Fixed-point results for generalized contraction in K-sequentially complete ordered dislocated fuzzy quasimetric spaces

Abstract

The ambition of this work is to introduce the notion of left (right) K-sequentially complete ordered dislocated fuzzy quasimetric spaces and to define a relevant Hausdorff metric on compact sets. A new approach, given in (Shoaib et al. in Filomat 34(2):323–338, 2020) has been used to obtain fixed-point results for multivalued mappings fulfilling generalized contraction in the latest framework. For the authenticity of our result, an example is formulated.

1 Introduction and preliminaries

One of the branch of functional analysis is fixed-point theory. Fixed-point theory plays a key role to find solutions of mathematical and engineering problems. The fixed-point results for multivalued mappings generalizes the results for single-valued mappings. Applications of the results for multivalued mappings can be seen in Nash equilibria, engineering, and game theory [6, 9, 10, 20, 27, 28, 30]. With the help of multivalued mapping, many results have been proved, for example, see [7, 19, 25, 26, 29, 36–38].

A solution for matrix equations was obtained by a fixed-point result in an ordered metric space that had been proved by Ran and Reurings [24]. In a complete ordered metric space, Altun et al. [3] proved a common fixed point for the mappings satisfying a new restriction of order. For more results in ordered spaces, see [1, 4, 5, 8, 15, 41]. The idea of fuzzy sets was given by Lotfi Zadeh for the first time in 1965 [43]. This concept has been extended in fuzzy functional analysis, fuzzy topology, fuzzy control theory, and decision making, see [18, 22, 23, 31]. One of the significant developments of fuzzy sets in fuzzy functional analysis is fuzzy quasimetric spaces, see [11, 13, 14, 32]. The symmetric condition was not assumed in the fuzzy quasimetric spaces. Arshad et al. [5] observed that there were mappings that had fixed points but there were no results to ensure the existence of a fixed point of such mappings. They introduced a contraction on a closed ball to achieve common fixed points for such mappings. For further results on a closed ball, see [33–37, 40]. Hitzler et al. [16] established a dislocated metric space and obtained some results, see also [17]. Recently, Poovaragavan et al. [21] introduced the concept of right complete dislocated quasi-G-fuzzy metric spaces and gave some results on a closed ball in these spaces. In this paper, we have introduced the concept of ordered dislocated fuzzy quasimetric spaces and dislocated Hausdorff fuzzy quasimetric spaces. We have used a new type of contraction on an intersection of an open ball and a sequence to obtain the common fixed point of multivalued mappings in left(right) K-sequentially complete dislocated fuzzy quasimetric spaces. Our results have extended the results of Altun et al. [3] and Shoaib et al. [39] in many ways. The idea of this manuscript is motived by [39], where an ordered left K-sequentially complete dislocated quasimetric space is replaced by a left K-sequentially complete ordered dislocated fuzzy quasimetric space. Theorem 2.1 is analogous to Theorem 2.2 in [39]. We give the following definitions and results that will be helpful to understand the paper.

Definition 1.1

([12])

A binary operation $\circledast :[0,1]\times {}[ 0,1]\longrightarrow {}[ 0,1]$ is a continuous triangular norm (t-norm) if the following axioms hold: $\mathcal{T}1$) $a\circledast b=b\circledast a$ and $a\circledast (b\circledast c)=(a\circledast b)\circledast c$; $\mathcal{T}2$) ⊛ is continuous; $\mathcal{T}3$) $a\circledast 1=1$ for all $a\in {}[ 0,1]$; $\mathcal{T}4$) $a\circledast b< c\circledast d$ whenever $a< c$ and $b< d$ for all $a,b,c,d\in {}[ 0,1]$.

Definition 1.2

([42])

Let Ψ be the class of all mappings $\mu :[0,1]\rightarrow {}[ 0,1]$ such that 1) μ is continuous and nondecreasing; 2) $\mu (t)>t$ for all $t\in (0,1)$.

Lemma 1.3

([42])

If $\mu \in \Psi$, then 1) $\mu (1)=1$; 2) $\lim_{k\rightarrow \infty }\mu ^{k}(t)=1$ for all $t\in (0,1)$.

Definition 1.4

([3])

Let $\mathcal{Y}$ be a nonempty set. Then, ⪯ is a partial order on $\mathcal{Y}$ if: (i) $x\preceq x$ for all $x\in \mathcal{Y}$; (ii) $x\preceq y$ and $y\preceq x$ implies $x=y$ for all $(x, y)\in \mathcal{Y}\times \mathcal{Y}$; (iii) $x\preceq y$ and $y\preceq z$ implies $x\preceq z$ for all $(x, y, z)\in \mathcal{Y}\times \mathcal{Y}\times \mathcal{Y}$. Let A ≠ϕ and $A\subseteq \mathcal{Y}$. Then, $x\preceq A$ iff $x\preceq a$, for all $a\in A$ and $x\succeq A$ iff $x\succeq a$, for all $a\in A$.

Definition 1.5

([2])

Let $\mathcal{Y}\neq \phi$ be an arbitrary set, ⊛ a continuous t-norm, and $\mathcal{F}_{d_{q}\text{ }}$a fuzzy set on $\mathcal{Y}\times \mathcal{Y}\times {}[ 0,\infty )$. The 3-tuple $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ is said to be a dislocated fuzzy quasimetric space, if $\mathcal{F}_{d_{q}}$ satisfies the following constraints for each $x,y,z\in X$ and $u,t>0$:

$\mathcal{F}1$) If $\mathcal{F}_{d_{q}}(x,y,u)=\mathcal{F}_{d_{q}}(y,x,u)=1$, then $x=y$;

$\mathcal{F}2$) $\mathcal{F}_{d_{q}}(x,y,u)\circledast \mathcal{F}_{d_{q}}(y,z,s)\leq \mathcal{F}_{d_{q}}(x,z,u+s)$.

For $x_{\circ }\in \mathcal{Y}$, $u>0$, $B_{\mathcal{F}_{dq}}(x_{\circ },r,u)=\{y\in \mathcal{Y}:\mathcal{F}_{d_{q}}(x_{ \circ },y,u)>1-r \wedge \mathcal{F}_{d_{q}}(y,x_{\circ },u)>1-r \}$ and $\overline{B_{\mathcal{F}_{dq}}(x_{\circ },r,u)}=\{y\in \mathcal{Y}:\mathcal{F}_{d_{q}}(x_{ \circ },y,u)\geq 1-r \wedge \mathcal{F}_{d_{q}}(y,x_{\circ },u)\geq 1-r \}$ are open and closed balls, respectively, in $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$.

Example 1.6

Let $\mathcal{Y}=[0,\infty )$. Then, $\mathcal{F}_{d_{q}}(x,y,u)=\frac{u}{u+x+2y}$ is a dislocated fuzzy quasimetric with $a\circledast b=\min \{a,b\}$. Note that $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ is neither a dislocated fuzzy metric space nor a fuzzy quasimetric space.

Definition 1.7

Let X be a nonempty set. Then, $({X},\preceq ,\mathcal{F}_{d_{q}})$ is called an ordered dislocated fuzzy quasimetric space if: (i) $\mathcal{F}_{d_{q}}$ is a dislocated fuzzy quasimetric on X and (ii) ⪯ is a partial order on X.

Definition 1.8

Let $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ be a dislocated fuzzy quasimetric space. A sequence $\{x_{k}\}$ in $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ is said to be

1) dislocated fuzzy quasiconvergent to a point $x\in \mathcal{Y}$, if $\lim_{k\rightarrow \infty }\mathcal{F}_{d_{q}}(x_{k},x,u)=1=\lim_{k\rightarrow \infty }\mathcal{F}_{d_{q}}(x,x_{k},u)$ for all $u>0$ or for any ε >0, there exists $k_{\circ }\in \mathbb{N}$, such that for all $k>k_{\circ }$, $\mathcal{F}_{d_{q}}(x_{k},x,u)<$ ε and $\mathcal{F}_{d_{q}}(x,x_{k},u)<$ ε. In this case x is called a limit of $\{x_{k}\}$.

2) a left (right) K-Cauchy sequence if for $k,m\in$ with k> m, $\lim_{k,m\rightarrow \infty }\mathcal{F}_{d_{q}}(x_{k},x_{m},u)=1$ $(\lim_{k,m\rightarrow \infty }\mathcal{F}_{d_{q}}(x_{m},x_{k},u)=1)$ for all $u>0$ or for any ε >0, there exists $k_{\circ }\in \mathbb{N}$, such that for all $k,m\in$ with k> $m\geq k_{\circ }$, $\mathcal{F}_{d_{q}}(x_{k},x_{m},u)<$ ε $(\mathcal{F}_{d_{q}}(x_{m},x_{k},u)<\varepsilon )$ for all $u>0$.

3) $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ is called left (right) K-sequentially complete if every left (right) K-Cauchy sequence in $\mathcal{Y}$ converges to a point $x\in \mathcal{Y}$.

Definition 1.9

Let $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ be a dislocated fuzzy quasimetric space. Then, for each $a\in \mathcal{Y}$, $B\subseteq \mathcal{Y}$, and $u>0$, define $\mathcal{F}_{d_{q}}(a,B,u)=\sup \{\mathcal{F}_{d_{q}}(a,b,u):b\in B \}$ and $\mathcal{F}_{d_{q}}(B,a,u)=\sup \{\mathcal{F}_{d_{q}}(b,a,u):b\in B \}$.

For a given fuzzy metric space $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$, $K_{0}(\mathcal{Y})$ will represent the set of nonempty compact subsets of $(\mathcal{Y},\tau _{\mathcal{F}})$, where $(\mathcal{Y},\tau _{\mathcal{F}})$ is a metrizable topological space, generated by a fuzzy metric space $(\mathcal{Y},\mathcal{F},\circledast )$.

Lemma 1.10

Let $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ be a dislocated fuzzy quasimetric space. Then, for each $a\in \mathcal{Y}$, $B\in K_{0}(\mathcal{Y})$ and $u>0$, there is $b_{0}\in B$ such that $\mathcal{F}_{d_{q}}(a,B,u)=\mathcal{F}_{d_{q}}(a,b_{0},u)$ and $\mathcal{F}_{d_{q}}(B,a,u)=\mathcal{F}_{d_{q}}(b_{0},a,u)$.

Lemma 1.11

Let $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ be a dislocated fuzzy quasimetric space. Then, for each $A\in K_{0}(\mathcal{Y})$ and for any nonempty subset B of $\mathcal{Y}$ and $u>0$, there exists $a_{0}\in A$ such that $\inf_{a\in A}\mathcal{F}_{d_{q}}(a,B,u)=\mathcal{F}_{d_{q}}(a_{0},B,u)$ and $\inf_{a\in A}\mathcal{F}_{d_{q}}(B,a,u)=\mathcal{F}_{d_{q}}(B,a_{0},u)$.

Definition 1.12

Let $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ be a dislocated fuzzy quasimetric space. We define a function $H_{d_{q}}$ on $K_{0}(\mathcal{Y})\times K_{0}(\mathcal{Y})\times (0,\infty )$ by

$$ H_{d_{q}}(A,B,u)=\min \Bigl\{ \inf_{a\in A} \mathcal{F}_{d_{q}}(a,B,u),\inf_{b\in B} \mathcal{F}_{d_{q}}(A,b,u)\Bigr\} . $$

$(K_{0}(\mathcal{Y}),H_{d_{q}},\circledast )$ is known as a dislocated Hausdorff fuzzy quasimetric space on $K_{0}(\mathcal{Y})$.

Lemma 1.13

Let $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ be a dislocated fuzzy quasimetric space and $(K_{0}(\mathcal{Y}),H_{d_{q}},\circledast )$ be a Hausdorff metric space on $K_{0}(\mathcal{Y})$. Then, for arbitrary $A,B\in K_{0}(\mathcal{Y})$ and for each $a\in A$, there exists $b_{a}\in B$ such that $H_{d_{q}}(A,B,u)\leq \mathcal{F}_{d_{q}}(a,b_{a},u)$ and $H_{d_{q}}(B,A,u)\leq \mathcal{F}_{d_{q}}(b_{a},a,u)$, where $\mathcal{F}_{d_{q}}(a,B,u)=\mathcal{F}_{d_{q}}(a,b_{a},u)$ and $\mathcal{F}_{d_{q}}(B,a,u)=\mathcal{F}_{d_{q}}(b_{a},a,u)$.

