New inequalities via Caputo-Fabrizio integral operator with applications

1.   Introduction and preliminaries

The most papular Banach contraction mapping principle (BCMP) ^[1] is the largest powerful fundamental fixed point result. This principle has a lot of applications in pure and applied mathematics (see ^[2,3,4]). In the past few decades, many authors extended and generalized the (BCMP) in several ways (see ^{[5,6,7,8,9,10]}). Ran and Reurings ^[11] obtained positive definite solutions of matrix equations using the aid of the Banach contraction principle in partially ordered sets. Nieto and Rodriguez-Lopez ^[12] also used partially ordered spaces and fixed point theorems to find solutions of some differential equations. Very recently, Wardowski ^[13] furnished the idea of an $F$-contraction, which is an extension of the (BCMP). Furthermore, common fixed point theorems for rational $F_{R}$ -contractive pairs of mappings with applications are announced in ^[14] as an extension of $F$-contractions in relation theoretic metric spaces. On the other hand, Matthews ^[15] introduced the notion of a partial metric space as a part of the study of semantics of dataflow network, and for more results in this direction see (^{[16,17,18,19,20]}). One of the latest extensions of a metric space and a partial metric space is initiated through the concept of a $m$-metric space ^[21], and some researchers work in this direction (see more ^{[22,23,24,25,26,27,28,29,30]}). In our article, we utilize two last notions to give an interesting type of generalized $F_{R}^{m}$-contractions in the frame of relation theoretic $m$-metric spaces and to prove some fixed point results.

Generally saying that, we generalize and extend some recent results in ^[31]$.$ We also extend the earlier mentioned results in the setting of relation theoretic $m$-metric spaces, that contain only the last two conditions imposed on the Wardowski function $F$ in the first section. Furthermore, the consequences of our main results improve and generalize some corresponding theorems appearing in the literature.

Our article consists four sections. In the first section, we recall some fundamental definitions and theorems concerning $m$-metric spaces and different types of $F$-contractions. In the second section, we define the notion of generalized $F_{R}^{m}$-contractions of rational type and generalized $F_{R}^{m}$-contractions of cyclic type. In the third section, we use the whole Wardowski function in the setting of $F_{R}^{m}$-contractions of rational type as consequences of main results in section Ⅱ. Using these ideas, we prove some new fixed point results in the frame of relation theoretic $m$-metric spaces and we present some examples to show that our obtained results are meaningful. In section Ⅳ, we present an application and we ensure the existence of a solution of a class of nonlinear matrix equations.

Throughout this article, $\mathbb{N}$ indicates a set of all natural numbers, $\mathbb{R}$ indicates set of real numbers and $\mathbb{R} ^{+}$ indicates set of positive real numbers, respectively. We also denote $\mathbb{N} _{0} = \mathbb{N} \cup \left\{ 0\right\}.$ Henceforth, $U$ will denote a non-empty set. Given a self mapping $\gamma :U\rightarrow U$. A Picard sequence based on an arbitrary $\zeta _{0}$ in $U$ is given by $\zeta _{\mu } = \gamma \left(\zeta _{\mu -1}\right) = \gamma ^{\mu }\left(\zeta _{0}\right)$ for all $\mu$ in $\mathbb{N},$ where $\gamma ^{\mu }$ denotes the $\mu ^{th}$-iteration of $\gamma.$

In 2013, the notion of a $m$-metric space was introduced by Asadi et al. ^[21]. They also extended the well known Banach contraction fixed point theorem from partial metric spaces to $m$-metric spaces. We start recalling some definitions and properties:

Definition 1.1. ^[21] Let $U\neq \emptyset$. The function $m:U\times U\rightarrow \mathbb{R} ^{+}$ is a $m$-metric on the set $U$ if for all $\zeta, \Im, \aleph \in U,$

$(i)$ $\zeta = \Im \Longleftrightarrow m\left(\zeta, \zeta \right) = m\left(\Im, \Im \right) = m\left(\zeta, \Im \right) \left(T_{0}{\text{ -separation axiom}}\right);$

$(ii)$ $m_{\zeta \Im }\leq m(\zeta, \Im)$ $\left({\text{minimum self distance axiom}}\right);$

$(iii)$ $m\left(\zeta, \Im \right) = m\left(\Im, \zeta \right)$ $\left({\text{symmetry}}\right);$

$(iv)$ $m\left(\zeta, \Im \right) -m_{\zeta \Im }\leq (m\left(\zeta, \aleph \right) -m_{\zeta \aleph })+(m\left(\aleph, \Im \right) -m_{\aleph \Im })$ $\left({\text{modified triangle inequality}}\right),$

where

Here, the pair $\left(U, m\right)$ is called a $m$-metric space.

On among the classical examples of $m$-metric spaces is the pair $\left(\zeta, m\right)$ where $U = \left\{ \zeta, \Im, \aleph \right\}$ and $m\left(\zeta, \zeta \right) = 1$, $m\left(\Im, \Im \right) = 9,$ $m\left(\aleph, \aleph \right) = 5.$ Other examples of $m$-metric spaces may be found, for instance in ^[21]. Clearly, each partial metric is a $m$ -metric space, but the converse does not hold (see ^[32,33,34]).

