Also,
\(m_{2 \mathcal{l}+ 1} \in [ \mathcal{P}_{c} \mathfrak{m}_{2 \mathcal{l}} ]_{\gamma _{(b_{2 \mathcal{l}} )}}\) for some
\(c\in \mathbb{N}^{o}\), so
\(\varphi ( \mathfrak{m}_{2 \mathcal{l}}, \mathfrak{m}_{2 \mathcal{l}+ 1} ) \geq 1\) and
\(m_{2 \mathcal{l}+ 2} \in [ \mathcal{Q}_{v} \mathfrak{m}_{2 \mathcal{l}+ 1} ]_{\delta ( b_{2 \mathcal{l}+ 1} )}\) for some
\(v\in \mathbb{N}^{e}\). Now, by using Lemma
1.10 and inequality (
2.1), we have
$$\begin{aligned}& \tau +\mathcal{F} \bigl( \mathcal{d} ( \mathfrak{m}_{2 \mathcal{l}+ 1}, \mathfrak{m}_{2 \mathcal{l}+ 2} ) \bigr) \\& \quad \leq \tau +\mathcal{F}\bigl( H_{\mathcal{d}} \bigl([ \mathcal{P}_{c} \mathfrak{m}_{2 \mathcal{l}} ]_{\gamma _{(b_{2 \mathcal{l}} )}},[ \mathcal{Q}_{v} \mathfrak{m}_{2 \mathcal{l}+ 1} ]_{\delta _{(b_{2 \mathcal{l}+ 1} )}}\bigr)\bigr)\leq \mathcal{F} \bigl( \mathcal{D} ( \mathfrak{m}_{2 \mathcal{l}}, \mathfrak{m}_{2 \mathcal{l}+ 1} ) \bigr)\\& \quad \leq \mathcal{F} \left( \max \begin{Bmatrix} \mathcal{d} ( \mathfrak{m}_{2 \mathcal{l},} \mathfrak{m}_{2 \mathcal{l}+ 1} ),\mathcal{d} ( \mathfrak{m}_{2 \mathcal{l}}, \mathfrak{m}_{2 \mathcal{l}+ 1} ),\mathcal{d} ( \mathfrak{m}_{2 \mathcal{l}+ 1}, \mathfrak{m}_{2 \mathcal{l}+ 2} ) \\ \mathcal{d} ( \mathfrak{m}_{2 \mathcal{l}}, \mathfrak{m}_{2 \mathcal{l}+ 2} )^{\frac{1}{2s}},\mathcal{d} ( \mathfrak{m}_{2 \mathcal{l}+ 1}, \mathfrak{m}_{2 \mathcal{l}+ 1} ) \end{Bmatrix} \right)^{\mathcal{k}}\\& \quad \leq \mathcal{F} \bigl( \max \bigl\{ \mathcal{d} ( \mathfrak{m}_{2 \mathcal{l},} \mathfrak{m}_{2 \mathcal{l}+ 1} ),\mathcal{d} ( \mathfrak{m}_{2 \mathcal{l}+ 1}, \mathfrak{m}_{2 \mathcal{l}+ 2} ) \bigr\} \bigr)^{\mathcal{k}}. \end{aligned}$$
Thus,
$$ \tau +\mathcal{Fd} ( \mathfrak{m}_{2 \mathcal{l}+ 1}, \mathfrak{m}_{2 \mathcal{l}+ 2} ) \leq \mathcal{F} \bigl( \mathcal{d} ( \mathfrak{m}_{2 \mathcal{l}}, \mathfrak{m}_{2 \mathcal{l}+ 1} ) \bigr)^{\mu}, $$
for each
\(\mathcal{l}\in \mathbb{N}\), where
\(\mu = \frac{1-s \mathcal{k}}{ s}\). As
\(\mathcal{F}\) is a strictly increasing function, then
$$ \mathcal{d} ( \mathfrak{m}_{2 \mathcal{l}+ 1}, \mathfrak{m}_{2 \mathcal{l}+ 2} ) < \mathcal{d} ( m_{2 \mathcal{l}}, m_{2 \mathcal{l}+ 1} )^{\mu}. $$
(2.3)
Similarly, if
\(\mathfrak{j}\) is even, we have
$$ \mathcal{d} ( \mathrm{m}_{2 \mathcal{l}+ 2}, \mathrm{m}_{2 \mathcal{l}+ 3} ) < \mathcal{d} ( \mathrm{m}_{2 \mathcal{l}+ 1}, \mathrm{m}_{2 \mathcal{l}+ 2} )^{\mu}. $$
(2.4)
We have,
$$ ( \mathfrak{m}_{j}, \mathfrak{m}_{\mathfrak{j} +1} ) < d ( \mathfrak{m}_{\mathfrak{j} -1}, \mathfrak{m}_{\mathfrak{j}} )^{\mu} \quad \text{for all }\mathfrak{j}\in N. $$
(2.5)
Therefore,
