{j=m}\mu ^{j}_{\beta } \\ \leq& r_{\beta }\lim_{m\rightarrow \infty }\mu ^{m}_{\beta }. \end{aligned}$$
This implies
$$\begin{aligned} \forall _{\beta \in \Omega } \Bigl\{ \lim _{m \rightarrow \infty } \sup_{n>m} \textsl{J}_{\beta } \bigl(z^{m},z^{n} \bigr)=0 \Bigr\} \quad \textrm{for all } \beta \in \Omega . \end{aligned}$$
(3.28)
Given that \((U,T)\) is thr left \(\mathcal{J}_{s ; \Omega }\)-admissible on U, hence using Definition 3.16, properties (3.26) and (3.28), we find \(z \in U\) such that
$$\begin{aligned} \lim_{m \rightarrow \infty }\textsl{J}_{\beta } \bigl(z,z^{m} \bigr)=0 \quad \textrm{for all } \beta \in \Omega . \end{aligned}$$
(3.29)
Now, taking \(v_{m}=z\) and \(u_{m}=z^{m}\) for \(m\in \mathbb{N}\), we observe that condition (3.4) and (3.6) hold for \(\{u_{m}\}\) and \(\{v_{m}\}\) in U by (3.28) and (3.29). Consequently, we get (3.8) by \((\mathcal{J}2)\) which gives
$$\begin{aligned} \lim_{m \rightarrow \infty }\textsl{q}_{\beta } \bigl(z,z^{m} \bigr)=\lim_{m \rightarrow \infty }\textsl{q}_{\beta }(v_{m},u_{m})=0 \quad \textrm{for all } \beta \in \Omega . \end{aligned}$$
Thus, we have \(z\in S^{L-\textsl{Q}_{s ; \Omega }}_{(z_{m}: m\in \mathbb{N})}=\{x \in U: \lim^{L-\textsl{Q}_{s ; \Omega }}_{m \rightarrow \infty }z^{m}=x \}\).
(II) Let \((U,T)\) be the left \(\mathcal{J}_{s ; \Omega }\)-admissible at a point \(z^{0}\in U\) and \(T^{[k]}\) be the left \(\textsl{Q}_{s ; \Omega }\)-quasi-closed on U, for some \(k\in \mathbb{N}\).
Let \(z^{0}\in U\) be arbitrary. Since \(S^{L-\textsl{Q}_{s ; \Omega }}_{(z^{m}: m\in \{0\}\cup \mathbb{N})} \neq \emptyset \) and for \(m\in \{0\}\cup \mathbb{N}\), we have
$$ z^{(m+1)k}\in T^{[k]} \bigl(z^{mk} \bigr). $$
By defining \(\{z_{m}=z^{m-1+k}: m\in \mathbb{N}\}\), we can write
$$ z_{m}\subset T^{[k]}(U) $$
and
$$ S^{L-\textsl{Q}_{s ; \Omega }}_{(z_{m}: m\in \{0\}\cup \mathbb{N})}=S^{L- \textsl{Q}_{s ; \Omega }}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}\neq \emptyset . $$
Also, its subsequences
$$ \bigl\{ y_{m}=z^{(m+1)k}:m\in \mathbb{N} \bigr\} \subset T^{[k]}(U) $$
and
$$ \bigl\{ x_{m}=z^{mk}:m\in \mathbb{N} \bigr\} \subset T^{[k]}(U) $$
satisfy
$$ y_{m}=T^{[k]}(x_{m}) \quad \textrm{for all } m \in \mathbb{N} $$
and are the left \(\textsl{Q}_{s ; \Omega }\)-convergent to each point \(z\in S^{L-\textsl{Q}_{s ; \Omega }}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}\). Now, since
$$ S_{({z_{m}: m\in \mathbb{N}})}^{L-\textsl{Q}_{s ; \Omega }}\subset S_{({y_{m}: m\in \mathbb{N}})}^{L-\textsl{Q}_{s ; \Omega }} \quad \text{and} \quad S_{({z_{m}: m\in \mathbb{N}})}^{L-\textsl{Q}_{s ; \Omega }}\subset S_{({x_{m}: m\in \mathbb{N}})}^{L-\textsl{Q}_{s ; \Omega }}, $$
using the assumption that \(T^{[k]}\) for some \(k\in \mathbb{N}\) is a left (right) \(\textsl{Q}_{s ; \Omega }\)-quasi-closed map on U, there exists \(z\in S^{L-\textsl{Q}_{s ; \Omega }}_{(z_{m}: m\in \{0\}\cup \mathbb{N})}=S^{L-\textsl{Q}_{s ; \Omega }}_{(z^{m}: m\in \{0\}\cup \mathbb{N})}\) such that \(z\in T^{[k]}(z)\). This completes the proof. □
We now extend the above theorems to the Banach type single-valued left (right)-contractions.
