Efficient Fixed-Point Method with Application to a Fractional Blood Flow Model

Abstract

This paper introduces two generalized frameworks, the extended bipolar parametric b-metricspace (EBPbMS) and the extended bipolar fuzzy b-metric space (EBFbMS), which unify and extend several existing bipolar and fuzzy metric structures. Within these settings, new fixed-point results are established for covariant and contravariant Meir–Keeler-type contractions. A fundamental correspondence between EBFbMSs and EBPbMSs is developed, providing a unified basis for analyzing convergence and stability in generalized metric environments. An illustrative example and an application to a fractional blood flow model confirm the effectiveness of the proposed approach and ensure the existence and uniqueness of the solution. These results demonstrate the capability of extended bipolar structures to model nonlinear fractional systems with memory effects.

1. Introduction and Preliminaries

The literature on fixed-point theory encompasses a wide spectrum of extensions of the classical Banach contraction principle. These developments primarily focus on introducing new types of contractive conditions and on generalizing the underlying metric structures. In particular, several notable generalizations of metric-type spaces have been proposed, including parametric metric spaces and bipolar metric spaces (see [1,2,3,4,5,6]).

Following these foundational works, numerous fixed-point theorems have been established by employing various contractive mappings within diverse generalized metric frameworks (see [7,8,9,10,11,12]).

Despite these advances, many existing structures treat the bipolar or parametric aspects in isolation, limiting their ability to describe systems where both positive and negative tendencies, or multi-parameter dependencies, coexist. In several nonlinear and fractional models, particularly those involving hereditary or memory effects, this interaction plays a crucial analytical role. For instance, bipolarity enables the simultaneous representation of attraction and repulsion forces, while the parametric extension provides a means to capture scale-dependent or contextual variations within the same space.

Motivated by these observations, the present paper introduces two unified frameworks, namely the extended bipolar parametric b-metric space (EBPbMS) and the extended bipolar fuzzy b-metric space (EBFbMS). These settings merge the flexibility of parametric generalizations with the dual-character structure of bipolar spaces, yielding a richer analytical environment for the study of generalized contractions. Within these frameworks, we establish several fixed-point results for covariant and contravariant Meir–Keeler-type contractions, supported by illustrative examples and an application to a fractional blood flow model. The proposed generalizations not only unify but also extend many existing results in the literature.

Definition 1

([13]). If the following conditions are satisfied, a function $\psi :[0,\infty )\to [0,\infty )$ is known as an altering distance function if

1. 

ψ is upper semi-continuous, non-decreasing;

2. 

${\left\{{\psi }^n\left(t\right)\right\}}_{n\in \mathbb{N}}$ approaches 0 as $n\to \infty ,$$t>0$;

3. 

$\psi \left(t\right)<t$ for every $t>0.$

We denote the set of the above functions ψ with $\Psi .$

In this work, we propose a new class of generalized bipolar metric spaces, termed extended bipolar parametric b-metric spaces (EBPbMSs)b, which extend and unify several existing bipolar and parametric metric frameworks.

Definition 2.

Let $\mathbb{A}$ and $\mathbb{B}$ be two nonempty sets such that $\mathbb{A}\cap \mathbb{B}\ne ⌀$, and let $\mathrm{\Omega }:[0,\infty )\to [0,\infty )$ be a continuous, strictly increasing function with $\mathrm{\Omega }\left(0\right)=0$ and $\mathrm{\Omega }\left(s\right)\ge s$ for all $s\ge 0$. A mapping $\varrho :\mathbb{A}\times \mathbb{B}\times (0,\infty )\to [0,\infty )$ is said to be an extended bipolar parametric b-metric on $\mathbb{A}\cup \mathbb{B}$ if, for every $\tau >0$, the following hold:

(ϱ₁)

For $\nu \in \mathbb{A}\cap \mathbb{B}$ and $\iota \in \mathbb{A}\cap \mathbb{B}$, $\varrho (\nu ,\iota ,\tau )=0$ if and only if $\nu =\iota$. For $\nu \in \mathbb{A}\setminus \mathbb{B}$ and $\iota \in \mathbb{B}\setminus \mathbb{A}$, $\varrho (\nu ,\iota ,\tau )>0$ for all $\tau >0$.

(ϱ₂)

$\varrho (\nu ,\iota ,\tau )=\varrho (\iota ,\nu ,\tau )$ whenever both sides are defined, i.e., for all $\nu ,\iota \in \mathbb{A}\cap \mathbb{B}$.

(ϱ₃)

For all $({\nu }_1,{\iota }_1),({\nu }_2,{\iota }_2)\in \mathbb{A}\times \mathbb{B}$,

$$\varrho ({\nu }_1,{\iota }_2,\tau )\le \mathrm{\Omega }\left(\varrho ({\nu }_1,{\iota }_1,\tau )+\varrho ({\nu }_2,{\iota }_1,\tau )+\varrho ({\nu }_2,{\iota }_2,\tau )\right).$$

Then $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ is called an extended bipolar parametric b-metric space (EBPbMS).

It is worth noting that

• When $\mathrm{\Omega }\left(\nu \right)=s\nu$, the EBPbMS reduces to the bipolar parametric b-metric space (BPbMS) (see [14]);
• When $\mathrm{\Omega }\left(\nu \right)=\nu$, it coincides with the bipolar parametric metric space (BPMS) (see [15]).

Definition 3.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be an EBPbMS with $\mathbb{A}\cap \mathbb{B}\ne ⌀$.

1. 

A point ι is called left if $\iota \in \mathbb{A}$, right if $\iota \in \mathbb{B}$, and central if $\iota \in \mathbb{A}\cap \mathbb{B}$.

2. 

A sequence $\left\{{\nu }_n\right\}\subset \mathbb{A}\cup \mathbb{B}$ is left if ${\nu }_n\in \mathbb{A}$ for all n, and right if ${\nu }_n\in \mathbb{B}$ for all n.

3. 

A sequence $\left\{{\nu }_n\right\}$ converges to a point ν if either (i) $\left\{{\nu }_n\right\}\subset \mathbb{A}$ and $\nu \in \mathbb{B}$ with $\varrho ({\nu }_n,\nu ,\tau )\to 0$ for all $\tau >0$, or (ii) $\left\{{\nu }_n\right\}\subset \mathbb{B}$ and $\nu \in \mathbb{A}$ with $\varrho (\nu ,{\nu }_n,\tau )\to 0$ for all $\tau >0$.

4. 

A pair sequence $\left\{({\nu }_n,{\iota }_n)\right\}\subset \mathbb{A}\times \mathbb{B}$ is called a bisequence.

5. 

The bisequence $\left\{({\nu }_n,{\iota }_n)\right\}$ is convergent if ${\nu }_n\to \nu$ and ${\iota }_n\to \iota$ in the above sense; it is biconvergent if, moreover, $\nu =\iota$ (so $\nu \in \mathbb{A}\cap \mathbb{B}$).

6. 

A bisequence $\left\{({\nu }_n,{\iota }_n)\right\}$ is Cauchy if for every $ϵ>0$ there exists $N\in \mathbb{N}$ such that

$$\varrho ({\nu }_n,{\iota }_m,\tau )<ϵ\mathit{for}\mathit{all}n,m\ge N\mathit{and}\mathit{all}\tau >0.$$

The space is complete if every Cauchy bisequence is convergent.

Definition 4.

Let $({\mathbb{A}}_1,{\mathbb{B}}_1,{\varrho }_1,{\mathrm{\Omega }}_1)$ and $({\mathbb{A}}_2,{\mathbb{B}}_2,{\varrho }_2,{\mathrm{\Omega }}_2)$ be two EBPbMSs and $T:({\mathbb{A}}_1,{\mathbb{B}}_1,{\varrho }_1,{\mathrm{\Omega }}_1)\to ({\mathbb{A}}_2,{\mathbb{B}}_2,{\varrho }_2,{\mathrm{\Omega }}_2)$ is a mapping which is referred to as

1. 

A covariant mapping if $T\left({\mathbb{A}}_1\right)\subseteq {\mathbb{B}}_1$ and $T\left({\mathbb{B}}_1\right)\subseteq {\mathbb{A}}_1$ and it is denoted by $T:({\mathbb{A}}_1,{\mathbb{B}}_1,{\varrho }_1,{\mathrm{\Omega }}_1)\rightrightarrows ({\mathbb{A}}_2,{\mathbb{B}}_2,{\varrho }_2,{\mathrm{\Omega }}_2).$

2. 

A contravariant mapping if $T\left({\mathbb{A}}_1\right)\subseteq {\mathbb{B}}_2$ and $T\left({\mathbb{B}}_1\right)\subseteq {\mathbb{A}}_2$ and it is written by $T:({\mathbb{A}}_1,{\mathbb{B}}_1,{\varrho }_1,{\mathrm{\Omega }}_1)\rightleftharpoons ({\mathbb{A}}_2,{\mathbb{B}}_2,{\varrho }_2,{\mathrm{\Omega }}_2).$

3. 

A left-continuous mapping at a point ${\nu }_0\in {\mathbb{A}}_1$ if for every sequence $\left\{{\iota }_n\right\}\subseteq {\mathbb{B}}_1$ with ${\iota }_n\to {\nu }_0$ we have $T\left({\iota }_n\right)\to T\left({\nu }_0\right)$ in $({\mathbb{A}}_2,{\mathbb{B}}_2,{\varrho }_2,{\mathrm{\Omega }}_2).$

4. 

A right-continuous mapping at a point ${\iota }_0\in {\mathbb{B}}_1$ if for any sequence $\left\{{\nu }_n\right\}\subseteq {\mathbb{A}}_1$ converging to ${\iota }_0$, it follows that $T\left({\nu }_n\right)\to T\left({\iota }_0\right)$ within $({\mathbb{A}}_2,{\mathbb{B}}_2,{\varrho }_2,{\mathrm{\Omega }}_2).$

5. 

A continuous mapping if it exhibits left-continuity at every point $\iota \in {\mathbb{A}}_1$ and right-continuity at every point $\iota \in {\mathbb{B}}_1$.

6. 

An orbital-left-continuous mapping if given ${\iota }_0\in {\mathbb{B}}_1$ and any sequence $\left\{k_n\right\}$ of positive integers, $\left\{T^{k_n}{\iota }_0\right\}\subseteq {\mathbb{B}}_1$ with $T^{k_n}{\iota }_0\to {\nu }_0$ we have $T\left(T^{k_n}{\iota }_0\right)\to T\left({\nu }_0\right)$ in $({\mathbb{A}}_2,{\mathbb{B}}_2,{\varrho }_2,{\mathrm{\Omega }}_2).$

7. 

An orbital-right-continuous mapping if given ${\nu }_0\in {\mathbb{A}}_1$ and any sequence $\left\{k_n\right\}$ of positive integers, $\left\{T^{k_n}{\nu }_0\right\}\subseteq {\mathbb{A}}_1$ with $T^{k_n}{\nu }_0\to {\iota }_0$ we have $T\left(T^{k_n}{\nu }_0\right)\to T\left({\iota }_0\right)$ in $({\mathbb{A}}_2,{\mathbb{B}}_2,{\varrho }_2,{\mathrm{\Omega }}_2).$

8. 

An orbital continuous mapping if it maintains orbital-left-continuity at every point $\iota \in {\mathbb{A}}_1$ and orbital-right-continuity at every point $\iota \in {\mathbb{B}}_1$.

Example 1.

Let $\mathcal{L}$ be the set of all Lebesgue measurable functions on $[0,1]$ such that ${\int }_0^1\left|f\left(\nu \right)\right|d\nu <\infty .$ Now, let $\mathbb{A}=\{f\in \mathcal{L}:f(\nu )\ge 0$ for all $\nu \in [0,{\displaystyle \frac{1}{2}}]$ and $f\left(\nu \right)\le 0$ for all $\nu \in ({\displaystyle \frac{1}{2}},1]\}$ and $\mathbb{B}=\{g\in \mathcal{L}:g(\nu )\le 0$ for all $\nu \in [0,{\displaystyle \frac{1}{2}}]$ and $g\left(\nu \right)\ge 0$ for all $\nu \in ({\displaystyle \frac{1}{2}},1]\}.$ Let $\varrho :\mathbb{A}\times \mathbb{B}\times (0,\infty )\to [0,\infty )$ be given by

$$\varrho (f,g,\tau )={\left[1+\tau {\int }_0^1|f\left(\nu \right)-g\left(\nu \right)|d\nu \right]}^3-1,$$

for all $f\in \mathbb{A}$, $g\in \mathbb{B}$ and for all $\tau >0.$ Then $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ is (EBPbMS) with the extended function $\mathrm{\Omega }={(1+s)}^3-1$ for all $s\ge 0.$ The conditions ($\varrho 1$) and ($\varrho 2$) of Definition 2 hold, and the condition ($\varrho 3$) remains to be proven. For all $(l,g),(f,h)\in \mathbb{A}\times \mathbb{B},\tau >0$, then we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \varrho (f,g,\tau )& ={\left[1+\tau {\int }_0^1|f\left(\nu \right)-g\left(\nu \right)|d\nu \right]}^3-1\hfill \\ & ={\left[1+\tau {\int }_0^1|f\left(\nu \right)-h\left(\nu \right)+h\left(\nu \right)-l\left(\nu \right)+l\left(\nu \right)-g\left(\nu \right)|d\nu \right]}^3-1\hfill \\ & \le {\left[1+\tau {\int }_0^1|f\left(\nu \right)-h\left(\nu \right)|+|h\left(\nu \right)-l\left(\nu \right)|+|l\left(\nu \right)-g\left(\nu \right)|d\nu \right]}^3-1\hfill \\ & \le {\left[1+\tau {\int }_0^1|f\left(\nu \right)-h\left(\nu \right)|d\nu \right]}^3+{\left[1+\tau {\int }_0^1|l\left(\nu \right)-h\left(\nu \right)|d\nu \right]}^3\hfill \\ & +{\left[1+\tau {\int }_0^1|l\left(\nu \right)-g\left(\nu \right)|d\nu \right]}^3-3\hfill \\ & \le [{\left[1+\tau {\int }_0^1|f\left(\nu \right)-h\left(\nu \right)|d\nu \right]}^3-1+{\left[1+\tau {\int }_0^1|l\left(\nu \right)-h\left(\nu \right)|d\nu \right]}^3-1\hfill \\ & +{\left[1+\tau {\int }_0^1|l\left(\nu \right)-g\left(\nu \right)|d\nu \right]}^3-1{]}^3-1\hfill \\ \hfill & =\mathrm{\Omega }\left[\varrho (f,h,\tau )+\varrho (l,h,\tau )+\varrho (l,g,\tau )\right].\hfill \end{array}\end{array}$$

Proposition 1.

