Coupled fixed point theorems on $\mathcal{C}^\star$-algebra valued bipolar metric spaces

1.   Introduction

In 2016, Mutlu and Gürdal ^[1] introduced the concepts of bipolar metric space and proved fixed point theorems.

Definition 1.1. ^[1] Let $\varGamma$ and $\varPsi$ be two non-void sets and $\varphi : \varGamma\times \varPsi\rightarrow \mathbb{R}^{+}$ be a function, such that

(a) $\varphi(\sigma, \zeta) = 0$ iff $\sigma = \zeta$, for all $(\sigma, \zeta)\in \varGamma\times \varPsi$.

(b) $\varphi(\sigma, \zeta) = \varphi(\zeta, \sigma)$, for all $\sigma, \zeta\in \varGamma\cap \varPsi$.

(c) $\varphi(\sigma, \zeta)\leq \varphi(\sigma, \omega)+\varphi(\sigma_{1}, \omega)+\varphi(\sigma_{1}, \zeta)$, for all $\sigma, \sigma_{1}\in \varGamma$ and $\omega, \zeta\in \varPsi$.

The pair $(\varGamma, \varPsi, \varphi)$ is called a bipolar metric space.

In bipolar metric spaces, a lot of significant work has been done (see ^{[2,3,4,5,6,7,8,9,10]}). In 2014, Ma, Jiang and Sun ^[11] introduced the notion of a $\mathcal{C}^{\star}$-algebra-valued metric space and proved fixed point theorem. In 2015, Batul and Kamran ^[12] proved fixed theorems on $\mathcal{C}^{\star}$-algebra-valued metric space. In the present paper, we introduce a new notion of $\mathcal{C}^{\star}$-algebra valued bipolar metric space and proved coupled fixed point theorems. The details on $\mathcal{C}^{\star}$-algebra are available in ^{[13,14,15,16,17]}.

An algebra $\mathbb{A}$, together with a conjugate linear involution map $\vartheta\longmapsto\vartheta^{\star}$, is called a ${\star}$-algebra if $(\vartheta\varpi)^{\star} = {\varpi^{\star}\vartheta^{\star}}$ and $(\vartheta^{\star})^{\star} = \vartheta$ for all $\vartheta, \varpi \in \mathbb{A}$. Moreover, the pair $(\mathbb{A}, {\star})$ is called a unital ${\star}$-algebra if ${\mathbb{A}}$ contains the identity element ${1_\mathbb{A}}$. By a Banach ${\star}$-algebra we mean a complete normed unital ${\star}$-algebra $(\mathbb{A}, {\star})$ such that the norm on ${\mathbb{A}}$ is submultiplicative and satisfies $\big\|\vartheta^{\star}\big\| = \big\|\vartheta\big\|$ for all $\vartheta \in \mathbb{A}$. Further, if for all $\vartheta\in\mathbb{A}$, we have $\big\|{\vartheta^{\star}}{\vartheta}\big\| = \big\|{\vartheta}\big\|^2$ in a Banach ${\star}$-algebra $(\mathbb{A}, {\star})$, then $\mathbb{A}$ is known as a $\mathcal{C}^{\star}$-algebra. A positive element of $\mathbb{A}$ is an element $\vartheta\in \mathbb{A}$ such that $\vartheta = \vartheta^{\star}$ and its spectrum $\mathfrak \sigma(\vartheta)\subset \mathbb{R}_{+}$, where $\mathfrak \sigma(\vartheta)$ = $\{\mathfrak{\upsilon} \in \mathbb{R} :\mathfrak{\upsilon}1_\mathbb{A}-\vartheta\, \text{is noninvertible}\}$. The set of all positive elements will be denoted by $\mathbb{A}_{+}$. Such elements allow us to define a partial ordering $\succeq$ on the elements of $\mathbb{A}$. That is,

If $\vartheta \in \mathbb{A}$ is positive, then we write $\vartheta\succeq 0_\mathbb{A}$, where $0_\mathbb{A}$ is the zero element of $\mathbb{A}$. Each positive element $\vartheta$ of a $\mathcal{C}^{\star}$-algebra $\mathbb{A}$ has a unique positive square root. From now on, by $\mathbb{A}$ we mean a unital $\mathcal{C}^{\star}$-algebra with identity element $1_\mathbb{A}$. Further, $\mathbb{A}_{+}$ = $\{\vartheta\in\mathbb{A} : \vartheta\succeq 0_\mathbb{A}\}$ and $(\vartheta^{\star}\vartheta)^{1/2}$ = $\big|\vartheta\big|$.

