Solution of a nonlinear fractional-order initial value problem via a $\mathscr{C}^{*}$-algebra-valued $\mathcal{R}$-metric space

Abstract

In this article, we prove new common fixed-point theorems on a $\mathscr{C}^{*}$-algebra-valued $\mathcal{R}$-metric space. An example is given based on our obtained results. To enhance our results, a strong application based on the fractional-order initial value problem is provided.

1 Introduction

The concept of $\mathscr{C}^{*}$-AVMS was outlined by Ma et al. in 2014, [1] and they proved some fixed-point results with a new contraction type. Many authors and researchers have generalized with a new type of outcome (see [2–5]).

Let $\mathcal{B}$ be the unital algebra with unit $\mathcal{I}$. The conjugate linear map $\delta \mapsto \delta ^{*}$ on $\mathcal{B}$ is such that $\delta ^{**}=\delta$ and $(\delta \eta )^{*}=\eta ^{*}\delta ^{*}$ for all $\delta , \eta \in \mathcal{B}$. The set of all bounded linear operators on a Hilbert space $\mathcal{H}$, under the norm topology $\mathcal{L}(\mathcal{H})$, is a $\mathscr{C}^{*}$-algebra. The concept of a cone metric space was outlined by Huang and Zhang in 2007 [6] and they replaced the set of real numbers by an ordered Banach space.

The CFP for commuting mappings in metric space was investigated by Jungck in 1966 [7]. Likewise, many fixed and CFP results were obtained in different types like cone metric space [8], uniform space [9], noncommutative Banach space [10], fuzzy metric space [11] and so on. Hussain et al. proved Suzuki–Berinde-type fixed-point theorems and the CFP theorem on a cone b-metric space in these works [12, 13], respectively. Khalehoghli, Rahimi and Gordji introduced the $\mathcal{R}$-metric space to prove the fixed-point theorem [14]. Wardowski proposed a new Banach contraction principle in a complete metric space to prove the fixed-point theorem [15]. Astha, Deepak and Choonkil proposed a $\mathscr{C}^{*}$ algebra-valued $\mathcal{R}$-metric space to prove a unique fixed-point theorem [16]. Afshari and Khoshvaghti proved a unique fixed-point theorem in an operator equation on the ordered Banach space [17]. Afshari et al. [18], used a fixed-point theorem to study a boundary value problem for a fractional differential equation in a b-metric space. Deuri and Das in [19] proved the fixed-point theorem in a newly constructed contraction operator. Chandra Deuri et al. [20] investigated the existence of a fractional integral equation by using the Darbo fixed-point theorem. Further, Das et al. [21], proved the fixed-point theorem based on the Darbo-type theorem. Researchers in [22], utilized the fixed-point theorem for discussing a generalized proportional fractional integral equation in a Banach space. Das and Deuri [23], proved the fixed-point theorem on a generalization of Darbo’s fixed-point theorem in a Banach space. The authors of [24, 25], established the qualitative properties of fractional differential equation in unbounded domains.

In this paper, we prove some CFP theorems on a $\mathscr{C}^{*}$-algebra-valued $\mathcal{R}$-metric space. Additionally, we established the uniqueness of a common solution for the fractional-order initial value problem. Throughout this paper, $\mathcal{B}$ will denote $\mathscr{C}^{*}$-algebra with unit $\mathcal{I}$ and $\mathcal{R}$ denotes a nonempty binary relation. $\mathscr{C}^{*}$-AVMS means a $\mathscr{C}^{*}$-algebra-valued metric space and $\mathscr{C}^{*}$-AV$\mathcal{R}$-MS means a $\mathscr{C}^{*}$-algebra-valued $\mathcal{R}$- metric space. CFP means Common Fixed Point.

2 Preliminaries

Definition 2.1

Let a nonvoid set be $\mathcal{X}$. Let the mapping $\varpi \colon \mathcal{X}\times \mathcal{X}\rightarrow \mathcal{B}$ be such that:

• (1) $0_{\mathcal{B}}\leq \varpi (\zeta , \vartheta )$ for all $\zeta , \vartheta \in \mathcal{X}$;

• (2) $\varpi (\zeta , \vartheta )=0_{\mathcal{B}}$ iff $\zeta =\vartheta$;

• (3) $\varpi (\zeta , \vartheta )=\varpi (\vartheta , \zeta )$ for all $\zeta , \vartheta \in \mathcal{X}$;

• (4) $\varpi (\zeta , \vartheta )\leq \varpi (\zeta , \nu )+\varpi (\nu , \vartheta )$ for all $\zeta , \vartheta , \nu \in \mathcal{X}$.

Then, $(\mathcal{X}, \mathcal{B}, \varpi )$ is called a $\mathscr{C}^{*}$-AVMS.

Definition 2.2

Let a nonvoid set be $\mathcal{X}$ defined a binary relation on $\mathcal{R}$, a sequence $\{\zeta _{\phi}\}_{\phi \in \mathbb{N}}\in \mathcal{X}$ is called a $\mathcal{R}$-sequence if $(\zeta _{\phi}, \zeta _{\phi +1})\in \mathcal{R}$ for all $\phi \in \mathbb{N}$.

