In this part, we present extended neutrosophic rectangular metric space and demonstrate some fixed point results.
Definition 4
Let \(\varDelta \neq \emptyset \) and \(\wp \colon \varDelta \times \varDelta \rightarrow [1, +\infty )\) be given non-comparable functions, ∗ be a continuous t-norm, ∘ be a continuous t-co-norm and Ω, Φ, Λ be neutrosophic sets. \(\varDelta \times \varDelta \times (0, +\infty )\) is said to be an extended neutrosophic rectangular metric on Δ if for any \(\psi , \lambda \in \varDelta \) and all distinct \(\upsilon ,\varGamma , \lambda \in \varDelta \), the following conditions are satisfied:
-
(A1)
\(\varOmega (\psi , \varGamma , \vartheta )+\varPhi (\psi , \varGamma , \vartheta )+\varLambda (\psi , \varGamma , \vartheta )\leq 3\);
-
(A2)
\(\varOmega (\psi , \varGamma , \vartheta )>0\);
-
(A3)
\(\varOmega (\psi , \varGamma , \vartheta )=1\) for all \(\vartheta >0\) if and only if \(\psi =\varGamma \);
-
(A4)
\(\varOmega (\psi , \varGamma , \vartheta )=\varOmega (\varGamma , \psi , \vartheta )\);
-
(A5)
\(\varOmega (\psi , \lambda , \wp (\psi , \lambda )(\vartheta +\varpi + \varsigma ))\geq \varOmega (\psi , \varGamma , \vartheta ) \ast \varOmega (\varGamma , \upsilon , \varpi )\ast \varOmega (\upsilon , \lambda , \varsigma )\);
-
(A6)
\(\varOmega (\psi , \varGamma , \cdot )\colon (0, +\infty ) \rightarrow [0, 1]\) is continuous and \(\lim_{\vartheta \rightarrow +\infty}\varOmega (\psi , \varGamma , \vartheta )=1\);
-
(A7)
\(\varPhi (\psi , \varGamma , \vartheta )<1\);
-
(A8)
\(\varPhi (\psi , \varGamma , \vartheta )=0\) for all \(\vartheta >0\) if and only if \(\psi =\varGamma \);
-
(A9)
\(\varPhi (\psi , \varGamma , \vartheta )=\varPhi (\varGamma , \psi , \vartheta )\);
-
(A10)
\(\varPhi (\psi , \lambda , \wp (\psi , \lambda )(\vartheta +\varpi + \varsigma )\leq \varPhi (\psi , \varGamma , \vartheta )\circ \varPhi (\varGamma , \upsilon , \varpi )\circ \varPhi ( \upsilon , \lambda , \varsigma )\);
-
(A11)
\(\varPhi (\psi , \varGamma , \cdot )\colon (0, +\infty )\rightarrow [0, 1]\) is continuous and \(\lim_{\vartheta \rightarrow +\infty}\varPhi (\psi , \varGamma , \vartheta )=0\);
-
(A12)
\(\varLambda (\psi , \varGamma , \vartheta )<1\);
-
(A13)
\(\varLambda (\psi , \varGamma , \vartheta )=0\) for all \(\vartheta >0\) if and only if \(\psi =\varGamma \);
-
(A14)
\(\varLambda (\psi , \varGamma , \vartheta )=\varLambda (\varGamma , \psi , \vartheta )\);
-
(A15)
\(\varLambda (\psi , \lambda , \wp (\psi , \lambda )(\vartheta + \varpi +\varsigma ))\leq \varLambda (\psi , \varGamma , \vartheta )\circ \varLambda (\varGamma , \upsilon , \varpi )\circ \varLambda (\upsilon , \lambda , \varpi )\);
-
(A16)
\(\varLambda (\psi , \varGamma , \cdot )\colon (0, +\infty ) \rightarrow [0, 1]\) is continuous and \(\lim_{\vartheta \rightarrow +\infty}\varLambda (\psi , \varGamma , \vartheta )=0\);
-
(A17)
If \(\vartheta \leq 0\), then \(\varOmega (\psi , \varGamma , \vartheta )=0\), \(\varPhi (\psi , \varGamma , \vartheta )=1\) and \(\mathcal{S}(\psi , \varGamma , \vartheta )=1\).
Then \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) is called an extended neutrosophic rectangular metric space (ENRMS).
Example 1
Let \(\varDelta =\{1, 2, 3,4\}\) and \(\wp \colon \varDelta \times \varDelta \rightarrow [1, +\infty )\) be a function given by \(\wp (\psi , \varGamma )=\psi +\varGamma +1\). Define \(\varOmega , \varPhi , \varLambda \colon \varDelta \times \varDelta \times (0, +\infty )\rightarrow [0, 1]\) as
$$\begin{aligned} \varOmega (\psi , \varGamma , \vartheta )&= \textstyle\begin{cases} 1, &\text{if } \psi =\varGamma \\ \frac{\vartheta}{\vartheta +\max \{\psi , \varGamma \}^{2}}, & \text{if otherwise}, \end{cases}\displaystyle \\ \varPhi (\psi , \varGamma , \vartheta )&= \textstyle\begin{cases} 0, &\text{if } \psi =\varGamma \\ \frac{\max \{\psi , \varGamma \}^{2}}{\vartheta +\max \{\psi , \varGamma \}^{2}}, &\text{if otherwise}, \end{cases}\displaystyle \end{aligned}$$
and
$$\begin{aligned} \varLambda (\psi , \varGamma , \vartheta )= \textstyle\begin{cases} 0, &\text{if } \psi =\varGamma \\ \frac{\max \{\psi , \varGamma \}^{2}}{\vartheta}, &\text{if otherwise}. \end{cases}\displaystyle \end{aligned}$$
Then \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) is an ENRMS with continuous t-norm \(\wp \ast \tau =\wp \tau \) and continuous t-co-norm, \(\wp \circ \bar{a}=\max \{\wp , \bar{a}\}\).
Here we prove (A5), (A10) and (A15), others are obvious.
