Fixed Point Results in C^$★$-Algebra-Valued Partial b-Metric Spaces with Related Application

Abstract

In this manuscript, we prove some fixed point theorems on C^🟉-algebra-valued partial b-metric spaces by using generalized contraction. We give support and suitable examples of our main results. Moreover, we present a generative application of the main results.

1. Introduction

Bakhtin [1] defined b-metric space, and Czerwik [2] established the fixed point results using the Banach contraction principle (see, for instance, [3,4,5] and references therein). Abdou et al. [6] illustrated a new concept of locally $\alpha \text{-}\psi \text{-}$contractive mapping, generalized $\alpha \text{-}\psi \text{-}$ rational contraction and established fixed point theorems for such mappings in the context of extended b-metric spaces. Gholidahneh et al. [7] demonstrated the concept of modular p-metric space and established some fixed point results for $\alpha \text{-}\overline{v}$-Meir–Keeler contractions in this space. Furthermore, they established a relationship between the fuzzy concept of Meir–Keeler and extended p-metrics with modular p-metrics and obtained fixed point results in triangular p-metric spaces with fuzzy concepts.

In 2014, Ma et al. [8] proved some fixed point theorems for self-maps with contractive or expansive conditions on $C^{🟉}$-algebra-valued metric spaces. Chandok et al. [9] presented the concept of $C^{🟉}$-algebra-valued partial metric space ($C^{🟉}$-AVPMS) and some fixed point results on such spaces using C-class functions in 2019. Mlaiki et al. [10] expanded the class of $C^{🟉}$-$A{V}_{b}MS$ ($C^{🟉}$-algebra-valued b-metric space) and $C^{🟉}$-AVPMS by introducing $C^{🟉}$-$AV{P}_{b}MS$ ($C^{🟉}$-algebra-valued partial b-metric space) and used it to prove fixed point results in 2021.

In this paper, we prove fixed point theorems for generalized contraction in $C^{🟉}$-$AV{P}_{b}MS$.

This paper consists of five sections, wherein Section 1 begins with an introduction. In Section 2 we first recall some definitions, lemma and theorem related to $C^{🟉}$-$AV{P}_{b}MS$ and discuss their related properties. In Section 3 we prove fixed point results as well as giving an example to support our main result. In Section 4 we apply our main result to examine the existence and uniqueness of a solution for the system of the Fredholm integral equation, and in the last section we present our conclusions.

2. Preliminaries

This section covers the basic definitions and properties of $C^{🟉}$-algebras [11,12] with the following important consequences Suppose that $\mathbb{B}$ is a unital algebra with unit $\mathfrak{I}$. An involution on $\mathbb{B}$ is a conjugate-linear map $\vartheta \mapsto {\vartheta }^{🟉}$ on $\mathbb{B}$ such that ${\vartheta }^{🟉🟉}=\vartheta \mathrm{and}{\left(\vartheta \varrho \right)}^{🟉}={\varrho }^{🟉}{\vartheta }^{🟉}\forall \vartheta ,\varrho \in \mathbb{B}.$ The pair $(\mathbb{B},🟉)$ is known as a 🟉-algebra. A Banach 🟉-algebra is a 🟉-algebra $\mathbb{B}$ together with a complete sub-multiplicative norm such that $\parallel {\vartheta }^{🟉}\parallel =\parallel \vartheta \parallel$ for all $\vartheta \in \mathbb{B}$. A $C^{🟉}$-algebra is a Banach 🟉-algebra with the property that $\parallel {\vartheta }^{🟉}{\vartheta \parallel =\parallel \vartheta \parallel }^2$ for all $\vartheta \in \mathbb{B}$. In this paper, we prove some fixed point theorems on $C^{🟉}$-algebra-valued partial b-metric spaces by using generalized contraction.

Throughout this paper, we denote a ${\mathcal{C}}^{🟉}$-algebra with unit $\mathfrak{I}$ by $\mathbb{B}$. Set ${\mathbb{B}}_{\mathfrak{h}}=\{\varrho \in \mathbb{B}:\varrho ={\varrho }^{🟉}\}$. Consider a positive element $\varrho \in \mathbb{B}$, i.e., $\varrho \ge \theta$ if $\varrho \in {\mathbb{B}}_{\mathfrak{h}}$ and $\varsigma \left(\varrho \right)\subset [0,\infty ),$ where $\varsigma \left(\varrho \right)$ is the spectrum of $\varrho$. We define a partial ordering on ${\mathbb{B}}_{\mathfrak{h}}$ as follows: $\varrho ⪯\xi$ iff $\xi -\varrho ⪰\theta$. Now, we denote the set $\{\varrho \in \mathbb{B}:\varrho ⪰\theta \}\mathrm{and}\left|\varrho \right|={\left({\varrho }^{🟉}\varrho \right)}^{\frac{1}{2}}$ by ${\mathbb{B}}_{+}$.

Lemma 1

([11,13]). Let $\mathbb{B}$ be a unital ${\mathcal{C}}^{🟉}$-algebra with a unit $\mathfrak{I}$.

1.

For each $\varrho \in {\mathbb{B}}_{+}$, we have $\varrho ⪯\mathfrak{I}\iff \parallel \varrho \parallel \le 1.$

2.

If $\vartheta \in {\mathbb{B}}_{+}$ with $\parallel \vartheta \parallel <\frac{1}{2}$, then $\mathfrak{I}-\vartheta$ is invertible and ${\parallel \vartheta (\mathfrak{I}-\vartheta )}^{-1}\parallel <1.$

3.

Assume that $\vartheta ,\varrho \in \mathbb{B}\mathrm{with}\vartheta ,\varrho ⪰\theta$ and $\vartheta \varrho =\varrho \vartheta$, then $\vartheta \varrho ⪰\theta .$

4.