Definition 2.3

A binary relation $\mathcal{R}$ on a metric space $(\mathcal{X}, \varpi )$ is called a $\mathcal{R}$-metric space and it is denoted by $(\mathcal{X}, \varpi , \mathcal{R})$.

Lemma 2.1

[26]

• 1. If $\{\eta _{\phi}\}_{\phi =1}^{\infty}\subseteq \mathcal{B}$ and $\lim_{\phi \rightarrow \infty}\eta _{\phi}=0_{\mathcal{B}}$, then for any $\delta \in \mathcal{B}$, $\lim_{\phi \rightarrow \infty}\delta ^{*} \eta _{\phi}\delta =0_{\mathcal{B}}$.

• 2. If $\delta , \eta \in \mathcal{B}_{\mathfrak{h}}$ and $\mathfrak{c}\in \mathcal{B}_{+}^{\prime }$, then $\delta \leq \eta$ deduces $\mathfrak{c}\delta \leq \mathfrak{c}\eta$, where $\mathcal{B}_{+}^{\prime }=\mathcal{B}_{+}\cap \mathcal{B}^{\prime }$.

• 3. Let $\{\zeta _{\phi}\}_{\phi =1}^{\infty}$ be a sequence in $\mathcal{X}$. If $\{\zeta _{\phi}\}$ converges to ζ and ϑ, respectively, then $\zeta =\vartheta$.

Definition 2.4

Let $(\mathcal{X}, \mathcal{B}, \varpi , \mathcal{R})$ be a $\mathscr{C}^{*}$-AV$\mathcal{R}$-MS, let a $\mathcal{R}$-sequence $\{\zeta _{\phi}\}_{\phi \in \mathbb{N}}\subset \mathcal{X}$ be said to be $\mathcal{R}$-Cauchy, if $\kappa >0$, we can find $\phi _{0}\in \mathbb{N}$ that satisfies $\|\varpi (\zeta _{\phi}, \zeta _{\mathfrak{m}})\|\leq \kappa$, $\forall \phi , \mathfrak{m}\geq \phi _{0}$.

Definition 2.5

Let $(\mathcal{X}, \mathcal{B}, \varpi , \mathcal{R})$ be a $\mathscr{C}^{*}$-AV$\mathcal{R}$-MS that is called a Complete $\mathscr{C}^{*}$-AV$\mathcal{R}$-MS, if every $\mathcal{R}$- Cauchy sequence with respect to $\mathcal{B}$ is convergent.

Definition 2.6

Let two mappings Ξ and Φ on $(\mathcal{X}, \mathcal{B}, \varpi )$ be a $\mathscr{C}^{*}$-AVMS be called compatible, if the sequence $\{\zeta _{\phi}\}_{\phi =1}^{\infty}\subseteq \mathcal{X}$, such that $\lim_{\phi \rightarrow \infty}\varXi \zeta _{\phi}=\lim_{\phi \rightarrow \infty}\varPhi \zeta _{\phi}=\sigma \in \mathcal{X}$, then $\varpi (\varXi \varPhi \zeta _{\phi}, \varPhi \varXi \zeta _{\phi}) \xrightarrow{\|\cdot \|_{\mathcal{B}}}0_{\mathcal{B}}$ ($\phi \rightarrow \infty$).

3 Main results

We prove our first result.

Theorem 3.1

Let $(\mathcal{X}, \mathcal{B}, \varpi , \mathcal{R})$ be a complete $\mathscr{C}^{*}$-AV$\mathcal{R}$-MS and let the two mappings $\varXi , \varPhi \colon \mathcal{X}\rightarrow \mathcal{X}$, such that

• (i) $\varXi (\mathcal{X})\subseteq \mathcal{X}$, $\quad \varPhi ( \mathcal{X})\subseteq \mathcal{X}$;

• (ii) Ξ, Φ are $\mathcal{R}$-preserving;

• (iii) We can find some $\zeta _{0}\in \mathcal{X}$ satisfying $(\zeta _{0}, \vartheta )\in \mathcal{R}$ for all $\vartheta \in \varXi (\mathcal{X})$;

• (iv) For all $\zeta , \vartheta \in \mathcal{X}$ with $(\zeta , \vartheta )\in \mathcal{R}$, there exists $\delta \in \mathcal{B}$, where $\|\delta \|<1$ such that

  $$\begin{aligned} \varpi (\varXi \zeta , \varPhi \vartheta )\leq \delta ^{*}\varpi ( \zeta , \vartheta )\delta , \quad \textit{for any } \zeta , \vartheta \in \mathcal{X}. \end{aligned}$$

Then, Ξ and Φ have a unique CFP.