Definition 3.23
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi-b-gauge space, let \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) is a left (right) \(\mathcal{J}_{s ; \Omega }\)-family on U, and let \(\zeta \in \{1,2\}\).
-
(c)
The left (right) b-distance \(\mathcal{D}^{L-\mathcal{J}_{s ; \Omega }}_{\zeta }=\{D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }, \beta \in \Omega \}(\mathcal{D}^{R- \mathcal{J}_{s ; \Omega }}_{\zeta }=\{D^{R-\mathcal{J}_{s ; \Omega }}_{ \zeta ;\beta }, \beta \in \Omega \})\) on U, where \(D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }:U\times U\rightarrow [0, \infty )\), \(\beta \in \Omega (D^{R-\mathcal{J}_{s ; \Omega }}_{\zeta ; \beta }: U\times U\rightarrow [0,\infty ), \beta \in \Omega )\) are defined for all \(\beta \in \Omega \) and for all \(u,v \in U\) as follows:
-
(c.1)
$$\begin{aligned}& D^{L-\mathcal{J}_{s ; \Omega }}_{1;\beta }(u,v)= \max \bigl\{ \textsl{J}_{\beta }(u,v), \textsl{J}_{\beta }(v,u)\bigr\} , \\& D^{L-\mathcal{J}_{s ; \Omega }}_{2;\beta }(u,v)= \textsl{J}_{\beta }(u,v), \quad \text{if } \mathcal{J}_{s ; \Omega }\in \mathbb{J}^{L}_{(U,\textsl{Q}_{s ; \Omega })}; \end{aligned}$$
-
(c.2)
$$\begin{aligned}& D^{R-\mathcal{J}_{s ; \Omega }}_{1;\beta }(u,v)= \max \bigl\{ \textsl{J}_{\beta }(u,v), \textsl{J}_{\beta }(v,u)\bigr\} , \\& D^{R-\mathcal{J}_{s ; \Omega }}_{2;\beta }(u,v)= \textsl{J}_{\beta }(u,v), \quad \text{if } \mathcal{J}_{s ; \Omega }\in \mathbb{J}^{R}_{(U,\textsl{Q}_{s ; \Omega })}. \end{aligned}$$
-
(d)
Let \(\mu =\{\mu _{\beta }\}_{\beta \in \Omega }\in [0,1)\). A single-valued map \(T:U\rightarrow U\) is \((\mathcal{D}^{L-\mathcal{J}_{s ; \Omega }}_{\zeta },\mu )\)-contraction on U (\((\mathcal{D}^{R-\mathcal{J}_{s ; \Omega }}_{\zeta },\mu )\)-contraction on U) if for all \(\beta \in \Omega \) and for all \(x,y \in U\):
-
(d.1)
\(D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }(T(x),T(y))\leq \mu _{ \beta }\textsl{J}_{\beta }(x,y)\), if \(\mathcal{J}_{s ; \Omega }\in \mathbb{J}^{L}_{(U,\textsl{Q}_{s ; \Omega })}\);
-
(d.2)
\(D^{R-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta }(T(x),T(y))\leq \mu _{ \beta }\textsl{J}_{\beta }(x,y)\), if \(\mathcal{J}_{s ; \Omega }\in \mathbb{J}^{R}_{(U,\textsl{Q}_{s ; \Omega })}\).
As a result of Definition 3.23 and Theorem 3.22, we now have the following theorem.
