Solving Fractional Boundary Value Problems with Nonlocal Mixed Boundary Conditions Using Covariant JS-Contractions

Abstract

This paper investigates the existence, uniqueness, and symmetry of solutions for $\Phi$–Atangana–Baleanu fractional differential equations of order $\mu \in (1,2]$ under mixed nonlocal boundary conditions. This is achieved through the use of covariant and contravariant $JS$-contractions within a generalized framework of a sequential extended bipolar parametric metric space. As a consequence, we obtain the results on covariant and contravariant Ćirić, Chatterjea, Kannan, and Reich contractions as corollaries. Additionally, we substantiate our fixed-point findings with specific examples and derive similar results in the setting of sequential extended fuzzy bipolar metric space.

1. Introduction

The notion of a metric space has been extensively generalized in the academic literature. One of the latest advances in this domain is the concept of a parametric space, as introduced by [1], which has subsequently been generalized in various manners, such as b-parametric metric space and extended parametric b-metric space; see [2,3,4,5]. On the other hand, Mutlu and Gürdal [6] defined bipolar metric spaces that formalize different types of distance. Many authors have published articles on fixed points in generalizations of bipolar metrics; see [7,8,9]. In this paper, we define a sequential extended bipolar parametric metric space (SEBPMS), which generalizes extended bipolar parametric b-metric space. Also, we generalize the work of [10,11] by defining covariant and contravariant $JS$-contractions in the SEBPMS setting and proving certain fixed-point theorems and many induced corollaries. By delineating a $\Omega$-triangular fuzzy bipolar set, we generalize our results to a new concept of sequential extended fuzzy bipolar metric spaces (SEFBMSs). Ultimately, inspired by studies of [12,13], we investigate the existence and uniqueness of solutions for fractional derivatives (FDs) of $\Phi \text{-}\mathcal{A}\mathcal{B}\mathcal{C}$ of order $1<\mu \le 2$ in the framework of non-local mixed boundary conditions (NMBCs).

Definition 1

([14]). Let $\mathfrak{P}$ and $\mathfrak{Q}$ be two nonempty sets and let $\mathfrak{S}:\mathfrak{P}\times \mathfrak{Q}\to [0,\infty ]$ be a function. Define the following sets:

$$\begin{array}{c}\hfill {\mathfrak{C}}_L(\mathfrak{P},\mathfrak{S},\mathfrak{q})=\{\left\{{\mathfrak{p}}_n\right\}\subset \mathfrak{P}:\underset{n\to \infty }{lim}\mathfrak{S}({\mathfrak{p}}_n,\mathfrak{q})=0\};\\ \hfill {\mathfrak{C}}_R(\mathfrak{Q},\mathfrak{S},\mathfrak{p})=\{\left\{{\mathfrak{q}}_n\right\}\subset \mathfrak{Q}:\underset{n\to \infty }{lim}\mathfrak{S}(\mathfrak{p},{\mathfrak{q}}_n)=0\},\end{array}$$

where $\mathfrak{S}$ meets the following criteria:

• ($\mathfrak{S}$1) $\mathfrak{S}(\mathfrak{p},\mathfrak{q})=0\Rightarrow \mathfrak{p}=\mathfrak{q}\in \mathfrak{P}\cap \mathfrak{Q}$;
• ($\mathfrak{S}$2) $\mathfrak{S}(\mathfrak{p},\mathfrak{q})=\mathfrak{S}(\mathfrak{q},\mathfrak{p}),\forall \mathfrak{p}\in \mathfrak{P},\mathfrak{q}\in \mathfrak{Q}$;
• ($\mathfrak{S}$3) for some $k>0,\forall {\mathfrak{p}}_1,{\mathfrak{p}}_2\in \mathfrak{P}$ and ${\mathfrak{q}}_1,{\mathfrak{q}}_2\in \mathfrak{Q}$; it follows that

  $$\begin{array}{c}\hfill \mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_2)\le k\underset{n\to \infty }{lim\; sup}[\mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_1)+\mathfrak{S}({\mathfrak{p}}_n,{\mathfrak{q}}_1)],\forall \left\{{\mathfrak{p}}_n\right\}\in {\mathfrak{C}}_L(\mathfrak{P},\mathfrak{S},{\mathfrak{q}}_2);\\ \hfill \mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_2)\le k\underset{n\to \infty }{lim\; sup}[\mathfrak{S}({\mathfrak{p}}_2,{\mathfrak{q}}_2)+\mathfrak{S}({\mathfrak{p}}_2,{\mathfrak{q}}_n)],\forall \left\{{\mathfrak{q}}_n\right\}\in {\mathfrak{C}}_R(\mathfrak{Q},\mathfrak{S},{\mathfrak{p}}_1).\end{array}$$

Hence the triplet $(\mathfrak{P},\mathfrak{Q},\mathfrak{S})$ is termed as a sequential bipolar metric space.

• Parvaneh et al., in their work [10], defined ${\Theta }^{\prime }$ as the set of control functions $\Theta :(0,\infty )\to (1,\infty )$ that meet the criteria:
  • ($\Theta$1) $\Theta$ is continuous and strictly increasing;
  • ($\Theta$2) for each sequence $\left\{{\varrho }_n\right\}\subseteq (0,\infty )$, ${lim}_{n\to \infty }\Theta \left({\varrho }_n\right)=1\iff {lim}_{n\to \infty }{\varrho }_n=0.$
• Parvaneh and Ghoncheh in [15] delineated $\tilde{\Omega }$ as the class of functions $\Omega :[0,\infty )\to [0,\infty )$ that are strictly increasing and continuous, satisfying the condition ${\Omega }^{-1}\left(s\right)\le s\le \Omega \left(s\right)$.

Definition 2.

Assume that $\mathfrak{P}$ and $\mathfrak{Q}$ are two nonempty sets, $\Omega \in \tilde{\Omega }$ and $\varrho :\mathfrak{P}\times \mathfrak{Q}\times (0,\infty )\to [0,\infty )$ is said to be an extended parametric bipolar b-metric space on $\mathfrak{P}\cup \mathfrak{Q}$ provided that $\forall \tau >0$:

• ($\varrho$1) $\varrho (\mathfrak{p},\mathfrak{q},\tau )=0\iff \mathfrak{p}=\mathfrak{q}$;
• ($\varrho$2) $\varrho (\mathfrak{p},\mathfrak{q},\tau )=\varrho (\mathfrak{q},\mathfrak{p},\tau ),\forall (\mathfrak{p},\mathfrak{q})\in \mathfrak{P}\cap \mathfrak{Q}$;
• ($\varrho$3) $\varrho ({\mathfrak{p}}_1,{\mathfrak{q}}_2,\tau )\le \Omega \left(\varrho ({\mathfrak{p}}_1,{\mathfrak{q}}_1,\tau )+\varrho ({\mathfrak{p}}_2,{\mathfrak{q}}_1,\tau )+\varrho ({\mathfrak{p}}_2,{\mathfrak{q}}_2,\tau )\right),\forall ({\mathfrak{p}}_1,{\mathfrak{q}}_2)\ne ({\mathfrak{p}}_2,{\mathfrak{q}}_1)\in \mathfrak{P}\times \mathfrak{Q}.$

Thus the quadruple $(\mathfrak{P},\mathfrak{Q},\varrho ,\Omega )$ defines what is known as an extended parametric bipolar b-metric space, (abbreviated as EPBbMS), governed by the control function Ω.

Remark 1.

It is important to mention the following remarks:

• If $\Omega \left(\mathfrak{p}\right)=s\mathfrak{p}$, the EPBbMS becomes parametric bipolar b-metric spaces (to sum up, PBbMSs).
• If $\Omega \left(\mathfrak{p}\right)=\mathfrak{p}$, it will revert back to a parametric bipolar metric space (abbreviated as PBMS).

2. Main Results

In this section, we present the concept of sequential extended bipolar parametric metric spaces (SEBPMSs), establish several fixed point theorems through $JS$-contractions, and obtain as corollaries the results on covariant and contravariant Ćirić, Chatterjea, Kannan, and Reich contractions. Additionally, we articulate and demonstrate fixed-point theorems within two categories of generalized metric spaces. Consider $\mathfrak{P}$ and $\mathfrak{Q}$ as two nonempty sets and define $\breve{\mathfrak{S}}:\mathfrak{P}\times \mathfrak{Q}\times (0,\infty )\to [0,\infty )$ as a function. Given $\mathfrak{p}\in \mathfrak{P}$ and $\mathfrak{q}\in \mathfrak{Q}$, we proceed to define the following sets:

$$\begin{array}{cc}\hfill {\tilde{\mathcal{C}}}_L(\mathfrak{P},\breve{\mathfrak{S}},\mathfrak{q})& =\{\left\{{\mathfrak{p}}_n\right\}\subset \mathfrak{P}:\underset{n\to \infty }{lim}\breve{\mathfrak{S}}({\mathfrak{p}}_n,\mathfrak{q},\tau )=0,\forall \tau >0\},\hfill \\ \hfill {\tilde{\mathcal{C}}}_R(\mathfrak{Q},\breve{\mathfrak{S}},\mathfrak{p})& =\{\left\{{\mathfrak{q}}_n\right\}\subset \mathfrak{Q}:\underset{n\to \infty }{lim}\breve{\mathfrak{S}}(\mathfrak{p},{\mathfrak{q}}_n,\tau )=0,\forall \tau >0\}.\hfill \end{array}$$

Definition 3.

Let $\mathfrak{P}$ and $\mathfrak{Q}$ be two nonempty sets and $\breve{\mathfrak{S}}:\mathfrak{P}\times \mathfrak{Q}\times (0,\infty )\to [0,\infty )$ be a function satisfying the following conditions for all $\tau >0$:

• ($\breve{\mathfrak{S}}$1) $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )=0$ implies $\mathfrak{p}=\mathfrak{q}\in \mathfrak{P}\cap \mathfrak{Q}$;
• ($\breve{\mathfrak{S}}$2) $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )=\breve{\mathfrak{S}}(\mathfrak{q},\mathfrak{p},\tau )$,$\forall \mathfrak{p},\mathfrak{q}\in \mathfrak{P}\cap \mathfrak{Q}$;
• ($\breve{\mathfrak{S}}$3) if there exists $\Omega \in \tilde{\Omega }$, such that for every distinct $({\mathfrak{p}}_1,{\mathfrak{q}}_1),({\mathfrak{p}}_2,{\mathfrak{q}}_2)\in \mathfrak{P}\times \mathfrak{Q}$,

  $$\begin{array}{c}\hfill \breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_2,\tau )\le \Omega \left(\underset{n\to \infty }{lim\; sup}[\breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_1,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_1,\tau )]\right),\forall \left\{{\mathfrak{p}}_n\right\}\in {\tilde{\mathcal{C}}}_L(\mathfrak{P},\breve{\mathfrak{S}},{\mathfrak{q}}_2);\\ \hfill \breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_2,\tau )\le \Omega \left(\underset{n\to \infty }{lim\; sup}[\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_2,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_n,\tau )]\right),\forall \left\{{\mathfrak{q}}_n\right\}\in {\tilde{\mathcal{C}}}_R(\mathfrak{Q},\breve{\mathfrak{S}},{\mathfrak{p}}_1).\end{array}$$

Then $\breve{\mathfrak{S}}$ is referred to as a sequential extended bipolar parametric metric, and the triple $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ is termed as a sequential extended bipolar parametric metric space (SEBPMS).

Definition 4.

Let $\mathfrak{Y}$ be a function from $({\mathfrak{P}}_1{\mathfrak{Q}}_1,{\breve{\mathfrak{S}}}_1)$ to $({\mathfrak{P}}_2,{\mathfrak{Q}}_2,{\breve{\mathfrak{S}}}_2)$; if $\mathfrak{Y}\left({\mathfrak{P}}_1\right)\subseteq {\mathfrak{P}}_2$ and $\mathfrak{Y}\left({\mathfrak{Q}}_1\right)\subseteq {\mathfrak{Q}}_2$, then $\mathfrak{Y}$ is called covariant mapping, and we denote this as $\mathfrak{Y}:({\mathfrak{P}}_1,{\mathfrak{Q}}_1,{\breve{\mathfrak{S}}}_1)\rightrightarrows ({\mathfrak{P}}_2,{\mathfrak{Q}}_2,{\breve{\mathfrak{S}}}_2).$ In contrast, if $\mathfrak{Y}\left({\mathfrak{P}}_1\right)\subseteq {\mathfrak{Q}}_2$ and $\mathfrak{Y}\left({\mathfrak{Q}}_1\right)\subseteq {\mathfrak{P}}_2$, then $\mathfrak{Y}$ is called contravariant mapping, and we denote this as $\mathfrak{Y}:({\mathfrak{P}}_1,{\mathfrak{Q}}_1,\breve{\mathfrak{S}})\rightleftharpoons ({\mathfrak{P}}_2,{\mathfrak{Q}}_2,{\breve{\mathfrak{S}}}_2).$

Definition 5.

Let $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ be an SEBPMS.

(i) 

The points of $\mathfrak{P},\mathfrak{Q},$ and $\mathfrak{P}\cap \mathfrak{Q}$ are said to be left points, right points, and central points, respectively, and sequences of left points, right points, and central points are called left sequences, right sequences, and bisequences, respectively;

(ii) 

If both $\left\{{\mathfrak{p}}_n\right\}$ and $\left\{{\mathfrak{q}}_n\right\}$ converge to the same central point, then $\left\{({\mathfrak{p}}_n,{\mathfrak{q}}_n)\right\}$ is said to be a biconvergent;

(iii) 

(a) A bisequence $\left\{({\mathfrak{p}}_n,{\mathfrak{q}}_n)\right\}$ is termed as a Cauchy bisequence if there exists $N\in \mathbb{N}$ such that ${lim}_{n,m\to \infty }\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_m,\tau )=0,\forall m,n\ge N,\tau >0$;

(b) 

If every Cauchy bisequence is biconvergent, then SEBPM is said to be complete.

Proposition 1.

Every extended parametric bipolar b-metric space $(\mathfrak{P},\mathfrak{Q},\varrho ,\Omega )$ is a sequential extended bipolar parametric metric space.

Proof. 

