Common φ-Fixed Point Results for S-Operator Pair in Symmetric M-Metric Spaces

Abstract

In this article, we define a new class of noncommuting self mappings known as the S-operator pair. Also, we provide the existence and uniqueness of common fixed point results involving the S-operator pair satisfying the $(F,\phi ,\psi ,Z)$-contractive condition in m-metric spaces, which unifies and generalizes most of the existing relevant fixed point theorems. Furthermore, the variables in the m-metric space are symmetric, which is significant for solving nonlinear problems in operator theory. In addition, examples are provided in order to illustrate the concepts and results presented herein. It has been demonstrated that the results can be applied to prove the existence of a solution to a system of integral equations, a nonlinear fractional differential equation and an ordinary differential equation for damped forced oscillations. Also, in the end, the satellite web coupling problem is solved.

1. Introduction

Over the last century, there has been rapid advancement in fixed point theory and its applications after the development of the Banach contraction principle [1] and the Brouwer fixed point theorem [2]. It has various applications in nonlinear analysis, game theory, economics, optimization theory, integral differential equations, dynamic system theory, signal and image processing, and other related fields of applied mathematics. For more details, we refer the reader to these references and papers [3,4,5,6]. The Banach contraction principle [1] (for short, BCP) is one of the fundamental principles of metric fixed point theory. It guarantees that any contraction mapping on complete metric space (for short, MS) has a unique fixed point. Several authors have explored the BCP in various directions to produce generalizations, extensions and applications of their findings. Exploring the fundamental properties of the new types of spaces is one of the most fascinating and interesting subjects. Thus, within the context of different generalized metric spaces, certain authors have demonstrated results related to fixed points, coincidence points and common fixed points, contributing to this area of study in this direction; for more references, consult [3,4,5,7,8]. In a MS $(W,l)$ with a self-mapping $J:W\to W,$ a point $v\in W$ is called a fixed point of J if $l(v,Jv)=0.$ A point $v\in W$ is called a coincidence point (for short, CP) [9] of J and L, where $J,L:W\to W$ if and only if $Lv=Jv$. If a point $v\in W$ exists, such that $Lv=Jv=w$, then a point $w\in W$ is referred as a point of coincidence [9]. On the other hand, a point $w\in W$ is considered as a common fixed point (for short, Cfp) [10] of J and L if and only if $Lw=Jw=w.$ The study of a Cfp of mappings satisfying various contraction conditions holds significant importance in fixed point theory and is of substantial interest to scholars. We consult the references included in [8,10,11,12] for an overview of Cfp theory, its associated results, applications and compared different contraction conditions.

Jleli et al. [13] gave the ideas of $\phi$-fixed point and $\left(F,\phi \right)$-contraction mappings and proved some $\phi$-fixed point results for such mappings in the framework of complete MS, and also deduced some fixed point results in the setting of complete partial metric spaces. In 2021, H. N. Saleh [5] established the definitions of point of $\phi$-coincidence and common $\phi$-fixed point and additionally proved a few associated results in the framework of MS and partial metric spaces. If $W\ne \varphi$, $J,L:W\to W$ and $\phi :W\to \left[0,\infty \right)$ is a given function, then $w\in W$ is said to be a point of $\phi$-coincidence of J and L if and only if $Lv=Jv=w$ and $\phi \left(w\right)=0.$ A point $w\in W$ is called a common $\phi$-fixed point of J and L if and only if $Lw=Jw=w$ and $\phi \left(w\right)=0.$

