A fixed point result on an extended neutrosophic rectangular metric space with application

Abstract

In this paper, we propose the notion of extended neutrosophic rectangular metric space and prove some fixed point results under contraction mapping. Finally, as an application of the obtained results, we prove the existence and uniqueness of the Caputo fractional differential equation.

1 Introduction

The foundation of fixed point theory consists of the notion of metric spaces and the Banach contraction principle [1]. The spaciousness of metric space is attracting thousands of academics with its axiomatic interpretation see [2–10]. There have been numerous metric space generalisations made recently. This demonstrates the elegance, allure and growth of the idea of metric spaces. The notion of fuzzy sets was proposed by Zadeh [11]. The term “fuzzy” appears to be widely used and frequently occurring in current research on the logical and set-theoretical foundations of mathematics. We believe that the primary reason for this rapid development is simple to understand. The world we live in is full of uncertainty because, for the most part, the data that come from our findings and measurements, the ideas we utilise and the information we gather from the environment are all imprecise and ambiguous. Therefore, any formal description of the real world, or parts of it, is always merely an idealisation and an approximation of the true reality. Fuzzy sets, fuzzy orderings, fuzzy languages and so on allow us to investigate and deal with the previously specified degree of uncertainty in a strictly formal and mathematical manner. The concept of fuzzy sets has succeeded in shifting a lot of mathematical structures within its concept. Schweizer and Sklar [12] defined the notion of continuous t-norms. Kramosil and Michalek [13] introduced the notion of fuzzy metric spaces. They applied the concept of fuzziness, via continuous t-norms, to classical notions of metric and metric spaces and compared the notions thus obtained with those resulting from some other, namely probabilistic, statistical generalisations of metric spaces. Garbiec [14] provided the fuzzy interpretation of Banach contraction principle in fuzzy metric spaces. Ur-Reham et al. [15] proved some $\alpha -\phi$-fuzzy cone contraction results with integral type application. Fuzzy metric spaces only deal with membership functions. An intuitionistic fuzzy metric space that is used to deal with both membership and non-membership functions was established by Park [16]. Konwar [17] presented the concept of an intuitionistic fuzzy b-metric space and proved several fixed point theorems. Kirişci and Simsek [18] introduced the notion of neutrosophic metric spaces that is used to deal with membership, non-membership and naturalness. Simsek and Kirişci [19] proved some amazing fixed point results in the context of neutrosophic metric spaces. Sowndrarajan et al. [20] proved some fixed point results in the setting of neutrosophic metric spaces. Itoh [21] proved an application regarding random differential equations in Banach spaces. Mlaiki [22] coined the concept of controlled metric spaces and proved several fixed point results for contraction mappings. Sezen [23] presented the notion of controlled fuzzy metric spaces and proved various contraction mapping results. Recently, Saleem et al. [24] introduced the concept of fuzzy double controlled metric spaces. For related articles, see [25, 26, 30–33]. In 2022, Uddin et al. [27] proved fixed point theorem on neutrosophic double controlled metric space. In 2022, Gunaseelan et al. [28] proposed neutrosophic rectangular triple controlled metric space and proved fixed point theorems.

In this paper, we introduce the notion of extended neutrosophic rectangular metric space and prove fixed point theorems. The main objectives of this paper are as follows:

• Introduce the notion of extended neutrosophic rectangular metric space;

• Prove several fixed point theorems for contraction mappings;

• Find the existence and uniqueness solution of the fractional differential equation with boundary conditions.

2 Preliminaries

In this section, we provide some definitions that will be helpful for readers to understand the main section.

Definition 1

[16] A binary operation $\ast \colon [0, 1]\times [0, 1]\rightarrow [0, 1]$ is called a continuous t-norm if:

• 1. $\wp \ast \tau =\tau \ast \wp$ for all $\wp , \tau \in [0, 1]$;

• 2. ∗ is continuous;

• 3. $\wp \ast 1=\wp$ for all $\wp \in [0, 1]$;

• 4. $(\wp \ast \tau )\ast \mu =\wp \ast (\tau \ast \mu )$ for all $\wp , \tau , \mu \in [0, 1]$;

• 5. If $\wp \leq \mu$ and $\tau \leq \mathfrak{d}$ with $\wp , \tau , \mu , \mathfrak{d}\in [0, 1]$, then $\wp \ast \tau \leq \mu \ast \mathfrak{d}$.