Let \(\psi =1\), \(\varGamma =2\), \(\upsilon =3\) and \(\lambda =4\). Then
$$\begin{aligned} \varOmega (1, 4, \vartheta +\varpi +\varsigma )= \frac{\vartheta +\varpi +\varsigma}{\vartheta +\varpi +\varsigma +\max \{1, 4\}^{2}}= \frac{\vartheta +\varpi +\varsigma}{\vartheta +\varpi +\varsigma +16}. \end{aligned}$$
On the other hand,
$$\begin{aligned} &\varOmega \biggl(1, 2, \frac{\vartheta}{\wp (1, 4)} \biggr)= \frac{\frac{\vartheta}{\wp (1, 4)}}{\frac{\vartheta}{\wp (1, 4)}+\max \{1, 2\}^{2}}= \frac{\frac{\vartheta}{6}}{\frac{\vartheta}{6}+4}= \frac{\vartheta}{\vartheta +24}, \\ &\varOmega \biggl(2, 3, \frac{\varpi}{\wp (1, 4)} \biggr)= \frac{\frac{\varpi}{\wp (1, 4)}}{\frac{\varpi}{\wp (1, 4)}+\max \{2, 3\}^{2}}= \frac{\frac{\varpi}{6}}{\frac{\varpi}{6}+9}=\frac{\varpi}{\varpi +54} \end{aligned}$$
and
$$\begin{aligned} \varOmega \biggl(3, 4, \frac{\varsigma}{\wp (1, 4)} \biggr)= \frac{\frac{\varsigma}{\wp (1, 4)}}{\frac{\varsigma}{\wp (1, 4)}+\max \{3, 4\}^{2}}= \frac{\frac{\varsigma}{6}}{\frac{\varsigma}{6}+16}= \frac{\varsigma}{\varsigma +96}. \end{aligned}$$
That is,
$$\begin{aligned} \frac{\vartheta +\varpi +\varsigma}{\vartheta +\varpi +\varsigma +16} \geq \frac{\vartheta}{\vartheta +24}\cdot \frac{\varpi}{\varpi +54}. \frac{\varsigma}{\varsigma +96}. \end{aligned}$$
Then it satisfies all \(\vartheta , \varpi , \varsigma >0\). Hence,
$$\begin{aligned} \varOmega (\psi , \lambda , \vartheta +\varpi +\varsigma )\geq \varOmega \biggl( \psi , \varGamma , \frac{\vartheta}{\wp (\psi , \lambda )} \biggr)\ast \varOmega \biggl( \varGamma , \upsilon , \frac{\varpi}{\wp (\psi , \lambda )} \biggr) \ast \varOmega \biggl(\upsilon , \lambda , \frac{\varsigma}{\wp (\psi , \lambda )} \biggr). \end{aligned}$$
Now,
$$\begin{aligned} \varPhi (1, 4, \vartheta +\varpi +\varsigma )= \frac{\max \{1, 4\}^{2}}{\vartheta +\varpi +\varsigma +\max \{1, 4\}^{2}}= \frac{16}{\vartheta +\varpi +\varsigma +16}. \end{aligned}$$
On the other hand,
$$\begin{aligned} &\varPhi \biggl(1, 2, \frac{\vartheta}{\wp (1, 4)} \biggr)= \frac{\max \{1, 2\}^{2}}{\frac{\vartheta}{\wp (1, 4)}+\max \{1, 2\}^{2}}= \frac{4}{\frac{\vartheta}{6}+4}=\frac{24}{\vartheta +24}, \\ &\varPhi \biggl(2, 3, \frac{\varpi}{\wp (1, 4)} \biggr)= \frac{\max \{2, 3\}^{2}}{\frac{\varpi}{\wp (1, 4)}+\max \{2, 3\}^{2}}= \frac{9}{\frac{\varpi}{6}+9}=\frac{54}{\varpi +54} \end{aligned}$$
and
$$\begin{aligned} \varPhi \biggl(3, 4, \frac{\varsigma}{\wp (1, 4)} \biggr)= \frac{\max \{3, 4\}^{2}}{\frac{\varsigma}{\wp (1, 4)}+\max \{3, 4\}^{2}}= \frac{16}{\frac{\varsigma}{6}+16}=\frac{96}{\varsigma +96}. \end{aligned}$$
That is,
$$\begin{aligned} \frac{16}{\vartheta +\varpi +\varsigma +16}\leq \max \biggl\{ \frac{24}{\vartheta +24}, \frac{54}{\varpi +54}, \frac{96}{\varsigma +96} \biggr\} . \end{aligned}$$
Then it satisfies all \(\vartheta , \varpi ,\varsigma >0\). Hence,
$$\begin{aligned} \varPhi (\psi , \lambda , \vartheta +\varpi +\varsigma )\leq \varPhi \biggl(\psi , \varGamma , \frac{\vartheta}{\wp (\psi , \lambda )} \biggr)\circ \varPhi \biggl(\upsilon , \lambda , \frac{\varpi}{\wp (\psi , \lambda )} \biggr)\circ \varPhi \biggl( \upsilon , \lambda , \frac{\varsigma}{\wp (\psi , \lambda )} \biggr). \end{aligned}$$
Now,
$$\begin{aligned} \varLambda (1, 3, \vartheta +\varpi +\varsigma )= \frac{\max \{1, 3\}^{2}}{\vartheta +\varpi +\varsigma}= \frac{9}{\vartheta +\varpi +\varsigma}. \end{aligned}$$
On the other hand,
$$\begin{aligned} &\varLambda \biggl(1, 2, \frac{\vartheta}{\wp (1, 4)} \biggr)= \frac{\max \{1, 2\}^{2}}{\frac{\vartheta}{\wp (1, 4)}}= \frac{4}{\frac{\vartheta}{6}}=\frac{24}{\vartheta}, \\ &\varLambda \biggl(2, 3, \frac{\varpi}{\wp (1, 4)} \biggr)= \frac{\max \{2, 3\}^{2}}{\frac{\varpi}{\wp (1, 4)}}= \frac{9}{\frac{\varpi}{6}}=\frac{54}{\varpi} \end{aligned}$$
and
$$\begin{aligned} \varLambda \biggl(3, 4, \frac{\varsigma}{\wp (1, 4)} \biggr)= \frac{\max \{3, 4\}^{2}}{\frac{\varsigma}{\wp (1, 4)}}= \frac{16}{\frac{\varsigma}{6}}=\frac{96}{\varsigma}. \end{aligned}$$
That is,
$$\begin{aligned} \frac{9}{\vartheta +\varpi +\varsigma}\leq \max \biggl\{ \frac{24}{\vartheta}, \frac{54}{\varpi}, \frac{96}{\varsigma} \biggr\} . \end{aligned}$$
Then it satisfies all \(\vartheta , \varpi >0\). Hence,
$$\begin{aligned} \varLambda (\psi , \lambda , \vartheta +\varpi +\varsigma )\leq \varLambda \biggl(\psi , \varGamma , \frac{\vartheta}{\wp (\psi , \lambda )} \biggr)\circ \varLambda \biggl( \varGamma , \upsilon , \frac{\varpi}{\wp (\psi , \lambda )} \biggr) \circ \varLambda \biggl(\upsilon , \lambda , \frac{\varpi}{\wp (\psi , \lambda )} \biggr). \end{aligned}$$
Hence \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) is an ENRMS.
Remark 1
The preceding example also satisfies for continuous t-norm \(\wp \ast \bar{a}=\min \{\wp , \bar{a}\}\) and continuous t-co-norm \(\wp \circ \bar{a}=\max \{\wp , \bar{a}\}\).
Definition 5
Let \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) be an ENRMS, an open ball is then defined \(\varLambda (\psi , \varrho , \vartheta )\) with centre ψ, radius \(\varrho , 0<\varrho <1\) and \(\vartheta >0\) as follows:
$$\begin{aligned} \varLambda (\psi , \varrho , \vartheta )= \bigl\{ \varGamma \in \varDelta \colon \varOmega (\psi , \varGamma , \vartheta )>1-\varrho , \varPhi ( \psi , \varGamma , \vartheta )< \varrho , \varLambda (\psi , \varGamma , \vartheta )< \varrho \bigr\} . \end{aligned}$$
Definition 6
Let \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) be an ENRMS and \(\{\psi _{\kappa}\}\) be a sequence in Δ. Then \(\{\psi _{\kappa}\}\) is said to be:
-
1.
Convergent if there exists \(\psi \in \varDelta \) such that
$$\begin{aligned} &\lim_{\kappa \rightarrow +\infty}\varOmega (\psi _{\kappa}, \psi , \vartheta )=1, \qquad \lim_{\kappa \rightarrow +\infty}\varPhi (\psi _{ \kappa}, \psi , \vartheta )=0, \\ &\lim_{\kappa \rightarrow +\infty} \varLambda (\psi _{\kappa}, \psi , \vartheta )=0\quad \text{for all } \vartheta >0; \end{aligned}$$
-
2.
Cauchy sequence if and only if for each \(\bar{a}>0\), \(\vartheta >0\), there exists \(\kappa _{0}\in \mathbb{N}\) such that
$$\begin{aligned} \varOmega (\psi _{\kappa}, \psi _{\kappa +\omega}, \vartheta )\geq 1- \bar{a}, \qquad \varPhi (\psi _{\kappa}, \psi _{\kappa +\omega}, \vartheta ) \leq \bar{a}, \qquad \varPhi (\psi _{\kappa}, \psi _{\kappa +\omega}, \vartheta )\leq \bar{a} \end{aligned}$$
for all \(\kappa , \pi \geq \kappa _{0}\).
If every Cauchy sequence is convergent in Δ, then \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) is called a complete ENRMS.
Lemma 1
Let \(\{\psi _{\kappa}\}\) be a Cauchy sequence in ENRMS \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) such that \(\psi _{\kappa}\neq \psi _{\pi}\) whenever \(\pi , \kappa \in \mathbb{N}\) with \(\kappa \neq \pi \). Then the sequence \(\{\psi _{\kappa}\}\) can converge to, at most, one limit point.