In an EBPbMS $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ every convergent Cauchy bisequence is biconvergent.

Proof. 

Let $\left\{({\nu }_n,{\iota }_n)\right\}$ be a Cauchy bisequence converging to $(\nu ,\iota )\in \mathbb{A}\times \mathbb{B}$ that ${\nu }_n\to \iota$ and ${\iota }_n\to \nu$ as $n\to \infty$. Then,

$$\begin{array}{cc}\hfill \varrho (\nu ,\iota ,\tau )\le & \mathrm{\Omega }[\varrho (\nu ,{\iota }_m,\tau )+\varrho ({\nu }_m,{\iota }_m,\tau )+\varrho ({\nu }_m,\iota ,\tau )],\hfill \\ & =\mathrm{\Omega }\left(0\right)=0\hfill \end{array}$$

(1)

Taking $m\to \infty$ on the right-hand side of (1), we get $\varrho (\nu ,\iota ,\tau )=0$, and therefore, $\nu \in \mathbb{A}\cap \mathbb{B}$. Hence, the bisequence $\{{\nu }_n,{\iota }_n\}$ is biconvergent. □

Remark 1.

Proposition 1 shows that if a Cauchy bisequence biconverges to some $\nu \in \mathbb{A}\times \mathbb{B}$ then $\varrho (\nu ,\nu ,\tau )=0$.

Topological Properties of EBPbMS

Definition 5

(Topology induced by $\varrho$). Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be an extended bipolar parametric b-metric space (EBPbMS) and fix $\tau >0$. For every $\nu \in \mathbb{A}$ and $\epsilon >0$, define the open ϱ-ball centered at ν by

$$B_{\varrho }(\nu ,\epsilon ,\tau )=\{\iota \in \mathbb{B}:\varrho (\nu ,\iota ,\tau )<\epsilon \}.$$

The collection

$${\mathcal{T}}_{\varrho }=\{U\subseteq \mathbb{A}\cup \mathbb{B}:\forall \nu \in U,\exists \epsilon >0,\tau >0\mathrm{such}\mathrm{that}{B}_{\varrho }(\nu ,\epsilon ,\tau )\subseteq U\}$$

is called the topology induced by ϱ on $\mathbb{A}\cup \mathbb{B}$.

Proposition 2.

If Ω is strictly increasing and continuous, then the topology ${\mathcal{T}}_{\varrho }$ is Hausdorff; that is, for any two distinct points $\nu ,\iota \in \mathbb{A}\cup \mathbb{B}$, there exist ${\epsilon }_1,{\epsilon }_2>0$ and $\tau >0$ such that $B_{\varrho }(\nu ,{\epsilon }_1,\tau )\cap B_{\varrho }(\iota ,{\epsilon }_2,\tau )=⌀.$

Remark 2.

When $\mathrm{\Omega }\left(s\right)=ks$ for some constant $k\ge 1$, the mapping $d(\nu ,\iota )={\displaystyle \frac{1}{k}}\varrho (\nu ,\iota ,1)$ defines a metric topologically equivalent to ϱ. Hence the space $(\mathbb{A}\cup \mathbb{B},{\mathcal{T}}_{\varrho })$ is metrizable. Moreover, if $\mathrm{\Omega }\left(s\right)=s$, the induced topology coincides with the standard bipolar metric topology, which in turn generalizes the classical metric topology.

Definition 6

(Pairwise topology induced by $\varrho$). Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be an EBPbMS and fix $\tau >0$. Define subbases

$${\mathcal{S}}_{\mathbb{A}}:=\left\{U(b,\epsilon ,\tau ):=\{a\in \mathbb{A}:\varrho (a,b,\tau )<\epsilon \}b\in \mathbb{B},\epsilon >0\right\},$$

$${\mathcal{S}}_{\mathbb{B}}:=\left\{V(a,\epsilon ,\tau ):=\{b\in \mathbb{B}:\varrho (a,b,\tau )<\epsilon \}a\in \mathbb{A},\epsilon >0\right\}.$$

Let ${\mathcal{T}}_{\mathbb{A}}$ (resp. ${\mathcal{T}}_{\mathbb{B}}$) be the topology on $\mathbb{A}$ (resp. on $\mathbb{B}$) generated by ${\mathcal{S}}_{\mathbb{A}}$ (resp. ${\mathcal{S}}_{\mathbb{B}}$). We call $(\mathbb{A},{\mathcal{T}}_{\mathbb{A}})$ and $(\mathbb{B},{\mathcal{T}}_{\mathbb{B}})$ the left and right bipolar topologies induced by ϱ.

Lemma 1

(Point-separation on each side). For any distinct $a_1,a_2\in \mathbb{A}$ and any fixed $\tau >0$, there exists $b\in \mathbb{B}$ and ${\epsilon }_1,{\epsilon }_2>0$ such that $a_1\in U(b,{\epsilon }_1,\tau )$, $a_2\notin U(b,{\epsilon }_1,\tau )$ and $a_2\in U(b,{\epsilon }_2,\tau )$, $a_1\notin U(b,{\epsilon }_2,\tau )$. An analogous statement holds for distinct $b_1,b_2\in \mathbb{B}$. Consequently, $(\mathbb{A},{\mathcal{T}}_{\mathbb{A}})$ and $(\mathbb{B},{\mathcal{T}}_{\mathbb{B}})$ are $T_1$.

Proof. 

Fix $\tau >0$. If $a_1\ne a_2$, then by $\left({\varrho }_1\right)$, there exists $b\in \mathbb{B}$ with $\varrho (a_1,b,\tau )\ne \varrho (a_2,b,\tau )$; otherwise, if $\varrho (a_1,b,\tau )=\varrho (a_2,b,\tau )$ for all b, then for any b and any $\epsilon >\varrho (a_1,b,\tau )$ we would have $a_1,a_2\in U(b,\epsilon ,\tau )$ and for any $0<\epsilon <\varrho (a_1,b,\tau )$ we would have $a_1,a_2\notin U(b,\epsilon ,\tau )$, contradicting $\left({\varrho }_1\right)$ by the standard separation via subbasic sets (one can refine by using two different b’s if needed). Without loss of generality, assume $\varrho (a_1,b,\tau )<\varrho (a_2,b,\tau )$. Choose ${\epsilon }_1$ with $\varrho (a_1,b,\tau )<{\epsilon }_1<\varrho (a_2,b,\tau )$; then $a_1\in U(b,{\epsilon }_1,\tau )$ and $a_2\notin U(b,{\epsilon }_1,\tau )$. Similarly, choose ${\epsilon }_2$ with $\varrho (a_2,b,\tau )<{\epsilon }_2$ and ${\epsilon }_2<\varrho (a_1,b,\tau )$ if the order is reversed. The case of $\mathbb{B}$ is analogous by symmetry of the construction of ${\mathcal{S}}_{\mathbb{B}}$. □

Proposition 3

(Hausdorffness on each side). Assume Ω is strictly increasing and continuous. Then $(\mathbb{A},{\mathcal{T}}_{\mathbb{A}})$ and $(\mathbb{B},{\mathcal{T}}_{\mathbb{B}})$ are Hausdorff spaces.

Proof. 

Fix distinct $a_1,a_2\in \mathbb{A}$ and $\tau >0$. By Lemma 1, pick $b\in \mathbb{B}$ and ${\epsilon }_1,{\epsilon }_2>0$ with $a_1\in U(b,{\epsilon }_1,\tau )$ and $a_2\notin U(b,{\epsilon }_1,\tau )$, and $a_2\in U(b,{\epsilon }_2,\tau )$ and $a_1\notin U(b,{\epsilon }_2,\tau )$. Then the open neighborhoods $U_1:=U(b,{\epsilon }_1,\tau )$ of $a_1$ and $U_2:=U(b,{\epsilon }_2,\tau )$ of $a_2$ are disjoint by construction. (Equivalently, one can consider the family of evaluation maps $e_{(b,\tau )}:\mathbb{A}\to [0,\infty )$, $e_{(b,\tau )}\left(a\right):=\varrho (a,b,\tau )$; these maps separate points by Lemma 1, hence the initial topology they generate is Hausdorff.) The proof for $(\mathbb{B},{\mathcal{T}}_{\mathbb{B}})$ is analogous. □

Corollary 1

(Hausdorffness of the topological sum). Let $(\mathbb{A},{\mathcal{T}}_{\mathbb{A}})$ and $(\mathbb{B},{\mathcal{T}}_{\mathbb{B}})$ be the left and right topologies induced by ϱ as above. Consider the disjoint topological sum $\mathbb{A}\bigsqcup \mathbb{B}$ endowed with the sum topology. If each of $(\mathbb{A},{\mathcal{T}}_{\mathbb{A}})$ and $(\mathbb{B},{\mathcal{T}}_{\mathbb{B}})$ is Hausdorff, then $\mathbb{A}\bigsqcup \mathbb{B}$ is Hausdorff.

Proof. 

Standard: the topological sum of Hausdorff spaces is Hausdorff. □

Definition 7

(Cross-separation condition (C)). We say that an EBPbMS satisfies (C) if for every $a\in \mathbb{A}\setminus \mathbb{B}$ and $b\in \mathbb{B}\setminus \mathbb{A}$ there exist $\tau >0$ and ${\epsilon }_A,{\epsilon }_B>0$ such that

$$U(b,{\epsilon }_A,\tau )\subseteq \mathbb{A}\setminus \mathbb{B}\mathit{and}V(a,{\epsilon }_B,\tau )\subseteq \mathbb{B}\setminus \mathbb{A}.$$

Proposition 4

(Hausdorffness on the union). Assume Ω is strictly increasing and continuous. Endow $\mathbb{A}\cup \mathbb{B}$ with the initial topology ${\mathcal{T}}_{\times }$ generated by the subbase $\mathcal{S}=\left\{U\right(b,\epsilon ,\tau \left)\right\}\cup \left\{V\right(a,\epsilon ,\tau \left)\right\}$. If $(\mathbb{A},{\mathcal{T}}_{\mathbb{A}})$ and $(\mathbb{B},{\mathcal{T}}_{\mathbb{B}})$ are Hausdorff and condition (C) holds, then $(\mathbb{A}\cup \mathbb{B},{\mathcal{T}}_{\times })$ is Hausdorff.

Proof. 

We must separate any two distinct points $x\ne y$ in $\mathbb{A}\cup \mathbb{B}$.

Case 1: $x,y\in \mathbb{A}$. Since $(\mathbb{A},{\mathcal{T}}_{\mathbb{A}})$ is Hausdorff, pick disjoint $U_x,U_y\in {\mathcal{T}}_{\mathbb{A}}$ with $x\in U_x$, $y\in U_y$. They are open in ${\mathcal{T}}_{\times }$ by definition.

Case 2: $x,y\in \mathbb{B}$. Analogous using $(\mathbb{B},{\mathcal{T}}_{\mathbb{B}})$.

Case 3: $x\in \mathbb{A}\setminus \mathbb{B}$ and $y\in \mathbb{B}\setminus \mathbb{A}$. By (C), choose $\tau >0$ and ${\epsilon }_A,{\epsilon }_B>0$ with $U(y,{\epsilon }_A,\tau )\subseteq \mathbb{A}\setminus \mathbb{B}$ and $V(x,{\epsilon }_B,\tau )\subseteq \mathbb{B}\setminus \mathbb{A}$. Then $x\in U(y,{\epsilon }_A,\tau )$, $y\in V(x,{\epsilon }_B,\tau )$ and the two opens are disjoint.

Case 4 (central/mixed): If one (or both) of $x,y$ lies in $\mathbb{A}\cap \mathbb{B}$, use the Hausdorffness of each side to find $U_x\in {\mathcal{T}}_{\mathbb{A}}$ for the x-side, and for the y-side, $V_y\in {\mathcal{T}}_{\mathbb{B}}$; then, if needed, shrink them using (C) so that $U_x\cap V_y=⌀$ (possible because (C) prevents any central points from belonging simultaneously to both small neighborhoods). Hence, in all cases, we obtain disjoint open neighborhoods separating x and y. □

Proposition 5

(Metrizability under a linear $\mathrm{\Omega }$). If $\mathrm{\Omega }\left(s\right)=ks$ for some constant $k\ge 1$, define $d_{\times }:{\left(\mathbb{A}\cup \mathbb{B}\right)}^2\to [0,\infty )$ by

$$d_{\times }(x,y):=\left\{\begin{array}{cc}0,\hfill & x=y,\hfill \\ {\displaystyle \frac{1}{k}}\varrho (x,y,1),\hfill & x\in \mathbb{A},y\in \mathbb{B},\hfill \\ {\displaystyle \frac{1}{k}}\varrho (y,x,1),\hfill & x\in \mathbb{B},y\in \mathbb{A},\hfill \\ \underset{b\in \mathbb{B}}{inf}\left({\displaystyle \frac{1}{k}}\varrho (x,b,1)+{\displaystyle \frac{1}{k}}\varrho (y,b,1)\right),\hfill & x,y\in \mathbb{A},\hfill \\ \underset{a\in \mathbb{A}}{inf}\left({\displaystyle \frac{1}{k}}\varrho (a,x,1)+{\displaystyle \frac{1}{k}}\varrho (a,y,1)\right),\hfill & x,y\in \mathbb{B}.\hfill \end{array}\right.$$

Then $d_{\times }$ is a metric on $\mathbb{A}\cup \mathbb{B}$. Moreover, the metric topologies induced by $d_{\times }$ restrict $\mathbb{A}$ and $\mathbb{B}$ to ${\mathcal{T}}_{\mathbb{A}}$ and ${\mathcal{T}}_{\mathbb{B}}$, respectively.