Proof

Let $\zeta _{0}\in \mathcal{X}$ and consider a $\mathcal{R}$-sequence $\{\zeta _{\phi}\}_{\phi =0}^{\infty}\subseteq \mathcal{X}$, such that $\zeta _{\phi}=\varPhi \zeta _{\phi -1}$, $\zeta _{\phi +1}=\varXi \zeta _{\phi}$, $\zeta _{\phi -1}=\varXi \zeta _{\phi -2}$. From condition (iv),

$$\begin{aligned} \varpi (\zeta _{\phi +1}, \zeta _{\phi})&=\varpi (\varXi \zeta _{\phi}, \varPhi \zeta _{\phi -1}) \\ &\leq \delta ^{*}\varpi (\zeta _{\phi}, \zeta _{\phi -1})\delta \\ &\leq \bigl(\delta ^{*}\bigr)^{2}\varpi (\zeta _{\phi -1}, \zeta _{\phi -2}) ( \delta )^{2} \\ &\vdots \\ &\leq \bigl(\delta ^{*}\bigr)^{\phi}\varpi (\zeta _{1}, \zeta _{0}) (\delta )^{ \phi}. \end{aligned}$$

Since, $\eta , \mathfrak{c}\in \mathcal{B}_{\mathfrak{h}}$, then $\eta \leq \mathfrak{c}$, which implies $\delta ^{*}\eta \delta \leq \delta ^{*}\mathfrak{c}\delta$.

Similarly,

$$\begin{aligned} \varpi (\zeta _{\phi}, \zeta _{\phi -1})&=\varpi (\varPhi \zeta _{ \phi -1}, \varXi \zeta _{\phi -2}) \\ &\leq \delta ^{*}\varpi (\zeta _{\phi -1}, \zeta _{\phi -2})\delta \\ &\vdots \\ &\leq \bigl(\delta ^{*}\bigr)\varpi (\zeta _{1}, \zeta _{0}) (\delta )^{\phi -1}, \end{aligned}$$

for any $\mathfrak{p}\in \mathbb{N}$, then by the triangle inequality,

$$\begin{aligned} \varpi (\zeta _{\phi +\mathfrak{p}})&\leq \varpi (\zeta _{\phi + \mathfrak{p}}, \zeta _{\phi +\mathfrak{p}-1})+\varpi (\zeta _{\phi + \mathfrak{p}-1}, \zeta _{\phi +\mathfrak{p}-2})+ \cdots +\varpi ( \zeta _{\phi +1}, \zeta _{\phi}) \\ &\leq \sum_{\upsilon =\phi}^{\phi +\mathfrak{p}-1}\bigl(\delta ^{*}\bigr)^{ \upsilon}\varpi (\zeta _{1}, \zeta _{0}) (\delta )^{\upsilon} \\ &\leq \sum_{\upsilon =\phi}^{\phi +\mathfrak{p}-1}\bigl(\delta ^{*}\bigr)^{ \upsilon}\eta ^{2}(\delta )^{\upsilon} \\ &\leq \sum_{\upsilon =\phi}^{\phi +\mathfrak{p}-1}\bigl(\delta ^{*}\bigr)^{ \upsilon}\eta \cdot \eta (\delta )^{\upsilon} \\ &\leq \sum_{\upsilon =\phi}^{\phi +\mathfrak{p}-1}\bigl(\eta \delta ^{ \upsilon}\bigr)^{*}\cdot \bigl(\eta \delta ^{\upsilon} \bigr) \\ &\leq \sum_{\upsilon =\phi}^{\phi +\mathfrak{p}-1} \bigl\vert \eta \delta ^{ \upsilon} \bigr\vert ^{2} \\ &\leq \sum_{\upsilon =\phi}^{\phi +\mathfrak{p}-1}\bigl\| \bigl|\eta \delta ^{ \upsilon}\bigr|^{2}\bigr\| 1_{\mathcal{B}} \\ &\leq \Vert \eta \Vert ^{2} 1_{\mathcal{B}}\sum _{\upsilon =\phi}^{\phi + \mathfrak{p}-1} \bigl\Vert \delta ^{\upsilon} \bigr\Vert \rightarrow 0_{\mathcal{B}} \quad \text{as } \phi \rightarrow \infty , \end{aligned}$$

where $1_{\mathcal{B}}$ is a unit element in $\mathcal{B}$ and $\varpi (\zeta _{1}, \zeta _{0})=\eta ^{2}$ for some $\eta \in \mathcal{B}$. From definition 2.5, we obtain that $\{\zeta _{\phi}\}_{\phi =1}^{\infty}$ is a Cauchy sequence in $\mathcal{X}$. We can find $\zeta \in \mathcal{X}$ satisfying $\lim_{\phi \rightarrow \infty}\zeta _{\phi}=\zeta$.

Now, using the triangle inequality

$$\begin{aligned} \varpi (\zeta , \varPhi \zeta )&\leq \varpi (\zeta , \zeta _{\phi})+ \varpi (\zeta _{\phi}, \varPhi \zeta ) \\ &\leq \varpi (\zeta , \zeta _{\phi})+\varpi (\varPhi \zeta _{\phi -1}, \varPhi \zeta ) \\ &\leq \varpi (\zeta , \zeta _{\phi})+\delta ^{*}\varpi ( \zeta _{\phi -1}, \zeta )\delta . \end{aligned}$$

Taking $\phi \rightarrow \infty$, the right-hand side approaches $0_{\mathcal{B}}$, by lemma 2.1 (condition 1), we obtain $\varPhi \zeta =\zeta$.