Proof
Contrarily, assume that \(\psi _{\kappa}\rightarrow \psi \) and \(\psi _{\kappa}\rightarrow \varGamma \) for \(\psi \neq \varGamma \). Then
$$ \lim_{\kappa \rightarrow +\infty}\varOmega (\psi _{\kappa}, \psi , \vartheta )=1,\qquad \lim_{\kappa \rightarrow +\infty}\varPhi (\psi _{ \kappa}, \psi , \vartheta )=0, \qquad \lim_{\kappa \rightarrow +\infty} \varLambda (\psi _{\kappa}, \psi , \vartheta )=0 $$
and
$$ \lim_{\kappa \rightarrow +\infty}\varOmega (\psi _{\kappa}, \varGamma , \vartheta )=1, \qquad \lim_{\kappa \rightarrow +\infty}\varPhi ( \psi _{\kappa}, \varGamma , \vartheta )=0,\qquad \lim_{\kappa \rightarrow + \infty}\varLambda (\psi _{\kappa}, \varGamma , \vartheta )=0 $$
for all \(\vartheta >0\). Suppose
$$\begin{aligned}& \varOmega (\psi , \varGamma , \vartheta )\geq \varOmega \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \varGamma )} \biggr)\ast \varOmega \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3\wp (\psi , \varGamma )} \biggr) \\& \hphantom{\varOmega (\psi , \varGamma , \vartheta )\geq} {}\ast \varOmega \biggl( \psi _{\kappa +1}, \varGamma , \frac{\vartheta}{3\wp (\psi , \varGamma )} \biggr) \\& \hphantom{\varOmega (\psi , \varGamma , \vartheta )}\rightarrow{} 1\ast 1\ast 1 \quad \text{as } \kappa \rightarrow +\infty , \\& \varPhi (\psi , \varGamma , \vartheta ) \leq \varPhi \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \varGamma )} \biggr) \circ \varPhi \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3\wp (\psi , \varGamma )} \biggr) \\& \hphantom{\varPhi (\psi , \varGamma , \vartheta ) \leq}{}\circ \varPhi \biggl( \psi _{\kappa +1}, \varGamma , \frac{\vartheta}{3\wp (\psi , \varGamma )} \biggr) \\& \hphantom{\varPhi (\psi , \varGamma , \vartheta )}\rightarrow{} 0\circ 0\circ 0\quad \text{as } \kappa \rightarrow +\infty , \\& \varLambda (\psi , \varGamma , \vartheta ) \leq \varLambda \biggl( \psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \varGamma )} \biggr)\circ \varLambda \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3\wp (\psi , \varGamma )} \biggr) \\& \hphantom{\varLambda (\psi , \varGamma , \vartheta ) \leq}{}\circ \varLambda \biggl(\psi _{\kappa +1}, \varGamma , \frac{\vartheta}{3\wp (\psi , \varGamma )} \biggr) \\& \hphantom{\varLambda (\psi , \varGamma , \vartheta )}\rightarrow{} 0\circ 0\circ 0 \quad \text{as } \kappa \rightarrow +\infty . \end{aligned}$$
That is, \(\varOmega (\psi , \varGamma , \vartheta )\geq 1\ast 1\ast 1=1\), \(\varPhi (\psi , \varGamma , \vartheta )\leq 0\circ 0\circ 0=0\) and \(\varLambda (\psi , \varGamma , \vartheta )\leq 0\circ 0\circ 0=0\). Hence \(\psi =\varGamma \), that is, the sequence converges to at most one limit point. □
Lemma 2
Let \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) be an ENRMS. If for some \(0<\theta <1\) and for any \(\psi , \varGamma \in \varDelta \), \(\vartheta >0\),
$$\begin{aligned} \begin{aligned}&\varOmega (\psi , \varGamma , \vartheta )\geq \varOmega \biggl(\psi , \varGamma , \frac{\vartheta}{\theta} \biggr),\qquad \varPhi (\psi , \varGamma , \vartheta )\leq \varPhi \biggl(\psi , \varGamma , \frac{\vartheta}{\theta} \biggr), \\ & \varLambda (\psi , \varGamma , \vartheta )\leq \varLambda \biggl(\psi , \varGamma , \frac{\vartheta}{\theta} \biggr), \end{aligned} \end{aligned}$$
(1)
then \(\psi =\varGamma \).
Proof
Condition (1) implies that
$$\begin{aligned} &\varOmega (\psi , \varGamma , \vartheta )\geq \varOmega \biggl(\psi , \varGamma , \frac{\vartheta}{\theta ^{\kappa}} \biggr), \qquad \varPhi (\psi , \varGamma , \vartheta )\leq \varPhi \biggl(\psi , \varGamma , \frac{\vartheta}{\theta ^{\kappa}} \biggr), \\ & \varLambda (\psi , \varGamma , \vartheta )\leq \varLambda \biggl(\psi , \varGamma , \frac{\vartheta}{\theta ^{\kappa}} \biggr), \end{aligned}$$
\(\kappa \in \mathbb{N}\), \(\vartheta >0\).
Now, we have
$$\begin{aligned} \varOmega (\psi , \varGamma , \vartheta )&\geq \lim_{\kappa \rightarrow +\infty} \varOmega \biggl(\psi , \varGamma , \frac{\vartheta}{\theta ^{\kappa}} \biggr)=1, \\ \varPhi (\psi , \varGamma , \vartheta )&\leq \lim_{\kappa \rightarrow +\infty}\varPhi \biggl(\psi , \varGamma , \frac{\vartheta}{\theta ^{\kappa}} \biggr)=0, \\ \varLambda (\psi , \varGamma , \vartheta )&\leq \lim_{\kappa \rightarrow +\infty} \varLambda \biggl(\psi , \varGamma , \frac{\vartheta}{\theta ^{\kappa}} \biggr)=0, \quad \vartheta >0. \end{aligned}$$
Also, by Definition 4 of (A3), (A8), (A13), we obtain \(\psi =\varGamma \). □
Theorem 1
Suppose that \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) is a complete ENRMS in the company of \(\wp \colon \varDelta \times \varDelta \rightarrow [1, +\infty )\) with \(0<\theta <1\) and suppose that
$$\begin{aligned} \begin{aligned}&\lim_{\vartheta \rightarrow +\infty}\varOmega (\psi , \varGamma , \vartheta )=1,\quad \lim _{\vartheta \rightarrow +\infty}\varPhi (\psi , \varGamma , \vartheta )=0 \quad \textit{and} \\ & \lim _{\vartheta \rightarrow +\infty}\varLambda (\psi , \varGamma , \vartheta )=0 \end{aligned} \end{aligned}$$
(2)
for all \(\psi , \varGamma \in \varDelta \) and \(\vartheta >0\). Let \(\nabla \colon \varDelta \rightarrow \varDelta \) be a mapping satisfying
$$\begin{aligned} &\varOmega (\nabla \psi , \nabla \varGamma , \theta \vartheta )\geq \varOmega ( \psi , \varGamma , \vartheta ), \\ &\varPhi (\nabla \psi , \nabla \varGamma , \theta \vartheta )\leq \varPhi (\psi , \varGamma , \vartheta ) \quad \textit{and}\quad \varLambda (\nabla \psi , \nabla \varGamma , \theta \vartheta )\leq \varLambda (\psi , \varGamma , \vartheta ) \end{aligned}$$
(3)
for all \(\psi , \varGamma \in \varDelta \) and \(\vartheta >0\). Further, suppose that for arbitrary \(\psi _{0}\in \varDelta \) and \(\kappa ,\omega \in \mathbb{N}\), we have
$$\begin{aligned} \wp (\psi _{\kappa},\psi _{\kappa +\omega} )< \frac{1}{\theta}. \end{aligned}$$
Then ∇ has a unique fixed point.