Define ${\mathbb{B}}^{\prime }=\{\vartheta \in \mathbb{B}:\vartheta \varrho =\varrho \vartheta ,\forall \varrho \in \mathbb{B}\}$. Let $\vartheta \in {\mathbb{B}}^{\prime }$. If $\varrho ,\mathfrak{c}\in \mathbb{B}$ with $\varrho ⪰\mathfrak{c}⪰\theta$, and $\mathfrak{I}-\vartheta \in {\mathbb{B}}_{+}^{{}^{\prime }}$ is an invertible operator, then

$$\begin{array}{c}\hfill {(\mathfrak{I}-\vartheta )}^{-1}\varrho \ge {(\mathfrak{I}-\vartheta )}^{-1}\mathfrak{c}.\end{array}$$

Ma et al. [14] presented in the sequel definition:

Definition 1.

Let $\mho \ne \varnothing$ and $\ell \in \mathbb{B}$ such that $\ell ⪰\mathfrak{I}$. A mapping $\mathsf{\Phi }:\mho \times \mho \to \mathbb{B}$ satisfies the following condition:

1.

$\theta ⪯\mathsf{\Phi }(\varrho ,\xi )$ for all $\mathsf{\Phi }(\varrho ,\xi )=\theta \iff \varrho =\xi ;$

2.

$\mathsf{\Phi }(\varrho ,\xi )=\mathsf{\Phi }(\xi ,\varrho )$;

3.

$\mathsf{\Phi }(\varrho ,\xi )⪯\ell \left[\mathsf{\Phi }\right(\varrho ,\varsigma )+\mathsf{\Phi }(\varsigma ,\xi \left)\right]$, ∀$\varrho ,\xi ,\varsigma \in \mho .$

Then Φ is called a $C^{🟉}\text{-}$algebra-valued b-metric space ($C^{🟉}\text{-}A{V}_{b}M$) on ℧ and $(\mho ,\mathbb{B},\mathsf{\Phi })$ is a $C^{🟉}\text{-}A{V}_{b}MS$.

Now, we remember that the definition of $C^{*}\text{-}AV{P}_{b}MS$ introduced by Mlaiki et al. [10].

Definition 2.

Let ℧ be a non-void set and $\ell \in \mathbb{B}$ such that $\ell \ge \mathfrak{I}$. A function $\mathsf{\Phi }:\mho \times \mho \to \mathbb{B}$ satisfies the following property:

1.

$\theta ⪯\mathsf{\Phi }(\varrho ,\xi )$, ∀$\varrho ,\xi \in \mho$ and $\mathsf{\Phi }(\varrho ,\varrho )=\mathsf{\Phi }(\xi ,\xi )=\mathsf{\Phi }(\varrho ,\xi )$ if $\varrho =\xi$;

2.

$\mathsf{\Phi }(\varrho ,\varrho )⪯\mathsf{\Phi }(\varrho ,\xi )$;

3.

$\mathsf{\Phi }(\varrho ,\xi )=\mathsf{\Phi }(\xi ,\varrho )$;

4.

$\mathsf{\Phi }(\varrho ,\xi )⪯\ell \left(\mathsf{\Phi }\right(\varrho ,\varsigma )+\mathsf{\Phi }(\varsigma ,\xi \left)\right)-\mathsf{\Phi }(\varsigma ,\varsigma )$, ∀$\varrho ,\xi ,\varsigma \in \mho$.

Then Φ is said to be a $C^{🟉}\text{-}AV{P}_{b}M$ on ℧ and $(\mho ,\mathbb{B},\mathsf{\Phi })$ is said to be a $C^{🟉}\text{-}AV{P}_{b}MS$.

Definition 3.

Let $(\mho ,\mathbb{B},\mathsf{\Phi })$ be a $C^{*}\text{-}AV{P}_{b}MS$. A sequence $\left\{{\varrho }_{\mathfrak{q}}\right\}$ in $(\mho ,\mathbb{B},\mathsf{\Phi })$ is said to be convergent (with respect to $\mathbb{B}$) to a point $\varrho \in \mho$ if $ϵ>0$, for each $\alpha \in \mathbb{N}$ satisfying $\left|\right|\mathsf{\Phi }({\varrho }_{\mathfrak{q}},\varrho )-\mathsf{\Phi }(\varrho ,\varrho )\left|\right|<ϵ$ for all $\mathfrak{q}>\alpha$.

Definition 4.

Let $(\mho ,\mathbb{B},\mathsf{\Phi })$ be a $C^{*}\text{-}AV{P}_{b}MS$. A sequence $\left\{{\varrho }_{\mathfrak{q}}\right\}$ in $(\mho ,\mathbb{B},\mathsf{\Phi })$ is said to be Cauchy (with respect to $\mathbb{B}$) if $\underset{\mathfrak{q}\to \infty }{lim}\mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\beta })$ exists and it is finite.

Definition 5.

Let $(\mho ,\mathbb{B},\mathsf{\Phi })$ be a $C^{*}\text{-}AV{P}_{b}MS$. A triplet $(\mho ,\mathbb{B},\mathsf{\Phi })$ is said to be complete $C^{*}\text{-}AV{P}_{b}MS$ if every Cauchy sequence is convergent to ϱ in ℧ such that

$$\begin{array}{c}\hfill \underset{\mathfrak{q},\beta \to \infty }{lim}\mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\beta })=\underset{\mathfrak{q}\to \infty }{lim}\mathsf{\Phi }({\varrho }_{\mathfrak{q}},\varrho )=\mathsf{\Phi }(\varrho ,\varrho ).\end{array}$$

Theorem 1

([10]). Let $(\mho ,\mathbb{B},\mathsf{\Phi })$ be a complete $C^{🟉}\text{-}AV{P}_{b}MS$ and $\Lambda :\mho \to \mho$ is a $C_b^{*}$-contraction. Then Λ has a unique fixed point $\varrho \in \mho$ such that $\mathsf{\Phi }(\varrho ,\varrho )=0_{\mathbb{B}}$.