Theorem 3.24
Let \((U,\textsl{Q}_{s ; \Omega })\) be a quasi-b-gauge space. Let \(\mathcal{J}_{s ; \Omega }=\{{{\textsl{J}}_{\beta }}: \beta \in \Omega \}\) be a left (right) \(\mathcal{J}_{s ; \Omega }\)-family of generalized quasi-pseudo-b-distances on U, and let \(\zeta \in \{1,2\}\). Moreover, assume that \(\mu =\{\mu _{\beta }\}_{\beta \in \Omega }\in [0,1)\) and \(T:U\rightarrow U\) is a \((\mathcal{D}^{L-\mathcal{J}_{s ; \Omega }}_{\zeta },\mu )\)-contraction on U (\((\mathcal{D}^{R-\mathcal{J}_{s ; \Omega }}_{\zeta },\mu )\)-contraction on U).
-
(I)
If \((U,T)\) is the left (right) \(\mathcal{J}_{s ; \Omega }\)-admissible at a point \(z^{0}\in U\), then there is a sequence \(\{z^{m}:m\in \{0\}\cup \mathbb{N}\}\) starting at \(z^{0}\in U\) with \(\{z^{m}=T^{[m]}(z^{0}): m\in \{0\}\cup \mathbb{N}\}\), a point \(z\in U\), and \(r=\{r_{\beta }\}_{\beta \in \Omega }\in (0,\infty )\) such that \(z^{m}\in B^{L-\mathcal{J}_{s ; \Omega }}(z^{0},r)(z^{m}\in B^{R- \mathcal{J}_{s ; \Omega }}(z^{0},r))\) for all \(m\in \mathbb{N}\) and \(\lim_{m\rightarrow \infty }^{L-\mathcal{J}_{s ; \Omega }}z_{m} = z\) (\(\lim_{m\rightarrow \infty }^{R-\mathcal{J}_{s ; \Omega }}z_{m} = z \)).
-
(II)
If \((U,T)\) is the left (right) \(\mathcal{J}_{s ; \Omega }\)-admissible at a point \(z^{0}\in U\) and if \(T^{[k]}\) for some \(k\in \mathbb{N}\) is a left (right) \(\textsl{Q}_{s ; \Omega }\)-quasi-closed map on U, then \(\mathrm{Fix} (T^{[k]})\) is non-empty, and there exists a sequence \(\{z^{m}:m\in {0}\cup \mathbb{N}\}\) starting at \(z^{0}\in U\) with \(\{z^{m}=T^{[m]}(z^{0}): m\in \{0\}\cup \mathbb{N}\}\), a point \(z\in \operatorname{Fix}(T^{[k]})\), and \(r=\{r_{\beta }\}_{\beta \in \Omega }\in (0,\infty )\) such that \(z^{m}\in B^{L-\mathcal{J}_{s ; \Omega }}(z^{0},r)(z^{m}\in B^{R- \mathcal{J}_{s ; \Omega }}(z^{0},r))\) for all \(m\in \mathbb{N}\), \(\lim_{m\rightarrow \infty }^{L-\mathcal{J}_{s ; \Omega }}z_{m} = z\) \((\lim_{m\rightarrow \infty }^{R-\mathcal{J}_{s ; \Omega }}z_{m} = z )\), and we have
$$\begin{aligned} \textsl{J}_{\beta } \bigl(z,T(z) \bigr)= \textsl{J}_{\beta } \bigl(T(z),z \bigr)=0, \quad \textrm{for all } \beta \in \Omega \textrm{ and for all } z\in \operatorname{Fix} \bigl(T^{[k]} \bigr). \end{aligned}$$
(3.30)
-
(III)
If \((U,\textsl{Q}_{s ; \Omega })\) is a Hausdorff space, and if \((U,T)\) at a point \(z^{0}\in U\) is lthe eft (right) \(\mathcal{J}_{s ; \Omega }\)-admissible, and if \(T^{[k]}\), for some \(k\in \mathbb{N}\), is a left (right) \(\textsl{Q}_{s ; \Omega }\)-quasi-closed map on U, then there exists a sequence \(\{z^{m}:m\in {0}\cup \mathbb{N}\}\) starting at \(z^{0}\in U\) with \(\{z^{m}=T^{[m]}(z^{0}): m\in \{0\}\cup \mathbb{N}\}\), a point \(z\in \operatorname{Fix}(T^{[k]})=\operatorname{Fix}(T)=\{z\}\) and \(r=\{r_{\beta }\}_{\beta \in \Omega }\in (0,\infty )\) such that \(z^{m}\in B^{L-\mathcal{J}_{s ; \Omega }}\) (\(z^{0},r)(z^{m}\in B^{R- \mathcal{J}_{s ; \Omega }}(z^{0},r)\)) for all \(m\in \mathbb{N}\), \(\lim_{m\rightarrow \infty }^{L-\mathcal{J}_{s ; \Omega }}z_{m} = z\) \((\lim_{m\rightarrow \infty }^{R-\mathcal{J}_{s ; \Omega }}z_{m} = z )\), and we have
$$\begin{aligned} \textsl{J}_{\beta }(z,z)=0 \quad \textrm{for all } \beta \in \Omega . \end{aligned}$$
(3.31)
Proof
We prove only (3.30) and (3.31).