Proof
Let \(\psi _{0}\in \varDelta \) and define a sequence \(\psi _{\kappa}\) by \(\psi _{\kappa}=\nabla ^{\kappa}\psi _{0}=\nabla \psi _{\kappa -1}\), \(\kappa \in \mathbb{N}\).
By utilising (2) for all \(\vartheta >0\), we obtain
$$\begin{aligned} &\varOmega (\psi _{\kappa}, \psi _{\kappa +1}, \theta \vartheta )= \varOmega (\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \theta \vartheta )\geq \varOmega (\psi _{\kappa -1}, \psi _{\kappa}, \vartheta ) \geq \varOmega \biggl(\psi _{\kappa -2}, \psi _{\kappa -1}, \frac{\vartheta}{\theta} \biggr) \\ &\hphantom{\varOmega (\psi _{\kappa}, \psi _{\kappa +1}, \theta \vartheta )}\geq \varOmega \biggl(\psi _{\kappa -3}, \psi _{\kappa -2}, \frac{\vartheta}{\theta ^{2}} \biggr)\geq \cdots \geq \varOmega \biggl( \psi _{0}, \psi _{1}, \frac{\vartheta}{\theta ^{\kappa -1}} \biggr), \\ &\varPhi (\psi _{\kappa}, \psi _{\kappa +1}, \theta \vartheta )= \varPhi (\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \theta \vartheta )\leq \varPhi (\psi _{\kappa -1}, \psi _{\kappa}, \vartheta )\leq \varPhi \biggl( \psi _{\kappa -2}, \psi _{\kappa -1}, \frac{\vartheta}{\theta} \biggr) \\ &\hphantom{\varPhi (\psi _{\kappa}, \psi _{\kappa +1}, \theta \vartheta )}\leq \varPhi \biggl(\psi _{\kappa -3}, \psi _{\kappa -2}, \frac{\vartheta}{\theta ^{2}} \biggr)\leq \cdots \leq \varPhi \biggl( \psi _{0}, \psi _{1}, \frac{\vartheta}{\theta ^{\kappa -1}} \biggr) \end{aligned}$$
and
$$\begin{aligned} \varLambda (\psi _{\kappa}, \psi _{\kappa +1}, \theta \vartheta )&= \varLambda (\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \vartheta ) \leq \varLambda (\psi _{\kappa -1}, \psi _{\kappa}, \vartheta )\leq \varLambda \biggl(\psi _{\kappa -2}, \psi _{\kappa -1}, \frac{\vartheta}{\theta} \biggr) \\ &\leq \varLambda \biggl(\psi _{\kappa -3}, \psi _{\kappa -2}, \frac{\vartheta}{\theta ^{2}} \biggr)\leq \cdots \leq \varLambda \biggl( \psi _{0}, \psi _{1}, \frac{\vartheta}{\theta ^{\kappa -1}} \biggr). \end{aligned}$$
We obtain
$$\begin{aligned} &\varOmega (\psi _{\kappa}, \psi _{\kappa +1}, \theta \vartheta ) \geq \varOmega \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{\theta ^{\kappa -1}} \biggr), \\ &\varPhi (\psi _{\kappa}, \psi _{\kappa +1}, \theta \vartheta )\leq \varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{\theta ^{\kappa -1}} \biggr)\quad \text{and} \\ & \varLambda (\psi _{\kappa}, \psi _{\kappa +1}, \theta \vartheta ) \leq \varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{\theta ^{\kappa -1}} \biggr). \end{aligned}$$
(4)
Using (A5), (A10) and (A15), we have the following cases:
Case 1. When \(\mathfrak{i}=2\pi +1\), i.e. \(\mathfrak{i}\) is odd, then
$$\begin{aligned} &\varOmega (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \geq \varOmega \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\ast \cdots \\ &\qquad {}\ast\varOmega \biggl(\psi _{\kappa +2\pi -1}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{\kappa +2\pi}, \psi _{\kappa +2\pi +1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr), \\ &\varPhi (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \leq \varPhi \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \cdots \\ &\qquad {}\circ\varPhi \biggl(\psi _{\kappa +2\pi -1}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{\kappa +2\pi}, \psi _{\kappa +2\pi +1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \end{aligned}$$
and
$$\begin{aligned} &\varLambda (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \leq \varLambda \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \cdots \\ &\qquad {}\circ\varLambda \biggl(\psi _{\kappa +2\pi -1}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{\kappa +2\pi}, \psi _{\kappa +2\pi +1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1}) \wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr). \end{aligned}$$
Using (4) in the above inequalities, we deduce
$$\begin{aligned} &\varOmega (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \geq \varOmega \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3\theta ^{\kappa -1}(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3\theta ^{\kappa}(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\ast \cdots \\ &\qquad {}\ast\varOmega \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}\theta ^{\kappa +2\pi -2}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}\theta ^{\kappa +2\pi -1}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr), \\ &\varPhi (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \leq \varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3\theta ^{\kappa -1}(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3\theta ^{\kappa}(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \cdots \\ &\qquad {}\circ\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}\theta ^{\kappa +2\pi -2}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}\theta ^{\kappa +2\pi -1}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr), \\ &\varLambda (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \leq \varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3\theta ^{\kappa -1}(\wp (\psi _{\kappa}, \psi _{\kappa +1}))} \biggr)\circ \varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3\theta ^{\kappa}(\wp (\psi _{\kappa +1}, \psi _{\kappa +2}))} \biggr) \\ &\qquad {}\circ \cdots \\ &\qquad {}\circ\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}\theta ^{\kappa +2\pi -2}(\wp (\psi _{\kappa +2\pi -1}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa +2}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}\theta ^{\kappa +2\pi -1}(\wp (\psi _{\kappa +2\pi}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa +2}, \psi _{\kappa +2\pi +1}))} \biggr). \end{aligned}$$
Case 2. When \(\mathfrak{i}=2\pi \), i.e. \(\mathfrak{i}\) is even, then
$$\begin{aligned} &\varOmega (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \geq \varOmega \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \ast \varOmega \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\ast \cdots \\ &\qquad {}\ast\varOmega \biggl(\psi _{\kappa +2\pi -3}, \psi _{\kappa +2\pi -2}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr), \\ &\varPhi (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \leq \varPhi \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \circ \varPhi \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \cdots \\ &\qquad {}\circ\varPhi \biggl(\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \end{aligned}$$
and
$$\begin{aligned} &\varLambda (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \leq \varLambda \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \circ \varLambda \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \cdots \\ &\qquad {}\circ\varLambda \biggl(\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi -3}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{\kappa +2\pi -3}, \psi _{\kappa +2\pi -2}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi}) \wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr). \end{aligned}$$
Using (4) in the above inequalities, we deduce
$$\begin{aligned} &\varOmega (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \geq \varOmega \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3\theta ^{\kappa -1}(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr)\ast \varOmega \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3\theta ^{\kappa}(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\ast \cdots \\ &\qquad {}\ast\varOmega \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}\theta ^{\kappa +2\pi -5}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}\theta ^{\kappa +2\pi -4}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}\theta ^{\kappa +2\pi -3}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr), \\ &\varPhi (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \leq \varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3\theta ^{\kappa -1}(\wp (\psi _{\kappa}, \psi _{\kappa +1}))} \biggr)\circ \varPhi \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3\theta ^{\kappa}(\wp (\psi _{\kappa +1}, \psi _{\kappa +2}))} \biggr) \\ &\qquad {}\circ \cdots \\ &\qquad {}\circ\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}\theta ^{\kappa +2\pi -5}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi -3})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa +2}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}\theta ^{\kappa +2\pi -4}(\wp (\psi _{\kappa +2\pi -3}, \psi _{\kappa +2\pi -2})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa +2}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}\theta ^{\kappa +2\pi -3}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa +2}, \psi _{\kappa +2\pi}))} \biggr) \end{aligned}$$