2.   Preliminaries

In this section, we extend Definition 1.1 to introduce the notion bipolar metric space in the setting of $\mathcal{C}^{*}$-algebra as follows.

Definition 2.1. Let $\mathbb{A}$ be a $\mathcal{C}^{*}$-algebra, and $\varGamma$, $\varPsi$ be two non-void sets. A mapping $\varphi : \varGamma \times \varPsi \to \mathbb{A}_{+}$ be a function such that

(a) $\varphi(\sigma, \zeta) = 0$ iff $\sigma = \zeta$, for all $(\sigma, \zeta)\in \varGamma\times \varPsi$.

(b) $\varphi(\sigma, \zeta) = \varphi(\zeta, \sigma)$, for all $\sigma, \zeta \in \varGamma\cap \varPsi$.

(c) $\varphi(\sigma, \zeta)\leq \varphi(\sigma, \gamma)+\varphi(\sigma_{1}, \gamma)+\varphi(\sigma_{1}, \zeta)$, for all $\sigma, \sigma_{1}\in \varGamma$ and $\gamma, \zeta\in \varPsi$.

The $4$-tuple $(\varGamma, \varPsi, \mathbb{A}, \varphi)$ is called a $\mathcal{C}^{\star}$-algebra valued bipolar metric space.

Lemma 2.2 (^[14,17]). Suppose that $\mathbb{A}$ is a unital $\mathcal{C}^{\star}$-algebra with a unit $I$.

(A1) For any $\sigma\in \mathbb{A}_{+}$, we have $\sigma\preceq I$ iff $||\sigma||\leq 1$.

(A2) If $\vartheta\in \mathbb{A}_{+}$ with $||\vartheta|| < \frac{1}{2}$, then $(I-\vartheta)$ is invertible and $||\vartheta(I-a)^{-1}|| < 1$.

(A3) Suppose that $\vartheta, \varpi\in \mathbb{A}$ with $\vartheta\varpi\succeq0_{\mathbb{A}}$ and $\vartheta\varpi = \varpi\vartheta$, then $\varpi \vartheta\succeq0_{\mathbb{A}}$.

(A4) By $\mathbb{A}'$ we denote the set $\{\vartheta\in \mathbb{A}:\vartheta\varpi = \varpi\vartheta, \forall\varpi\in \mathbb{A}\}$. Let $\vartheta\in \mathbb{A}'$, if $\varpi, \mathfrak{c}\in \mathbb{A}$ with $\varpi\succeq \mathfrak{c}\succeq 0_{\mathbb{A}}$, and $I-\vartheta\in \mathbb{A}'_{+}$ is an invertible operator, then

(A5) If $\varpi, \mathfrak{c}\in \mathbb{A}_\mathfrak{h} = \{\sigma\in \mathbb{A}:\sigma = \sigma^{\star}\}$ and $\vartheta\in \mathbb{A}$, then $\varpi\preceq \mathfrak{c}$ implies that $\vartheta^{\star}\varpi\vartheta\preceq\vartheta^{\star}\mathfrak{c}\vartheta$.

(A6) $0_{\mathbb{A}}\preceq\vartheta\preceq\varpi$, then $||\vartheta||\leq ||\varpi||$.

Notice that in a $\mathcal{C}^{\star}$-algebra, if $0_{\mathbb{A}}\preceq \vartheta, \varpi$, one cannot conclude that $0_{\mathbb{A}}\preceq \vartheta\varpi$.

Definition 2.3. Let $(\varGamma_{1}, \varPsi_{1}, \mathbb{A}, \varphi)$ and $(\varGamma_{2}, \varPsi_{2}, \mathbb{A}, \varphi)$ be pairs of sets and given a function $\varPhi:\varGamma_{1}\cup\varPsi_{1}\to \varGamma_{2}\cup\varPsi_{2}$.