Every $m$-metric $m$ on $U$ generates a $T_{0}$ topology $\tau _{m}\left({\text{say}}\right)$ on $U$ which has a base of collection of $m$-open balls

where

If $m$ is a $m$-metric space on $U$, then the functions $m^{w},$ $m^{s}:U\times U\rightarrow \mathbb{R} ^{+}$ given by:

define ordinary metrics on $U$. It is easy to see that $m^{w}$ and $m^{s}$ are equivalent metrics on $U.$

Definition 1.2. According to ^[21],

$\left(i\right)$ a sequence $\{\zeta _{\mu }\}$ in a $m$-metric space $\left(U, m\right)$ converges with respect to $\tau _{m}$ to $\zeta$ if and only if

$(ii)$ a sequence $\{\zeta _{\mu }\}$ in a $m$-metric space $\left(U, m\right)$ is called $m$-Cauchy if $\lim_{\mu, \nu \rightarrow \infty }\left(m\left(\zeta _{\mu }, \zeta _{\nu }\right) -m_{\zeta _{\mu }\zeta _{\nu }}\right)$ and $\lim_{\mu, \nu \rightarrow \infty }\left(M_{\zeta _{\mu }, \zeta _{\nu }}-m_{\zeta _{\mu }\zeta _{\nu }}\right)$ exist and are finite,

$(iii)$ $\left(U, m\right)$ is said to be complete if every $m$ -Cauchy sequence $\left\{ \zeta _{\mu }\right\}$ in $U$ is $m$-convergent to $\zeta$ with respect to $\tau _{m}$ in $U$ such that

$\left(iv\right)$ $\{\zeta _{\mu }\}$ is a Cauchy sequence in $\left(U, m\right)$ if and only if it is a Cauchy sequence in the metric space $\left(U, m^{w}\right),$

$\left(v\right)$ $\left(U, m\right)$ is complete if and only if $\left(U, m^{w}\right)$ is complete.

Consider a function $F:\left(0, \infty \right) \rightarrow R$ so that:

$\left(F_{1}\right)$ $F\left(\zeta \right) < F\left(\Im \right)$ for all $\zeta < \Im,$

$\left(F_{2}\right)$ for each sequence $\left\{ \varpi _{\mu }\right\} \subseteq \left(0, \infty \right)$, $\lim_{\mu \rightarrow \infty }\varpi _{\mu } = 0$ iff $\lim_{\mu \rightarrow \infty }F\left(\varpi _{\mu }\right) = -\infty,$

$\left(F_{3}\right)$ there exists $p\in \left(0, 1\right)$ such that $\lim_{\varpi _{\mu } \rightarrow 0^{+}}\varpi ^{p}F\left(\varpi \right) = 0.$

According to ^[13], denote by $\nabla \left(F\right)$ the collection of functions $F:\left(0, \infty \right) \rightarrow R$ satisfying $\left(F_{2}\right)$ and $\left(F_{3}\right)$. Take also

Example 1.1. ^[13] The following below functions belong to $\Pi \left(F\right)$:

$(1)$ $F\left(s\right) = \ln s$,

$(2)$ $F\left(s\right) = s+\ln s$,

$(3)$ $F\left(s\right) = \ln \left(s^{2}+s\right)$,

$(4)$ $F\left(s\right) = -\frac{1}{\sqrt{s}}$,

for all $s > 0.$

Example 1.2. The following functions are not strictly increasing and belong to $\nabla \left(F\right) :$

$(1)$ $F\left(s\right) = 100\ln \left(\frac{s}{2}+\sin s\right)$,

$(2)$ $F\left(s\right) = \sin s+\ln s$,

$(3)$ $F\left(s\right) = \sin s-\frac{1}{\sqrt{s}}$,

for all $s > 0$.

Let $\gamma$ be a self-mapping on a $mm$-space $U$. The following are some valuable notations that are useful for the rest.

$\left(i\right)$ $\left(\gamma \right) _{Fix}$ is the set of all fixed points of $\gamma$,

$\left(ii\right)$ $\Theta \left(\Psi, {S}\right) = \left\{ \zeta \in U:\left(\zeta, \gamma \left(\zeta \right) \right) \in {R}\right\},$

$\left(iii\right)$ $\digamma \left(\zeta, \Im, \nabla \right)$ is thefashion of all paths in $\nabla$ from $\zeta$ to $\Im.$

Altun et al. ^[35] gave two fixed point results for multivalued $F$ -contractions on $mm$-spaces. We ensure the existence of fixed point results for generalized $F_{R}^{m}$-contractions by using the concept given in ^[35] to the metric space setup. The motivation of this study is to solve nonlinear matrix equations. First, inspired by Altun et al. ^[31] and Wardowski ^[13], we give the following concepts.

Theoretic relations have been used in many research articles, for examples see ^[36]. A non-empty subset $R$ of $U^{2}$ is said to be a relation on the $m$-metric space $\left(U, m\right)$ if ${R = }\left\{ \left(\zeta, \Im \right) \in U^{2}:\zeta, \Im \in U\right\}.$ If $\left(\zeta, \Im \right) \in R$, then we say that $\zeta \preceq \Im$ $\left(\zeta {\text{ precede }}\Im \right)$ under $R$ denoted by $\left(\zeta, \Im \right) \in { R, }$ and the inverse of $R$ is denoted as $R^{-1} = \left\{ \left(\zeta, \Im \right) \in U^{2}:\left(\Im, \zeta \right) \in R\right\}.$ Set ${S = R\cup R} ^{-1}\subseteq U^{2}$. Consequently, we illustrate another relation on $U$ denoted $S^{\ast }$ and is given as $\left(\zeta, \Im \right) \in S^{\ast }\Leftrightarrow \left(\Im, \zeta \right) \in S$ and $\Omega \neq \Im.$

Definition 1.3. ^[36] Let $U\neq \emptyset$ and $R$ be a binary relation on $U$. Then $R$ is transitive if $\left(\zeta, \xi \right) \in R$ and $\left(\xi, \Im \right) \in R$ $\Rightarrow$ $\left(\zeta, \Im \right) \in R$, for all $\zeta, \Im, \xi \in U$.