Definition 2

[16] A binary operation $\circ \colon [0, 1]\times [0, 1]\rightarrow [0, 1]$ is called a continuous t-co-norm if:

• 1. $\wp \circ \tau =\tau \circ \wp$ for all $\wp , \tau \in [0, 1]$;

• 2. ∘ is continuous;

• 3. $\wp \circ 0=0$ for all $\wp \in [0, 1]$;

• 4. $(\wp \circ \tau )\circ \mu =\wp \circ (\tau \circ \mu )$ for all $\wp , \tau , \mu \in [0, 1]$;

• 5. If $\wp \leq \mu$ and $\mu \leq \mathfrak{d}$ with $\wp , \tau , \mu , \mathfrak{d}\in [0, 1]$, then $\wp \circ \tau \leq \mu \circ \mathfrak{d}$.

Definition 3

[28] Let $\varDelta \neq \emptyset$ and $\wp , \Gamma ,\eta \colon \varDelta \times \varDelta \rightarrow [1, + \infty )$ 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 a neutrosophic rectangular triple controlled metric on Δ if for any $\psi , \lambda \in \varDelta$ and all distinct $\upsilon , \varGamma \in \varDelta \backslash \{\psi ,\lambda \}$, the following conditions are satisfied:

• 1. $\varOmega (\psi , \varGamma , \vartheta )+\varPhi (\psi , \varGamma , \vartheta )+\varLambda (\psi , \varGamma , \vartheta )\leq 3$;

• 2. $\varOmega (\psi , \varGamma , \vartheta )>0$;

• 3. $\varOmega (\psi , \varGamma , \vartheta )=1$ for all $\vartheta >0$ if and only if $\psi =\varGamma$;

• 4. $\varOmega (\psi , \varGamma , \vartheta )=\varOmega (\varGamma , \psi , \vartheta )$;

• 5. $\varOmega (\psi , \lambda , \vartheta +\varpi +\varsigma )\geq \varOmega (\psi , \varGamma , \frac{\vartheta}{\wp (\psi , \varGamma )} )\ast \varOmega ( \varGamma , \upsilon , \frac{\varpi}{\Gamma (\varGamma , \upsilon )} )\ast \varOmega (\upsilon , \lambda , \frac{\varsigma}{\eta (\upsilon , \lambda )} )$;

• 6. $\varOmega (\psi , \varGamma , \cdot )\colon (0, +\infty ) \rightarrow [0, 1]$ is continuous and $\lim_{\vartheta \rightarrow +\infty}\varOmega (\psi , \varGamma , \vartheta )=1$;

• 7. $\varPhi (\psi , \varGamma , \vartheta )<1$;

• 8. $\varPhi (\psi , \varGamma , \vartheta )=0$ for all $\vartheta >0$ if and only if $\psi =\varGamma$;

• 9. $\varPhi (\psi , \varGamma , \vartheta )=\varPhi (\varGamma , \psi , \vartheta )$;

• 10. $\varPhi (\psi , \lambda , \vartheta +\varpi +\varsigma )\leq \varPhi (\psi , \varGamma , \frac{\vartheta}{\wp (\psi , \varGamma )} )\circ \varPhi ( \varGamma , \upsilon , \frac{\varpi}{\Gamma (\varGamma , \upsilon )} )\circ \varPhi (\upsilon , \lambda , \frac{\varsigma}{\eta (\upsilon , \lambda )} )$;

• 11. $\varPhi (\psi , \varGamma , \cdot )\colon (0, +\infty )\rightarrow [0, 1]$ is continuous and $\lim_{\vartheta \rightarrow +\infty}\varPhi (\psi , \varGamma , \vartheta )=0$;

• 12. $\varLambda (\psi , \varGamma , \vartheta )<1$;

• 13. $\varLambda (\psi , \varGamma , \vartheta )=0$ for all $\vartheta >0$ if and only if $\psi =\varGamma$;

• 14. $\varLambda (\psi , \varGamma , \vartheta )=\varLambda (\varGamma , \psi , \vartheta )$;

