Existence and Uniqueness of Fixed Point Results in Non-solid C*-algebra-Valued Bipolar b-Metric Spaces

1. Introduction

The notion of distance, as a natural extension of one of the oldest and natural quantitative concepts, can be thought of as the length of a gap. Metric spaces are regarded as the one of the most significant mathematical frameworks for the study of distance, in that they are intuitive and simple structures that allow us to consider distances between points of a set. These were formalized in 1906 by Fréchet [1]. And in 1922, Banach [2] gave a constructive method to obtain a fixed point for a self-map in metric spaces.

In 1990, Murphy [3] pioneered the novel $C^{★}\text{-}\mathit{algebra}$ and operator theory (see [4,5] and references therein). In 2014, in the spades of novel generalizations, Ma [6] obtained $C^{★}\text{-}\mathit{algebra}$-valued metric spaces and later proved fixed point results in the setting of $b-$metric spaces. The Banach contraction principle has seen further generalizations in the settings of cone metric spaces. One can see Perov [28] and Zabrejko [27] for foundational works, and the reintroduction by Huang and Zhang [29] as the so-called cone metric spaces. In this context, the cone’s solidness is a desirable property for the establishment of fixed point theorems. In 2023, Xu, Cheng, and Han [5] introduced a novel approach to fixed point theory by establishing common fixed point results in cone b-metric spaces over Banach algebras, without requiring the solidness of the underlying cone, only maintaining normality of the cone. Their work focused on contractions with vector-valued coefficients and introduced a kind of new convergence of sequences, termed it wrtn-convergence, that is, ’with respect to the norm’ convergence.

In some cases distances arise between elements of two different sets, rather than between points of a unique set. Such type of distances are abundant in mathematics and applied sciences. And so in 2016, Mutlu and Gürdal [20] formalized these distances under the name bipolar metric spaces and later [25] defined $\alpha -\psi$ contractive mappings and multivalued mappings, respectively, and established fixed point theorems in the context of bipolar metric spaces. In 2021, Gaba, Aphane, and Ayidi [12] introduced $(\alpha -BK)$-contractractions in Bipolar metric spaces and recovered well-known classical results.

Motivated by the above results, we give fixed point results for $C^{★}\text{-}\mathit{algebra}$-valued bipolar b-metric spaces without requiring the solidity of the underlying cone under ($\alpha -BK$)–type contractions.

2. Preliminaries

The following basic concepts from literature in $C^{★}\text{-}\mathit{algebra}$ are necessary for proving results.

A complex algebra $\mathcal{H}$, together with a conjugate linear involution map $p\mapsto p^{★}$, is called a ★-algebra if ${\left(pq\right)}^{★}=q^{★}{p}^{★}$ and ${\left(p^{★}\right)}^{★}=p$ for all $p,q\in \mathcal{H}$. Moreover, the pair $(\mathcal{H},★)$ is called a unital ★-algebra if $\mathcal{H}$ contains an identity element $1_{\mathcal{H}}$.

By a Banach ★-algebra, we mean a complete normed unital ★-algebra $(\mathcal{H},★)$ such that the norm on $\mathcal{H}$ is submultiplicative and satisfies $\parallel p^{★}\parallel =\parallel p\parallel$ for all $p\in \mathcal{H}$. Further, if for all $p\in \mathcal{H}$, we have $\parallel p^{★}{p\parallel =\parallel p\parallel }^2$ in a Banach ★-algebra $(\mathcal{H},★)$, then $\mathcal{H}$ is known as a $C^{★}$-algebra.

A positive element of $\mathcal{H}$ is an element $pv$ such that $p=p^{★}$ and its spectrum satisfies $\sigma \left(p\right)\subset \mathbb{R}$, where

$$\sigma \left(p\right)=\{\lambda \in \mathbb{C}:\lambda 1_{\mathcal{H}}-p\mathrm{is}\mathrm{non}\text{-}\mathrm{invertible}\}.$$

The set of all positive elements will be denoted by ${\mathcal{H}}_{+}$. Such elements allow us to define a partial ordering ⪰ on the elements of $\mathcal{H}$. That is,

$$q⪰p\iff q-p\in {\mathcal{H}}_{+}.$$

If $p\in \mathcal{H}$ is positive, then we write $p⪰0_{\mathcal{H}}$, where $0_{\mathcal{H}}$ is the zero element of $\mathcal{H}$. Each positive element p of a $C^{★}$-algebra $\mathcal{H}$ has a unique positive square root denoted by $p^{1/2}$ in $\mathcal{H}$. From now on, by $\mathcal{H}$ we mean a unital $C^{★}$-algebra with identity element $1_{\mathcal{H}}$. Further,

$${\mathcal{H}}_{+}=\{p\in \mathcal{H}:p⪰0_{\mathcal{H}}\},\mathrm{and}{\left(p^{*}p\right)}^{1/2}=p.$$

Definition 1.

[17] A subset ${\mathcal{H}}_{+}$ of $\mathcal{H}$ is called a cone if:

(1)

${\mathcal{H}}_{+}$ is non-empty, closed, and $\{0_{\mathcal{H}},1_{\mathcal{H}}\}\subset {\mathcal{H}}_{+}$,

(2)

$\alpha x+\beta y\in {\mathcal{H}}_{+},\forall \alpha ,\beta \in {\mathbb{R}}_{+}$, and $x,y\in {\mathcal{H}}_{+},$

(3)

$x·x=x^2\in {\mathcal{H}}_{+},$

(4)

$x+(-x)=0_{\mathcal{H}}$.

For a given cone ${\mathcal{H}}_{+}\subset \mathcal{H}$, we can define a partial ordering ⪯ with respect to ${\mathcal{H}}_{+}$ by $x⪯y$ if and only if $y-x\in {\mathcal{H}}_{+}$. We denote $x\prec y$ if $x⪯y$ and $x\ne y$, while $x\ll y$ will stand for $y-x\in \mathit{int}\left({\mathcal{H}}_{+}\right)$, where $\mathit{int}\left({\mathcal{H}}_{+}\right)$ denotes the interior of ${\mathcal{H}}_{+}$.

The cone ${\mathcal{H}}_{+}$ is called normal if there is a number $M>0$ such that for all $x,y\in \mathcal{H}$, $0_{\mathcal{H}}⪯x⪯y$ implies $\parallel x\parallel \le M\parallel y\parallel$. The least positive number satisfying the above is called the normal constant of ${\mathcal{H}}_{+}$.

Definition 2.

[3] Let $\mathcal{H}$ be a unital $C^{★}$-algebra, and let ${\mathcal{H}}_{+}$ denote the positive cone of $\mathcal{H}$, consisting of all positive elements $a\in \mathcal{H}$ such that $a=x^{*}x$ for some $x\in \mathcal{H}$. A map $F:{\mathcal{H}}_{+}\to {\mathcal{H}}_{+}$ is called a positive function if it satisfies the following properties:

• For all $z\in {\mathcal{H}}_{+}$, $F\left(z\right)\in {\mathcal{H}}_{+}$.
• For all $z_1,z_2\in {\mathcal{H}}_{+}$, if $z_1⪯z_2$, then $F\left(z_1\right)⪯F\left(z_2\right)$, where ⪯ denotes the partial order induced by the positive cone.
• $F\left(0_{\mathcal{H}}\right)=0_{\mathcal{H}}$, where $0_{\mathcal{H}}$ is the zero element of ${\mathcal{H}}_{+}$.

Example 1.

Let ${\mathcal{H}}_{+}$ be the positive cone of a unital $C^{★}$-algebra $\mathcal{H}$. Define the functions $F_{\mathcal{H}},G_{\mathcal{H}}:{\mathcal{H}}_{+}\to {\mathcal{H}}_{+}$ as follows:

$$F_{\mathcal{H}}\left(z\right)=z^2,G_{\mathcal{H}}\left(z\right)=z+z^2,\forall z\in {\mathcal{H}}_{+}.$$

For both functions, condition $\left(1\right)$ and $\left(2\right)$ follow directly from the definition. For condition 3, when $z=0_{\mathcal{H}}$, we have $G_{\mathcal{H}}\left(0\right)=F_{\mathcal{H}}\left(0\right)=0_{\mathcal{H}}$, when $z\ne 0_{\mathcal{H}},$ the inequality $F_{\mathcal{H}}\left(z\right)\prec G_{\mathcal{H}}$ holds because $G_{\mathcal{H}}$ dominates $F_{\mathcal{H}}$ for any $z\in {\mathcal{H}}_{+}$ and $z\ne 0_{\mathcal{H}}$.

Definition 3.