Similarly,

$$\begin{aligned} \varpi (\varXi \zeta , \zeta )&=\varpi (\varXi \zeta , \varPhi \zeta ) \\ &\leq \delta ^{*}\varpi (\zeta , \zeta )\delta \\ &=0_{\mathcal{B}}. \end{aligned}$$

We have,

$$\begin{aligned} \varpi (\varXi \zeta , \zeta )=0_{\mathcal{B}}, \end{aligned}$$

which means, $\varXi \zeta =\zeta$.

Let us take another fixed point $\vartheta \in \mathcal{X}$ such that $\varXi \vartheta =\varPhi \vartheta =\vartheta$, From condition (iv) of Theorem 3.1:

$$\begin{aligned} \varpi (\zeta , \vartheta )=\varpi (\varXi \zeta , \varPhi \vartheta ) \leq \delta ^{*}\varpi (\zeta , \vartheta )\delta , \end{aligned}$$

with $\|\delta \|<1$, such that

$$\begin{aligned} 0&\leq \bigl\Vert \varpi (\zeta , \vartheta ) \bigr\Vert \leq \Vert \delta \Vert ^{2} \bigl\Vert \varpi ( \zeta , \vartheta ) \bigr\Vert \\ &\leq \bigl\Vert \varpi (\zeta , \vartheta ) \bigr\Vert . \end{aligned}$$

Thus, $\|\varpi (\zeta , \vartheta )\|=0$ and $\varpi (\zeta , \vartheta )=0_{\mathcal{B}}$, which gives $\zeta =\vartheta$. Hence, Ξ and Φ have a unique CFP in $\mathcal{X}$. □

Here, we prove our second result.

Theorem 3.2

Let $(\mathcal{X}, \mathcal{B}, \varpi , \mathcal{R})$ be a complete $\mathscr{C}^{*}$-AV$\mathcal{R}$-MS and let the two mapping $\varXi , \varPhi \colon \mathcal{X}\rightarrow \mathcal{X}$ such that

• (i) $\varXi (\mathcal{X})\subseteq \mathcal{X}$, $\quad \varPhi ( \mathcal{X})\subseteq \mathcal{X}$;

• (ii) Ξ, Φ is $\mathcal{R}$-preserving;

• (iii) We can find some $\zeta _{0}\in \mathcal{X}$ satisfying $(\zeta _{0}, \vartheta )\in \mathcal{R}$ for all $\vartheta \in \varXi (\mathcal{X})$;

• (iv) For all $\zeta , \vartheta \in \mathcal{R}$ with $(\zeta , \vartheta )\in \mathcal{R}$, there exist $\delta \in \mathcal{B}$, where $\|\delta \|<1$ such that

  $$\begin{aligned} \varpi (\varXi \zeta , \varXi \vartheta )\leq \delta \varpi (\varXi \zeta , \varPhi \zeta )+\delta \varpi (\varXi \vartheta , \varPhi \vartheta ). \end{aligned}$$

Then, Ξ and Φ have a unique CFP.

Proof

Let $\zeta _{0}\in \mathcal{X}$ and consider a $\mathcal{R}$-sequence $\{\zeta _{\phi}\}_{\phi =0}^{\infty}\subseteq \mathcal{X}$ such that $\varPhi \zeta _{\phi}=\zeta _{\phi +1}$, and $\varPhi \zeta _{\phi +1}=\zeta _{\phi +2}$, then

$$\begin{aligned}& \begin{aligned} \varpi (\zeta _{\phi +2}, \zeta _{\phi +1})&=\varpi (\varPhi \zeta _{ \phi +1}, \varPhi \zeta _{\phi}) \\ &\leq \delta \varpi (\varXi \zeta _{\phi +1}, \varPhi \zeta _{\phi +1})+ \delta \varpi (\varXi \zeta _{\phi}, \varPhi \zeta _{\phi}) \\ &\leq \delta \varpi (\zeta _{\phi +1}, \zeta _{\phi +2})+\delta \varpi (\zeta _{\phi}, \zeta _{\phi +1}) \\ &\leq \delta \varpi (\zeta _{\phi +2}, \zeta _{\phi +1})+\delta \varpi (\zeta _{\phi +1}, \zeta _{\phi}), \end{aligned} \\& \varpi (\zeta _{\phi +2}, \zeta _{\phi +1})-\delta \varpi (\zeta _{ \phi +2}, \zeta _{\phi +1})=\delta \varpi (\zeta _{\phi +1}, \zeta _{ \phi}), \\& (1_{\mathcal{B}}-\delta )\varpi (\zeta _{\phi +2}, \zeta _{\phi +1})= \delta \varpi (\zeta _{\phi +1}, \zeta _{\phi}), \\& \varpi (\zeta _{\phi +2}, \zeta _{\phi +1})\leq \frac{\delta}{(1_{\mathcal{B}}-\delta )}\varpi (\zeta _{\phi +1}, \zeta _{\phi}), \\& \varpi (\zeta _{\phi +2}, \zeta _{\phi +1})\leq \eta \varpi ( \zeta _{ \phi +1}, \zeta _{\phi}), \quad \text{where } \eta = \frac{\delta}{(1_{\mathcal{B}}-\delta )}. \end{aligned}$$