$$ \mathcal{d} ( \mathfrak{m}_{j}, \mathfrak{m}_{\mathfrak{j} +1} ) < \mathcal{d} ( \mathfrak{m}_{\mathfrak{j} -1}, \mathfrak{m}_{\mathfrak{j}} )^{\mu} < \mathcal{d} ( \mathfrak{m}_{\mathfrak{j} -2}, \mathfrak{m}_{\mathfrak{j} -1} )^{\mu ^{2}} < \cdots < \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{\mathfrak{j}}. $$
(2.6)
Now,
$$\begin{aligned} \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{\mathfrak{j} +1} ) &\leq \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{s}. \mathcal{d} ( \mathfrak{m}_{1}, \mathfrak{m}_{2} )^{s^{2}}. \mathcal{d} ( \mathfrak{m}_{1}, \mathfrak{m}_{2} )^{s^{3}}. \cdots .\mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{s^{\mathfrak{j} +1}}\\ &\leq \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{s} .\mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{\mu s^{2}} .\mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{\mu ^{2} s^{3}} . \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{\mu ^{3} s^{4}}\\ &\quad \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{\mu ^{4} s^{5}} .\cdots .\mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{\mu ^{\mathfrak{j}} s^{\mathfrak{j} +1}}\\ &\leq \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{s( \mu ^{0} + s\mu ^{1} + s^{2} \mu ^{2} + s^{2} \mu ^{2} + s^{3} \mu ^{3} +\cdots + s^{\mathfrak{j}} \mu ^{\mathfrak{j}} )}\\ &\leq \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{s ( \frac{1}{1-s\mu} )}. \end{aligned}$$
Then, we have
$$ \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{\mathfrak{j} +1} ) \leq \mathcal{r}^{\frac{ ( 1-s ( \mu ) ) \times s}{s\times ( 1-s ( \mu ) )}} \leq \mathcal{r}. $$
This shows that
\(\mathfrak{m}_{\mathfrak{j} +1} \in \overline{B_{\mathcal{d} \mathfrak{m}} (\mathfrak{m}_{0},\mathcal{r} )} \). It follows that for all
\(\mathfrak{n}\in \mathbb{N}\), by induction,
\(\mathfrak{m}_{\mathfrak{n}} \in \overline{B_{\mathcal{d} \mathfrak{m}} (\mathfrak{m}_{0},\mathcal{r} )}\). Also,
\(\varphi ( \mathfrak{m}_{\mathfrak{n}}, \mathfrak{m}_{\mathfrak{n} +1} ) \geq 1\), for any
\(\mathfrak{n}\in \mathbb{N} \cup \{ 0 \} \). Now, we deduce
$$ \mathcal{d} ( \mathfrak{m}_{\mathfrak{n}}, \mathfrak{m}_{\mathfrak{n} +1} ) < \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{\mu ^{\mathfrak{n}}},\quad \text{for all } \mathfrak{n}\in \mathbb{N} $$
(2.7)
Now,
\(\mathcal{e},\mathcal{g}(\mathcal{g}>\mathcal{e})\) be the positive integer, then