It is evident that conditions ($\breve{\mathfrak{S}}1$) and ($\breve{\mathfrak{S}}2$) as specified in Definition 3 are met. Let ${\mathfrak{p}}_1,{\mathfrak{p}}_2\in \mathfrak{P},{\mathfrak{q}}_1,{\mathfrak{q}}_2\in \mathfrak{Q}$, $\left\{{\mathfrak{p}}_n\right\}\in {\tilde{\mathcal{C}}}_L(\mathfrak{P},\breve{\mathfrak{S}},{\mathfrak{q}}_2)$ and $\left\{{\mathfrak{q}}_n\right\}\in {\tilde{\mathcal{C}}}_R(\mathfrak{Q},\breve{\mathfrak{S}},{\mathfrak{p}}_1)$.

$$\begin{array}{cc}\hfill \breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_2,\tau )& \le \Omega \left(\breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_1,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_1,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_2,\tau )\right)\hfill \\ & \le \Omega \left(\underset{n\to \infty }{lim\; sup}\left[\breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_1,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_1,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_2,\tau )\right]\right)\hfill \\ & \le \Omega \left(\underset{n\to \infty }{lim\; sup}\left[\breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_1,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_1,\tau )\right]\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_2,\tau )& \le \Omega \left(\breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_n,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_n,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_2,\tau )\right)\hfill \\ & \le \Omega (\underset{n\to \infty }{lim\; sup}\left[\breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_n,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_n,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_2,\tau )\right])\hfill \\ & =\Omega \left(\underset{n\to \infty }{lim\; sup}[\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_n,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_2,\tau )]\right).\hfill \end{array}$$

for all $n\ge 1,\tau >0$. Hence, ($\breve{\mathfrak{S}}3$) is satisfied. □

Example 1.

Let $\mathfrak{S}:\mathfrak{P}\times \mathfrak{Q}\to [0,\infty )$ be an SBM function and $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )=e^{\tau \mathfrak{S}(\mathfrak{p},\mathfrak{q})}-1$. We show that $\breve{\mathfrak{S}}$ is an SEBPMS with $\Omega \left(\sigma \right)=e^{k\sigma }-1$, for some $k>0$. Obviously, conditions ($\breve{\mathfrak{S}}1)$ and $\left(\breve{\mathfrak{S}}2\right)$ of Definition 3 are satisfied. On the other hand, for ${\mathfrak{p}}_1,{\mathfrak{p}}_2\in \mathfrak{P}$, ${\mathfrak{q}}_1,{\mathfrak{q}}_2\in \mathfrak{Q}$, and $\forall \tau >0,$

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_2,t)& =e^{\tau {\mathfrak{S}}_b({\mathfrak{p}}_1,{\mathfrak{q}}_2)}-1\hfill \\ \hfill & \le e^{\tau k{lim\; sup}_{n\to \infty }[\mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_1)+{\mathfrak{S}}_b({\mathfrak{p}}_n,{\mathfrak{q}}_1)]}-1,\hfill \\ & \le e^{e^{k{lim\; sup}_{n\to \infty }[\tau \mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_1)-1+\tau \mathfrak{S}({\mathfrak{p}}_n,{\mathfrak{q}}_1)-1]}}-1\hfill \\ & =e^{k{lim\; sup}_{n\to \infty }[\breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_1,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_1,\tau )]}-1\hfill \\ & =\Omega \left(\underset{n\to \infty }{lim\; sup}[\breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_1,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_1,\tau )]\right),\hfill \end{array}\end{array}$$

$\forall \left\{{\mathfrak{p}}_n\right\}\in C_L(\mathfrak{P},\mathfrak{S},{\mathfrak{q}}_2)\Rightarrow \forall \left\{{\mathfrak{p}}_n\right\}\in {\tilde{\mathcal{C}}}_L(\mathfrak{P},\breve{\mathfrak{S}},{\mathfrak{q}}_2)$.

• Similarly,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \breve{\mathfrak{S}}({\mathfrak{p}}_1,{\mathfrak{q}}_2,t)& =e^{\tau \mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_2)}-1\hfill \\ \hfill & \le e^{\tau k{lim\; sup}_{n\to \infty }[\mathfrak{S}({\mathfrak{p}}_2,{\mathfrak{q}}_n)+\mathfrak{S}({\mathfrak{p}}_2,{\mathfrak{q}}_2)]}-1,\hfill \\ & \le e^{e^{k{lim\; sup}_{n\to \infty }[\tau \mathfrak{S}({\mathfrak{p}}_2,{\mathfrak{q}}_n)-1+\tau \mathfrak{S}({\mathfrak{p}}_2,{\mathfrak{q}}_2)-1]}}-1\hfill \\ & =e^{k{lim\; sup}_{n\to \infty }[\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_n,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_2,\tau )]}-1\hfill \\ & =\Omega \left(\underset{n\to \infty }{lim\; sup}[\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_n,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_2,{\mathfrak{q}}_2,\tau )]\right),\hfill \end{array}\end{array}$$

$\forall \left\{{\mathfrak{q}}_n\right\}\in C_R(\mathfrak{Q},\mathfrak{S},{\mathfrak{p}}_1)\Rightarrow \forall \left\{{\mathfrak{q}}_n\right\}\in {\tilde{\mathcal{C}}}_R(\mathfrak{Q},\breve{\mathfrak{S}},{\mathfrak{q}}_2)$. Thus, condition $\left(\breve{\mathfrak{S}}3\right)$ from Definition 3 is fulfilled, confirming that $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ qualifies as an SEBPMS.

Proposition 2.

Let $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ be an SEBPMS. If a central point $\mathfrak{p}$ is a limit of a (right or left) sequence $\left\{{\mathfrak{p}}_n\right\}$ such that $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{p},\tau )=0,\forall \tau >0$, then it is the unique limit of this sequence.

Proof. 

Let $\left\{{\mathfrak{p}}_n\right\}\subseteq \mathfrak{P}$ be a left sequence converging to $\mathfrak{p}$ in $\mathfrak{P}\cap \mathfrak{Q}$, with $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{p},\tau )=0$ for every $\tau >0$. Assuming that $\mathfrak{q}\in \mathfrak{Q}$ is another limit point of this sequence, the following is established.

$$\begin{array}{cc}\hfill \breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\le & \Omega \left(\underset{n\to \infty }{lim\; sup}[\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{p},\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_n,\mathfrak{p},\tau )]\right),\hfill \\ & =\Omega \left(0\right)=0.\hfill \end{array}$$

(1)

Hence, $\mathfrak{p}=\mathfrak{q}$. Therefore, $\mathfrak{p}$ is the unique limit of $\left\{{\mathfrak{q}}_n\right\}$. Analogously, $\left\{{\mathfrak{p}}_n\right\}\subseteq \mathfrak{P}$ is a right sequence converging to $\mathfrak{q}$ in $\mathfrak{P}\cap \mathfrak{Q}$, where $\breve{\mathfrak{S}}(\mathfrak{q},\mathfrak{q},\tau )=0,\forall \tau >0.$ □

Proposition 3.

In an SEBPMS $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$, every convergent Cauchy bisequence $\left\{({\mathfrak{p}}_n,{\mathfrak{q}}_n)\right\}$ is biconvergent to a central limit $\mathfrak{p}$ with $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{p},\tau )=0$.

Proof. 

Let $\left\{({\mathfrak{p}}_n,{\mathfrak{q}}_n)\right\}$ be a Cauchy bisequence that converges to $(\mathfrak{p},\mathfrak{q})\in \mathfrak{P}\times \mathfrak{Q}$ that ${\mathfrak{p}}_n\to \mathfrak{q}$ and ${\mathfrak{q}}_n\to \mathfrak{p}$ as $n\to \infty$. Then,

$$\begin{array}{cc}\hfill \breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\le & \Omega \left(\underset{m\to \infty }{lim\; sup}[\breve{\mathfrak{S}}(\mathfrak{p},{\mathfrak{q}}_m,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_m,{\mathfrak{q}}_m,\tau )]\right)\hfill \\ & =\Omega \left(0\right)=0.\hfill \end{array}$$

(2)

As m tends to infinity on the right-hand side of Equation (2); it can be deduced that $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )=0$, thereby indicating that $\mathfrak{p}$ is a central limit. Consequently, the bisequence $\left\{({\mathfrak{p}}_n,{\mathfrak{q}}_n)\right\}$ is biconvergent to $\mathfrak{p}$ with $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{p},\tau )=0$. □

Definition 6.

Let $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ be an SEBPMS. Then, $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightrightarrows \mathfrak{P}\cup \mathfrak{Q}$ is called covariant $JS$-contraction if there are $\Theta \in {\Theta }^{\prime }$ and ${\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4\ge 0$ with ${\sum }_{i=1}^4{\mathfrak{a}}_i<1$, such that the following inequality holds:

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )\right)& \le {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )\right))}^{{\mathfrak{a}}_2}\hfill \\ \hfill & \times {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )\right))}^{{\mathfrak{a}}_3}\times {\left(\Theta \left[{\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )+\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4},\hfill \end{array}$$

(3)

$\forall \mathfrak{p}\in \mathfrak{P},\mathfrak{q}\in \mathfrak{Q}$, $\tau >0.$

Theorem 1.

Let $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ denote a complete SEBPMS, and let $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightrightarrows \mathfrak{P}\cup \mathfrak{Q}$ be a covariant $JS$-contraction. Assume the existence of $({\mathfrak{p}}_0,{\mathfrak{q}}_0)\in \mathfrak{P}\times \mathfrak{Q}$ such that $\Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0))=sup\{\breve{\mathfrak{S}}({\mathfrak{p}}_i,{\mathfrak{q}}_j,\tau ):i,j=1,2,\cdots :{\mathfrak{q}}_n=\mathfrak{Y}{\mathfrak{q}}_{n-1},{\mathfrak{p}}_n=\mathfrak{Y}{\mathfrak{p}}_{n-1},\tau >0\}<\infty .$ Then, $\mathfrak{Y}$ has a unique fixed point in $\mathfrak{P}\cap \mathfrak{Q}$.

Proof. 

Consider $({\mathfrak{p}}_0,{\mathfrak{q}}_0)\in \mathfrak{P}\times \mathfrak{Q}$; we subsequently define

$$\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^n,({\mathfrak{p}}_0,{\mathfrak{q}}_0))=sup\{\breve{\mathfrak{S}}({\mathfrak{p}}_{n-1+i},{\mathfrak{q}}_{n-1+j},\tau ):i,j=1,2,\cdots ,\tau >0\},\forall n\ge 1.$$

Then, for $\forall i,j\in \mathbb{N}$, we obtain

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n+i},{\mathfrak{q}}_{n+j},\tau )\right)& =\Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}{\mathfrak{p}}_{n-1+i},\mathfrak{Y}{\mathfrak{q}}_{n-1+j},\tau )\right)\hfill \\ & \le {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n-1+i},{\mathfrak{q}}_{n-1+i},\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}{\mathfrak{p}}_{n-1+i},{\mathfrak{q}}_{n+j},\tau )\right))}^{{\mathfrak{a}}_2}\hfill \\ & \times {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n-1+i},\mathfrak{Y}{\mathfrak{q}}_{n-1+j},\tau )\right))}^{{\mathfrak{a}}_3}\hfill \\ & \times {\left(\Theta \left[{\displaystyle \frac{\mathfrak{S}(\mathfrak{Y}{\mathfrak{p}}_{n-1+i},{\mathfrak{q}}_{n+j},\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_{n-1+i},\mathfrak{Y}{\mathfrak{q}}_{n-1+j},\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}\hfill \\ & ={(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n-1+i},{\mathfrak{q}}_{n-1+i},\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n+i},{\mathfrak{q}}_{n+j},\tau )\right))}^{{\mathfrak{a}}_2}\hfill \\ & \times {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n-1+i},{\mathfrak{q}}_{n+j},\tau )\right))}^{{\mathfrak{a}}_3}\hfill \\ & \times {\left(\Theta \left[{\displaystyle \frac{\breve{\mathfrak{S}}({\mathfrak{p}}_{n+i},{\mathfrak{q}}_{n+j},\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_{n-1+i},{\mathfrak{q}}_{n+j},\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}\hfill \\ & \le (\Theta {(\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^n,({\mathfrak{p}}_0,{\mathfrak{q}}_0))))}^{{\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4}\mathrm{for}\mathrm{any}n\in \mathbb{N}.\hfill \end{array}$$

(4)

As $\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^n,({\mathfrak{p}}_0,{\mathfrak{q}}_0))\le \Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,{\mathfrak{q}}_0))<\infty ,$ from (4), it follows that $\forall n\ge 1$,

$$\begin{array}{cc}\hfill \Theta (\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^{n+1},({\mathfrak{p}}_0,{\mathfrak{q}}_0))& =\Theta (sup\breve{\mathfrak{S}}({\mathfrak{Y}}^{n+i}{\mathfrak{p}}_0,{\mathfrak{Y}}^{n+j}{\mathfrak{q}}_0,\tau ))\hfill \\ & \le {\left(\Theta (\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^n,({\mathfrak{p}}_0,{\mathfrak{q}}_0)))\right)}^{{\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4}\hfill \\ & \le {\left(\Theta (\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^{n-1},({\mathfrak{p}}_0,{\mathfrak{q}}_0)))\right)}^{{({\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4)}^2}\hfill \\ & ⋮\hfill \\ & \le {\left(\Theta (\Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,{\mathfrak{q}}_0)))\right)}^{{({\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4)}^n}.\hfill \end{array}$$

Then, for any $m>n\ge 1$, it follows that

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_m,\tau )\right)& =\Theta \left(\breve{\mathfrak{S}}({\mathfrak{Y}}^n{\mathfrak{p}}_0,{\mathfrak{Y}}^m{\mathfrak{q}}_0,\tau )\right)\hfill \\ & =\Theta \left(\breve{\mathfrak{S}}({\mathfrak{Y}}^{(n-1)+1}{\mathfrak{p}}_0,{\mathfrak{Y}}^{(n-1)+(m-n)}{\mathfrak{q}}_0,\tau )\right)\hfill \\ & \le \Theta (\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^n,({\mathfrak{p}}_0,{\mathfrak{q}}_0)))\hfill \\ & \le {\left(\Theta (\Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,{\mathfrak{q}}_0)))\right)}^{{({\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4)}^n}\to 1\mathrm{as}n\to \infty .\hfill \end{array}$$