On contrary, suppose that there exist \(\beta _{0}\in \Omega \) and \(z\in \operatorname{Fix}(T^{[k]})\) such that \(\textsl{J}_{\beta _{0}}(z,T(z))>0\). Indeed, \(z=T^{[2k]}(z)\), \(T(z)=T^{[2k]}(T(z))\) and for \(\zeta \in \{1,2\}\), by Definition (3.23),
$$\begin{aligned} 0&< \textsl{J}_{\beta _{0}} \bigl(z,T(z) \bigr)=\textsl{J}_{\beta _{0}} \bigl(T^{[2k]}(z),T^{[2k]} \bigl(T(z) \bigr) \bigr) \\ &\leq D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta _{0}} \bigl(T^{[2k]}(z),T^{[2k]} \bigl(T(z) \bigr) \bigr) \\ &\leq \mu _{\beta _{0}}\textsl{J}_{\beta _{0}} \bigl(T^{[2k-1]}(z),T^{[2k-1]} \bigl(T(z) \bigr) \bigr) \\ &\leq \mu _{\beta _{0}}D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta _{0}} \bigl(T^{[2k-1]}(z),T^{[2k-1]} \bigl(T(z) \bigr) \bigr) \\ &\leq \mu ^{2}_{\beta _{0}}\textsl{J}_{\beta _{0}} \bigl(T^{[2k-2]}(z),T^{[2k-2]} \bigl(T(z) \bigr) \bigr) \leq \cdots \\ &\leq \mu ^{2k}_{\beta _{0}}\textsl{J}_{\beta _{0}} \bigl(z,T(z) \bigr)< \textsl{J}_{\beta _{0}} \bigl(z,T(z) \bigr), \end{aligned}$$
which is a contradiction.
Now, suppose that there exist \(\beta _{0}\in \Omega \) and \(z\in \operatorname{Fix}(T^{[k]})\) such that \(\textsl{J}_{\beta _{0}}(T(z),z)>0\). Then, Definition 3.23 and the fact that \(z=T^{[k]}(z)=T^{[2k]}(z)\) imply that for \(\zeta \in \{1,2\}\),
$$\begin{aligned} 0< {}&\textsl{J}_{\beta _{0}} \bigl(T(z),z \bigr)=\textsl{J}_{\beta _{0}} \bigl(T^{[k+1]}(z),T^{[2k]}(z) \bigr) \\ \leq{}& s_{\beta _{0}}\textsl{J}_{\beta _{0}} \bigl(T^{[k+1]}(z),T^{[k+2]}(z) \bigr)+s^{2}_{ \beta _{0}}\textsl{J}_{\beta _{0}} \bigl(T^{[k+2]}(z),T^{[k+3]}(z) \bigr) \\ &{}+\cdots + s^{k-2}_{\beta _{0}}\textsl{J}_{\beta _{0}} \bigl(T^{[2k-1]}(z),T^{[2k]}(z) \bigr) \\ \leq{}& s_{\beta _{0}}D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta _{0}} \bigl(T^{[k+1]}(z),T^{[k+2]}(z) \bigr)+s^{2}_{ \beta _{0}}D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ;\beta _{0}} \bigl(T^{[k+2]}(z),T^{[k+3]}(z) \bigr) \\ &{}+\cdots +s^{k-2}_{\beta _{0}}D^{L-\mathcal{J}_{s ; \Omega }}_{\zeta ; \beta _{0}} \bigl(T^{[2k-1]}(z),T^{[2k]}(z) \bigr) \\ \leq{}& s_{\beta _{0}} \mu ^{k+1}_{\beta _{0}}\textsl{J}_{\beta _{0}} \bigl(z,T(z) \bigr)+s^{2}_{ \beta _{0}} \mu ^{k+2}_{\beta _{0}} \textsl{J}_{\beta _{0}} \bigl(z,T(z) \bigr) \\ &{}+\cdots +s^{k-2}_{\beta _{0}} \mu ^{2k-1}_{\beta _{0}} \textsl{J}_{ \beta _{0}} \bigl(z,T(z) \bigr)=0, \end{aligned}$$
which is impossible. Thus, property (3.30) holds.