Definition 1.4. ^[36] Let $U\neq \emptyset$. A sequence $\zeta _{\mu }\in U$ is called $R$-preserving, if $\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R.$

Definition 1.5. ^[36] Let $U\neq \emptyset$ and $\gamma :U\rightarrow U$. A binary relation $R$ on $U$ is called $\gamma$-closed if for any $\zeta, \Im$ in $U$, we deduce $\left(\zeta, \Im \right) \in R\Rightarrow \left(\gamma \left(\zeta \right), \gamma \left(\Im \right) \right) \in R.$

2.   Main results

We begin with the following definitions.

Definition 2.1. We say that $(U, m, R)$ is regular if for each sequence $\left\{ \zeta _{\mu }\right\}$ in $U$,

Definition 2.2. A relation theoretic $m$-metric space $\left(U, m, R\right)$ is said to be $R$-complete if for an $R$-preserving $m$-Cauchy sequence $\left\{ \zeta _{\mu }\right\}$ in $U$, there exists some $\zeta$ in $U$ such that

Definition 2.3. Let $\left(U, m\right)$ be a $m$-metric space endowed with a binary relation $R$ on $U$ and $\gamma$ be a self-mapping on $U$. Then, $\gamma$ is said to be a $F_{R}^{m}$-contractions, if there exist $F_{R}^{m}\in \Pi \left(\nabla \right)$ and $\xi > 0,$ such that

for all $\zeta, \Im \in U$ with $\left(\zeta, \Im \right) \in S^{\ast }.$

Now, we introduce the concept of a generalized rational type $F_{R}^{m}$-contraction.

Definition 2.4. Let $\left(U, m\right)$ be a $m$-metric space endowed with a binary relation $R$ on $U$. Let $\gamma :U\rightarrow U$ be a self-mapping on $U$. It is called a generalized rational type $F_{R}^{m}$-contraction if there are $F_{R}^{m}\in \nabla \left(F\right)$ and $\xi > 0$ such that

for all $\zeta, \Im \in U$ with $\left(\zeta, \Im \right) \in S^{\ast }.$

Theorem 2.1. Let $\left(U, m\right)$ be a complete $m$-metric space with a binary relation $R$ on $U$ and $\gamma$ be a self-mapping on $U$ such that:

$\left(i\right)$ the class $\Theta \left(\gamma, R\right)$ is nonempty;

$\left(ii\right)$ $R$ is $\gamma$-closed;

$\left(iii\right)$ the mapping $\gamma$ is $R$-continuous;

$\left(iv\right)$ $\gamma$ is a generalized rational type $F_{R}^{m}$-contraction mapping.

Then $\gamma$ possesses a fixed point in $U$.

Proof. Let $\zeta _{0}\in \Theta \left(\left[ \gamma, R\right] \right)$. We define a sequence $\left\{ \zeta _{\mu}\right\}$ by $\zeta _{\mu +1} = \gamma \left(\zeta _{\mu }\right) = \gamma ^{\mu }\left(\zeta _{0}\right)$ for each $\mu \in \mathbb{N}.$ If there is $\mu _{0}$ in $\mathbb{N}$ so that $\gamma \left(\zeta _{\mu _{0}}\right) = \zeta _{\mu _{0}}$, then $\gamma$ has a fixed point $\zeta _{\mu _{0}}$ and the proof is complete. Let $\zeta _{\mu +1}\neq \zeta _{\mu }$ for all $\mu$ in $\mathbb{N}$, so $m\left(\zeta _{\mu +1}, \zeta _{\mu }\right) > 0.$ Since $\left(\gamma \left(\zeta _{0}\right), \zeta _{0}\right) \in S^{\ast }$, using $\gamma$-closedness of $R$, we get $\left(\gamma \left(\zeta _{\mu +1}\right), \zeta _{\mu}\right) \in S^{\ast }$. Then using the fact that $\gamma$ is a generalized rational type $F_{R}^{m}$-contraction mapping, one writes

If $\max \left\{ m\left(\zeta _{\mu }, \zeta _{\mu -1}\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu }, \zeta _{\mu +1}\right),$ then from $\left(2.3\right),$ we have

which is a contradiction. Thus, $\max \left\{ m\left(\zeta _{\mu }, \zeta _{\mu -1}\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu }, \zeta _{\mu -1}\right)$ and so from $\left(2.3\right),$ we have

Denote $\delta _{\mu } = m\left(\zeta _{\mu }, \zeta _{\mu +1}\right).$ We have $\delta _{\mu } > 0$ for all $\mu \in \mathbb{N}$ and using $\left(2.4\right)$ we deduce that

It implies that $\lim_{\mu \rightarrow \infty }F_{R}^{m}\left(\delta _{\mu }\right) = -\infty,$ then by $\left(F_{2}\right),$ we have $\lim_{\mu \rightarrow \infty }\delta _{\mu } = 0.$ Due to $\left(F_{3}\right)$, there exists $k\in \left(0, 1\right)$ such that $\lim_{\mu \rightarrow \infty }\delta _{\mu }^{k}F_{R}^{m}\left(\delta _{\mu }\right) = 0.$

From $\left(2.4\right)$ the following is true for all $\mu \in \mathbb{N},$

Letting $\mu \rightarrow \infty$ in $\left(2.6\right)$, we get

From $\left(2.7\right)$, there exists $\mu _{1}\in \mathbb{N}$ so that $\mu \delta _{n}^{k}\leq 1$ for all $\mu \geq \mu _{1},$ then we deduce

We claim that $\left\{ \zeta _{\mu }\right\}$ is a $m$-Cauchy sequence in the $m$-metric space. Let $\nu, \mu \in \mathbb{N}$ such that $\nu > \mu \geq \mu _{1}.$ Using the triangle inequality of a $m$-metric space, one writes