2 Main result

Let $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ be a dislocated fuzzy quasimetric space, $\mathfrak{e}_{\circ }\in \mathcal{Y}$ and $\mathcal{T}:\mathcal{Y}\rightarrow K_{\circ }(\mathcal{Y})$ be a multivalued mapping on $\mathcal{Y}$. As $\mathcal{T}\mathfrak{e}_{\circ }$ is a compact set, there exists $\mathfrak{e}_{1}\in \mathcal{T}\mathfrak{e}_{\circ }$ such that $\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathcal{T}\mathfrak{e}_{\circ },u)=\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},u)$ and $\mathcal{F}_{d_{q}}(\mathcal{T}\mathfrak{e}_{\circ },\mathfrak{e}_{\circ },u)=\mathcal{F}_{d_{q}}(\mathfrak{e}_{1}, \mathfrak{e}_{\circ },u)$. Now, for $\mathfrak{e}_{1}\in \mathcal{Y}$, there exists $\mathfrak{e}_{2}\in \mathcal{T}\mathfrak{e}_{1}$ such that $\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathcal{T}\mathfrak{e}_{1},u)=\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{2},u)$ and $\mathcal{F}_{d_{q}}(\mathcal{T}\mathfrak{e}_{1},\mathfrak{e}_{1},u)=\mathcal{F}_{d_{q}}(\mathfrak{e}_{2},\mathfrak{e}_{1},u)$. Continuing this process, we construct a sequence $\mathfrak{e}_{k}$ of points in $\mathcal{Y}$ such that $\mathfrak{e}_{k+1}\in \mathcal{T}\mathfrak{e}_{k}$, $\mathcal{F}_{d_{q}}(\mathfrak{e}_{k},\mathcal{T}\mathfrak{e}_{k},u)=\mathcal{F}_{d_{q}}(\mathfrak{e}_{k},\mathfrak{e}_{k+1},u)$ and $\mathcal{F}_{d_{q}}(\mathcal{T}\mathfrak{e}_{k},\mathfrak{e}_{k},u)=\mathcal{F}_{d_{q}}(\mathfrak{e}_{k+1}, \mathfrak{e}_{k},u)$. We denote this iterative sequence by $\{\mathcal{YT}(\mathfrak{e}_{k})\}$ and say that $\{\mathcal{YT}(\mathfrak{e}_{k})\}$ is a sequence in $\mathcal{Y}$ generated by $\mathfrak{e}_{\circ }$.

Theorem 2.1

Let $(\mathcal{Y},\preceq ,\mathcal{F}_{d_{q}},\circledast )$ be a left K-sequentially complete ordered dislocated fuzzy quasimetric space with $a\ast b=\min \{a,b\}$. Let $(K_{0}(\mathcal{Y}),H_{d_{q}},\circledast )$ be a dislocated Hausdorff fuzzy quasimetric space on $K_{0}(\mathcal{Y})$. Let $S,\mathcal{T}:\mathcal{Y}\rightarrow K_{\circ }(\mathcal{Y})$ be multivalued mappings. Assume that the following assertions hold: (i) There exist $\mu \in \Psi$, $\mathfrak{e}_{\circ }\in \mathcal{Y}$ and $r>0$ such that for every $\mathfrak{e},f\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{ \circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k})\}$ with $\mathfrak{e}\succeq S\mathfrak{e}$, $f\preceq Sf$, we have

$$ \min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u),H_{d_{q}}( \mathcal{T}f,\mathcal{T}\mathfrak{e},u)\bigr\} \geq \mu \bigl(D( \mathfrak{e},f,u)\bigr), $$

for all $u>0$, where

$$ D(\mathfrak{e},f,u)=\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e},f,u), \mathcal{F}_{d_{q}}(\mathfrak{e},\mathcal{T}\mathfrak{e},u), \mathcal{F}_{d_{q}}(f,\mathcal{T}f,u)\bigr\} . $$

(ii) If $\mathfrak{e}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{ \circ })$,

$$ \mathcal{F}_{d_{q}}(\mathfrak{e},\mathcal{T}\mathfrak{e},u)= \mathcal{F}_{d_{q}}(\mathfrak{e},f,u)\textit{ and } \mathcal{F}_{d_{q}}( \mathcal{T}\mathfrak{e},\mathfrak{e},u)= \mathcal{F}_{d_{q}}(f,\mathfrak{e},u), $$

then (iia) If $\mathfrak{e}\preceq S\mathfrak{e}$, then $f\succeq Sf$. (iib) If $\mathfrak{e}\succeq S\mathfrak{e}$, then $f\preceq Sf$. (iii) The set $G(S)=\{\mathfrak{e}:\mathfrak{e}\preceq S\mathfrak{e}\textit{ and } \mathfrak{e}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\}$ is closed and contains $\mathfrak{e}_{\circ }$. (iv) For $k\in \cup \{0\}$, we have

$$ \underset{p=0}{\overset{k}{\circledast }}\min \biggl\{ \mu ^{p} \biggl(\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ },\frac{u_{\circ }}{2^{p+1}}\biggr)\biggr), \mu ^{p}\biggl(\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{\circ },\mathfrak{e}_{1}, \frac{u_{\circ }}{2^{p+1}} \biggr)\biggr)\biggr\} >1-r. $$

Then, the subsequence $\{\mathfrak{e}_{2k}\}$ of $\{\mathcal{YT}(\mathfrak{e}_{k})\}$ is a sequence in $G(S)$ and $\mathfrak{e}_{2k}\rightarrow \mathfrak{e}^{\divideontimes }\in G(S)$ and $\mathcal{F}_{d_{q}}(\mathfrak{e}^{\divideontimes },\mathfrak{e}^{\divideontimes },u)=1$. Also, if the inequality (i) holds for $\mathfrak{e}^{\divideontimes }$, then S and $\mathcal{T}$ have a common fixed point $\mathfrak{e}^{\divideontimes }$ in $B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })$.

Proof

Since $\mathfrak{e}_{\circ }\in G(S)$, $\mathrm{(iii)}$ implies that $\mathfrak{e}_{\circ }\preceq S\mathfrak{e}_{\circ }$ and $\mathfrak{e}_{\circ }\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })$. Consider the sequence $\{\mathcal{YT}(\mathfrak{e}_{k})\}$. Then, there exists $\mathfrak{e}_{1}\in \mathcal{T}\mathfrak{e}_{\circ }$ such that

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathcal{T} \mathfrak{e}_{ \circ },u)=\mathcal{F}_{d_{q}}( \mathfrak{e}_{\circ },\mathfrak{e}_{1},u) \quad \text{and}\quad \mathcal{F}_{d_{q}}(\mathcal{T}\mathfrak{e}_{\circ }, \mathfrak{e}_{\circ },u)=\mathcal{F}_{d_{q}}( \mathfrak{e}_{1}, \mathfrak{e}_{\circ },u). $$

Now, $\mathrm{(iia)}$ implies that $\mathfrak{e}_{1}\succeq S\mathfrak{e}_{1}$. By using the property of the t-norm and (iv), we have

$$\begin{aligned}& \min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},u_{ \circ }),\mathcal{F}_{d_{q}}( \mathfrak{e}_{1},\mathfrak{e}_{\circ },u_{ \circ }) \bigr\} \\& \quad= \min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},u_{\circ }),\mathcal{F}_{d_{q}}( \mathfrak{e}_{1},\mathfrak{e}_{ \circ },u_{\circ }) \bigr\} \circledast 1 \\& \quad\geq \min \biggl\{ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},\frac{u_{\circ }}{2}\biggr), \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ },\frac{u_{\circ }}{2}\biggr)\biggr\} \circledast \min \biggl\{ \mu \biggl(\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ },\mathfrak{e}_{1},\frac{u_{\circ }}{2^{2}} \biggr)\biggr), \\& \qquad{}\mu \biggl(\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ }, \frac{u_{\circ }}{2^{2}}\biggr)\biggr)\biggr\} \\& \quad= \underset{p=0}{\overset{1}{\circledast }}\min \biggl\{ \mu ^{p} \biggl( \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ }, \frac{u_{\circ }}{2^{p+1}}\biggr)\biggr),\mu ^{p}\biggl(\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1}, \frac{u_{\circ }}{2^{p+1}}\biggr)\biggr)\biggr\} \\& \quad> 1-r. \end{aligned}$$

It follows that $\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{1},u_{\circ })>1-r$ and $\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{\circ },u_{\circ })>1-r$. Hence, $\mathfrak{e}_{1}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })$. Also,

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathcal{T} \mathfrak{e}_{1},u)= \mathcal{F}_{d_{q}}( \mathfrak{e}_{1},\mathfrak{e}_{2},u)\quad \text{and}\quad \mathcal{F}_{d_{q}}(\mathcal{T}\mathfrak{e}_{1}, \mathfrak{e}_{1},u)=\mathcal{F}_{d_{q}}(\mathfrak{e}_{2},\mathfrak{e}_{1},u). $$

As $\mathfrak{e}_{1}\succeq S\mathfrak{e}_{1}$, $\mathrm{(iib)}$ implies $\mathfrak{e}_{2}\preceq S\mathfrak{e}_{2}$. By the triangle inequality, we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{2},u_{\circ }) \geq \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1}, \frac{u_{\circ }}{2}\biggr)\circledast \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{2}, \frac{u_{\circ }}{2}\biggr). $$

(2.1)

By Lemma 1.13, we have

$$\begin{aligned} \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{2},\frac{u}{2}\biggr) \geq &H_{d_{q}}\biggl(\mathcal{T}\mathfrak{e}_{\circ },\mathcal{T} \mathfrak{e}_{1},\frac{u}{2}\biggr) \\ \geq &\min \biggl\{ H_{d_{q}}\biggl(\mathcal{T}\mathfrak{e}_{\circ }, \mathcal{T}\mathfrak{e}_{1},\frac{u}{2} \biggr),H_{d_{q}}\biggl(\mathcal{T}\mathfrak{e}_{1}, \mathcal{T}\mathfrak{e}_{\circ },\frac{u}{2}\biggr) \biggr\} . \end{aligned}$$

As $\mathfrak{e}_{\circ }$, $\mathfrak{e}_{1}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k}) \}$, $\mathfrak{e}_{1}\succeq S\mathfrak{e}_{1}$ and $\mathfrak{e}_{\circ }\preceq S\mathfrak{e}_{\circ }$, by (i), we have

$$\begin{aligned} \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{2},\frac{u}{2}\biggr) \geq &\mu \biggl(D \biggl(\mathfrak{e}_{1},\mathfrak{e}_{\circ }, \frac{u}{2}\biggr)\biggr) \\ =&\mu \biggl(\min \biggl\{ \mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{1},\mathfrak{e}_{ \circ },\frac{u}{2}\biggr),\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{\circ },\mathcal{T} \mathfrak{e}_{\circ }, \frac{u}{2}\biggr),\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{1}, \mathcal{T}\mathfrak{e}_{1}, \frac{u}{2}\biggr)\biggr\} \biggr) \\ =&\mu \biggl(\min \biggl\{ \mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{1},\mathfrak{e}_{ \circ },\frac{u}{2}\biggr),\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{\circ },\mathfrak{e}_{1},\frac{u}{2}\biggr),\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{1},\mathfrak{e}_{2}, \frac{u}{2} \biggr)\biggr\} \biggr). \end{aligned}$$