[6] Consider a unital $C^{★}\text{-}\mathit{algebra}$ $\mathcal{H}$ with a unital $I_{\mathcal{H}}$, a set $\mathsf{\Phi }\ne \varnothing$, and $I_{\mathcal{H}}⪯b\in {\mathcal{H}}_{+}$. A distance function $\phi :\mathsf{\Phi }\times \mathsf{\Phi }\to {\mathcal{H}}_{+}$ with the following

• $\phi (\sigma ,\nu )=0_{\mathcal{H}}$ if and only if $\sigma =\nu$, for all $(\sigma ,\nu )\in \mathsf{\Phi }\times \mathsf{\Phi };$
• $\phi (\sigma ,\nu )=\phi (\nu ,\sigma )$ for all $\sigma ,\nu \in \mathsf{\Phi };$
• $\phi (\sigma ,\nu )⪯b\left(\phi (\sigma ,h)+\phi (h,\nu )\right),$ for all $\sigma ,\nu ,h\in \mathsf{\Phi }.$

Then $(\mathsf{\Phi },\mathcal{H},\phi )$ is known as $C^{★}\text{-}\mathit{algebra}$ valued b–metric space.

Definition 4

([20]). Let Φ and Ψ be two non-empty subsets of a set V, and let $\phi :\mathsf{\Phi }\times \mathsf{\Psi }\to {\mathbb{R}}_{+}$ be a function such that:

• $\phi (\sigma ,\nu )=0$ if and only if $\sigma =\nu$;
• $\phi (\sigma ,\nu )=\phi (\nu ,\sigma )$, for all $\sigma ,\nu \in \mathsf{\Phi }\cap \mathsf{\Psi }$;
• $\phi (\sigma ,\nu )\le \phi (\sigma ,{\nu }_1)+\phi ({\sigma }_1,{\nu }_1)+\phi ({\sigma }_1,\nu )$, for all $\sigma ,{\sigma }_1\in \mathsf{\Phi }$ and $\nu ,{\nu }_1\in \mathsf{\Psi }$.

Then, the triplet $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ is called a bipolar metric space.

In 2022, Mani, Gnanaprakasam, Haq, Baloch, and jarad [31], gave the following:

Definition 5.

[31] Consider a unital $C^{★}\text{-}\mathit{algebra}$ $\mathcal{H}$ with a unital $I_{\mathcal{H}}$, two sets $\mathsf{\Phi },\mathsf{\Psi }\ne \varnothing$, and $I⪯b\in {\mathcal{H}}_{+}$. A distance function $\phi :\mathsf{\Phi }\times \mathsf{\Psi }\to {\mathcal{H}}_{+}$ with the following

• $\phi (\sigma ,\nu )=0_{\mathcal{H}}$ if and only if $\sigma =\nu$, for all $(\sigma ,\nu )\in \mathsf{\Phi }\times \mathsf{\Psi }$;
• $\phi (\sigma ,\nu )=\phi (\nu ,\sigma )$, for all $\sigma ,\nu \in \mathsf{\Phi }\cap \mathsf{\Psi }$;
• $\sigma ({\sigma }_1,{\nu }_2)⪯\sigma ({\sigma }_1,{\nu }_1)+\phi ({\sigma }_2,{\nu }_1)+\phi ({\sigma }_2,{\nu }_2),$ for all $({\sigma }_1,{\nu }_1),({\sigma }_2,{\nu }_2)\in \mathsf{\Phi }\times \mathsf{\Psi }$.

is called $C^{★}\text{-}\mathit{algebra}$ bipolar metric and $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ is called $C^{★}\text{-}\mathit{algebra}$ valued bipolar metric space.

Definition 6

([32]). Let $(\mathsf{\Phi },\mathsf{\Psi },\phi )$ be a bipolar metric space. A function $\phi :\mathsf{\Phi }\times \mathsf{\Psi }\to {\mathcal{H}}_{+}$ is called a bipolar b-metric if there exists a constant $b\ge 1$ such that:

• $\phi (\sigma ,\nu )=0$ if and only if $\sigma =\nu$;
• $\phi (\sigma ,\nu )=\phi (\nu ,\sigma )$, for all $\sigma ,\nu \in \mathsf{\Phi }\cap \mathsf{\Psi }$;
• $\phi ({\sigma }_1,{\nu }_2)\le b[\phi ({\sigma }_1,{\nu }_1)+\phi ({\sigma }_2,{\nu }_1)+\phi ({\sigma }_2,{\nu }_2)]$, for all ${\sigma }_1,{\sigma }_2\in \mathsf{\Phi }$ and ${\nu }_1,{\nu }_2\in \mathsf{\Psi }$.

Then $(\mathsf{\Phi },\mathsf{\Psi },{\mathcal{H}}_{+},\phi )$ is called a bipolar b-metric space.

Remark 1.

The space is said to be joint if $\mathsf{\Phi }\cap \mathsf{\Psi }\ne \varnothing$ otherwise disjoint.

Example 2.

[32] Consider $\mathsf{\Phi }=(-\infty ,0],\mathsf{\Psi }=[0,\infty ),\mathcal{H}={\mathcal{M}}_2\left(\mathbb{R}\right)$ and $\phi :\mathsf{\Phi }\times \mathsf{\Psi }\to \mathcal{H}$ as $\phi (\sigma ,\nu )=diag\{c_1\left(\right|\sigma -{\nu \left|\right)}^p,c_2\left(\right|\sigma -{\nu \left|\right)}^p$} where $p>1$ and $c_1,c_2>0.$

It can be easily verified that conditions 1 and 2 of definition 1.4 holds.

Using $|{\sigma }_1-{\nu }_2|p\le 2^{2p}\left(\right|{\sigma }_1-{\nu }_1{|}^p+|{\sigma }_2-{\nu }_1{|}^p+|{\sigma }_2-{\nu }_2{|}^p)$, one can prove that:

$$\phi ({\sigma }_1,{\nu }_2)\le b[\phi ({\sigma }_1,{\nu }_1)+\phi ({\sigma }_2,{\nu }_1)+\phi ({\sigma }_2,{\nu }_2)],$$

where $b=2^{2p}I$.

Thus, $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ is a complete $C^{*}$-algebra valued bipolar b-metric space.

If we take ${\nu }_1=-{\displaystyle \frac{1}{2}}$, ${\nu }_2=0$, ${\nu }_1=0$, and ${\nu }_2={\displaystyle \frac{1}{2}}$, then:

$$\phi ({\sigma }_1,{\nu }_2)⪰\phi ({\sigma }_1,{\nu }_1)+\phi ({\sigma }_1,{\nu }_1)+\phi ({\sigma }_2,{\nu }_2),$$

for all $({\sigma }_1,{\nu }_1),({\sigma }_2,{\nu }_2)\in \mathsf{\Phi }\times \mathsf{\Psi }$. So, it is not a $C^{*}$-algebra valued bipolar metric space.

Definition 7.

Let $({\mathsf{\Phi }}_1,{\mathsf{\Psi }}_1,\mathcal{H},{\phi }_1)$ and $({\mathsf{\Phi }}_2,{\mathsf{\Psi }}_2,\mathcal{H},{\phi }_2)$ be $C^{★}$-algebra valued bipolar b-metric spaces, where ${\phi }_1$ and ${\phi }_2$ are $C^{★}$-algebra valued metrics on $({\mathsf{\Phi }}_1\times {\mathsf{\Psi }}_1)$ and $({\mathsf{\Phi }}_2\times {\mathsf{\Psi }}_2)$, respectively.

(1)

A function $T:{\mathsf{\Phi }}_1\cup {\mathsf{\Psi }}_1\to {\mathsf{\Phi }}_2\cup {\mathsf{\Psi }}_2$ is called a contravariant map from $({\mathsf{\Phi }}_1,{\mathsf{\Psi }}_1)$ to $({\mathsf{\Phi }}_2,{\mathsf{\Psi }}_2)$, and denoted by $f:({\mathsf{\Phi }}_1,{\mathsf{\Psi }}_1)\rightleftharpoons ({\mathsf{\Phi }}_2,{\mathsf{\Psi }}_2)$, if :

$$T\left({\mathsf{\Phi }}_1\right)\subseteq {\mathsf{\Psi }}_2\mathit{and}T\left({\mathsf{\Psi }}_1\right)\subseteq {\mathsf{\Phi }}_2.$$

(2)

Moreover, if ${\phi }_1$ and ${\phi }_2$ are $C^{★}$-algebra valued bipolar b-metrics on $({\mathsf{\Phi }}_1\times {\mathsf{\Psi }}_1)$ and $({\mathsf{\Phi }}_2\times {\mathsf{\Psi }}_2)$, respectively, then the notation:

$$T:({\mathsf{\Phi }}_1,{\mathsf{\Psi }}_1,{\phi }_1)⤨({\mathsf{\Phi }}_2,{\mathsf{\Psi }}_2,{\phi }_2)$$

denotes a contravariant map between $C^{★}$-algebra valued bipolar b-metric spaces, where the distances ${\phi }_1$ and ${\phi }_2$ respect the $C^{★}$-algebra structure.

Definition 8.