The convergence of the series $\sum_{i = \mu }^{\infty }\frac{1}{i^{\frac{1}{k} }}$ yields that $m\left(\zeta _{\mu }, \zeta _{\nu }\right) -m_{\zeta _{\mu }, \zeta _{\nu }}\rightarrow 0.$ Thus, $\left\{ \zeta _{\mu }\right\}$ is a $M$-Cauchy sequence in $\left(U, m\right)$. Since $\left(U, m, R\right)$ is $R$-complete, there exists $\zeta \in U$ such that $\left\{ \zeta _{\mu }\right\}$ converges to $\zeta$ with respect to $t_{\kappa }$, that is, $m\left(\zeta _{\mu }, \zeta \right) -m_{\zeta _{\mu }, \zeta }\rightarrow 0$ as $\mu \rightarrow \infty.$ Now, the $R$-continuity of $\gamma$ implies that

Hence, $\zeta$ is a fixed point of $\gamma.$

Example 2.1. Let $U = \left[ 0, \infty \right)$ and $m$ be defined by $m\left(\zeta, \Im \right) = \min \left\{ \zeta, \Im \right\}$ for all $\zeta, \Im \in U$. $\left(U, m\right)$ is a complete $m$-metric space. Consider the sequence $\left\{ z_{\mu }\right\} \subseteq U$ given by $z_{\mu } = \frac{\mu \left(\mu +1\right) \left(2\mu +1\right) }{6}$ for all $\mu \geq 2.$ Set an binary relation on $U$ denoted by $R$ given by $R = \left\{ \left(z_{1}, z_{1}\right), \left(z_{\mu -1}, z_{\mu }\right) :\mu = 2, 3, ...100\right\}.$ Now, give $\gamma :U\rightarrow U$ as

Obviously, $R$ is $\gamma$-closed and $\gamma$ is continuous. Choosing $\zeta = z_{\mu }$ and $\Im = z_{\mu +1}$ (for $\mu = 1, 2, 3, \cdots, 100)$, for first condition of $F$ (which is $(F_{1})$), we have

and

Now, for $\mu = 2, 3, 4, \cdots, 100,$ we have

implies that

Let

In view of Table 1 and Figure 1, since the function $\left\{ f\left(\mu \right) \right\} _{\mu \geq 2}$ is decreasing and discontinuous, the smallest value in $\left(2.11\right)$ is $5.02.$ Therefore, the Eq $\left(2.10\right)$ holds for $0 < \xi < 5.$ So

for all $\zeta, \Im \in U$ such that $\left(\zeta, \Im \right) \in {\bf{S }}^{\ast }$ with $mm$-space. Hence, $\gamma$ is a generalized rational type $F_{R}^{m}$-contraction mapping with $0 < \xi < 5.$ Generally, we can say that $\gamma$ has infinite $\left(F.Ps\right)$.

Theorem 2.2. Theorem $2.1$ remains true if the condition $\left(ii\right)$ is replaced by the following:

$\left(i\right)$ $\left(ii\right) ^{^{\prime }},$

$\left(ii\right)$ $\left(X, \kappa, \nabla \right)$ is regular.

Proof. It is a same argument as Theorem $2.1$. Here, the sequence $\left\{ \zeta _{\mu }\right\}$ is $m$-Cauchy and converges to some $\Omega$ in $U$ such that $m\left(\zeta _{\mu }, \zeta \right) -m_{\zeta _{\mu }, \zeta }$ as limit $\mu \rightarrow \infty$ which implies that

As $\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in {R, }$ then $\left(\zeta _{\mu }, \zeta \right) \in R$ for all $\mu \in \mathbb{N}$. Set $\mu = \left\{ \mu \in N:\gamma \left(\zeta _{\mu }\right) = \gamma \left(\zeta \right) \right\}.$ We will take two cases depending on $\mu$

C-1. If $\mu$ is a finite set, then there exists $\mu _{0}$ in $\mathbb{N},$ so that $\gamma \left(\zeta _{\mu }\right) \neq \gamma \left(\zeta \right)$ for every $\mu \geq \mu _{0}.$ In particular, $\left(\zeta _{\mu }, \zeta \right) \in S^{\ast }$ and $\left(\gamma \left(\zeta _{\mu }\right), \gamma \left(\zeta \right) \right) \in S^{\ast },$ then for all $\mu \geq \mu _{0}$,

Since $\lim_{\mu \rightarrow \infty }m\left(\zeta _{\mu }, \zeta \right) = 0$ implies that $\lim_{\mu \rightarrow \infty }F_{R}^{m}\left(m\left(\zeta _{\mu }, \zeta \right) \right) = -\infty,$ one writes $\lim_{\mu \rightarrow \infty }F_{R}^{m}\left(m\left(\gamma \left(\zeta _{\mu }\right), \gamma \left(\zeta \right) \right) \right) = -\infty.$ Therefore, $\lim_{\mu \rightarrow \infty }m\left(\gamma \left(\zeta _{\mu }\right), \gamma \left(\zeta \right) \right) = 0,$ which yields that $\gamma \left(\zeta \right) = \zeta$, that is, $\zeta$ is a fixed point of $\gamma.$

C-2. If $\mu$ is an infinite set, then there exists a subsequence $\left\{ \zeta _{\mu _{k}}\right\}$ of $\left\{ \zeta _{\mu }\right\}$ so that $\zeta _{\mu _{k}+1} = \gamma \left(\zeta _{\mu _{k}}\right) = \gamma \left(\zeta \right)$ for $k\in \mathbb{N},$ so $\gamma \left(\zeta _{\mu _{k}}\right) \rightarrow \gamma \left(\zeta \right)$ with respect to $t_{m}$ as $\zeta _{\mu }$ converges $\zeta,$ then $\gamma \left(\zeta \right) = \zeta$, i.e., $\gamma$ has a fixed point. Hence, the proof is complete.

Now, we prove a result of uniqueness.