In 1976, Jungck [10] established a common fixed point theorem (for short, Cfpt) for commuting mappings (for short, Cmaps) as a generalization of the contraction theorem of Banach. Sessa [8] proposed the idea of weakly commuting mappings (for short, W-Cmaps) and established a Cfpt for self mappings in the setting of a complete MS. Jungck [12] defined that the compatible mappings (for short, Comp-maps) are an extension of the Cmaps. Moreover, properties of the Comp-maps are derived and applied to acquire results. In 1997, Jungck and Rhoades [14] provided the concept of weakly compatible mappings (for short, WComp-maps) and proved fixed point results for set valued non-continuous functions. Al-Thagafi and Shahzad [15] proved the Cfpt for new classes of noncommuting self mappings known as occasionally weakly compatible mappings (for short, OWComp-maps). Additionally, they employed them to derive various types of invariant approximation results that complemented, extended and assembled the well-known results from the literature. Pathak and Hussain [16] proposed a notion of the P-operator pair (for short, P-Op), which comprises WComp-maps and OWComp-maps as a proper class. Hussain et al. [17] presented a new class of noncommuting mapping called the $JH$-operator pair (for short, $JH$-Op) and acquired Cfpt for such mappings. Pathak and Deepmala [16] demonstrated the existence of Cfp solutions for a new class of noncommuting mapping known as the $PD$-operator pair (for short, $PD$-Op), which is distinct from P-Op and $JH$-Op. In recent years, many writers have used these ideas to produce Cfpt for different classes of mappings.

Symmetry is a key feature of Banach spaces and is closely related to fixed point problems [18]. Globally, renowned researchers are observing and researching this phenomenon. The unwavering interest in this area of research originates from its practical applications in various fields. Now, note that symmetry is a mapping on some object, W, that is supposed to be structured onto itself in such a way that the structure remains preserved. Saleem et al. [19] and Sain [20] proposed many ways this mapping can be obtained. By using the concept of symmetry, Neugebaner [18] obtained various applications of a layered compression–expansion fixed point theorem to find the of solutions of a second-order difference equation with Dirichlet boundary conditions.

In recent years, various generalizations of standard MS have emerged. For instance, a partial metric space is one of the most influential generalizations of ordinary MS. In 1994, Matthews [21] proposed the notion of partial metric space (for short, p-MS) by keeping the symmetric condition of the space and replacing the equality $l(v,v)=0$ in the definition of the metric with the inequality $l(v,v)\le l(v,w)$ for all $v,w\in W$. Therefore, the self distance of any point of the space may not be zero. In fact, the motivation for presenting the idea of a p-MS was to acquire an appropriate mathematical model in computation theory. Also, he acquired various results in p-MS. Particularly, he investigated the development of the Banach fixed point result in the setting of p-MS. Later on, various mathematicians studied the existence and uniqueness of a fixed point for nonlinear mapping satisfying many contractive conditions in the framework of p-MS. Based on the result of Matthews, in 2014, Asadi et al. [22] extended the p-MS to an m-metric space (for short, m-MS) and provided several examples to demonstrate how the notion of m-MS is a true generalization of p-MS. They proposed the symmetric m-metric space by replacing the inequality $l(v,v)\le l(v,w)$ in the definition of p-MS with $min\left\{l(v,v),l(w,w)\right\}\le l(v,w)$ because $l(w,w)=0$ may become $l(w,w)\ne 0.$ Additionally, they demonstrated the validity of some of the main theorems by generalized contractions for obtaining fixed points for mappings.

In this paper, we introduce a new class of noncommuting self mappings known as the S-operator pair. This class contain the Cmaps, WComp-map, OWComp-map, $JH$-Op, P-Op and $PD$-Op as proper subclasses, as illustrated in remarks and examples. For this new class, we establish some common $\phi$-fixed point results satisfying $(F,\phi ,\psi ,Z)$-contraction in the context of m-MS $(W,m),$ which is more general than ordinary metric space. Our results unify, extend and complement fixed point theorems due to Jungck [10], Jungck and Rhoades [14], Jungck [9], Pathak and Deepmala [16], Hussain et al. [17], H.K. Pathak and N. Hussain [23] and many others. Some corollaries and related $\phi$-fixed point results are also provided to prove the validity of our results. After that, examples and applications of our results for the existence of a solution to a system of integral equations, a nonlinear fractional differential equation and an ordinary differential equation for damped forced oscillations is also given in this paper. In the end, we find the solution of the satellite web coupling problem. In the sequel, $\mathbb{R},{\mathbb{R}}^{+}$ and $\mathbb{N}$ represent the set of all real numbers, positive real numbers and natural numbers, respectively.

2. Preliminaries

In this section, we will now present some fundamental definitions, notions, examples, lemmas and remarks pertaining to our major finding in this part.