Next, we show that property (3.31) holds.
If the space \((U,\textsl{Q}_{s ; \Omega })\) is the Hausdorff one, then Proposition (3.11) and property (3.30) suggest that \(T(z)=z\) for all \(z\in \operatorname{Fix}(T^{[k]})\) and \(\textsl{J}_{\beta }(z,z)\leq s_{\beta }\textsl{J}_{\beta }(z,T(z))+s_{ \beta }\textsl{J}_{\beta }(T(z),z)=0\), for all \(\beta \in \Omega \) and for all \(z\in \operatorname{Fix}(T^{[k]})\). Thus, \(\operatorname{Fix}(T^{[k]})=\operatorname{Fix}(T)\) and for all \(z\in \operatorname{Fix}(T^{[k]})=\operatorname{Fix}(T)\), we have \(\textsl{J}_{\beta }(z,z)=0\).
To prove \(\operatorname{Fix}(T)\) is a singleton, on the contrary, let \(y,z\in \operatorname{Fix}(T)\) and \(y\neq z\). Then, Proposition (3.11) implies there exists \(\beta _{0}\in \Omega \) such that \(\textsl{J}_{\beta _{0}}(y,z)> 0 \vee \textsl{J}_{\beta _{0}}(z,y)>0 \). Obviously, for \(\zeta \in \{1,2\}\), we then have
$$\begin{aligned} & \bigl[\textsl{J}_{\beta _{0}}(y,z)> 0 \wedge \textsl{J}_{\beta _{0}}(y,z)= \textsl{J}_{\beta _{0}} \bigl(T(y),T(z) \bigr)\leq D^{L-\mathcal{J}_{s ; \Omega }}_{ \zeta ;\beta _{0}} \bigl(T(y),T(z) \bigr) \\ &\quad \leq \mu _{\beta _{0}}\textsl{J}_{\beta _{0}}(y,z)< \textsl{J}_{ \beta _{0}}(y,z) \bigr]\vee \bigl[\textsl{J}_{\beta _{0}}(z,y)> 0 \wedge \textsl{J}_{\beta _{0}}(z,y) \\ &\quad = \textsl{J}_{\beta _{0}} \bigl(T(z),T(y) \bigr)\leq D^{L- \mathcal{J}_{s ; \Omega }}_{\zeta ;\beta _{0}} \bigl(T(z),T(y) \bigr) \\ &\quad \leq \mu _{\beta _{0}}\textsl{J}_{\beta _{0}}(z,y)< \textsl{J}_{ \beta _{0}}(z,y) \bigr], \end{aligned}$$
which is impossible. Hence, we obtain \(\operatorname{Fix}(T)=\{z\}\). The theorem is proved. □
Remark 3.25
The proof of the right case in above theorems is based on an analogous technique.