Theorem 2.3. Following Theorems 2.1 and 2.2 $\gamma$ possesses a unique fixed point if $\digamma \left(\zeta, \Im, \nabla \right) \neq \emptyset,$ for all $\zeta, \Im \in \left(\gamma \right) _{Fix}.$

Proof. Let $\zeta, \Im \in \left(\gamma \right) _{Fix}$ such that $\zeta \neq \Im.$ Since $\digamma \left(\zeta, \Im, \nabla \right) \neq \emptyset,$ there exists a path $\left(\left\{ a_{0}, a_{1}, ...a_{\mu }\right\} \right)$ of some finite length $\mu$ in $\nabla$ from $\Omega$ to $\Im$ $\left({\text{with }}a_{s}\neq a_{s+1}{\text{ for all }}s\in \left[ 0, p-1\right] \right)$. Then $a_{0} = \zeta$, $a_{k} = \Im,$ $\left(a_{s}, a_{s+1}\right) \in S^{\ast }$ for every $s\in \left[ 0, p-1\right].$ As $a_{s}\in \gamma \left(U\right),$ $\gamma \left(a_{s}\right) = a_{s}$ for all $s\in \left[ 0, p-1\right]$ we deduce that

It is a contradiction. Hence, $\gamma$ possesses a unique fixed point.

Now, we say that $\gamma :U\rightarrow U$ has the property $P$ if

In this theorem, we use above condition having property $P$.

Theorem 2.4. Let $\left(U, m\right)$ be a complete $m$-metric space with a binary relation $R$ on $U$ and $\gamma$ be a self-mapping such that:

$\left(i\right)$ the class $\Theta \left(\gamma, R\right)$ is nonempty,

$\left(ii\right)$ the binary relation $R$ is $\gamma$-closed,

$\left(iii\right)$ $\gamma$ is $R$-continuous,

$\left(iv\right)$ there are $F_{R}^{m}\in \nabla \left(F\right)$ and $\xi > 0$ so that

for all $\zeta \in U,$ with $\left(\gamma \left(\zeta \right), \gamma ^{2}\left(\zeta \right) \right) \in S^{\ast }.$

Then $\gamma$ has a fixed point. Furthermore, if

$\left(v\right)$ $\left(iv\right) ^{^{\prime }}$;

$\left(vi\right)$ $\zeta \in \left(\gamma ^{\mu }\right) _{Fix}$ $\left(for \;some\;\mu \in \mathbb{N} \right)$ which implies that $\left(\zeta, \gamma \left(\zeta \right) \right) \in R,$

then $\gamma$ has a property $P.$

Proof. Let $\zeta _{0}\in \Theta \left(\left[ \gamma, R\right] \right)$, i.e., $\left(\zeta _{0}, \gamma \left(\zeta _{0}\right) \right) \in {\bf{ R}}$ $,$therefore using assumption $\left(ii\right)$, we get $\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R$ for each $\mu \in \mathbb{N}$. Denote $\zeta _{\mu +1} = \gamma \left(\zeta _{\mu }\right) =$ $\gamma ^{\mu +1}\left(\zeta _{0}\right),$ for all $\mu \in \mathbb{N}.$ If there exists $\mu _{0}\in \mathbb{N}$ so that $\gamma \left(\zeta _{\mu _{0}}\right) = \zeta _{\mu _{0}},$ then $\gamma$ has a fixed point $\zeta _{\mu _{0}}$ and it completes the proof. Otherwise, assume that $\zeta _{\mu +1}\neq \zeta _{\mu }$ for every $\mu \in \mathbb{N}$. Then $\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R$ $\left({\rm{for \;all}}\;\mu \in \mathbb{N} \right)$. Continuing this process and using the assumption $\left(iv\right)$, we deduce $\left({\rm{for \;all}}\;\mu \in \mathbb{N} \right)$

Assume that $\max \left\{ m\left(\zeta _{\mu -1}, \zeta _{\mu }\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu }, \zeta _{\mu +1}\right),$ then we get

It is a contradiction. Hence, $\max \left\{ m\left(\zeta _{\mu -1}, \zeta _{\mu }\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu -1}, \zeta _{\mu }\right),$ and so

This yields that $\left({\rm{ for\; all}}\;\mu \in \mathbb{N} \right)$

By applying limit as $\mu$ goes to $\infty$ in above equation, we deduce $\lim_{\mu \rightarrow \infty }F_{R}^{m}\left(m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right) = -\infty.$ Since $F_{R}^{m}\in \nabla \left(F\right),$ we deduce that $\lim_{\mu \rightarrow \infty }m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) = 0.$ Using $(F_{3})$, there is $k\in \left(0, 1\right)$ so that

Now, from $\left(2.9\right)$, we have

Letting $\mu \rightarrow \infty$ in $\left(2.10\right)$, we get $\lim_{\mu \rightarrow \infty }\left(m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right) ^{k} = 0.$ There is $\mu _{1}$ in $\mathbb{N}$ so that

That is,

Now, for $\nu > \mu > \mu _{1}$, we have

Since the series $\sum_{i = \mu }^{\nu -1}\frac{1}{i^{\frac{1}{k}}}$ is convergent, i.e., $m\left(\zeta _{\mu }, \zeta _{\nu }\right) -m_{\zeta _{\mu }, \zeta _{\nu }}$ converges to $0$, the sequence $\left\{ \zeta _{\mu }\right\}$ is a $m$-Cauchy sequence. Since $\left(U, m, R\right)$ is $R$-complete and $\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R$ for all $\mu \in \mathbb{N},$ $\left\{ \zeta _{\mu }\right\}$ converges to $\zeta \in U.$ Now, using the $R$-continuity of $\gamma$, we deduce that