(B1) If $\varPhi(\varGamma_{1})\subseteq \varGamma_{2}$ and $\varPhi(\varPsi_{1})\subseteq \varPsi_{2}$, then $\varPhi$ is called a covariant map, or a map from $(\varGamma_{1}, \varPsi_{1}, \mathbb{A}, \varphi_{1})$ to $(\varGamma_{2}, \varPsi_{2}, \mathbb{A}, \varphi_{2})$ and this is written as

(B2) If $\varPhi(\varGamma_{1})\subseteq \varPsi_{2}$ and $\varPhi(\varPsi_{1})\subseteq \varGamma_{2}$, then $\varPhi$ is called a contravariant map from $(\varGamma_{1}, \varPsi_{1}, \mathbb{A}, \varphi_{1})\, \, to\, \, (\varGamma_{2}, \varPsi_{2}, \mathbb{A}, \varphi_{2})$ and this is denoted as

Definition 2.4. Let $(\varGamma, \varPsi, \mathbb{A}, \varphi)$ be a $\mathcal{C}^{\star}$-algebra valued bipolar metric space.

(C1) A point $\sigma\in\varGamma\cup\varPsi$ is said to be a left point if $\sigma\in \varGamma$, a right point if $\sigma\in \varPsi$ and a central point if both hold. Similarly, a sequence$\{\sigma_{\alpha}\}$ on the set $\varGamma$ and a sequence $\{\zeta_{n}\}$ on the set $\varPsi$ are called a left and right sequence with respect to $\mathbb{A}$, respectively.

(C2) A sequence $\{\sigma_{\alpha}\}$ converges to a point $\zeta$ with respect to $\mathbb{A}$ iff $\{\sigma_{\alpha}\}$ is a left sequence, $\zeta$ is a right point and $\lim\limits_{\alpha\to \infty}\varphi(\sigma_{\alpha}, \zeta) = 0_{\mathbb{A}}$ or $\{\sigma_{\alpha}\}$ is a right sequence, $\zeta$ is a left point and $\lim\limits_{\alpha\to \infty}\varphi(\zeta, \sigma_{\alpha}) = 0_{\mathbb{A}}$.

(C3) A bisequence $(\{\sigma_{n}\}, \{\zeta_{n}\})$ is a sequence on the set $\varGamma \times \varPsi$. If the sequence $\{\sigma_{n}\}$ and $\{\zeta_{n}\}$ are convergent with respect to $\mathbb{A}$, then the bisequence $(\{\sigma_{n}\}, \{\zeta_{n}\})$ is said to be convergent with respect to $\mathbb{A}$. $(\{\sigma_{n}\}, \{\zeta_{n}\})$ is a Cauchy bisequence with respect to $\mathbb{A}$ if $\lim\limits_{\alpha, \beta\to \infty}\varphi(\sigma_{\alpha}, \zeta_{\beta}) = 0_{\mathbb{A}}$, hence biconvergent with respect to $\mathbb{A}$.

(C4) $(\varGamma, \varPsi, \mathbb{A}, \varphi)$ is complete, if every Cauchy bisequence with respect to $\mathbb{A}$ is convergent in $\varGamma \times \varPsi$.

3.   Main results

Theorem 3.1. Let $(\varGamma, \varPsi, \mathbb{A}, \varphi)$ be a complete $\mathcal{C}^{\star}$-algebra valued bipolar metric space. Suppose

is a covariant mapping such that

where $\upsilon\in \mathbb{A}$ with $2||\upsilon||^{2} < 1$. Then the function

has a unique coupled fixed point.