By induction,

$$\begin{aligned} \varpi (\zeta _{\phi +2}, \zeta _{\phi +1})\leq \eta ^{\phi}\varpi ( \zeta _{1}, \zeta _{0}). \end{aligned}$$

For $\phi >\mathfrak{m}$,

$$\begin{aligned} \varpi (\zeta _{\phi +1}, \zeta _{\mathfrak{m}})&\leq \varpi (\zeta _{ \phi +1}, \zeta _{\phi})+\varpi (\zeta _{\phi}, \zeta _{\phi -1})+ \cdots +\varpi (\zeta _{\mathfrak{m}+1}, \zeta _{\mathfrak{m}}) \\ &\leq \bigl(\eta ^{\phi}+\eta ^{\phi -1}+\cdots +\eta ^{\mathfrak{m}}\bigr) \varpi (\zeta _{1}, \zeta _{0}) \\ &\leq \bigl\Vert \eta ^{\phi}+\eta ^{\phi -1}+\cdots +\eta ^{\mathfrak{m}} \bigr\Vert \bigl\Vert \varpi (\zeta _{1}, \zeta _{0}) \bigr\Vert 1_{\mathcal{B}} \\ &\leq \bigl\Vert \eta ^{\phi} \bigr\Vert + \bigl\Vert \eta ^{\phi -1} \bigr\Vert +\cdots + \bigl\Vert \eta ^{ \mathfrak{m}} \bigr\Vert \bigl\Vert \varpi (\zeta _{1}, \zeta _{0}) \bigr\Vert 1_{\mathcal{B}} \\ &\leq \frac{ \Vert \eta \Vert ^{\mathfrak{m}}}{1- \Vert \eta \Vert } \bigl\Vert \varpi (\zeta _{1}, \zeta _{0}) \bigr\Vert 1_{\mathcal{B}}. \end{aligned}$$

Hence, $\{\zeta _{\phi}\}_{\phi =0}^{\infty}$ is a Cauchy sequence in $\mathcal{R}$-sequence. We can find $\mathfrak{q}\in \mathcal{X}$ satisfying $\lim_{\phi \rightarrow \infty}\zeta _{\phi}= \mathfrak{q}$. By condition (iv),

$$\begin{aligned}& \varpi (\zeta _{\phi +1}, \mathfrak{q}) =\varpi (\varPhi \zeta _{\phi}, \varXi \mathfrak{q}) \\& \hphantom{\varpi (\zeta _{\phi +1}, \mathfrak{q})} \leq \delta \varpi (\varPhi \zeta _{\phi}, \varXi \zeta _{\phi})+ \delta \varpi (\varXi \mathfrak{q}, \varPhi \mathfrak{q}) \\& \hphantom{\varpi (\zeta _{\phi +1}, \mathfrak{q})} \leq \delta \varpi (\varPhi \zeta _{\phi}, \varXi \mathfrak{q})+ \delta \varpi (\varXi \mathfrak{q}, \varXi \zeta _{\phi})+\delta \varpi ( \varXi \mathfrak{q}, \varPhi \zeta _{\phi})+\delta \varpi ( \varPhi \zeta _{\phi}, \varPhi \mathfrak{q}) \\& \hphantom{\varpi (\zeta _{\phi +1}, \mathfrak{q})} \leq 2\delta \varpi (\varPhi \zeta _{\phi}, \varXi \mathfrak{q})+ \delta \varpi (\varXi \mathfrak{q}, \varXi \zeta _{\phi})+\delta \varpi ( \varPhi \zeta _{\phi}, \varPhi \mathfrak{q}), \\& (1_{\mathcal{B}}-2\delta )\varpi (\zeta _{\phi +1}, \mathfrak{q}) \leq \delta \varpi (\varXi \mathfrak{q}, \varXi \zeta _{\phi})+ \delta \varpi (\varPhi \zeta _{\phi}, \varPhi \mathfrak{q}). \end{aligned}$$

Since $\|\delta \|<1$, then $1_{\mathcal{B}}-2\delta$ is invertible:

$$\begin{aligned} \varpi (\zeta _{\phi +1}, \mathfrak{q})\leq \frac{\delta}{(1_{\mathcal{B}}-2\delta )}\varpi ( \varXi \mathfrak{q}, \varXi \zeta _{\phi})+\frac{\delta}{(1_{\mathcal{B}}-2\delta )} \varpi ( \varPhi \zeta _{\phi}, \varPhi \mathfrak{q}), \end{aligned}$$

then $\lim_{\phi \rightarrow \infty}\zeta =\mathfrak{q}$. Let us choose $\varXi \mathfrak{q}=\varPhi \mathfrak{q}$. Hence, Ξ and Φ have a coincidence point in $\mathcal{X}$.