• 15. $\varLambda (\psi , \lambda , \vartheta +\varpi +\varsigma )\leq \varLambda (\psi , \varGamma , \frac{\vartheta}{\wp (\psi , \varGamma )} )\circ \varLambda (\varGamma , \upsilon , \frac{\varpi}{\Gamma (\varGamma , \upsilon )} )\circ \varLambda (\upsilon , \lambda , \frac{\varpi}{\eta (\upsilon , \lambda )} )$;

• 16. $\varLambda (\psi , \varGamma , \cdot )\colon (0, +\infty ) \rightarrow [0, 1]$ is continuous and $\lim_{\vartheta \rightarrow +\infty}\varLambda (\psi , \varGamma , \vartheta )=0$;

• 17. 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 a neutrosophic rectangular triple controlled metric space.

3 Main results

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 ∇.

4 Application to fractional differential equations

This section is devoted to finding a solution of the following fractional differential equation consisting of Caputo fractional derivative. Further details can be found in [29].

$$\begin{aligned} \mathcal{D}^{\delta}_{0+}\psi (\tau )+\mathfrak{g} \bigl(\tau , \psi (\tau ) \bigr)=0, \quad 0< \tau < 1, \end{aligned}$$

(12)

where $1<\delta \leq 2$, $\psi (0)+\psi{'}(0)=0$, $\psi (1)+\psi{'}(1)=0$ are the boundary conditions with $\mathfrak{g}\colon [0, 1]\times [0, \infty )\rightarrow [0, \infty )$ being continuous. Define Ω, Φ and Λ given by

$$\begin{aligned} &\varOmega \bigl(\psi (\tau ), \varGamma (\tau ), \vartheta \bigr)=\sup _{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\vartheta}{\vartheta + \vert \psi (\tau )-\varGamma (\tau ) \vert ^{2}} \quad \text{for all } \psi , \varGamma \in \varDelta \text{ and } \vartheta >0, \\ &\varPhi \bigl(\psi (\tau ), \varGamma (\tau ), \vartheta \bigr)=1-\sup _{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\vartheta}{\vartheta + \vert \psi (\tau )-\varGamma (\tau ) \vert ^{2}} \quad \text{for all } \psi , \varGamma \in \varDelta \text{ and } \vartheta >0 \end{aligned}$$

and

$$\begin{aligned} \varLambda \bigl(\psi (\tau ), \varGamma (\tau ), \vartheta \bigr)&=\sup _{ \tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{ \vert \psi (\tau )-\varGamma (\tau ) \vert ^{2}}{\vartheta} \quad \text{for all } \psi , \varGamma \in \varDelta \text{ and } \vartheta >0, \end{aligned}$$

with continuous t-norm and continuous t-co-norm defined by $\hat{e}\ast \breve{a}=\hat{e}\cdot \breve{a}$ and $\hat{e}\circ \breve{a}=\max \{\hat{e}, \breve{a}\}$, respectively. Define $\wp \colon \varDelta \times \varDelta \rightarrow [1, +\infty )$ as

$$\begin{aligned} \wp (\psi , \varGamma )&=\psi +\varGamma +1. \end{aligned}$$

Then $(\varDelta , \varOmega , \varPhi , \varLambda , \ast , \circ )$ is a complete ENRMS. Note that $\psi \in \varDelta$ solves (12) whenever $\psi \in \varDelta$ is the solution of

$$\begin{aligned} \psi (\tau )={}&\frac{1}{\Gamma (\delta )} \int _{0}^{1}(1-\zeta )^{ \delta -1}(1-\tau ) \mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)\,\mathit{d} \zeta \\ &{}+\frac{1}{\Gamma (\delta -1)} \int _{0}^{1}(1-\zeta )^{\delta -2}(1- \tau ) \mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)\,\mathit{d}\zeta \\ &{}+ \frac{1}{\Gamma (\delta )} \int _{0}^{\tau}(\tau -\zeta )^{\delta -1} \mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)\,\mathit{d}\zeta . \end{aligned}$$