Example 3.26
Let \(U=[0,6]\), and let \(\textsl{Q}_{s ; \Omega }=\{q\}\), where q is a quasi-pseudo-b-metric on U defined for all \(u,v\in U\) by
$$\begin{aligned} q(u,v) = \textstyle\begin{cases} 0 & \text{if }u\geq v, \\ (u-v)^{2} & \text{if }u< v. \end{cases}\displaystyle \end{aligned}$$
(3.32)
Let \(G= [0,3)\cup (3,6]\) be a subset of U. Let \(\mathcal{J}_{s ; \Omega }=\{\textsl{J}\}\), where \(\textsl{J}:U\times U\rightarrow [0,\infty )\) is defined for all \(u,v\in U\) by
$$\begin{aligned} \textsl{J}(u,v) = \textstyle\begin{cases} q(u,v) & \text{if }G\cap \{u,v\}=\{u,v\}, \\ 40 & \text{if }G\cap \{u,v\}\neq \{u,v\}. \end{cases}\displaystyle \end{aligned}$$
(3.33)
The set-valued map T is defined by
$$\begin{aligned} T(u) = \textstyle\begin{cases} [4;6] & \text{for }u\in [0,3)\cup (3,6], \\ [5,6] & \text{for }u=3. \end{cases}\displaystyle \end{aligned}$$
(3.34)
-
(I.1)
\(\mathcal{J}_{s ; \Omega }\) is not symmetric. Indeed, \(\textsl{J}(4,0)= 0\) and \(\textsl{J}(0,4)= 16\).
-
(I.2)
\((U,\textsl{Q}_{s ; \Omega })\) is a quasi-b-gauge space, and \(\mathcal{J}_{s ; \Omega }\in \mathbb{J}^{L}_{(U,\textsl{Q}_{s ; \Omega })}\cap \mathbb{J}^{R}_{(U,\textsl{Q}_{s ; \Omega })}\). See Example 3.12.
-
(I.3)
The property \(T : U\rightarrow Cl^{L-\textsl{Q}_{s ; \Omega }}(U)\) \((T : U\rightarrow Cl^{R-\textsl{Q}_{s ; \Omega }}(U))\) holds. This follows from (3.32) and Definition 3.14 and 3.13(C).
-
(I.4)
\(T : U\rightarrow Cl^{L-\textsl{Q}_{s ; \Omega }}(U)\) is a \((\mathcal{D}^{L-\mathcal{J}_{s ; \Omega }}_{1}, \mu = \frac{1}{10})\)-contraction on U, i.e., for all \(u,v \in U\) \(D^{L-\mathcal{J}_{s ; \Omega }}_{1}(T(u),T(v))\leq \mu \textsl{J}(u,v)\), where \(D^{L-\mathcal{J}_{s ; \Omega }}_{1}(U,V)= \max \{\sup_{u\in U} \textsl{J}(u,V), \sup_{v\in V}\textsl{J}(U,v)\}\), \(U,V \in 2^{U}\).
Denoting \(D^{L-\mathcal{J}_{s ; \Omega }}_{1}=D_{1}\), we prove this in the following subcases:
-
(I.4.1)
If \(u,v\in [0,3)\cup (3,6]\), this implies \(u,v\in G\), \(T(u)=T(v)=[4,6]=E \subset G\), and by (3.32) for all \(e\in E\), we have \(\inf \{\textsl{J}(e,f): f\in E\}=\textsl{J}(e,e)=q(e,e)=0\). Thus, \(D_{1}(T(u),T(v))=0\leq \mu \textsl{J}(u,v)\).
-
(I.4.2)
If \(u\in [0,3)\cup (3,6]\) and \(v=3\), then \(u\in G\), \(v\notin G\), \(\textsl{J}(u,v)=40\), \(T(u)=[4,6]=E \subset G\), \(T(v)=[5,6] =F \subset G\) and by (3.32), \(e\in E\) suggests
$$\begin{aligned} \inf \bigl\{ \textsl{J}(e,f)=q(e,f): f\in F \bigr\} = \textstyle\begin{cases} 4 & \text{whenever }e\in [4;5], \\ 0 & \text{whenever }e\in [5;6]. \end{cases}\displaystyle \end{aligned}$$
Whereas, \(f\in F\) inferred \(\inf \{\textsl{J}(e,f)=q(e,f): e\in E\}=0\). Thus, \(D_{1}(T(u),T(v))=4=\mu \textsl{J}(u,v)\).