and
$$\begin{aligned} &\varLambda (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \leq \varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3\theta ^{\kappa -1}(\wp (\psi _{\kappa}, \psi _{\kappa +1}))} \biggr)\circ \varLambda \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3\theta ^{\kappa}(\wp (\psi _{\kappa +1}, \psi _{\kappa +2}))} \biggr) \\ &\qquad {}\circ \cdots \\ &\qquad {} \circ\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}\theta ^{\kappa +2\pi -5} (\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi -3})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa +2}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}\theta ^{\kappa +2\pi -4}(\wp (\psi _{\kappa +2\pi -3}, \psi _{\kappa +2\pi -2})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa +2}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}\theta ^{\kappa +2\pi -3}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa +2}, \psi _{\kappa +2\pi}))} \biggr). \end{aligned}$$
Since \(\kappa ,\omega \in \mathbb{N}\), we have
$$\begin{aligned} \wp (\psi _{\kappa},\psi _{\kappa +\omega} )< \frac{1}{\theta}. \end{aligned}$$
Therefore, from (2), for each case \(\kappa \rightarrow +\infty \), we deduce
$$\begin{aligned} &\lim_{\kappa \rightarrow +\infty}\varOmega (\psi _{\kappa}, \psi _{ \kappa +\mathfrak{i}}, \vartheta )=1\ast 1\ast \cdots \ast 1=1, \\ &\lim_{\kappa \rightarrow +\infty}\varPhi (\psi _{\kappa}, \psi _{ \kappa +\mathfrak{i}}, \vartheta )=0\circ 0\circ \cdots \circ 0=0 \end{aligned}$$
and
$$\begin{aligned} \lim_{\kappa \rightarrow +\infty}\varLambda (\psi _{\kappa}, \psi _{ \kappa +\mathfrak{i}}, \vartheta )=0\circ 0\circ \cdots \circ 0&=0. \end{aligned}$$
Therefore, \(\{\psi _{\kappa}\}\) is a Cauchy sequence. Since \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) is complete, there exists
$$\begin{aligned} \lim_{\kappa \rightarrow +\infty}\psi _{\kappa}=\psi . \end{aligned}$$
Using (A5), (A10), (A15) and (2), we get
$$\begin{aligned} &\varOmega (\psi , \nabla \psi , \vartheta ) \\ &\quad \geq \varOmega \biggl( \psi , \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr)\ast \varOmega \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\qquad {}\ast\varOmega \biggl(\psi _{\kappa +1},\nabla \psi , \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\quad =\varOmega \biggl(\psi , \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr)\ast \varOmega \biggl(\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\qquad {}\ast\varOmega \biggl(\nabla \psi _{\kappa},\nabla \psi , \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\quad \geq \varOmega \biggl(\psi , \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \ast \varOmega \biggl(\psi _{\kappa -1}, \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{\kappa},\psi , \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\quad \rightarrow 1\ast 1\ast 1=1 \quad \text{as } \kappa \rightarrow + \infty , \\ &\varPhi (\psi , \nabla \psi , \vartheta ) \\ &\quad \leq \varPhi \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \circ \varPhi \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{\kappa +1}, \nabla \psi , \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\quad =\varPhi \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr)\circ \varPhi \biggl(\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\nabla \psi _{\kappa}, \nabla \psi , \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\quad \leq \varPhi \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr)\circ \varPhi \biggl(\psi _{\kappa -1}, \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{\kappa},\psi , \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\quad \rightarrow 0\circ 0\circ 0=0 \quad \text{as } \kappa \rightarrow + \infty \end{aligned}$$
and
$$\begin{aligned} &\varLambda (\psi , \nabla \psi , \vartheta ) \\ &\quad \leq \varLambda \biggl( \psi , \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr)\circ \varLambda \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\qquad {}\circ\varLambda \biggl(\psi _{\kappa +1}, \nabla \psi , \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\quad =\varLambda \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr)\circ \varLambda \biggl(\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\nabla \psi _{\kappa}, \nabla \psi , \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\quad \leq \varLambda \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \circ \varLambda \biggl(\psi _{\kappa -1}, \psi _{\kappa}, \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{\kappa},\psi , \frac{\vartheta}{3(\wp (\psi , \nabla \psi ))} \biggr) \\ &\quad \rightarrow 0\circ 0\circ 0=0\quad \text{as } \kappa \rightarrow + \infty . \end{aligned}$$
Hence, \(\nabla \psi =\psi \). Let \(\nabla \mu =\mu \) for some \(\mu \in \varDelta \), then
$$\begin{aligned} &1\geq \varOmega (\mu , \psi , \vartheta )=\varOmega (\nabla \mu , \nabla \psi , \vartheta )\geq \varOmega \biggl(\mu , \psi , \frac{\vartheta}{\theta} \biggr)= \varOmega \biggl(\nabla \mu , \nabla \psi , \frac{\vartheta}{\theta} \biggr) \\ &\hphantom{1}\geq \varOmega \biggl(\mu , \psi , \frac{\vartheta}{\theta ^{2}} \biggr)\geq \cdots \geq \varOmega \biggl(\mu , \psi , \frac{\vartheta}{\theta ^{\kappa}} \biggr)\rightarrow 1 \quad \text{as } \kappa \rightarrow +\infty , \\ &0\leq \varPhi (\mu , \psi , \vartheta )=\varPhi (\nabla \mu , \nabla \psi , \vartheta )\leq \varPhi \biggl(\mu , \psi , \frac{\vartheta}{\theta} \biggr)=\varPhi \biggl(\nabla \mu , \nabla \psi , \frac{\vartheta}{\theta} \biggr) \\ &\hphantom{0}\leq \varPhi \biggl(\mu , \psi , \frac{\vartheta}{\theta ^{2}} \biggr) \leq \cdots \leq \varPhi \biggl(\mu , \psi , \frac{\vartheta}{\theta ^{\kappa}} \biggr)\rightarrow 0 \quad \text{as } \kappa \rightarrow +\infty \end{aligned}$$
and
$$\begin{aligned} 0&\leq \varLambda (\mu , \psi , \vartheta )=\varLambda (\nabla \mu , \nabla \psi , \vartheta )\leq \varLambda \biggl(\mu , \psi , \frac{\vartheta}{\theta} \biggr)= \varLambda \biggl(\nabla \mu , \nabla \psi , \frac{\vartheta}{\theta} \biggr) \\ &\leq \varLambda \biggl(\mu , \psi , \frac{\vartheta}{\theta ^{2}} \biggr)\leq \cdots \leq \varLambda \biggl(\mu , \psi , \frac{\vartheta}{\theta ^{\kappa}} \biggr)\rightarrow 0 \quad \text{as } \kappa \rightarrow +\infty \end{aligned}$$
by using (A3), (A8) and (A13), \(\psi =\mu \). Therefore, ∇ has a unique fixed point. □
Definition 7
Let \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) be an ENRMS. A map \(\nabla \colon \varDelta \rightarrow \varDelta \) is an ENRC(extended neutrosophic rectangular contraction) if there exists \(0<\theta <1\) such that
$$\begin{aligned} &\frac{1}{\varOmega (\mathcal{P}\psi , \mathcal{P}\varGamma , \vartheta )}-1 \leq \theta \biggl[\frac{1}{\varOmega (\psi , \varGamma , \vartheta )}-1 \biggr] \end{aligned}$$
(5)
$$\begin{aligned} &\varPhi (\mathcal{P}\psi , \mathcal{P}\varGamma , \vartheta )\leq \theta \varPhi (\psi , \varGamma , \vartheta ) \end{aligned}$$
(6)
and
$$\begin{aligned} \varLambda (\mathcal{P}\psi , \mathcal{P}\varGamma , \vartheta )\leq \theta \varLambda (\psi , \varGamma , \vartheta ) \end{aligned}$$
(7)
for all \(\psi , \varGamma \in \varDelta \) and \(\vartheta >0\).
Now, we prove the theorem for ENRC.