Finally, we will prove that $\left(\gamma ^{\mu }\right) _{Fix} = \left(\gamma \right) _{Fix}$ where $\mu \in \mathbb{N}.$ Assume on contrary that $\zeta \in \left(\gamma ^{\mu }\right) _{Fix}$ and $\zeta \notin \left(\gamma \right) _{Fix}$ for some $\mu \in \mathbb{N}$. Then $m\left(\zeta, \gamma \left(\zeta \right) \right) > 0$, $\left(\zeta, \gamma \left(\zeta \right) \right) \in R\left({\rm{from \;condition}}\;\left(iv\right) ^{^{\prime }}\right).$ From assumption $\left(ii\right)$ we obtain $\left(\gamma ^{\mu }\left(\zeta \right), \gamma ^{\mu +1}\left(\zeta \right) \right) \in R$ for all $\mu \in \mathbb{N}$. Assumption $\left(iv\right)$ implies that

Taking $\mu \rightarrow \infty$ in above inequality, we obtain $F_{R}^{m}\left(m\left(\zeta, \gamma \left(\zeta \right) \right) \right) = -\infty,$ a contradiction. So, $\left(\gamma ^{\mu }\right) _{Fix} = \left(\gamma \right) _{Fix}$ for any $\mu \in \mathbb{N}.$

Corollary 2.1. Let $\left(U, m\right)$ be a complete $m$-metric space with a binary relation $R$ on $U$ and $\gamma$ be a self-mapping such that:

$\left(i\right)$ the class $\Theta \left(\gamma, R\right)$ is nonempty;

$\left(ii\right)$ the binary relation $R$ is $\gamma$-closed;

$\left(iii\right)$ $\gamma$ is $R$-continuous;

$\left(iv\right)$ $\gamma$ is a $F_{R}^{m}$-contraction mapping.

Then $\gamma$ possesses a fixed point in $U$.

Here, we use the definition of $F$-contractions with the standard conditions $\left(i-iii\right).$

Definition 2.5. Given a $mm$-space $\left(U, m\right)$ and a binary relation $R$ on $U.$ Suppose that

We say that a self-mapping $\gamma :U\rightarrow U$ is a rational type $F_{R}^{m}$-contraction if there exists $F_{R}^{m}\in \Pi \left(\nabla \right)$ such that

for all $\left(\zeta, \Im \right) \in \Xi.$

Theorem 2.5. Let $\left(U, m\right)$ be a complete $m$-metric space, $R$ be a binary relation on $U$ and $\gamma$ be a self-mapping on $U.$ Assume that:

$\left(i\right)$ the class $\Theta \left(\gamma, R\right)$ is non-empty;

$\left(ii\right)$ the binary relation $R$ is $\gamma$-closed;

$\left(iii\right)$ $\gamma$ is $R$-continuous;

$\left(iv\right)$ $\gamma$ is a rational type $F_{R}^{m}$ -contraction mapping.

Then $\gamma$ possesses a fixed point in $U$.

Proof. Let $\zeta _{0}\in \Theta \left(\left[ \gamma, R\right] \right)$, i.e., $\left(\left[ \zeta _{0}, \gamma \left(\zeta _{0}\right) \right] \right) \in R$ $\ $. We define a sequence $\left\{ \zeta _{\mu +1}\right\}$ given as $\zeta _{\mu +1} = \gamma \left(\zeta _{\mu }\right) =$ $\gamma ^{\mu +1}\left(\zeta _{0}\right)$. We have $\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R$ for all $\mu$ in $\mathbb{N}$. If there exists $\mu _{0}\ $ in $\mathbb{N}$ such that $\gamma \left(\zeta _{\mu _{0}}\right) = \zeta _{\mu _{0}},$ then $\zeta _{\mu _{0}}$ is a fixed point of $\gamma$ and the proof is finished. Now, assume that $\zeta _{\mu +1}\neq \zeta _{\mu }$ for all $\mu \in \mathbb{N}$. Then $\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \in R\left({\text{for all }}\mu \in \mathbb{N} \right)$. Using the condition $\left(iv\right)$, we deduce $\left({\text{for all }}\mu \in \mathbb{N} \right)$

By $(F_{1}),$ we have $\max \left\{ m\left(\zeta _{\mu -1}, \zeta _{\mu }\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu }, \zeta _{\mu +1}\right),$ then we get a contradiction. Thus, $\max \left\{ m\left(\zeta _{\mu }, \zeta _{\mu -1}\right), m\left(\zeta _{\mu }, \zeta _{\mu +1}\right) \right\} = m\left(\zeta _{\mu }, \zeta _{\mu -1}\right)$ and so from $\left(2.3\right)$ we have

The proof of Theorem 2.1 is complete.

Corollary 2.2. Let $\left(U, m\right)$ be a complete $m$-metric space, $R$ be a binary relation on $U$ and $\gamma$ be a self-mapping on $U.$ Assume that:

$\left(i\right)$ the class $\Theta \left(\gamma, R\right)$ is non-empty;

$\left(ii\right)$ $R$ is $\gamma$-closed;

$\left(iii\right)$ $\gamma$ is $R$-continuous;

$\left(iv\right)$ $\gamma$ is a $F_{R}^{m}$-contraction mapping.

Then $\gamma$ possesses a fixed point in $U$.