[31] Let $({\mathsf{\Phi }}_1,{\mathsf{\Psi }}_1,\mathcal{H},{\phi }_1)$ and $({\mathsf{\Phi }}_2,{\mathsf{\Psi }}_2,\mathcal{H},{\phi }_2)$ be two $C^{★}$-algebra valued bipolar metric spaces,

(1)

T is called left continuous at a point ${\sigma }_0\in {\mathsf{\Phi }}_1$ if, for every $ϵ>0$, there exists a $\delta >0$ such that:

$$\parallel {\phi }_2(T{\sigma }_0,T\nu )\parallel <ϵ\mathit{whenever}\parallel {\phi }_1({\sigma }_0,\nu )\parallel <\delta ,\nu \in {\mathsf{\Psi }}_1.$$

(2)

T is called right continuous at a point ${\nu }_0\in {\mathsf{\Pi }}_1$ if, for every $ϵ>0$, there exists a $\delta >0$ such that:

$$\parallel {\sigma }_2(T\sigma ,T{\nu }_0)\parallel <ϵ\mathit{whenever}\parallel {\sigma }_1(\sigma ,{\nu }_0)\parallel <\delta ,\sigma \in {\mathsf{\Phi }}_1.$$

(3)

T is called continuous if it is left continuous at every ${\sigma }_0\in {\mathsf{\Phi }}_1$ and right continuous at every ${\nu }_0\in {\mathsf{\Pi }}_1$.

(4)

A map T is called a continuous contravariant map if it satisfies the same continuity conditions as a covariant map, with:

$$T\left({\mathsf{\Phi }}_1\right)\subseteq {\mathsf{\Psi }}_2\mathit{and}T\left({\mathsf{\Psi }}_1\right)\subseteq {\mathsf{\Phi }}_2.$$

Definition 9.

Let $F:{\mathcal{A}}_{+}\to {\mathcal{A}}_{+}$ be a positive function on the positive cone ${\mathcal{A}}_{+}$ of a $C^{*}$-algebra $\mathcal{A}$. Then F satisfies the combined condition:

$$F(a+b)=F\left(a\right)+F\left(b\right),\mathit{for}\mathit{all}a,b\in {\mathcal{A}}_{+}\mathit{with}ab=ba,$$

and

$$F\left(\lambda a\right)=\lambda F\left(a\right),\mathit{for}\mathit{all}\lambda \ge 0\mathit{and}a\in {\mathcal{A}}_{+}.$$

Equivalently, for $a,b\in {\mathcal{A}}_{+}$ with $ab=ba$, and $\lambda ,\mu \ge 0$,

$$F(\lambda a+\mu b)=\lambda F\left(a\right)+\mu F\left(b\right).$$

This linear combination ensures that F is additive and homogeneous on commuting elements in the positive cone.

Definition 10.

Let $F:{\mathcal{A}}_{+}\to {\mathcal{A}}_{+}$ be a positive function on the positive cone ${\mathcal{A}}_{+}$ of a $C^{*}$-algebra $\mathcal{A}$. Then F satisfies the linearity condition:

$$F(\lambda a+\mu b)=\lambda F\left(a\right)+\mu F\left(b\right),$$

for all $a,b\in {\mathcal{A}}_{+}$ such that $ab=ba$ (commutative elements), and for all $\lambda ,\mu \ge 0$.

In this paper, motivated by the above results, we prove fixed point theorems on $C^{★}-algebra\text{-}\mathrm{valued}$ metric spaces in the following settings:

• (C1) The underlying cone is non-solid. Unlike many prior results that assume solidness of the cone, this work addresses cases where the cone lacks interior points. Non-solid cones can lead to situations where convergence results fail, as observed in certain applications in quantum mechanics.
• (C2) The contractions considered are $(F_{\mathcal{H}}-G_{\mathcal{H}})$-contractions, extending the applicability of contraction mappings in this context.
• (C3) Applications to the Ulam-Hyers Stability problem are provided, illustrating the utility of the theoretical results in practical and analytical settings.

3. Main Results

Definition 11.

Let $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ be a $C^{★}$-algebra-valued bipolar b-metric space in a non-solid cone ${\mathcal{H}}_{+}\subseteq \mathcal{H}$. Let $F_{\mathcal{H}},G_{\mathcal{H}}:{\mathcal{H}}_{+}\to {\mathcal{H}}_{+}$, where $F_{\mathcal{H}}⪯G_{\mathcal{H}}$ for all $z\in {\mathcal{H}}_{+}$, and $F_{\mathcal{H}}\left(z\right)=G_{\mathcal{H}}\left(z\right)$ if and only if $z=0_{\mathcal{H}}$, then:

• Elements of Φ are left elements, of Ψ are right elements, and of $\mathsf{\Phi }\cup \mathsf{\Psi }$ are central elements.
• A left sequence $\left\{{\sigma }_n\right\}\subseteq \mathsf{\Phi }$ converges to $\nu \in \mathsf{\Psi }$ if and only if :

  $$\parallel F_{\mathcal{H}}\phi ({\sigma }_n,\nu )\parallel \to 0_{\mathcal{H}}\mathit{as}n\to \infty .$$
• A right sequence $\left\{{\nu }_n\right\}\subseteq \mathsf{\Psi }$ converges to $\sigma \in \mathsf{\Phi }$ if and only if:

  $$\parallel F_{\mathcal{H}}\phi ({\nu }_n,\sigma )\parallel \to 0_{\mathcal{H}}\mathit{as}n\to \infty .$$
• A bisequence is a pair of sequences $(\left\{{\sigma }_n\right\},\left\{{\nu }_n\right\})$ on $\mathsf{\Phi }\times \mathsf{\Psi }$.
• A bisequence $(\left\{{\sigma }_n\right\},\left\{{\nu }_n\right\})$ is convergent if both $\left\{{\sigma }_n\right\}$ and $\left\{{\nu }_n\right\}$ converge to a common point $z\in \mathsf{\Phi }\cap \mathsf{\Psi }$. This is called biconvergence.
• A bisequence $(\left\{{\sigma }_n\right\},\left\{{\nu }_n\right\})$ is a Cauchy bisequence if and only if:

  $$\parallel F_{\mathcal{H}}\phi ({\sigma }_n,{\nu }_m)\parallel \to 0_{\mathcal{H}}\mathit{as}n,m\to \infty .$$
• The space $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ is $wrtn$-complete if every Cauchy bisequence is convergent.

Remark 2.

Here, $wrtn$ means ’ with respect to the norm of $\mathcal{H}$.’ Convergence is defined with respect to the norm inherent in $\mathcal{H}.$

Proposition 1.

In a $C^{★}-algebra$ valued bipolar b-metric space, every convergent bisequence is a Cauchy bisequence.

Proof. 

Let $(\left\{{\sigma }_n\right\},\left\{{\nu }_n\right\})$ be a biconvergent bisequence in a $C^{★}$-algebra valued bipolar b-metric space $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$, which converges to some $z\in \mathsf{\Phi }\cap \mathsf{\Psi }$, then:

$$\phi ({\sigma }_n,{\nu }_m)⪯b\left(\phi ({\sigma }_n,z)+\phi (z,z)+\phi (z,{\nu }_m)\right).$$

By normality condition, we have:

$$\parallel \phi ({\sigma }_n,{\nu }_m)\parallel ⪯bM\left(\parallel \phi ({\sigma }_n,z)+\phi (z,z)+\phi (z,{\nu }_m)\parallel \right)\le bM\left(\parallel \phi ({\sigma }_n,z)\parallel +\parallel \phi (z,z)\parallel +\parallel \phi (z,{\nu }_m)\parallel \right)$$

Since $\left\{{\sigma }_n\right\}\to z\in \mathsf{\Psi }$ and $\left\{{\nu }_m\right\}\to z\in \mathsf{\Phi }$, then

$$\parallel \phi ({\sigma }_n,z)\parallel \to 0_{\mathcal{H}},\mathrm{and}\parallel \phi ({\nu }_m,z)\parallel \to 0_{\mathcal{H}},\mathrm{as}n,m\to \infty .$$

Additionally, $\parallel \phi (z,z)\parallel =0_{\mathcal{H}}.$ Thus $\parallel \phi ({\sigma }_n,{\nu }_m)\parallel \to 0$ as $n,m\to \infty .$ Hence, $(\left\{{\sigma }_n\right\},\left\{{\nu }_n\right\})$ is a Cauchy bisequence. □

Proposition 2.

In $C^{★}-algebra$ valued bipolar b-metric space, every convergent Cauchy bisequence is biconvergent.

Proof. 