Theorem 2
Let \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) be a complete ENRMS with \(\wp \colon \varDelta \times \varDelta \rightarrow [1, +\infty )\) and suppose that
$$\begin{aligned} \lim_{\vartheta \rightarrow +\infty}\varOmega (\psi , \varGamma , \vartheta )=1,\qquad \lim _{\vartheta \rightarrow +\infty}\varPhi (\psi , \varGamma , \vartheta )=0 \quad \textit{and}\quad \lim _{\vartheta \rightarrow +\infty}\varLambda (\psi , \varGamma , \vartheta )=0 \end{aligned}$$
(8)
for all \(\psi , \varGamma \in \varDelta \) and \(\vartheta >0\). Let \(\nabla \colon \varDelta \rightarrow \varDelta \) be an ENRC. Further, suppose that for an arbitrary \(\psi _{0}\in \varDelta \) and \(\kappa ,\omega \in \mathbb{N}\), we have
$$\begin{aligned} \wp (\psi _{\kappa},\psi _{\kappa +\omega} )< \frac{1}{\theta}. \end{aligned}$$
Then ∇ has a unique fixed point.
Proof
Let \(\psi _{0}\) be a point of Δ and define a sequence \(\psi _{\kappa}\) by \(\psi _{\kappa}=\nabla ^{\kappa}\psi _{0}=\nabla \psi _{\kappa -1}\), \(\kappa \in \mathbb{N}\). By using (5), (6) and (7) for all \(\vartheta >0\), \(\kappa >\omega \), we deduce
$$\begin{aligned}& \frac{1}{\varOmega (\psi _{\kappa}, \psi _{\kappa +1}, \vartheta )}-1 \\& \qquad = \frac{1}{\varOmega (\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \vartheta )}-1 \leq \theta \biggl[ \frac{1}{\varOmega (\psi _{\kappa -1}, \psi _{\kappa}, \vartheta )} \biggr]= \frac{\theta}{\varOmega (\psi _{\kappa -1}, \psi _{\kappa}, \vartheta )}- \theta \\& \quad \Rightarrow \quad \frac{1}{\varOmega (\psi _{\kappa}, \psi _{\kappa +1}, \vartheta )} \\& \hphantom{\quad \Rightarrow \quad }\quad \leq \frac{\theta}{\varOmega (\psi _{\kappa -1}, \psi _{\kappa}, \vartheta )}+(1- \theta )\leq \frac{\theta ^{2}}{\varOmega (\psi _{\kappa -2}, \psi _{\kappa -1}, \vartheta )}+ \theta (1-\theta )+(1-\theta ). \end{aligned}$$
Carrying on in this manner, we deduce
$$\begin{aligned} \frac{1}{\varOmega (\psi _{\kappa}, \psi _{\kappa +1}, \vartheta )}& \leq \frac{\theta ^{\kappa}}{\varOmega (\psi _{0}, \psi _{1}, \vartheta )}+ \theta ^{\kappa -1}(1-\theta )+ \theta ^{\kappa -2}(1-\theta )+\cdots + \theta (1-\theta )+(1-\theta ) \\ &\leq \frac{\theta ^{\kappa}}{\varOmega (\psi _{0}, \psi _{1}, \vartheta )}+ \bigl( \theta ^{\kappa -1}+\theta ^{\kappa -2}+ \cdots +1 \bigr) (1-\theta ) \\ &\leq \frac{\theta ^{\kappa}}{\varOmega (\psi _{0}, \psi _{1}, \vartheta )}+ \bigl(1- \theta ^{\kappa} \bigr). \end{aligned}$$
We obtain
$$\begin{aligned} &\frac{1}{\frac{\theta ^{\kappa}}{\varOmega (\psi _{0}, \psi _{1}, \vartheta )}+(1-\theta ^{\kappa})} \leq \varOmega (\psi _{\kappa}, \psi _{\kappa +1}, \vartheta ) , \end{aligned}$$
(9)
$$\begin{aligned} &\varPhi (\psi _{\kappa}, \psi _{\kappa +1}, \vartheta )=\varPhi ( \nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \vartheta )\leq \theta \varPhi (\psi _{\kappa -1}, \psi _{\kappa}, \vartheta )= \varPhi (\nabla \psi _{\kappa -2}, \nabla \psi _{\kappa -1}, \vartheta ) \\ &\hphantom{\varPhi (\psi _{\kappa}, \psi _{\kappa +1}, \vartheta )}\leq \theta ^{2}\varPhi (\psi _{\kappa -2}, \psi _{\kappa -1}, \vartheta )\leq \cdots \leq \theta ^{\kappa}\varPhi (\psi _{0}, \psi _{1}, \vartheta ) \end{aligned}$$
(10)
and
$$\begin{aligned} \varLambda (\psi _{\kappa}, \psi _{\kappa +1}, \vartheta )&= \varLambda (\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \vartheta )\leq \theta \varLambda (\psi _{\kappa -1}, \psi _{\kappa}, \vartheta )=\varLambda ( \nabla \psi _{\kappa -2}, \nabla \psi _{ \kappa -1}, \vartheta ) \\ &\leq \theta ^{2}\varLambda (\psi _{\kappa -2}, \psi _{\kappa -1}, \vartheta )\leq \cdots \leq \theta ^{\kappa}\varLambda (\psi _{0}, \psi _{1}, \vartheta ). \end{aligned}$$
(11)
Using (A5), (A10) and (A15), we have the following cases:
Case 1. When \(\mathfrak{i}=2\pi +1\), i.e. \(\mathfrak{i}\) is odd, then
$$\begin{aligned} &\varOmega (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \geq \varOmega \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\ast\varOmega \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr)\ast \cdots \\ &\qquad {}\ast\varOmega \biggl(\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi -1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{\kappa +2\pi -1}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{\kappa +2\pi}, \psi _{\kappa +2\pi +1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1}) \cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr), \\ &\varPhi (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \leq \varPhi \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr)\circ \varPhi \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \cdots \\ &\qquad {} \circ\varPhi \biggl(\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi -1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{\kappa +2\pi -1}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{\kappa +2\pi}, \psi _{\kappa +2\pi +1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \end{aligned}$$
and
$$\begin{aligned} &\varLambda (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \leq \varLambda \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ\varLambda \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \circ \cdots \\ &\qquad {} \circ\varLambda \biggl(\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi -1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{\kappa +2\pi -1}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{\kappa +2\pi}, \psi _{\kappa +2\pi +1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr). \end{aligned}$$
Using (4) in the above inequalities, we deduce
$$\begin{aligned} &\varOmega (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \geq \frac{1}{\frac{\theta ^{\kappa}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} )}+(1-\theta ^{\kappa})} \ast \frac{1}{\frac{\theta ^{\kappa +1}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} )}+(1-\theta ^{\kappa +1})} \ast \cdots \\ &\qquad {}\ast \frac{1}{\frac{\theta ^{\kappa +2\pi -2}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} )}+(1-\theta ^{\kappa +2\pi -2})} \\ &\qquad {}\ast \frac{1}{\frac{\theta ^{\kappa +2\pi -1}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi + 1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} )}+(1-\theta ^{\kappa +2\pi -1})} \\ &\qquad {}\ast \frac{1}{\frac{\theta ^{\kappa +2\pi}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} )}+(1-\theta ^{\kappa +2\pi})}, \end{aligned}$$
$$\begin{aligned} &\varPhi (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \leq \theta ^{\kappa}\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ\theta ^{\kappa +1}\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr)\circ \cdots \\ &\qquad {}\circ\theta ^{\kappa +2\pi -2}\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1}) \wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \theta ^{\kappa +2\pi -1}\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1}) \wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \theta ^{\kappa +2\pi}\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1}) \wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \end{aligned}$$
and
$$\begin{aligned} &\varLambda (\psi _{\kappa}, \psi _{\kappa +2\pi +1}, \vartheta ) \\ &\quad \leq \theta ^{\kappa}\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ\theta ^{\kappa +1}\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr)\circ \cdots \\ &\qquad {}\circ\theta ^{\kappa +2\pi -2}\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \theta ^{\kappa +2\pi -1}\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr) \\ &\qquad {}\circ \theta ^{\kappa +2\pi}\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi}(\wp (\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi +1})\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi +1}))} \biggr). \end{aligned}$$
Case 2. When \(\mathfrak{i}=2\pi \), i.e. \(\mathfrak{i}\) is even, then
$$\begin{aligned} &\varOmega (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \geq \varOmega \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \ast \varOmega \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\ast \cdots \\ &\qquad {}\ast\varOmega \biggl(\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi -3}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{\kappa +2\pi -3}, \psi _{\kappa +2\pi -2}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\ast \varOmega \biggl(\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp\ (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr), \\ &\varPhi (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \leq \varPhi \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \circ \varPhi \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{\kappa +2}, \psi _{\kappa +3}, \frac{\vartheta}{3^{2}(\wp (\psi _{\kappa +2}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr)\circ \cdots \\ &\qquad {} \circ\varPhi \biggl(\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi -3}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{\kappa +2\pi -3}, \psi _{\kappa +2\pi -2}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varPhi \biggl(\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \end{aligned}$$