Proof. 

(i) Non-negativity and symmetry are clear by definition and $\left({\varrho }_1\right)$–$\left({\varrho }_2\right)$ (noting the cross-terms). If $x\ne y$ and $x,y\in \mathbb{A}$ (resp. in $\mathbb{B}$), then by $\left({\varrho }_1\right)$ there exists $b\in \mathbb{B}$ (resp. $a\in \mathbb{A}$) such that at least one of the summands in the infimum is $>0$, hence $d_{\times }(x,y)>0$. If $x\in \mathbb{A},y\in \mathbb{B}$, then $d_{\times }(x,y)={\displaystyle \frac{1}{k}}\varrho (x,y,1)>0$ when $x\ne y$ by $\left({\varrho }_1\right)$.

(ii) Triangle inequality. We check the representative cases; the others are similar. Let $x\in \mathbb{A},z\in \mathbb{B}$ and arbitrary y. If $y\in \mathbb{A}$, choose $b\in \mathbb{B}$ so that

$$d_{\times }(x,y)+ϵ>\frac{1}{k}\varrho (x,b,1)+\frac{1}{k}\varrho (y,b,1).$$

By $\left({\varrho }_3\right)$ with $\mathrm{\Omega }\left(s\right)=ks$,

$$\varrho (x,z,1)\le k\left(\varrho (x,b,1)+\varrho (y,b,1)+\varrho (y,z,1)\right).$$

Divideby k and take the infimum over b to get

$$d_{\times }(x,z)\le d_{\times }(x,y)+d_{\times }(y,z)+ϵ.$$

Let $ϵ\downarrow 0$. The remaining mixed cases $x,y,z$ in $\mathbb{A}$ or $\mathbb{B}$ follow from the same estimate applied to appropriate bridges $b\in \mathbb{B}$ or $a\in \mathbb{A}$ and taking infima.

(iii) Topology agreement. If $x\in \mathbb{A}$ and $b\in \mathbb{B}$, then

$$\{x\in \mathbb{A}:d_{\times }(x,b)<\epsilon \}=\{x\in \mathbb{A}:\varrho (x,b,1)<k\epsilon \}=U\left(b,k\epsilon ,1\right),$$

so the $d_{\times }$-open sets on $\mathbb{A}$ agree with ${\mathcal{T}}_{\mathbb{A}}$. The proof on $\mathbb{B}$ is analogous. □

Proposition 6.

In an EBPbMS $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$, every convergent Cauchy bisequence is biconvergent: if $\left\{({\nu }_n,{\iota }_n)\right\}\subset \mathbb{A}\times \mathbb{B}$ is Cauchy and there exist $\nu \in \mathbb{A}$, $\iota \in \mathbb{B}$ with ${\iota }_n\to \nu$ and ${\nu }_n\to \iota$; then, necessarily, $\nu =\iota \in \mathbb{A}\cap \mathbb{B}$.

Proof. 

Fix $\tau >0$. For any $ϵ>0$, since $\left\{({\nu }_n,{\iota }_n)\right\}$ is Cauchy, there exists N such that $\varrho ({\nu }_n,{\iota }_m,\tau )<ϵ$ for all $n,m\ge N$. Using $\left({\varrho }_3\right)$ for ${\nu }_1=\nu$, ${\iota }_1={\iota }_m$, ${\nu }_2={\nu }_n$, ${\iota }_2=\iota$, we obtain

$$\varrho (\nu ,\iota ,\tau )\le \mathrm{\Omega }\left(\varrho (\nu ,{\iota }_m,\tau )+\varrho ({\nu }_n,{\iota }_m,\tau )+\varrho ({\nu }_n,\iota ,\tau )\right).$$

Let $n,m\to \infty$. Because ${\iota }_m\to \nu$ and ${\nu }_n\to \iota$, the first and third terms tend to be 0; the middle term is $<ϵ$ for $n,m\ge N$. By continuity and monotonicity of $\mathrm{\Omega }$, the right-hand side tends to $\mathrm{\Omega }\left(0\right)=0$ (note that from ${\mathrm{\Omega }}^{-1}\left(0\right)\le 0\le \mathrm{\Omega }\left(0\right)$ and monotonicity, we get $\mathrm{\Omega }\left(0\right)=0$). Hence $\varrho (\nu ,\iota ,\tau )=0$ for all $\tau >0$ and thus, by $\left({\varrho }_1\right)$, $\nu =\iota \in \mathbb{A}\cap \mathbb{B}$. □

Remark 3.

Proposition 6 yields uniqueness of the bipolar limit for Cauchy bisequences. Together with Proposition 5, when $\mathrm{\Omega }\left(s\right)=ks$, the convergence in $(\mathbb{A},{\mathcal{T}}_{\mathbb{A}})$ and $(\mathbb{B},{\mathcal{T}}_{\mathbb{B}})$ agrees with the metric convergence in $(\mathbb{A}\cup \mathbb{B},d_{\times })$.

Remark 4

(Reduction to classical cases). If $\mathrm{\Omega }\left(s\right)=s$ and $\mathbb{A}=\mathbb{B}=X$, then ϱ becomes an ordinary metric on X, and all fixed-point results obtained in this setting reduce to the classical Banach contraction principle and its well-known extensions. Thus, EBPbMS generalizes both the b–metric and bipolar frameworks while preserving their topological consistency.

2. Main Results

We now present a generalized covariant and contravariant $\alpha$-$\psi$-Meir-Keeler contractive in EBPbMSs and establish fixed-point theorems for these functions in EBPbMSs.

Definition 8.

Assume $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ is an EBPbMS, $T:\mathbb{A}\cup \mathbb{B}\rightrightarrows \mathbb{A}\cup \mathbb{B}$ is a contravariant mapping and $\alpha :\mathbb{A}\times \mathbb{B}\times (0,\infty )\to [0,\infty )$ is a defined function. We define T as α-orbital admissible mapping if for all $(\nu ,\iota )\in \mathbb{A}\times \mathbb{B},$ and $\tau >0,$

$$\alpha (\nu ,\iota ,\tau )\ge 1\mathit{implies}\alpha (T\nu ,T\iota ,\tau )\ge 1$$

(2)

Definition 9.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be an EBPbMS and $T:\mathbb{A}\cup \mathbb{B}\rightrightarrows \mathbb{A}\cup \mathbb{B}$ be a covariant mapping. Also, suppose that $\alpha :\mathbb{A}\times \mathbb{B}\times (0,\infty )\to [0,\infty )$ and $\psi \in \Psi .$ Then T is referred to as a generalized covariant α-ψ-Meir-Keeler contractive mapping if for each $ϵ>0$ there exists $\delta >0,$ such that

$$\psi \left(ϵ\right)\le {\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}(\nu ,\iota ,\tau )\right)\right)<\psi \left(ϵ\right)+\delta \Rightarrow \alpha (\nu ,T\iota ,\tau )\alpha (T\nu ,\iota ,\tau )\varrho (T\nu ,T\iota ,\tau )<\psi \left(ϵ\right),$$

(3)

where

$$\mathbb{M}(\nu ,\iota ,\tau )=max\left\{\varrho (\nu ,\iota ,\tau ),\varrho (\nu ,T\iota ,\tau )),\varrho (T\nu ,\iota ,\tau )\right\},$$

for all $(\nu ,\iota )\in \mathbb{A}\times \mathbb{B},\tau >0$.

Definition 10.

Assume $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ is an EBPbMS, $T:\mathbb{A}\cup \mathbb{B}\rightleftharpoons \mathbb{A}\cup \mathbb{B}$ is a contravariant mapping and $\alpha :\mathbb{A}\times \mathbb{B}\times (0,\infty )\to [0,\infty )$ is a defined function. We define T as α-orbital admissible mapping if

for all $(\nu ,\iota )\in \mathbb{A}\times \mathbb{B},$ and $\tau >0,$

$$\alpha (\nu ,T\nu ,\tau )\ge 1\mathit{implies}\alpha (T^2\nu ,T\nu ,\tau )\ge 1,$$

(4)

and

$$\alpha (T\iota ,\iota ,\tau )\ge 1\mathit{implies}\alpha (T\iota ,T^2\iota ,\tau )\ge 1.$$

(5)

Definition 11.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be an EBPbMS and $T:\mathbb{A}\cup \mathbb{B}\rightleftharpoons \mathbb{A}\cup \mathbb{B}$ represent contravariant mapping. Also, suppose that $\alpha :\mathbb{A}\times \mathbb{B}\times (0,\infty )\to [0,\infty )$ and $\psi \in \Psi .$ Then T is referred to as a generalized contravariant $-\alpha$-ψ-Meir-Keeler contractive mapping if for each $ϵ>0$ there exists $\delta >0,$ such that

$$\psi \left(ϵ\right)\le {\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}(\nu ,\iota ,\tau )\right)\right)<ϵ+\delta \Rightarrow \alpha (\nu ,T\nu ,\tau )\alpha (T\iota ,\iota ,\tau )\psi \left(\varrho (T\iota ,T\nu ,\tau )\right)<\psi \left(ϵ\right),$$

(6)

where

$$\mathbb{M}(\nu ,\iota ,\tau )=max\left\{\varrho (\nu ,\iota ,\tau ),\varrho (\nu ,T\nu ,\tau )),\varrho (T\iota ,\iota ,\tau )\right\},$$

for all $(\nu ,\iota )\in \mathbb{A}\times \mathbb{B},\tau >0$.

Remark 5.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be an EBPbMS with Ω continuous and strictly increasing, and let $T:\mathbb{A}\cup \mathbb{B}\rightleftharpoons \mathbb{A}\cup \mathbb{B}$ be a generalized contravariant α–ψ–Meir–Keeler contractive mapping. Define the pointwise threshold:

$$\eta (\nu ,\iota ,\tau ):={\mathrm{\Omega }}^{-1}\left(\psi \left(M\right(\nu ,\iota ,\tau \left)\right)\right).$$

Then, for every $\tau >0$ and all admissible pairs with $\alpha (\nu ,\iota ,\tau )\ge 1$,

$$\nu \ne \iota ⟹\varrho (T\iota ,T\nu ,\tau )<\eta (\nu ,\iota ,\tau ),\nu =\iota ⟹\varrho (T\iota ,T\nu ,\tau )\le \eta (\nu ,\iota ,\tau ).$$

Proof. 

Fix $\tau >0$ and $(\nu ,\iota )$ with $\alpha (\nu ,\iota ,\tau )\ge 1$. Set $\eta :=\eta (\nu ,\iota ,\tau )={\mathrm{\Omega }}^{-1}\left(\psi \left(M(\nu ,\iota ,\tau )\right)\right)$. By definition of $\eta$ and the monotonicity of $\mathrm{\Omega }$ and $\psi$,

$$\psi \left(M\right(\nu ,\iota ,\tau \left)\right)=\psi \left(\mathrm{\Omega }\right(\eta \left)\right).$$

By the generalized contravariant $\alpha$–$\psi$–Meir–Keeler contractive condition (applied to the pair $(\nu ,\iota )$), we obtain

$$\psi \left(\varrho (T\nu ,T\iota ,\tau )\right)<\psi \left(\mathrm{\Omega }\left(\eta \right)\right)=\psi \left(\mathrm{\Omega }\left(\eta \right(\nu ,\iota ,\tau \left)\right)\right).$$

Since $\psi$ is non-decreasing and $\mathrm{\Omega }$ is strictly increasing, it follows that $\varrho (T\nu ,T\iota ,\tau )<\mathrm{\Omega }\left(\eta \right)=\mathrm{\Omega }\left({\mathrm{\Omega }}^{-1}\left(\psi \left(M(\nu ,\iota ,\tau )\right)\right)\right)$. Hence $\varrho (T\nu ,T\iota ,\tau )<\eta (\nu ,\iota ,\tau )$ when $\nu \ne \iota$ (strict MK-inequality), and if $\nu =\iota$, the same reasoning yields the non-strict bound $\varrho (T\nu ,T\iota ,\tau )\le \eta (\nu ,\iota ,\tau )$. □

Theorem 1.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be a complete EBPbMS. Suppose that $T:\mathbb{A}\cup \mathbb{B}\rightleftharpoons \mathbb{A}\cup \mathbb{B}$ is a generalized contravariant α-ψ-Meir-Keeler contractive mapping. Should the subsequent conditions be satisfied,

1. 

T is α-orbital admissible;

2. 

For some ${\nu }_0\in \mathbb{A}$, it holds that $\alpha ({\nu }_0,T{\nu }_0,\tau )\ge 1$;

3. 

T is an orbital continuous;

then T has a fixed point.

Proof. 