Hence, by ($\Theta 2)$, the bisequence $\left\{({\mathfrak{p}}_n,{\mathfrak{q}}_n)\right\}$ forms a Cauchy sequence within a complete SEBPMS; this pair sequence converges and, according to Proposition 3, biconverges to a unique limit $\zeta$ in $\mathfrak{P}\cap \mathfrak{Q}$, for which $\breve{\mathfrak{S}}(\zeta ,\zeta ,\tau )=0$. Moreover,

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_n,\mathfrak{Y}\zeta ,\tau )\right)=& \Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}{\mathfrak{p}}_{n-1},\mathfrak{Y}\zeta ,\tau )\right)\hfill \\ & \le {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n-1},\zeta ,\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_n,\zeta ,\tau )\right))}^{{\mathfrak{a}}_2}\times \hfill \\ \hfill & {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}{\mathfrak{p}}_{n-1},\mathfrak{Y}\zeta ,\tau )\right))}^{{\mathfrak{a}}_3}\times {\left(\Theta \left[{\displaystyle \frac{\breve{\mathfrak{S}}({\mathfrak{p}}_n,\zeta ,\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_{n-1},\mathfrak{Y}\zeta ,\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}.\hfill \end{array}$$

Taking $n\to \infty$ and using $(\Theta 1)$, we obtain the following result:

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}(\zeta ,\mathfrak{Y}\zeta ,\tau )\right)& \le {(\Theta \left(\breve{\mathfrak{S}}(\zeta ,\mathfrak{Y}\zeta ,\tau )\right))}^{{\mathfrak{a}}_3+{\mathfrak{a}}_4}.\hfill \end{array}$$

We deduce that $\mathfrak{Y}\zeta =\zeta .$ Now, let $\mu \in \mathfrak{P}$ be a fixed point of $\mathfrak{Y}$ such that $\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )<\infty$; then,

$$\begin{array}{cc}\hfill 1<\Theta \left(\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )\right)=& \Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}\mu ,\mathfrak{Y}\zeta ,\tau )\right)\hfill \\ & \le {(\Theta \left(\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}(\mu ,\mathfrak{Y}\zeta ,\tau )\right))}^{{\mathfrak{a}}_2}\times \hfill \\ \hfill & {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}\mu ,\zeta ,\tau )\right))}^{{\mathfrak{a}}_3}\times {\left(\Theta \left[{\displaystyle \frac{\breve{\mathfrak{S}}(\mu ,\mathfrak{Y}\zeta ,\tau )+\breve{\mathfrak{S}}(\mathfrak{Y}\mu ,\zeta ,\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}\hfill \\ & ={(\Theta \left(\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )\right))}^{{\mathfrak{a}}_2}\times \hfill \\ \hfill & {(\Theta \left(\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )\right))}^{{\mathfrak{a}}_3}\times {\left(\Theta \left[{\displaystyle \frac{\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )+\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}\hfill \\ & ={(\Theta \left(\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )\right))}^{{\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4}\hfill \\ & <\Theta \left(\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )\right),\hfill \end{array}$$

which is a contradiction. So, $\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )=0$, i.e., $\zeta =\mu .$ A similar conclusion holds whenever $\zeta \in \mathfrak{Q}$. □

Corollary 1.

Let $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ denote a complete SEBPMS and let $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightrightarrows \mathfrak{P}\cup \mathfrak{Q}$ be a covariant function satisfying the following:

$$\begin{array}{cc}\hfill \sqrt{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}& \le {\mathfrak{a}}_1\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )}+{\mathfrak{a}}_2\sqrt{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )}\hfill \\ \hfill & +{\mathfrak{a}}_3\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}+{\mathfrak{a}}_4\sqrt{{\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )+(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}{2}}},\hfill \end{array}$$

(5)

$\forall \mathfrak{p}\in \mathfrak{P},\mathfrak{q}\in \mathfrak{Q}$, $\tau >0$ and ${\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4\ge 0$ with ${\sum }_{i=1}^4{\mathfrak{a}}_i<1.$ Assume the existence of $({\mathfrak{p}}_0,{\mathfrak{q}}_0)\in \mathfrak{P}\times \mathfrak{Q}$ such that $\Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0))=sup\{\breve{\mathfrak{S}}({\mathfrak{p}}_i,{\mathfrak{q}}_j,\tau ):i,j=1,2,\cdots :{\mathfrak{q}}_n=\mathfrak{Y}{\mathfrak{q}}_{n-1},{\mathfrak{p}}_n=\mathfrak{Y}{\mathfrak{p}}_{n-1},\tau >0\}<\infty .$ Then, $\mathfrak{Y}$ has a unique fixed point in $\mathfrak{P}\cap \mathfrak{Q}$.

Proof. 

Putting $\Theta \left(s\right)=e^{\sqrt{s}},$ where $\Theta \in {\Theta }^{\prime }$ in Theorem 1, we then have (5). □

Remark 2.

The condition (5) is equivalent to

$$\begin{array}{cc}\hfill \breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )& \le {{\mathfrak{a}}_1}^2\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )+{{\mathfrak{a}}_2}^2\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )\hfill \\ \hfill & +{{\mathfrak{a}}_3}^2\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )+{{\mathfrak{a}}_4}^2{\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )+\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}{2}}\hfill \\ & +2{\mathfrak{a}}_1{\mathfrak{a}}_2\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )}\hfill \\ & +2{\mathfrak{a}}_1{\mathfrak{a}}_3\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}\hfill \\ & +2{\mathfrak{a}}_1{\mathfrak{a}}_4\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau ){\displaystyle \frac{[\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )+\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )]}{2}}}\hfill \\ & +2{\mathfrak{a}}_2{\mathfrak{a}}_3\sqrt{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}\hfill \\ & +2{\mathfrak{a}}_2{\mathfrak{a}}_4\sqrt{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau ){\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )+\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}{2}}}\hfill \\ & +2{\mathfrak{a}}_3{\mathfrak{a}}_4\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau ){\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )+(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}{2}}},\hfill \end{array}$$

$\forall \mathfrak{p}\in \mathfrak{P},\mathfrak{q}\in \mathfrak{Q}$, $\tau >0$ and ${\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4\ge 0$ with ${\sum }_{i=1}^4{\mathfrak{a}}_i<1.$

Remark 3.

From Corollary 1 and Remark 2, we obtain the following results:

• A covariant extension of a Kannan result by taking ${\mathfrak{a}}_1={\mathfrak{a}}_4=0;$
• A covariant Chatterjea-type result by taking ${\mathfrak{a}}_1={\mathfrak{a}}_2={\mathfrak{a}}_3=0,{\mathfrak{a}}_4\in [0,{\displaystyle \frac{1}{2}});$
• A covariant extension of a Reich result by taking ${\mathfrak{a}}_4=0;$
• A covariant Ćirića-type result by taking ${\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+2{\mathfrak{a}}_4<1$ and $\Theta \left(s\right)=e^s,$ where $\Theta \in {\Theta }^{\prime },$ in Theorem 1.

Example 2.

Consider $\mathfrak{P}=\{(\xi ,\eta )\in {\mathbb{R}}^2:{\xi }^2+{\eta }^2\le 4\}$, $\mathfrak{Q}=\{(\xi ,\eta )\in {\mathbb{R}}^2:{(\xi -{\displaystyle \frac{1}{2}})}^2-{\eta }^2\ge {\displaystyle \frac{1}{4}}\}$ and $\breve{\mathfrak{S}}(({\mathfrak{a}}_1,{\mathfrak{b}}_1),({\mathfrak{c}}_1,{\mathfrak{d}}_1),\tau )=e^{\tau |{\mathfrak{b}}_1-{\mathfrak{d}}_1|}-1,\forall ({\mathfrak{a}}_1,{\mathfrak{b}}_1)\in \mathfrak{P},({\mathfrak{c}}_1,{\mathfrak{d}}_1)\in \mathfrak{Q}$ and $\tau >0$. Then, $\breve{\mathfrak{S}}$ is a complete SEBPMS on $\mathfrak{P}\cup \mathfrak{Q}$ with $\Omega \left(\sigma \right)=exp\left(\sigma \right)-1,\forall \sigma \ge 0$. Define $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightrightarrows \mathfrak{P}\cup \mathfrak{Q}$ by

$$\mathfrak{Y}(\xi ,\eta )=(0,sin{\displaystyle \frac{\xi }{2}}),\forall (\xi ,\eta )\in \mathfrak{P}\cup \mathfrak{Q}$$

(6)

Clearly, $\mathfrak{Y}$ is a covariant $JS$-contraction. For any $(\mathfrak{p},\mathfrak{q})=:\left(\right(\mathfrak{a},\mathfrak{b}),(\mathfrak{c},\mathfrak{d}\left)\right)\in \mathfrak{P}\times \mathfrak{Q},$ it follows that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Delta (\breve{\mathfrak{S}},\mathfrak{Y},(\mathfrak{p},\mathfrak{q}))& =sup\{\breve{\mathfrak{S}}({\mathfrak{p}}_i,{\mathfrak{q}}_j,\tau );i,j\in \mathbb{N},\tau >0\}\hfill \\ & =sup\{\breve{\mathfrak{S}}(\mathfrak{Y}{\mathfrak{p}}_{i-1},\mathfrak{Y}{\mathfrak{q}}_{j-1},\tau );i,j\in \mathbb{N},\tau >0\}\hfill \\ & =sup\left\{exp\left(\tau \right|sin{\displaystyle \frac{{\mathfrak{c}}_j}{2}}-sin{\displaystyle \frac{{\mathfrak{d}}_j}{2}}\left|\right)-1\right\}<\infty .\hfill \end{array}\end{array}$$

Let $\Theta =e^{\sigma },\sigma >0$; this implies that

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )\right)& =e^{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}\hfill \\ & =e^{(exp(\tau |sin{\displaystyle \frac{{\mathfrak{c}}_j}{2}}-sin{\displaystyle \frac{{\mathfrak{d}}_j}{2}}|)-1)}\hfill \\ & \le e^{(exp(\tau |{\displaystyle \frac{{\mathfrak{c}}_j}{2}}-{\displaystyle \frac{{\mathfrak{d}}_j}{2}}|)-1)}\hfill \\ & \le {\left[e^{(exp\left(\tau \right|{\mathfrak{d}}_j-{\mathfrak{b}}_i\left|\right)-1)}\right]}^{{\displaystyle \frac{1}{2}}}\hfill \\ & ={(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\right))}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

We conclude that ${\mathfrak{a}}_1={\displaystyle \frac{1}{2}},{\mathfrak{a}}_2={\mathfrak{a}}_3={\mathfrak{a}}_4=0.$ Therefore, the conditions of Theorem 1 are satisfied and $\mathfrak{Y}$ has a unique fixed point $(0,0)\in \mathfrak{P}\cap \mathfrak{Q}$.

Definition 7.

Let $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ be an SEBPMS. Then, $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightleftharpoons \mathfrak{P}\cup \mathfrak{Q}$ is called contravariant $JS$-contraction, whenever there are $\Theta \in {\Theta }^{\prime }$ and ${\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4\ge 0$ with ${\sum }_{i=1}^4{\mathfrak{a}}_i<1$ such that the following inequality holds:

$$\begin{array}{cc}\hfill \Theta (\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{Y}\mathfrak{p},\tau ))& \le {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )\right))}^{{\mathfrak{a}}_2}\hfill \\ \hfill & \times {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau )\right))}^{{\mathfrak{a}}_3}\times \Theta {\left(\left[{\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )+\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4},\hfill \end{array}$$

(7)

$\forall \mathfrak{p}\in \mathfrak{P},\mathfrak{q}\in \mathfrak{Q}$, $\tau >0.$

Theorem 2.

Let $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ be a complete SEBPMS; $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightleftharpoons \mathfrak{P}\cup \mathfrak{Q}$ is a continuous contravariant $JS$-contraction such that there exists $({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0)\in \mathfrak{P}\times \mathfrak{Q}$ with $\Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0))=sup\{\breve{\mathfrak{S}}({\mathfrak{p}}_i,{\mathfrak{q}}_j,\tau ):i,j=1,2,\cdots :{\mathfrak{q}}_n=\mathfrak{Y}{\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}=\mathfrak{Y}{\mathfrak{q}}_n,\tau >0\}<\infty .$ Then, $\mathfrak{Y}$ has a unique fixed point in $\mathfrak{P}\cap \mathfrak{Q}$.

Proof. 