If $\min \{\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{\circ }, \frac{u}{2}),\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{1}, \frac{u}{2}),\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{2},\frac{u}{2}) \}=\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{2},\frac{u}{2})$, then a contradiction arises due to the fact that $\mu (u)>u$. Hence, we have

$$ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{2},\frac{u}{2}\biggr) \geq \mu \biggl(\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{ \circ },\frac{u}{2}\biggr),\mathcal{F}_{d_{q}} \biggl(\mathfrak{e}_{\circ },\mathfrak{e}_{1}, \frac{u}{2}\biggr)\biggr\} \biggr). $$

(2.2)

Using (2.2) in (2.1), we have

$$ \begin{gathered} \begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{2},u_{\circ }) \geq {}&\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1}, \frac{u_{\circ }}{2}\biggr)\circledast \mu \biggl( \min \biggl\{ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ },\frac{u_{\circ }}{2}\biggr), \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{ \circ },\mathfrak{e}_{1},\frac{u_{\circ }}{2}\biggr)\biggr\} \biggr) \\ \geq {}&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},\frac{u_{\circ }}{2}\biggr), \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ },\frac{u_{\circ }}{2}\biggr)\biggr\} \\ &{}\circledast \min \biggl\{ \mu \biggl(\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{\circ },\mathfrak{e}_{1},\frac{u_{\circ }}{2^{2}}\biggr)\biggr),\mu \biggl(\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{1},\mathfrak{e}_{\circ }, \frac{u_{\circ }}{2^{2}}\biggr)\biggr)\biggr\} \\ ={}&\underset{p=0}{\overset{1}{\circledast }}\min \biggl\{ \mu ^{p} \biggl( \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1}, \frac{u_{\circ }}{2^{p+1}}\biggr)\biggr),\mu ^{p}\biggl(\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ }, \frac{u_{\circ }}{2^{p+1}}\biggr)\biggr)\biggr\} . \end{aligned} \\ \mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{2},u_{\circ })>1-r. \end{gathered} $$

(2.3)

Now, by the triangular inequality, we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2},\mathfrak{e}_{\circ },u_{\circ }) \geq \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{2}, \mathfrak{e}_{1}, \frac{u_{\circ }}{2}\biggr)\circledast \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{ \circ },\frac{u_{\circ }}{2}\biggr). $$

(2.4)

By Lemma 1.13, we have

$$\begin{aligned} \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{2}, \mathfrak{e}_{1},\frac{u}{2}\biggr) \geq &H_{d_{q}}\biggl(\mathcal{T}\mathfrak{e}_{1},\mathcal{T} \mathfrak{e}_{ \circ },\frac{u}{2}\biggr) \\ \geq &\min \biggl\{ H_{d_{q}}\biggl(\mathcal{T}\mathfrak{e}_{1}, \mathcal{T} \mathfrak{e}_{\circ },\frac{u}{2} \biggr),H_{d_{q}}\biggl(\mathcal{T}\mathfrak{e}_{\circ }, \mathcal{T}\mathfrak{e}_{1},\frac{u}{2}\biggr) \biggr\} . \end{aligned}$$

As $\mathfrak{e}_{\circ }$, $\mathfrak{e}_{1}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k}) \},\mathfrak{e}_{1}\succeq S\mathfrak{e}_{1}$ and $\mathfrak{e}_{\circ }\preceq S\mathfrak{e}_{\circ }$, by (i), we have

$$\begin{aligned} F_{d_{q}}\biggl(e_{2},e_{1}, \frac{u}{2}\biggr) \geq &\mu \biggl(D\biggl(e_{1},e_{\circ }, \frac{u}{2}\biggr)\biggr) \\ =&\mu \biggl(\min \biggl\{ \mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{1},\mathfrak{e}_{ \circ },\frac{u}{2}\biggr),\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{1},\mathfrak{e}_{2}, \frac{u}{2} \biggr),\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1}, \frac{u}{2}\biggr)\biggr\} \biggr). \end{aligned}$$

If $\min \{\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{\circ }, \frac{u}{2}),\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{2}, \frac{u}{2}),\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{1}, \frac{u}{2})\}=\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{2},\frac{u}{2})$, then by (2.2)

$$ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{2}, \mathfrak{e}_{1},\frac{u}{2}\biggr) \geq \mu (\min \{ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{ \circ },\frac{u}{2}\biggr),\mathcal{F}_{d_{q}} \biggl(\mathfrak{e}_{\circ },\mathfrak{e}_{1}, \frac{u}{2}\biggr). $$

If $\min \{\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{\circ }, \frac{u}{2}),\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{2}, \frac{u}{2}),\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{1}, \frac{u}{2})\}=\min \{\mathcal{F}_{d_{q}}(\mathfrak{e}_{1},\mathfrak{e}_{\circ }, \frac{u}{2}),\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{1}, \frac{u}{2})\}$, then in both cases, we have

$$ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{2}, \mathfrak{e}_{1},\frac{u}{2}\biggr) \geq \mu \biggl(\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{ \circ },\frac{u}{2}\biggr),\mathcal{F}_{d_{q}} \biggl(\mathfrak{e}_{\circ },\mathfrak{e}_{1}, \frac{u}{2}\biggr)\biggr\} \biggr). $$

(2.5)

By inserting (2.5) into (2.4), we have

$$ \begin{gathered} \begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2},\mathfrak{e}_{\circ },u_{\circ }) \geq {}&\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ }, \frac{u_{\circ }}{2}\biggr)\circledast \mu \biggl( \min \biggl\{ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ },\frac{u_{\circ }}{2}\biggr), \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{ \circ },\mathfrak{e}_{1},\frac{u_{\circ }}{2}\biggr)\biggr\} \biggr) \\ \geq {}&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{ \circ },\frac{u_{\circ }}{2}\biggr), \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},\frac{u_{\circ }}{2}\biggr)\biggr\} \\ &{}\circledast \min \biggl\{ \mu \biggl(\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{1},\mathfrak{e}_{ \circ },\frac{u_{\circ }}{2^{2}}\biggr)\biggr),\mu \biggl(\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{ \circ },\mathfrak{e}_{1}, \frac{u_{\circ }}{2^{2}}\biggr)\biggr)\biggr\} \\ ={}&\underset{p=0}{\overset{1}{\circledast }}\min \biggl\{ \mu ^{p} \biggl( \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ }, \frac{u_{\circ }}{2^{p+1}}\biggr)\biggr),\mu ^{p}\biggl(\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1}, \frac{u_{\circ }}{2^{p+1}}\biggr)\biggr)\biggr\} , \end{aligned} \\ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2},\mathfrak{e}_{\circ },u_{\circ })>1-r. \end{gathered} $$

(2.6)

From (2.3) and (2.6), it follows that $\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{2},u_{\circ })>1-r$ and $\mathcal{F}_{d_{q}}(\mathfrak{e}_{2},\mathfrak{e}_{\circ },u_{\circ })>1-r$. Hence, $\mathfrak{e}_{2}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })$. Also,

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2},\mathcal{T} \mathfrak{e}_{2},u)= \mathcal{F}_{d_{q}}( \mathfrak{e}_{2},\mathfrak{e}_{3},u)\quad \text{and}\quad \mathcal{F}_{d_{q}}(\mathcal{T}\mathfrak{e}_{2}, \mathfrak{e}_{2},u)=\mathcal{F}_{d_{q}}(\mathfrak{e}_{3},\mathfrak{e}_{2},u). $$

As $\mathfrak{e}_{2}\preceq S\mathfrak{e}_{2}$, by $\mathrm{(iib),}$ we have $\mathfrak{e}_{3}\succeq S\mathfrak{e}_{3}$. Let $\mathfrak{e}_{3},\mathfrak{e}_{4},\ldots,\mathfrak{e}_{s}\in B_{\mathcal{F}_{dq}}( \mathfrak{e}_{\circ },r,u_{\circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k})\}$, $\mathfrak{e}_{s}\preceq S\mathfrak{e}_{s}$ and $\mathfrak{e}_{s-1}\succeq S\mathfrak{e}_{s-1}$ for some $s\in \mathbb{N}$, where $s=2p$ and $p=1,2,3,\ldots,\frac{s}{2}$. By using Lemma 1.13, we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u) \geq &H_{d_{q}}( \mathcal{T}\mathfrak{e}_{2p-1},\mathcal{T} \mathfrak{e}_{2p},u) \\ \geq &\min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e}_{2p-1}, \mathcal{T} \mathfrak{e}_{2p},u),H_{d_{q}}( \mathcal{T}\mathfrak{e}_{2p},\mathcal{T} \mathfrak{e}_{2p-1},u) \bigr\} . \end{aligned}$$

As $\mathfrak{e}_{2p-1}$, $\mathfrak{e}_{2p}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k}) \}$, $\mathfrak{e}_{2p-1}\succeq S\mathfrak{e}_{2p-1}$ and $\mathfrak{e}_{2p}\preceq S\mathfrak{e}_{2p}$, by (i), we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u) \geq & \mu \bigl(D(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p},u)\bigr) \\ =&\mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p},u),\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-1},\mathcal{T}\mathfrak{e}_{2p-1},u), \\ &\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p},\mathcal{T} \mathfrak{e}_{2p},u) \bigr\} \bigr) \\ =&\mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p},u),\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u)\bigr\} \bigr). \end{aligned}$$

If $\min \{\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u),\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u)\}= \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u)$, then a contradiction arises due to the fact that $\mu (u)>u$. Therefore,

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u) \geq \mu \bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p},u)\bigr), $$

(2.7)

which implies that

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u) \geq \min \bigl\{ \mu \bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u)\bigr),\mu \bigl( \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p}, \mathfrak{e}_{2p-1},u)\bigr)\bigr\} . $$

(2.8)

Now, by Lemma 1.13, we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u) \geq &H_{d_{q}}( \mathcal{T}\mathfrak{e}_{2p-2},\mathcal{T} \mathfrak{e}_{2p-1},u) \\ \geq &\min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e}_{2p-2}, \mathcal{T} \mathfrak{e}_{2p-1},u),H_{d_{q}}( \mathcal{T}\mathfrak{e}_{2p-1},\mathcal{T} \mathfrak{e}_{2p-2},u) \bigr\} . \end{aligned}$$

As $\mathfrak{e}_{2p-1}$, $\mathfrak{e}_{2p-2}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k}) \},\mathfrak{e}_{2p-1}\succeq S\mathfrak{e}_{2p-1}$ and $\mathfrak{e}_{2p-2}\preceq S\mathfrak{e}_{2p-2}$, by (i), we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u) \geq & \mu \bigl(D(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p-2},u)\bigr) \\ =&\mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p-2},u),\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u), \\ &\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2}, \mathfrak{e}_{2p-1},u)\bigr\} \bigr). \end{aligned}$$

If $\min \{\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1},\mathfrak{e}_{2p-2},u),\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u), \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2},\mathfrak{e}_{2p-1},u)\}=\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u)$, then a contradiction arises due to the fact that $\mu (u)>u$, so

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u) \geq \mu \bigl( \min \bigl\{ \mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-1},\mathfrak{e}_{2p-2},u), \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2}, \mathfrak{e}_{2p-1},u)\bigr\} \bigr). $$

Apply μ on both sides. Since μ is a nondecreasing function,

$$\begin{aligned}& \begin{gathered} \mu \bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p},u)\bigr) \geq \mu ^{2}\bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p-2},u),\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2},\mathfrak{e}_{2p-1},u) \bigr)\bigr\} ,\\ \mu \bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p},u)\bigr) \geq \min \bigl\{ \mu ^{2}\bigl( \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p-2},u) \bigr),\mu ^{2}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-2}, \mathfrak{e}_{2p-1},u)\bigr) \bigr\} . \end{gathered} \end{aligned}$$

(2.9)

By inserting (2.9) into (2.7), we obtain

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u) \geq \min \bigl\{ \mu ^{2}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-1},\mathfrak{e}_{2p-2},u)\bigr), \mu ^{2}\bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2}, \mathfrak{e}_{2p-1},u)\bigr) \bigr\} . $$

(2.10)