Proof. Let $\sigma_{0}, \zeta_{0} \in \varGamma$ and $\mathfrak{u}_{0}, \mathfrak{v}_{0}\in \varPsi$. For each $\alpha\in \mathbb{N}$, define

and

Then $(\{\sigma_{\alpha}\}, \{\zeta_{\alpha}\})$ and $(\{\mathfrak{u}_{\alpha}\}, \{\mathfrak{v}_{\alpha}\})$ are bisequences on $(\varGamma, \varPsi, \mathbb{A}, \varphi)$. Then, for each $\alpha\in \mathbb{N}$,

where

Similarly, we get

Now,

By Lemma 2.2 (A5), we have

On the other hand,

where

Similarly, we get

Now,

By Lemma 2.2 (A5), we have

Moreover,

where

Similarly, we get

Now,

By Lemma 2.2 (A5), we have

Now,

and

for each $\alpha, \beta\in \mathbb{N}$, $\alpha < \beta$. Then,

and

Therefore, $(\{\sigma_{\alpha}\}, \{\mathfrak{u}_{\alpha}\})$ and $(\{\zeta_{\alpha}\}, \{\mathfrak{v}_{\alpha}\})$ are Cauchy bisequences in $\varGamma\times\varPsi$ with respect to $\mathbb{A}$. By completeness of $(\varGamma, \varPsi, \mathbb{A}, \varphi)$, there exist $\sigma, \zeta\in \varGamma$ and $\mathfrak{u}, \mathfrak{v}\in \varPsi$ with

Since $(\{\sigma_{\alpha}\}, \{\mathfrak{u}_{\alpha}\})$ and $(\{\zeta_{\alpha}\}, \{\mathfrak{v}_{\alpha}\})$ are Cauchy bisequences, we derive that

Then,

As $\alpha\to \infty$, we have

Then,

Hence, $\varPhi(\sigma, \zeta) = \mathfrak{u}$. Similarly, we can derive $\varPhi(\zeta, \sigma) = \mathfrak{v}$, $\varPhi(\mathfrak{u}, \mathfrak{v}) = \sigma$ and $\varPhi(\mathfrak{v}, \mathfrak{u}) = \zeta$. On the other hand, we derive that

and

So, $\sigma = \mathfrak{u}$ and $\zeta = \mathfrak{v}$. Therefore, $(\sigma, \zeta)\in \varGamma^{2}\cap\varPsi^{2}$ is a coupled fixed point of $\varPhi$. Let $(\mathfrak{e}, \mathfrak{f})\in \varGamma^{2}\cup\varPsi^{2}$ is a another coupled fixed point $\varPhi$. If $(\mathfrak{e}, \mathfrak{f})\in \varGamma^{2}$, then

and

which implies that

Then,

Since $2||\upsilon||^{2} < 1$, we derive that

Therefore, $\mathfrak{e} = \sigma$ and $\mathfrak{f} = \zeta$. Similarly, if $(\mathfrak{e}, \mathfrak{f})\in \varPsi^{2}$, then $\mathfrak{e} = \sigma$ and $\mathfrak{f} = \zeta$. Then $(\sigma, \zeta)$ is a unique coupled fixed point of $\varPhi$.

Example 3.2. Let $\varGamma = [0, 1]$, $\varPsi = \{0\}\cup\mathbb{N}-\{1\}$, $\mathbb{A}_{+} = \mathcal{M}_{2}(\mathbb{C})$ and the map $\varphi : \varGamma \times \varPsi \to \mathbb{A}_{+}$ is defined by

for all $\sigma \in \varGamma$ and $\mathfrak{u}\in \varPsi$, where $\Bbbk \geq 0$ is a constant. Let $\preceq$ be the partial order on $\mathbb{A}$ given by

Then $(\varGamma, \varPsi, \mathbb{A}, \varphi)$ is a complete $C^{\star}$-algebra-valued bipolar metric space. Define

by

$\forall \; \sigma, \zeta \in \varGamma^{2}\cup \varPsi^{2}$. Then

for all $\sigma, \zeta\in \varGamma, \mathfrak{u}, \mathfrak{v} \in \varPsi$, where

and $||\upsilon|| = \frac{1}{3} < \frac{1}{\sqrt{2}}$. All the conditions of Theorem 3.1 are fulfilled and $\varPhi$ has a unique fixed point $(0, 0)$.

Theorem 3.3. Let $(\varGamma, \varPsi, \mathbb{A}, \varphi)$ be a complete $\mathcal{C}^{\star}$-algebra valued bipolar metric space. Suppose

is a covariant mapping such that

where $\upsilon\in \mathbb{A}$ with $2||\upsilon||^{2} < 1$. Then the function

has a unique coupled fixed point.