Notation 1.

The following notations are essential in the sequel:

1

$\phi$-fp $=\phi -$fixed point;

2

$\phi$-Poc = point of $\phi$-coincidence;

3

$\phi$-Cfp = common $\phi$-fixed point;

4

$Poc(J,L)=$ the set of all the points at which J and L coincide;

5

$Cp(J,L)=$ the set of all CPs;

6

$F\left(J\right)=$ represents the collection of fixed points of $J;$

7

$m_{vw}=min\left\{m(v,v),m(w,w)\right\}$;

8

$M_{vw}=max\left\{m(v,v),m(w,w)\right\};$

9

$D\left(Poc(J,L)\right)=sup\left\{l(v,w):v,w\in Poc(J,L)\right\};$

10

$D\left(Cp(J,L)\right)=sup\left\{l(v,w):v,w\in Cp(J,L)\right\}.$

In 1976, Jungck [10] proposed the notion of Cmaps and proved Cfpt for such mappings in the framework of MS, defined as follows.

Definition 1

([10]). Suppose $(W,l)$ is a MS, and J and L are self mappings on $W,$ then a pair $(J,L)$ is said to be Cmaps if and only if

$$JLv=LJv,\mathit{for}\mathit{all}v\in W.$$

In 1982, Sessa [8] generalized Cmaps by calling self mappings J and L of a MS $(J,L)$ a weakly commuting pair, defined as follows.

Definition 2

([8]). Suppose $(W,l)$ is a MS, and J and L are self mappings on $W,$ then a pair $(J,L)$ is said to be W-Cmaps if and only if

$$JLv=LJv,\mathit{for}\mathit{some}v\in W.$$

Remark 1.

Every pair of Cmaps are WCmaps. Instead, the converse is invalid.

In 1986, Jungck [12] gave the generalization of Cmaps and W-Cmaps known as Comp-maps, defined as follows.

Definition 3

([12]). Suppose $(W,l)$ is a MS, and J and L are self mappings on $W,$ then a pair $(J,L)$ is said to be Comp-maps if and only if

$$\underset{w\to \infty }{lim}l(JL{v}_w,LJ{v}_w)=0,$$

If $\left\{v_w\right\}$ is a sequence in $W,$ then ${lim}_{w\to \infty }J{v}_w={lim}_{w\to \infty }L{v}_w=w$ for some $w\in W.$

Remark 2.

Every pair of W-Cmaps are Comp-maps. Instead, the converse is invalid.

In 1997, in order to prove fixed point results for set valued non-continuous functions, Jungck and Rhoades [14] gave the concept of WComp-maps, defined as follows.

Definition 4

([14]). Suppose $(W,l)$ is a MS, and J and L are self mappings on $W,$ then a pair $(J,L)$ is said to be WComp-map if and only if

$$JLv=LJv,\mathit{for}\mathit{all}v\in Cp(J,L).$$

Remark 3.

Every pair of Comp-maps are WComp-maps. Instead, the converse is invalid.

Jungck [9] established Cfpt for new classes of noncommuting mappings known as OWComp-maps, defined as follows.

Definition 5

([9]). Suppose $(W,l)$ is a MS, and J and L are self mappings on $W,$ then a pair $(J,L)$ is said to be OWComp-maps if and only if

$$JLv=LJv,\mathit{for}\mathit{some}v\in Cp(J,L).$$

Remark 4.

If J and L are WComp-maps, then they are OWComp-maps. Instead, the converse is invalid in general.

In 2010, Pathak [16] proved Cfpt for new classes of noncommuting mappings known as $P-$Op, defined as follows.

Definition 6

([16]). Suppose $(W,l)$ is a MS, and J and L are self mappings on $W,$ then a pair $(J,L)$ is said to be P-Op if and only if there is a point $v\in X$, such that $v\in Cp(J,L)$ and

$$l(v,Jv)\le D\left(Cp\right(J,L\left)\right).$$

Remark 5.

Every pair of OWComp-maps are P-Op. Instead, the converse is invalid. If the self mappings J and L of W are WComp-map, then $L\left(Cp(J,L)\right)\subseteq Cp(J,L)$, and hence J and L are P-Op. Clearly, OWComp-maps and nontrivial WComp-maps J and L, which do have a CP, are P-Op.