Assume ${\nu }_0\in \mathbb{A}$ satisfying $\alpha ({\nu }_0,T{\nu }_0,\tau )\ge 1$. Define the sequences $\left\{{\nu }_n\right\}$ and $\left\{{\iota }_n\right\}$ where ${\iota }_n=T{\nu }_n$ and ${\nu }_{n+1}=T{\iota }_n$ for each $n\in \mathbb{N}$. It is evident that $\left\{({\nu }_n,{\iota }_n)\right\}$ forms a bisequence. Given that T is $\alpha$-orbital admissible, it follows that

$$\begin{array}{c}\hfill \alpha ({\nu }_0,{\iota }_0,\tau )=\alpha ({\nu }_0,T{\nu }_0,\tau )\ge 1\Rightarrow \alpha (T^2{\nu }_0,T{\nu }_0,\tau )=\alpha ({\nu }_1,{\iota }_0,\tau )\ge 1,\\ \hfill \alpha ({\nu }_1,{\iota }_0,\tau )=\alpha (T{\iota }_0,{\iota }_0,\tau )\ge 1\Rightarrow \alpha (T{\iota }_0,{\mathfrak{T}}^2{\iota }_0,\tau )=\alpha ({\nu }_1,{\iota }_1,\tau )\ge 1,\\ \hfill \alpha ({\nu }_1,{\iota }_1,\tau )=\alpha ({\nu }_1,T{\nu }_1,\tau )\ge 1\Rightarrow \alpha (T^2{\nu }_1,T{\nu }_1,\tau )=\alpha ({\nu }_2,{\iota }_1,\tau )\ge 1,\\ \hfill \alpha ({\nu }_2,{\iota }_1,\tau )=\alpha (T{\iota }_1,{\iota }_1,\tau )\ge 1\Rightarrow \alpha (T{\iota }_1,T^2{\iota }_1,\tau )=\alpha ({\nu }_2,{\iota }_2,\tau )\ge 1.\end{array}$$

Proceeding in this manner, we obtain

$$\alpha ({\nu }_n,{\iota }_n,\tau )\ge 1\mathrm{and}\alpha ({\nu }_{n+1},{\iota }_n,\tau )\ge 1,\forall n\in \mathbb{N}\mathrm{and}\tau >0.$$

(7)

Referencing Remark 5 and (7), we derive the following relationship:

$$\begin{array}{cc}\hfill \psi \left(\varrho ({\nu }_n,{\iota }_n,\tau )\right)=& \psi \left(\varrho (T{\iota }_{n-1},T{\nu }_n,\tau )\right)\le \alpha ({\nu }_n,T{\nu }_n,\tau )\alpha (T{\iota }_{n-1},{\iota }_{n-1},\tau )\psi \left(\varrho (T{\iota }_{n-1},T{\nu }_n,\tau )\right)\hfill \\ \hfill <& {\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}({\nu }_n,{\iota }_{n-1},\tau )\right)\right)\hfill \\ \hfill =& {\mathrm{\Omega }}^{-1}\left(\psi \left(max\left\{\varrho ({\nu }_n,{\iota }_{n-1},\tau ),\varrho ({\nu }_n,T{\nu }_n,\tau ),\varrho (T{\iota }_{n-1},{\iota }_{n-1},\tau )\right\}\right)\right)\hfill \\ \hfill =& {\mathrm{\Omega }}^{-1}\left(\psi (max\left\{\varrho ({\nu }_n,{\iota }_{n-1},\tau ),\varrho ({\nu }_n,{\iota }_n,\tau ),\varrho ({\nu }_n,{\iota }_{n-1},\tau )\right\})\right)\hfill \\ \hfill \le & {\mathrm{\Omega }}^{-1}\left(\psi (max\left\{\varrho ({\nu }_n,{\iota }_n,\tau ),\varrho ({\nu }_n,{\iota }_{n-1},\tau )\right\})\right).\hfill \end{array}$$

If we suppose that $\psi \left(\varrho ({\nu }_n,{\iota }_n,\tau )\right)>\varrho ({\nu }_n,{\iota }_{n-1},\tau )$, then $\psi \left(\varrho ({\nu }_n,{\iota }_n,\tau )\right)<{\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho ({\nu }_n,{\iota }_n,\tau )\right)\right)\le \psi \left(\varrho ({\nu }_n,{\iota }_n,\tau )\right)<\varrho ({\nu }_n,{\iota }_n,\tau ),$ which is a contradiction. Hence

$$\psi \left(\varrho ({\nu }_n,{\iota }_n,\tau )\right)\le {\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho ({\nu }_n,{\iota }_{n-1},\tau )\right)\right)\le \psi \left(\varrho ({\nu }_n,{\iota }_{n-1},\tau )\right).$$

(8)

That is

$$\varrho ({\nu }_n,{\iota }_n,\tau )\le \varrho ({\nu }_n,{\iota }_{n-1},\tau ),$$

(9)

for every $n\in \mathbb{N}$. Likewise, through the application of Remark 5 and (7), we can readily derive

$$\psi \left(\varrho ({\nu }_{n+1},{\iota }_n,\tau )\right)\le {\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho ({\nu }_n,{\iota }_n,\tau )\right)\right)\le \psi \left(\varrho ({\nu }_n,{\iota }_n,\tau )\right).$$

(10)

That is, for all $n\in \mathbb{N}$ and $\tau >0$,

$$\varrho ({\nu }_{n+1},{\iota }_n,\tau )\le \varrho ({\nu }_n,{\iota }_n,\tau ).$$

(11)

Combining (8) and (10), we deduce that

$$\begin{array}{cc}\hfill \psi \left(\varrho ({\nu }_n,{\iota }_n,\tau )\right)\le \psi \left(\varrho ({\nu }_n,{\iota }_{n-1},\tau )\right)& \le {\psi }^2\left(\varrho ({\nu }_{n-1},{\iota }_{n-1},\tau )\right)\hfill \end{array}$$

(12)

$$\begin{array}{cc}& ⋮\hfill \end{array}$$

(13)

$$\begin{array}{cc}& \le {\psi }^n\left(\varrho ({\nu }_0,{\iota }_0,\tau )\right)\hfill \end{array}$$

(14)

for every $n\in \mathbb{N}$. Using the property (ii) of $\psi$, it is clear that

$$\underset{n\to \infty }{lim}\varrho ({\nu }_n,{\iota }_n,\tau )=0.$$

(15)

Similarly, we can easily get

$$\underset{n\to \infty }{lim}\varrho ({\nu }_{n+1},{\iota }_n,\tau )=0.$$

(16)

We now demonstrate that the sequence $\left\{({\nu }_n,{\iota }_n)\right\}$ forms a Cauchy bisequence. Given that both $\varrho ({\nu }_n,{\iota }_n,\tau )$ and $\varrho ({\nu }_{n+1},{\iota }_n,\tau )$ approach zero for any $\tau >0$ as n approaches infinity, and taking $ϵ,$ there exists $\delta$ such that the condition (6) is satisfied. Without any loss of generality, we take $\delta <ϵ$ and establish the existence of $N_1,N_2\in \mathbb{N}$ such that

$$\begin{array}{cc}\hfill \varrho ({\nu }_n,{\iota }_n,\tau )& <{\displaystyle \frac{{\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right)}{3}}\mathrm{for}\mathrm{all}n\ge N_1,\mathrm{and}\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill \varrho ({\nu }_{n+1},{\iota }_n,\tau )& <{\displaystyle \frac{{\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right)}{3}}\mathrm{for}\mathrm{all}n\ge N_2.\hfill \end{array}$$

(18)

Note that ${\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right)\le \psi \left(\delta \right)<\delta .$

Now, choose $N=max\{N_1,N_2\}$; we prove

$$\psi \left(\varrho ({\nu }_{n+l},{\iota }_n,\tau )\right)<\psi \left(ϵ\right),$$

(19)

and

$$\psi \left(\varrho ({\nu }_n,{\iota }_{n+l},\tau )\right)<\psi \left(ϵ\right),$$

(20)

for all $n\ge \mathbb{N},$ where $N=max\{N_1,N_2\}$ and all $l\ge 1$

Initially, we employ mathematical induction to establish (19), specifically $\psi \left(\varrho ({\nu }_{n+l},{\iota }_n,\tau )\right)<\psi \left(ϵ\right).$ According to (18), it is evident that the inequality is valid when $l=1.$ Assuming the statement holds for $l=k,$

$$\psi \left(\varrho ({\nu }_{n+k},{\iota }_n,\tau )\right)<\psi \left(ϵ\right),\mathrm{for}\mathrm{all}n\ge N.$$

(21)

Now, we have to prove

$$\psi \left(\varrho ({\nu }_{n+k+1},{\iota }_n,\tau )\right)<\psi \left(ϵ\right),\mathrm{for}\mathrm{all}n\ge N.$$

(22)

By using the Definition of EBPbMS, (17), (18), and (21), we get

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho ({\nu }_n,{\iota }_{n+k},\tau )\right)\right)& \le \psi (\varrho ({\nu }_n,{\iota }_n,\tau )+\varrho ({\nu }_{n+k},{\iota }_n,\tau )+\varrho ({\nu }_{n+k},{\iota }_{n+k},\tau ))\hfill \\ & <\varrho ({\nu }_n,{\iota }_n,\tau )+\varrho ({\nu }_{n+k},{\iota }_n,\tau )+\varrho ({\nu }_{n+k},{\iota }_{n+k},\tau )\hfill \\ & \le {\displaystyle \frac{{\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right)}{3}}+\psi \left(ϵ\right)+{\displaystyle \frac{{\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right)}{3}}\hfill \\ & <\psi \left(ϵ\right)+\delta .\hfill \end{array}$$

(23)

Therefore,

$${\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}({\nu }_n,{\iota }_{n+k},\tau )\right)\right)\le \psi \left(ϵ\right)+\delta .$$

If ${\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}({\nu }_n,{\iota }_{n+k},\tau )\right)\right)\ge \psi \left(ϵ\right)$, then according to (6), it follows that

$$\begin{array}{cc}\hfill \psi \left(\varrho ({\nu }_{n+k+1},{\iota }_n,\tau )\right)& =\psi \left(\varrho (T{\iota }_{n+k},T{\nu }_n,\tau )\right)\hfill \\ & \le \alpha ({\nu }_{n+k+1},{\iota }_{n+k+1},\tau )\alpha ({\nu }_{n+1},{\iota }_n,\tau )\psi \left(\varrho ({\nu }_{n+k+1},{\iota }_n)\right)\hfill \\ & <\alpha ({\nu }_{n+k+1},{\iota }_{n+k+1},\tau )\alpha ({\nu }_{n+1},{\iota }_n,\tau )\varrho (T{\iota }_{n+k},T{\nu }_n)\hfill \\ & <\psi \left(ϵ\right).\hfill \end{array}$$

Hence, (19) holds. If ${\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}({\nu }_n,{\iota }_{n+k},\tau )\right)\right)<\psi \left(ϵ\right)$, then according to Remark 5, it follows that

$$\begin{array}{cc}\hfill \psi \left(\varrho ({\nu }_{n+k+1},{\iota }_n,\tau )\right)& \le \alpha ({\nu }_n,{\iota }_n,\tau )\alpha ({\nu }_{n+k+1},{\iota }_{n+k},\tau )\psi \left(\varrho ({\nu }_{n+k+1},{\iota }_n)\right)\hfill \\ & <\alpha ({\nu }_n,{\iota }_n,\tau )\alpha ({\nu }_{n+k+1},{\iota }_{n+k},\tau )\varrho (T{\iota }_{n+k},T{\nu }_n)\hfill \\ & \le {\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}({\nu }_n,{\iota }_{n+k},\tau )\right)\right)\hfill \\ & <\psi \left(ϵ\right).\hfill \end{array}$$

Thus, (19) is valid for every $l\in \mathbb{N}$. Hence,

$$\varrho ({\nu }_n,{\iota }_m,\tau )<ϵ\mathrm{for}\mathrm{all}n>m\ge N.$$

(24)

Once more, through the application of mathematical induction, we establish (20). Using the Definition of EBPbMS, (17), (18), and (21), we get

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho ({\nu }_n,{\iota }_{n+1},\tau )\right)\right)& \le \psi \left(\varrho ({\nu }_{n+k},{\iota }_n,\tau )\right)+\varrho ({\nu }_{n+1},{\iota }_n,\tau )+\varrho ({\nu }_{n+1},{\iota }_{n+1},\tau ))\hfill \\ & <\varrho ({\nu }_{n+k},{\iota }_n,\tau ))+\varrho ({\nu }_{n+1},{\iota }_n,\tau )+\varrho ({\nu }_{n+1},{\iota }_{n+1},\tau )\hfill \\ & \le {\displaystyle \frac{{\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right)}{3}}+{\displaystyle \frac{{\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right)}{3}}+{\displaystyle \frac{{\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right)}{3}}\hfill \\ & ={\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right),\hfill \end{array}$$

which implies that $\psi \left(\varrho ({\nu }_n,{\iota }_{n+1},\tau )\right)<\psi \left(ϵ\right).$ Assume that the statement holds for some $l=k$,

$$\psi \left(\varrho ({\nu }_n,{\iota }_{n+k},\tau )\right)<\psi \left(ϵ\right),\mathrm{for}\mathrm{all}n\ge N.$$

(25)

Now, using the Definition of EBPbMS, (18), and (25), we get

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}^{-1}\psi \left(\varrho ({\nu }_{n+k+1},{\iota }_{n-1},\tau )\right)& <{\mathrm{\Omega }}^{-1}\varrho ({\nu }_{n+k+1},{\iota }_{n-1},\tau ))\hfill \\ & \le \varrho ({\nu }_{n+k+1},{\iota }_{n+k},\tau )+\varrho ({\nu }_n,{\iota }_{n+k},\tau )+\varrho ({\nu }_n,{\iota }_{n-1},\tau ))\hfill \\ & \le {\displaystyle \frac{{\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right)}{3}}+\psi \left(ϵ\right)+{\displaystyle \frac{{\mathrm{\Omega }}^{-1}\left(\psi \left(\delta \right)\right)}{3}}\hfill \\ & <\psi \left(ϵ\right)+\delta .\hfill \end{array}$$

(26)

Therefore,

$${\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}({\nu }_{n+k+1},{\iota }_{n-1},\tau )\right)\right)\le \psi \left(ϵ\right)+\delta$$

If ${\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}({\nu }_{n+k+1},{\iota }_{n-1},\tau )\right)\right)\ge \psi \left(ϵ\right)$, then, according to condition (6), it follows that

$$\begin{array}{cc}\hfill \psi \left(\varrho ({\nu }_n,{\iota }_{n+k+1},\tau )\right)& =\psi \left(\varrho (T{\iota }_{n-1},T{\nu }_{n+k+1},\tau )\right)\hfill \\ & \le \alpha ({\nu }_{n+k+1},T{\nu }_{n+k+1},\tau )\alpha (T{\iota }_{n-1},{\iota }_{n-1},\tau )\psi \left(\varrho ({\nu }_n,{\iota }_{n+k+1})\right)\hfill \\ & <\alpha ({\nu }_{n+k+1},{\iota }_{n+k+1},\tau )\alpha ({\nu }_n,{\iota }_{n-1},\tau )\varrho (T{\iota }_{n-1},T{\nu }_{n+k+1})\hfill \\ & <\psi \left(ϵ\right).\hfill \end{array}$$