Proof. Let $\sigma_{0}, \zeta_{0} \in \varGamma$ and $\mathfrak{u}_{0}, \mathfrak{v}_{0}\in \varPsi$. For each $\alpha\in \mathbb{N}$, define $\varPhi(\sigma_{\alpha}, \mathfrak{u}_{\alpha}) = \sigma_{\alpha+1}$, $\varPhi(\zeta_{\alpha}, \mathfrak{v}_{\alpha}) = \zeta_{\alpha+1}$, $\varPhi(\mathfrak{u}_{\alpha}, \sigma_{\alpha}) = \mathfrak{u}_{\alpha+1}$ and $\varPhi(\mathfrak{v}_{\alpha}, \zeta_{\alpha}) = \mathfrak{v}_{\alpha+1}$. Then $(\{\sigma_{\alpha}\}, \{\mathfrak{u}_{\alpha}\})$ and $(\{\zeta_{\alpha}\}, \{\mathfrak{v}_{\alpha}\})$ are bisequences on $(\varGamma, \varPsi, \mathbb{A}, \varphi)$. Then, for each $\alpha\in \mathbb{N}$,

Let

for all $\alpha\in \mathbb{N}$. Then

By Lemma 2.2 (A5), we have

Let

for all $\alpha\in \mathbb{N}$. Then

By Lemma 2.2 (A5), we have

On the other hand,

and

for all $\alpha\in \mathbb{N}$. Let

for all $\alpha\in \mathbb{N}$. Then

By Lemma 2.2 (A5), we have

Now,

and

for each $\alpha, \beta\in \mathbb{N}$, $\alpha < \beta$. Then,

and

Therefore, $(\{\sigma_{\alpha}\}, \{\mathfrak{v}_{\alpha}\})$ and $(\{\zeta_{\alpha}\}, \{\mathfrak{u}_{\alpha}\})$ are Cauchy bisequences in $\varGamma\times\varPsi$ with respect to $\mathbb{A}$. By completeness of $(\varGamma, \varPsi, \mathbb{A}, \varphi)$, there exist $\sigma, \zeta\in \varGamma$ and $\mathfrak{u}, \mathfrak{v}\in \varPsi$ with

Then for given $\epsilon > 0$, there exists $\alpha_{1}\in \mathbb{N}$ with $\varphi(\sigma_{\alpha}, \mathfrak{v}) < \frac{\epsilon}{2}$ for all $\alpha\geq \alpha_{1}$. Since $(\{\sigma_{\alpha}\}, \{\mathfrak{v}_{\alpha}\})$ and $(\{\zeta_{\alpha}\}, \{\mathfrak{u}_{\alpha}\})$ are Cauchy bisequences, we derive that

Then,

As $\alpha\to \infty$, we have

Then,

Hence, $\varPhi(\sigma, \mathfrak{u}) = \mathfrak{v}$. Similarly, we can derive $\varPhi(\mathfrak{u}, \sigma) = \zeta$, $\varPhi(\zeta, \mathfrak{v}) = \mathfrak{u}$ and $\varPhi(\mathfrak{v}, \zeta) = \sigma$. On the other hand, we derive that

and

So, $\sigma = \mathfrak{v}$ and $\zeta = \mathfrak{u}$. Therefore, $(\sigma, \mathfrak{u})\in (\varGamma\times \varPsi)\cap(\varPsi\times\varGamma)$ is a coupled fixed point of $\varPhi$. As in the proof of the Theorem 3.1, one can easily prove uniqueness part.