Hussain et al. [17] established a new class of noncommuting mappings called $JH$-Op, defined as follows.

Definition 7

([17]). Suppose $(W,l)$ is a MS, and J and L are self mappings on $W,$ then a pair $(J,L)$ is said to be $JH$-Op if and only if there is a point $w=Lv=Jv$ in $Poc(J,L)$, such that

$$l(w,v)\le D\left(Poc\right(J,L\left)\right).$$

Remark 6.

If J and L are $JH$-Op, then they are not OWComp-maps and not WComp-maps.

H.K. Pathak and Deepmala [23] defined $PD$-Op of single valued mappings and proved Cfpt for this class of mappings under relaxed conditions.

Definition 8

([23]). Suppose $(W,l)$ is a MS, and J and L are self mappings on $W,$ then a pair $(J,L)$ is said to be $PD$-Op if there is a point $v\in X$, such that $v\in Cp(J,L)$ and

$$l(JLv,LJv)\le D\left(Poc\right(J,L\left)\right).$$

Remark 7.

Every pair of Cmaps, nontrivial WComp-maps and nontrivial OWComp-maps are $PD-$Op. In addition, $PD$-Op is a weaker form of W-Cmaps and Comp-maps, but the reverse implication is invalid. $PD$-Op is not the same as P-Op or $JH$-Op.

On the other hand, Matthews [21] established the notion of p-MS and demonstrated the validity of the BCP in such spaces. After that, Asadi and Karapinar [22] presented the concept of a m-MS and established fixed point theorems within their context. Using a few examples, they showed how m-MS is a true generalization of p-MS. We also refer to [4,24] for more findings in this direction.

Matthews [21] provided the definition of p-MS, which is as follows:

Definition 9

([21]). Let W$\ne \varphi .$ The mapping $\rho :W\times W\to [0,\infty )$ is a partial metric on W if each of the following conditions are satisfied for all $v,w\in W$, $x\in W\setminus \left\{v,w\right\}$:

(1)

$\rho (v,v)=\rho (w,w)=\rho (v,w)\iff v=w;$

(2)

$0\le \rho (v,v)\le \rho (v,w);$

(3)

$\rho (v,w)=\rho (w,v);$

(4)

$\rho (v,w)\le \rho (v,x)+\rho (x,w)-\rho (x,x).$

Then the pair $(W,\rho )$ is known as a p-MS.

It follows from (1) and (2) that $v=w$ if $\rho (v,w)=0.$ In general, nevertheless, the contrary is not true.

Asadi et al. [22] provided the following notion of m-MS as an extension of p-MS, which is defined as follows:

Definition 10

([22]). Let W$\ne \varphi .$ The mapping $m:W\times W\to [0,\infty )$ is a m-metric on W if each of the following conditions are satisfied for all $v,w\in W$, $x\in W\setminus \left\{v,w\right\}$:

(m1)

$m(v,v)=m(w,w)=m(v,w)\iff v=w;$

(m2)

$0\le m_{vw}\le m(v,w);$

(m3)

$m(v,w)=m(w,v);$

(m4)

$m(v,w)-m_{vw}\le m(v,x)-m_{vx}+m(x,w)-m_{xw}.$

Then the pair $(W,m)$ is known as $m-$MS.

Lemma 1

([22]). Every ρ-metric is a m-metric, but the converse does not hold in general.

Definition 11

([22]). Let $(W,m)$ be a m-MS and there exists a sequence $\left\{v_t\right\}$ in $W.$ Then:

(1)

A sequence $\left\{v_t\right\}$ is m-convergent that converges to $v\in W$ if and only if $v_t\to v$ as $t\to \infty$. In this particular case, we can write

$$\underset{t\to \infty }{lim}\left(m(v_t,v)-m_{v_{t}v}\right)=0.$$

(2)

A sequence $\left\{v_t\right\}$ is an m-Cauchy sequence if and only if

$$\underset{t,k\to \infty }{lim}\left(m(v_t,v_k)-m_{v_t{v}_k}\right)\mathit{and}\underset{t,k\to \infty }{lim}\left(M_{v_t{v}_k}-m_{v_t{v}_k}\right),$$

exist and are finite.