Assume $\mathfrak{p}\in \mathcal{X}$ such that $\varXi \mathfrak{p}=\varPhi \mathfrak{p}$, and by using condition (iv), we obtain

$$\begin{aligned} \varpi (\varPhi \mathfrak{p}, \varPhi \mathfrak{q})=\varpi (\varXi \mathfrak{p}, \varXi \mathfrak{q})\leq \delta \varpi (\varXi \mathfrak{p}, \varPhi \mathfrak{p})+\delta \varpi (\varXi \mathfrak{q}, \varPhi \mathfrak{q}), \end{aligned}$$

which shows that $\|\varpi (\varPhi \mathfrak{p}, \varPhi \mathfrak{q})\|=0$, then

$$\begin{aligned} \varPhi \mathfrak{p}=\varPhi \mathfrak{q}. \end{aligned}$$

Similarly,

$$\begin{aligned} \varXi \mathfrak{p}=\varXi \mathfrak{q}. \end{aligned}$$

Hence, Ξ and Φ have a unique CFP in $\mathcal{X}$. □

Example 3.3

Let $\mathcal{X}=\mathbb{R}$ and $\mathcal{B}=\mathcal{M}_{2}(\mathbb{R})$. Define relation $\mathcal{R}$ on $\mathcal{X}$ as $(\zeta , \vartheta )\in \mathbb{R}$ iff $\zeta , \vartheta \geq 0$ and $\varpi (\zeta ,\vartheta )=\left[\begin{array}{cc}|\zeta -\vartheta {|}^2& 0\\ 0& \upsilon |\zeta -\vartheta {|}^2\end{array}\right]$, where $\zeta , \vartheta \in \mathbb{R}$ and $\upsilon \geq 0$ is a constant. Then, $(\mathcal{X}, \mathcal{B}, \varpi , \mathcal{R})$ is a complete $\mathscr{C}^{*}$-AV$\mathcal{R}$-MS:

$$\begin{aligned} \varXi \zeta = \textstyle\begin{cases} 2-\frac{1}{\zeta}, &\zeta \in [0, \frac{5}{4}), \\ 2, &\zeta \in (\frac{5}{4}, 3], \end{cases}\displaystyle \qquad \varPhi \zeta = \textstyle\begin{cases} \frac{2}{\zeta ^{2}}, &\zeta \in [0, 1), \\ \zeta , &\zeta \in (1, 3]. \end{cases}\displaystyle \end{aligned}$$

Clearly, Ξ and Φ are $\mathcal{R}$-preserving. First, the set of their coincidence points is singleton $\{2\}$, and then we have Ξ and Φ commute at this point. Thereby, Ξ and Φ are weak compatible.

Let the sequence $\{\zeta _{\phi}\}\subseteq \mathcal{X}$ such that $\zeta _{\phi}=1-\phi \in \mathcal{X}$, hence,

$$\begin{aligned} \varXi \zeta _{\phi}=2-\frac{1}{1-\phi}=\frac{1-2\phi}{1-\phi}, \qquad \varPhi \zeta _{\phi}=\frac{2}{(1-\phi )^{2}}. \end{aligned}$$

Then, $\lim_{\phi \rightarrow \infty}\varXi \zeta _{\phi}= \lim_{\phi \rightarrow \infty}\varPhi \zeta _{\phi}=3$,

$$\begin{aligned} \varpi (\varXi \zeta _{\phi}, 3)&=\varpi \biggl(\frac{1-2\phi}{1-\phi}, 3 \biggr)= \begin{bmatrix} \vert \frac{\phi -2}{1-\phi} \vert ^{2} &0 \\ 0 &\upsilon \vert \frac{\phi -2}{1-\phi} \vert ^{2} \end{bmatrix} \xrightarrow{ \Vert \cdot \Vert _{\mathcal{B}}}0_{\mathcal{B}}, \quad \text{as } \phi \rightarrow \infty , \\ \varpi (\varPhi \zeta _{\phi}, 3)&=\varpi \biggl( \frac{2}{(1-\phi )^{2}}, 3 \biggr)= \begin{bmatrix} \vert \frac{3\phi -1}{1-\phi} \vert ^{2} &0 \\ 0 &\upsilon \vert \frac{3\phi -1}{1-\phi} \vert \end{bmatrix} \xrightarrow{ \Vert \cdot \Vert _{\mathcal{B}}}0_{\mathcal{B}}, \quad \text{as } \phi \rightarrow \infty . \end{aligned}$$

However,

$$\begin{aligned} \varpi (\varXi \varPhi \zeta _{\phi}, \varPhi \varXi \zeta _{\phi})&= \varpi \biggl(\varXi \biggl(\frac{1-2\phi}{1-\phi} \biggr), \varPhi \biggl(\frac{2}{(1-\phi )^{2}} \biggr) \biggr) \\ &=\varpi (3, 2) \\ &= \begin{bmatrix} 1 &0 \\ 0 &\upsilon \end{bmatrix}, \end{aligned}$$

which means $\varpi (\varXi \varPhi \zeta _{\phi}, \varPhi \varXi \zeta _{\phi}) \nrightarrow 0_{\mathcal{B}}$. Hence, Ξ and Φ have a unique CFP.