Hence, (20) holds. If ${\mathrm{\Omega }}^{-1}\left(\mathrm{\Omega }\left(\mathbb{M}({\nu }_{n+k+1},{\iota }_{n-1},\tau )\right)\right)<\psi \left(ϵ\right)$ then by Remark 5, we have

$$\begin{array}{cc}\hfill \psi \left(\varrho ({\nu }_n,{\iota }_{n+k+1},\tau )\right)& \le \alpha ({\nu }_{n+k+1},{\iota }_{n+k+1},\tau )\alpha ({\nu }_n,{\iota }_{n-1},\tau )\psi \left(\varrho ({\nu }_n,{\iota }_{n+k+1})\right)\hfill \\ & <\alpha ({\nu }_{n+k+1},T{\nu }_{n+k+1},\tau )\alpha (T{\iota }_{n-1},{\iota }_{n-1},\tau )\varrho (T{\iota }_{n-1},T{\nu }_{n+k+1})\hfill \\ & {\mathrm{\Omega }}^{-1}\left(\mathrm{\Omega }\left(\mathbb{M}({\nu }_{n+k+1},{\iota }_{n-1},\tau )\right)\right)\hfill \\ & <\psi \left(ϵ\right).\hfill \end{array}$$

Therefore, (20) is valid for every $l\in \mathbb{N}$. Hence,

$$\varrho ({\nu }_n,{\iota }_m,\tau )<ϵ\mathrm{for}\mathrm{all}m>n\ge N.$$

(27)

From (24) and (27), we can say that $\left\{({\nu }_n,{\iota }_n)\right\}$ is a Cauchy bisequence. Since $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ is a complete EBPbMS, then $\left\{({\nu }_n,{\iota }_n)\right\}$ biconverges. That is, there exists $\iota \in \mathbb{A}\cap \mathbb{B}$ such that $\left\{{\nu }_n\right\}\to \iota$ and $\left\{{\iota }_n\right\}\to \iota$ as $n\to \infty$. As (1), $\left\{{\nu }_n\right\}\to \iota$ implies ${\iota }_n=T{\nu }_n\to T\iota$. Combining ${\iota }_n=T{\nu }_n\to T\iota$ with ${\iota }_n\to \iota$ and Proposition 1, we have $T\iota =\iota .$ □

In the subsequent Theorem, we exclude continuity and introduce a novel condition to derive fixed-point results.

Theorem 2.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be a complete EBPbMS. Suppose that $T:\mathbb{A}\cup \mathbb{B}\rightleftharpoons \mathbb{A}\cup \mathbb{B}$ is a generalized contravariant α-ψ-Meir-Keeler contractive mapping. Should the subsequent conditions be satisfied,

1. 

T is α-orbital admissible;

2. 

There exists ${\nu }_0\in \mathbb{A}$ such that $\alpha ({\nu }_0,T{\nu }_0,\tau )\ge 1$, for all $\tau >0$;

3. 

If $\left\{({\nu }_n,{\iota }_n)\right\}$ is a bisequence such that $\alpha ({\nu }_n,{\iota }_n,\tau )\ge 1$ for all $n\in \mathbb{N}$ and for all $\tau >0$ and ${\iota }_n\to \iota \in \mathbb{A}\cap \mathbb{B}$ as $n\to \infty ,$ then $\alpha (T\iota ,\iota ,\tau )\ge 1$ for all $\tau >0$.

Then T has a fixed point.

Proof. 

From the proof of Theorem 1, we deduce that the sequence $\left\{({\nu }_n,{\iota }_n)\right\}$ forms a Cauchy bisequence. Since $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ is a complete EBPbMS, then $\left\{({\nu }_n,{\iota }_n)\right\}$ is biconvergent. Hence, there exist $\iota \in \mathbb{A}\cap \mathbb{B}$ such that ${\nu }_n\stackrel{\varrho }{\to }\iota ,{\iota }_n\stackrel{\varrho }{\to }\iota .$ From condition (3), we get $\alpha (T\iota ,\iota ,\tau )\ge 1$. Now, applying the condition ($\varrho 3$) of Definition 2, (6), and (7) we get

$$\begin{array}{cc}\hfill \varrho (\iota ,T\iota ,\tau )& \le \mathrm{\Omega }\left[\varrho (\iota ,T{\nu }_n,\tau )+\varrho (T{\iota }_n,T{\nu }_n,\tau )+\varrho (T{\iota }_n,T\iota ,\tau )\right]\hfill \\ & \le \mathrm{\Omega }\left[\varrho (\iota ,T{\nu }_n,\tau )+\varrho (T{\iota }_n,T{\nu }_n,\tau )+\alpha ({\nu }_{n+1},T{\nu }_{n+1},\tau )\alpha (T\iota ,\iota ,\tau )\varrho (T{\iota }_n,T\iota ,\tau )\right]\hfill \\ & <\mathrm{\Omega }\left[\varrho (\iota ,T{\nu }_n,\tau )+\varrho (T{\iota }_n,T{\nu }_n,\tau )+{\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}(\iota ,{\iota }_n,\tau )\right)\right)\right]\hfill \end{array}$$

where

$$\mathbb{M}(\iota ,{\iota }_n,\tau )=max\left\{\varrho (\iota ,{\iota }_n,\tau ),\varrho (\iota ,T\iota ,\tau ),\varrho (T{\iota }_n,{\iota }_n,\tau )\right\}.$$

Letting $n\to \infty$ in the above inequality

$$\varrho (\iota ,T\iota ,\tau )<\psi \left(\varrho \right(\iota ,T\iota ,\tau \left)\right)<\varrho (\iota ,T\iota ,\tau ).$$

Since $\psi$ is continuous, we get $\varrho (T\iota ,\iota ,\tau )=0.$ Hence, $T\iota =\iota .$ □

For the covariant case, the proofs of the subsequent theorems are identical to those of Theorems 1 and 2 and are thus omitted.

Theorem 3.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be a complete EBPbMS. Suppose that $T:\mathbb{A}\cup \mathbb{B}\rightrightarrows \mathbb{A}\cup \mathbb{B}$ is a generalized covariant α-ψ-Meir-Keeler contractive mapping. The following conditions must be satisfied:

1. 

T is α-orbital admissible;

2. 

For some ${\nu }_0\in \mathbb{A},{\iota }_0\in \mathbb{B}\Rightarrow \alpha ({\nu }_0,T{\nu }_0,\tau )\ge 1,\alpha ({\nu }_0,{\iota }_0,\tau )\ge 1;$

3. 

T is an orbital continuous.

Then T has a fixed point.

Theorem 4.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be a complete EBPbMS. Suppose that $T:\mathbb{A}\cup \mathbb{B}\rightrightarrows \mathbb{A}\cup \mathbb{B}$ is a generalized covariant α-ψ-Meir-Keeler contractive mapping. Should the subsequent conditions be satisfied,

1. 

T is α-orbital admissible;

2. 

For some ${\nu }_0\in \mathbb{A},{\iota }_0\in \mathbb{B}\Rightarrow \alpha ({\nu }_0,T{\nu }_0,\tau )\ge 1,\alpha ({\nu }_0,{\iota }_0,\tau )\ge 1;$

3. 

If $\left\{({\nu }_n,{\iota }_n)\right\}$ is a bisequence such that $\alpha ({\nu }_n,{\iota }_n,\tau )\ge 1$ for all $n\in \mathbb{N}$ and for all $\tau >0$ and ${\iota }_n\to \iota \in \mathbb{A}\cap \mathbb{B}$ as $n\to \infty ,$ then $\alpha (T\iota ,\iota ,\tau )\ge 1$ for all $\tau >0$;

Then T has a fixed point.

Theorem 5.

By incorporating hypothesis (H) into Theorems 1–4, a unique fixed point is obtained.

(H) 

If $\iota =T\iota$, then $\alpha (\iota ,T\iota ,\tau )\ge 1$ for all $\tau >0$.

Proof. 

Assuming hypothetically that T possesses two different fixed points, $\iota$ and $\nu$, as stipulated by the hypothesis (H), $\alpha (\iota ,T\iota ,\tau ),\alpha (\nu ,T\nu ,\tau )\ge 1,$ for all $\tau >0$. Now, by Remark 5 we have

$$\begin{array}{cc}\hfill \varrho (\iota ,\nu ,\tau )=\varrho (T\iota ,T\nu ,\tau )& <{\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}(\iota ,\nu ,\tau )\right)\right),\mathrm{for}\mathrm{all}\tau >0.\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \mathbb{M}(\iota ,\nu ,\tau )& =max\left\{\varrho (\iota ,\nu ,\tau ),\varrho (\iota ,T\iota ,\tau ),\varrho (T\nu ,\nu ,\tau )\right\}\hfill \\ & =max\left\{\varrho (\iota ,\nu ,\tau ),\varrho (\iota ,\iota ,\tau ),\varrho (\nu ,\nu ,\tau )\right\}\hfill \\ & =\varrho (\iota ,\nu ,\tau ),\hfill \end{array}$$

which leads to a contradiction, thereby implying that $\iota =\nu .$ □

Example 2.

Let $\mathbb{A}=(-\infty ,0]\times [0,\infty ),\mathbb{B}=[0,\infty )\times [0,\infty )$ be endowed with the EBPbM

$$\varrho (({\nu }_1,{\nu }_2),({\iota }_1,{\iota }_2),\tau )=\left\{\begin{array}{cc}{e}^{\left({\displaystyle \frac{|{\nu }_1|+|{\iota }_1|}{\tau }}\right)},\hfill & \mathit{if}|{\nu }_1|\ne {\iota }_1,\hfill \\ 0\hfill & \mathit{if}|{\nu }_1|={\iota }_1,\hfill \end{array}\right.$$

for all $({\nu }_1,{\nu }_2)\in \mathbb{A},({\iota }_1,{\iota }_2)\in \mathbb{B}$$\mathrm{\Omega }\left(s\right)=e^s,s\ge 0$, let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be a complete EBPbMS. Define $\psi \left(s\right)={\displaystyle \frac{s}{2}},s\ge 0$,$T:\mathbb{A}\cup \mathbb{B}\rightleftharpoons \mathbb{A}\cup \mathbb{B}$ by

$$T\left(({\nu }_1,{\nu }_2)\right)=\left\{\begin{array}{cc}(2{\nu }_1^2+1,2{\nu }_2^2+1)\hfill & \mathit{if}({\nu }_1,{\nu }_2)\in (-\infty ,0)\times [0,\infty ),\hfill \\ ({\displaystyle \frac{-1}{200}}{\nu }_1^2,{\displaystyle \frac{1}{200}}{\nu }_2^2)\hfill & \mathit{if}({\nu }_1,{\nu }_2)\in [0,1]\times [0,\infty ),\hfill \\ (-3{\nu }_1+1,3{\nu }_2-1),\hfill & \mathit{if}({\nu }_1,{\nu }_2)\in (1,2)\times [0,\infty )\hfill \\ (-6{\nu }_1^8,6{\nu }_2^8),\hfill & \mathit{if}({\nu }_1,{\nu }_2)\in (2,+\infty )\times [0,\infty ),\hfill \end{array}\right.$$

and $\alpha :\mathbb{A}\cup \mathbb{B}\times (0,\infty )\to [0,\infty )$ by

$$\alpha (({\nu }_1,{\nu }_2),({\iota }_1,{\iota }_2),\tau )=\left\{\begin{array}{cc}1,\hfill & \mathit{if}({\nu }_1,{\nu }_2),({\iota }_1,{\iota }_2)\in [0,1]\times [0,\infty )\hfill \\ 0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

It is obvious that T represents α-orbital admissible mapping. If $\{({{\nu }_1}_n,{{\nu }_2}_n),({{\iota }_1}_n,{{\iota }_2}_n)\}$ in $\mathbb{A}\times \mathbb{B}$ such that $\alpha (({{\nu }_1}_n,{{\nu }_2}_n),({{\iota }_1}_n,{{\iota }_2}_n),\tau )\ge 1$ with $\left\{({{\iota }_1}_n,{{\iota }_2}_n)\right\}\to (u_1,u_2)$ as $n\to \infty ,$ then $\left\{({{\iota }_1}_n,{{\iota }_2}_n)\right\}\in [0,1]\times [0,\infty )$ for all $n\in \mathbb{N}$ and so $(u_1,u_2)\in [0,1]\times [0,\infty ).$ This ensures that $\alpha (({{\nu }_1}_n,{{\nu }_2}_n),({{\iota }_1}_n,{{\iota }_2}_n),\tau )\ge 1$ for all $n\in \mathbb{N}.$ Clearly $\alpha ((u_1,u_2),T(u_1,u_2),\tau )\ge 1.$ Let $\alpha (({\nu }_1,{\nu }_2),({\iota }_1,{\iota }_2),\tau )\ge 1,$ $\psi \left(ϵ\right)\le {\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}(({\nu }_1,{\nu }_2),({\iota }_1,{\iota }_2),\tau )\right)\right)<\psi \left(ϵ\right)+\delta$ where $ϵ>0$ is arbitrary and

$$\delta <200lnϵ-ϵ.$$

Then $({\nu }_1,{\nu }_2),({\iota }_1,{\iota }_2)\in [0,1]\times [0,\infty ),{\nu }_1\ne {\iota }_1,$ where,

$$\mathbb{M}(\nu ,\iota ,\tau )=max\left\{exp\left({\displaystyle \frac{|{\nu }_1|+|{\iota }_1|}{\tau }}\right),exp\left({\displaystyle \frac{|{\nu }_1|+\frac{1}{200}{\nu }_1^2}{\tau }}\right),exp\left({\displaystyle \frac{\frac{1}{200}{\iota }_1^2+\left|{\iota }_1\right|}{\tau }}\right)\right\}.$$

