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. (1)

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

  2. (2)

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

  3. (3)

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

  4. (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. 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. 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. 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

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (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

  1. (i)

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

  2. (ii)

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

  3. (iii)

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

  4. (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 ϖ(ζ,ϑ)=[ | ζ − ϑ | 2 0 0 υ | ζ − ϑ | 2 ], 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 ϖ(ζ,ϑ)=[ | ζ − ϑ 0 0 υ | ζ − ϑ | ], 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:

  1. (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.

  2. (ii)

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

  3. (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.