Now, by Lemma 1.13, we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2},\mathfrak{e}_{2p-1},u) \geq &H_{dq}(\mathcal{T}\mathfrak{e}_{2p-3}, \mathcal{T}\mathfrak{e}_{2p-2},u) \\ \geq &\min \bigl\{ H_{dq}(\mathcal{T}\mathfrak{e}_{2p-3}, \mathcal{T}\mathfrak{e}_{2p-2},u),H_{dq}( \mathcal{T}\mathfrak{e}_{2p-2}, \mathcal{T}\mathfrak{e}_{2p-3},u) \bigr\} . \end{aligned}$$

As $\mathfrak{e}_{2p-3},\mathfrak{e}_{2p-2}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k}) \}$, $\mathfrak{e}_{2p-3}\succeq S\mathfrak{e}_{2p-3}$ and $\mathfrak{e}_{2p-2}\preceq S\mathfrak{e}_{2p-2}$, by $\mathrm{(i)}$, we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2},\mathfrak{e}_{2p-1},u) \geq \mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u),\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-2},\mathfrak{e}_{2p-1},u) \bigr\} \bigr). $$

If $\min \{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-3},\mathfrak{e}_{2p-2},u),\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2},\mathfrak{e}_{2p-1},u) \} =\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2}, \mathfrak{e}_{2p-1},u)$, then a contradiction arises. Therefore,

$$\begin{aligned}& \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2},\mathfrak{e}_{2p-1},u) \geq \mu \bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u)\bigr), \end{aligned}$$

(2.11)

$$\begin{aligned}& \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2},\mathfrak{e}_{2p-1},u) \geq \mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u),\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-2},\mathfrak{e}_{2p-3},u) \bigr\} \bigr), \\& \mu ^{2}\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2}, \mathfrak{e}_{2p-1},u) \geq \mu ^{3}\bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u), \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2},\mathfrak{e}_{2p-3},u) \bigr\} \bigr). \end{aligned}$$

(2.12)

Now, by Lemma 1.13, we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1},\mathfrak{e}_{2p-2},u) \geq &H_{dq}(\mathcal{T}\mathfrak{e}_{2p-2}, \mathcal{T}\mathfrak{e}_{2p-3},u) \\ \geq &\min \bigl\{ H_{dq}(\mathcal{T}\mathfrak{e}_{2p-3}, \mathcal{T}\mathfrak{e}_{2p-2},u),H_{dq}( \mathcal{T}\mathfrak{e}_{2p-2}, \mathcal{T}\mathfrak{e}_{2p-3},u) \bigr\} . \end{aligned}$$

As $\mathfrak{e}_{2p-3},\mathfrak{e}_{2p-2}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k}) \}$, $\mathfrak{e}_{2p-3}\succeq S\mathfrak{e}_{2p-3}$ and $\mathfrak{e}_{2p-2}\preceq S\mathfrak{e}_{2p-2}$, by $\mathrm{(i)}$, we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1},\mathfrak{e}_{2p-2},u) \geq \mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u),\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-2},\mathfrak{e}_{2p-1},u) \bigr\} \bigr). $$

By using inequality (2.11), we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1},\mathfrak{e}_{2p-2},u) \geq & \mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u),\mu \bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u)\bigr) \bigr\} \bigr) \\ =&\mu \bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u)\bigr). \end{aligned}$$

This implies that

$$ \mu ^{2}\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p-2},u) \geq \mu ^{2}\bigl(\mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u),\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-2},\mathfrak{e}_{2p-3},u) \bigr\} \bigr)\bigr). $$

(2.13)

Combining inequalities (2.10), (2.12), and (2.13), we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u) \geq \min \bigl\{ \mu ^{3}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u)\bigr), \mu ^{3}\bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2}, \mathfrak{e}_{2p-3},u)\bigr) \bigr\} \text{.} $$

(2.14)

Following the patterns of inequalities (2.8), (2.10), and (2.14), we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u) \geq \min \bigl\{ \mu ^{2p}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{\circ }, \mathfrak{e}_{1},u)\bigr), \mu ^{2p}\bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{1}, \mathfrak{e}_{ \circ },u)\bigr) \bigr\} . $$

(2.15)

Now, by Lemma 1.13, we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+1},\mathfrak{e}_{2p},u) \geq &H_{d_{q}}( \mathcal{T}\mathfrak{e}_{2p},\mathcal{T} \mathfrak{e}_{2p-1},u) \\ \geq &\min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e}_{2p-1}, \mathcal{T} \mathfrak{e}_{2p},u),H_{d_{q}}( \mathcal{T}\mathfrak{e}_{2p},\mathcal{T} \mathfrak{e}_{2p-1},u) \bigr\} . \end{aligned}$$

As $\mathfrak{e}_{2p-1}$, $\mathfrak{e}_{2p}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k}) \},\mathfrak{e}_{2p-1}\succeq S\mathfrak{e}_{2p-1}$ and $\mathfrak{e}_{2p}\preceq S\mathfrak{e}_{2p}$, by (i), we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+1},\mathfrak{e}_{2p},u) \geq & \mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p},u),\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u), \\ &\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p}, \mathfrak{e}_{2p+1},u)\bigr\} \bigr) \\ =&\mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p},u),\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p},\mathfrak{e}_{2p+1},u)\bigr\} \bigr). \end{aligned}$$

Using (2.7), we have

$$\begin{aligned}& \begin{gathered} \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+1},\mathfrak{e}_{2p},u) \geq \mu \bigl( \min \bigl\{ \mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u),\mu \bigl( \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p},u)\bigr)\bigr\} \bigr)\\ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+1},\mathfrak{e}_{2p},u) \geq \mu \bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-1}, \mathfrak{e}_{2p},u)\bigr), \end{gathered} \end{aligned}$$

(2.16)

which implies that

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+1},\mathfrak{e}_{2p},u) \geq \min \bigl\{ \mu \bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-1},\mathfrak{e}_{2p},u)\bigr),\mu \bigl( \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p}, \mathfrak{e}_{2p-1},u)\bigr)\bigr\} . $$

(2.17)

By (2.9) and (2.16), we obtain

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+1},\mathfrak{e}_{2p},u) \geq \min \bigl\{ \mu ^{2}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-1},\mathfrak{e}_{2p-2},u)\bigr), \mu ^{2}\bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-2}, \mathfrak{e}_{2p-1},u)\bigr) \bigr\} . $$

(2.18)

Combining (2.12), (2.13), and (2.18), we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+1},\mathfrak{e}_{2p},u) \geq \min \bigl\{ \mu ^{3}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p-2},\mathfrak{e}_{2p-3},u)\bigr), \mu ^{3}\bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p-3}, \mathfrak{e}_{2p-2},u)\bigr) \bigr\} . $$

Continuing in this way, we obtain

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+1},\mathfrak{e}_{2p},u) \geq \min \bigl\{ \mu ^{2p}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{1},\mathfrak{e}_{ \circ },u)\bigr),\mu ^{2p}\bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},u)\bigr)\bigr\} . $$

(2.19)

If $s=2p+1$, $p=1,2,\ldots,\frac{s-1}{2}$, $\mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+1},\mathcal{T}\mathfrak{e}_{2p+1},u)=\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p+1},\mathfrak{e}_{2p+2},u)$ and $\mathcal{F}_{d_{q}}(\mathcal{T}\mathfrak{e}_{2p+1},\mathfrak{e}_{2p+1},u)=\mathcal{F}_{d_{q}}( \mathfrak{e}_{2p+2},\mathfrak{e}_{2p+1},u)$. By using the same procedure as above, we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+1},\mathfrak{e}_{2p+2},u) \geq \min \bigl\{ \mu ^{2p+1}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{1},\mathfrak{e}_{ \circ },u)\bigr),\mu ^{2p+1}\bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},u)\bigr)\bigr\} $$

(2.20)

and

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{2p+2},\mathfrak{e}_{2p+1},u) \geq \min \bigl\{ \mu ^{2p+1}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{1},\mathfrak{e}_{ \circ },u)\bigr),\mu ^{2p+1}\bigl(\mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},u)\bigr)\bigr\} . $$

(2.21)

By combining (2.15) and (2.20), we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{s},\mathfrak{e}_{s+1},u) \geq \min \bigl\{ \mu ^{s}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{1},\mathfrak{e}_{\circ },u)\bigr), \mu ^{s}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{\circ },\mathfrak{e}_{1},u)\bigr)\bigr\} , \quad \text{for some }s\in $$

(2.22)

and by combining (2.19) and (2.21), we obtain

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{s+1},\mathfrak{e}_{s},u) \geq \min \bigl\{ \mu ^{s}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{1},\mathfrak{e}_{\circ },u)\bigr), \mu ^{s}\bigl(\mathcal{F}_{d_{q}}( \mathfrak{e}_{\circ },\mathfrak{e}_{1},u)\bigr)\bigr\} , \quad \text{for some }s\in . $$

(2.23)

By using the triangular inequality, $\mathrm{(iv)}$ and (2.22), we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{s+1},u_{ \circ }) \geq &\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},\frac{u_{\circ }}{2}\biggr)\circledast \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{2},\frac{u_{\circ }}{2^{2}}\biggr)\circledast \cdots\circledast \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{s-1},\mathfrak{e}_{s},\frac{u_{\circ }}{2^{s}} \biggr) \\ &{}\circledast \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{s}, \mathfrak{e}_{s+1}, \frac{u_{\circ }}{2^{s}}\biggr) \\ \geq &\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1}, \frac{u_{\circ }}{2}\biggr)\circledast \cdots \circledast \mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{s-1}, \mathfrak{e}_{s},\frac{u_{\circ }}{2^{s}}\biggr)\circledast \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{s}, \mathfrak{e}_{s+1}, \frac{u_{\circ }}{2^{s+1}}\biggr) \\ \geq &\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1},\frac{u_{\circ }}{2}\biggr), \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ },\frac{u_{\circ }}{2}\biggr)\biggr\} \circledast \\ &\min \biggl\{ \mu \biggl(\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{\circ },\mathfrak{e}_{1},\frac{u_{\circ }}{2^{2}}\biggr)\biggr),\mu \biggl(\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{1},\mathfrak{e}_{\circ }, \frac{u_{\circ }}{2^{2}}\biggr)\biggr)\biggr\} \circledast \cdots \circledast \\ &\min \biggl\{ \mu ^{s}\biggl(\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{\circ }, \mathfrak{e}_{1},\frac{u_{\circ }}{2^{s+1}}\biggr)\biggr),\mu ^{s}\biggl( \mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{1},\mathfrak{e}_{\circ },\frac{u_{\circ }}{2^{s+1}}\biggr)\biggr)\biggr\} \\ =&\underset{p=0}{\overset{s}{\circledast }}\min \biggl\{ \mu ^{p} \biggl( \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{\circ }, \mathfrak{e}_{1}, \frac{u_{\circ }}{2^{p+1}}\biggr)\biggr),\mu ^{p}\biggl(\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ }, \frac{u_{\circ }}{2^{p+1}}\biggr)\biggr)\biggr\} . \end{aligned}$$

Hence, by $\mathrm{(iv)}$, we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{\circ },\mathfrak{e}_{s+1},u_{ \circ })>1-r. $$

(2.24)

Similarly, by the triangular inequality, $\mathrm{(iv)}$ and (2.23), we have

$$ \mathcal{F}_{d_{q}}(\mathfrak{e}_{s+1},\mathfrak{e}_{\circ },u_{ \circ })>1-r. $$

(2.25)

From (2.24) and (2.25), $\mathfrak{e}_{s+1}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })$. Hence, by mathematical induction, we have $\mathfrak{e}_{k}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{ \circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k})\}$, $\mathfrak{e}_{2k}\preceq S\mathfrak{e}_{2k}$ and $\mathfrak{e}_{2k+1}\succeq S\mathfrak{e}_{2k+1}$ for all k∈. Also, from (iii) $\mathfrak{e}_{2k}\in G(S)$. Now, inequalities (2.22) and (2.23) will be held for all s∈. Now, for $k,m\in$ with $k< m$, we have