-
(I.4.3)
If \(u=3\) and \(v\in [0,3)\cup (3,6]\), then \(u\notin G\), \(v\in G\), \(\textsl{J}(u,v)=40\), \(T(u)=[5,6]=E \subset G\), \(T(v)=[4,6]=F \subset G\). As a result, by (3.32), \(e\in E\) implies \(\inf \{\textsl{J}(e,f)=q(e,f): f\in F\}=0\). Further, by (3.32) \(f\in F\) suggests \(\inf \{\textsl{J}(e,f): e\in E\}=0\). Thus, \(D_{1}(T(u),T(v))=0\leq \mu \textsl{J}(u,v)\).
-
(I.4.4)
If \(u=v=3\), then \(\textsl{J}(u,v)=36\), \(T(u)=T(v)=[5,6]=E \subset G\) and for all \(e\in E\) \(\inf \{\textsl{J}(e,f)=q(e,f): f\in E\}=q(e,e)=0\). Therefore, \(D_{1}(T(u),T(v))=0< \mu \textsl{J}(u,v)\).
-
(I.5)
To prove that there exists \(v\in T(u)\) such that \(\textsl{J}(u,v)<\textsl{J}(u,T(u))+\gamma \), for all \(u\in U\) and for all \(\gamma \in (0,\infty )\), we observe the following subcases:
-
(I.5.1)
If \(u\in [0,3)\cup (3,4)\) and \(v=4\in T(u)=[4,6]\), then \(\textsl{J}(u,v)=q(u,v)=(u-v)^{2}\), \(\textsl{J}(u,T(u))= (u-v)^{2}\) and \(\textsl{J}(u,v)<\textsl{J}(u,T(u))+\gamma \) for all \(\gamma \in (0,\infty )\).
-
(I.5.2)
If \(u\in [4,6]\) and \(v=4\in T(u)=[4,6]\), then \(\textsl{J}(u,v)=q(u,v)=0\), \(\textsl{J}(u,T(u))=0\) and \(\textsl{J}(u,v)<\textsl{J}(u,T(u))+\gamma \) for all \(\gamma \in (0,\infty )\).
-
(I.5.3)
If \(u=3\) and \(v\in T(u)=[5,6]\), then \(\textsl{J}(u,v)=\textsl{J}(u,T(u))=40\) and \(\textsl{J}(u,v)<\textsl{J}(u,T(u))+\gamma \) for all \(\gamma \in (0,\infty )\).
-
(I.6)
Let \((U,T)\) be left \(\mathcal{J}_{s ; \Omega }\)-admissible at U. We show that if \(z^{0}\in U\), and \(\{z^{m}: m\in \{0\}\cup \mathbb{N}\}\) fulfils the properties
$$\begin{aligned} z^{m+1}\in T \bigl(z^{m} \bigr), \quad \textrm{for all } m \in \{0\}\cup \mathbb{N} \end{aligned}$$
(3.35)
and
$$\begin{aligned} \lim_{m\rightarrow \infty }\sup_{n>m} \textsl{J} \bigl(z^{m},z^{n} \bigr)=0, \end{aligned}$$
(3.36)
then
$$\begin{aligned} \lim_{m\rightarrow \infty }\textsl{J} \bigl(z,z^{m} \bigr)=0 \quad \textrm{where }z=6. \end{aligned}$$
(3.37)
In fact, we observe
$$\begin{aligned} T^{[m]}(U)= [4;6]\subset G \quad \textrm{for }m \geq 2. \end{aligned}$$
(3.38)
We can also write (3.36) in the form that there exists \(m_{0}\in \mathbb{N}\) such that for all \(\epsilon >0\) and for all \(n>m\geq m_{0}\), we have \(\textsl{J}(z^{m},z^{n})<\epsilon \) and so, in particular in view of (3.38), (3.32), and (3.33), this implies that there exists \(m_{1}\geq m_{0}\) such that for all \(0<\epsilon \) and for all \(n>m\geq m_{1}\), we have
$$\begin{aligned} \textsl{J} \bigl(z^{m},z^{n} \bigr)= q \bigl(z^{m},z^{n} \bigr)=0 < \epsilon . \end{aligned}$$
(3.39)
From (3.38), (3.39), (3.32), and (3.33), we conclude that \(z^{m}\geq z^{m+1}\) for all \(m\geq m_{1}\), and since \(6\geq z^{m}\) for all m and \(6\in G\), we have \(\lim_{m\rightarrow \infty }q(z,z^{m})=0\) where \(z=6\) and this implies (3.37). Thus, \((U,T)\) is the left \(\mathcal{J}_{s ; \Omega }\)-admissible at U.