(3)

An m-MS is called m-complete if every m-Cauchy sequence $\left\{v_t\right\}$ in W converges with respect to $t_m$ ( topology induced by m ) to a point $v\in W$, such that

$$\underset{t\to \infty }{lim}\left(m(v_t,v)-m_{v_{t}v}\right)=0\mathit{and}\underset{t\to \infty }{lim}\left(M_{v_{t}v}-m_{v_{t}v}\right)=0.$$

Remark 8

([22]). For every $v,w,i\in (W,m),$ we have

1.

$0\le M_{vw}+m_{vw}=m(v,v)+m(w,w);$

2.

$0\le M_{vw}-m_{vw}=\left[m(v,v)-m(w,w)\right];$

3.

$M_{vw}-m_{vw}\le \left(M_{vi}-m_{vi}\right)+\left(M_{iw}-m_{iw}\right).$

Definition 12

([25]). Consider $\mathsf{\Psi }$ to be the collection of all functions $\psi :[0,\infty )\to [0,\infty )$, which are called $\left(c\right)$-comparison functions, satisfying each of the following properties:

(i)

ψ is monotone increasing, i.e., $p_1\le p_2$ implies $\psi \left(p_1\right)\le \psi \left(p_2\right);$

(ii)

${\displaystyle \sum _{e=1}^{+\infty }}{\psi }^e\left(p\right)<\infty$ for all $p>0,$ where ${\psi }^e$ is an $e-$iterate of $\psi .$

Remark 9.

Note that, if $\psi \in \mathsf{\Psi }$ is a $\left(c\right)-$comparison function, then

(1)

$\psi \left(0\right)=0$;

(2)

$\psi \left(p\right)<p,$ for all $p>0$;

(3)

${\displaystyle \sum _{e=1}^{+\infty }}{\psi }^e\left(p\right)<\infty$ implies ${lim}_{e\to \infty }{\psi }^e\left(p\right)=0,$ for all $p\in \left(0,\infty \right).$

In 2014, Jleli et al. [13] provided the following notion of control functions.

Definition 13.

Consider $\mathcal{F}$ to be the collection of all functions $F:{[0,\infty )}^3\to [0,\infty )$ which are called control functions and satisfy the following conditions:

(F1)

$max\left\{u,x\right\}\le F(u,x,i),$ for all $u,x,i\in [0,\infty );$

(F2)

$F(p_1,p_2,p_3)=0$ if and only if $p_{\mathfrak{i}}=0$for all $i=1,2,3;$

(F3)

F is continuous.

The set of such control functions is denoted by $\mathcal{F}$. Some examples of control functions are given below:

Example 1

([13]). Let $r=\left\{1,2,3\right\}.$ Define $F_r:{[0,\infty )}^3\to [0,\infty )$ for all $r\in \left\{1,2,3\right\}$ as follows:

$$F_1(u,x,i)=u+x+i,F_2(u,x,i)=max\left\{u+x\right\}+i\mathit{and}{F}_3(u,x,i)=u+u^2+x+i,$$

for all $u,x,i\in [0,\infty ).$ Clearly, $F_1,F_2,F_3\in \mathcal{F}.$

In [13], the idea of $\left(F,\phi \right)$-contraction mapping was introduced and the existence of the fixed point was studied for such mappings.

Definition 14

([13]). Let $(W,l)$ be a complete MS and $\phi :W\to \left[0,\infty \right).$ A mapping $J:W\to W$ is said to be a $(F,\phi )-$contraction mapping if there exist $F\in \mathcal{F}$ and $k\in (0,1)$, such that

$$F\left(l\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right)\le \psi \left(l(v,w),\phi \left(v\right),\phi \left(w\right)\right),\mathit{for}\mathit{all}v,w\in W.$$

Definition 15

([3]). Let J and L be self mappings on a nonempty set W, such that $J\left(W\right)\subset L\left(W\right)$ with $v_0\in W.$ Let sequence $\left\{L{v}_q\right\},$ such that

$$L{v}_{q+1}=J{v}_q,q=0,1,2,\dots$$

Then, the sequence $\left\{L{v}_q\right\}$ with initial point $v_0$ is known as the $J-$sequence.