4 Application

Consider the nonlinear fractional-order initial value problem (FIVP) of the form

$$ \begin{aligned} &\mathcal{D}_{0}^{\alpha}\zeta ( \sigma )=\kappa \zeta (\varrho )+ \mathfrak{g}\bigl(\varrho , \zeta (\varrho ) \bigr),\quad \sigma \geq 0, \\ &\zeta (0)=\mu , \end{aligned} $$

(4.1)

where $0<\alpha \leq 1$ is the fractional order, κ is a nonnegative real constant, and μ is a real constant. The nonlinear term is $\mathfrak{g}$ and it is continuous for every $\sigma \in \mathbb{R}^{\mathfrak{n}}$. (For more details see [27]).

The solution of equation (4.1) is

$$\begin{aligned} \zeta (\sigma )=\mu +\frac{1}{\Gamma (\alpha )} \int _{0}^{\sigma}( \sigma -\varrho )^{\alpha -1}\bigl[\kappa \cdot \zeta (\varrho )+ \mathfrak{g}\bigl(\varrho , \zeta (\varrho )\bigr)\bigr]\,d\varrho . \end{aligned}$$

Let $\mathcal{X}=\{e\in \mathcal{C}(\mathcal{I}, \mathbb{R}) \colon e(\sigma )>0, \forall \sigma \in \mathcal{I}\}$ and $\mathcal{B}=\mathcal{M}_{2}(\mathbb{R})$. Define relation $\mathcal{R}$ on $\mathcal{X}$ as $(\zeta , \vartheta )\in \mathcal{R}$ iff $\zeta , \vartheta \geq 0$ and $\varpi (\zeta ,\vartheta )=\left[\begin{array}{cc}|\zeta -\vartheta & 0\\ 0& \upsilon |\zeta -\vartheta |\end{array}\right]$, where $\zeta , \vartheta \in \mathcal{R}$ and $\upsilon \geq 0$ is a constant. Then, $(\mathcal{X}, \mathcal{B}, \varpi , \mathcal{R})$ is a complete $\mathscr{C}^{*}$-AV$\mathcal{R}$-MS.

Theorem 4.1

Assume the nonlinear fractional-order initial value problem as given in (4.1). Suppose that the following condition is satisfied:

• (i) Consider that the solutions of the nonlinear fractional-order initial value problem (4.1) are

  $$\begin{aligned}& \zeta (\sigma )=\mu +\frac{1}{\Gamma (\alpha )} \int _{0}^{\sigma}( \sigma -\varrho )^{\alpha -1}\bigl[\kappa \cdot \zeta (\varrho )+ \mathfrak{g}_{1} \bigl(\varrho , \zeta (\varrho )\bigr)\bigr]\,d\varrho , \\& \zeta (\sigma )=\mu +\frac{1}{\Gamma (\alpha )} \int _{0}^{\sigma}( \sigma -\varrho )^{\alpha -1}\bigl[\kappa \cdot \zeta (\varrho )+ \mathfrak{g}_{2} \bigl(\varrho , \zeta (\varrho )\bigr)\bigr]\,d\varrho , \end{aligned}$$

  where $\mathfrak{g}_{1}$, $\mathfrak{g}_{2}$ are nonnegative real constants.

• (ii) There exist a constant $\mathcal{L}\in \mathbb{R}^{+}$ and $\kappa >0$ such that $|\mathfrak{g}(\sigma , e)-\mathfrak{g}(\sigma , l)| \leq \frac{\mathcal{L}}{\kappa}|e-l|$,

• (iii) There exists $0<\alpha \leq 1$ such that $\frac{\sigma ^{\alpha}}{\Gamma (\alpha )\mathcal{L}}<1$.

Then, the nonlinear fractional-order initial value value problem (4.1), has a unique common solution.

Proof

Define $\varXi , \Phi \colon \mathcal{X}\rightarrow \mathcal{X}$ by

$$\begin{aligned}& \varXi \zeta (\sigma )=\mu +\frac{1}{\Gamma (\alpha )} \int _{0}^{ \sigma}(\sigma -\varrho )^{\alpha -1}\bigl[\kappa \cdot \zeta (\varrho )+ \mathfrak{g}_{1} \bigl(\varrho , \zeta (\varrho )\bigr)\bigr]\,d\varrho , \\& \Phi \zeta (\sigma )=\mu +\frac{1}{\Gamma (\alpha )} \int _{0}^{\sigma}( \sigma -\varrho )^{\alpha -1}\bigl[\kappa \cdot \zeta (\varrho )+ \mathfrak{g}_{2} \bigl(\varrho , \zeta (\varrho )\bigr)\bigr]\,d\varrho . \end{aligned}$$