Let $({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0)\in \mathfrak{P}\times \mathfrak{Q}$; subsequently, we define

$$\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^n,({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0))=sup\{\breve{\mathfrak{S}}({\mathfrak{p}}_{n-1+i},{\mathfrak{q}}_{n-1+j},\tau ):i,j=1,2,\cdots ,\tau >0\},\forall n\ge 1.$$

Then, $\forall i,j\in \mathbb{N}$, we obtain

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n+i},{\mathfrak{q}}_{n+j},\tau )\right)& =\Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}{\mathfrak{q}}_{n-1+i},\mathfrak{Y}{\mathfrak{p}}_{n+j},\tau )\right)\hfill \\ & \le {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n+j},{\mathfrak{q}}_{n-1+i},\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n+j},\mathfrak{Y}{\mathfrak{p}}_{n+j},\tau )\right))}^{{\mathfrak{a}}_2}\hfill \\ & \times {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}{\mathfrak{q}}_{n-1+i},{\mathfrak{q}}_{n-1+i},\tau )\right))}^{{\mathfrak{a}}_3}\hfill \\ & \times \Theta {\left(\left[{\displaystyle \frac{\breve{\mathfrak{S}}({\mathfrak{p}}_{n+j},\mathfrak{Y}{\mathfrak{p}}_{n+j},\tau )+\breve{\mathfrak{S}}(\mathfrak{Y}{\mathfrak{q}}_{n-1+j},{\mathfrak{q}}_{n-1+j},\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}\hfill \\ & \le {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n+j},{\mathfrak{q}}_{n-1+i},\tau )\right))}^{{\mathfrak{a}}_1}{(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n+j},{\mathfrak{q}}_{n+j},\tau )\right))}^{{\mathfrak{a}}_2}\hfill \\ & \times {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_{n+j},{\mathfrak{q}}_{n-1+i},\tau )\right))}^{{\mathfrak{a}}_3}\hfill \\ & \times \Theta {\left(\left[{\displaystyle \frac{\breve{\mathfrak{S}}({\mathfrak{p}}_{n+j},{\mathfrak{q}}_{n+j},\tau )+\breve{\mathfrak{S}}({\mathfrak{p}}_{n+j},{\mathfrak{q}}_{n-1+i},\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}\hfill \\ & \le {\left(\Theta (\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^n,({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0)))\right)}^{{\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4},\forall n\ge 1.\hfill \end{array}$$

(8)

As $\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^n,({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0))\le \Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0))<\infty ,\forall n\ge 1$, then from (8), it follows that

$$\begin{array}{cc}\hfill \Theta (\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^{n+1},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0)))& =\Theta (sup\breve{\mathfrak{S}}({\mathfrak{Y}}^{n+i}{\mathfrak{p}}_0,{\mathfrak{Y}}^{n+j}{\mathfrak{q}}_0,\tau ))\hfill \\ & \le {\left(\Theta (\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^n,({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0)))\right)}^{{\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4}\hfill \\ & \le {\left(\Theta (\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^{n-1},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0)))\right)}^{{({\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4)}^2}\hfill \\ \hfill ⋮\\ & \le {\left(\Theta (\Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0)\left)\right)\right)}^{{({\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4)}^n}.\hfill \end{array}$$

$\forall m>n\ge 1$, it follows that:

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}({\mathfrak{p}}_n,{\mathfrak{q}}_m,\tau )\right)& =\Theta \left(\breve{\mathfrak{S}}({\mathfrak{Y}}^n{\mathfrak{p}}_0,{\mathfrak{Y}}^m{\mathfrak{p}}_0,\tau )\right)\hfill \\ & =\Theta \left(\breve{\mathfrak{S}}({\mathfrak{Y}}^{(n-1)+1}{\mathfrak{p}}_0,{\mathfrak{Y}}^{(n-1)+(m-n)}{\mathfrak{p}}_0,\tau )\right)\hfill \\ & \le \Theta (\Delta (\breve{\mathfrak{S}},{\mathfrak{Y}}^{n-1},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0)))\hfill \\ & \le {\left(\Theta (\Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0)))\right)}^{{({\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+{\mathfrak{a}}_4)}^n}\to 1\mathrm{as}n\to \infty .\hfill \end{array}$$

Hence, the sequence $\left\{({\mathfrak{p}}_n,{\mathfrak{q}}_n)\right\}$ forms a Cauchy sequence within $\mathfrak{P}\cup \mathfrak{Q}$. Given the completeness of $\mathfrak{P}\cup \mathfrak{Q}$, this sequence converges, and according to Proposition 3, it biconverges to a certain $\zeta$ in $\mathfrak{P}\cap \mathfrak{Q}$ for which $\breve{\mathfrak{S}}(\zeta ,\zeta ,\tau )=0$. Furthermore,

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}({\mathfrak{q}}_n,\mathfrak{Y}\zeta ,\tau )\right)=& \Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}{\mathfrak{q}}_{n-1},\mathfrak{Y}\zeta ,\tau )\right)\hfill \\ & \le {(\Theta \left(\breve{\mathfrak{S}}(\zeta ,{\mathfrak{q}}_{n-1},\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{q}}_{n-1},\mathfrak{Y}{\mathfrak{q}}_{n-1},\tau )\right))}^{{\mathfrak{a}}_2}\times \hfill \\ \hfill & {(\Theta \left(\breve{\mathfrak{S}}(\zeta ,\mathfrak{Y}\zeta ,\tau )\right))}^{{\mathfrak{a}}_3}\times {\left(\Theta \left[{\displaystyle \frac{\breve{\mathfrak{S}}({\mathfrak{q}}_{n-1},\mathfrak{Y}{\mathfrak{q}}_{n-1},\tau )+\breve{\mathfrak{S}}(\zeta ,\mathfrak{Y}\zeta ,\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}\hfill \\ & ={(\Theta \left(\breve{\mathfrak{S}}(\zeta ,{\mathfrak{q}}_{n-1},\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}({\mathfrak{q}}_{n-1},{\mathfrak{p}}_n,\tau )\right))}^{{\mathfrak{a}}_2}\times \hfill \\ \hfill & {(\Theta \left(\breve{\mathfrak{S}}(\zeta ,\mathfrak{Y}\zeta ,\tau )\right))}^{{\mathfrak{a}}_3}\times {\left(\Theta \left[{\displaystyle \frac{\breve{\mathfrak{S}}({\mathfrak{q}}_{n-1},{\mathfrak{p}}_n,\tau )+\breve{\mathfrak{S}}(\zeta ,\mathfrak{Y}\zeta ,\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}.\hfill \end{array}$$

Taking $n\to \infty$ and using $(\Theta 1)$, we obtain

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}(\zeta ,\mathfrak{Y}\zeta ,\tau )\right)& \le {(\Theta \left(\breve{\mathfrak{S}}(\zeta ,\mathfrak{Y}\zeta ,\tau )\right))}^{{\mathfrak{a}}_3+{\mathfrak{a}}_4},\hfill \end{array}$$

We deduce that $\mathfrak{Y}\zeta =\zeta .$

Now, if $\mu \in \mathfrak{P}\cap \mathfrak{Q}$ is a fixed point of $\mathfrak{Y}$ such that $\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )<\infty$, then

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )\right)=& \Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}\mu ,\mathfrak{Y}\zeta ,\tau )\right)\le {(\Theta \left(\breve{\mathfrak{S}}(\zeta ,\mu ,\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}(\zeta ,\mathfrak{Y}\zeta ,\tau )\right))}^{{\mathfrak{a}}_2}\hfill \\ \hfill & \times {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}\mu ,\mu ,\tau )\right))}^{{\mathfrak{a}}_3}\times \Theta {\left(\left[{\displaystyle \frac{\breve{\mathfrak{S}}(\zeta ,\mathfrak{Y}\zeta ,\tau )+\breve{\mathfrak{S}}(\mathfrak{Y}\mu ,\mu ,\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}\hfill \\ & ={(\Theta \left(\breve{\mathfrak{S}}(\zeta ,\mu ,\tau )\right))}^{{\mathfrak{a}}_1}\times {(\Theta \left(\breve{\mathfrak{S}}(\zeta ,\zeta ,\tau )\right))}^{{\mathfrak{a}}_2}\hfill \\ \hfill & \times {(\Theta \left(\breve{\mathfrak{S}}(\mu ,\mu ,\tau )\right))}^{{\mathfrak{a}}_3}\times \Theta {\left(\left[{\displaystyle \frac{\breve{\mathfrak{S}}(\zeta ,\zeta ,\tau )+\breve{\mathfrak{S}}(\mu ,\mu ,\tau )}{2}}\right]\right)}^{{\mathfrak{a}}_4}\hfill \\ & ={(\Theta \left(\breve{\mathfrak{S}}(\zeta ,\mu ,\tau )\right))}^{{\mathfrak{a}}_1}\hfill \\ & <\Theta \left(\breve{\mathfrak{S}}(\zeta ,\mu ,\tau )\right)=\Theta \left(\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )\right),\hfill \end{array}$$

which is a contradiction. So, $\breve{\mathfrak{S}}(\mu ,\zeta ,\tau )=0,$ i.e., $\zeta =\mu .$ A similar conclusion holds whenever $\zeta \in \mathfrak{Q}$. □

Corollary 2.

Let $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ denote a complete SEBPMS, and let $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightleftharpoons \mathfrak{P}\cup \mathfrak{Q}$ be a continuous contravariant satisfying the following:

$$\begin{array}{cc}\hfill \sqrt{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}& \le {\mathfrak{a}}_1\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )}+{\mathfrak{a}}_2\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )}\hfill \\ \hfill & +{\mathfrak{a}}_3\sqrt{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau )}+{\mathfrak{a}}_4\sqrt{{\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )+\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau )}{2}}},\hfill \end{array}$$

(9)

$\forall \mathfrak{p}\in \mathfrak{P},\mathfrak{q}\in \mathfrak{Q}$, $\tau >0$ and ${\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4\ge 0$ with ${\sum }_{i=1}^4{\mathfrak{a}}_i<1.$ Assume the existence of $({\mathfrak{p}}_0,{\mathfrak{q}}_0)\in \mathfrak{P}\times \mathfrak{Q}$ such that $\Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0))=sup\{\breve{\mathfrak{S}}({\mathfrak{p}}_i,{\mathfrak{q}}_j,\tau ):i,j=1,2,\cdots :{\mathfrak{q}}_n=\mathfrak{Y}{\mathfrak{q}}_{n-1},{\mathfrak{p}}_n=\mathfrak{Y}{\mathfrak{p}}_{n-1},\tau >0\}<\infty .$ Then, $\mathfrak{Y}$ has a unique fixed point in $\mathfrak{P}\cap \mathfrak{Q}$.

Proof. 

Putting $\Theta \left(s\right)=e^{\sqrt{s}},$ where $\Theta \in {\Theta }^{\prime }$ in Theorem 2, we then have (9). □

Remark 4.

Condition (9) is equivalent to

$$\begin{array}{cc}\hfill \breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )& \le {{\mathfrak{a}}_1}^2\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )+{{\mathfrak{a}}_2}^2\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )\hfill \\ \hfill & +{{\mathfrak{a}}_3}^2\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau )+{{\mathfrak{a}}_4}^2{\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )+\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau )}{2}}\hfill \\ & +2{\mathfrak{a}}_1{\mathfrak{a}}_2\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )}\hfill \\ & +2{\mathfrak{a}}_1{\mathfrak{a}}_3\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau )}\hfill \\ & +2{\mathfrak{a}}_1{\mathfrak{a}}_4\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau ){\displaystyle \frac{[\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )+\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau )]}{2}}}\hfill \\ & +2{\mathfrak{a}}_2{\mathfrak{a}}_3\sqrt{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )\breve{\mathfrak{S}}(\mathfrak{q},\mathfrak{q},\tau )}\hfill \\ & +2{\mathfrak{a}}_2{\mathfrak{a}}_4\sqrt{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau ){\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )+\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau )}{2}}}\hfill \\ & +2{\mathfrak{a}}_3{\mathfrak{a}}_4\sqrt{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau ){\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},\tau )+\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},\tau )}{2}}},\hfill \end{array}$$

(10)

$\forall \mathfrak{p}\in \mathfrak{P},\mathfrak{q}\in \mathfrak{Q}$, $\tau >0$ and ${\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4\ge 0$ with ${\sum }_{i=1}^4{\mathfrak{a}}_i<1.$

Remark 5.

From Corollary 2 and Remark 4, we obtain the following results:

• A contravariant extension of a Kannan result by taking ${\mathfrak{a}}_1={\mathfrak{a}}_4=0;$
• A contravariant Chatterjea-type result by taking ${\mathfrak{a}}_1={\mathfrak{a}}_2={\mathfrak{a}}_3=0,{\mathfrak{a}}_4\in [0,{\displaystyle \frac{1}{2}});$
• A contravariant extension of a Reich result by taking ${\mathfrak{a}}_4=0;$
• A contravariant Ćirića-type result by taking ${\mathfrak{a}}_1+{\mathfrak{a}}_2+{\mathfrak{a}}_3+2{\mathfrak{a}}_4<1$ and $\Theta \left(s\right)=e^s,$ where $\Theta \in {\Theta }^{\prime },$ in Theorem 2.

Example 3.

Consider $\mathfrak{P}=[-16,0]$, $\mathfrak{Q}=[0,16]$ and $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )=e^{\tau |\mathfrak{p}-\mathfrak{q}|}-1,\forall \mathfrak{p}\in \mathfrak{P},\mathfrak{q}\in \mathfrak{Q}$ and $\tau >0$. Then, $\breve{\mathfrak{S}}$ is a complete SEBPMS on $\mathfrak{P}\cup \mathfrak{Q}$ with $\Omega \left(\sigma \right)=e^{\sigma }-1,\forall \sigma \ge 0$. Define $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightleftharpoons \mathfrak{P}\cup \mathfrak{Q}$ by $\mathfrak{Y}\left(u\right)=-{\displaystyle \frac{u}{4}}.$ For any $({\mathfrak{p}}_0,{\mathfrak{q}}_0)\in \mathfrak{P}\times \mathfrak{Q},$ it follows that

$$\begin{array}{cc}\hfill \Delta (\breve{\mathfrak{S}},\mathfrak{Y},({\mathfrak{p}}_0,{\mathfrak{q}}_0))& =sup\{\breve{\mathfrak{S}}({\mathfrak{p}}_i,{\mathfrak{q}}_j,\tau );i,j\in \mathbb{N},\tau >0\}\hfill \\ & =sup\{\breve{\mathfrak{S}}(\mathfrak{Y}{\mathfrak{q}}_{i-1},\mathfrak{Y}{\mathfrak{p}}_j,\tau );i,j\in \mathbb{N},\tau >0\}\hfill \\ & =sup\{e^{\tau |-{\displaystyle \frac{{\mathfrak{q}}_{i-1}}{4}}-(-{\displaystyle \frac{{\mathfrak{p}}_j}{4}})|}-1\}<\infty ,\hfill \end{array}$$

$\mathfrak{Y}$ is a contravariant $JS$-contraction as $\mathfrak{Y}\left(\right[-16,0\left]\right)=[0,4]\subset [0,16]$ and $\mathfrak{Y}\left(\right[16,0\left]\right)=[-4,0]\subset [-16,0]$. Further, for $\Theta =e^{\sigma e^{\sigma }},\sigma >0$, we obtain

$$\begin{array}{cc}\hfill \Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )\right)& =e^{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )e^{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}}\hfill \\ & =e^{exp\{\tau |-{\displaystyle \frac{\mathfrak{p}}{4}}-(-{\displaystyle \frac{\mathfrak{q}}{4}})|-1\}{e}^{exp\{\tau |-{\displaystyle \frac{\mathfrak{p}}{4}}-(-{\displaystyle \frac{\mathfrak{q}}{4}})|-1\}}}\hfill \\ & \le {\left[e^{exp\left\{\tau \right|\mathfrak{p}-\mathfrak{q}|-1\}{e}^{exp\left\{\tau \right|\mathfrak{p}-\mathfrak{q}|-1\}}}\right]}^{{\displaystyle \frac{1}{4}}}\hfill \\ & ={(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\right))}^{{\displaystyle \frac{1}{4}}}\hfill \\ & \le {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\right))}^{{\displaystyle \frac{1}{4}}}\times {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )\right))}^{{\displaystyle \frac{1}{12}}}\hfill \\ \hfill & \times {(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )\right))}^{{\displaystyle \frac{1}{2}}}\times {\left(\Theta \left[{\displaystyle \frac{\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},\tau )+\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},\tau )}{2}}\right]\right)}^{{\displaystyle \frac{1}{8}}}.\hfill \end{array}$$

Taking ${\mathfrak{a}}_1={\displaystyle \frac{1}{4}},{\mathfrak{a}}_2={\displaystyle \frac{1}{12}},{\mathfrak{a}}_3={\displaystyle \frac{1}{2}},{\mathfrak{a}}_4={\displaystyle \frac{1}{8}},$ the conditions of Theorem 2 are satisfied, and $\mathfrak{Y}$ has a unique fixed point $0\in \mathfrak{P}\cap \mathfrak{Q}$.