Let $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ be a $C^{★}-algebra$ valued bipolar b-metric space, and $(\left\{{\sigma }_n\right\},\left\{{\nu }_n\right\})$ be a convergent Cauchy bisequence, such that ${\sigma }_n\to \nu \in \mathsf{\Psi }$ and ${\nu }_n\to \sigma \in \mathsf{\Phi }$, then

$$\phi (\sigma ,\nu )⪯b\left(\phi (\sigma ,{\nu }_n)+\phi ({\nu }_n,{\sigma }_n)+\phi ({\sigma }_n,\nu )\right).$$

By normality condition, we have:

$$\parallel (\phi (\sigma ,\nu )\parallel \le bM\left(\parallel \phi (\sigma ,{\nu }_n)+\phi ({\nu }_n,{\sigma }_n)+\phi ({\sigma }_n,\nu )\right)\parallel \le bM\left(\parallel \phi (\sigma ,{\nu }_n)\parallel +\parallel \phi ({\nu }_n,{\sigma }_n)\parallel +\phi ({\sigma }_n,\nu )\right)\parallel .$$

Since $(\left\{{\sigma }_n\right\},\left\{{\nu }_n\right\})$ is a convergent Cauchy bisequence, we have $\parallel \phi ({\nu }_n,{\sigma }_n)\parallel \to 0\mathrm{as}n\to \infty ,$ and $\left\{{\sigma }_n\right\}\to \nu$ and $\left\{{\nu }_n\right\}\to \sigma$, we also have:

$$\parallel (\phi (\sigma ,{\nu }_n)\parallel \to 0\mathrm{and}\parallel (\phi ({\sigma }_n,\nu )\parallel \to 0\mathrm{as}n\to \infty .$$

Thus $\parallel \phi (\sigma ,\nu )\parallel \to 0$ as $n\to \infty .$ This implies $\phi (\sigma ,\nu )=0_{\mathcal{H}}$. Hence, $\sigma =\nu .$ □

Definition 12.

Let $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ be $C^{★}$-algebra valued bipolar b-metric space and $T:(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H}\phi )⤨(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H}\phi )$ be a contravariant map such that there exists constants ${\alpha }_1,{\alpha }_2,{\alpha }_3\ge 0$, with ${\alpha }_1+{\alpha }_2+{\alpha }_3<1$. Then T is called $(F_{\mathcal{H}}-G_{\mathcal{H}}$)- contraction if there exits two positive functions $F_{\mathcal{H}},G_{\mathcal{H}}:{\mathcal{H}}_{+}\to {\mathcal{H}}_{+}$ such that

$$F_{\mathcal{H}}\left(\phi (T\sigma ,T\nu )\right)⪯G_{\mathcal{H}}\left({\alpha }_1\phi (\sigma ,\nu )+{\alpha }_2\phi (\sigma ,T\sigma )+{\alpha }_3\phi (T\nu ,\nu )\right),$$

whenever $(\sigma ,\nu )\in \mathsf{\Phi }\times \mathsf{\Psi }$ and $\phi (T\sigma ,T\nu )>0_{\mathcal{H}}.$

Theorem 1.

Let $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ be a $wrtn$-complete $C^{★}$-algebra-valued bipolar b-metric space, where:

(i)

$\phi :(\mathsf{\Phi }\times \mathsf{\Psi })\to {\mathcal{H}}_{+}$ is the $C^{★}$-algebra-valued bipolar b-metric,

(ii)

${\mathcal{H}}_{+}$ is the positive cone of a unital $C^{★}$-algebra $\mathcal{H}$,

(iii)

the cone ${\mathcal{H}}_{+}$ isnon-solid(i.e., it has an empty interior, $\mathrm{Int}\left({\mathcal{H}}_{+}\right)=\varnothing$).

Suppose a mapping $T:\mathsf{\Phi }\cup \mathsf{\Psi }\to \mathsf{\Phi }\cup \mathsf{\Psi }$ satisfies the following conditions:

(1)

There exist constants ${\alpha }_1,{\alpha }_2,{\alpha }_3\ge 0$ such that:

$${\alpha }_1+{\alpha }_2+{\alpha }_3<1.$$

(2)

There exist positive functions $F_{\mathcal{H}},G_{\mathcal{H}}:{\mathcal{H}}_{+}\to {\mathcal{H}}_{+}$ such that:

$$F_{\mathcal{H}}\left(z\right)⪯G_{\mathcal{H}}\left(z\right),\forall z\in {\mathcal{H}}_{+}\setminus \left\{0_{\mathcal{H}}\right\},$$

and:

$$F_{\mathcal{H}}\left(0_{\mathcal{H}}\right)=G_{\mathcal{H}}\left(0_{\mathcal{H}}\right)=0_{\mathcal{H}}.$$

(3)

For all $\sigma ,\nu \in \mathsf{\Phi }\cup \mathsf{\Psi }$, the contraction condition holds:

$$F_{\mathcal{H}}\left(\phi (T\nu ,T\sigma )\right)⪯G_{\mathcal{H}}\left({\alpha }_1\phi (\sigma ,\nu )+{\alpha }_2\phi (\sigma ,T\sigma )+{\alpha }_3\phi (T\nu ,\nu )\right).$$

Then, the mapping T has a unique fixed point $z\in \mathsf{\Phi }\cap \mathsf{\Psi }$ such that:

$$Tz=z.$$

Proof. 

Define ${\sigma }_{n+1}=T{\nu }_n$ and ${\nu }_n=T{\sigma }_n$. Then

$$\phi ({\sigma }_n,{\nu }_n)=\phi (T{\nu }_{n-1},T{\sigma }_n).$$

Applying the contraction, we get:

$$\begin{array}{cc}\hfill F_{\mathcal{H}}\left(\phi (T{\nu }_{n-1},T{\sigma }_n)\right)& ⪯G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_n,{\nu }_{n-1})+{\alpha }_2\phi ({\sigma }_n,T{\sigma }_n)+{\alpha }_3\phi (T{\nu }_{n-1},{\nu }_{n-1})\right)\hfill \\ \hfill & =G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_n,{\nu }_{n-1})+{\alpha }_2\phi ({\sigma }_n,{\nu }_n)+{\alpha }_3\phi ({\sigma }_n,{\nu }_{n-1})\right)\hfill \\ \hfill & =G_{\mathcal{H}}\left(({\alpha }_1+{\alpha }_3)\phi ({\sigma }_n,{\nu }_{n-1})+{\alpha }_2\phi ({\sigma }_n,{\nu }_n)\right).\hfill \end{array}$$

Therefore,

$$F_{\mathcal{H}}\left(\phi ({\sigma }_n,{\nu }_n)\right)⪯G_{\mathcal{H}}\left(({\alpha }_1+{\alpha }_2)\phi ({\sigma }_n,{\nu }_{n-1})+{\alpha }_2\phi ({\sigma }_n,{\nu }_n)\right),$$

(1)

for all integers $n\ge 1.$ Therefore, we consider:

$$\phi ({\sigma }_n,{\nu }_{n-1})=\phi ({\sigma }_{n-1},{\nu }_{n-2}).$$

Applying the contraction again, we get:

$$\begin{array}{cc}\hfill F_{\mathcal{H}}\left(\phi (T{\nu }_{n-2},T{\sigma }_{n-1})\right)& ⪯G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_{n-1},{\nu }_{n-2})+{\alpha }_2\phi ({\sigma }_{n-1},T{\sigma }_{n-1})+{\alpha }_3\phi (T{\nu }_{n-2},{\nu }_{n-2})\right)\hfill \\ \hfill & =G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_{n-1},{\nu }_{n-2})+{\alpha }_2\phi ({\sigma }_{n-1},{\nu }_{n-1})+{\alpha }_3\phi ({\sigma }_{n-1},{\nu }_{n-2})\right)\hfill \\ \hfill & =G_{\mathcal{H}}\left(({\alpha }_1+{\alpha }_3)\phi ({\sigma }_{n-1},{\nu }_{n-2})+{\alpha }_2\phi ({\sigma }_{n-1},{\nu }_{n-1})\right).\hfill \end{array}$$

Therefore, by monotonicity of $F_{\mathcal{H}}$ and $G_{\mathcal{H}}$, we have:

$$\phi ({\sigma }_n,{\nu }_{n-1})⪯F_{\mathcal{H}}^{-1}\left(G_{\mathcal{H}}\left(({\alpha }_1+{\alpha }_3)\phi ({\sigma }_{n-1},{\nu }_{n-2})+{\alpha }_2\phi ({\sigma }_{n-1},{\nu }_{n-1})\right)\right).$$

(2)

Substituting $\left(2\right)$ into $\left(1\right)$, we get

$$F_{\mathcal{H}}\left(\phi ({\sigma }_n,{\nu }_n)\right)⪯G_{\mathcal{H}}\left(({\alpha }_1+{\alpha }_3)F_{\mathcal{H}}^{-1}\left(G_{\mathcal{H}}\left(({\alpha }_1+{\alpha }_3)\phi ({\sigma }_{n-1},{\nu }_{n-2})+{\alpha }_2\phi ({\sigma }_{n-1},{\nu }_{n-1})\right)\right)+{\alpha }_2\phi ({\sigma }_n,{\nu }_n)\right).$$

(3)