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e}_{k},\mathfrak{e}_{k+m},u) \geq & \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{k}, \mathfrak{e}_{k+1},\frac{u}{2}\biggr)\circledast \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{k+1}, \mathfrak{e}_{k+2}, \frac{u}{2^{2}}\biggr) \\ &{}\circledast \cdots\circledast\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{k+m-2}, \mathfrak{e}_{k+m-1}, \frac{u}{2^{m-1}}\biggr)\circledast \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{k+m-1}, \mathfrak{e}_{k+m},\frac{u}{2^{m-1}}\biggr) \\ \geq &\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{k}, \mathfrak{e}_{k+1}, \frac{u}{2^{m}}\biggr)\circledast \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{k+1}, \mathfrak{e}_{k+2},\frac{u}{2^{m}}\biggr) \\ &{}\circledast \cdots\circledast\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{k+m-2}, \mathfrak{e}_{k+m-1}, \frac{u}{2^{m}}\biggr)\circledast \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{k+m-1}, \mathfrak{e}_{k+m},\frac{u}{2^{m}}\biggr) \\ \geq &\underset{p=0}{\overset{m-1}{\circledast }}\biggl\{ \mu ^{k+p}\biggl( \mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{\circ },\mathfrak{e}_{1},\frac{u}{2^{m}} \biggr)\biggr), \mu ^{k+p}\biggl(\mathcal{F}_{d_{q}} \biggl(\mathfrak{e}_{1},\mathfrak{e}_{\circ }, \frac{u}{2^{m}}\biggr)\biggr)\biggr\} . \end{aligned}$$

By Lemma 1.3,

$$ \lim_{k\rightarrow \infty }\mu ^{k+p}\biggl(\mathcal{F}_{d_{q}} \biggl(\mathfrak{e}_{\circ },\mathfrak{e}_{1}, \frac{u}{2^{m}}\biggr)\biggr)=1,\quad \text{for every }p\in \{0,1, \ldots,m-1\} $$

and

$$ \lim_{k\rightarrow \infty }\mu ^{k+p}\biggl(\mathcal{F}_{d_{q}} \biggl(\mathfrak{e}_{1},\mathfrak{e}_{\circ }, \frac{u}{2^{m}}\biggr)\biggr)=1, \quad \text{for every }p\in \{0,1, \ldots,m-1\}. $$

Hence, $\mathcal{F}_{d_{q}}(\mathfrak{e}_{k},\mathfrak{e}_{k+m},u) \rightarrow 1$ as $k\rightarrow \infty$. Thus, we proved that $\{\mathcal{YT}(\mathfrak{e}_{k})\}$ is a left K-Cauchy sequence in $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$. As $(\mathcal{Y},\mathcal{F}_{d_{q}},\circledast )$ is left K-sequentially complete, so $\{\mathcal{YT}(\mathfrak{e}_{k})\}\rightarrow \mathfrak{e}^{\divideontimes } \mathfrak{\in \mathcal{Y}}$. As $\{\mathfrak{e}_{2k}\}$ is a subsequence of $\{\mathcal{YT}(\mathfrak{e}_{k})\}$, so $\mathfrak{e}_{2k}\rightarrow \mathfrak{e}^{\divideontimes }\mathfrak{.}$ Also, $\{\mathfrak{e}_{2k}\}$ is a sequence in $G(S)$ and $G(S)$ is closed, so $\mathfrak{e}^{\divideontimes }\in G(S)$ and therefore $\mathfrak{e}^{\divideontimes }\preceq S\mathfrak{e}^{\divideontimes }$. Also,

$$ \lim_{k\rightarrow \infty }\mathcal{F}_{d_{q}}\bigl( \mathfrak{e}_{2k},\mathfrak{e}^{\divideontimes },u\bigr)=1= \lim_{k\rightarrow \infty }\mathcal{F}_{d_{q}}\bigl( \mathfrak{e}^{\divideontimes },\mathfrak{e}_{2k},u\bigr). $$

(2.26)

Now,

$$\begin{aligned}& \mathcal{F}_{d_{q}}\bigl(\mathfrak{e}^{\divideontimes }, \mathfrak{e}^{\divideontimes },u\bigr) \geq \mathcal{F}_{d_{q}} \biggl(\mathfrak{e}^{\divideontimes },\mathfrak{e}_{2k}, \frac{u}{2}\biggr)\circledast \mathcal{F}_{d_{q}} \biggl(\mathfrak{e}_{2k},\mathfrak{e}^{\divideontimes }, \frac{u}{2}\biggr) \\& \lim_{k\rightarrow \infty }\mathcal{F}_{d_{q}}\bigl( \mathfrak{e}^{\divideontimes },\mathfrak{e}^{\divideontimes },u\bigr) \geq 1 \circledast 1=1. \end{aligned}$$

Hence,

$$ \mathcal{F}_{d_{q}}\bigl(\mathfrak{e}^{\divideontimes }, \mathfrak{e}^{\divideontimes },u\bigr)=1. $$

(2.27)

Now,

$$\begin{aligned} \mathcal{F}_{d_{q}}\bigl(\mathfrak{e}_{2k+2},\mathcal{T} \mathfrak{e}^{\divideontimes },u\bigr) \geq &H_{d_{q}}\bigl( \mathcal{T}\mathfrak{e}_{2k+1},\mathcal{T} \mathfrak{e}^{\divideontimes },u\bigr) \\ \geq &\min \bigl\{ H_{d_{q}}\bigl(\mathcal{T}\mathfrak{e}_{2k+1}, \mathcal{T} \mathfrak{e}^{\divideontimes },u\bigr),H_{d_{q}} \bigl(\mathcal{T}\mathfrak{e}^{ \divideontimes },\mathcal{T} \mathfrak{e}_{2k+1},u\bigr)\bigr\} . \end{aligned}$$

By assumption, inequality (i) holds for $\mathfrak{e}^{\divideontimes }$, also $\mathfrak{e}\succeq S\mathfrak{e}_{2k+1}$ and $\mathfrak{e}^{\divideontimes }\preceq S\mathfrak{e}^{\divideontimes }$. Hence,

$$\begin{aligned} \mathcal{F}_{d_{q}}\bigl(\mathfrak{e}_{2k+2},\mathcal{T} \mathfrak{e}^{\divideontimes },u\bigr) \geq &\mu \bigl(D\bigl( \mathfrak{e}_{2k+1}, \mathfrak{e}^{\divideontimes },u\bigr) \bigr) \\ =&\mu \bigl(\min \bigl\{ \mathcal{F}_{d_{q}}\bigl(\mathfrak{e}_{2k+1}, \mathfrak{e}^{\divideontimes },u\bigr),\mathcal{F}_{d_{q}}( \mathfrak{e}_{2k+1}, \mathfrak{e}_{2k+2},u), \mathcal{F}_{d_{q}}\bigl(\mathfrak{e}^{\divideontimes }, \mathcal{T}\mathfrak{e}^{\divideontimes },u\bigr)\bigr\} \bigr). \end{aligned}$$

Letting $k\rightarrow \infty$ and by using (2.27), we obtain

$$ \mathcal{F}_{d_{q}}\bigl(\mathfrak{e}^{\divideontimes },\mathcal{T} \mathfrak{e}^{\divideontimes },u\bigr)\geq \mu \bigl( \mathcal{F}_{d_{q}}\bigl(\mathfrak{e}^{\divideontimes }, \mathcal{T}\mathfrak{e}^{\divideontimes },u\bigr)\bigr). $$

By definition of μ, we obtain

$$ \mathcal{F}_{d_{q}}\bigl(\mathfrak{e}^{\divideontimes },\mathcal{T} \mathfrak{e}^{\divideontimes },u\bigr)=1. $$

(2.28)

$$\begin{aligned} \mathcal{F}_{d_{q}}\bigl(\mathcal{T}\mathfrak{e}^{\divideontimes }, \mathfrak{e}^{\divideontimes },u\bigr) \geq &\mathcal{F}_{d_{q}} \biggl(\mathcal{T} \mathfrak{e}^{\divideontimes },\mathfrak{e}_{2k+2}, \frac{u}{2}\biggr)\circledast \mathcal{F}_{d_{q}} \biggl(\mathfrak{e}_{2k+2},\mathfrak{e}^{\divideontimes }, \frac{u}{2}\biggr) \\ \geq &H_{d_{q}}\biggl(\mathcal{T}\mathfrak{e}^{\divideontimes }, \mathcal{T}\mathfrak{e}_{2k+1},\frac{u}{2}\biggr) \circledast \mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{2k+2}, \mathfrak{e}^{\divideontimes },\frac{u}{2}\biggr) \\ \geq &\min \biggl\{ H_{d_{q}}\biggl(\mathcal{T}\mathfrak{e}^{\divideontimes }, \mathcal{T}\mathfrak{e}_{2k+1},\frac{u}{2} \biggr),H_{d_{q}}\biggl(\mathcal{T}\mathfrak{e}_{2k+1},\mathcal{T}\mathfrak{e}^{\divideontimes },\frac{u}{2}\biggr)\biggr\} \\ &{}\circledast\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{2k+2}, \mathfrak{e}^{ \divideontimes },\frac{u}{2}\biggr). \end{aligned}$$

By assumption, inequality $\mathrm{(i)}$ holds for $\mathfrak{e}^{\divideontimes }$, also $\mathfrak{e}_{2k+1}\succeq S\mathfrak{e}_{2k+1}$ and $\mathfrak{e}^{\divideontimes }\preceq S\mathfrak{e}^{\divideontimes }$. Hence,

$$\begin{aligned} \mathcal{F}_{d_{q}}\bigl(\mathcal{T}\mathfrak{e}^{\divideontimes }, \mathfrak{e}^{\divideontimes },u\bigr) \geq &\mu \biggl(D\biggl( \mathfrak{e}_{2k+1}, \mathfrak{e}^{\divideontimes }, \frac{u}{2}\biggr)\biggr)\circledast \mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{2k+2},\mathfrak{e}^{\divideontimes }, \frac{u}{2}\biggr) \\ =&\mu \biggl(\min \biggl\{ \mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{2k+1},\mathfrak{e}^{\divideontimes }, \frac{u}{2}\biggr),\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{2k+1},\mathfrak{e}_{2k+2}, \frac{u}{2}\biggr) \\ &\mathcal{F}_{d_{q}}\biggl(\mathfrak{e}^{\divideontimes },\mathcal{T} \mathfrak{e}^{\divideontimes },\frac{u}{2}\biggr)\biggr\} \biggr)\circledast \mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{2k+2}, \mathfrak{e}^{\divideontimes },\frac{u}{2}\biggr). \end{aligned}$$

Letting $k\rightarrow \infty$, and by using (2.26) and (2.28), we obtain

$$ \mathcal{F}_{d_{q}}\bigl(\mathcal{T}\mathfrak{e}^{\divideontimes }, \mathfrak{e}^{\divideontimes },u\bigr)\geq \mu \biggl( \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}^{\divideontimes }, \mathcal{T}\mathfrak{e}^{\divideontimes }, \frac{u}{2}\biggr) \biggr)=\mu (1). $$

This implies that $\mathcal{F}_{d_{q}}(\mathcal{T}\mathfrak{e}^{\divideontimes },\mathfrak{e}^{\divideontimes },u)=1$. Hence, $\mathfrak{e}^{\divideontimes }\in \mathcal{T}\mathfrak{e}^{\divideontimes }$. Also,

$$ \mathfrak{e}^{\divideontimes }\preceq S\mathfrak{e}^{\divideontimes }, $$

(2.29)

and

$$ \mathcal{F}_{d_{q}}\bigl(\mathcal{T}\mathfrak{e}^{\divideontimes }, \mathfrak{e}^{\divideontimes },u\bigr)=\mathcal{F}_{d_{q}}\bigl( \mathfrak{e}^{ \divideontimes },\mathcal{T}\mathfrak{e}^{\divideontimes },u \bigr)=\mathcal{F}_{d_{q}}\bigl( \mathfrak{e}^{\divideontimes }, \mathfrak{e}^{\divideontimes },u\bigr)=1. $$