Theorem 3

Consider the operator $\nabla \colon \varDelta \rightarrow \varDelta$ as

$$\begin{aligned} \nabla \psi (\tau )={}&\frac{1}{\Gamma (\delta )} \int _{0}^{1}(1-\zeta )^{ \delta -1}(1-\tau ) \mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)\,\mathit{d} \zeta \\ &{}+\frac{1}{\Gamma (\delta -1)} \int _{0}^{1}(1-\zeta )^{\delta -2}(1- \tau ) \mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)\,\mathit{d}\zeta \\ &{} + \frac{1}{\Gamma (\delta )} \int _{0}^{\tau}(\tau -\zeta )^{\delta -1} \mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)\,\mathit{d}\zeta . \end{aligned}$$

Suppose that the conditions:

• (i) for all $\psi , \varGamma \in \varDelta$, $\mathfrak{g}\colon [0, 1]\times [0, \infty )\rightarrow [0, \infty )$ and $\theta \in (0,1)$ satisfies

  $$\begin{aligned} \bigl\vert \mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)-\mathfrak{g} \bigl( \zeta , \varGamma ( \zeta ) \bigr) \bigr\vert \leq \sqrt{\theta} \bigl\vert \psi (\zeta )-\varGamma (\zeta ) \bigr\vert ; \end{aligned}$$
• (ii)
  $$\begin{aligned} \sup_{\tau \in (0, 1)} \biggl\vert \frac{1-\tau}{\Gamma (\delta +1)}+ \frac{1-\tau}{\Gamma (\delta )}+ \frac{\tau ^{\delta}}{\Gamma (\delta +1)} \biggr\vert ^{2}=\eta < 1, \end{aligned}$$

hold. Then equation (1) has a unique solution.

Proof

Let $\psi , \varGamma \in \varDelta$ and consider

$$\begin{aligned} \bigl\vert \nabla \psi (\tau )-\nabla \varGamma (\tau ) \bigr\vert ^{2}={}& \biggl\vert \frac{1}{\Gamma (\delta )} \int _{0}^{1}(1-\zeta )^{\delta -1}(1-\tau ) \bigl( \mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)-\mathfrak{g} \bigl(\zeta , \varGamma ( \zeta ) \bigr) \bigr)\,\mathit{d}\zeta \\ &{}+\frac{1}{\Gamma (\delta -1)} \int _{0}^{1}(1-\zeta )^{\delta -2}(1- \tau ) \bigl(\mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)-\mathfrak{g} \bigl(\zeta , \varGamma (\zeta ) \bigr) \bigr)\,\mathit{d}\zeta \\ &{}+\frac{1}{\Gamma (\delta )} \int _{0}^{\tau}(\tau -\zeta )^{\delta -1} \bigl( \mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)-\mathfrak{g} \bigl(\zeta , \varGamma ( \zeta ) \bigr) \bigr)\,\mathit{d}\zeta \biggr\vert ^{2} \\ \leq{}& \biggl(\frac{1}{\Gamma (\delta )} \int _{0}^{1}(1-\zeta )^{ \delta -1}(1-\tau ) \bigl\vert \bigl(\mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)- \mathfrak{g} \bigl(\zeta , \varGamma (\zeta ) \bigr) \bigr) \bigr\vert \,\mathit{d}\zeta \\ &{}+\frac{1}{\Gamma (\delta -1)} \int _{0}^{1}(1-\zeta )^{\delta -2}(1- \tau ) \bigl\vert \bigl(\mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)-\mathfrak{g} \bigl(\zeta , \varGamma (\zeta ) \bigr) \bigr) \bigr\vert \,\mathit{d}\zeta \\ &{}+\frac{1}{\Gamma (\delta )} \int _{0}^{\tau}(\tau -\zeta )^{\delta -1} \bigl\vert \bigl(\mathfrak{g} \bigl(\zeta , \psi (\zeta ) \bigr)-\mathfrak{g} \bigl( \zeta , \varGamma (\zeta ) \bigr) \bigr) \bigr\vert \,\mathit{d}\zeta \biggr)^{2} \\ \leq{}& \biggl(\frac{1}{\Gamma (\delta )} \int _{0}^{1}(1-\zeta )^{ \delta -1}(1-\tau ) \theta ^{\frac{1}{2}} \bigl\vert \psi (\zeta )-\varGamma ( \zeta ) \bigr\vert \,\mathit{d}\zeta \\ &{}+\frac{1}{\Gamma (\delta -1)} \int _{0}^{1}(1-\zeta )^{\delta -2}(1- \tau ) \theta ^{\frac{1}{2}} \bigl\vert \psi (\zeta )-\varGamma (\zeta ) \bigr\vert \,\mathit{d}\zeta \\ &{}+\frac{1}{\Gamma (\delta )} \int _{0}^{\tau}(\tau -\zeta )^{\delta -1} \theta ^{\frac{1}{2}} \bigl\vert \psi (\zeta )-\varGamma (\zeta ) \bigr\vert \,\mathit{d} \zeta \biggr)^{2} \\ ={}&\theta \bigl\vert \psi (\tau )-\varGamma (\tau ) \bigr\vert ^{2} \biggl( \frac{1}{\Gamma (\delta )} \int _{0}^{1}(1-\zeta )^{\delta -1}(1-\tau ) \,\mathit{d}\zeta \\ &{}+\frac{1}{\Gamma (\delta -1)} \int _{0}^{1}(1-\zeta )^{\delta -2}(1- \tau ) \,\mathit{d}\zeta +\frac{1}{\Gamma (\delta )} \int _{0}^{\tau}( \tau -\zeta )^{\delta -1} \,\mathit{d}\zeta \biggr)^{2} \\ ={}&\theta \bigl\vert \psi (\tau )-\varGamma (\tau ) \bigr\vert ^{2} \biggl( \frac{1-\tau}{\Gamma (\delta +1)}+\frac{1-\tau}{\Gamma (\delta )}+ \frac{\tau ^{\delta}}{\Gamma (\delta +1)} \biggr)^{2} \\ \leq{}& \theta \bigl\vert \psi (\tau )-\varGamma (\tau ) \bigr\vert ^{2}\sup_{\tau \in (0, 1)} \biggl(\frac{1-\tau}{\Gamma (\delta +1)}+ \frac{1-\tau}{\Gamma (\delta )}+ \frac{\tau ^{\delta}}{\Gamma (\delta +1)} \biggr)^{2} \\ ={}&\eta \theta \bigl\vert \psi (\tau )-\varGamma (\tau ) \bigr\vert ^{2} \\ \leq{}& \theta \bigl\vert \psi (\tau )-\varGamma (\tau ) \bigr\vert ^{2}. \end{aligned}$$