WOLG, suppose ${\nu }_1\le {\iota }_1$. Now, let ${\displaystyle \frac{{\nu }_1+{\iota }_1}{\tau }}<ϵ+\delta$. Thus, it follows that

$$\begin{array}{cc}\hfill \psi \left(\varrho (T({\nu }_1,{\nu }_2),T({\iota }_1,{\iota }_2),\tau )\right)& ={\displaystyle \frac{1}{3}}\varrho (({\displaystyle \frac{-1}{200}}{\nu }_1^2,{\displaystyle \frac{1}{200}}{\nu }_2^2),\left(\right|{\displaystyle \frac{-1}{200}}|{\iota }_1^2,{\displaystyle \frac{1}{200}}{\iota }_2^2),\tau )\hfill \\ & ={\displaystyle \frac{1}{3}}{exp}^{({\displaystyle \frac{|\frac{-1}{200}{\nu }_1^2|+|\frac{-1}{200}{\iota }_1^2|}{\tau }})}\hfill \\ & \le {\displaystyle \frac{1}{3}}{exp}^{({\displaystyle \frac{|{\nu }_1|+|{\iota }_1|}{200\tau }})}\hfill \\ & \le {\displaystyle \frac{1}{3}}{exp}^{({\displaystyle \frac{ϵ+\delta }{200}})}\hfill \\ & <{\displaystyle \frac{1}{3}}{exp}^{({\displaystyle \frac{ϵ+200lnϵ-ϵ}{200}})}\hfill \\ & <{\displaystyle \frac{1}{3}}ϵ.\hfill \end{array}$$

Otherwise, $\alpha (({\nu }_1,{\nu }_2),({\iota }_1,{\iota }_2),\tau )=0$ and evidently

$$\alpha (\nu ,T\nu ,\tau )\alpha (T\iota ,\iota ,\tau )\varrho (T\iota ,T\nu ,\tau )<ϵ.$$

Hence, $T:(\mathbb{A},\mathbb{B})\rightleftharpoons (\mathbb{A},\mathbb{B})$ is a generalized contravariant α-ψ-Meir-Keeler mapping. Thus, all the conditions of Theorems 1 and 5 hold and $(0,0)$ is a unique fixed point of $T.$

Definition 12.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be an EBPbMS. Assume that the mapping $T:(\mathbb{A},\mathbb{B})\rightleftharpoons (\mathbb{A},\mathbb{B})$ is contravariant, and for any $ϵ>0$, there is $\delta \left(ϵ\right)>0$ such that

$$ϵ\le {\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho (\nu ,\iota ,\tau )\right)\right)<ϵ+\delta \Rightarrow \alpha (\nu ,T\nu ,\tau )\alpha (T\iota ,\iota ,\tau )\varrho (T\iota ,T\nu ,\tau )<ϵ,$$

(28)

for all $(\nu ,\iota )\in \mathbb{A}\times \mathbb{B}$ and $\tau >0$ and $\alpha :\mathbb{A}\times \mathbb{B}\times (0,\infty )\to \mathbb{R}$. Then T is said to be contravariant α-ψ-Meir-Keeler contractive mapping.

Remark 6.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be an EBPbMS, and $T:\mathbb{A}\cup \mathbb{B}\rightleftharpoons \mathbb{A}\cup \mathbb{B}$ satisfies (28); then,

$$\varrho (T\iota ,T\nu ,\tau )<{\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho (\nu ,\iota ,\tau )\right)\right)$$

when $\nu \ne \iota$ for all $\tau >0$ with $\alpha (\nu ,\iota ,\tau )\ge 1.$

If $\nu =\iota$ then

$$\varrho (T\iota ,T\nu ,\tau )\le {\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho (\nu ,\iota ,\tau )\right)\right),$$

for all $\tau >0$ with $\alpha (\nu ,\iota ,\tau )\ge 1$.

Proof. 

Since $\nu \ne \iota$ we have $\varrho (\nu ,\iota ,\tau )>0$ for all $\tau >0$. Assume $\delta >0$ and $ϵ={\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho (\nu ,\iota ,\tau )\right)\right).$ Then

$${\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho (\nu ,\iota ,\tau )\right)\right)<{\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho (\nu ,\iota ,\tau )\right)\right)+\delta <ϵ+\delta$$

and so from (28), we have

$$\varrho (T\nu ,T\iota ,\tau )\le \alpha (\nu ,T\nu ,\tau )\alpha (T\iota ,\iota ,\tau )\varrho (T\iota ,T\nu ,\tau )<ϵ={\mathrm{\Omega }}^{-1}\left(\varrho (\nu ,\iota ,\tau )\right).$$

□

Example 3

(Application of Theorem 1 to $T\left(x\right)={\displaystyle \frac{x}{1+x}}$). Let $X=[0,\infty )$ and define $T:X\to X$ by

$$T\left(x\right)={\displaystyle \frac{x}{1+x}}.$$

Clearly, T has at least the fixed point $x^{*}=0$ (since $T\left(x\right)=x$ iff $x=0$). We show that T satisfies all the hypotheses of Theorem 1 in a complete extended bipolar parametric b-metric setting and hence has a (unique) fixed point by the theorem.

Step 0: The EBPbMS structure and completeness. Let $\mathbb{A}=\mathbb{B}=X$, and for $\tau >0$, set

$$\varrho (x,y,\tau ):=e^{\tau |x-y|}-1,\mathrm{\Omega }\left(s\right):=e^s-1(s\ge 0).$$

Then for all $x,y,z\in X$ and $\tau >0$,

$$\varrho (x,z,\tau )=e^{\tau |x-z|}-1\le e^{\tau \left(\right|x-y|+|y-z\left|\right)}-1=\mathrm{\Omega }\left(\underset{=\varrho (x,y,\tau )}{\underbrace{(e^{\tau |x-y|}-1)}}+\underset{=\varrho (y,z,\tau )}{\underbrace{(e^{\tau |y-z|}-1)}}\right),$$

so $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ is an EBPbMS. Moreover, the map $|x-y|\mapsto e^{\tau |x-y|}-1$ is an increasing homeomorphism $[0,\infty )\to [0,\infty )$ for each fixed $\tau >0$; hence Cauchy/biconvergent behavior in $\varrho (·,·,\tau )$ is equivalent to that in the standard metric $|x-y|$, and since $(X,|·\left|\right)$ is complete, the EBPbMS is complete.

Step 1: $\alpha$-admissibility and orbital continuity. Choose $\alpha \equiv 1$ on $\mathbb{A}\times \mathbb{B}\times (0,\infty )$. Then T is trivially α-orbital admissible, and T is continuous on X, so T is orbital continuous as required in Theorem 1. We also have $\alpha ({\nu }_0,T{\nu }_0,\tau )=1\ge 1$ for any ${\nu }_0\in X$ and $\tau >0$.

Step 2: A Meir–Keeler-type trigger for T under $\varrho$.We prove a Meir–Keeler implication in the sense of Theorem 1 with $\psi \left(t\right)=t/2$. For any $x,y\in X$,

$$|T\left(x\right)-T\left(y\right)|=|{\displaystyle \frac{x}{1+x}}-{\displaystyle \frac{y}{1+y}}|={\displaystyle \frac{|x-y|}{(1+x)(1+y)}}.$$

Fix $\tau >0$ and put $d=|x-y|\ge 0$. Then

$${\displaystyle \frac{\varrho (Tx,Ty,\tau )}{\varrho (x,y,\tau )}}={\displaystyle \frac{e^{\tau \left|T\right(x)-T(y\left)\right|}-1}{e^{\tau d}-1}}={\displaystyle \frac{e^{\tau \frac{d}{(1+x)(1+y)}}-1}{e^{\tau d}-1}}.$$

Subject to the constraint $|x-y|=d$, the product $(1+x)(1+y)$ is minimized when $min\{x,y\}=0$ and $max\{x,y\}=d$; hence, $(1+x)(1+y)\ge 1+d$, and so

$${\displaystyle \frac{\varrho (Tx,Ty,\tau )}{\varrho (x,y,\tau )}}\le {\displaystyle \frac{e^{\tau \frac{d}{1+d}}-1}{e^{\tau d}-1}}=:q_{\tau }\left(d\right).$$

For each fixed $\tau >0$, $q_{\tau }\left(d\right)$ is strictly less than 1 for every $d>0$, and $q_{\tau }$ is decreasing on $[D,\infty )$ for any $D>0$. In particular, given any radius $\epsilon >0$, define

$$D\left(\epsilon \right):={\displaystyle \frac{1}{\tau }}ln(1+\epsilon ),q\left(\epsilon \right):=q_{\tau }\left(D\left(\epsilon \right)\right)={\displaystyle \frac{{(1+\epsilon )}^{\frac{1}{1+D\left(\epsilon \right)}}-1}{\epsilon }}\in (0,1).$$

Then for all pairs with $\varrho (x,y,\tau )\ge \epsilon$ (equivalently $d\ge D\left(\epsilon \right)$) we have

$$\varrho (Tx,Ty,\tau )\le q\left(\epsilon \right)\varrho (x,y,\tau ),\mathit{with}q\left(\epsilon \right)\in (0,1).$$

(29)

Now choose $\psi \left(t\right)=t/2\in \Psi$. Since ${\mathrm{\Omega }}^{-1}\left(s\right)=ln(1+s)$ and ψ is increasing, the implication

$$\psi \left(\epsilon \right)\le {\mathrm{\Omega }}^{-1}\left(\psi \left(\mathbb{M}\right(x,y,\tau \left)\right)\right)<\psi \left(\epsilon \right)+\delta ⟹\varrho (Tx,Ty,\tau )<\psi \left(\epsilon \right)$$

follows by taking $\delta >0$ as arbitrary and using (29) with $\mathbb{M}$ as in Theorem 1: indeed, $\mathbb{M}(x,y,\tau )\ge \varrho (x,y,\tau )$; hence, the left premise forces $\varrho (x,y,\tau )\ge \epsilon$, and then (29) gives $\varrho (Tx,Ty,\tau )\le q\left(\epsilon \right)\varrho (x,y,\tau )\le q\left(\epsilon \right)\mathbb{M}(x,y,\tau )$. Since $q\left(\epsilon \right)\in (0,1)$ and $\psi \left(t\right)=t/2$, we obtain $\psi \left(\varrho (Tx,Ty,\tau )\right)<\psi \left(\epsilon \right)$. Thus, T satisfies the generalized α-ψ-Meir–Keeler condition of Theorem 1 with $\alpha \equiv 1$.

Step 3: Applying Theorem 1. We have verified that

• $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ is a complete EBPbMS;
• T is (trivially) α-orbital admissible with $\alpha \equiv 1$ and T is orbital continuous;
• The generalized α-ψ-Meir–Keeler trigger holds with $\psi \left(t\right)=t/2$.

Hence, all assumptions of Theorem 1 are satisfied, so T has a fixed point in X.

Step 4: Uniqueness (Theorem 5). Condition (H) holds automatically here (since $\alpha \equiv 1$). Thus, Theorem 5 applies, yielding the uniqueness of the fixed point. Since $T\left(x\right)=x$ iff $x=0$, we conclude that the fixed point is unique and equals $x^{*}=0$.

Conclusion 1.

The map $T\left(x\right)={\displaystyle \frac{x}{1+x}}$ is not a Banach contraction in any classical b-metric on X, yet in the present EBPbMS it satisfies a genuine Meir–Keeler-type decrease depending on the radius $\epsilon$ (as encapsulated by $q\left(\epsilon \right)<1$ in (29)). Therefore, by Theorems 1 and 5, T admits a unique fixed point $x^{*}=0$.

Corollary 2.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be a complete EBPbMS. Suppose that $T:(\mathbb{A},\mathbb{B})\rightleftharpoons (\mathbb{A},\mathbb{B})$ is a generalized contravariant α-ψ-Meir-Keeler mapping. Should the subsequent conditions be satisfied,

1. 

T is α-orbital admissible;

2. 

For every $\tau >0$, it holds that $\alpha ({\nu }_0,T{\nu }_0,\tau )\ge 1$, for some ${\nu }_0$;

3. 

T is continuous;

4. 

Condition $\left(H\right)$ holds;

Then, T has a unique fixed point.

Proof. 

The proof is obvious from the Theorems 1 and 5 due to the fact that

$$\varrho (\nu ,\iota ,\tau )\le \mathbb{M}(\nu ,\iota ,\tau )$$

for all $\nu \in \mathbb{A},\iota \in \mathbb{B}$ and for all $\tau >0.$ □

Corollary 3.

Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be a complete EBPbMS. Suppose that $T:(\mathbb{A},\mathbb{B})\rightleftharpoons (\mathbb{A},\mathbb{B})$ is a generalized contravariant α-ψ-Meir-Keeler mapping. Should the subsequent criteria be satisfied,

1. 

T is α-orbital admissible;

2. 

There exists ${\nu }_0\in \mathbb{A}$ such that $\alpha ({\nu }_0,T{\nu }_0,\tau )\ge 1$, for all $\tau >0$;

3. 

If $\left\{({\nu }_n,{\iota }_n)\right\}$ is a bisequence such that ${\iota }_n\to \iota \in \mathbb{A}\cap \mathbb{B}$, $\alpha ({\nu }_n,{\iota }_n,\tau )\ge 1$ for all $n\in \mathbb{N}$ and for all $\tau >0$, and as $n\to \infty ,$ then $\alpha (T\iota ,\iota ,\tau )\ge 1$ for all $\tau >0$;

4. 

Condition $\left(H\right)$ holds.

Therefore, T possesses a unique fixed point.

Proof. 

The proof is straightforward by referencing Theorems 2 and 5 replacing $\mathbb{M}(\nu ,\iota ,\tau )$ with $\varrho (\nu ,\iota ,\tau )$ for all $\nu \in \mathbb{A},\iota \in \mathbb{B}$ and for all $\tau >0.$ □

3. Relation Between Extended Bipolar Parametric b-Metric and Extended Fuzzy Bipolar b-Metric Spaces

Definition 13

([16]). A binary operation $*:[0,1]\times [0,1]\to [0,1]$ is termed a continuous t-norm (CTN) if * is commutative, associative, $a*1=a$ and for all $a,b,c,d\in [0,1],a*b\le c*d$, where $a\le c$ and $b\le d.$

Example 4

(Examples of CTN). 