Similarly, by expanding the recursive terms $\phi ({\sigma }_{n-1},{\nu }_{n-2})$ and $\phi ({\sigma }_{n-1},{\nu }_{n-1})$, we get:

$$\phi ({\sigma }_{n-1},{\nu }_{n-2})⪯F_{\mathcal{H}}^{-1}\left(G_{\mathcal{H}}\left(({\alpha }_1+{\alpha }_3)\phi ({\sigma }_{n-2},{\nu }_{n-3})+{\alpha }_2\phi ({\sigma }_{n-2},{\nu }_{n-2})\right)\right),$$

(4)

and

$$\phi ({\sigma }_{n-1},{\nu }_{n-1})⪯F_{\mathcal{H}}^{-1}\left(G_{\mathcal{H}}\left(({\alpha }_1+{\alpha }_3)\phi ({\sigma }_{n-2},{\nu }_{n-3})+{\alpha }_2\phi ({\sigma }_{n-1},{\nu }_{n-2})\right)\right).$$

(5)

Substituting $\left(4\right)$ and $\left(5\right)$ into $\left(3\right)$, we get:

$$\begin{array}{cc}\hfill F_{\mathcal{H}}\left(\phi ({\sigma }_n,{\nu }_n)\right)& ⪯G_{\mathcal{H}}({({\alpha }_1+{\alpha }_3)}^2{F}_{\mathcal{H}}^{-1}\left(G_{\mathcal{H}}\left(({\alpha }_1+{\alpha }_3)\phi ({\sigma }_{n-2},{\nu }_{n-3})+{\alpha }_2\phi ({\sigma }_{n-2},{\nu }_{n-2})\right)\right)\hfill \\ \hfill & +({\alpha }_1+{\alpha }_3){\alpha }_2{F}_{\mathcal{H}}^{-1}\left(G_{\mathcal{H}}\left(\phi ({\sigma }_{n-1},{\nu }_{n-1})\right)\right)+{\alpha }_2\phi ({\sigma }_n,{\nu }_n)).\hfill \end{array}$$

(6)

Clearly, for $k\ge 0$, the general pattern emerges as:

$$\phi ({\sigma }_n,{\nu }_n)⪯\sum _{k=0}^n{({\alpha }_1+{\alpha }_3)}^k{\alpha }_2\phi ({\sigma }_{n-k},{\nu }_{n-k}).$$

(7)

As $({\alpha }_1+{\alpha }_2)<1$, it follows that:

$$\phi ({\sigma }_n,{\nu }_n)⪯{\alpha }_2\sum _{k=0}^n{({\alpha }_1+{\alpha }_3)}^k\phi ({\sigma }_0,{\nu }_0),$$

(8)

and as $n\to \infty$ the geometric series converges:

$$\sum _{k=0}^{\infty }{({\alpha }_1+{\alpha }_3)}^k={\displaystyle \frac{1}{1-({\alpha }_1+{\alpha }_3)}}.$$

(9)

Thus

$$\phi ({\sigma }_n,{\nu }_n)⪯{\displaystyle \frac{{\alpha }_2\phi ({\sigma }_0,{\nu }_0)}{1-({\alpha }_1+{\alpha }_3)}}.$$

(10)

By setting

$$\lambda ={\displaystyle \frac{{\alpha }_2\phi ({\sigma }_0,{\nu }_0)}{1-({\alpha }_1+{\alpha }_3)}}$$

we see that $\lambda <1.$ Therefore, for all positive integers $m,n\in \mathbb{N}$,

• if $m>n$, from the recursive definitions:

  $${\sigma }_{n+1}=T{\nu }_n,{\nu }_m=T{\sigma }_{m-1}$$

  we have

  $$F_{\mathcal{H}}\left(\phi ({\sigma }_n,{\nu }_m)\right)⪯G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_n,{\nu }_{m-1})+{\alpha }_2\phi ({\sigma }_n,{\nu }_m)+{\alpha }_3\phi ({\sigma }_{n-1},{\nu }_m)\right)$$

  (11)

  and

  $$F_{\mathcal{H}}\left(\phi ({\sigma }_n,{\nu }_{m-1})\right)⪯G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_n,{\nu }_{m-2})+{\alpha }_2\phi ({\sigma }_n,{\nu }_{m-1})+{\alpha }_3\phi ({\sigma }_{n-1},{\nu }_{m-1})\right).$$

  (12)
  Substituting $\left(12\right)$ into $(11$ we get:

  $$\begin{array}{cc}\hfill F_{\mathcal{H}}\left(\phi ({\sigma }_n,{\nu }_m)\right)& ⪯G_{\mathcal{H}}({\alpha }_1{F}_{\mathcal{H}}^{-1}\left(G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_n,{\nu }_{m-2})+{\alpha }_2\phi ({\sigma }_n,{\nu }_{m-1})+{\alpha }_3\phi ({\sigma }_{n-1},{\nu }_{m-1})\right)\right)\hfill \\ \hfill & +{\alpha }_2\phi ({\sigma }_n,{\nu }_m)+{\alpha }_3\phi ({\sigma }_{n-1},{\nu }_m)).\hfill \\ \hfill & =\sum _{j=0}^{m-n}{({\alpha }_1+{\alpha }_3)}^j{\alpha }_2\phi ({\sigma }_0,{\nu }_0),\hfill \end{array}$$
  Therefore

  $$F_{\mathcal{H}}\left(\phi ({\sigma }_n,{\nu }_m)\right)⪯\sum _{j=0}^{m-n}{({\alpha }_1+{\alpha }_3)}^j{\alpha }_2\phi ({\sigma }_0,{\nu }_0)$$

  (13)
  After k recursive substitutions:

  $$\phi ({\sigma }_n,{\nu }_m)⪯\sum _{j=0}^{m-n}{({\alpha }_1+{\alpha }_3)}^j{\alpha }_2\phi ({\sigma }_0,{\nu }_0).$$
  For $({\alpha }_1+{\alpha }_3)<1$, the geometric series converges:

  $$\sum _{j=0}^{\infty }{({\alpha }_1+{\alpha }_3)}^j={\displaystyle \frac{1}{1-({\alpha }_1+{\alpha }_3)}}.$$
  Thus, as $m\to \infty$, we have

  $$\phi ({\sigma }_n,{\nu }_m)⪯{\displaystyle \frac{{\alpha }_2\phi ({\sigma }_0,{\nu }_0)}{1-({\alpha }_1+{\alpha }_3)}}.$$
• If $m<n$,

  $$F_{\mathcal{H}}\left(\phi ({\sigma }_n,{\nu }_m)\right)⪯G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_n,{\nu }_{m-1})+{\alpha }_2\phi ({\sigma }_n,{\nu }_m)+{\alpha }_3\phi ({\sigma }_{n-1},{\nu }_m)\right),$$

  (14)

  and

  $$F_{\mathcal{H}}\left(\phi ({\sigma }_{n-1},{\nu }_m)\right)⪯G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_{n-1},{\nu }_{m-1})+{\alpha }_2\phi ({\sigma }_{n-1},{\nu }_m)+{\alpha }_3\phi ({\sigma }_{n-2},{\nu }_m)\right).$$

  (15)
  Substituting $\left(15\right)$ into $\left(14\right)$ we get:

  $$\begin{array}{cc}\hfill F_{\mathcal{H}}\left(\phi ({\sigma }_n,{\nu }_m)\right)& ⪯G_{\mathcal{H}}({\alpha }_1\phi ({\sigma }_n,{\nu }_{m-1})+{\alpha }_2\phi ({\sigma }_n,{\nu }_m)\hfill \\ \hfill & +{\alpha }_3{F}_{\mathcal{H}}^{-1}\left(G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_{n-1},{\nu }_{m-1})+{\alpha }_2\phi ({\sigma }_{n-1},{\nu }_m)+{\alpha }_3\phi ({\sigma }_{n-2},{\nu }_m)\right)\right)).\hfill \end{array}$$

  (16)
  After k recursive expansions:

  $$\phi ({\sigma }_n,{\nu }_m)⪯\sum _{j=0}^{n-m}{({\alpha }_1+{\alpha }_3)}^j{\alpha }_2\phi ({\sigma }_0,{\nu }_0),$$

  (17)
  For $({\alpha }_1+{\alpha }_3)<1$, the geometric series converges:

  $$\sum _{j=0}^{\infty }{({\alpha }_1+{\alpha }_3)}^j={\displaystyle \frac{1}{1-({\alpha }_1+{\alpha }_3)}}.$$
  Thus, as $n\to \infty$:

  $$\phi ({\sigma }_n,{\nu }_m)⪯{\displaystyle \frac{{\alpha }_2\phi ({\sigma }_0,{\nu }_0)}{1-({\alpha }_1+{\alpha }_3)}}.$$

  (18)