Then, from (ii)

$$ \mathfrak{e}^{\divideontimes }\succeq S\mathfrak{e}^{\divideontimes }. $$

(2.30)

From (2.29) and (2.30), we have $\mathfrak{e}^{\divideontimes }\preceq S\mathfrak{e}^{\divideontimes }\preceq \mathfrak{e}^{\divideontimes }$. This implies

$$ \mathfrak{e}^{\divideontimes }\preceq f\preceq \mathfrak{e}^{ \divideontimes }\quad \text{for all }f \in S\mathfrak{e}^{\divideontimes }. $$

Therefore,

$$ \mathfrak{e}^{\divideontimes }=f \quad \text{for all }f\in S\mathfrak{e}^{\divideontimes }. $$

Hence, $\mathfrak{e}^{\divideontimes }$ is a common fixed point of $\mathcal{T}$ and S. □

Example 2.2

Let $\mathcal{Y}=[0,+\infty )$ and let $\mathcal{F}_{d_{q}}(\mathfrak{e},f,u)=\frac{u}{u+\mathfrak{e}+2f}$ for all $\mathfrak{e},f,u\in \mathcal{Y}$. Let $\mathcal{R}$ be the binary relation on $\mathcal{Y}$ defined by

$$\begin{aligned} \mathcal{R} =& \bigl\{ (\mathfrak{e},\mathfrak{e}):\mathfrak{e}\in \mathcal{Y} \bigr\} \cup \biggl\{ \biggl(\mathfrak{e}, \frac{\mathfrak{e}}{5}\biggr): \mathfrak{e}\in \biggl\{ 0,\frac{3}{7}, \frac{3}{7\times 25}, \frac{3}{7\times 25^{2}},\frac{3}{7\times 25^{3}},\ldots\biggr\} \biggr\} \\ &{}\cup \biggl\{ \biggl(\frac{\mathfrak{e}}{5},\mathfrak{e}\biggr):\mathfrak{e} \in \biggl\{ 0,\frac{3}{35},\frac{3}{35\times 25} \frac{3}{35\times 25^{2}},\ldots \biggr\} \biggr\} . \end{aligned}$$

Consider the partial order on $\mathcal{Y}$ defined by

$$ (\mathfrak{e},f)\in \mathcal{Y}\times \mathcal{Y},\text{ } \mathfrak{e}\preceq f\text{ if and only }(\mathfrak{e},f)\in R. $$

Then, $(\mathcal{Y},\preceq ,\mathcal{F}_{d_{q}},\circledast )$ is a left K-sequentially complete ordered dislocated fuzzy quasimetric space with $a\circledast b=\min \{a,b\}$. Define the pair of mappings $\mathcal{T},S:\mathcal{Y}\rightarrow \mathcal{Y}$ by

$$ \mathcal{T}(\mathfrak{e})=\textstyle\begin{cases} {}[ \frac{\mathfrak{e}}{7},\frac{\mathfrak{e}}{5}]&\text{if } \mathfrak{e}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\cap \{ \mathcal{YT}(\mathfrak{e}_{k})\}, \\ {}[ 3\mathfrak{e},6\mathfrak{e}]&\text{if }\mathfrak{e}\notin B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\cap \{ \mathcal{YT}(\mathfrak{e}_{k})\}\end{cases} $$

and

$$ S(\mathfrak{e})=\textstyle\begin{cases} \{\frac{\mathfrak{e}}{5}\} &\text{if }\mathfrak{e}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{ \circ })\cap \{\mathcal{YT}(\mathfrak{e}_{k})\}, \\ {}[ 7\mathfrak{e}+5,8\mathfrak{e}+9]&\text{if }\mathfrak{e}\notin B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ },r,u_{\circ })\cap \{ \mathcal{YT}(\mathfrak{e}_{k})\}.\end{cases} $$

Define a nondecreasing mapping $\mu :[0,1]\rightarrow {}[ 0,1]$ by

$$ \mu (u)=\sqrt{u}. $$

Observe that in this case, we have

$$\begin{aligned} A =& \{ \mathfrak{e}:\mathfrak{e}\preceq S\mathfrak{e} \} = \biggl\{ 0,\frac{3}{7},\frac{3}{7\times 25},\frac{3}{7\times 25^{2}}, \frac{3}{7\times 25^{3}},\ldots\biggr\} , \\ B =& \{ f:f\succeq Sf \} =\biggl\{ 0,\frac{3}{35}, \frac{3}{35\times 25}\frac{3}{35\times 25^{2}},\ldots\biggr\} . \end{aligned}$$

Let $\mathfrak{e}_{\circ }=\frac{3}{7}$, $r=\frac{3}{4}$ and $u=1$

$$\begin{aligned} B_{\mathcal{F}_{dq}}\biggl(\frac{3}{7},\frac{3}{4},1\biggr) =& \biggl\{ f: \mathcal{F}_{d_{q}}\biggl( \frac{3}{7},f,1\biggr)>1-\frac{3}{4}\wedge \mathcal{F}_{d_{q}}\biggl(f, \frac{3}{7},1\biggr)>1- \frac{3}{4} \biggr\} \\ =& \biggl\{ f:\frac{1}{1+\frac{3}{7}+2f}>\frac{1}{4}\wedge \frac{1}{1+f+2(\frac{3}{7})}>\frac{1}{4} \biggr\} \\ =& \{ f:28>10+14f\wedge 28>13+7f \} \\ =& \biggl\{ f:\frac{9}{7}>f\wedge \frac{15}{7}>f \biggr\} =\biggl[0, \frac{9}{7}\biggr). \end{aligned}$$

Then,

$$\begin{aligned} G(S) =& \bigl\{ \mathfrak{e}:\mathfrak{e}\preceq S\mathfrak{e} \text{ and }\mathfrak{e}\in B_{\mathcal{F}_{dq}}( \mathfrak{e}_{\circ ,}r,u_{ \circ }) \bigr\} \\ =& \biggl\{ 0,\frac{3}{7},\frac{3}{7\times 25}, \frac{3}{7\times 25^{2}},\frac{3}{7\times 25^{3}},\ldots \biggr\} \cap {}\biggl[ 0, \frac{9}{7}\biggr) \\ =& \biggl\{ 0,\frac{3}{7},\frac{3}{7\times 25}, \frac{3}{7\times 25^{2}},\frac{3}{7\times 25^{3}},\ldots \biggr\} . \end{aligned}$$

$G(S)$ contains $\frac{3}{7}$ and is also a closed set. Now, $\frac{3}{7\times 25^{k-1}}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ ,}r,u_{ \circ })$, where k∈:

$$\begin{aligned} \mathcal{F}_{d_{q}}(\mathfrak{e},\mathcal{T}\mathfrak{e},u) =& \mathcal{F}_{d_{q}}\biggl(\frac{3}{7\times 25^{k-1}},\mathcal{T} \biggl( \frac{3}{7\times 25^{k-1}}\biggr),u\biggr) \\ =&\mathcal{F}_{d_{q}}\biggl(\frac{3}{7\times 25^{k-1}}, \frac{3}{5\times 7\times 25^{k-1}},u\biggr) \\ =&\mathcal{F}_{d_{q}}\biggl(\frac{3}{7\times 25^{k-1}}, \frac{3}{35\times 25^{k-1}},u\biggr), \\ \mathcal{F}_{d_{q}}(\mathcal{T}\mathfrak{e},\mathfrak{e},u) =& \mathcal{F}_{d_{q}}\biggl(\mathcal{T}\biggl( \frac{3}{7\times 25^{k-1}}\biggr), \frac{3}{7\times 25^{k-1}},u\biggr) \\ =&\mathcal{F}_{d_{q}}\biggl(\frac{3}{5\times 7\times 25^{k-1}}, \frac{3}{7\times 25^{k-1}},u\biggr) \\ =&\mathcal{F}_{d_{q}}\biggl(\frac{3}{35\times 25^{k-1}}, \frac{3}{7\times 25^{k-1}},u\biggr). \end{aligned}$$

Also, $(\frac{3}{7\times 25^{k-1}},\frac{3}{5\times 7\times 25^{k-1}})\in \mathcal{R}$, so $\frac{3}{7\times 25^{k-1}}\preceq S(\frac{3}{7\times 25^{k-1}})$. As $(\frac{3}{5\times 35\times 25^{k-1}},\frac{3}{35\times 25^{k-1}}) \in \mathcal{R}$ so $\frac{3}{35\times 25^{k-1}}\succeq S(\frac{3}{35\times 25^{k-1}})$. Hence, condition (iia) is satisfied. Now $\frac{3}{35\times 25^{k-1}}\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{ \circ ,}r,u_{\circ })$, where k∈.

$$\begin{aligned}& \begin{aligned} \mathcal{F}_{d_{q}}\biggl(\frac{3}{35\times 25^{k-1}},\mathcal{T}\biggl( \frac{3}{35\times 25^{k-1}}\biggr),u\biggr) &=\mathcal{F}_{d_{q}}\biggl( \frac{3}{35\times 25^{k-1}},\frac{3}{5\times 35\times 25^{k-1}},u\biggr) \\ &=\mathcal{F}_{d_{q}}\biggl(\frac{3}{35\times 25^{k-1}}, \frac{3}{25\times 7\times 25^{k-1}},u\biggr) \\ &=\mathcal{F}_{d_{q}}\biggl(\frac{3}{35\times 25^{k-1}}, \frac{3}{7\times 25^{k}},u\biggr), \end{aligned} \\& \begin{aligned} \mathcal{F}_{d_{q}}\biggl(\mathcal{T}\biggl(\frac{3}{35\times 25^{k-1}} \biggr),\biggl( \frac{3}{35\times 25^{k-1}}\biggr),u\biggr) &=\mathcal{F}_{d_{q}} \biggl( \frac{3}{5\times 35\times 25^{k-1}},\frac{3}{35\times 25^{k-1}},u\biggr) \\ &=\mathcal{F}_{d_{q}}\biggl(\frac{3}{25\times 7\times 25^{k-1}}, \frac{3}{35\times 25^{k-1}},u\biggr) \\ &=\mathcal{F}_{d_{q}}\biggl(\frac{3}{7\times 25^{k}}, \frac{3}{35\times 25^{k-1}},u\biggr). \end{aligned} \end{aligned}$$

Also, $(\frac{3}{5\times 35\times 25^{k-1}},\frac{3}{35\times 25^{k-1}}) \in \mathcal{R}$ so $\frac{3}{35\times 25^{k-1}}\succeq S(\frac{3}{35\times 25^{k-1}})$. As $(\frac{3}{7\times 25^{k}},\frac{3}{5\times 7\times 25^{k}})\in \mathcal{R}$ so $\frac{3}{7\times 25^{k}}\preceq S(\frac{3}{7\times 25^{k}})$. Hence, condition (iia) is satisfied.

$$\begin{aligned} B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ ,}r,u_{\circ })\cap \mathcal{YT}(\mathfrak{e}_{k}) =& \biggl\{ \frac{3}{7},\frac{3}{7\times 5}, \frac{3}{7\times 5^{2}}, \frac{3}{7\times 5^{3}},\ldots \biggr\} \\ =& \biggl\{ \frac{3}{7},\frac{3}{35},\frac{3}{7\times 25}, \frac{3}{35\times 25},\frac{3}{7\times 25^{2}},\frac{3}{35\times 25^{2}}\ldots \biggr\} . \end{aligned}$$

Now, for $\mathfrak{e},f\notin B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ ,}r,u_{ \circ })\cap \mathcal{YT}(\mathfrak{e}_{k})$. Let $\mathfrak{e}=5$, $f=6$ and $u=\frac{1}{2}$.