Inspired by Theorem 1, we prove fixed point theorems for generalized contractions in $C^{*}\text{-}AV{P}_{b}MS$ with an application.

3. Main Results

Now, we prove fixed point theorems for generalized contractions in $C^{*}\text{-}AV{P}_{b}MS$.

Theorem 2.

Let $(\mho ,\mathbb{B},\mathsf{\Phi })$ be a complete $C^{🟉}\text{-}AV{P}_{b}MS$. Suppose the mapping $\Lambda :\mho \to \mho$ satisfying the condition:

$$\begin{array}{c}\hfill \mathsf{\Phi }(\Lambda \varrho ,\Lambda \xi )\le \gamma \left(\mathsf{\Phi }\right(\Lambda \varrho ,\xi )+\mathsf{\Phi }(\Lambda \xi ,\varrho \left)\right),\forall \varrho ,\xi \in \mho ,\end{array}$$

where $\gamma \in {\mathbb{B}}_{+}^{{}^{\prime }}$ and $\parallel \gamma \ell \parallel <\frac{1}{2}.$ Then Λ a unique fixed point $\varrho \in \mho$ such that $\mathsf{\Phi }(\varrho ,\varrho )=\theta$.

Proof. 

If we assume $\gamma =\theta$, then $\Lambda$ maps ℧ into a single point. Thus, without loss of generality, we assume that $\gamma \ne \theta$. Notice that for $\gamma \in {\mathbb{B}}_{+}^{{}^{\prime }},\gamma (\mathsf{\Phi }(\Lambda \varrho ,\xi )+\mathsf{\Phi }(\Lambda \xi ,\varrho ))\ge \theta$. Choose ${\varrho }_0\in \mho$ and set ${\varrho }_{\mathfrak{q}+1}=\Lambda {\varrho }_{\mathfrak{q}}={\Lambda }^{\mathfrak{q}+1}{\varrho }_0,\mathfrak{q}=1,2,....$, and $\mathsf{\Phi }({\varrho }_1,{\varrho }_0)={\gamma }_0$. Then

$$\begin{array}{cc}\hfill \mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},{\varrho }_{\mathfrak{q}})& =\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\Lambda {\varrho }_{\mathfrak{q}-1})\hfill \\ & ⪯\gamma (\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}-1})+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}-1},{\varrho }_{\mathfrak{q}}))\hfill \\ & =\gamma (\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\Lambda {\varrho }_{\mathfrak{q}-2})+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}-1},\Lambda {\varrho }_{\mathfrak{q}-1}))\hfill \\ & ⪯\gamma \ell (\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\Lambda {\varrho }_{\mathfrak{q}-1})+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}-1},\Lambda {\varrho }_{\mathfrak{q}-2}))\hfill \\ & -\gamma \mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}-1},\Lambda {\varrho }_{\mathfrak{q}-1})+\gamma \mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}-1},\Lambda {\varrho }_{\mathfrak{q}-1})\hfill \\ & =\gamma \ell \left(\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\Lambda {\varrho }_{\mathfrak{q}-1})\right)+\gamma \ell \left(\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}-1},\Lambda {\varrho }_{\mathfrak{q}-2})\right)\hfill \\ \hfill & =\gamma \ell \left(\mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},{\varrho }_{\mathfrak{q}})\right)+\gamma \ell \left(\mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}-1})\right).\hfill \end{array}$$

By Lemma 1,

$$\begin{array}{c}\hfill (\mathfrak{I}-\gamma \ell )\mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},{\varrho }_{\mathfrak{q}})\le \gamma \ell \mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}-1}).\end{array}$$

Since $\ell ,\gamma \in {\mathbb{B}}_{+}^{{}^{\prime }}$ with $\parallel \gamma \ell \parallel <\frac{1}{2}$ and $\ell ⪰\mathfrak{I},$ we have $\mathfrak{I}-\gamma \ell ⪯\mathfrak{I}-\gamma$ and furthermore ${(\mathfrak{I}-\gamma \ell )}^{-1}\in {\mathbb{B}}_{+}^{{}^{\prime }}$ with ${\parallel (\mathfrak{I}-\gamma \ell )}^{-1}\gamma \ell \parallel <1$ by Lemma 1. Therefore,

$$\begin{array}{c}\hfill \mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},{\varrho }_{\mathfrak{q}})⪯{(\mathfrak{I}-\gamma \ell )}^{-1}\gamma \ell \mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}-1})=\chi \mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}-1}),\end{array}$$

where $\chi ={(\mathfrak{I}-\gamma \ell )}^{-1}\gamma \ell$.