3. SEBPMSs to SEFMSs

In this section, we establish a linkage lemma, which can be applied to obtain fixed-point outcomes in $\Omega$-triangular sequential extended fuzzy bipolar metric spaces (SEFMSs). Let $\mathfrak{P}$ and $\mathfrak{Q}$ be two non-empty sets and let $\mathcal{B}:\mathfrak{P}\times \mathfrak{Q}\times (0,\infty )\to [0,1]$ be a function. For $\mathfrak{p}\in \mathfrak{P}$ and $\mathfrak{q}\in \mathfrak{Q}$, let us define the following sets:

$$\begin{array}{c}\hfill {\mathcal{C}}_L(\mathfrak{P},\mathcal{B},\mathfrak{q})=\{\left\{{\mathfrak{p}}_n\right\}\subset \mathfrak{P}:\underset{n\to \infty }{lim}\mathcal{B}({\mathfrak{p}}_n,\mathfrak{q},t)=1,\forall t>0\},\\ \hfill {\mathcal{C}}_R(\mathfrak{Q},\mathcal{B},\mathfrak{p})=\{\left\{{\mathfrak{q}}_n\right\}\subset \mathfrak{Q}:\underset{n\to \infty }{lim}\mathcal{B}(\mathfrak{p},{\mathfrak{q}}_n,t)=1,\forall t>0\}.\end{array}$$

Definition 8.

Let $\mathfrak{P}$ and $\mathfrak{Q}$ be two non-empty sets and $\mathcal{B}:\mathfrak{P}\times \mathfrak{Q}\times (0,\infty )\to [0,1],\forall t,s>0$ satisfy the following conditions:

• ($\mathcal{B}$1) $\mathcal{B}(\mathfrak{p},\mathfrak{q},t)>0$;
• ($\mathcal{B}$2) $\mathcal{B}(\mathfrak{p},\mathfrak{q},t)=1$ implies $\mathfrak{p}=\mathfrak{q}\in \mathfrak{P}\cap \mathfrak{Q};$
• ($\mathcal{B}$3) $\mathcal{B}(\mathfrak{p},\mathfrak{q},t)=\mathcal{B}(\mathfrak{q},\mathfrak{p},t),\forall \mathfrak{p},\mathfrak{q}\in \mathfrak{P}\cap \mathfrak{Q}$;
• ($\mathcal{B}$4) if there exists a function $\Omega \in \tilde{\Omega }$, such that $\forall {\mathfrak{p}}_1,{\mathfrak{p}}_2\in \mathfrak{P}$ and ${\mathfrak{q}}_1,{\mathfrak{q}}_2\in \mathfrak{Q}$, it follows that

  $$\mathcal{B}({\mathfrak{p}}_1,{\mathfrak{q}}_2,\Omega (t+s))\ge \underset{n\to \infty }{lim\; sup}[\mathcal{B}({\mathfrak{p}}_1,{\mathfrak{q}}_1,t)\ast \mathcal{B}({\mathfrak{p}}_n,{\mathfrak{q}}_1,s)],\forall \left\{{\mathfrak{p}}_n\right\}\in {\mathcal{C}}_L(\mathfrak{P},\mathcal{B},{\mathfrak{q}}_2);$$

  $$\mathcal{B}({\mathfrak{p}}_1,{\mathfrak{q}}_2,\Omega (t+s))\ge \underset{n\to \infty }{lim\; sup}[\mathcal{B}({\mathfrak{p}}_2,{\mathfrak{q}}_2,s)\ast \mathcal{B}({\mathfrak{p}}_2,{\mathfrak{q}}_n,t)],\forall \left\{{\mathfrak{q}}_n\right\}\in {\mathcal{C}}_R(\mathfrak{Q},\mathcal{B},{\mathfrak{p}}_1);$$
• ($\mathcal{B}$5) $\mathcal{B}(\mathfrak{p},\mathfrak{q},.):(0,\infty )\to [0,1]$ is left continuous and ${lim}_{t\to \infty }\mathcal{B}(\mathfrak{p},\mathfrak{q},t)=1.$

Then, $\mathcal{B}$ is called a sequential extended bipolar fuzzy metric and the quintuple $(\mathfrak{P},\mathfrak{Q},\mathcal{B},\Omega ,\ast )$ is called a sequential fuzzy bipolar metric space.

Definition 9.

Let $(\mathfrak{P},\mathfrak{Q},\mathcal{B},\Omega ,\ast )$ be an SEFBMS

(a) 

A bisequence $\left\{({\mathfrak{p}}_n,{\mathfrak{q}}_n)\right\}$ is termed a Cauchy bisequence if there exists $N\in \mathbb{N}$ such that ${lim}_{n,m\to \infty }\mathcal{B}({\mathfrak{p}}_n,{\mathfrak{q}}_m,t)=1,\forall m,n\ge N,\ge N,\tau >0$;

(b) 

If every Cauchy bisequence is biconvergent, then $(\mathfrak{P},\mathfrak{Q},\mathcal{B},\Omega ,\ast )$ is said to be a complete SEFBPM

Definition 10.

Let $(\mathfrak{P},\mathfrak{Q},\mathcal{B},\Omega ,\ast ),$ be an SEFBMS. It is called a $\Omega$-triangular whenever $\forall {\mathfrak{p}}_1,{\mathfrak{p}}_2\in \mathfrak{P}$ and ${\mathfrak{q}}_1,{\mathfrak{q}}_2\in \mathfrak{Q},t>0$

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{\mathcal{B}({\mathfrak{p}}_1,{\mathfrak{q}}_2,t)}}-1\le \Omega \left[{\displaystyle \frac{1}{\mathcal{B}({\mathfrak{p}}_1,{\mathfrak{q}}_1,t)}}-1+{\displaystyle \frac{1}{{lim\; sup}_{n\to \infty }\mathcal{B}({\mathfrak{p}}_n,{\mathfrak{q}}_1,t)}}-1\right];\end{array}$$

$\forall \left\{{\mathfrak{p}}_n\right\}\in {\mathcal{C}}_L(\mathfrak{P},\mathcal{B},{\mathfrak{q}}_2)$,and

$${\displaystyle \frac{1}{\mathcal{B}({\mathfrak{p}}_1,{\mathfrak{q}}_2,t)}}-1\le \Omega \left[{\displaystyle \frac{1}{\mathcal{B}({\mathfrak{p}}_2,{\mathfrak{q}}_2,t)]}}-1+{\displaystyle \frac{1}{{lim\; sup}_{n\to \infty }[\mathcal{B}({\mathfrak{p}}_2,{\mathfrak{q}}_n,t)}}-1\right];$$

$\forall \left\{{\mathfrak{q}}_n\right\}\in {\mathcal{C}}_R(\mathfrak{Q},\mathcal{B},{\mathfrak{p}}_1).$

Example 4.

Let $(\mathfrak{P},\mathfrak{Q},\mathfrak{S})$ be a sequential bipolar metric space with constant $k=1$. Let $\mathcal{B}:\mathfrak{P}\times \mathfrak{Q}\times (0,\infty )\to [0,1]$ be defined by

$$\mathcal{B}(\mathfrak{p},\mathfrak{q},t)={\displaystyle \frac{{\Omega }^{-1}\left(t\right)}{{\Omega }^{-1}\left(t\right)+\mathfrak{S}(\mathfrak{p},\mathfrak{q})}},$$

(11)

where $\Omega \in \tilde{\Omega }$, with product t-norm. We show that $\mathcal{B}$ is an SEFBMS.

Proof. 

We check only $(\mathcal{B}4$), because verifying the other conditions is standard. Given that $\Omega$ is a strictly increasing continuous function, it follows that $\forall {\mathfrak{p}}_1,{\mathfrak{p}}_2\in \mathfrak{P}$, ${\mathfrak{q}}_1,{\mathfrak{q}}_2\in \mathfrak{Q}$ is arbitrary such that $\left\{{\mathfrak{p}}_n\right\}\in {\mathcal{C}}_L(\mathfrak{P},\mathcal{B},{\mathfrak{q}}_2)$ and $\left\{{\mathfrak{q}}_n\right\}\in {\mathcal{C}}_R(\mathfrak{Q},\mathcal{B},{\mathfrak{p}}_1)$. We can see by (11) that $\mathcal{B}({\mathfrak{p}}_n,\mathfrak{q},t)=1$, which implies $\mathfrak{S}({\mathfrak{p}}_n,\mathfrak{q})=0$, so we conclude that $\left\{{\mathfrak{p}}_n\right\}\in {\mathfrak{C}}_L(\mathfrak{P},\mathfrak{S},{\mathfrak{q}}_2)$. Similarly, it is established that $\left\{{\mathfrak{q}}_n\right\}\in {\mathfrak{C}}_R(\mathfrak{Q},\mathfrak{S},{\mathfrak{p}}_1)$. Next, we demonstrate that, $\forall t,s>0$

$$\begin{array}{cc}\hfill \mathcal{B}({\mathfrak{p}}_1,{\mathfrak{q}}_2,\Omega (t+s))& ={\displaystyle \frac{{\Omega }^{-1}\Omega (t+s)}{{\Omega }^{-1}\Omega (t+s)+\mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_2)}}\hfill \\ & \ge {\displaystyle \frac{t+s}{t+s+{lim\; sup}_{n\to \infty }[\mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_1)+\mathfrak{S}({\mathfrak{p}}_n,{\mathfrak{q}}_1)]}}\hfill \\ & \ge \underset{n\to \infty }{lim\; sup}\left({\displaystyle \frac{t+s}{t+s+[\mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_1)+\mathfrak{S}({\mathfrak{p}}_n,{\mathfrak{q}}_1)]}}\right)\hfill \\ & =\underset{n\to \infty }{lim\; sup}\left({\displaystyle \frac{1}{1+\frac{[\mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_1)+\mathfrak{S}({\mathfrak{p}}_n,{\mathfrak{q}}_1)]}{s+t}}}\right)\hfill \\ & \ge \underset{n\to \infty }{lim\; sup}\left({\displaystyle \frac{1}{1+\frac{\mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_1)}{s}}}·{\displaystyle \frac{1}{1+\frac{{\mathfrak{S}}_b({\mathfrak{p}}_n,{\mathfrak{q}}_1)}{t}}}\right)\hfill \\ & =\underset{n\to \infty }{lim\; sup}\left({\displaystyle \frac{s}{s+\mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_1)}}·{\displaystyle \frac{t}{t+\mathfrak{S}({\mathfrak{p}}_n,{\mathfrak{q}}_1)}}\right)\hfill \\ & \ge \underset{n\to \infty }{lim\; sup}\left({\displaystyle \frac{{\Omega }^{-1}\left(s\right)}{{\Omega }^{-1}\left(s\right)+\mathfrak{S}({\mathfrak{p}}_1,{\mathfrak{q}}_1)}}·{\displaystyle \frac{{\Omega }^{-1}\left(t\right)}{{\Omega }^{-1}\left(t\right)+\mathfrak{S}({\mathfrak{p}}_n,{\mathfrak{q}}_1)}}\right)\hfill \\ & =\underset{n\to \infty }{lim\; sup}[\mathcal{B}({\mathfrak{p}}_1,{\mathfrak{q}}_1,t)\ast \mathcal{B}({\mathfrak{p}}_n,{\mathfrak{q}}_1,s)].\hfill \end{array}$$

Similarly, we can prove

$$\mathcal{B}({\mathfrak{p}}_1,{\mathfrak{q}}_2,\Omega (t+s))\ge \underset{n\to \infty }{lim\; sup}[\mathcal{B}({\mathfrak{p}}_2,{\mathfrak{q}}_n,t)\ast \mathcal{B}({\mathfrak{p}}_2,{\mathfrak{q}}_2,s)],\forall \left\{{\mathfrak{q}}_n\right\}\in {\mathcal{C}}_R(\mathfrak{Q},\mathcal{B},{\mathfrak{p}}_1).$$

□

Lemma 1.

Given that $\mathcal{B}$ is an Ω-triangular SEFBMS, it follows that $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},t)=\Omega \left[{\displaystyle \frac{1}{\mathcal{B}(\mathfrak{p},\mathfrak{q},t)}}-1\right]$ constitutes an SEBMS.

As an application of the Lemma 1 and the results recognized in Section 2, we can deduce the subsequent results in SEFBPMSs.

Definition 11.

Let $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ be an SEFBPMS and $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightrightarrows \mathfrak{P}\cup \mathfrak{Q}$ be a $\Theta$-contravariant mapping whenever there are $\Theta \in {\Theta }^{\prime }$ and ${\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4\ge 0$ with ${\sum }_{i=1}^4{\mathfrak{a}}_i<1$, such that the following conditions hold

$$\begin{array}{cc}\hfill \Theta (\Omega \left[{\displaystyle \frac{1}{\mathcal{B}(\mathfrak{Y}\mathfrak{p},\mathfrak{Y}\mathfrak{q},t)}}-1\right])& \le {(\Theta (\Omega \left[{\displaystyle \frac{1}{\mathcal{B}(\mathfrak{p},\mathfrak{q},t)}}-1\right]))}^{{\mathfrak{a}}_1}\times {(\Theta (\Omega \left[{\displaystyle \frac{1}{\mathcal{B}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},t)}}-1\right]\left)\right)}^{{\mathfrak{a}}_2}\hfill \\ \hfill & \times {(\Theta (\Omega \left[{\displaystyle \frac{1}{\mathcal{B}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},t)}}-1\right]\left)\right)}^{{\mathfrak{a}}_3}\hfill \\ & \times {\left(\Theta \left[{\displaystyle \frac{\Omega \left[\frac{1}{\mathcal{B}(\mathfrak{Y}\mathfrak{p},\mathfrak{q},t)}-1\right]+\Omega \left[\frac{1}{\mathcal{B}(\mathfrak{p},\mathfrak{Y}\mathfrak{q},t)}-1\right]}{2}}\right]\right)}^{{\mathfrak{a}}_4},\hfill \end{array}$$

(12)

$\forall \mathfrak{p}\in \mathfrak{P},\mathfrak{q}\in \mathfrak{Q}$, $t>0.$

Theorem 3.