Since $\lambda <1,$ this means that $\phi ({\sigma }_n,{\nu }_m)$ can be made arbitrarily small by larger values of m and n, and hence $\left(\left\{{\sigma }_n\right\},\left\{{\nu }_m\right\}\right)$ is a Cauchy bisequence with respect to $\mathcal{H}$. Since $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ is complete, $\left(\left\{{\sigma }_n\right\},\left\{{\nu }_m\right\}\right)$ converges, and as a convergent Cauchy bisequence, in particular it biconverges, it follows that ${\sigma }_n\to z$ and ${\nu }_m\to z$, where $z\in \mathsf{\Phi }\cap \mathsf{\Psi }$. Since T is continuous, $T{\sigma }_n\to Tz$. Therefore, $Tz=z$. Hence z is fixed point of $T.$ If $u\in \mathsf{\Psi }$ is any other fixed point of T, then

$$0_{\mathcal{H}}⪯\phi (z,u)=\phi (Tz,Tu)$$

and

$$\begin{array}{cc}\hfill F_{\mathcal{H}}\left(\phi (Tz,Tu)\right)& ⪯G_{\mathcal{H}}\left({\alpha }_1\phi (u,z)+{\alpha }_2\phi (u,Tu)+{\alpha }_3\phi (Tz,z)\right)\hfill \\ \hfill & ={\alpha }_1{G}_{\mathcal{H}}\left(\phi (u,z)\right).\hfill \end{array}$$

This implies $F_{\mathcal{H}}\left(\phi (z,u)\right)⪯{\alpha }_1{G}_{\mathcal{H}}\left(\phi (u,z)\right)$, which is not true by monotonicity of $F_{\mathcal{H}}$ and $G_{\mathcal{H}}$, unless $\phi (z,u)=\phi (u,z)=0_{\mathcal{H}}$. Hence, $z=u$. Similar arguments hold if $u\in \mathsf{\Phi }$. □

4. Consequences

Corollary 1

($(\alpha ,\mathbf{BK})$-Type-Contractions). Let $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ be a $wrtn$-complete $C^{★}$-algebra-valued bipolar b-metric space, and let $T:(\mathsf{\Phi },\mathsf{\Psi },\phi )⤨(\mathsf{\Phi },\mathsf{\Psi },\phi )$ be a contravariant map satisfying the contraction condition:

$$F_{\mathcal{H}}\left(\phi (T\nu ,T\sigma )\right)⪯G_{\mathcal{H}}\left({\alpha }_1\phi (\sigma ,\nu )+{\alpha }_2\phi (\sigma ,T\sigma )+{\alpha }_3\phi (T\nu ,\nu )\right),$$

where $F_{\mathcal{H}},G_{\mathcal{H}}:{\mathcal{H}}_{+}\to {\mathcal{H}}_{+}$. By setting $F_{\mathcal{H}}\left(z\right)=z$ and $G_{\mathcal{H}}\left(z\right)={\alpha }_1{z}_1+{\alpha }_2{z}_2+{\alpha }_3{z}_3$, where $z=(z_1,z_2,z_3)$ and $\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3)$ are tuplets of elements in ${\mathcal{H}}_{+}$ satisfying ${\alpha }_1+{\alpha }_2+{\alpha }_3\prec 1(\mathrm{with}\mathrm{respect}\mathrm{to}\mathrm{the}\mathrm{ordering}\mathrm{in}{\mathcal{H}}_{+})$, the contraction becomes:

$$\phi (T\nu ,T\sigma )⪯{\alpha }_1\phi (\sigma ,\nu )+{\alpha }_2\phi (\sigma ,T\sigma )+{\alpha }_3\phi (T\nu ,\nu ),$$

for all $\sigma ,\nu \in \mathsf{\Phi }\cup \mathsf{\Psi }$. Then function $T:(\mathsf{\Phi },\mathsf{\Psi },\phi )\to (\mathsf{\Phi },\mathsf{\Psi },\phi )$ has a unique fixed point.

Corollary 2

(Vector-Valued Contractions). Let $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ be a $wrtn$-complete $C^{★}$-algebra-valued bipolar b-metric space, and let $T:(\mathsf{\Phi },\mathsf{\Psi },\phi )⤨(\mathsf{\Phi },\mathsf{\Psi },\phi )$ be a contravariant map. Suppose T satisfies the contraction condition:

$$F_{\mathcal{H}}\left(\phi (T\nu ,T\sigma )\right)⪯G_{\mathcal{H}}\left(k·\phi (\nu ,\sigma )\right),$$

where $F_{\mathcal{H}}$, $G_{\mathcal{H}}:{\mathcal{H}}_{+}\to {\mathcal{H}}_{+}$ are positive functions of a unital $C^{★}$-algebra $\mathcal{H}$, $k\in {\mathcal{H}}_{+}$ is a vector-valued contraction coefficient satisfying $r\left(k\right)<1$, where $r\left(k\right)$ is the spectral radius of k. By setting $F_{\mathcal{H}}\left(z\right)=z$ and $G_{\mathcal{H}}\left(z\right)=k·z$, the contraction condition simplifies to:

$$\phi \left(T\nu ,T\sigma \right)⪯k·\phi (\nu ,\sigma ),$$

for all $\nu ,\sigma \in \mathsf{\Phi }\cup \mathsf{\Psi }$. Then T has a unique fixed point.

Corollary 3

(Kannan-Type Contractions). Let $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ be a $wrtn$-complete $C^{★}$-algebra-valued bipolar b-metric space, and let $T:(\mathsf{\Phi },\mathsf{\Psi },\phi )\to (\mathsf{\Phi },\mathsf{\Psi },\phi )$ be a contravariant map. Suppose T satisfies the contraction condition:

$$F_{\mathcal{H}}\left(\phi (T\nu ,T\sigma )\right)⪯G_{\mathcal{H}}\left(\alpha ·\left(\phi \right(T\nu ,\nu )+\phi (T\sigma ,\sigma \left)\right)\right),$$

where $F_{\mathcal{H}},G_{\mathcal{H}}:{\mathcal{H}}_{+}\to {\mathcal{H}}_{+}$ are positive functions of a unital $C^{★}$-algebra $\mathcal{H}$, and $\alpha \in {\mathcal{H}}_{+}$, $r\left(\alpha \right)<{\displaystyle \frac{1}{2}}$, $\nu ,\sigma \in \mathsf{\Phi }\cup \mathsf{\Psi }$. By setting $F_{\mathcal{H}}\left(z\right)=z$ and $G_{\mathcal{H}}\left(z\right)=\alpha ·z$, the contraction condition becomes:

$$\phi (T\nu ,T\sigma )⪯\alpha ·\left(\phi (T\nu ,\nu )+\phi (T\sigma ,\sigma )\right),$$

for all $\nu ,\sigma \in \mathsf{\Phi }\cup \mathsf{\Psi }$. Then T has a unique fixed point.

5. Examples

Example 3.

Let $\mathsf{\Phi }=(-\infty ,0]$, $\mathsf{\Psi }=[0,\infty )$, and $\mathcal{H}=M_2\left(\mathbb{C}\right)$, the space of $2\times 2$ complex matrices, with ${\mathcal{H}}_{+}=\{A\in \mathcal{H}:A=A^{*},A⪰0\}$. Define the bipolar b-metric map $\phi :\mathsf{\Phi }\times \mathsf{\Psi }\to {\mathcal{H}}_{+}$ by $\phi (\sigma ,\nu )=\left[\begin{array}{cc}{3|\sigma -\nu |}^2& 0\\ 0& {4|\sigma -\nu |}^2\end{array}\right]$ and the mapping $T:\mathsf{\Phi }\cup \mathsf{\Psi }\to \mathsf{\Phi }\cup \mathsf{\Psi }$ by $T\left(\sigma \right)=-{\displaystyle \frac{\sigma }{7}}.$ For any $\sigma ,\nu \in \mathsf{\Phi }\cup \mathsf{\Psi }$, the contraction condition is given by:

$$\parallel F_{\mathcal{H}}\left(\phi (T\sigma ,T\nu )\right)\parallel \le \parallel G_{\mathcal{H}}({\alpha }_1\phi (\sigma ,\nu )+{\alpha }_2\phi (\sigma ,T\sigma )+{\alpha }_3\phi (T\nu ,\nu ))\parallel .$$

One can easily verify that $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ is $wrtn$-complete $C^{★}\text{-}\mathit{algebra}$ valued bipolar metric space, where $F_{\mathcal{H}}\left(A\right)=A^2,G_{\mathcal{H}}\left(A\right)=A,\mathrm{and}{\alpha }_1+{\alpha }_2+{\alpha }_3<1.$ More so, we have:

$$\begin{array}{cc}\hfill \phi (T\sigma ,T\nu )& =\left[\begin{array}{cc}3{\left|{\displaystyle \frac{\sigma }{7}}-{\displaystyle \frac{\nu }{7}}\right|}^2& 0\\ 0& 4{\left|{\displaystyle \frac{\sigma }{7}}-{\displaystyle \frac{\nu }{7}}\right|}^2\end{array}\right]\hfill \\ \hfill & =\left[\begin{array}{cc}{\displaystyle \frac{3}{49}}{\left|\sigma -\nu \right|}^2& 0\\ 0& {\displaystyle \frac{4}{49}}{\left|\sigma ,\nu \right|}^2\end{array}\right]\hfill \end{array}$$