(1) 

: $*(a,b)=a·b$;

(2) 

: $*(a,b)=min\{a,b\}$;

(3) 

: $*(a,b)=max\{a+b-1,0\}$.

Definition 14

([17]). Let $\mathbb{A}$ and $\mathbb{B}$ be two nonempty sets and $s\ge 1.$ A quadruple $(\mathbb{A},\mathbb{B},\mathcal{B},*)$ is called a fuzzy bipolar b-metric space (in short, FBMS), where * is a CTN and $\mathcal{B}$ is a fuzzy set on $\mathbb{A}\times \mathbb{B}\times (0,\infty )$, if for all $\rho ,\delta ,\tau >0$:

1. 

$\mathcal{B}(\nu ,\iota ,\tau )>0$ for all $(\nu ,\iota )\in (\mathbb{A},\mathbb{B})$;

2. 

$\mathcal{B}(\nu ,\iota ,\tau )=1$ if and only if $\nu =\iota$ for $\iota \in \mathbb{A}$ and $\iota \in \mathbb{B}$;

3. 

$\mathcal{B}(\nu ,\iota ,\tau )=\mathcal{B}(\iota ,\nu ,\tau )$ for all $\nu ,\iota \in \mathbb{A}\cap \mathbb{B}$;

4. 

$\mathcal{B}({\nu }_1,{\iota }_2,s(\tau +\rho +\delta )\ge \mathcal{B}({\nu }_1,{\iota }_1,\tau )*\mathcal{B}({\nu }_2,{\iota }_1,\rho )*\mathcal{B}({\nu }_2,{\iota }_2,\delta )$ for all ${\nu }_1,{\nu }_2\in \mathbb{A},{\iota }_1,{\iota }_2\in \mathbb{B}$;

5. 

$\mathcal{B}(\nu ,\iota ,.):[0,\infty )\to [0,1]$ is left continuous;

6. 

$\mathcal{B}(\nu ,\iota ,.)$ is non-decreasing for all $\nu \in \mathbb{A}$ and $\iota \in \mathbb{B}$.

Now, we introduce the concept of extended fuzzy bipolar b-metric space as the following:

Definition 15.

Let $\mathbb{A}$ and $\mathbb{B}$ be two nonempty sets and $\mathrm{\Omega }:[0,\infty )\to [0,\infty )$ be a strictly increasing continuous function with ${\mathrm{\Omega }}^{-1}\left(s\right)\le s\le \mathrm{\Omega }\left(s\right)$. A quadruple $(\mathbb{A},\mathbb{B},\mathcal{B},*,\mathrm{\Omega })$ is called an extended fuzzy bipolar b- metric space (in short, EFBbMS), where * is a CTN and $\mathcal{B}$ is a fuzzy set on $\mathbb{A}\times \mathbb{B}\times (0,\infty )$, if for all $\rho ,\delta ,\tau >0$:

1. 

$\mathcal{B}(\nu ,\iota ,\tau )>0$ for all $(\nu ,\iota )\in \mathbb{A}\times \mathbb{B}$;

2. 

$\mathcal{B}(\nu ,\iota ,\tau )=1$ if and only if $\nu =\iota$ for $\iota \in \mathbb{A}$ and $\iota \in \mathbb{B}$;

3. 

$\mathcal{B}(\nu ,\iota ,\tau )=\mathcal{B}(\iota ,\nu ,\tau )$ for all $\nu ,\iota \in \mathbb{A}\cap \mathbb{B}$;

4. 

$\mathcal{B}({\nu }_1,{\iota }_2,\mathrm{\Omega }(\tau +\rho +\delta )\ge \mathcal{B}({\nu }_1,{\iota }_1,\tau )*\mathcal{B}({\nu }_2,{\iota }_1,\rho )*\mathcal{B}({\nu }_2,{\iota }_2,\delta )$ for all ${\nu }_1,{\nu }_2\in X,{\iota }_1,{\iota }_2\in \mathbb{B}$;

5. 

$\mathcal{B}(\nu ,\iota ,.):[0,\infty )\to [0,1]$ is left continuous;

6. 

$\mathcal{B}(\nu ,\iota ,.)$ is non-decreasing for all $\nu \in \mathbb{A}$ and $\iota \in \mathbb{B}$.

Definition 16.

The EFBbMS $(\mathbb{A},\mathbb{B},\mathcal{B},*,\mathrm{\Omega })$ is called Ω-rectangular whenever

$${\displaystyle \frac{1}{\mathcal{B}({\nu }_1,{\iota }_2,\tau )}}-1\le \mathrm{\Omega }\left[{\displaystyle \frac{1}{\mathcal{B}({\nu }_1,{\iota }_1,\tau )}}-1+{\displaystyle \frac{1}{\mathcal{B}({\nu }_2,{\iota }_1,\tau )}}-1+{\displaystyle \frac{1}{\mathcal{B}({\nu }_2,{\iota }_2,\tau )}}\right]$$

for all ${\nu }_1,{\nu }_2\in \mathbb{A},{\iota }_1,{\iota }_2\in \mathbb{B}$ and for all $\tau >0$.

Remark 7.

Let $(\mathbb{A},\mathbb{B},\mathcal{B},*,\mathrm{\Omega })$ be a Ω-rectangular EFBbMS. Define the mapping $\varrho :\mathbb{A}\times \mathbb{B}\times (0,\infty )\to [0,\infty )$ by $\varrho (\nu ,\iota ,\tau )={\displaystyle \frac{1}{\mathcal{B}(\nu ,\iota ,\tau )}}-1$. Then ϱ is EBPbMS.

Next, we introduce the concept of a generalized covariant and contravariant $\alpha$-$\psi$-$\mathcal{B}$-Meir-Keeler contraction mapping.

Definition 17.

Let $(\mathbb{A},\mathbb{B},\mathcal{B},\ast ,\mathrm{\Omega })$ be a rectangular EFBbMS. Suppose that $T:(\mathbb{A}\cup \mathbb{B})\rightrightarrows (\mathbb{A}\cup \mathbb{B})$ is a mapping, $\psi \in \Psi$ and for each $ϵ>0$, there is a 0 such that

$$ϵ\le {\mathrm{\Omega }}^{-1}\left(\psi \left({\mathbb{M}}_{\mathcal{B}}(\nu ,\iota ,\tau )\right)\right)<ϵ+\delta \Rightarrow \alpha (\nu ,T\iota ,\tau )\alpha (T\nu ,\iota ,\tau )\left[{\displaystyle \frac{1}{\varrho (T\nu ,T\iota ,\tau )}}-1\right]<ϵ,$$

(30)

where

$$\begin{array}{cc}\hfill {\mathbb{M}}_{\mathcal{B}}(\nu ,\iota ,\tau )=max\{& {\displaystyle \frac{1}{\varrho (\nu ,\iota ,\tau )}}-1,{\displaystyle \frac{1}{\varrho (\nu ,T\iota ,\tau )}}-1,{\displaystyle \frac{1}{\varrho (T\nu ,\iota ,\tau )}}-1\},\hfill \end{array}$$

for all $(\nu ,\iota )\in \mathbb{A}\times \mathbb{B},\tau >0$. Then, T is said to be a generalized covariant α-ψ-$\mathcal{B}$-Meir-Keeler contraction.

Definition 18.

Let $(\mathbb{A},\mathbb{B},\mathcal{B},\ast ,\mathrm{\Omega })$ be a rectangular EFBbMS. Suppose that $T:(\mathbb{A}\cup \mathbb{B})\rightleftharpoons (\mathbb{A}\cup \mathbb{B})$ is a mapping, $\psi \in \Psi$ and for each $ϵ>0$, there is a $\delta >0$ such that

$$ϵ\le {\mathrm{\Omega }}^{-1}\left(\psi \left({\mathbb{M}}_{\mathcal{B}}(\nu ,\iota ,\tau )\right)\right)<ϵ+\delta \Rightarrow \alpha (\nu ,T\nu ,\tau )\alpha (T\iota ,\iota ,\tau )\left[{\displaystyle \frac{1}{\varrho (T\iota ,T\nu ,\tau )}}-1\right]<ϵ,$$

(31)

where

$$\begin{array}{cc}\hfill {\mathbb{M}}_{\mathcal{B}}(\nu ,\iota ,\tau )=max\{& {\displaystyle \frac{1}{\varrho (\nu ,\iota ,\tau )}}-1,{\displaystyle \frac{1}{\varrho (\nu ,T\nu ,\tau )}}-1,{\displaystyle \frac{1}{\varrho (T\iota ,\iota ,\tau )}}-1\},\hfill \end{array}$$

for all $(\nu ,\iota )\in \mathbb{A}\times \mathbb{B},\tau >0$. Then, T is said to be a generalized covariant α-ψ-$\mathcal{B}$-Meir-Keeler contraction.

Theorem 6.

Let $(\mathbb{A},\mathbb{B},\mathcal{B},\ast ,\mathrm{\Omega })$ be a Ω-rectangular EFBbMS. Suppose that $T:(\mathbb{A}\cup \mathbb{B})\rightrightarrows (\mathbb{A}\cup \mathbb{B})$ is a generalized covariant and α-ψ-$\mathcal{B}$-Meir-Keeler contraction. If the following conditions are satisfied,

1. 

T is α-orbital admissible;

2. 

T is continuous;

3. 

The condition $\left(H\right)$ holds.

Then, T has a unique fixed point.

Proof. 

We establish $\varrho (\nu ,\iota ,\tau )={\displaystyle \frac{1}{\mathcal{B}(\nu ,\iota ,\tau )}}-1$ for every $\iota \in \mathbb{A}$, $\iota \in \mathbb{B}$ where $\tau >0$. Then, by Definition 16, $\varrho (\nu ,\iota ,\tau )$ is an EBPbMS. Hence, all of the conditions of Theorems 3 and 5 hold, and T has a unique fixed point. □

Theorem 7.

Let $(\mathbb{A},\mathbb{B},\mathcal{B},\ast ,\mathrm{\Omega })$ be a Ω-rectangular space with the control function Ω. Suppose that $T:(\mathbb{A}\cup \mathbb{B})\rightrightarrows (\mathbb{A}\cup \mathbb{B})$ is a generalized covariant and α-ψ-$\mathcal{B}$-Meir-Keeler contraction. Should these conditions be met,

1. 

T is α-orbital admissible;

2. 

There exists ${\nu }_0\in \mathbb{A}$ such that $\alpha ({\nu }_0,T{\nu }_0,\tau )\ge 1$, for all $\tau >0$;

3. 

If $\left\{({\nu }_n,{\iota }_n)\right\}$ is a bisequence such that $\alpha ({\nu }_n,{\iota }_n,\tau )\ge 1$ for all $n\in \mathbb{N}$ and for all $\tau >0$ and ${\iota }_n\to \iota \in \mathbb{A}\cap \mathbb{B}$ as $n\to \infty$, then $\alpha (T\iota ,\iota ,\tau )\ge 1$ for all $\tau >0$;

4. 

The condition $\left(H\right)$ holds.

Then, T has a unique fixed point.

Proof. 

We define $\varrho (\nu ,\iota ,\tau )={\displaystyle \frac{1}{\mathcal{B}(\nu ,\iota ,\tau )}}-1$ for every $\iota \in \mathbb{A},\iota \in \mathbb{B}$ where $t>0.$ Then by Definition 16, $\varrho (\nu ,\iota ,\tau )$ is an EBPbMS. Therefore, all the prerequisites of Theorem 4 and Theorem 5 are satisfied, and the operator T possesses a unique fixed point. □

Theorem 8.

Let $(\mathbb{A},\mathbb{B},\mathcal{B},\ast ,\mathrm{\Omega })$ represent a Ω-rectangular EFBbMS with a control function Ω. Assume that $T:(\mathbb{A}\cup \mathbb{B})\rightleftharpoons (\mathbb{A}\cup \mathbb{B})$ is a generalized contravariant α-ψ-$\mathcal{B}$-Meir-Keeler contraction mapping. If the following conditions are satisfied:

1. 

T is α-orbital admissible;

2. 

T is $\mathcal{B}$-continuous;

3. 

The condition $\left(H\right)$ holds.

Then, T has a unique fixed point.

Proof. 

We establish $\varrho (\nu ,\iota ,\tau )={\displaystyle \frac{1}{\mathcal{B}(\nu ,\iota ,\tau )}}-1$ for every $\iota \in \mathbb{A},\iota \in \mathbb{B}$ where $\tau >0.$ Then by Definition 16, $\varrho (\nu ,\iota ,\tau )$ is an EBPbMS. Hence, all of the conditions of Theorems 1 and 5 hold, and T has a unique fixed point. □

Theorem 9.

Let $(\mathbb{A},\mathbb{B},\mathcal{B},\ast ,\mathrm{\Omega })$ be a Ω-rectangular. Suppose that $T:(\mathbb{A}\cup \mathbb{B})\rightleftharpoons (\mathbb{A}\cup \mathbb{B})$ is a generalized contravariant α-ψ-$\mathcal{B}$-Meir-Keeler contraction mapping. Should the following conditions be met:

1. 

T is α-orbital admissible;

2. 

There exists ${\nu }_0\in \mathbb{A}$ such that $\alpha ({\nu }_0,T{\nu }_0,\tau )\ge 1$, $\alpha ({\nu }_0,{\nu }_0,\tau )\ge 1$ for all $\tau >0$;

3. 

If $\left\{({\nu }_n,{\iota }_n)\right\}$ is a bisequence such that $\alpha ({\nu }_n,{\iota }_n,\tau )\ge 1$ for all $n\in \mathbb{N}$ and for all $\tau >0$, and ${\iota }_n\to \iota \in \mathbb{A}\cap \mathbb{B}$ as $n\to \infty$, then $\alpha (T\iota ,\iota ,\tau )\ge 1$ for all $\tau >0$;

4. 

Condition $\left(H\right)$ holds.