Let $(\mathfrak{P},\mathfrak{Q},\mathcal{B},\Omega ,\ast )$ be a complete SEFBMS and $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightrightarrows \mathfrak{P}\cup \mathfrak{Q}$ be a $\Theta$-contravariant such that there exists $({\mathfrak{p}}_0,{\mathfrak{q}}_0)\in \mathfrak{P}\times \mathfrak{Q}$ such that $\Delta (\mathcal{B},\mathfrak{Y},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0))=sup\{\Omega \left[{\displaystyle \frac{1}{\mathcal{B}({\mathfrak{p}}_i,{\mathfrak{q}}_j,t)}}-1\right]:i,j=1,2,\cdots :{\mathfrak{q}}_n=\mathfrak{Y}{\mathfrak{q}}_{n-1},{\mathfrak{p}}_n=\mathfrak{Y}{\mathfrak{p}}_{n-1},t>0\}<\infty .$ Then, $\mathfrak{Y}$ has a unique fixed point in $\mathfrak{P}\cup \mathfrak{Q}$.

Proof. 

We define $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},t)=\Omega \left[{\displaystyle \frac{1}{\mathcal{B}(\mathfrak{p},\mathfrak{q},t)}}-1\right],\forall (\mathfrak{p},\mathfrak{q})\in \mathfrak{P}\times \mathfrak{Q},t>0.$ Then, by Remark 1 $\mathcal{B}$, it is a $\Omega$-triangular SEFBMS, and by Theorem 1, the proof is complete. □

Definition 12.

Let $(\mathfrak{P},\mathfrak{Q},\mathcal{B})$ be an SEFBMS and $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightleftharpoons \mathfrak{P}\cup \mathfrak{Q}$ be a contravariant $JS$-contraction whenever there are $\Theta \in {\Theta }^{\prime }$ and ${\mathfrak{a}}_1,{\mathfrak{a}}_2,{\mathfrak{a}}_3,{\mathfrak{a}}_4\ge 0$ with ${\sum }_{i=1}^4{\mathfrak{a}}_i<1$ such that the following conditions hold:

$$\begin{array}{cc}\hfill \Theta (\breve{\mathfrak{S}}(\mathfrak{Y}\mathfrak{q},\mathfrak{Y}\mathfrak{p},\tau )& \le {(\Theta (\Omega \left[{\displaystyle \frac{1}{\mathcal{B}(\mathfrak{p},\mathfrak{q},t)}}-1\right]))}^{{\mathfrak{a}}_1}\times {(\Theta (\Omega \left[{\displaystyle \frac{1}{\mathcal{B}(\mathfrak{p},\mathfrak{Y}\mathfrak{p},t)}}-1\right]))}^{{\mathfrak{a}}_2}\hfill \\ \hfill & \times {(\Theta (\Omega \left[{\displaystyle \frac{1}{\mathcal{B}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},t)}}-1\right]))}^{{\mathfrak{a}}_3}\hfill \\ & \times \Theta {\left(\left[{\displaystyle \frac{\Omega \left[\frac{1}{\mathcal{B}(\mathfrak{p},\mathfrak{Y}\Xi ,t)}-1\right]+\Omega \left[\frac{1}{\mathcal{B}(\mathfrak{Y}\mathfrak{q},\mathfrak{q},t)}-1\right]}{2}}\right]\right)}^{{\mathfrak{a}}_4},\hfill \end{array}$$

(13)

$\forall \mathfrak{p}\in \mathfrak{P},\mathfrak{q}\in \mathfrak{Q}$, $t>0.$

Theorem 4.

Let $(\mathfrak{P},\mathfrak{Q},\mathcal{B},\Omega ,\ast )$ be a complete SEFBPMS and $\mathfrak{Y}:\mathfrak{P}\cup \mathfrak{Q}\rightleftharpoons \mathfrak{P}\cup \mathfrak{Q}$ be a contravariant $JS$-contraction such that there exists $({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0)\in \mathfrak{P}\times \mathfrak{Q}$ such that $\Delta (\mathcal{B},\mathfrak{Y},({\mathfrak{p}}_0,\mathfrak{Y}{\mathfrak{p}}_0))=sup\{\Omega \left[{\displaystyle \frac{1}{\mathcal{B}({\mathfrak{p}}_i,{\mathfrak{q}}_j,t)}}-1\right]:i,j=1,2,\cdots :{\mathfrak{q}}_n=\mathfrak{Y}{\mathfrak{p}}_n,{\mathfrak{p}}_{n+1}=\mathfrak{Y}{\mathfrak{q}}_n,\tau >0\}<\infty .$ Then, $\mathfrak{Y}$ has at least one fixed point in $\mathfrak{P}\cup \mathfrak{Q}$. Furthermore, if $\mathfrak{p}$ and $\mathfrak{q}$ are two fixed points of $\mathfrak{Y}$ in $\mathfrak{P}\cup \mathfrak{Q}$ with $\mathcal{B}(\mathfrak{p},\mathfrak{q},t)<1$, then $\mathfrak{p}=\mathfrak{q}.$

Proof. 

We define $\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},t)=\Omega \left[{\displaystyle \frac{1}{\mathcal{B}(\mathfrak{p},\mathfrak{q},t)}}-1\right]\forall (\mathfrak{p},\mathfrak{q})\in \mathfrak{P}\times \mathfrak{Q},t>0.$ Then, by Remark 1 $\mathcal{B}$, it is an $\Omega$-triangular SEFBMS and by Theorem 2, the proof is complete. □

4. $\Phi \text{-}\mathcal{A}\mathcal{B}\mathcal{C}$ Fractional Differential Equations of Order $\mathbf{1}<\mathbf{\mu }\le \mathbf{2}$ with NMB Conditions

In fractional calculus, two main categories of nonlocal operators have emerged. The first category includes operators with singular kernels, such as Caputo [16], Riemann–Liouville [16], Katugampola [17], Hadamard [16], and Hilfer [18] operators. Additionally, these operators have been extended to incorporate weighted fractional operators with respect to a different function [19,20,21]. In fact, the singularity of kernels sometimes caused problems in numerical analysis. For this reason, a second type of nonlocal operators appeared, which created a new fractional operator with a nonsingular kernel, introduced by Caputo and Fabrizio [22] in 2015. Subsequently, Atangana and Baleanu $\left(\mathcal{AB}\right)$ [23] proposed a novel nonsingular fractional operator through a Mittag-Leffler function. These operators have garnered significant interest from researchers for investigating diverse problems and their applications [24]. Later on, Fernandez and Baleanu [25] expanded the $\mathcal{AB}$ operator to differentiation and integration concerning an alternative function, naming it $\Phi \text{-}\mathcal{A}\mathcal{B}\mathcal{C}$ in the Caputo framework. Recently, Abdeljawad and collaborators [12] have generalized this concept to encompass higher-order fractional derivatives. Inspired by the preceding discussion, this section examines the existence and uniqueness of solutions for the $\Phi \text{-}\mathcal{A}\mathcal{B}\mathcal{C}$ fractional differential equation of order $1<\mu \le 2$ under nonlocal mixed boundary conditions, as presented below.

$$\left\{\begin{array}{c}{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }u\left(t\right)=\mathfrak{H}(t,u\left(t\right)),t\in \mathfrak{J}=[\mathfrak{a},\mathfrak{b}],\hfill \\ u\left(\mathfrak{a}\right)=0,\hfill \\ \mathfrak{K}=\sum _{i=1}^m{\gamma }_{i}u\left({\mathfrak{q}}_i\right)+\sum _{j=1}^n{\lambda }_j{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{{\beta }_j;\Phi }u\left({\mathfrak{p}}_j\right)+\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r;\Phi }u\left({\zeta }_r\right)\hfill \end{array}\right.$$

(14)

where $u\in C^1(\mathfrak{J},\mathbb{R})$ is a continuous function, ${}^{\mathcal{ABC}}\mathcal{D}_{0^{+}}^{\mu ;\Phi }(.),{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{{\beta }_j;\Phi }$ denotes the $\Phi \text{-}\mathcal{A}\mathcal{B}\mathcal{C}$ fractional derivative of order $\mu ,{\beta }_j$, respectively; $(1<{\beta }_j<\mu \le 2),j=1,2,\cdots ,n,1<\mu \le 2,$${}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r}$ is the $\Phi \text{-}\mathcal{A}\mathcal{B}\mathcal{C}$ fractional integral of order ${\delta }_r>1$ for $r=1,2,\cdots ,k,$ for the given constant ${\gamma }_i,{\lambda }_j,{\sigma }_r,\mathfrak{K}\in \mathbb{R};$ the points ${\mathfrak{q}}_j,{\mathfrak{p}}_j,{\zeta }_r\in [\mathfrak{a},\mathfrak{b}],i,1,2,\cdots ,m$ and $\mathfrak{H}:[\mathfrak{a},\mathfrak{b}]\times \mathbb{R}\to \mathbb{R}$ are a continuous function. We encourage the reader to review the following studies [26,27,28,29,30].

Remark 6.

Diverse symmetric configurations of the function $\Phi$ and the parameter μ give rise to symmetric systems; refer to [31]. For example:

1. 

If $\Phi \left(s\right)=s$, then the system (14) reduces to the following form:

$$\left\{\begin{array}{c}{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{\mu ;s}u\left(t\right)=\mathfrak{H}(t,u\left(t\right)),t\in \mathfrak{J}=[\mathfrak{a},\mathfrak{b}],\hfill \\ u\left(\mathfrak{a}\right)=0,\hfill \\ \mathfrak{K}=\sum _{i=1}^m{\gamma }_{i}u\left({\mathfrak{q}}_i\right)+\sum _{j=1}^n{\lambda }_j{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{{\beta }_j;s}u\left({\mathfrak{p}}_j\right)+\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r;s}u\left({\zeta }_r\right)\hfill \end{array}\right.$$

(15)

2. 

When $\Phi \left(s\right)=logs$, the system (14) simplifies to the form below:

$$\left\{\begin{array}{c}{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{\mu ;logs}u\left(t\right)=\mathfrak{H}(t,u\left(t\right)),t\in \mathfrak{J}=[\mathfrak{a},\mathfrak{b}],\hfill \\ u\left(\mathfrak{a}\right)=0,\hfill \\ \mathfrak{K}=\sum _{i=1}^m{\gamma }_{i}u\left({\mathfrak{q}}_i\right)+\sum _{j=1}^n{\lambda }_j{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{{\beta }_j;logs}u\left({\mathfrak{p}}_j\right)+\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r;logs}u\left({\zeta }_r\right)\hfill \end{array}\right.$$

(16)

3. 

If $\Phi \left(s\right)=s^q$ with $q>0$, then the system in Equation (14) simplifies to the form below:

$$\left\{\begin{array}{c}{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{\mu ;s^q}u\left(t\right)=\mathfrak{H}(t,u\left(t\right)),t\in \mathfrak{J}=[\mathfrak{a},\mathfrak{b}],\hfill \\ u\left(\mathfrak{a}\right)=0,\hfill \\ \mathfrak{K}=\sum _{i=1}^m{\gamma }_{i}u\left({\mathfrak{q}}_i\right)+\sum _{j=1}^n{\lambda }_j{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{{\beta }_j;s^q}u\left({\mathfrak{p}}_j\right)+\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r;s^q}u\left({\zeta }_r\right).\hfill \end{array}\right.$$

(17)

Definition 13

([12]). Consider $\Phi :[\mathfrak{a},\mathfrak{b}]\to {\mathbb{R}}^{+}$ to be an increasing function with ${\Phi }^{\prime }\left(t\right)\ne 0,\forall t\in [\mathfrak{a},\mathfrak{b}],g^{\left(n\right)}\in {\mathcal{H}}^1(\mathfrak{a},\mathfrak{b})$ and $\mu \in (n,n+1],\theta =\mu -n,n=0,1,2,\cdots .$ Thus, the μth left-sided $\Phi \text{-}\mathcal{A}\mathcal{B}\mathcal{C}$ fractional derivative can be expressed as:

$$\begin{array}{cc}\hfill \left({}^{\mathcal{ABC}}\mathcal{D}_{\mathfrak{a}}^{\mu ;\Phi }g\right)\left(t\right)& =\left({}^{\mathcal{ABC}}\mathcal{D}_{\mathfrak{a}}^{\mu ;\Phi }{g}_{\Phi }^{\left(n\right)}\right)\left(t\right)\hfill \\ & ={\displaystyle \frac{\mathcal{N}\left(\theta \right)}{1-\theta }}{\int }_{\mathfrak{a}}^t{\Phi }^{\prime }\left(s\right){\mathbb{E}}_{\theta }\left({\displaystyle \frac{\theta }{\theta -1}}{(\Phi \left(t\right)-\Phi \left(s\right))}^{\theta }\right)g_{\Phi }^{(n+1)}\left(s\right)ds\hfill \\ & ={\displaystyle \frac{\mathcal{N}(\mu -n)}{n+1-\mu }}{\int }_{\mathfrak{a}}^t{\Phi }^{\prime }\left(s\right){\mathbb{E}}_{\mu }-n\left({\displaystyle \frac{\mu -n}{\mu -n-1}}{(\Phi \left(t\right)-\Phi \left(s\right))}^{\mu -n}\right)g_{\Phi }^{(n+1)}\left(s\right)ds,\hfill \end{array}$$

where $g_{\Phi }^{\left(n\right)}\left(t\right)={\left({\displaystyle \frac{1}{{\Phi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^{n}g\left(t\right)$ and $g_{\Phi }^{\left(0\right)}\left(t\right)=g\left(t\right).$ If $\mu =m\in \mathbb{N}$, then $\left({}^{\mathcal{ABC}}\mathcal{D}_{\mathfrak{a}}^{\mu ;\Phi }g\right)\left(t\right)=g_{\Phi }^{\left(m\right)}\left(t\right)$, and $\mathcal{N}\left(\mu \right)$ is a normalization function with $\mathcal{N}\left(0\right)=\mathcal{N}\left(1\right)=1$ and ${\mathbb{E}}_{\mu }$ is called the Mittag-Leffler function, defined by

$${\mathbb{E}}_{\mu }\left(s\right)=\sum _{i=0}^{\infty }{\displaystyle \frac{r^i}{\mathsf{\Gamma }(\mu i+1)}},$$