Applying the contractraction gives:

$$F_{\mathcal{H}}\left(\phi (T\sigma ,T\nu )\right)=\phi {(T\sigma ,T\nu )}^2=\left[\begin{array}{cc}{\left({\displaystyle \frac{3}{49}}{|\sigma -\nu |}^2\right)}^2& 0\\ 0& {\left({\displaystyle \frac{4}{49}}{|\sigma -\nu |}^2\right)}^2\end{array}\right].$$

For the right-hand side of the contraction we have:

$${\alpha }_1\phi (\sigma ,\nu )=\left[\begin{array}{cc}3{\alpha }_1{|\sigma -\nu |}^2& 0\\ 0& 4{\alpha }_1{|\sigma -\nu |}^2\end{array}\right].$$

$${\alpha }_2\phi (\sigma ,T\sigma )=\left[\begin{array}{cc}3{\alpha }_2{|\sigma -T\sigma |}^2& 0\\ 0& 4{\alpha }_2{|\sigma -T\sigma |}^2\end{array}\right].$$

$${\alpha }_3\phi (T\nu ,\nu )=\left[\begin{array}{cc}3{\alpha }_3{|T\nu -\nu |}^2& 0\\ 0& 4{\alpha }_3{|T\nu -\nu |}^2\end{array}\right].$$

Summing these terms gives:

$$A=\left[\begin{array}{cc}3\left({\alpha }_1\right|\sigma -\nu {|}^2+{\alpha }_2|\sigma -T\sigma {|}^2+{\alpha }_3|T\nu -\nu {|}^2)& 0\\ 0& 4\left({\alpha }_1\right|\sigma -\nu {|}^2+{\alpha }_2|\sigma -T\sigma {|}^2+{\alpha }_3|T\nu -\nu {|}^2)\end{array}\right],$$

where $A={\alpha }_1\phi (\sigma ,\nu )+{\alpha }_2\phi (\sigma ,T\sigma )+{\alpha }_3\phi (T\nu ,\nu ).$ Applying $G_{\mathcal{H}}$ yields:

$$\begin{array}{cc}\hfill G_{\mathcal{H}}\left(A\right)& =\left[\begin{array}{cc}3\left({\alpha }_1\right|\sigma -\nu {|}^2+{\alpha }_2|\sigma -T\sigma {|}^2+{\alpha }_3|T\nu -\nu {|}^2)& 0\\ 0& 4\left({\alpha }_1\right|\sigma -\nu {|}^2+{\alpha }_2|\sigma -T\sigma {|}^2+{\alpha }_3|T\nu -\nu {|}^2)\end{array}\right]\hfill \\ \hfill & =A.\hfill \end{array}$$

We observe that each term in $F_{\mathcal{H}}\left(\phi (T\sigma ,T\nu )\right)$ involves ${\left|\sigma -\nu \right|}^4$ grows quadratically, while the right-hand side involves a linear combination ${\left|\sigma ,\nu \right|}^2,{\left|\sigma -T\sigma \right|}^2$, and ${\left|T\nu -\nu \right|}^2$, and since ${\alpha }_1+{\alpha }_2+{\alpha }_3<1$, and T is continuous, the contraction is satisfied for all $\sigma ,\nu \in \mathsf{\Phi }\cup \mathsf{\Psi }.$ So, all the assumptions of Theorem 4.4 have occurred. Hence, T has a unique fixed point. The fixed point equation follows as:

$$-{\displaystyle \frac{z}{7}}=z\Rightarrow z=0.$$

Thus, $z=0$ is the unique fixed point. However, as if T were not contravariant, Theorem 4.4 cannot be applied.

6. Applications

6.1. Ulam-Hyers Stability Problem for Non-Solid Cones

Let $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ be a $wrtn$-complete $C^{★}$-algebra-valued bipolar b-metric space, where ${\mathcal{H}}_{+}$ is a non-solid cone of the $C^{★}$-algebra $\mathcal{H}$. A mapping $T:\mathsf{\Phi }\cup \mathsf{\Psi }\to \mathsf{\Phi }\cup \mathsf{\Psi }$ is said to satisfy the Ulam-Hyers stability if for any $ϵ>0$ and any $ϵ$-solution ${\sigma }_0\in \mathsf{\Phi }\cup \mathsf{\Psi }$ satisfying:

$$\parallel F_{\mathcal{H}}\phi (T{\sigma }_0,{\sigma }_0)\parallel \le ϵ,$$

(19)

there exist a unique ${\sigma }^{*}\in \mathsf{\Phi }\cap \mathsf{\Psi }$ such that

$$T{\sigma }^{*}={\sigma }^{*},\mathrm{and}$$

(20)

$$\parallel F_{\mathcal{H}}\left(\phi ({\sigma }_0,{\sigma }^{*})\right)\parallel \le Mϵ,\mathrm{for}\mathrm{some}M>0.$$

(21)

Any point ${\sigma }_0\in \mathsf{\Phi }\cup \mathsf{\Psi }$, which is a solution of equation (19), is called an $ϵ$-solution of the mapping T.

Theorem 2.

Let $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ be a $wrtn$-complete $C^{★}$-algebra-valued bipolar b-metric space, where ${\mathcal{H}}_{+}$ is a non-solid cone of the $C^{★}$-algebra $\mathcal{H}$ and $T:\mathsf{\Phi }\cup \mathsf{\Psi }\to \mathsf{\Phi }\cup \mathsf{\Psi }$ be a contravariant mapping satisfying definition (12), then the fixed point equation (20) of T is Ulam-Hyers stable.

Proof. 

Suppose ${\sigma }_0\in \mathsf{\Phi }\cup \mathsf{\Psi }$ is an $ϵ$-solution, that is:

$$\parallel F_{\mathcal{H}}\left(\phi (T{\sigma }_0,{\sigma }_0)\right)\parallel \le ϵ,$$

and define the sequence $\left\{{\sigma }_n\right\}$ such that ${\sigma }_{n+1}=T{\sigma }_n$, for $n\ge 0$.

By the contraction condition (12), we have:

$$F_{\mathcal{H}}\left(\phi ({\sigma }_{n+1},{\sigma }_n)\right)⪯G_{\mathcal{H}}\left({\alpha }_1\phi ({\sigma }_n,{\sigma }_{n-1})+{\alpha }_2\phi ({\sigma }_n,{\sigma }_{n+1})+{\alpha }_3\phi ({\sigma }_{n+1},{\sigma }_n)\right).$$

Expanding recursively for k-steps, we obtain:

$$F_{\mathcal{H}}\left(\phi ({\sigma }_{n+k},{\sigma }_n)\right)⪯G_{\mathcal{H}}\left(\sum _{j=0}^{k-1}{\alpha }_1^j\phi ({\sigma }_1,{\sigma }_0)\right).$$

Since ${\alpha }_1+{\alpha }_2+{\alpha }_3<1$, the geometric series converges:

$$\sum _{j=0}^{\infty }{\alpha }_1^j={\displaystyle \frac{1}{1-{\alpha }_1}}.$$

Thus:

$$F_{\mathcal{H}}\left(\phi ({\sigma }_{n+k},{\sigma }_n)\right)⪯G_{\mathcal{H}}\left({\displaystyle \frac{\phi ({\sigma }_1,{\sigma }_0)}{1-{\alpha }_1}}\right),$$

and the corresponding norm satisfies:

$$\parallel F_{\mathcal{H}}\left(\phi ({\sigma }_{n+k},{\sigma }_n)\right)\parallel \to 0\mathrm{as}n\to \infty .$$

Since $\left\{{\sigma }_n\right\}$ is a Cauchy sequence, and $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ is $wrtn$-complete, the sequence $\left\{{\sigma }_n\right\}$ converges to a unique fixed point ${\sigma }^{*}\in \mathsf{\Phi }\cap \mathsf{\Psi }$. By continuity of T, this limit satisfies:

$$T{\sigma }^{*}={\sigma }^{*}.$$

Finally, using the contraction condition and the $ϵ$-solution property, we have:

$$\parallel F_{\mathcal{H}}\left(\phi ({\sigma }_0,{\sigma }^{*})\right)\parallel \le \parallel F_{\mathcal{H}}\left(\phi (T{\sigma }_0,{\sigma }_0)\right)\parallel ·\sum _{j=0}^{\infty }{\alpha }_1^j.$$

Simplifying this yields the Ulam-Hyers stability inequality:

$$\parallel F_{\mathcal{H}}\left(\phi ({\sigma }_0,{\sigma }^{*})\right)\parallel \le {\displaystyle \frac{ϵ}{1-{\alpha }_1}}.$$

This concludes the proof. □

Theorem 3.