-
(I.7)
To prove \((U,T)\) is a left \(\textsl{Q}_{s ; \Omega }\)-quasi-closed map in U, suppose \((w_{m}: m\in \mathbb{N})\subset T(U)\) is a left \(\textsl{Q}_{s ; \Omega }\)-converging sequence in U. Now, as \([4,6]\subset Cl^{L-\textsl{Q}_{s ; \Omega }}(U)\), there exists \(w\in T(U)=[4,6]\) such that \(\lim_{m\rightarrow \infty }q(w,w_{m})=0\).
Equivalently, there exist \(w\in T(U)=[4,6]\) and \(m_{0}\) such that \(q(w,w_{m})< \epsilon \) for all \(\epsilon >0\) and for all \(m\geq m_{0}\), and thus, by (3.33) and (3.32), there exist \(w\in T(U)=[4,6]\) and \(m_{1}\geq m_{0}\) such that \(q(w,w_{m})=0 < \epsilon \), for all \(0< \epsilon \) and \(m\geq m_{1,}\) or analogously there exist \(w\in T(U)=[4,6]\) and \(m_{1}\) such that \(w\geq w_{m}\) for all \(m\geq m_{1}\). Obviously, then \([w,6]\subset S^{L-\textsl{Q}_{s ; \Omega }}_{(w_{m}:m\in \mathbb{N})}\). The consideration above implies that if \((x_{m}: m\in \mathbb{N})\) and \((y_{m}: m\in \mathbb{N})\) are fixed and arbitrary subsequences of \(\{w_{m}: m\in \mathbb{N}\}\) fulfilling \(y_{m}\in T(x_{m})\) for all \(m\in \mathbb{N}\), then there exists \(m_{1}\) such that \(x_{m}\in [4;6]\wedge y_{m}\in T(x_{m})\wedge z\geq x_{m}\wedge z \geq y_{m}\wedge z\in T(z)\) for all \(m\geq m_{1}\) and for all \(z\in [w,6]\).
-
(I.8)
From (I.1)–(I.7), we observe that all the hypotheses of Theorem 3.22 hold in the left case.
Thus, we have \(\mathrm{Fix} (T)=[4;6]\) and declare that if \(z^{0}\in U\), \(z^{1}\in T(z^{0})\), \(z^{2}\in T(z^{1})\) and \(w\in [4;6]\) are fixed and arbitrary and \(z^{m}=w\) for all \(m\geq 3\), then the sequence \(\{z^{m}: m\in \{0\}\cup \mathbb{N}\}\), beginning at \(z^{0}\) and left \(\textsl{Q}_{s ; \Omega }\)-converging to each point z, satisfies \(z\in T(z)\).
Remark 3.27
Let a quasi-b-gauge space \((U,\textsl{Q}_{s ; \Omega })\) and a family \(\mathcal{J}_{s ; \Omega }=\{\textsl{J}\}\) be as defined in Example 3.26.
-
(a)
(3.32) implies that q is a quasi-pseudo-b-metric, where \(s=2\), and q is not a quasi-pseudo-metric. This implies that \((U,\textsl{Q}_{s ; \Omega })\) is a quasi-b-gauge space but not a quasi-gauge space. Hence a quasi-b-gauge space becomes a more general space than a quasi-gauge space.
-
(b)
From cases I.4.2 and I.4.3, it follows that \(4=D^{L-\mathcal{J}_{s ; \Omega }}_{1}(E,F)\neq D^{L-\mathcal{J}_{s ; \Omega }}_{1}(E,F)=0\) for \(F=[5;6]\) and \(E=[4;6]\).
-
(c)
We see that \(D^{L-\mathcal{J}_{s ; \Omega }}_{1}(E,E)\neq 0\) if \(E=\{3\}\).