(18)

where $Re\left(\mu \right)>0,r\in \mathbb{C}.$

Definition 14

([12]). Consider $g\in {\mathcal{H}}^1(\mathfrak{a},\mathfrak{b})$ and $\mu \in (n,n+1],\theta =\mu -n,n=0,1,2,\cdots .$ Thus, the μth left-sided $\Phi \text{-}\mathcal{A}\mathcal{B}\mathcal{C}$ fractional integral can be expressed as

$$\begin{array}{cc}\hfill \left({}^{\mathcal{AB}}\mathcal{I}_{\mathfrak{a}}^{\mu ;\Phi }g\right)\left(t\right)& =\left({}^{\mathcal{RL}}\mathcal{I}_{\mathfrak{a}}^{n;\Phi }{}^{\mathcal{AB}}\mathcal{I}_{\mathfrak{a}}^{\mu ;\Phi }g\right)\left(t\right)=\left({}^{\mathcal{AB}}\mathcal{I}_{\mathfrak{a}}^{\mu ;\Phi }{}^{\mathcal{RL}}\mathcal{I}_{\mathfrak{a}}^{n;\Phi }g\right)\left(t\right)\hfill \\ & ={\displaystyle \frac{n+1-\mu }{\mathcal{N}(\mu -n)}}{}^{\mathcal{RL}}\mathcal{I}_{\mathfrak{a}}^{n;\Phi }g\left(t\right)+{\displaystyle \frac{\mu -n}{\mathcal{N}(\mu -n)}}{}^{\mathcal{RL}}\mathcal{I}_{\mathfrak{a}}^{\mu ;\Phi }g\left(t\right)\hfill \end{array}$$

where ${}^{\mathcal{RL}}\mathcal{I}_{\mathfrak{a}}^{n;\Phi }$ is defined as:

$$\left({}^{\mathcal{RL}}\mathcal{I}_{\mathfrak{a}}^{n;\Phi }g\right)\left(t\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(n\right)}}{\int }_{\mathfrak{a}}^{\mathfrak{b}}{\Phi }^{\prime }\left(s\right){(\Phi \left(t\right)-\Phi \left(s\right))}^{n-1}g\left(s\right)ds,t>\mathfrak{a},$$

where $\mathsf{\Gamma }\left(n\right)={\int }_0^{+\infty }{e}^{-s}{s}^{n-1}ds,n>0.$

Proposition 4

([12]). Let $g\in {\tilde{\mathcal{C}}}^n([\mathfrak{a},\mathfrak{b}],\mathbb{R})$ and $\Phi \in {\tilde{\mathcal{C}}}^n([\mathfrak{a},\mathfrak{b}],{\mathbb{R}}^{+}).$ For $\mu \in (n,n+1],\beta \ge n+1,\theta =\mu -n,n=0,1,\cdots$, the subsequent relationships are established:

1. 

$\left({}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }g\right)\left(t\right)=g\left(t\right)-g\left(\mathfrak{a}\right){\mathbb{E}}_{\mu -n}\left({\displaystyle \frac{\mu -n}{\mu -n-1}}{(\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right))}^{\mu -n}\right)$;

2. 

$\left({}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }g\right)\left(t\right)=g\left(\mathfrak{t}\right)-{\sum }_{k=0}^n{\displaystyle \frac{g_{\Phi }^{\left(k\right)}\left(\mathfrak{a}\right)}{k!}}{(\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right))}^k$.

Proposition 5

([12]). Let $\Phi \in {\tilde{\mathcal{C}}}^n([\mathfrak{a},\mathfrak{b}],{\mathbb{R}}^{+}),{\Phi }^{\prime }\left(t\right)\ne 0.$ For $\mu \in (n,n+1],\beta \ge n+1,\sigma >0,\theta =\mu -n,n=0,1,\cdots$, the subsequent relationships are established:

1. 

${}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }{[\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right)]}^{\beta }=\left\{\begin{array}{cc}{\displaystyle \frac{\mathcal{N}(\mu -n)}{n+1-\mu }}{\sum }_{i=0}^{\infty }{\left({\displaystyle \frac{\mu -n}{\mu -n-1}}\right)}^i{\displaystyle \frac{\mathsf{\Gamma }(\beta +1)[\Phi \left(t\right)-\Phi {\left(\mathfrak{a}\right)}^{i(\mu -n)+\beta -n}]}{\mathsf{\Gamma }\left(i\right(\mu -n)+\beta -n+1)}}\hfill & \mathrm{if}\beta \ge n+1\hfill \\ 0\hfill & \mathrm{if}\beta \le n;\hfill \end{array}\right.$

2. 

$\left({}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }1\right)u={\displaystyle \frac{(n+1-\mu ){[\Phi \left(u\right)-\Phi \left(\mathfrak{a}\right)]}^n}{\mathcal{N}(\mu -n)\mathsf{\Gamma }(n+1)}}+{\displaystyle \frac{(\mu -n){[\Phi \left(u\right)-\Phi \left(\mathfrak{a}\right)]}^{\mu }}{\mathcal{N}(\mu -n)\mathsf{\Gamma }(n+1)}};$

3. 

$({}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }{[\Phi \left(u\right)-\Phi \left(\mathfrak{a}\right)]}^{\sigma }={\displaystyle \frac{(n+1-\mu )\mathsf{\Gamma }(\sigma +1){[\Phi \left(u\right)-\Phi \left(\mathfrak{a}\right)]}^{\sigma +n}}{\mathcal{N}(\mu -n)\mathsf{\Gamma }(n+\sigma +1)}}+{\displaystyle \frac{(\mu -n)\mathsf{\Gamma }(\sigma +1){[\Phi \left(u\right)-\Phi \left(\mathfrak{a}\right)]}^{\sigma +\mu }}{\mathcal{N}(\mu -n)\mathsf{\Gamma }(\sigma +1+\mu )}}.$

Lemma 2.

Suppose $\mathfrak{H}:[\mathfrak{a},\mathfrak{b}]\times \mathbb{R}\to \mathbb{R}$ is continuous and $\Xi ={\sum }_{i=1}^m{\gamma }_i(\Phi \left({\mathfrak{q}}_i\right)-\Phi \left(\mathfrak{a}\right))+{\sum }_{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r;\Phi }(\Phi \left({\zeta }_r\right)-\Phi \left(\mathfrak{a}\right))\ne 0.$ Then, the corresponding fractional integral equation of (14) is given by:

$$\begin{array}{cc}\hfill u\left(t\right)=& {\displaystyle \frac{1}{\Xi }}[\mathfrak{K}-\sum _{i=1}^m{\gamma }_i{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}({\mathfrak{q}}_i,u\left({\mathfrak{q}}_i\right))-\sum _{j=1}^n{\lambda }_j{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu -{\beta }_j;\Phi }\mathfrak{H}({\mathfrak{p}}_j,u\left({\mathfrak{p}}_j\right))\hfill \\ & -\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r+\mu ;\Phi }\mathfrak{H}({\zeta }_r,u\left({\zeta }_r\right))](\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right))+{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}(t,u\left(t\right)).\hfill \end{array}$$

(19)

Proof. 

Utilizing the operator ${}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }$ on both sides of Equation (14) and referencing the relevant Proposition 4, we obtain

$$u\left(t\right)={\mathfrak{a}}_0+{\mathfrak{c}}_1(\Phi \left(t\right)-\Phi \left(\mathfrak{c}\right))+{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}(t,u\left(t\right))$$

(20)

where ${\mathfrak{c}}_0,{\mathfrak{c}}_1$ are arbitrary constants. Then, the boundary condition $u\left(\mathfrak{a}\right)=0$ in (20) yields ${\mathfrak{c}}_0=0.$ As a result, Equation (20) is rewritten as follows:

$$u\left(t\right)={\mathfrak{c}}_1(\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right))+{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}(t,u\left(t\right)).$$

(21)

Now, we substitute Equation (21) into the second boundary condition:

$$\mathfrak{K}=\sum _{i=1}^m{\gamma }_{i}u\left({\mathfrak{q}}_i\right)+\sum _{j=1}^n{\lambda }_j{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{{\beta }_j;\Phi }u\left({\mathfrak{p}}_j\right)+\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r;\Phi }u\left({\zeta }_r\right),$$

(22)

we obtain the following:

$$\begin{array}{cc}\hfill \mathfrak{K}& =\sum _{i=1}^m{\gamma }_i[{\mathfrak{c}}_1(\Phi \left({\mathfrak{q}}_i\right)-\Phi \left(\mathfrak{a}\right))+{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}({\mathfrak{q}}_i,u\left({\mathfrak{q}}_i\right))]\hfill \\ & +\sum _{j=1}^n{\lambda }_j{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{{\beta }_j;\Phi }[{\mathfrak{c}}_1(\Phi \left({\mathfrak{p}}_j\right)-\Phi \left(\mathfrak{a}\right))+{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}({\mathfrak{p}}_j,u\left({\mathfrak{p}}_j\right))]\hfill \\ & +\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r;\Phi }[{\mathfrak{c}}_1(\Phi \left({\zeta }_r\right)-\Phi \left(\mathfrak{a}\right))+{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}({\zeta }_r,u\left({\zeta }_r\right))]\hfill \\ & =\sum _{i=1}^m{\gamma }_i{\mathfrak{c}}_1(\Phi \left({\mathfrak{q}}_i\right)-\Phi \left(\mathfrak{a}\right))+\sum _{i=1}^m{\gamma }_i{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}({\mathfrak{q}}_i,u\left({\mathfrak{q}}_i\right))\hfill \\ & +\sum _{j=1}^n{\lambda }_j{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{{\beta }_j;\Phi }{\mathfrak{c}}_1(\Phi \left({\mathfrak{p}}_j\right)-\Phi \left(\mathfrak{a}\right))+\sum _{j=1}^n{\lambda }_j{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{{\beta }_j;\Phi }{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}({\mathfrak{p}}_j,u\left({\mathfrak{p}}_j\right))\hfill \\ & +\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r;\Phi }{\mathfrak{c}}_1(\Phi \left({\zeta }_r\right)-\Phi \left(a\right))+\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r;\Phi }{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}({\zeta }_r,u\left({\zeta }_r\right))\hfill \\ & ={\mathfrak{c}}_1[\sum _{i=1}^m{\gamma }_i(\Phi \left({\mathfrak{q}}_i\right)-\Phi \left(\mathfrak{a}\right))+\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r;\Phi }(\Phi \left({\zeta }_r\right)-\Phi \left(\mathfrak{a}\right))]\hfill \\ & +\sum _{i=1}^m{\gamma }_i{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}({\mathfrak{q}}_i,u\left({\mathfrak{q}}_i\right))+\sum _{j=1}^n{\lambda }_j{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu -{\beta }_j;\Phi }\mathfrak{H}({\mathfrak{p}}_j,u\left({\mathfrak{p}}_j\right))\hfill \\ & +\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r+\mu ;\Phi }\mathfrak{H}({\zeta }_r,u\left({\zeta }_r\right)),\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill {\mathfrak{c}}_1& ={\displaystyle \frac{1}{\Xi }}[\mathfrak{K}-\sum _{i=1}^m{\gamma }_i{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}({\mathfrak{q}}_i,u\left({\mathfrak{q}}_i\right))-\sum _{j=1}^n{\lambda }_j{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu -{\beta }_j;\Phi }\mathfrak{H}({\mathfrak{p}}_j,u\left({\mathfrak{p}}_j\right))\hfill \\ & -\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r+\mu ;\Phi }\mathfrak{H}({\zeta }_r,u\left({\zeta }_r\right))]\hfill \end{array}.$$

(23)

Substituting the constant ${\mathfrak{c}}_1$ into (21), we obtain the integral Equation (19). □

Using Lemma 2, we define the operator $\mathfrak{Y}:U\to U$ by

$$\begin{array}{cc}\hfill \mathfrak{Y}u\left(t\right)=& {\displaystyle \frac{1}{\Xi }}[\mathfrak{K}-\sum _{i=1}^m{\gamma }_i{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}({\mathfrak{q}}_i,u\left({\mathfrak{q}}_i\right))-\sum _{j=1}^n{\lambda }_j{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu -{\beta }_j;\Phi }\mathfrak{H}({\mathfrak{p}}_j,u\left({\mathfrak{p}}_j\right))\hfill \\ & -\sum _{r=1}^k{\sigma }_r{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r+\mu ;\Phi }\mathfrak{H}({\zeta }_r,u\left({\zeta }_r\right))](\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right))+{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\mathfrak{H}(t,u\left(t\right)).\hfill \end{array}$$

(24)

Furthermore, to simplify calculations, we assume

$$\begin{array}{cc}\hfill {\Xi }^{*}& =[\sum _{i=1}^m\left|{\gamma }_i\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\hfill \\ & +\sum _{j=1}^n\left|{\lambda }_j\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu -{\beta }_j;\Phi }+\sum _{r=1}^k\left|{\sigma }_r\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r+\mu ;\Phi }]|\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right)|\hfill \\ & +|\Xi |\left({\displaystyle \frac{(2-\mu )[\Phi (\mathfrak{b})-\Phi (\mathfrak{a}\left)\right]}{\mathcal{N}(\mu -1)\mathsf{\Gamma }\left(2\right)}}+{\displaystyle \frac{(\mu -1){[\Phi \left(\mathfrak{b}\right)-\Phi \left(\mathfrak{a}\right)]}^{\mu }}{\mathcal{N}(\mu -1)\mathsf{\Gamma }\left(2\right)}}\right).\hfill \end{array}$$

(25)

Theorem 5.