Example 2.2. Let $U = \left[ 0, 1\right]$ and $m$ be a relation theoretic $m$-metric defined by

We define the binary relation

$\left(U, m\right)$ is a complete $m$-metric space with a binary relation. Define a mapping $\gamma :U\rightarrow U$ by

Obviously, $\Re$ is $\gamma$-closed, also and $\gamma$ is $\Re$ -continuous. Define $F_{\Re }^{m}:\left(0, \infty \right) \rightarrow R$ by

Assume that $\left(\zeta, \Im \right) \in \Xi = \left\{ \left(\zeta, \Im \right) \in S^{\ast }:m\left(\gamma \left(\zeta \right), \gamma \left(\Im \right) \right) > 0\right\}.$ Therefore, for all $\zeta, \Im \in U,$ with $0 < \zeta < 1, \Im = 1,$ we have

Now, consider $Z_{A} = \max \left\{ \begin{array}{c} m\left(\zeta, 1\right), m\left(\zeta, \gamma \left(\zeta \right) \right), m\left(\Im, \gamma \left(\Im \right) \right), \\ \frac{m\left(\zeta, \gamma \left(\zeta \right) \right) \left[ 1+m\left(\Im, \gamma \left(\Im \right) \right) \right] }{1+m\left(\Omega, 1\right) } \end{array} \right\}$

From Table Table.2, $\gamma$ is a rational type $F_{R}^{m}$ -contraction mapping with $\xi = 2.$ Moreover, there is $\zeta _{0} = 0.1$ in $U$ so that$\left(\xi _{0, }\gamma \left(\xi _{0}\right) \right) \in S^{\ast }$ and the class $\Theta \left(\gamma, {R} \right)$ is nonempty. Hence, all conditions of Theorem 2.5 hold, and therefore $\gamma$ has a fixed point.

3.   Some fixed point results for cyclic contractions

In ^[37], Kirk et al. gave the concept of a cyclic contraction, which is the extension of the Banach contraction. It is utilized in the following theorem.

Theorem 3.1. Suppose that $\left(U, m\right)$ is a compete $m$-metric space, $G$, $H$ are two nonempty closed subsets of $U\ $ and $\gamma :U\rightarrow U$ verifies the following conditions:

$\left(i\right)$ $\gamma \left(B\right) \subseteq D$ and $\gamma \left(D\right) \subseteq B;$

$\left(ii\right)$ there exists a constant $k\in \left(0, 1\right)$ such that

Then $B\cap D$ is nonempty and there is $\zeta \in B\cap D$ a fixed point of $\gamma$.

By Theorems 2.1 and 3.1, we obtain successive fixed point results for cyclic rational type $F_{R}^{m}$- generalized contraction mappings.

Theorem 3.2. Let $\left(U, m\right)$ be a complete $m$-metric space, $G$ and $H$ be two nonempty closed subsets of $U$ and $\gamma :U\rightarrow U$ be an operator. Assume that the successive axioms hold:

$\left(i\right)$ $\gamma \left(G\right) \subseteq H$ and $\gamma \left(H\right) \subseteq G;$

$\left(ii\right)$ there exist $F_{R}^{m}\in \nabla \left(F\right)$ and $\xi > 0$ such that

for all $\zeta$ in $G$ and $\Im$ in $H$.

Then there is $\zeta \in G\cap H$ a fixed point of $\gamma$.

Proof. $Z = G\cup H$ is closed, so $Z$ is a closed subspace of $U$. Therefore, $\left(U, m\right)$ is a complete $m$-metric space. Set a binary relation on $Z$ denoted by $R$ given as

It means that

Set ${S = R\cup R}^{-1}$ an asymmetric relation. Directly, $\left(U, m, S\right)$ is regular. Assume $\left\{ \zeta _{\mu }\right\} \in Z$ is any sequence and $\zeta \in Z$ so that

and

Using the definition of ${\bf{S, }}$ we obtain

Immediately, the product fashion $Z\times Z$ involves a $mm$-space $m$ given as

Since $\left(U, m\right)$ is a complete $m$-metric space, we obtain $\left(Z\times Z, m\right)$ is complete. Furthermore, $G\times H$ and $H\times G$ are closed in $\left(Z\times Z, m\right),$ because $G$ and $H$ are closed in $\left(U, m\right).$ Letting $\mu \rightarrow \infty$ in $\left(2.11\right)$, we have $\left(\zeta, \zeta \right) \in \left(B\times D\right) \cup \left(D\times B\right).$ This implies that $\zeta \in B\cap D.$ Furthermore, from Eq $\left(2.11\right)$, we have $\zeta _{\mu }\in B\cup D.$ Thus, we get $\left(\zeta _{\mu }, \zeta \right) \in S\left({\text{for all }}\mu \in \mathbb{N} \right).$ Therefore, our assertions hold. Furthermore, since $\gamma$ is a self-mapping and from condition $\left(i\right)$, we obtain for all $\zeta, \Im \in U,$

The binary relation $R$ is $\gamma$-closed. As $B\neq \emptyset$, there exists $\zeta _{0}\in B,$ such that $\gamma \left(\zeta _{0}\right) \in D$ that is $\left(\zeta _{0}, \gamma \left(\zeta _{0}\right) \right) \in R.$ Therefore, all the hypotheses of Theorem $2.2$ are satisfied. Hence, $\left(\gamma \right) _{Fix}$ $\neq \emptyset$ and also$\left(\gamma \right) _{Fix}\subseteq B\cap D$. Finally, as $\left(\zeta, \Im \right) \in R$ for all $\zeta, \Im \in G\cap H$, $G\cap H$ is $\nabla$-directed. Hence, the main conditions of Theorem $\ 2.2$ are satisfied, so $\gamma$ has a unique fixed point. It finishes the proof.