$$\begin{aligned}& \begin{aligned} \mathcal{d} ( \mathfrak{m}_{\mathcal{e}}, \mathfrak{m}_{\mathcal{g}} ) &\leq \mathcal{d} ( \mathfrak{m}_{\mathcal{e}}, \mathfrak{m}_{\mathcal{e}+ 1} )^{s} .\mathcal{d} ( \mathfrak{m}_{\mathcal{e}+ 1}, \mathfrak{m}_{\mathcal{e}+ 2} )^{s^{2}} .\cdots .\mathcal{d} ( \mathfrak{m}_{\mathcal{g}- 1}, \mathfrak{m}_{\mathcal{g}} )^{s^{\mathcal{g}} \mu ^{\mathcal{g}- 1}}\\ &\leq \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{s \mu ^{\mathcal{e}}} .\mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{\mu ^{\mathcal{e}+ 1} s^{2}} .\cdots .\mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{s^{z} \mu ^{\mathcal{g}- 1}} \quad \text{(by (2.7))} \\ &\leq \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{(s \mu ^{\mathcal{e}} + s^{2} \mu ^{\mathcal{e}+ 1} + s^{3} \mu ^{\mathcal{e}+ 2} + s^{4} \mu ^{\mathcal{e}+ 3} +\cdots + s^{\mathcal{g}} \mu ^{\mathcal{g}- 1} )} \\ & < \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{(s \mu ^{\mathcal{e}} + s^{2} \mu ^{\mathcal{e}+ 1} + s^{3} \mu ^{\mathcal{e}+ 2} +\cdots )}, \end{aligned} \\& \mathcal{d} ( m_{\mathcal{e}}, m_{\mathcal{g}} ) < \mathcal{d} ( \mathfrak{m}_{0}, \mathfrak{m}_{1} )^{ ( \frac{s \mu ^{\mathcal{e}}}{1-s\mu} )}. \end{aligned}$$
As
\(\mathcal{e},\mathcal{g}\rightarrow +\infty \), then
\(\mathcal{d} ( m_{\mathcal{e}}, m_{\mathcal{g}} ) \rightarrow 1\). Therefore,
\(\{ \mathcal{Q}_{e} \mathcal{P}_{o} ( \mathfrak{m}_{g} ) \} \) is a Cauchy sequence in
\(\overline{B_{\mathcal{d} \mathfrak{m}} (\mathfrak{m}_{0},\mathcal{r} )}\). So, there is a
\(\mathfrak{m}^{*} \in \overline{B_{\mathcal{d} \mathfrak{m}} (\mathfrak{m}_{0},\mathcal{r} )}\) and
\(\{ \mathcal{Q}_{e} \mathcal{P}_{o} ( \mathfrak{m}_{g} ) \} \rightarrow \mathfrak{m}^{*}\) such that
\(\mathcal{g}\rightarrow +\infty \). Then,
$$ \lim_{\mathcal{g}\rightarrow +\infty} \bigl( \mathfrak{m}_{\mathcal{g}}, \mathfrak{m}^{*} \bigr) =1. $$
(2.8)
Now, by using the inequality, we have
$$ \mathcal{d} \bigl( \mathfrak{m}^{*},\bigl[ \mathcal{Q}_{e} \mathfrak{m}^{*} \bigr]_{\delta ( \mathfrak{m}^{*} )} \bigr) \leq \mathcal{d} \bigl( \mathfrak{m}^{*}, \mathfrak{m}_{2z+1} \bigr)^{s}. \mathcal{d} \bigl( \mathfrak{m}_{2z+1},\bigl[ \mathcal{Q}_{e} \mathfrak{m}^{*} \bigr]_{\delta ( \mathfrak{m}^{*} )} \bigr)^{s}. $$
Using Lemma
1.10 and inequality (
2.1), we obtain
$$ \mathcal{d} \bigl( \mathfrak{m}^{*},\bigl[ \mathcal{Q}_{e} \mathfrak{m}^{*} \bigr]_{\gamma ( \mathfrak{m}^{*} )} \bigr) \leq \mathcal{d} \bigl( \mathfrak{m}^{*}, \mathfrak{m}_{2 \mathcal{g}+ 1} \bigr)^{s}. H_{\mathcal{d}} \bigl( [ \mathcal{P}_{o} \mathfrak{m}_{2 \mathcal{g}} ]_{\gamma (2 \mathcal{g})}, \bigl[ \mathcal{Q}_{e} \mathfrak{m}^{*} \bigr]_{\delta ( \mathfrak{m}^{*} )} \bigr)^{s}. $$
(2.9)
By supposition
\(\varphi ( \mathfrak{m}_{\mathcal{g}}, \mathfrak{m}^{*} ) \geq 1\). Suppose that