Let $\mathfrak{H}:\mathfrak{J}\times \mathbb{R}\to \mathbb{R}$ be a continuous function. Suppose that

$$|\mathfrak{H}(t,u\left(t\right))-\mathfrak{H}(t,v\left(t\right))|\le \mathcal{L}|u-v|,\forall u,v\in C^1(\mathfrak{J},\mathbb{R}),\mathcal{L}>0.$$

(26)

If ${\displaystyle \frac{{\Xi }^{*}\mathcal{L}}{\Xi }}<1$, then the problem (14) has a unique solution in $\mathfrak{P}\cap \mathfrak{Q}.$

Proof. 

Let $U=C^1(\mathfrak{J},\mathbb{R})$ be a Banach space of all continuous functions $\mathfrak{p}:\mathfrak{J}\to \mathbb{R}$ endowed with the supnorm $\parallel \mathfrak{p}\parallel ={sup}_{t\in [0,\mathfrak{Y}]}\left\{\right|\mathfrak{p}\left(t\right)|:t\in [0,\mathfrak{Y}]\}.$ Let $\mathfrak{P}=\{\mathfrak{p}\in U:\parallel \mathfrak{p}\parallel \le {\mathfrak{r}}_1,{\mathfrak{r}}_1>0\}$ and $\mathfrak{Q}=\{\mathfrak{p}\in U:\parallel \mathfrak{p}\parallel \le {\mathfrak{r}}_2,{\mathfrak{r}}_2\ge {\mathfrak{r}}_1>0\},$ with ${\displaystyle \frac{\mathfrak{K}|\Phi (\mathfrak{b})-\Phi (\mathfrak{a}\left)\right|+\mathcal{M}{\Xi }^{*}}{|\Xi |-\mathcal{L}{\Xi }^{*}}}\le {\mathfrak{r}}_i,i=1,2,$) and $\mathcal{M}=sup\left\{\right|\mathfrak{H}(t,0)|:t\in \mathfrak{J}\}.$ Then, triplet $(\mathfrak{P},\mathfrak{Q},\breve{\mathfrak{S}},\Omega )$ is a complete SEBPMS with respect to the SEBPM $\breve{\mathfrak{S}}:\mathfrak{P}\times \mathfrak{Q}\to [0,\infty )$, defined by

$$\begin{array}{c}\hfill \breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )=e^{\tau {sup}_{t\in [0,1]}|\mathfrak{p}\left(t\right)-\mathfrak{q}\left(t\right)|}-1,\forall (\mathfrak{p}\left(t\right),\mathfrak{q}\left(t\right))\in \mathfrak{P}\times \mathfrak{Q},\tau >0,\end{array}$$

• for $\Omega \left(\sigma \right)=e^{\sigma }-1,\forall \sigma \ge 0,\Theta \left(t\right):=e^{\sqrt{t}}$.
• Let $u\left(t\right)\in \mathfrak{P}$:

$$\begin{array}{cc}\hfill \left|\mathfrak{Y}u\right(t\left)\right|& \le {\displaystyle \frac{1}{|\Xi |}}\left[\right|\mathfrak{K}|+\sum _{i=1}^m\left|{\gamma }_i\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\left|\mathfrak{H}({\mathfrak{q}}_i,u\left({\mathfrak{q}}_i\right))\right|\hfill \\ & +\sum _{j=1}^n\left|{\lambda }_j\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu -{\beta }_j;\Phi }\left|\mathfrak{H}({\mathfrak{p}}_j,u\left({\mathfrak{p}}_j\right))\right|\hfill \\ & +\sum _{r=1}^k\left|{\sigma }_r\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r+\mu ;\Phi }|\mathfrak{H}({\zeta }_r,u\left({\zeta }_r\right))\left|\right]|\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right)|+{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\left|\mathfrak{H}(t,u\left(t\right))\right|\hfill \\ & \le {\displaystyle \frac{1}{|\Xi |}}\left[\right|\mathfrak{K}|+\sum _{i=1}^m\left|{\gamma }_i\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\left(\right|\mathfrak{H}({\mathfrak{q}}_i,u\left({\mathfrak{q}}_i\right))-\mathfrak{H}({\mathfrak{q}}_i,0)|+|\mathfrak{H}({\mathfrak{q}}_i,0)\left|\right)\hfill \\ & +\sum _{j=1}^n\left|{\lambda }_j\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu -{\beta }_j;\Phi }\left(\right|\mathfrak{H}({\mathfrak{p}}_j,u\left({\mathfrak{p}}_j\right))-\mathfrak{H}({\mathfrak{p}}_j,0)|+|\mathfrak{H}({\mathfrak{p}}_j,0)\left|\right)\hfill \\ & +\sum _{r=1}^k\left|{\sigma }_r\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r+\mu ;\Phi }\left(\right|\mathfrak{H}({\zeta }_r,u\left({\zeta }_r\right))-\mathfrak{H}({\zeta }_r,0)|+|\mathfrak{H}({\zeta }_r,0)\left|\right)\left]\right|\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right)|\hfill \\ & +{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\left(\right|\mathfrak{H}(t,u\left(t\right))-\mathfrak{H}(t,0)|+|\mathfrak{H}(t,0)\left|\right)\hfill \\ & \le {\displaystyle \frac{\mathfrak{K}|\Phi (\mathfrak{b})-\Phi (\mathfrak{a}\left)\right|}{|\Xi |}}+{\displaystyle \frac{\mathcal{L}{\mathfrak{r}}_1+\mathcal{M}}{|\Xi |}}\left(\right[\sum _{i=1}^m\left|{\gamma }_i\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\left({\mathfrak{q}}_i\right)\hfill \\ & +\sum _{j=1}^n\left|{\lambda }_j\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu -{\beta }_j;\Phi }\left({\mathfrak{p}}_j\right)+\sum _{r=1}^k\left|{\sigma }_r\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r+\mu ;\Phi }\left({\zeta }_r\right)\left]|\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right)|\right)\hfill \\ & +(\mathcal{L}{\mathfrak{r}}_1+\mathcal{M}){}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\left(\mathfrak{Y}\right)\hfill \\ & \le {\displaystyle \frac{\mathfrak{K}|\Phi (\mathfrak{b})-\Phi (\mathfrak{a}\left)\right|}{|\Xi |}}+\left({\displaystyle \frac{\mathcal{L}{\mathfrak{r}}_1+\mathcal{M}}{|\Xi |}}\right){\Xi }^{*}\le {\mathfrak{r}}_1,\hfill \end{array}$$

by taking the norm on $[a,\mathfrak{b}],$ we obtain $\parallel \mathfrak{Y}u\left(t\right)\parallel \le r_1$, i.e., $\mathfrak{Y}\left(\mathfrak{P}\right)\subseteq \mathfrak{P}.$ Similarly, we prove $\mathfrak{Y}\left(\mathfrak{Q}\right)\subseteq \mathfrak{Q}.$ Now,

$$\begin{array}{cc}\hfill \left|\mathfrak{Y}u\right(t)-\mathfrak{Y}v(t\left)\right|& \le {\displaystyle \frac{1}{|\Xi |}}[\sum _{i=1}^m\left|{\gamma }_i\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }|\mathfrak{H}({\mathfrak{q}}_i,u\left({\mathfrak{q}}_i\right))-\mathfrak{H}({\mathfrak{q}}_i,v\left({\mathfrak{q}}_i\right))|\hfill \\ & +\sum _{j=1}^n{\lambda }_j{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu -{\beta }_j;\Phi }|\mathfrak{H}({\mathfrak{p}}_j,u\left({\mathfrak{p}}_j\right))-\mathfrak{H}({\mathfrak{p}}_j,v\left({\mathfrak{p}}_j\right))|\hfill \\ & +\sum _{r=1}^k\left|{\sigma }_r\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r+\mu ;\Phi }\left|\mathfrak{H}\right({\zeta }_r,u\left({\zeta }_r\right))-\mathfrak{H}({\zeta }_r,v\left({\zeta }_r\right))|]|\Phi \left(t\right)-\Phi \left(a\right)|\hfill \\ & +{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }|\mathfrak{H}(t,u\left(t\right))-\mathfrak{H}(t,v\left(t\right))|\hfill \\ & \le {\displaystyle \frac{\mathcal{L}|u-v|}{|\Xi |}}[\sum _{i=1}^m\left|{\gamma }_i\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }+\sum _{j=1}^n{\left|{\lambda }_j\right|}^{\mathcal{AB}}{\mathcal{I}}_{{\mathfrak{a}}^{+}}^{\mu -{\beta }_j;\Phi }+\hfill \\ & \sum _{r=1}^k\left|{\sigma }_r\right|{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\delta }_r+\mu ;\Phi }\left]\right|\Phi \left(t\right)-\Phi \left(\mathfrak{a}\right)|+{\mathcal{L}|u-v|}^{\mathcal{AB}}{\mathcal{I}}_{{\mathfrak{a}}^{+}}^{\mu ;\Phi }\hfill \\ & \le {\displaystyle \frac{{\Xi }^{*}\mathcal{L}|u-v|}{\Xi }},\alpha ={\displaystyle \frac{{\Xi }^{*}\mathcal{L}}{\Xi }}<1,\hfill \end{array}$$

(27)

therefore,

$$\begin{array}{cc}\hfill \Theta [e^{{sup}_{t\in [\mathfrak{a},\mathfrak{b}]}\left(\tau \right|\mathfrak{Y}u\left(t\right)-\mathfrak{Y}v\left(t\right)\left|\right)}-1]& \le \Theta [e^{{sup}_{t\in [\mathfrak{a},\mathfrak{b}]}\left({\displaystyle \frac{{\Xi }^{*}\mathcal{L}\left|u\right(t)-v(t\left)\right|}{\Xi }}\right)}-1]\hfill \\ & =\Theta [e^{{sup}_{t\in [\mathfrak{a},\mathfrak{b}]}\left(\alpha \right|u\left(t\right)-v\left(t\right)\left|\right)}-1],\alpha <1\hfill \\ & \le \Theta [e^{{sup}_{t\in [\mathfrak{a},\mathfrak{b}]}\left|u\right(t)-v(t\left)\right|}{-1]}^{\sqrt{\alpha }}\hfill \\ & ={(\Theta \left(\breve{\mathfrak{S}}(\mathfrak{p},\mathfrak{q},\tau )\right))}^{{\mathfrak{a}}_1},{\mathfrak{a}}_1=\sqrt{\alpha }<1.\hfill \end{array}$$

(28)

by Corollary 2, where ${\mathfrak{a}}_2={\mathfrak{a}}_3={\mathfrak{a}}_4=0$. Hence, $\mathfrak{Y}$ has exactly one solution. □

Example 5.

Consider the following problem:

$$\left\{\begin{array}{c}{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{1.8;log\left(t\right)}u\left(t\right)=sin\left(t^2\right)-{\displaystyle \frac{tcos\left(t\right)u\left(t\right)}{36(1+u(t\left)\right)}},t\in \mathfrak{J}=[{\displaystyle \frac{1}{2}},3],\hfill \\ u\left({\displaystyle \frac{1}{2}}\right)=0,\hfill \\ 3.75={\displaystyle \frac{1}{3}}u\left({\displaystyle \frac{5}{2}}\right)+{\displaystyle \frac{1}{4}}{}^{\mathcal{ABC}}\mathcal{D}_{{\mathfrak{a}}^{+}}^{{\displaystyle \frac{7}{4}};{\displaystyle \frac{u}{3}}}u\left({\displaystyle \frac{8}{3}}\right)+{\displaystyle \frac{1}{2}}{}^{\mathcal{AB}}\mathcal{I}_{{\mathfrak{a}}^{+}}^{{\displaystyle \frac{5}{3}};{\displaystyle \frac{u}{3}}}u\left({\displaystyle \frac{3}{2}}\right),\hfill \end{array}\right.$$

(29)

here ${\beta }_1={\displaystyle \frac{7}{4}},{\delta }_1={\displaystyle \frac{5}{3}},2,{\mathfrak{q}}_1={\displaystyle \frac{5}{2}},{\mathfrak{p}}_1={\displaystyle \frac{8}{3}},{\zeta }_1={\displaystyle \frac{3}{2}},{\gamma }_1={\displaystyle \frac{1}{3}},{\lambda }_1={\displaystyle \frac{1}{4}},{\sigma }_1={\displaystyle \frac{1}{2}},\mathfrak{H}(t,u\left(t\right))=sin\left(t^2\right)-{\displaystyle \frac{tcos\left(t\right)u\left(t\right)}{36(1+u(t\left)\right)}}.$

Now, we check the conditions of Theorem 5 as follows:

$$\begin{array}{cc}\hfill \left|\mathfrak{H}\right(t,u\left(t\right))-\mathfrak{H}(t,v\left(t\right)\left)\right|& =|sin\left(t^2\right)-{\displaystyle \frac{tcos\left(t\right)u\left(t\right)}{36(1+u(t\left)\right)}}-sin\left(t^2\right)-{\displaystyle \frac{tcos\left(t\right)v\left(t\right)}{36(1+v(t\left)\right)}}|\hfill \\ & \le {\displaystyle \frac{1}{12}}|u\left(t\right)-v\left(t\right)|,\hfill \end{array}$$

then, $\mathcal{L}={\displaystyle \frac{1}{3}}$; by taking $\mathcal{N}(\mu -1)=1$ we find ${\Xi }^{*}\cong 0.363,\Xi \cong 0.23305.$ Hence, in view of Theorem 5, the system (29) possesses a unique solution in $[{\displaystyle \frac{1}{2}},3].$

5. Conclusions

This research was devoted to introduce and study SEBPMSs that generalize EBPMSs and established the existence and uniqueness of fixed points through covariant and contravariant $JS$-contractions. Furthermore, we demonstrate the applicability of theorems in SEBPMS to derive new fixed point outcomes in $\Omega$-triangular SEFBMSs. The practical relevance of our results was highlighted by detailed examples, a clarification of the implications, and an application to $\Phi \text{-}\mathcal{A}\mathcal{B}\mathcal{C}$ fractional differential equations of order $1<\mu \le 2$ under NMB conditions (14) and their symmetric cases. Our study provides significant insights into the essential theories and uses of fixed-point theory in various metric spaces, such as Menger probabilistic bipolar metric spaces [32]. Indeed, regarding a different function $\Phi$ employed in this research, the $\Phi \text{-}\mathcal{A}\mathcal{B}\mathcal{C}$ fractional operator is more generalized, making our framework broader and addressing numerous new and existing issues in the field. It would be intriguing to explore the problem with an integral term for future work, considering NMB conditions.