4.   Application

In this section, we illustrate how to guarantee existence of a solution of a matrix type equation. We shall use the following notations. Let $A\left(\mu \right)$ be the set of all $\mu \times \mu$ complex matrices, let $H\left(\mu \right) \subseteq A\left(\mu \right)$ be the family of all $\mu \times \mu$ Hermitian matrices, let $G\left(\mu \right) \subseteq A\left(\mu \right)$ be the set of all $\mu \times \mu$ positive definite matrices, $H^{+}\left(\mu \right) \subseteq F\left(\mu \right)$ be the set of all $\mu \times \mu$ positive semidefinite matrices. For $\Lambda$ in $G\left(\mu \right),$ we will also denote $\Lambda \succ 0.$ Furthermore, $\Lambda \succeq 0$ means that $\Lambda$ in $H^{+}\left(\mu \right)$ $.$ As a different notation for $\Lambda -\Delta \succeq 0$ and $\Lambda -\Delta \succ 0,$ we will denote $\Lambda \succeq \Delta$ and$\Lambda \succ \Delta$, respectively. Also, for each $\ \Lambda, \Delta$ in $A\left(\mu \right)$ there is a greatest lower bound and least upper bound, see ^[38]$.$ In addition, take

such that

We use the $m$-metric induced by the trace norm $\left\Vert.\right\Vert _{tr}$ given as $\left\Vert Q\right\Vert _{tr} = \sum_{i = 1}^{\mu }\Xi _{i}\left(Q\right),$ where $\Xi _{i}\left(Q\right),$ $i = 1, 2, ..., \mu$ are the singular values of $Q$ in $A\left(\mu \right).$ The set $H\left(\mu \right)$ endowed with this norm is a complete $m$-metric space. Moreover, we see that

Consider the following nonlinear matrix equation

where $\vartheta$ is a positive definite matrix, $Q_{1}, Q_{2}, \cdots, Q_{m}$ are $\mu \times \mu$ matrices and $\Xi$ is an order persevering continuous map from $H\left(\mu \right)$ to $G\left(\mu \right)$. Then, $F_{R}^{m}\in \nabla \left(F\right)$ and $\left(A\left(\mu \right), m\right)$ is a complete $mm$-space, where

In this section, we prove the existence of the positive definite solution to the nonlinear matrix Eq $\left(4.1\right).$

Theorem 4.1. Assume that there are positive real numbers $C$ and $\xi$ such that:

$\left(i\right)$ for each $\Lambda, \Delta$ in $H\left(\mu \right)$ such that $\left(\Lambda, \Delta \right)$ in $\preceq$ with $\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Lambda \right) Q_{i}\neq \sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Delta \right) Q_{i}$,

$\left(ii\right)$ there exists a positive number $N$ for which $\sum\limits_{i = 1}^{\mu }Q_{i}Q_{i}^{\ast } < CI_{\mu }$ and $\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Lambda \right) Q_{i} > 0.$

Then the matrix Eq $(4.1)$ has a solution$.$ Furthermore, the iteration

where $\Lambda _{0}$ in $F\left(\mu \right)$ satisfies $\Lambda _{0}\preceq \vartheta +\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(S_{\mu -1}\right) Q_{i}$, converges in the sense of trace norm $\left\Vert.\right\Vert _{tr}$ to the solution of the matrix Eq $\left(4.1\right).$

Proof. We define the mapping $\gamma :H\left(\mu \right) \rightarrow H\left(\mu \right)$ and $F_{R}^{m}:R^{+}\rightarrow R$ by

and set

Then, $\gamma$ is well defined and $\preceq$ is a relation under $R$, and $\preceq$ on $F\left(\mu \right)$ is $\gamma$-closed. $F_{R}^{m}\left(a\right) = -\frac{1}{\sqrt{a}}$ for all $a\in R^{+}$. Furthermore, a fixed point of $\gamma$ is a positive solution of $\left(4.1\right).$ Now, we want to prove that $\gamma$ is a $F_{R}^{m}$-contraction mapping with $\xi$ Let $\left(\Lambda, \Delta \right) \in \varpi = \left\{ \left(\left(\Lambda, \Delta \right) \in R:\Xi \left(\Lambda \right) \neq \Xi \left(\Delta \right) \right) \right\}$ which implies that $\Lambda \prec \Delta.$ Since $\Xi$ is an order preserving mapping, we deduce that $\Xi \left(\Lambda \right) \prec \Xi \left(\Delta \right).$ We have

and so

This implies that

and then

Consequently,

Now, we get

This shows that $\gamma$ is a $F_{R}^{m}$-contraction. Using $\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\vartheta \right) Q_{i}\succ 0$, we deduce that $\vartheta \preceq \gamma \left(\vartheta \right).$ This means that $\vartheta$ in $H^{+}\left(\gamma, \preceq \right).$ From Corollary $2.2$, there exists $\Lambda _{0}\in H\left(\mu \right)$ such that $\gamma \left(\Lambda _{0}\right) =$ $\Lambda _{0}$. Hence, the matrix Eq $\left(4.1\right)$ has a solution$.$

Example 4.1. Now, consider the matrix equation

where

and Define $F_{R}^{m}:R^{+}\rightarrow R$ by

for all $a\in R^{+},$ and $\Xi :H\left(\mu \right) \rightarrow H\left(\mu \right)$ is given by $\Xi \left(\Lambda \right) = \frac{\Lambda }{3}.$ Then, all conditions of Corollary 2.2 are satisfied for $N = \frac{6}{10}$ by using the iterative sequence

After some iterations, we get the approximation solution

Hence, all the conditions of Theorem $4.1$ are satisfied$.$

5.   Conclusions

In this paper, a relation theoretic $M$-metric fixed point algorithm under rational type $\ F_{R}^{m}$-contractions $\left({\text{respectively, rational type}}\ {\text{generalized }}F_{R}^{m}{\text{-contractions}}\right)$ is proposed to solve the nonlinear matrix equation $\Lambda = S+\sum\limits_{i = 1}^{\mu }Q_{i}^{\ast }\Xi \left(\Lambda \right) Q_{i}.$ Some numerical comparison experiments with existing algorithms are presented within given tables and figures. Analogously, this proposed work can be extended to generalized distance spaces, such as symmetric spaces, $m_{b}m$ -spaces $rmm$-spaces, $rm_{b}m$-spaces, $pm$-spaces, $p_{b}m$-spaces, etc. Some problems of fixed point results could be studied in near future.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors would like to thank the Deanship of Scientific Research at Umm Al-Qura University for supporting this work grant code: 23UQU4331214DSR001.

Conflict of interest

The authors declare that they have no conflict of interest.