and
$$\begin{aligned} &\varLambda (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \leq \varLambda \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \circ \varLambda \biggl(\psi _{\kappa +1}, \psi _{\kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \cdots \\ &\qquad {}\circ\varLambda \biggl(\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi -3}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{\kappa +2\pi -3}, \psi _{\kappa +2\pi -2}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \varLambda \biggl(\psi _{\kappa +2\pi -2}, \psi _{\kappa +2\pi}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi +1})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr). \end{aligned}$$
Using (4) in the above inequalities, we deduce
$$\begin{aligned} &\varOmega (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \geq \frac{1}{\frac{\theta ^{\kappa}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} )}+(1-\theta ^{\kappa})} \ast \frac{1}{\frac{\theta ^{\kappa +1}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} )}+(1-\theta ^{\kappa +1})} \\ &\qquad {}\ast \frac{1}{\frac{\theta ^{\kappa +2}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{2}(\wp (\psi _{\kappa +2}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} )}+(1-\theta ^{\kappa +2})} \ast \cdots \ast \\ &\qquad {}\ast \frac{1}{\frac{\theta ^{\kappa +2\pi -4}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} )}+(1-\theta ^{\kappa +2\pi -4})} \\ &\qquad {}\ast \frac{1}{\frac{\theta ^{\kappa +2\pi -3}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} )}+(1-\theta ^{\kappa +2\pi -3})} \\ &\qquad {}\ast \frac{1}{\frac{\theta ^{\kappa +2\pi -2}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} )}+(1-\theta ^{\kappa +2\pi -2})}, \\ &\varPhi (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \leq \theta ^{\kappa}\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \theta ^{\kappa +1}\varPhi \biggl(\psi _{\kappa +1}, \psi _{ \kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \circ \cdots \\ & \qquad {}\circ\theta ^{\kappa +2\pi -4}\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \theta ^{\kappa +2\pi -3}\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \theta ^{\kappa +2\pi -2}\varPhi \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr), \\ &\varLambda (\psi _{\kappa}, \psi _{\kappa +2\pi}, \vartheta ) \\ &\quad \leq \theta ^{\kappa}\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \theta ^{\kappa +1}\varLambda \biggl(\psi _{\kappa +1}, \psi _{ \kappa +2}, \frac{\vartheta}{3(\wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \circ \cdots \\ &\qquad {}\circ \theta ^{\kappa +2\pi -4}\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi}) \cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \theta ^{\kappa +2\pi -3}\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi}) \cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr) \\ &\qquad {}\circ \theta ^{\kappa +2\pi -2}\varLambda \biggl(\psi _{0}, \psi _{1}, \frac{\vartheta}{3^{\pi -1}(\wp (\psi _{\kappa +2\pi -4}, \psi _{\kappa +2\pi})\wp (\psi _{\kappa +2\pi -6}, \psi _{\kappa +2\pi})\cdots \wp (\psi _{\kappa}, \psi _{\kappa +2\pi}))} \biggr). \end{aligned}$$
Since \(\kappa ,\omega \in \mathbb{N}\), we have
$$\begin{aligned} \wp (\psi _{\kappa},\psi _{\kappa +\omega} )< \frac{1}{\theta}. \end{aligned}$$
Therefore, from (8), for each case \(\kappa \rightarrow +\infty \), we deduce that
$$\begin{aligned} \lim_{\kappa \rightarrow +\infty}\varOmega (\psi _{\kappa}, \psi _{ \kappa +\omega}, \vartheta )&=1\ast 1\ast \cdots \ast =1, \\ \lim_{\kappa \rightarrow +\infty}\varPhi (\psi _{\kappa}, \psi _{ \kappa +\omega}, \vartheta )&=0\circ 0\circ \cdots \circ 0=0 \end{aligned}$$
and
$$\begin{aligned} \lim_{\kappa \rightarrow +\infty}\varLambda (\psi _{\kappa}, \psi _{ \kappa +\omega}, \vartheta )&=0\circ 0\circ \cdots \circ 0=0. \end{aligned}$$
Therefore, \(\{\psi _{\kappa}\}\) is a Cauchy sequence. Since \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) is complete, there exists
$$\begin{aligned} \lim_{\kappa \rightarrow +\infty}\psi _{\kappa}=\psi . \end{aligned}$$
From (A5), (A10) and (A15), we get
$$\begin{aligned} \frac{1}{\varOmega (\nabla \psi _{\kappa}, \nabla \psi , \vartheta )}-1& \leq \theta \biggl[ \frac{1}{\varOmega (\psi _{\kappa}, \psi , \vartheta )}-1 \biggr]= \frac{\theta}{\varOmega (\psi _{\kappa}, \psi , \vartheta )}-\theta \\ &\Rightarrow \frac{1}{\frac{\theta}{\varOmega (\psi _{\kappa}, \psi , \vartheta )}+(1-\theta )} \leq \varOmega (\nabla \psi _{\kappa}, \nabla \psi , \vartheta ). \end{aligned}$$
Using the above inequality, we obtain
$$\begin{aligned} &\begin{aligned}\varOmega (\psi , \nabla \psi , \vartheta )\geq{}& \varOmega \biggl( \psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr)\ast \varOmega \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ &{}\ast \varOmega \biggl(\psi _{\kappa +1}, \nabla \psi , \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ \geq{}& \varOmega \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \ast \varOmega \biggl(\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ &{}\ast \varOmega \biggl(\nabla \psi _{\kappa}, \nabla \psi , \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ \geq {}&\varOmega \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{(3\wp (\psi , \nabla \psi )} \biggr) \ast \frac{1}{\frac{\theta ^{\kappa}}{\varOmega (\psi _{0}, \psi _{1}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )})+(1-\theta ^{\kappa})}} \\ &{}\ast \frac{1}{\frac{\theta}{\varOmega (\psi _{\kappa}, \psi , \frac{\vartheta}{3\wp (\psi , \nabla \psi )} )+(1-\theta )}} \\ \rightarrow{}& 1\ast 1\ast 1=1 \quad \text{as } \kappa \rightarrow + \infty , \end{aligned} \\ &\begin{aligned} \varPhi (\psi , \nabla \psi , \vartheta )\leq{}& \varPhi \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \circ \varPhi \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ &{}\circ \varPhi \biggl(\psi _{\kappa +1}, \nabla \psi , \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ \leq{}& \varPhi \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \circ \varPhi \biggl(\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ &{}\circ \varPhi \biggl(\nabla \psi _{\kappa}, \nabla \psi , \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ \leq{}& \varPhi \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \circ \theta ^{ \kappa -1}\varPhi \biggl(\psi _{\kappa -1}, \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ &{}\circ \theta \varPhi \biggl(\psi _{\kappa}, \psi , \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ \rightarrow{}& 0\circ 0\circ 0=0 \quad \text{as } \kappa \rightarrow + \infty \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \varLambda (\psi , \nabla \psi , \vartheta )\leq{}& \varLambda \biggl( \psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr)\circ \varLambda \biggl(\psi _{\kappa}, \psi _{\kappa +1}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ &{}\circ \varLambda \biggl(\psi _{\kappa +1}, \nabla \psi , \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ \leq{}& \varLambda \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \circ \varLambda \biggl(\nabla \psi _{\kappa -1}, \nabla \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ &{}\circ \varLambda \biggl(\nabla \psi _{\kappa}, \nabla \psi , \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ \leq{}& \varLambda \biggl(\psi , \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \circ \theta ^{ \kappa -1}\varLambda \biggl(\psi _{\kappa -1}, \psi _{\kappa}, \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ &{}\circ \theta \varLambda \biggl(\psi _{\kappa}, \psi , \frac{\vartheta}{3\wp (\psi , \nabla \psi )} \biggr) \\ \rightarrow{}& 0\circ 0\circ 0=0\quad \text{as } \kappa \rightarrow + \infty . \end{aligned}$$
Hence, \(\nabla \psi =\psi \). Let \(\nabla \mu =\mu \) for some \(\mu \in \varDelta \), then
$$\begin{aligned} \frac{1}{\varOmega (\psi , \mu , \vartheta )}-1&= \frac{1}{\varOmega (\nabla \psi , \nabla \mu , \vartheta )}-1 \\ &\leq \theta \biggl[\frac{1}{\varOmega (\psi , \mu , \vartheta )}-1 \biggr]< \frac{1}{\varOmega (\psi , \mu , \vartheta )}-1, \end{aligned}$$
which is a contradiction.