\(\mathcal{d} ( \mathfrak{m}^{*}, [ \mathcal{Q}_{e} \mathfrak{m}^{*} ]_{\delta ( \mathfrak{m}^{*} )} ) >0\) and
p is a non-negative integer that exists, such that
\(\mathcal{d} ( \mathfrak{m}_{\mathfrak{n}}, [ \mathcal{Q}_{e} \mathfrak{m}^{*} ]_{\delta ( \mathfrak{m}^{*} )} ) >0\), for all
\(\mathcal{g}\geq p\), we have
$$\begin{aligned}& \mathcal{d} \bigl( \mathfrak{m}^{*},[ \mathcal{Q}_{e} \mathfrak{m}^{*} ]_{\gamma ( \mathfrak{m}^{*} )} \bigr) \\& \quad < \mathcal{d} ( \mathfrak{m}^{*}, \mathfrak{m}_{2 \mathcal{g}+ 1} )^{s}. \left( \max \begin{Bmatrix} \mathcal{d} ( \mathfrak{m}_{2 \mathcal{g},} \mathfrak{m}^{*} ),\mathcal{d} ( \mathfrak{m}_{2 \mathcal{g}}, [ \mathcal{Q}_{e} \mathfrak{m}^{*} ]_{\delta ( \mathfrak{m}^{*} )} ),\mathcal{d} ( \mathfrak{m}_{2 \mathcal{g}+ 1}, [ \mathcal{Q}_{e} \mathfrak{m}^{*} ]_{\delta ( \mathfrak{m}^{*} )} )\\ \mathcal{d} ( \mathfrak{m}_{2z}, \mathfrak{m}_{2 \mathcal{g}+ 1} )^{\frac{1}{2s}},\mathcal{d} ( \mathfrak{m}_{2 \mathcal{g}+ 1}, \mathfrak{m}_{2 \mathcal{g}+ 2} ) \end{Bmatrix}^{p} \right)^{s} . \end{aligned}$$
(2.10)
By taking
\(\lim\mathfrak{n}\rightarrow +\infty \) and inequality (
2.8) from both sides of (
2.9), we have a result that is not generally true,
\(\mathcal{d} ( \mathfrak{m}^{*},[ \mathcal{Q}_{e} \mathfrak{m}^{*} ]_{\gamma ( \mathfrak{m}^{*} )} ) <\mathcal{d} ( \mathfrak{m}^{*}, [ \mathcal{Q}_{e} \mathfrak{m}^{*} ]_{\gamma ( \mathfrak{m}^{*} )} )^{ps}\). Our assumption is not true because
\(ps<1\). Therefore,
\(\mathcal{d} ( \mathfrak{m}^{*},[ \mathcal{Q}_{e} \mathfrak{m}^{*} ]_{\gamma ( \mathfrak{m}^{*} )} ) = 1\) or
\(\mathfrak{m}^{*} \in [ \mathcal{Q}_{e} \mathfrak{m}^{*} ]_{\gamma ( \mathfrak{m}^{*} )}\). Similarly, using Lemma
1.10 and inequality (
2.8), we can obtain either
\(\mathcal{d} ( \mathfrak{m}^{*}, [ \mathcal{P}_{o} \mathfrak{m}^{*} ]_{\gamma ( \mathfrak{m}^{*} )} ) = 1\) or
\(\mathfrak{m}^{*} \in [ \mathcal{P}_{o} \mathfrak{m}^{*} ]_{\gamma ( \mathfrak{m}^{*} )}\). Therefore, in
\(\overline{B_{\mathcal{d} \mathfrak{m}} (\mathfrak{m}_{0},\mathcal{r} )}\),
\(\mathcal{P}_{o}\) and
\(\mathcal{Q}_{e}\) admit a fuzzy F
\(\mathscr{P}\) that is
\(\mathfrak{m}^{*}\). Now, using the above multiplicative triangular inequality, we get
$$ \mathcal{d} \bigl( \mathfrak{m}^{*}, \mathfrak{m}^{*} \bigr) \leq \bigl[ \mathcal{d} \bigl( \mathfrak{m}^{*}, \bigl[ \mathcal{Q}_{e} \mathfrak{m}^{*} \bigr]_{\delta ( \mathfrak{m}^{*} )} \bigr) .\mathcal{d} \bigl( \bigl[ \mathcal{Q}_{e} \mathfrak{m}^{*} \bigr]_{\delta ( \mathfrak{m}^{*} )}, \mathfrak{m}^{*} \bigr) \bigr]^{s}. $$
This indicate that
\(\mathcal{d} ( \mathfrak{m}^{*}, \mathfrak{m}^{*} ) =1\). □