For any $\beta \ge 1\mathrm{and}\sigma \ge 1,$ we have

$$\begin{array}{cc}\hfill \mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta })& ⪯\ell [\mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta +\sigma -1})+\mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta })]-\mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta +\sigma -1})\hfill \\ & ⪯\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta +\sigma -1})+\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta })\hfill \\ & ⪯\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta +\sigma -1})+{\ell }^2[\mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta +\sigma -2})+\mathsf{\Phi }({\varrho }_{\beta +\sigma -2},{\varrho }_{\beta })]\hfill \\ & -\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma -2},{\varrho }_{\beta +\sigma -2})\hfill \\ & ⪯\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta +\sigma -1})+{\ell }^2\mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta +\sigma -2})+{\ell }^2\mathsf{\Phi }({\varrho }_{\beta +\sigma -2},{\varrho }_{\beta })\hfill \\ & ⪯\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta +\sigma -1})+{\ell }^2\mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta +\sigma -2})+\cdots \hfill \\ & +{\ell }^{\sigma -1}\mathsf{\Phi }({\varrho }_{\beta +2},{\varrho }_{\beta +1})+{\ell }^{\sigma -1}\mathsf{\Phi }({\varrho }_{\beta +1},{\varrho }_{\beta })\hfill \\ & ⪯\ell {\left(\chi \right)}^{\beta +\sigma -1}{\gamma }_0+{\ell }^2{\left(\chi \right)}^{\beta +\sigma -2}{\gamma }_0+{\ell }^3{\left(\chi \right)}^{\beta +\sigma -3}{\gamma }_0+\cdots \hfill \\ & +{\ell }^{\sigma -1}{\left(\chi \right)}^{\beta +1}{\gamma }_0+{\ell }^{\sigma -1}{\left(\chi \right)}^{\beta }{\gamma }_0\hfill \\ & =\sum _{\alpha =1}^{\sigma -1}{\ell }^{\alpha }{\left(\chi \right)}^{\beta +\sigma -\alpha }{\gamma }_0+{\ell }^{\sigma -1}{\left(\chi \right)}^{\beta }{\gamma }_0\hfill \\ & =\sum _{\alpha =1}^{\sigma -1}|{\gamma }_0^{\frac{1}{2}}{\chi }^{\frac{\beta +\sigma -\alpha }{2}}{\ell }^{\frac{\alpha }{2}}{|}^2+{|{\gamma }_0^{\frac{1}{2}}{\ell }^{\frac{\sigma -1}{2}}{\chi }^{\frac{\beta }{2}}|}^2\hfill \\ & ⪯\parallel {\gamma }_0\parallel \sum _{\alpha =1}^{\sigma -1}{\parallel \chi \parallel }^{\beta +\sigma -\alpha }{\parallel \ell \parallel }^{\alpha }\mathfrak{I}+{\parallel \ell \parallel }^{\sigma -1}{\parallel \chi \parallel }^{\beta }\parallel {\gamma }_0\parallel \mathfrak{I}\hfill \\ & ⪯\frac{\parallel {\gamma }_0{\parallel \parallel \ell \parallel }^{\sigma }{\parallel \chi \parallel }^{\beta +1}}{\parallel \ell \parallel -\parallel \chi \parallel }\mathfrak{I}+{\parallel \ell \parallel }^{\sigma -1}{\parallel \chi \parallel }^{\beta }\parallel {\gamma }_0\parallel \mathfrak{I}\hfill \\ & \to \theta (\beta \to \infty ).\hfill \end{array}$$

This implies that $\left\{{\varrho }_{\mathfrak{q}}\right\}$ is a Cauchy sequence in $\mathbb{B}$. By the completeness of $(\mho ,\mathbb{B},\mathsf{\Phi }),$ we can find $\varrho \in \mho$ satisfying ${lim}_{\mathfrak{q}\to \infty }{\varrho }_{\mathfrak{q}}=\varrho$ and

$$\begin{array}{c}\hfill \underset{\mathfrak{q},\beta \to \infty }{lim}\mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\beta })=\underset{\mathfrak{q}\to \infty }{lim}\mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}})=\underset{\mathfrak{q}\to \infty }{lim}\mathsf{\Phi }({\varrho }_{\mathfrak{q}},\varrho )=\mathsf{\Phi }(\varrho ,\varrho )=\theta .\end{array}$$

So,

$$\begin{array}{cc}\hfill \mathsf{\Phi }(\mathbb{T}\varrho ,\varrho )& ⪯\ell [\mathsf{\Phi }(\Lambda \varrho ,\Lambda {\varrho }_{\mathfrak{q}})+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\varrho )]-\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}})\hfill \\ \hfill & ⪯\ell [\mathsf{\Phi }(\Lambda \varrho ,\Lambda {\varrho }_{\mathfrak{q}})+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\varrho )]\hfill \\ \hfill & ⪯\ell [\gamma (\mathsf{\Phi }(\Lambda \varrho ,{\varrho }_{\mathfrak{q}})+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\varrho ))+\mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},\varrho )]\hfill \\ \hfill & ⪯\ell \gamma \ell (\mathsf{\Phi }(\Lambda \varrho ,{\varrho }_{\mathfrak{q}})+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\varrho ))+\ell \gamma \mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},\varrho )+\ell \mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},\varrho ).\hfill \end{array}$$

This is equivalent to

$$\begin{array}{cc}\hfill (\mathfrak{I}-{\ell }^2\gamma )\mathsf{\Phi }(\Lambda \varrho ,\varrho )& ⪯{\ell }^2\gamma \mathsf{\Phi }(\varrho ,{\varrho }_{\mathfrak{q}})+(\ell \gamma +\ell )\mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},\varrho ).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \parallel \mathsf{\Phi }(\Lambda \varrho ,\varrho )\parallel & \le \parallel {(\mathfrak{I}-{\ell }^2\gamma )}^{-1}{\ell }^2\gamma \parallel \parallel \mathsf{\Phi }(\varrho ,{\varrho }_{\mathfrak{q}})\parallel +\parallel {(\mathfrak{I}-{\ell }^2\gamma )}^{-1}(\ell \gamma +\ell )\parallel \parallel \mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},\varrho )\parallel \hfill \\ & \to \theta (\mathfrak{q}\to \infty ).\hfill \end{array}$$

Therefore, $\Lambda \varrho =\varrho$.