$$\begin{aligned}& \begin{aligned} H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u) &= H_{d_{q}} \biggl( \mathcal{T}(5),\mathcal{T}(6),\frac{1}{2}\biggr) \\ &=H_{d_{q}}\biggl([15,30],[18,36],\frac{1}{2}\biggr) \\ &=\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(15,36,\frac{1}{2} \biggr),\mathcal{F}_{d_{q}}\biggl(30,18,\frac{1}{2} \biggr)\biggr\} \\ &=\min \biggl\{ \frac{1}{175},\frac{1}{133}\biggr\} =0.005714, \end{aligned}\\& \begin{aligned} H_{d_{q}}(\mathcal{T}f,\mathcal{T}\mathfrak{e},u) &= H_{d_{q}} \biggl( \mathcal{T}(6),\mathcal{T}(5),\frac{1}{2}\biggr) \\ &=\min \{\mathcal{F}_{d_{q}}\biggl(18,30,\frac{1}{2}\biggr), \mathcal{F}_{d_{q}}\biggl(36,15,\frac{1}{2}\biggr) \\ &=\min \biggl\{ \frac{1}{181},\frac{1}{151}\biggr\} =0.00552486. \end{aligned}\\& \min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u),H_{d_{q}}( \mathcal{T}f,\mathcal{T}\mathfrak{e},u)\bigr\} =\min \{0.005714,0.00552486\}=0.00552486. \\& \begin{aligned} \mu \bigl(D(\mathfrak{e},f,u)\bigr) &=\mu \biggl\{ \mathcal{F}_{d_{q}} \biggl(5,6, \frac{1}{2}\biggr),\mathcal{F}_{d_{q}} \biggl(5,30,\frac{1}{2}\biggr),\mathcal{F}_{d_{q}}\biggl(6,36, \frac{1}{2}\biggr)\biggr\} \\ &=\mu \biggl\{ \frac{1}{35},\frac{1}{131},\frac{1}{157} \biggr\} =\mu \biggl\{ \frac{1}{157}\biggr\} =0.079749, \end{aligned}\\& \min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u),H_{d_{q}}( \mathcal{T}f,\mathcal{T}\mathfrak{e},u)\bigr\} \ngeq \mu \bigl(D( \mathfrak{e},f,u)\bigr). \end{aligned}$$

Hence, contraction does not hold on the whole space $\mathcal{Y}$. Now, for $\mathfrak{e}$, $f\in B_{\mathcal{F}_{dq}}(\mathfrak{e}_{\circ ,}r,u_{\circ })\cap \mathcal{YT}(\mathfrak{e}_{k})$ with $\mathfrak{e}\succeq S\mathfrak{e}$ and $f\preceq Sf$, $\mathfrak{e}\in B$ and $f\in A$. In general, for some $k,m\in$.

Case (i): Let $k\leq m$, $\mathfrak{e}=\frac{3}{35\times 25^{m-1}}$, $f=\frac{3}{7\times 25^{k-1}}$ $u>0$. We have

$$\begin{aligned}& \begin{aligned} H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u) ={}&H_{d_{q}} \biggl( \mathcal{T}\biggl(\frac{3}{35\times 25^{m-1}}\biggr),\mathcal{T} \biggl(\frac{3}{7\times 25^{k-1}}\biggr),u\biggr) \\ ={}&H_{d_{q}}\biggl(\biggl[\frac{3}{7\times 35\times 25^{m-1}}, \frac{3}{5\times 35\times 25^{m-1}} \biggr], \\ &{}\biggl[\frac{3}{7\times 7\times 25^{k-1}}, \frac{3}{5\times 7\times 25^{k-1}}\biggr],u\biggr) \\ ={}&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(\frac{3}{7\times 35\times 25^{m-1}}, \frac{3}{5\times 7\times 25^{k-1}},u\biggr), \\ &{}\mathcal{F}_{d_{q}}\biggl(\frac{3}{5\times 35\times 25^{m-1}}, \frac{3}{7\times 7\times 25^{k-1}},u\biggr)\biggr\} , \end{aligned} \\& H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u)=\min \biggl\{ \frac{u}{u+\frac{3}{7\times 35\times 25^{m-1}}+\frac{6}{35\times 25^{k-1}}}, \frac{u}{u+\frac{3}{5\times 35\times 25^{m-1}}+\frac{6}{7\times 7\times 25^{k-1}}} \biggr\} , \\& \begin{aligned} H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u) ={}&\min \biggl\{ \frac{245\times 25^{m-1}u}{245\times 25^{m-1}u+3+42\times 25^{m-k}}, \\ &{} \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+21+150\times 25^{m-k}} \biggr\} \\ ={}&\min \biggl\{ \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+3(5+70\times 25^{m-k})}, \\ &{} \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+3(7+50\times 25^{m-k})} \biggr\} \\ ={}& \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+15+210\times 25^{m-k}} \\ ={}& \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+15(1+14\times 25^{m-k})}. \end{aligned} \end{aligned}$$

(2.31)

Now,

$$\begin{aligned}& \begin{aligned} H_{d_{q}}(\mathcal{T}f,\mathcal{T}\mathfrak{e},u) ={}&H_{d_{q}} \biggl( \mathcal{T}\biggl(\frac{3}{7\times 25^{k-1}}\biggr),\mathcal{T} \biggl(\frac{3}{35\times 25^{m-1}}\biggr),u\biggr) \\ ={}&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(\frac{3}{49\times 25^{k-1}}, \frac{3}{175\times 25^{m-1}},u\biggr), \\ &{}\mathcal{F}_{d_{q}}\biggl(\frac{3}{35\times 25^{k-1}}, \frac{3}{245\times 25^{m-1}},u\biggr)\biggr\} , \end{aligned} \\& H_{d_{q}}(\mathcal{T}f,\mathcal{T}\mathfrak{e},u)=\min \biggl\{ \frac{u}{u+\frac{3}{49\times 25^{k-1}}+\frac{6}{175\times 25^{m-1}}}, \frac{u}{u+\frac{3}{35\times 25^{k-1}}+\frac{6}{245\times 25^{m-1}}}\biggr\} , \\& \begin{aligned} H_{d_{q}}(\mathcal{T}f,\mathcal{T}\mathfrak{e},u) ={}&\min \biggl\{ \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+3(25\times 25^{m-k}+14)}, \\ &{} \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+3(35\times 25^{m-k}+10)} \biggr\} \\ ={}& \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+105\times 25^{m-k}+30} \\ ={}& \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+15(2+7\times 25^{m-k})}. \end{aligned} \end{aligned}$$

(2.32)

Also,

$$\begin{aligned} \min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u),H_{d_{q}}( \mathcal{T}f,\mathcal{T}\mathfrak{e},u)\bigr\} =&\min \biggl\{ \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+15(1+14\times 25^{m-k})}, \\ & {}\frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+15(2+7\times 25^{m-k})} \biggr\} \\ =& \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+15+210\times 25^{m-k}}. \end{aligned}$$

Now, for $\mathfrak{e}\succeq S\mathfrak{e}$, $f\preceq Sf$, we have

$$\begin{aligned}& \begin{gathered} \begin{aligned} D(\mathfrak{e},f,u) ={}&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl( \frac{3}{35\times 25^{m-1}},\frac{3}{7\times 25^{k-1}},u\biggr), \mathcal{F}_{d_{q}}\biggl( \frac{3}{35\times 25^{m-1}}, \\ &{}\biggl[\frac{3}{7\times 35\times 25^{m-1}}, \frac{3}{5\times 35\times 25^{m-1}}\biggr],u \biggr), \\ &{}\mathcal{F}_{d_{q}}\biggl(\frac{3}{7\times 25^{k-1}},\biggl[ \frac{3}{7\times 7\times 25^{k-1}},\frac{3}{5\times 7\times 25^{k-1}}\biggr],u\biggr) \biggr\} , \end{aligned}\\ D(\mathfrak{e},f,u)=\min \begin{Bmatrix} \mathcal{F}_{d_{q}}\biggl( \frac{3}{35\times 25^{m-1}}, \frac{3}{7\times 25^{k-1}},u\biggr), \mathcal{F}_{d_{q}}\biggl(\frac{3}{35\times 25^{m-1}}, \frac{3}{175\times 25^{m-1}},u \biggr), \\ \mathcal{F}_{d_{q}}\biggl(\frac{3}{7\times 25^{k-1}}, \frac{3}{35\times 25^{k-1}},u \biggr)\end{Bmatrix} , \\ \begin{aligned} D(\mathfrak{e},f,u) ={}&\min \biggl\{ \frac{35\times 25^{m-1}u}{35\times 25^{m-1}u+3+30\times 25^{m-k}}, \frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+21}, \\ &{}\frac{35\times 25^{k-1}u}{35\times 25^{k-1}u+21}\biggr\} \\ ={}&\frac{35\times 25^{m-1}u}{35\times 25^{m-1}u+3+30\times 25^{m-k}}, \end{aligned}\\ \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+15+210\times 25^{m-k}} \geq \sqrt{ \frac{35\times 25^{m-1}u}{35\times 25^{m-1}u+3+30\times 25^{m-k}}}, \\ \min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u),H_{d_{q}}( \mathcal{T}f,\mathcal{T}\mathfrak{e},u)\bigr\} \geq \mu \bigl(D( \mathfrak{e},f,u)\bigr). \end{gathered} \end{aligned}$$

(2.33)

Case (ii): For $k>m$, $\mathfrak{e}=\frac{3}{35\times 25^{m-1}}$, $f= \frac{3}{7\times 25^{k-1}}$ and $u>0$. From (2.31), we have

$$\begin{aligned} H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u) =&\min \biggl\{ \frac{u}{u+\frac{3}{7\times 35\times 25^{m-1}}+\frac{6}{5\times 7\times 25^{k-1}}}, \\ &{} \frac{u}{u+\frac{3}{5\times 35\times 25^{m-1}}+\frac{6}{7\times 7\times 25^{k-1}}} \biggr\} \\ =&\min \biggl\{ \frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+15\times 25^{k-m}+210}, \\ &{} \frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+21\times 25^{k-m}+150} \biggr\} \\ =& \frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+21\times 25^{k-m}+150}. \end{aligned}$$

From (2.32)

$$\begin{aligned} H_{d_{q}}(\mathcal{T}f,\mathcal{T}\mathfrak{e},u) =&\min \biggl\{ \frac{u}{u+\frac{3}{7\times 7\times 25^{k-1}}+\frac{6}{5\times 35\times 25^{m-1}}}, \frac{u}{u+\frac{3}{5\times 7\times 25^{k-1}}+\frac{6}{7\times 35\times 25^{m-1}}} \biggr\} \\ =&\min \biggl\{ \frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+75+42\times 25^{k-m}}, \\ & {}\frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+105+30\times 25^{k-m}} \biggr\} \\ =& \frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+75+42\times 25^{k-m}}. \end{aligned}$$

Also,

$$\begin{aligned}& \min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u),H_{d_{q}}(\mathcal{T}f,\mathcal{T}\mathfrak{e},u)\bigr\} \\& \quad =\min \biggl\{ \frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+21\times 25^{k-m}+150}, \frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+75+42\times 25^{k-m}} \biggr\} \\& \quad = \frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+75+42\times 25^{k-m}}. \end{aligned}$$

Now, from (2.33), we have

$$\begin{aligned} D(\mathfrak{e},f,u) =&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl( \frac{3}{35\times 25^{m-1}},\frac{3}{7\times 25^{k-1}},u\biggr), \\ &{}\mathcal{F}_{d_{q}}\biggl(\frac{3}{35\times 25^{m-1}}, \frac{3}{5\times 35\times 25^{m-1}},u\biggr),\\ &{}\mathcal{F}_{d_{q}}\biggl( \frac{3}{7\times 25^{k-1}},\frac{3}{5\times 7\times 25^{k-1}},u\biggr)\biggr\} \\ =&\min \biggl\{ \frac{35\times 25^{k-1}u}{35\times 25^{k-1}u+3\times 25^{k-m}+30},\frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+21},\\ &{} \frac{35\times 25^{k-1}u}{35\times 25^{k-1}u+21}\biggr\} . \end{aligned}$$