3. Main Results

In this section, we provide the notion of a new class of a noncommuting pair of mappings known as the “S-operator pair”. Also, we show that there exists a $\phi$-Poc for such a pair of mappings within the framework of $m-$MS. Then we use this conclusion to derive the Cfp of a pair $(J,L)$ of mappings, where J is a $(F,\phi ,\psi ,Z)$-contraction with respect to L.

We now provide the definition of an S-operator pair, defined as follows.

Definition 16.

Let $(W,l)$ be a MS, and J and L are self mappings on $W.$ The pair $(J,L)$ is known as an $S-$Operator pair (for short, $S-$Op) if and only if there exists a point $w=Lv=Jv$ in $Poc(J,L)$, such that

$$l(w,Lw)\le D\left(Poc\right(J,L\left)\right),\mathit{for}\mathit{all}w,v\in W.$$

Example 2.

Let $W=\mathbb{R}$ and define $l:W\times W\to [0,\infty )$ by $l(v,w)=\left|v-w\right|$. Define $J,L:W\to W$ by $Jv=v^3$ and $Lv={\displaystyle \frac{v^2}{4}}.$ Since $Cp(J,L)=\left\{0,{\displaystyle \frac{1}{4}}\right\},$ $Poc(J,L)=\left\{0,{\displaystyle \frac{1}{64}}\right\},$ $D\left(Cp(J,L)\right)={\displaystyle \frac{1}{4}}$ and $D\left(Poc(J,L)\right)={\displaystyle \frac{1}{64}}.$ Clearly, $(J,L)$ is $S-$Op.

Remark 10.

Every pair of Cmaps, nontrivial WComp-maps and nontrivial OWComp-maps are S-Op, but the reverse implications do not always hold true.

Example 3.

Let $W=\mathbb{R}$ and define $l:W\times W\to [0,\infty )$ by $l(v,w)=\left|v-w\right|$. Define $J,L:W\to W$ by $Jv=v^2$ and $Lv={\displaystyle \frac{v}{2}}$. Since $Cp(J,L)=\left\{0,{\displaystyle \frac{1}{2}}\right\},$ $Poc(J,L)=\left\{0,{\displaystyle \frac{1}{4}}\right\},$ $D\left(Cp(J,L)\right)={\displaystyle \frac{1}{2}}$ and $D\left(Poc(J,L)\right)={\displaystyle \frac{1}{4}}.$ Clearly, $(J,L)$ is $S-$Op, $P-$Op, $JH-$Op and $PD-$Op, but not Cmaps, not WComp-map and not OWComp-map.

Remark 11.

Every S-Op is $PD$-Op but every $JH$-Op, P-Op and $PD$-Op are not S-Op.

We take into consideration the following example for this.

Example 4.

Let $W=\left[0,\infty \right)$ be endowed with the Euclidean metric. Define $J,L:W\to W$ by

$$Jv=v\mathrm{and}Lv=v^2,\mathit{for}\mathit{all}v\ne 0,$$

and

$$J0=L0=3.$$

Since $Cp(J,L)=\left\{0,1\right\},$ $Poc(J,L)=\left\{1,3\right\},$ $D\left(Cp\right(J,L\left)\right)=1$ and $D\left(Poc\right(J,L\left)\right)=2$. Clearly, the pair $(J,L)$ is $PD$-Op, P-Op and $JH$-Op, but not S-Op.

Example 5.

Let $W=[0,\infty )$ and defined $l:W\times W\to \left[0,\infty \right)$ by $l(v,w)=|v-w|.$ Define $J,L:W\to W$ by

$$Jv=\left\{\begin{array}{c}2,\mathrm{if}v=0,\\ 3v,\mathrm{if}0<v\le 1,\\ {\displaystyle \frac{v}{2}},\mathrm{if}1<v<\infty .\end{array}\right.\mathit{and}Lv=\left\{\begin{array}{c}v+2,\mathrm{if}0\le v<2,\\ v,\mathrm{if}2\le v<\infty .\end{array}\right.$$

Note that $Cp(J,L)=\left\{0,1\right\},$ $Poc(J,L)=\left\{2,3\right\}$ and $D\left(Cp\right(J,L\left)\right)=1$ and $D\left(Poc\right(J,L\left)\right)=1$. Clearly the pair $(J,L)$ is S-Op, but not Cmaps, not WComp-map and not OWComp-map. Also note that the pair $(J,L)$ is neither P-Op nor $JH$-Op.