Now if $\xi (\ne \varrho )$ is another fixed point of $\Lambda$, then

$$\begin{array}{cc}\hfill \theta ⪯\mathsf{\Phi }(\varrho ,\xi )& =\mathsf{\Phi }(\Lambda \varrho ,\Lambda \xi )\hfill \\ \hfill & ⪯\gamma \left(\mathsf{\Phi }\right(\Lambda \varrho ,\xi )+\mathsf{\Phi }(\Lambda \xi ,\varrho \left)\right)\hfill \\ & =\gamma \left(\mathsf{\Phi }\right(\varrho ,\xi )+\mathsf{\Phi }(\xi ,\varrho \left)\right).\hfill \end{array}$$

That is,

$$\begin{array}{c}\hfill \mathsf{\Phi }(\varrho ,\xi )⪯{(\mathfrak{I}-\gamma )}^{-1}\gamma \mathsf{\Phi }(\Lambda \varrho ,\Lambda \xi ).\end{array}$$

Since ${\parallel \gamma (\mathfrak{I}-\gamma )}^{-1}\parallel <1,$

$$\begin{array}{cc}\hfill 0& \le \parallel \mathsf{\Phi }(\varrho ,\xi )\parallel =\parallel \mathsf{\Phi }(\Lambda \varrho ,\Lambda \xi )\parallel \hfill \\ & {\le \parallel (\mathfrak{I}-\gamma )}^{-1}\gamma \mathsf{\Phi }(\varrho ,\xi )\parallel \hfill \\ & {\le \parallel (\mathfrak{I}-\gamma )}^{-1}\gamma \parallel \parallel \mathsf{\Phi }(\varrho ,\xi )\parallel \hfill \\ & <\parallel \mathsf{\Phi }(\varrho ,\xi )\parallel .\hfill \end{array}$$

This means that

$$\begin{array}{c}\hfill \mathsf{\Phi }(\varrho ,\xi )=\theta \iff \varrho =\xi .\end{array}$$

□

Theorem 3.

Let $(\mho ,\mathbb{B},\mathsf{\Phi })$ be a complete $C^{🟉}\text{-}AV{P}_{b}MS$. Suppose the mapping $\Lambda :\mho \to \mho$ satisfying the following condition:

$$\begin{array}{c}\hfill \mathsf{\Phi }(\Lambda \varrho ,\Lambda \xi )⪯\gamma \left(\mathsf{\Phi }\right(\Lambda \varrho ,\varrho )+\mathsf{\Phi }(\Lambda \xi ,\xi \left)\right),\forall \varrho ,\xi \in \mho ,\end{array}$$

where $\gamma \in {\mathbb{B}}_{+}^{{}^{\prime }}$ and $\parallel \gamma \parallel <\frac{1}{2}.$ Then Λ has a unique fixed point in ℧.

Proof. 

We assume that $\gamma \ne \theta$, without loss of generality. Notice that for $\gamma \in {\mathbb{B}}_{+}^{{}^{\prime }}$, $\gamma \left(\mathsf{\Phi }\right(\Lambda \varrho ,\varrho )+\mathsf{\Phi }(\Lambda \xi ,\xi \left)\right)\ge \theta$. Choose ${\varrho }_0\in \mho$ and set ${\varrho }_{\mathfrak{q}+1}=\Lambda {\varrho }_{\mathfrak{q}}={\Lambda }^{\mathfrak{q}+1}{\varrho }_0,\mathfrak{q}=1,2,....$ and $\mathsf{\Phi }({\varrho }_1,{\varrho }_0)={\gamma }_0.$ Then

$$\begin{array}{cc}\hfill \mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},{\varrho }_{\mathfrak{q}})& =\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\Lambda {\varrho }_{\mathfrak{q}-1})\hfill \\ \hfill & ⪯\gamma (\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}})+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}-1},{\varrho }_{\mathfrak{q}-1}))\hfill \\ \hfill & =\gamma (\mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},{\varrho }_{\mathfrak{q}})+\mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}-1})).\hfill \end{array}$$

Thus,

$$\begin{array}{c}\hfill \mathsf{\Phi }({\varrho }_{\mathfrak{q}+1},{\varrho }_{\mathfrak{q}})⪯{(\mathfrak{I}-\gamma )}^{-1}\gamma \mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}-1})=\chi \mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}-1}),\end{array}$$

where $\chi ={(\mathfrak{I}-\gamma )}^{-1}\gamma$.

For any $\beta \ge 1$ and $\sigma \ge 1,$ we have

$$\begin{array}{cc}\hfill \mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta })& ⪯\ell [\mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta +\sigma -1})+\mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta })]-\mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta +\sigma -1})\hfill \\ \hfill & ⪯\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta +\sigma -1})+\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta })\hfill \\ \hfill & ⪯\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta +\sigma -1})+{\ell }^2[\mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta +\sigma -2})+\mathsf{\Phi }({\varrho }_{\beta +\sigma -2},{\varrho }_{\beta })]\hfill \\ \hfill & -\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma -2},{\varrho }_{\beta +\sigma -2})\hfill \\ \hfill & ⪯\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta +\sigma -1})+{\ell }^2\mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta +\sigma -2})+{\ell }^2\mathsf{\Phi }({\varrho }_{\beta +\sigma -2},{\varrho }_{\beta })\hfill \\ \hfill & ⪯\ell \mathsf{\Phi }({\varrho }_{\beta +\sigma },{\varrho }_{\beta +\sigma -1})+{\ell }^2\mathsf{\Phi }({\varrho }_{\beta +\sigma -1},{\varrho }_{\beta +\sigma -2})+\cdots \hfill \\ \hfill & +{\ell }^{\sigma -1}\mathsf{\Phi }({\varrho }_{\beta +2},{\varrho }_{\beta +1})+{\ell }^{\sigma -1}\mathsf{\Phi }({\varrho }_{\beta +1},{\varrho }_{\beta })\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ⪯\ell {\left(\chi \right)}^{\beta +\sigma -1}{\gamma }_0+{\ell }^2{\left(\chi \right)}^{\beta +\sigma -2}{\gamma }_0+{\ell }^3{\left(\chi \right)}^{\beta +\sigma -3}{\gamma }_0+\cdots \hfill \\ \hfill & +{\ell }^{\sigma -1}{\left(\chi \right)}^{\beta +1}{\gamma }_0+{\ell }^{\sigma -1}{\left(\chi \right)}^{\beta }{\gamma }_0\hfill \\ & =\sum _{\alpha =1}^{\sigma -1}{\ell }^{\alpha }{\left(\chi \right)}^{\beta +\sigma -\alpha }{\gamma }_0+{\ell }^{\sigma -1}{\left(\chi \right)}^{\beta }{\gamma }_0\hfill \\ \hfill & =\sum _{\alpha =1}^{\sigma -1}|{\gamma }_0^{\frac{1}{2}}{\chi }^{\frac{\beta +\sigma -\alpha }{2}}{\ell }^{\frac{\alpha }{2}}{|}^2+{|{\gamma }_0^{\frac{1}{2}}{\ell }^{\frac{\sigma -1}{2}}{\chi }^{\frac{\beta }{2}}|}^2\hfill \\ \hfill & ⪯\parallel {\gamma }_0\parallel \sum _{\alpha =1}^{\sigma -1}{\parallel \chi \parallel }^{\beta +\sigma -\alpha }{\parallel \ell \parallel }^{\alpha }\mathfrak{I}+{\parallel \ell \parallel }^{\sigma -1}{\parallel \chi \parallel }^{\beta }\parallel {\gamma }_0\parallel \mathfrak{I}\hfill \\ \hfill & ⪯\frac{\parallel {\gamma }_0{\parallel \parallel \ell \parallel }^{\sigma }{\parallel \chi \parallel }^{\beta +1}}{\parallel \ell \parallel -\parallel \chi \parallel }\mathfrak{I}+{\parallel \ell \parallel }^{\sigma -1}{\parallel \chi \parallel }^{\beta }\parallel {\gamma }_0\parallel \mathfrak{I}\hfill \\ & \to \theta (\beta \to \infty ).\hfill \end{array}$$