Clearly, Ξ and Φ are $\mathcal{R}$-preserving. For all $(\zeta , \vartheta )\in \mathcal{R}$, one has

$$\begin{aligned} &\varpi (\varXi \zeta , \varPhi \vartheta ) \\ &\quad = \begin{bmatrix} \vert \varXi \zeta -\varPhi \vartheta \vert &0 \\ 0 &\upsilon \vert \varXi \zeta -\varPhi \vartheta \vert \end{bmatrix} \\ &\quad = \begin{bmatrix} \vert \mu +\frac{1}{\Gamma (\alpha )}\int _{0}^{\sigma}(\sigma - \varrho )^{\alpha -1}[\kappa \cdot \zeta (\varrho ) \\ +\mathfrak{g}_{1}(\varrho , \zeta (\varrho ))]\,d\varrho \\ -\mu -\frac{1}{\Gamma (\alpha )}\int _{0}^{\sigma}(\sigma -\varrho )^{ \alpha -1}[\kappa \cdot \vartheta (\varrho ) \\ +\mathfrak{g}_{2}(\varrho , \vartheta (\varrho ))]\,d\varrho \vert &0 \\ 0 &\upsilon \vert \mu +\frac{1}{\Gamma (\alpha )}\int _{0}^{\sigma}( \sigma -\varrho )^{\alpha -1}[\kappa \cdot \zeta (\varrho ) \\ &+\mathfrak{g}_{1}(\varrho , \zeta (\varrho ))]\,d\varrho \\ &-\mu -\frac{1}{\Gamma (\alpha )}\int _{0}^{\sigma}(\sigma -\varrho )^{ \alpha -1}[\kappa \cdot \vartheta (\varrho ) \\ &+\mathfrak{g}_{2}(\varrho , \vartheta (\varrho ))]\,d\varrho \vert \end{bmatrix} \\ &\quad = \begin{bmatrix} \vert \frac{1}{\Gamma (\alpha )} [\int _{0}^{\sigma}(\sigma - \varrho )^{\alpha -1}[\kappa \cdot \zeta (\varrho )+\mathfrak{g}_{1}( \varrho , \zeta (\varrho ))]\,d\varrho \\ -\int _{0}^{\sigma}(\sigma -\varrho )^{\alpha -1}[\kappa \cdot \vartheta (\varrho )+\mathfrak{g}_{2}(\varrho , \vartheta (\varrho ))] \,d\varrho ] \vert &0 \\ 0 &\upsilon \vert \frac{1}{\Gamma (\alpha )} [\int _{0}^{\sigma}( \sigma -\varrho )^{\alpha -1}[\kappa \cdot \zeta (\varrho )+ \mathfrak{g}_{1}(\varrho , \zeta (\varrho ))]\,d\varrho \\ & -\int _{0}^{\sigma}(\sigma -\varrho )^{\alpha -1}[\kappa \cdot \vartheta (\varrho )+\mathfrak{g}_{2}(\varrho , \vartheta (\varrho ))] \,d\varrho ] \vert \end{bmatrix} \\ &\quad \leq \begin{bmatrix} \frac{1}{\Gamma (\alpha )}\mathcal{L} \Vert \zeta -\vartheta \Vert \vert \int _{0}^{ \sigma}(\sigma -\varrho )^{\alpha -1}\,d\varrho \vert &0 \\ 0 &\frac{\upsilon}{\Gamma (\alpha )}\mathcal{L} \Vert \zeta -\vartheta \Vert \vert \int _{0}^{\sigma}(\sigma -\varrho )^{\alpha -1}\,d\varrho \vert \end{bmatrix} \\ &\quad = \begin{bmatrix} \frac{1}{\Gamma (\alpha )}\mathcal{L} \Vert \zeta -\vartheta \Vert \frac{\sigma ^{\alpha}}{\varrho} &0 \\ 0 &\frac{\upsilon}{\Gamma (\alpha )}\mathcal{L} \Vert \zeta -\vartheta \Vert \frac{\sigma ^{\alpha}}{\varrho} \end{bmatrix} \\ &\quad \leq \biggl(\frac{\sigma ^{\varrho}}{\Gamma (\alpha )\varrho} \biggr) \mathcal{L} \begin{bmatrix} \Vert \zeta -\vartheta \Vert &0 \\ 0 &\upsilon \Vert \zeta -\vartheta \Vert \end{bmatrix}, \end{aligned}$$

which implies that

$$\begin{aligned} \varpi (\varXi \zeta , \varPhi \vartheta )\leq \mathcal{P}\varpi ( \zeta , \vartheta ), \quad \text{where } \mathcal{P}= \biggl( \frac{\sigma ^{\alpha}}{\Gamma (\alpha )\varrho} \biggr)\mathcal{L}< 1. \end{aligned}$$

Therefore, all the hypothesis of Theorem 3.1 are satisfied. Hence, Ξ and Φ have a unique common solution. □

5 Conclusion

In this paper, we proved some CFP theorems on $\mathscr{C}^{*}$-AV$\mathcal{R}$-MS. In addition, based on our obtained results an example was provided. Specifically, an application of a fractional-order initial value problem was presented.