Now, we will provide the notion of the $(F,\phi ,\psi ,Z)$-generalized contraction for the pair of mappings J and L in the setting of m-MS as follows.

Definition 17.

Let $(W,m)$ be a m-MS, $J,L:W\to W$ and $F\in \mathcal{F}$. A mapping J is known as a $(F,\phi ,\psi ,Z)$-contraction with respect to L, if there is a lower semicontinuous function $\phi :W\to \left[0,\infty \right)$, such that

$$F\left(m\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right)\le \mathit{\psi }\left(Z(v,w)\right),$$

(1)

where

$$Z(v,w)=max\left\{\begin{array}{c}F\left(m\left(Lv,Lw\right),\phi \left(Lv\right),\phi \left(Lw\right)\right),\\ F\left(m\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right),\\ F\left(m\left(Lw,Jw\right),\phi \left(Lw\right),\phi \left(Jw\right)\right),\\ {\displaystyle \frac{1}{2}}\left[\begin{array}{c}F\left(m\left(Lv,Lw\right),\phi \left(Lv\right),\phi \left(Lw\right)\right)\\ +F\left(m\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right)\end{array}\right]\end{array}\right\},$$

for all $v,w\in W$ and $\psi \in \mathsf{\Psi }$.

By taking $Z(v,w)=F$$\left(m\left(Lv,Lw\right),\phi \left(Lv\right),\phi \left(Lw\right)\right)$ in (1), we obtain the following definition.

Definition 18.

Let $(W,m)$ be a m-MS, $J,L:W\to W$ and $F\in \mathcal{F}$. A mapping J is known as a $(F,\phi ,\psi )$-contraction with respect to L, if there is a lower semicontinuous function $\phi :W\to \left[0,\infty \right)$, such that

$$F\left(m\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right)\le \mathit{\psi }\left(F\left(m\left(Lv,Lw\right),\phi \left(Lv\right),\phi \left(Lw\right)\right)\right),$$

(2)

for all $v,w\in W$ and $\psi \in \mathsf{\Psi }$.

• By taking $L=I$ in (2), we obtain the mapping in [26].

Now, we are ready to state and prove our main result.

Theorem 1.

Let $(W,m)$ be a m-complete m-MS and $J,L:W\to W,$ $F\in \mathcal{F}$ with $J\left(W\right)\subseteq L\left(W\right)$. Suppose that one of the subspaces $JW$ and $LW$ is closed in W and the pair $(J,L)$ is a S-Op. Then J and L has a unique φ-Cfp provided that J is a $(F,\phi ,\psi ,Z)$-contraction with respect to L.

Proof. 

Suppose that $v_0\in W.$ As $J\left(W\right)\subseteq L\left(W\right),$ we obtain a sequence $\left\{J{v}_w\right\}$ by $L{v}_{w+1}=J{v}_w$ with initial point $v_0.$ Suppose that $J{v}_{w_0}=J{v}_{w_0+1},$ for some $w_0\in \mathbb{N}.$ Then $J{v}_{w_0}=J{v}_{w_0+1}=L{v}_{w_0+1}$ implies that $v_{w_0+1}$ is a CP. For any $w\ge 0,$ put $v=v_w$ and $w=v_{w+1}$ in condition (1) to obtain

$$F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right)\le \mathit{\psi }\left(Z\left(v_w,v_{w+1}\right)\right),$$