This implies $\left\{{\varrho }_{\mathfrak{q}}\right\}$ is a Cauchy sequence in $\mathbb{B}$. By the completeness of $(\mho ,\mathbb{B},\mathsf{\Phi }),$ we can find $\varrho \in \mho$ satisfying ${lim}_{\mathfrak{q}\to \infty }{\varrho }_{\mathfrak{q}}=\varrho$ and

$$\begin{array}{c}\hfill \underset{\mathfrak{q},\beta \to \infty }{lim}\mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\beta })=\underset{\mathfrak{q}\to \infty }{lim}\mathsf{\Phi }({\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}})=\underset{\mathfrak{q}\to \infty }{lim}\mathsf{\Phi }({\varrho }_{\mathfrak{q}},\varrho )=\mathsf{\Phi }(\varrho ,\varrho )=\theta .\end{array}$$

So,

$$\begin{array}{cc}\hfill \mathsf{\Phi }(\mathbb{T}\varrho ,\varrho )& ⪯\ell [\mathsf{\Phi }(\Lambda \varrho ,\Lambda {\varrho }_{\mathfrak{q}})+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\varrho )]\hfill \\ & ⪯\ell [\gamma (\mathsf{\Phi }(\Lambda \varrho ,\varrho )+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}})+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\varrho )]\hfill \\ & =\ell \gamma (\mathsf{\Phi }(\Lambda \varrho ,\varrho )+\mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}}))+\ell \mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\varrho ).\hfill \end{array}$$

This is equivalent to

$$\begin{array}{cc}\hfill \mathsf{\Phi }(\Lambda \varrho ,\varrho )& ⪯{(\mathfrak{I}-\ell \gamma )}^{-1}\ell \gamma \mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\Lambda {\varrho }_{\mathfrak{q}-1})+{(\mathfrak{I}-\ell \gamma )}^{-1}\ell \mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\varrho ).\hfill \end{array}$$

Thus,

$$\begin{array}{cc}\hfill \parallel \mathsf{\Phi }(\Lambda \varrho ,\varrho )\parallel & {\le \parallel (\mathfrak{I}-\ell \gamma )}^{-1}\ell \gamma \parallel \parallel \mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},{\varrho }_{\mathfrak{q}})\parallel +\parallel {(\mathfrak{I}-\ell \gamma )}^{-1}\ell \parallel \parallel \mathsf{\Phi }(\Lambda {\varrho }_{\mathfrak{q}},\varrho )\parallel \hfill \\ & \to 0(\mathfrak{q}\to \infty ).\hfill \end{array}$$

It follows that $\Lambda \varrho =\varrho$. Hence, $\varrho$ is a fixed point of $\Lambda$. Let $\xi (\ne \varrho )$ be a other fixed point of $\Lambda$, then

$$\begin{array}{c}\hfill \theta ⪯\mathsf{\Phi }(\varrho ,\xi )=\mathsf{\Phi }(\Lambda \varrho ,\Lambda \xi )⪯\gamma \left(\mathsf{\Phi }\right(\Lambda \varrho ,\varrho )+\mathsf{\Phi }(\Lambda \xi ,\xi \left)\right)=\theta .\end{array}$$

Hence, $\varrho =\xi .$ □

Example 1.

Let $\mho =[0,1]$ and $\mathbb{B}={\mathcal{M}}_2\left(\mathbb{C}\right)$ and a mapping $\mathsf{\Phi }:\mho \times \mho \to \mathbb{B}$ is defined by

$$\mathsf{\Phi }(\varrho ,\xi )=\left[\begin{array}{c}\begin{array}{cc}{|\varrho -\xi |}^2& 0\\ 0& {\mathbb{W}|\varrho -\xi |}^2\end{array}\end{array}\right]+\left[\begin{array}{c}\begin{array}{cc}max{\{\varrho ,\xi \}}^2& 0\\ 0& \mathbb{W}max{\{\varrho ,\xi \}}^2\end{array}\end{array}\right],$$