Let $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ be a $C^{*}$-algebra-valued bipolar b-metric space, where ${\mathcal{H}}_{+}$ is a non-solid cone in the $C^{*}$-algebra $\mathcal{H}$. Consider the Fredholm integral operator:

$$T\left(x\right)\left(t\right)=q\left(t\right)+\lambda {\int }_{{\mathcal{H}}_1\cup {\mathcal{H}}_2}G(t,s,x\left(s\right))ds,$$

(22)

where $T:\mathsf{\Phi }\cup \mathsf{\Psi }\to \mathsf{\Phi }\cup \mathsf{\Psi }$, and assume the following:

• The kernel $G:({\mathcal{H}}_1\cup {\mathcal{H}}_2)\times ({\mathcal{H}}_1\cup {\mathcal{H}}_2)\times {\mathcal{H}}_{+}\to {\mathcal{H}}_{+}$ satisfies:

  (a)

  $G(t,s,x\left(s\right))\in {\mathcal{H}}_{+}$ for all $t,s\in {\mathcal{H}}_1\cup {\mathcal{H}}_2$ and $x\left(s\right)\in {\mathcal{H}}_{+}$,

  (b)

  $\parallel G(t,s,\sigma \left(s\right))-G(t,s,\sigma \left(s\right))\parallel \le {\displaystyle \frac{1}{2}}\parallel \theta (t,s)\parallel \parallel \sigma \left(s\right)-\sigma \left(s\right)\parallel$, where $\theta :({\mathcal{H}}_1\cup {\mathcal{H}}_2)\times ({\mathcal{H}}_1\cup {\mathcal{H}}_2)\to {\mathcal{H}}_{+}$ is continuous.

• The function θ satisfies:

  $$\underset{t\in {\mathcal{H}}_1\cup {\mathcal{H}}_2}{sup}{\int }_{{\mathcal{H}}_1\cup {\mathcal{H}}_2}\parallel \theta (t,s)\parallel ds\le 1.$$
• $q\in L^{\infty }({\mathcal{H}}_1,{\mathcal{H}}_{+})\cap L^{\infty }({\mathcal{H}}_2,{\mathcal{H}}_{+})$, with:

  $$\parallel q\left(t\right)\parallel \le M,\forall t\in {\mathcal{H}}_1\cup {\mathcal{H}}_2.$$
• The operator T satisfies:

  $$\parallel F_{\mathcal{H}}\left(\phi (Tx,Ty)\right)\parallel \le {\alpha }_1\parallel G_{\mathcal{H}}\left(\phi (x,y)\right)\parallel ,$$
  where $F_{\mathcal{H}}\left(x\right)=x^2$, $G_{\mathcal{H}}\left(x\right)=x$, and $0<{\alpha }_1<1$.

Then integral equation has a unique solution in $L^{\infty }({\mathcal{H}}_1,{\mathcal{H}}_{+})\cap L^{\infty }({\mathcal{H}}_2,{\mathcal{H}}_{+}).$

Proof. 

Let $\mathsf{\Phi }=L^{\infty }\left({\mathcal{H}}_1\right)$ and $\mathsf{\Psi }=L^{\infty }\left({\mathcal{H}}_2\right)$ be two normed linear spaces of essentially bounded measurable functions, where ${\mathcal{H}}_1,{\mathcal{H}}_2\subset [a,b]$ are disjoint Lebesgue measurable subsets such that $\mu ({\mathcal{H}}_1\cup {\mathcal{H}}_2)<\infty$, $\mathcal{H}=L^2({\mathcal{H}}_1\cup {\mathcal{H}}_2,{\mathcal{H}}_{+})$ be the space of square-integrable functions with values in the positive cone ${\mathcal{H}}_{+}$ of the $C^{*}$-algebra $\mathcal{H}$. Define the bipolar b-metric map $\phi :\mathsf{\Phi }\times \mathsf{\Psi }\to L\left(\mathcal{H}\right)$ as:

$$\phi (\sigma ,\nu )=\underset{t\in {\mathcal{H}}_1\cup {\mathcal{H}}_2}{sup}{\parallel \sigma \left(t\right)-\nu \left(t\right)\parallel }_{\mathcal{H}},$$

where $\sigma \in \mathsf{\Phi }$, $\nu \in \mathsf{\Psi }$, and ${\parallel ·\parallel }_{\mathcal{H}}$ is the norm in $\mathcal{H}$. Then $(\mathsf{\Phi },\mathsf{\Psi },\mathcal{H},\phi )$ is a complete $C^{*}$-algebra-valued bipolar b-metric space. Define $T:\mathsf{\Phi }\cup \mathsf{\Psi }\to \mathsf{\Phi }\cup \mathsf{\Psi }$ by $\left(22\right)$ above. Then, let $x=\sigma \in \mathsf{\Phi }$. By definition, this implies $\sigma :{\mathcal{H}}_1\to {\mathcal{H}}_{+}.$ Therefore,

$$T\left(\sigma \right)\left(t\right)=q\left(t\right)+\lambda {\int }_{{\mathcal{h}}_1\cup {\mathcal{H}}_2}G(t,s,\sigma \left(s\right))ds,t\in {\mathcal{H}}_2.$$

Since $\sigma \left(s\right)\in \mathsf{\Phi }$, the kernel $G(t,s,\sigma (s\left)\right)$ alternates the domain of evaluation from ${\mathcal{H}}_1$ into ${\mathcal{H}}_2$, ensuring $T\left(\sigma \right)\left(t\right)\in \mathsf{\Psi },\mathrm{for}\mathrm{all}t\in {\mathcal{H}}_2.$ Thus $T\left(x\right)\in \mathsf{\Phi }$ when $x\in \mathsf{\Phi }.$

Let $x=\nu \in \mathsf{\Psi }$. By definition, this implies $\nu :{\mathcal{H}}_2\to {\mathcal{H}}_{+}.$ Therefore,

$$T\left(\nu \right)\left(t\right)=q\left(t\right)+\lambda {\int }_{{\mathcal{H}}_1\cup {\mathcal{H}}_2}G(t,s,\nu \left(s\right))ds,t\in {\mathcal{H}}_1.$$

Since $\nu \left(s\right)\in \mathsf{\Psi }$, the kernel $G(t,s,\nu (s\left)\right)$ alternates the domain of evaluation from $H_2$ into ${\mathcal{H}}_1$, ensuring: $T\left(\nu \right)\left(t\right)\in \mathsf{\Phi },\mathrm{for}\mathrm{all}t\in H_1.$ Therefore, T is contravariant.

For any $\sigma \in L^{\infty }\left({\mathcal{H}}_1\right),\nu \in L^{\infty }\left({\mathcal{H}}_2\right),$

$$\begin{array}{cc}\hfill \phi \left(T\nu ,T\sigma \right)& =\underset{t\in {\mathcal{H}}_1\cup {\mathcal{H}}_2}{sup}{\parallel T\left(\nu \right)\left(t\right)-T\left(\sigma \right)\left(t\right)\parallel }_{\mathcal{H}}\hfill \\ \hfill & =\underset{t\in {\mathcal{H}}_1\cup {\mathcal{H}}_2}{sup}\parallel \lambda {\int }_{{\mathcal{H}}_1\cup {\mathcal{H}}_2}\left(G(t,s,\nu (s\left)\right)-G(t,s,\sigma (s\left)\right)\right)ds{\parallel }_{\mathcal{H}}\hfill \\ \hfill & \le \underset{t\in {\mathcal{H}}_1\cup {\mathcal{H}}_2}{sup}\lambda ·{\displaystyle \frac{1}{2}}{\int }_{{\mathcal{H}}_1\cup {\mathcal{H}}_2}{\parallel \theta (t,s)\parallel \parallel \nu \left(s\right)-\sigma \left(s\right)\parallel }_{\mathcal{H}}ds\hfill \\ \hfill & \le {\displaystyle \frac{\lambda }{2}}\phi (\nu ,\sigma ),\hfill \end{array}$$

therefore, $\phi \left(T\nu ,T\sigma \right)\le {\displaystyle \frac{\lambda }{2}}\phi (\nu ,\sigma )$ and $\parallel F_{\mathcal{H}}\phi (T\nu ,T\sigma )\parallel \le {\displaystyle \frac{\lambda }{2}}\parallel \phi (G_{\mathcal{H}}\nu ,\sigma )\parallel$, where $0<\lambda <2.$ Since ${\displaystyle \frac{\lambda }{2}}<1$, the mapping T is a contravariant contraction. Now, all the conditions of Theorem 3.4 are satisfied. Hence, the integral equation has a unique solution. □

7. Conclusions

In this paper, we established the existence and uniqueness of fixed point results under the $C^{*}—\mathit{algebra}$- valued bipolar b-metric spaces, where the underlying cone is non-solid. To achieve this, we used ($F_{\mathcal{H}}-G_{\mathcal{H}})$-contractions, which are positive and monotone functions. The obtained results were used to extend the Ulam - Hyers’ stability problem and Fredholm integral equations. Some examples were given to demonstrate our research results. It will be an open problem to generalize our outcomes to other types of new contractive conditions in this context.