Assume that

$$ D(\mathfrak{e},f,u)= \frac{35\times 25^{k-1}u}{35\times 25^{k-1}u+3\times 25^{k-m}+30}. $$

Then,

$$\begin{aligned}& \frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+75+42\times 25^{k-m}} \geq \sqrt{ \frac{35\times 25^{k-1}u}{35\times 25^{k-1}u+3\times 25^{k-m}+30}}, \\& \min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u),H_{d_{q}}( \mathcal{T}f,\mathcal{T}\mathfrak{e},u)\bigr\} \geq \mu \bigl(D( \mathfrak{e},f,u)\bigr). \end{aligned}$$

Assume that

$$ D(\mathfrak{e},f,u)= \frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+21}. $$

Then,

$$\begin{aligned}& \frac{1225\times 25^{k-1}u}{1225\times 25^{k-1}u+75+42\times 25^{k-m}} \geq \sqrt{\frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+21}}, \\& \min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u),H_{d_{q}}( \mathcal{T}f,\mathcal{T}\mathfrak{e},u)\bigr\} \geq \mu \bigl(D( \mathfrak{e},f,u)\bigr). \end{aligned}$$

Case (iii): For $\mathfrak{e}=0$, $f=\frac{3}{7\times 25^{k-1}}$, $u>0$

$$\begin{aligned} H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u) =&H_{d_{q}} \biggl( \mathcal{T}(0),\mathcal{T}\biggl(\frac{3}{7\times 25^{k-1}}\biggr),u \biggr) \\ =&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(0,\frac{3}{5\times 7\times 25^{k-1}},u \biggr),\mathcal{F}_{d_{q}}\biggl(0,\frac{3}{7\times 7\times 25^{k-1}},u \biggr)\biggr\} \\ =&\min \biggl\{ \frac{245\times 25^{k-1}u}{245\times 25^{k-1}u+42}, \frac{245\times 25^{k-1}u}{245\times 25^{k-1}u+30}\biggr\} \\ =&\frac{245\times 25^{k-1}u}{245\times 25^{k-1}u+42}. \end{aligned}$$

Also,

$$\begin{aligned} H_{d_{q}}(\mathcal{T}f,\mathcal{T}\mathfrak{e},u) =&H_{d_{q}} \biggl( \mathcal{T}\biggl(\frac{3}{7\times 25^{k-1}}\biggr),\mathcal{T}(0),u \biggr) \\ =&H_{d_{q}} \biggl\{ \biggl[\frac{3}{7\times 7\times 25^{k-1}}, \frac{3}{5\times 7\times 25^{k-1}}\biggr],[0,0],u \biggr\} \\ =&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl( \frac{3}{7\times 7\times 25^{k-1}},0,u \biggr),\mathcal{F}_{d_{q}}\biggl(\frac{3}{5\times 7\times 25^{k-1}},0,u \biggr) \biggr\} \\ =&\min \biggl\{ \frac{49\times 25^{k-1}u}{49\times 25^{k-1}u+3}, \frac{35\times 25^{k-1}u}{35\times 25^{k-1}u+3} \biggr\} \\ =&\min \biggl\{ \frac{245\times 25^{k-1}u}{245\times 25^{k-1}u+15}, \frac{245\times 25^{k-1}u}{245\times 25^{k-1}u+21} \biggr\} \\ =&\frac{245\times 25^{k-1}u}{245\times 25^{k-1}u+21}. \end{aligned}$$

Now, we simplify the left-hand side of inequality (i).

$$\begin{aligned} \min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u),H_{d_{q}}( \mathcal{T}f,\mathcal{T}\mathfrak{e},u)\bigr\} =&\min \biggl\{ \frac{245\times 25^{k-1}u}{245\times 25^{k-1}u+42}, \frac{245\times 25^{k-1}u}{245\times 25^{k-1}u+21}\biggr\} \\ =&\frac{245\times 25^{k-1}u}{245\times 25^{k-1}u+42}. \end{aligned}$$

Also,

$$\begin{aligned} D(\mathfrak{e},f,u) =&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(0, \frac{3}{7\times 25^{k-1}},u\biggr),\mathcal{F}_{d_{q}}(0,0,u),\\ &{} \mathcal{F}_{d_{q}}\biggl( \frac{3}{7\times 25^{k-1}},\frac{3}{5\times 7\times 25^{k-1}},u \biggr)\biggr\} \\ =&\min \biggl\{ \frac{35\times 25^{k-1}u}{35\times 25^{k-1}u+30},1, \frac{35\times 25^{k-1}u}{35\times 25^{k-1}u+21}\biggr\} \\ =&\frac{35\times 25^{k-1}u}{35\times 25^{k-1}u+30}. \end{aligned}$$

Hence,

$$ \frac{245\times 25^{k-1}u}{245\times 25^{k-1}u+42}\geq \mu \biggl( \frac{35\times 25^{k-1}u}{35\times 25^{k-1}u+30}\biggr). $$

Case (iv): $\mathfrak{e}=\frac{3}{35\times 25^{m-1}}$, $f=0$

$$\begin{aligned} H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u) =&H_{d_{q}} \biggl( \mathcal{T}\biggl(\frac{3}{35\times 25^{m-1}}\biggr),\mathcal{T}(0),u \biggr) \\ =&H_{d_{q}} \biggl( \biggl[\frac{3}{7\times 35\times 25^{m-1}}, \frac{3}{5\times 35\times 25^{m-1}}\biggr],[0,0],u \biggr) \\ =&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl( \frac{3}{7\times 35\times 25^{m-1}},0,u \biggr),\mathcal{F}_{d_{q}}\biggl(\frac{3}{5\times 35\times 25^{m-1}},0,u \biggr) \biggr\} \\ =&\min \biggl\{ \frac{245\times 25^{m-1}u}{245\times 25^{m-1}u+3}, \frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+3} \biggr\} \\ =&\min \biggl\{ \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+15}, \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+21} \biggr\} \\ =&\frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+21}, \end{aligned}$$

$$\begin{aligned} H_{d_{q}}(\mathcal{T}f,\mathcal{T}\mathfrak{e},u) =&H_{d_{q}} \biggl( \mathcal{T}(0),\mathcal{T}\biggl(\frac{3}{35\times 25^{m-1}}\biggr),u \biggr) \\ =&H_{d_{q}} \biggl( [0,0],\biggl[\frac{3}{7\times 35\times 25^{m-1}}, \frac{3}{5\times 35\times 25^{m-1}}\biggr],u \biggr) \\ =&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl(0, \frac{3}{5\times 35\times 25^{m-1}},u \biggr),\mathcal{F}_{d_{q}}\biggl(0,\frac{3}{7\times 35\times 25^{m-1}},u \biggr) \biggr\} \\ =&\min \biggl\{ \frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+6}, \frac{245\times 25^{m-1}u}{245\times 25^{m-1}u+6} \biggr\} \\ =&\min \biggl\{ \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+42}, \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+30} \biggr\} \\ =&\frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+42}. \end{aligned}$$

Now, we simplify the left-hand side of inequality (i).

$$\begin{aligned}& \min \bigl\{ H_{d_{q}}(\mathcal{T}\mathfrak{e},\mathcal{T}f,u),H_{d_{q}}( \mathcal{T}f,\mathcal{T}\mathfrak{e},u)\bigr\} \\& \quad =\min \biggl\{ \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+21}, \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+42} \biggr\} \\& \quad =\frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+42}. \end{aligned}$$

Also,

$$\begin{aligned} D(\mathfrak{e},f,u) =&\min \biggl\{ \mathcal{F}_{d_{q}}\biggl( \frac{3}{35\times 25^{m-1}},0,u\biggr),\mathcal{F}_{d_{q}}\biggl( \frac{3}{35\times 25^{m-1}}, \frac{3}{5\times 35\times 25^{m-1}},u\biggr), \\ &\mathcal{F}_{d_{q}}(0,0,u)\biggr\} \\ =&\min \biggl\{ \frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+15}, \frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+21},1 \biggr\} \\ =&\frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+21}. \end{aligned}$$

Hence,

$$\begin{aligned} \frac{1225\times 25^{m-1}u}{1225\times 25^{m-1}u+42} \geq &\sqrt{ \frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+21}} \\ =&\mu \biggl(\frac{175\times 25^{m-1}u}{175\times 25^{m-1}u+21}\biggr). \end{aligned}$$

Case (v) The contraction is trivially held for $\mathfrak{e}=0$ and $f=0$. Now,

$$\begin{aligned} & \underset{p=0}{\overset{k}{\circledast }}\min \biggl\{ \mu ^{p} \biggl( \mathcal{F}_{d_{q}}\biggl(\mathfrak{e}_{1}, \mathfrak{e}_{\circ },\frac{1}{2^{p+1}}\biggr)\biggr), \mu ^{p}\biggl(\mathcal{F}_{d_{q}}\biggl( \mathfrak{e}_{\circ },\mathfrak{e}_{1}, \frac{1}{2^{p+1}}\biggr)\biggr)\biggr\} \\ &\quad= \underset{p=0}{\overset{k}{\circledast }}\min \biggl\{ \mu ^{p} \biggl( \mathcal{F}_{d_{q}}\biggl(\frac{3}{35}, \frac{3}{7},\frac{1}{2^{p+1}}\biggr)\biggr),\mu ^{p} \biggl( \mathcal{F}_{d_{q}}\biggl(\frac{3}{7}, \frac{3}{35},\frac{1}{2^{p+1}}\biggr)\biggr)\biggr\} \\ &\quad= \underset{p=0}{\overset{k}{\circledast }}\min \biggl\{ \mu ^{p} \biggl( \frac{\frac{1}{2^{p+1}}}{\frac{1}{2^{p+1}}+\frac{3}{35}+\frac{6}{7}}\biggr),\mu ^{p}\biggl( \frac{\frac{1}{2^{p+1}}}{\frac{1}{2^{p+1}}+\frac{3}{7}+\frac{6}{35}}\biggr)\biggr\} \\ &\quad= \underset{p=0}{\overset{k}{\circledast }}\min \biggl\{ \mu ^{p} \biggl( \frac{35}{35+3\times 2^{p+1}+30\times 2^{p+1}}\biggr),\mu ^{p}\biggl( \frac{35}{35+15\times 2^{p+1}+6\times 2^{p+1}}\biggr)\biggr\} \\ &\quad= \min \biggl\{ \biggl(\frac{35}{101}\biggr),\biggl(\frac{35}{77} \biggr)\biggr\} \circledast \min \biggl\{ \mu \biggl( \frac{35}{167}\biggr), \mu \biggl(\frac{35}{119}\biggr)\biggr\} \circledast \biggl\{ \mu ^{2}\biggl(\frac{35}{299}\biggr), \mu ^{2}\biggl( \frac{35}{203}\biggr)\biggr\} \\ &\qquad{}\circledast \cdots\circledast\min \biggl\{ \mu ^{k}\biggl(\frac{35}{35+3\times 2^{k+1}+30\times 2^{k+1}}\biggr), \mu ^{k}\biggl(\frac{35}{35+15\times 2^{k+1}+6\times 2^{k+1}}\biggr)\biggr\} \\ &\quad= \frac{35}{101}\circledast \mu \biggl(\frac{35}{167}\biggr) \circledast \mu ^{2}\biggl( \frac{35}{299}\biggr)\circledast \cdots\circledast \mu ^{k}\biggl( \frac{35}{35+3\times 2^{k+1}+30\times 2^{k+1}}\biggr)= \frac{35}{101} \\ &\quad> \frac{1}{4}=1-\frac{3}{4}=1-r. \end{aligned}$$

Hence, all the constraints of Theorem 2.1 are satisfied. Hence, S and $\mathcal{T}$ have a common fixed point and it is 0.

Remark 2.3

By taking six proper subsets of $D(\mathfrak{e},f,u)$ instead of $D(\mathfrak{e},f,u)$, we can obtain six new theorems as corollaries of Theorem 2.1.

Remark 2.4

Fixed-point results in right K-sequentially quasimetric spaces can be obtained in a similar way.