where $\mathbb{W}\ge 0$ is a constant. For any $\mathcal{B}\in \mathbb{B}$, we denote its norm as, $\left|\right|\mathbb{B}\left|\right|={max}_{1\le \mathfrak{i}\le 4}\left\{\right|{\mathfrak{a}}_{\mathfrak{i}}\left|\right\}$. Then, $(\mho ,\mathbb{B},\mathsf{\Phi })$ is a complete $C^{🟉}\text{-}AV{P}_{b}MS$. Define a mapping $\Lambda :\mho \to \mho$ by $\Lambda \left(\varrho \right)=\frac{\varrho }{2}$ for all $\varrho \in \mho$. Observe that

$$\begin{array}{c}\hfill \mathsf{\Phi }(\Lambda \varrho ,\Lambda \xi )⪯\gamma \left(\mathsf{\Phi }\right(\Lambda \varrho ,\varrho )+\mathsf{\Phi }(\Lambda \xi ,\xi \left)\right),\forall \varrho ,\xi \in \mho ,\end{array}$$

which satisfies

$$\gamma =\left[\begin{array}{c}\begin{array}{cc}\frac{\sqrt{2}}{2}& 0\\ 0& \frac{\sqrt{2}}{2}\end{array}\end{array}\right]$$

and $\left|\right|\gamma \left|\right|=\frac{\sqrt{2}}{2}=\frac{1}{\sqrt{2}}<\frac{1}{2}$. Therefore, all the postulates of Theorem 3 are fulfilled and Λ has the unique fixed point $\varrho =0$.

Example 2.

Let $\mathbb{B}={\mathbb{R}}^2$ and $\mho =[0,\infty )$. Let ⪯ be the partial order on $\mathbb{B}$ given by

$$\begin{array}{c}\hfill ({\mathfrak{a}}_1,{\mathfrak{b}}_1)⪯({\mathfrak{a}}_2,{\mathfrak{b}}_2)\iff {\mathfrak{a}}_1\le {\mathfrak{a}}_2\mathrm{a}nd{\mathfrak{b}}_1\le {\mathfrak{b}}_2\end{array}$$

with the norm $\left|\right|({\mathfrak{a}}_1,{\mathfrak{b}}_1)\left|\right|=max\left\{\right|{\mathfrak{a}}_1|,|{\mathfrak{b}}_1\left|\right\}$. Define

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_{\mathfrak{b}}:\mho \times \mho \to \mathbb{B},\end{array}$$

is defined by

$$\begin{array}{c}\hfill {\mathsf{\Phi }}_{\mathfrak{b}}(\varrho ,\xi )=({(\varrho -\xi )}^2,0)+(max{\{\varrho ,\xi \}}^2,0).\end{array}$$

(1)

Then $(\mho ,\mathbb{B},\mathsf{\Phi })$ is a complete $C^{🟉}\text{-}AV{P}_{b}MS$. Define a mapping $\Lambda :\mho \to \mho$ by $\Lambda \left(\varrho \right)=1-2^{-\varrho }$ for all $\varrho \in \mho$. Observe that

$$\begin{array}{c}\hfill \mathsf{\Phi }(\Lambda \varrho ,\Lambda \xi )⪯\gamma \left(\mathsf{\Phi }\right(\Lambda \varrho ,\varrho )+\mathsf{\Phi }(\Lambda \xi ,\xi \left)\right),\forall \varrho ,\xi \in \mho ,\end{array}$$

which satisfies $\gamma =(\frac{1}{3},0)$ and $\left|\right|\gamma \left|\right|<\frac{1}{2}$. Therefore, all the postulates of Theorem 3 are fulfilled and Λ has the unique fixed point $\varrho =0$.

4. Application

We consider the Fredholm integral equation:

$$\begin{array}{c}\hfill \varrho \left(\pounds \right)={\int }_{\mathcal{B}}\mathcal{Q}(\pounds ,\hslash ,\varrho (\hslash ))d\hslash +\delta \left(\pounds \right),\pounds ,\hslash \in \mathcal{B},\end{array}$$

(2)

where $\mathcal{B}$ is a measurable, $\mathcal{Q}:\mathcal{B}\times \mathcal{B}\times \mathbb{R}\to \mathbb{R}$ and $\delta \in {\mathcal{L}}^{\infty }\left(\mathcal{B}\right)$. Let $\mho ={\mathcal{L}}^{\infty }\left(\mathcal{B}\right)$, $\mathcal{W}={\mathcal{L}}^2\left(\mathcal{B}\right)$ and $\mathcal{L}\left(\mathcal{W}\right)=\mathbb{B}$. Define a mapping $\rho :\mho \times \mho \to \mathbb{B}$ by

$$\begin{array}{c}\hfill \rho (\delta ,\mathfrak{w})={\pi }_{{|\delta -\mathfrak{w}|}^2}+\mathfrak{I},\end{array}$$

for all $\delta ,\mathfrak{w},\mathfrak{I}\in \mho$ with $\left|\right|\lambda \left|\right|=\mathfrak{w}<1$, where ${\pi }_{♭}:\mathcal{W}\to \mathcal{W}$ is the multiplicative operator, defined by

$$\begin{array}{c}\hfill {\pi }_{♭}\left(\psi \right)=♭·\psi .\end{array}$$

Theorem 4.

For all $\varrho ,\xi \in \mho$, suppose that

1.

$\exists \kappa :\mathcal{B}\times \mathcal{B}\to \mathbb{R}$ be a continuous function and $\mathfrak{w}\in (0,1)$ such that

$$\begin{array}{cc}\hfill \left|\mathcal{Q}\right(\pounds ,\hslash ,\varrho (\hslash ))-\mathcal{Q}(\pounds ,\hslash ,\xi \left(p\right)\left)\right|& \le \mathfrak{w}|\kappa (\pounds ,\hslash )|(|{\int }_{\mathcal{B}}\mathcal{Q}(\pounds ,\hslash ,\varrho (\hslash ))d\hslash +\delta \left(\pounds \right)-\xi (\hslash )|\hfill \\ \hfill & +|{\int }_{\mathcal{B}}\mathcal{Q}(\pounds ,\hslash ,\xi (\hslash ))d\hslash +\delta \left(\pounds \right)-\varrho (\hslash )|+\mathfrak{I}-{\mathfrak{w}}^{-1}\mathfrak{I})\hfill \end{array}$$

for all $\pounds ,\hslash \in \mathcal{B}$;

2.