So, we have

$$\begin{aligned} \bigl\vert \nabla \psi (\tau )-\nabla \varGamma (\tau ) \bigr\vert ^{2}\leq \theta \bigl\vert \psi (\tau )-\varGamma (\tau ) \bigr\vert ^{2}. \end{aligned}$$

Now, for all $\psi , \varGamma \in \varDelta$, we deduce

$$\begin{aligned} &\varOmega \bigl(\nabla \psi (\tau ), \nabla \varGamma (\tau ), \theta \vartheta \bigr)=\sup_{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\theta \vartheta}{\theta \vartheta + \vert \nabla \psi (\tau )-\nabla \varGamma (\tau ) \vert ^{2}} \\ &\hphantom{\varOmega \bigl(\nabla \psi (\tau ), \nabla \varGamma (\tau ), \theta \vartheta \bigr)}\geq \sup_{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\theta \vartheta}{\theta \vartheta +\theta \vert \psi (\tau )-\varGamma (\tau ) \vert ^{2}} \\ &\hphantom{\varOmega \bigl(\nabla \psi (\tau ), \nabla \varGamma (\tau ), \theta \vartheta \bigr)}=\sup_{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\vartheta}{\vartheta + \vert \psi (\tau )-\varGamma (\tau ) \vert ^{2}} \\ &\hphantom{\varOmega \bigl(\nabla \psi (\tau ), \nabla \varGamma (\tau ), \theta \vartheta \bigr)}=\varOmega \bigl(\psi (\tau ), \varGamma (\tau ), \vartheta \bigr), \\ &\varPhi \bigl(\nabla \psi (\tau ), \nabla \varGamma (\tau ), \theta \vartheta \bigr)=1-\sup_{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\theta \vartheta}{\theta \vartheta + \vert \nabla \psi (\tau )-\nabla \varGamma (\tau ) \vert ^{2}} \\ &\hphantom{\varPhi \bigl(\nabla \psi (\tau ), \nabla \varGamma (\tau ), \theta \vartheta \bigr)}\leq 1-\sup_{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\theta \vartheta}{\theta \vartheta +\theta \vert \psi (\tau )-\varGamma (\tau ) \vert ^{2}} \\ &\hphantom{\varPhi \bigl(\nabla \psi (\tau ), \nabla \varGamma (\tau ), \theta \vartheta \bigr)}=1-\sup_{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\vartheta}{\vartheta + \vert \psi (\tau )-\varGamma (\tau ) \vert ^{2}} \\ &\hphantom{\varPhi \bigl(\nabla \psi (\tau ), \nabla \varGamma (\tau ), \theta \vartheta \bigr)}=\varPhi \bigl(\psi (\tau ), \varGamma (\tau ), \vartheta \bigr) \end{aligned}$$