$$\begin{aligned} \varPhi (\psi , \mu , \vartheta )=\varPhi (\nabla \psi , \nabla \mu , \vartheta )\leq \theta \varPhi (\psi , \mu , \vartheta )< \varPhi ( \psi , \mu , \vartheta ), \end{aligned}$$
which is a contradiction and
$$\begin{aligned} \varLambda (\psi , \mu , \vartheta )=\varLambda (\nabla \psi , \nabla \mu , \vartheta )\leq \theta \varLambda (\psi , \mu , \vartheta )< \varLambda (\psi , \mu , \vartheta ), \end{aligned}$$
which is a contradiction. Therefore, \(\varOmega (\psi , \mu , \vartheta )=1\), \(\varPhi (\psi , \mu , \vartheta )=0\) and \(\varLambda (\psi , \mu , \vartheta )=0\), hence, \(\psi =\mu \). Hence, ∇ has a unique fixed point. □
Example 2
Let \(\varDelta =[0, 1]\) and \(\wp \colon \varDelta \times \varDelta \rightarrow [1, +\infty )\) be a function given by
$$\begin{aligned} \wp (\psi , \varGamma )= \textstyle\begin{cases} 1 &\text{if } \psi =\varGamma , \\ \frac{1+\max \{\psi , \varGamma \}}{1+\min \{\psi , \varGamma \}} & \text{if } \psi \neq \varGamma . \end{cases}\displaystyle \end{aligned}$$
Define \(\varOmega , \varPhi , \varLambda \colon \varDelta \times \varDelta \times (0, +\infty )\rightarrow [0, 1]\) as
$$\begin{aligned} &\varOmega (\psi , \varGamma , \vartheta )= \frac{\vartheta}{\vartheta + \vert \psi -\varGamma \vert ^{2}}, \\ &\varPhi (\psi , \varGamma , \vartheta )= \frac{ \vert \psi -\varGamma \vert }{\vartheta + \vert \psi -\varGamma \vert ^{2}}, \\ &\varLambda (\psi , \varGamma , \vartheta )= \frac{ \vert \psi -\varGamma \vert ^{2}}{\vartheta}. \end{aligned}$$
Then \((\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )\) is a complete ENRMS with continuous t-norm \(\wp \ast \tau =\wp \tau \) and continuous t-co-norm \(\wp \circ \tau =\max \{\wp , \tau \}\).
Define \(\nabla \colon \varDelta \rightarrow \varDelta \) by \(\nabla (\psi )=\frac{1-3^{-\psi}}{7}\) and take \(\theta \in [\frac{1}{2}, 1)\), then
$$\begin{aligned} &\varOmega (\nabla \psi , \nabla \varGamma , \theta \vartheta )= \varOmega \biggl(\frac{1-3^{-\psi}}{7}, \frac{1-3^{-\varGamma}}{7}, \theta \vartheta \biggr) \\ &\hphantom{\varOmega (\nabla \psi , \nabla \varGamma , \theta \vartheta )}= \frac{\theta \vartheta}{\theta \vartheta + \vert \frac{1-3^{-\psi}}{7}-\frac{1-3^{-\varGamma}}{7} \vert ^{2}}= \frac{\theta \vartheta}{\theta \vartheta +\frac{ \vert 3^{-\psi}-3^{-\varGamma} \vert ^{2}}{49}} \\ &\hphantom{\varOmega (\nabla \psi , \nabla \varGamma , \theta \vartheta )}\geq \frac{\theta \vartheta}{\theta \vartheta +\frac{ \vert \psi -\varGamma \vert ^{2}}{49}}= \frac{49\theta \vartheta}{49\theta \vartheta + \vert \psi -\varGamma \vert } \geq \frac{\vartheta}{\vartheta + \vert \psi -\varGamma \vert }= \varOmega (\psi , \varGamma , \vartheta ), \\ &\varPhi (\nabla \psi , \nabla \varGamma , \theta \vartheta )= \varPhi \biggl( \frac{1-3^{-\psi}}{7}, \frac{1-3^{-\varGamma}}{7}, \theta \vartheta \biggr) \\ &\hphantom{\varPhi (\nabla \psi , \nabla \varGamma , \theta \vartheta )}= \frac{ \vert \frac{1-3^{-\psi}}{7}-\frac{1-3^{-\varGamma}}{7} \vert ^{2}}{\theta \vartheta + \vert \frac{1-3^{-\psi}}{7}-\frac{1-3^{-\varGamma}}{7} \vert ^{2}}= \frac{\frac{ \vert 3^{-\psi}-3^{-\varGamma} \vert ^{2}}{49}}{\theta \vartheta +\frac{ \vert 3^{-\psi}-3^{-\varGamma} \vert ^{2}}{49}} \\ &\hphantom{\varPhi (\nabla \psi , \nabla \varGamma , \theta \vartheta )}= \frac{ \vert 3^{-\psi}-3^{-\varGamma} \vert ^{2}}{49\theta \vartheta + \vert 3^{-\psi}-3^{-\varGamma} \vert ^{2}} \\ &\hphantom{\varPhi (\nabla \psi , \nabla \varGamma , \theta \vartheta )}\leq \frac{ \vert \psi -\varGamma \vert ^{2}}{49\theta \vartheta + \vert \psi -\varGamma \vert ^{2}} \leq \frac{ \vert \psi -\varGamma \vert ^{2}}{\vartheta + \vert \psi -\varGamma \vert ^{2}}= \varPhi (\psi , \varGamma , \vartheta ) \end{aligned}$$
and
$$\begin{aligned} \varLambda (\nabla \psi , \nabla \varGamma , \theta \vartheta )&= \varLambda \biggl(\frac{1-3^{-\psi}}{7}, \frac{1-3^{-\varGamma}}{7}, \theta \vartheta \biggr) \\ &= \frac{ \vert \frac{1-3^{-\psi}}{7}-\frac{1-3^{-\varGamma}}{7} \vert ^{2}}{\theta \vartheta}= \frac{\frac{ \vert 3^{-\psi}-3^{-\varGamma} \vert ^{2}}{49}}{\theta \vartheta} \\ &=\frac{ \vert 3^{-\psi}-3^{-\varGamma} \vert ^{2}}{49\theta \vartheta}\leq \frac{ \vert \psi -\varGamma \vert }{49\theta \vartheta}\leq \frac{ \vert \psi -\varGamma \vert }{\vartheta}=\varLambda (\psi , \varGamma , \vartheta ). \end{aligned}$$
As a result, all of the conditions of Theorem 1 are satisfied, and 0 is the only fixed point for ∇.