${sup}_{\pounds \in \mathcal{B}}{\int }_{\mathcal{B}}\left|\kappa (\pounds ,\hslash )\right|d\hslash \le 1$.

Then the integral Equation (2) has a unique solution in ℧.

Proof. 

Define $\Lambda :\mho \to \mho$ by

$$\begin{array}{c}\hfill \Lambda \varrho \left(\pounds \right)={\int }_{\mathcal{B}}\mathcal{Q}(\pounds ,\hslash ,\varrho (\hslash ))d\hslash +\delta \left(\pounds \right),\forall \pounds ,\hslash \in \mathcal{B}.\end{array}$$

Set $\lambda =\mathfrak{w}\mathfrak{I}$. Then $\lambda \in \mathbb{B}$. For any $\varkappa \in \mathcal{W}$, we have

$$\begin{array}{cc}\hfill \left|\right|\mathsf{\Phi }(\Lambda \varrho ,\Lambda \xi )\left|\right|& =\underset{\left|\right|\varkappa \left|\right|=1}{sup}({\pi }_{{|\Lambda \varrho -\Lambda \xi |}^2+\mathfrak{I}}\varkappa ,\varkappa )\hfill \\ \hfill & =\underset{\left|\right|\varkappa \left|\right|=1}{sup}{\int }_{\mathcal{B}}{\left(\right|\Lambda \varrho -\Lambda \xi |}^2+\mathfrak{I})\varkappa \left(\pounds \right)\overline{\varkappa \left(\pounds \right)}d\pounds \hfill \\ \hfill & \le \underset{\left|\right|z\left|\right|=1}{sup}{\int }_{\mathcal{B}}{\left[{\int }_{\mathcal{B}}|\mathcal{Q}(\pounds ,\hslash ,\varrho (\hslash ))-\mathcal{Q}(\pounds ,\hslash ,\xi (\hslash ))|\right]}^{2}d\hslash {\left|z\left(\pounds \right)\right|}^{2}d\pounds \hfill \\ \hfill & +\underset{\left|\right|\varkappa \left|\right|=1}{sup}{\int }_{\mathcal{B}}{\int }_{\mathcal{B}}d\hslash {\left|\varkappa \left(\pounds \right)\right|}^{2}d\pounds \mathfrak{I}\hfill \\ \hfill & \le \underset{\left|\right|\varkappa \left|\right|=1}{sup}{\int }_{\mathcal{B}}[{\int }_{\mathcal{B}}\mathfrak{w}\left|\kappa (\pounds ,\hslash )\right|\left(\right|{\int }_{\mathcal{B}}\mathcal{Q}(\pounds ,\hslash ,\varrho (\hslash ))d\hslash +\delta \left(\pounds \right)-\xi (\hslash )|\hfill \\ \hfill & +|{\int }_{\mathcal{B}}\mathcal{Q}(\pounds ,\hslash ,\xi (\hslash ))d\hslash +\delta \left(\pounds \right)-\varrho (\hslash )|+\mathfrak{I}-{\mathfrak{w}}^{-1}\mathfrak{I})d\hslash {]}^2{\left|z\left(\pounds \right)\right|}^{2}d\pounds +\mathfrak{I}\hfill \\ \hfill & \le {\mathfrak{w}}^2\underset{\left|\right|z\left|\right|=1}{sup}{\int }_{\mathcal{B}}{\left[{\int }_{\mathcal{B}}\left|\kappa (\pounds ,\hslash )\right|d\hslash \right]}^2{\left|z\left(\pounds \right)\right|}^{2}d\pounds \left(\right||\Lambda \varrho -{\xi \left|\right|}_{\infty }^2+\left|\right|\Lambda \xi -{\varrho \left|\right|}_{\infty }^2)\hfill \\ \hfill & \le \mathfrak{w}\underset{\pounds \in \mathcal{B}}{sup}{\int }_{\mathcal{B}}\left|\kappa (\pounds ,\hslash )\right|d\hslash \underset{\left|\right|z\left|\right|=1}{sup}{\int }_{\mathcal{B}}{\left|z\left(\pounds \right)\right|}^{2}d\pounds \left(\right||\Lambda \varrho -{\xi \left|\right|}_{\infty }^2+\left|\right|\Lambda \xi -{\varrho \left|\right|}_{\infty }^2)\hfill \\ \hfill & \le \mathfrak{w}\left[\right||\Lambda \varrho -\xi |{|}_{\infty }^2+\left|\right|\Lambda \xi -\varrho \left|{|}_{\infty }^2\right]\hfill \\ \hfill & =\left|\right|\lambda \left|\right|\left[\right|\left|\mathsf{\Phi }\right(\Lambda \varrho ,\xi \left)\right||+|\left|\mathsf{\Phi }\right(\Lambda \xi ,\varrho \left)\right|\left|\right].\hfill \end{array}$$

Hence all the hypotheses of Theorem 2 are fulfilled, and thus Equation (2) has a unique solution. □

5. Conclusions

In this paper, we presented fixed point theorems for generalized contractions on $C^{*}\text{-}AV{P}_{b}MS$. The examples and applications on $C^{*}\text{-}AV{P}_{b}MS$ are presented to strengthen our main results. Samreen et al. [15] proved fixed point theorems on extended b-metric spaces. It is an interesting open problem to prove fixed theorems on $C^{🟉}$-algebra-valued extended partial b-metric spaces. Arabnia Firozjah et al. [16] proved fixed point results on cone b-metric spaces over Banach algebras. Furthermore, it is an interesting open problem to prove fixed theorems on $C^{🟉}$-algebra-valued cone b-metric spaces.