and

$$\begin{aligned} \varLambda \bigl(\nabla \psi (\tau ), \nabla \varGamma (\tau ), \theta \vartheta \bigr)&=\sup_{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\theta \vartheta}{\theta \hat{r}+ \vert \nabla \psi (\tau )-\nabla \varGamma (\tau ) \vert ^{2}} \\ &\leq \sup_{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\theta \hat{r}}{\theta \hat{r}+\theta \vert \psi (\tau )-\varGamma (\tau ) \vert ^{2}} \\ &=\sup_{\tau \in [\mathfrak{c}, \mathfrak{a}]} \frac{\hat{r}}{\hat{r}+ \vert \psi (\tau )-\varGamma (\tau ) \vert ^{2}} \\ &=\varOmega \bigl(\psi (\tau ), \varGamma (\tau ), \hat{r} \bigr). \end{aligned}$$

As a result, all of the conditions of Theorem 1 are satisfied and operator ∇ has a unique fixed point. □

Example 3

According to equation (12), we consider

$$ \textstyle\begin{cases} \mathcal{D}^{\delta}_{0+} \psi ( \tau ) + \frac{\sqrt{6} \ln ( \tau +1)\cos (\psi ( \tau ))}{\tau ^{2} +1 } = 0,& 0 \leq \tau \leq 1, \\ \psi (0)+\psi{'}(0)=0, \qquad \psi (1)+\psi{'}(1)=0, & \end{cases} $$

(13)

with three cases $\delta = \{ \frac{25}{19}, \frac{7}{4}, \frac{35}{19} \}$, where $\mathfrak{g} ( \tau , \psi ( \tau ))= \frac{\sqrt{6} \ln ( \tau +1)\cos (\psi ( \tau ))}{\tau ^{2} +1 }$. Then, for ψ, $\varGamma \in \varDelta =[0,1]$, we have

$$\begin{aligned} \bigl\vert \mathfrak{g} \bigl( \tau , \psi ( \tau ) \bigr) - \mathfrak{g} \bigl( \tau , \varGamma ( \tau ) \bigr) \bigr\vert ={}& \biggl\vert \frac{\sqrt{6} \ln ( \tau +1)\cos (\psi ( \tau ))}{ \tau ^{2} +1 } \\ &{} - \frac{\sqrt{6} \ln ( \tau +1) \cos (\varGamma ( \tau ) )}{\tau ^{2} +1 } \biggr\vert \\ ={}& \biggl\vert \frac{\sqrt{6} \ln ( \tau +1)}{ \tau ^{2} +1 } \biggr\vert \bigl\vert \cos \bigl(\psi ( \tau ) \bigr)- \cos \bigl(\varGamma ( \tau ) \bigr) \bigr\vert \\ \leq{}& \sqrt{6} \ln 2 \bigl\vert \psi ( \tau )- \varGamma ( \tau ) \bigr\vert = \sqrt{\theta} \bigl\vert \psi ( \tau )- \varGamma ( \tau ) \bigr\vert , \end{aligned}$$

where $\theta = (\sqrt{6} \ln 2 )^{2}$. Therefore, all the conditions of Theorem 3 are true. Hence, ∇ has a unique fixed point.

5 Conclusion

This paper introduced the concept of ENRMS, as well as various new types of fixed point theorems that can be proved in this novel environment. Furthermore, we offered a non-trivial example to show that the proposed solutions are viable. We have complemented our work with an application that shows how the developed approach outperforms the literature-based methods. It is an interesting open problem to prove a coupled fixed point under this space.