Example 3.4. Let $\varGamma = \{0, 1, 2, 7\}$, $\varPsi = \{0, \frac{1}{4}, \frac{1}{2}, 3\}$, $\mathbb{A}_{+} = \mathcal{M}_{2}(\mathbb{C})$ and the map $\varphi : \varGamma \times \varPsi \to \mathbb{A}_{+}$ is defined by

for all $\sigma \in \varGamma$ and $\mathfrak{u}\in \varPsi$, where $\Bbbk \geq 0$ is a constant. Let $\preceq$ be the partial order on $\mathbb{A}$ given by

Then $(\varGamma, \varPsi, \mathbb{A}, \varphi)$ is a complete $C^{\star}$-algebra-valued bipolar metric space. Define

by

for all $\sigma, \zeta\in (\varGamma\times\varPsi) \cup (\varPsi\times\varGamma)$. Then

for all $\sigma, \zeta\in \varGamma$ and $\mathfrak{u}, \mathfrak{v} \in \varPsi$, where

and $||\upsilon|| = \frac{1}{5} < \frac{1}{\sqrt{2}}$. All the conditions of Theorem 3.3 are fulfilled and $\varPhi$ has a unique fixed point $(0, 0)$.

4.   Applications

As an application of Theorem 3.1, we find an existence and uniqueness result for a type of following system of Fredholm integral equations.

Theorem 4.1. Let us consider the system of Fredholm integral equations

where $\mathcal{E}_{1}\cup\mathcal{E}_{2}$ is a Lebesgue measurable set. Suppose

(T1) $\mathcal{G}:(\mathcal{E}^{2}_{1}\cup\mathcal{E}^{2}_{2})\times \mathbb{R}\times \mathbb{R}\to[0, \infty)$ and $\delta\in L^{\infty}(\mathcal{E}_{1})\cup L^{\infty}(\mathcal{E}_{2})$.

(T2) There exists a continuous function $\kappa:\mathcal{E}^{2}_{1}\times\mathcal{E}^{2}_{2}\to \mathbb{R}$ and $\theta\in (0, 1)$, such that

for all $\mu, p\in \mathcal{E}_{1}\cup\mathcal{E}_{2}$.

(T3) $\sup_{\mu\in \mathcal{E}_{1}\cup\mathcal{E}_{2}}\int_{\mathcal{E}_{1}\cup\mathcal{E}_{2}}|\kappa(\mu, p)|dp\leq 1$.

Then the integral equation has a unique solution in $L^{\infty}(\mathcal{E}_{1})\cup L^{\infty}(\mathcal{E}_{2})$.

Proof. Let $\varGamma = L^{\infty}(\mathcal{E}_{1})$ and $\varPsi = L^{\infty}(\mathcal{E}_{2})$ be two normed linear spaces, where $\mathcal{E}_{1}, \mathcal{E}_{2}$ are Lebesgue measurable sets and $m(\mathcal{E}_{1}\cup\mathcal{E}_{2}) < \infty$. Let $\mathcal{H} = L^{2}(\mathcal{E}_{1})\cup L^{2}(\mathcal{E}_{2})$. Consider $\varphi:\varGamma\times \varPsi\to L(\mathcal{H})$ defined by $\varphi(\sigma, \zeta) = \pi_{|\sigma-\zeta|}$, where $\pi_{\mathfrak{h}}:\mathcal{H}\to \mathcal{H}$ is the multiplication operator defined by $\pi_{\mathfrak{h}}(\omega) = \mathfrak{h}.\omega$ for $\omega\in \mathcal{H}$. Then $(\varGamma, \varPsi, \mathbb{A}, \varphi)$ is a complete $\mathcal{C}^{\star}$-algebra valued bipolar metric space.

Define the covariant mapping $\varPhi:\varGamma^{2}\cup \varPsi^{2}\to \varGamma\cup \varPsi$ by

Set $\tau = \theta I$, then $\tau\in L(\mathcal{H})_{+}$ and $||\tau|| = \theta < 1$. For any $\omega\in \mathcal{H}$, we have

Therefore, all the conditions of Theorem 3.1 are fulfilled. Hence, the integral equation (4.1) has a unique solution.

5.   Conclusions

In this paper, we introduced the notion of a $\mathcal{C}^{\star}$-algebra valued bipolar metric space and proved coupled fixed point theorems. An illustrative example is provided that show the validity of the hypothesis and the degree of usefulness of our findings.

Acknowledgments

The authors would like to thank the anonymous referees for their valuable comments and suggestions, which led to considerable improvement of the article. The work was supported by the Higher Education Commission of Pakistan.

Conflict of interest

The authors declare no conflicts of interest.