Then, T has a unique fixed point.

Proof. 

We define $\varrho (\nu ,\iota ,\tau )={\displaystyle \frac{1}{\mathcal{B}(\nu ,\iota ,\tau )}}-1$ for every $\iota \in \mathbb{A},\iota \in \mathbb{B}$ where $\tau >0.$ Then by Definition 16, $\varrho (\nu ,\iota ,\tau )$ is an EBPbMS. Therefore, all the prerequisites of Theorems 2 and 5 are satisfied, and the operator T possesses a unique fixed point. □

4. Application to the Fractional Blood Flow Model

Fractional differential equations (FDEs) serve as fundamental instruments for describing a wide range of complex dynamical phenomena encountered across disciplines such as finance, viscoelasticity, engineering, population dynamics, and various applied sciences. In contrast to traditional integer-order formulations, fractional models possess an intrinsic capability to represent memory and hereditary characteristics of physical and biological systems, thereby providing more precise and realistic depictions of dynamic behavior.

In the context of hemodynamics, fractional derivatives capture the hereditary and viscoelastic properties of blood flow more accurately than classical integer-order models. The memory effect implies that the present velocity profile depends not only on the current pressure gradient but also on the past deformation and viscosity history of the arterial wall. This enables fractional operators to model the non-Newtonian and elastic behavior of blood, providing a realistic representation of pulsatile flow through arteries and capillaries.

These models have found successful applications in diverse areas, including economics, aerodynamics, hemodynamics, physics, and image processing (see [18] for detailed references). Further examples demonstrating the broad applicability of fractional calculus can be found in [19].

Although numerous analytical and numerical techniques have been developed to address fractional differential equations [20], deriving exact analytical solutions for nonlinear fractional-order systems continues to be a formidable challenge [21].

Let $A\left(\right[0,1],\mathbb{R})$ denote a Banach space endowed with the supremum norm

$$\parallel u_1-u_2{\parallel }_{\infty }=\underset{r\in [0,1]}{max}|u_1\left(r\right)-u_2\left(r\right)|.$$

(32)

Within this framework, consider the nonlinear fractional Volterra–Fredholm integro-differential equation

$${}^{c}D^{\varsigma }u\left(r\right)=h\left(r\right)u\left(r\right)+p\left(r\right)+{\int }_0^r{R}_1(r,w)S_1\left(u\left(w\right)\right)dw+{\int }_0^1{R}_2(r,w)S_2\left(u\left(w\right)\right)dw,$$

(33)

subject to the initial conditions

$$y^{\left(j\right)}\left(0\right)={\beta }_j,j=0,1,\dots ,k-1,k\in {\mathbb{Z}}^{+},$$

(34)

where ${}^{c}D^{\varsigma }$ denotes the Caputo fractional derivative, $n-1<\varsigma \le n$ with $n\in \mathbb{N}$, and $u:A\to \mathbb{R}$ is an unknown continuous function. The kernel functions $R_i:A\times A\to \mathbb{R}$ are continuous, and the nonlinearities $S_i:\mathbb{R}\to \mathbb{R}$ ($i=1,2$) satisfy the Lipschitz condition.

Hamoud et al. [22] demonstrated that problem (33) is equivalent to an integral equation of the form given in Lemma 2, ensuring the existence and uniqueness of a solution corresponding to the fractional blood flow model.

Lemma 2

([22]). For some initial function $e_0\left(r\right)\in (A,R)$, finding the solution to (33) and (34) is equivalent to solving the integral equation.

$$\begin{array}{cc}\hfill u\left(r\right)& =u_0\left(r\right)+{\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_0^r{(r-w)}^{\varsigma -1}h\left(w\right)u\left(w\right)dw+{\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_0^r{(r-w)}^{\varsigma -1}p\left(w\right)dw\hfill \\ & +{\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_0^r{(r-w)}^{\varsigma -1}\left({\int }_0^w{R}_1(w,\lambda )S_1\left(u\left(\lambda \right)\right)d\lambda +{\int }_0^w{R}_2(w,\lambda )S_2\left(u\left(\lambda \right)\right)d\lambda \right)dw.\hfill \end{array}$$

Theorem 10

(Unique fixed point for the fractional operator T). Let $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$ be a complete extended bipolar parametric b-metric space (EBPbMS) with

$$\varrho (\nu ,\iota ,\tau )=2^{p-1}\underset{r\in [0,1]}{sup}{u}^{-\tau r}{|\nu \left(r\right)-\iota \left(r\right)|}^p,\mathrm{\Omega }\left(s\right)=2^{p-1}s,p\ge 1,s\ge 0,\tau >0.$$

Define $T:C\left(\right[0,1],\mathbb{R})\to C\left(\right[0,1],\mathbb{R})$ by

$$\begin{array}{cc}\hfill \left(Tu\right)\left(r\right)& =u_0\left(r\right)+{\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_0^r{(r-w)}^{\varsigma -1}h\left(w\right)u\left(w\right)dw+{\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_0^r{(r-w)}^{\varsigma -1}p\left(w\right)dw\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_0^r{(r-w)}^{\varsigma -1}\left({\int }_0^w{R}_1(w,\lambda )S_1\left(u\left(\lambda \right)\right)d\lambda +{\int }_0^w{R}_2(w,\lambda )S_2\left(u\left(\lambda \right)\right)d\lambda \right)dw,\hfill \end{array}$$

(35)

where $e_0,h,p\in C([0,1],\mathbb{R})$, the kernels $R_i\in C\left({[0,1]}^2\right)$, and the nonlinearities $S_i$ are Lipschitz with constants $L_{S_i}$. Let

$$L_h={\parallel h\parallel }_{\infty },K_i=\underset{w\in [0,1]}{sup}{\int }_0^w\left|R_i(w,\lambda )\right|d\lambda ,i=1,2,$$

and assume the smallness condition

$$\kappa ={\displaystyle \frac{1}{\Gamma (\varsigma +1)}}\left(L_h+K_1{L}_{S_1}+K_2{L}_{S_2}\right)<1.$$

(36)

If T satisfies the following:

1. 

T is contravariant on $(\mathbb{A}\cup \mathbb{B})$;

2. 

$\alpha (\nu ,\iota ,\tau )\equiv 1$ (so T is trivially α-orbital admissible);

3. 

T is continuous and condition $\left(H\right)$ holds.

Then, T is a ψ–Meir–Keeler contractive mapping (with $\alpha =1$) on $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$. Consequently, T admits a unique fixed point $u^{*}\in \mathbb{A}\cup \mathbb{B}$, corresponding to the unique solution of the fractional blood flow model.

Proof. 

Step 1 (Well-definedness of T).

The integrands appearing in (35) are continuous on compact domains, hence the integral operator T maps $C\left(\right[0,1],\mathbb{R})$ into itself. By the Dominated Convergence Theorem, T is continuous; therefore,

$$T:C\left(\right[0,1],\mathbb{R})\to C\left(\right[0,1],\mathbb{R})$$

is well-defined and continuous.

Step 2 (Contravariance/invariance).

By assumption, $(\mathbb{A},\mathbb{B})$ represent the “polar” subsets of the EBPbMS, and T acts contravariantly:

$$u\in \mathbb{A}\Rightarrow Tu\in \mathbb{B},u\in \mathbb{B}\Rightarrow Tu\in \mathbb{A}.$$

Thus, T preserves the structure of the bipolar system.

Step 3 (Banach-type estimate for T). Let $u_1,u_2\in \mathbb{A}\times \mathbb{B}$ and fix $r\in [0,1]$. Using (35), we obtain

$$\begin{array}{cc}\hfill |T{u}_1\left(r\right)-T{u}_2\left(r\right)|& \le {\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_0^r{(r-w)}^{\varsigma -1}\left|h\left(w\right)\right||u_1\left(w\right)-u_2\left(w\right)|dw+\hfill \\ \hfill & {\displaystyle \frac{1}{\Gamma \left(\varsigma \right)}}{\int }_0^r{(r-w)}^{\varsigma -1}\left({\int }_0^w|R_1(w,\lambda )|L_{S_1}|u_1\left(\lambda \right)-u_2\left(\lambda \right)|d\lambda +{\int }_0^w|R_2(w,\lambda )|L_{S_2}|u_1\left(\lambda \right)-u_2\left(\lambda \right)|d\lambda \right)dw.\hfill \end{array}$$

Taking the supremum over $r\in [0,1]$ and noting that ${\displaystyle {\int }_0^r{(r-w)}^{\varsigma -1}dw=\frac{r^{\varsigma }}{\varsigma }\le \frac{1}{\varsigma }}$, we derive

$$\parallel T{u}_1-T{u}_2{\parallel }_{\infty }\le {\displaystyle \frac{1}{\Gamma (\varsigma +1)}}\left(L_h+K_1{L}_{S_1}+K_2{L}_{S_2}\right)\parallel u_1-u_2{\parallel }_{\infty }=\kappa {\parallel u_1-u_2\parallel }_{\infty },$$

where $\kappa \in (0,1)$ by the smallness condition (36). Hence, T is a strict contraction on the Banach space $(C([0,1],\mathbb{R}),\parallel ·{\parallel }_{\infty })$.

Step 4 (Translation to the EBPbMS metric). For any $\tau >0$, the bipolar parametric distance satisfies

$$\varrho (T{u}_1,T{u}_2,\tau )=2^{p-1}\underset{r\in [0,1]}{sup}{u}^{-\tau r}|T{u}_1\left(r\right)-T{u}_2\left(r\right){|}^p\le 2^{p-1}{\kappa }^p\underset{r\in [0,1]}{sup}{u}^{-\tau r}{|u_1\left(r\right)-u_2\left(r\right)|}^p={\kappa }^p\varrho (u_1,u_2,\tau ),$$

(37)

since $u^{-\tau r}\le 1$. Thus, T is contractive in the EBPbMS sense with contraction factor ${\kappa }^p\in (0,1)$.

Step 5 (Verification of the $\psi$–Meir–Keeler condition with $\alpha =1$).

Let $ϵ>0$ be arbitrary. Choose $\delta \left(ϵ\right)>0$ such that

$$(ϵ+\delta \left(ϵ\right)){\kappa }^p<ϵ⟺\delta \left(ϵ\right)<ϵ\left({\displaystyle \frac{1}{{\kappa }^p}}-1\right),$$

which is possible because ${\kappa }^p<1$. Assume

$$ϵ\le {\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho (u_1,u_2,\tau )\right)\right)<ϵ+\delta \left(ϵ\right).$$

By monotonicity of $\psi$ and ${\mathrm{\Omega }}^{-1}$, it follows that $\varrho (u_1,u_2,\tau )<\mathrm{\Omega }\left({\psi }^{-1}(ϵ+\delta \left(ϵ\right))\right)$. Using (37), we get

$$\varrho (T{u}_1,T{u}_2,\tau )\le {\kappa }^p\varrho (u_1,u_2,\tau )<{\kappa }^p(ϵ+\delta \left(ϵ\right))<ϵ.$$

Since $\alpha \equiv 1$, the Meir–Keeler trigger condition

$$ϵ\le {\mathrm{\Omega }}^{-1}\left(\psi \left(\varrho (u_1,u_2,\tau )\right)\right)<ϵ+\delta \left(ϵ\right)\Rightarrow \alpha (u_1,T{u}_1,\tau )\alpha (T{u}_2,u_2,\tau )\varrho (T{u}_1,T{u}_2,\tau )<ϵ$$

is fulfilled. Hence, T is a $\psi$–Meir–Keeler contraction on $(\mathbb{A},\mathbb{B},\varrho ,\mathrm{\Omega })$.

Hence, all the assumptions of Corollary 2 are satisfied, and therefore, T admits a unique fixed point $u^{*}\in \mathbb{A}\cup \mathbb{B}$. Consequently, $u^{*}$ represents the unique solution of fractional blood flow models (33) and (34). □

5. Conclusions and Future Works

This study introduced and analyzed a new generalized framework, the extended bipolar parametric b-metric space (EBPbMS), and established several fixed-point results for contractive mappings defined within this setting. The developed theoretical results were successfully applied to a nonlinear fractional Volterra–Fredholm integro-differential equation that models blood flow dynamics. This model captures the non-Newtonian and memory-dependent behavior of arterial flow through the use of Caputo fractional derivatives and nonlinear integral kernels.

From a practical standpoint, the fractional blood flow model reflects the hereditary and viscoelastic properties of blood, where the present flow profile depends not only on current pressure gradients but also on the historical deformation of vessel walls. By incorporating memory effects, fractional calculus provides a more accurate and physiologically realistic description of pulsatile flow in arteries and microvascular systems.

By defining an appropriate fractional operator T and demonstrating that it satisfies the $\psi$–Meir–Keeler contractive condition in a complete EBPbMS, we proved the existence and uniqueness of the solution to the fractional blood flow model. The proof relies on extending the Banach and Meir–Keeler principles to the bipolar parametric setting and verifying that the corresponding integral operator is a strict contraction under a suitable smallness condition. This guarantees both the mathematical stability and convergence of the proposed operator framework.

Moreover, the developed fixed-point theory provides a natural analytical basis for assessing the convergence of iterative approximation schemes, such as the Variational Iteration Method and Picard-type processes, which are widely employed for solving nonlinear fractional systems. The flexibility of the EBPbMS structure also opens promising directions for future research, particularly in modeling fractional Micro-Electro-Mechanical Systems (MEMS), where nonlinear damping, electrostatic forces, and memory kernels can be studied under similar contraction principles.

Future Works. Potential future investigations include

• Extending the EBPbMS framework to non-Archimedean or neutrosophic settings;
• Developing fuzzy and bipolar–neutrosophic analogues for uncertain fractional systems;
• Applying the established results to other classes of nonlinear fractional systems, such as fractional MEMS oscillation and hybrid biological models;
• Exploring new contraction types, including F-contractions and rational-type mappings, within the EBPbMS structure.

Overall, the obtained results highlight the analytical strength and wide applicability of the extended bipolar framework in connecting abstract fixed-point theory with real-world fractional dynamics.