(3)

where

$$\begin{array}{ccc}\hfill Z\left(v_w,v_{w+1}\right)& =& max\left\{\begin{array}{c}F\left(m\left(L{v}_w,L{v}_{w+1}\right),\phi \left(L{v}_w\right),\phi \left(L{v}_{w+1}\right)\right),\\ F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right),\\ F\left(m\left(L{v}_{w+1},J{v}_{w+1}\right),\phi \left(L{v}_{w+1}\right),\phi \left(J{v}_{w+1}\right)\right),\\ {\displaystyle \frac{1}{2}}\left[\begin{array}{c}F\left(m\left(L{v}_w,L{v}_{w+1}\right),\phi \left(L{v}_w\right),\phi \left(L{v}_{w+1}\right)\right)\\ +F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(S{v}_w\right),\phi \left(J{v}_{w+1}\right)\right)\end{array}\right]\end{array}\right\}\hfill \\ & =& max\left\{\begin{array}{c}F\left(m\left(L{v}_w,L{v}_{w+1}\right),\phi \left(L{v}_w\right),\phi \left(L{v}_{w+1}\right)\right),\\ F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right),\\ F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right),\\ {\displaystyle \frac{1}{2}}\left[\begin{array}{c}F\left(m\left(L{v}_w,L{v}_{w+1}\right),\phi \left(L{v}_w\right),\phi \left(L{v}_{w+1}\right)\right)\\ +F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right)\end{array}\right]\end{array}\right\}\hfill \\ & \le & max\left\{\begin{array}{c}F\left(m\left(L{v}_w,L{v}_{w+1}\right),\phi \left(L{v}_w\right),\phi \left(L{v}_{w+1}\right)\right),\\ F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right)\end{array}\right\}.\hfill \end{array}$$

If we take $Z(v_w,v_{w+1})=F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right),$ then from (3) we obtain

$$\begin{array}{ccc}\hfill F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right)& \le & \mathit{\psi }\left(F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right)\right)\hfill \\ & <& F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(S{v}_w\right),\phi \left(J{v}_{w+1}\right)\right),\hfill \end{array}$$

which contradicts it. Thus, we obtain

$$\begin{array}{ccc}\hfill F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right)& \le & \mathit{\psi }\left(F\left(m\left(L{v}_w,L{v}_{w+1}\right),\phi \left(L{v}_w\right),\phi \left(L{v}_{w+1}\right)\right)\right)\hfill \\ & =& \mathit{\psi }\left(F\left(m\left(J{v}_{w-1},J{v}_w\right),\phi \left(J{v}_{w-1}\right),\phi \left(J{v}_w\right)\right)\right)\hfill \\ & \le & \mathit{\psi }\left(\mathit{\psi }\left(F\left(m\left(L{v}_{w-1},L{v}_w\right),\phi \left(L{v}_{w-1}\right),\phi \left(L{v}_w\right)\right)\right)\right)\hfill \\ & =& {\mathit{\psi }}^2\left(F\left(m\left(J{v}_{w-2},J{v}_{w-1}\right),\phi \left(J{v}_{w-2}\right),\phi \left(J{v}_{w-1}\right)\right)\right)\hfill \\ & & ⋮\hfill \\ & \le & {\mathit{\psi }}^{w-1}\left(\psi \left(F\left(m\left(L{v}_1,L{v}_2\right),\phi \left(L{v}_1\right),\phi \left(L{v}_2\right)\right)\right)\right)\hfill \\ & =& {\mathit{\psi }}^w\left(F\left(m\left(J{v}_0,J{v}_1\right),\phi \left(J{v}_0\right),\phi \left(J{v}_1\right)\right)\right).\hfill \end{array}$$

By using induction, we obtain

$$F\left(m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right),\phi \left(J{v}_{w+1}\right)\right)\le {\mathit{\psi }}^w\left(F\left(m\left(J{v}_0,J{v}_1\right),\phi \left(J{v}_0\right),\phi \left(J{v}_1\right)\right)\right).$$

From condition (F1) of definition (13), it may be concluded that

$$max\left\{m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right)\right\}\le {\mathit{\psi }}^w\left(F\left(m\left(J{v}_0,J{v}_1\right),\phi \left(J{v}_0\right),\phi \left(J{v}_1\right)\right)\right).$$

(4)

By assuming $max\left\{m\left(J{v}_w,J{v}_{w+1}\right),\phi \left(J{v}_w\right)\right\}=m\left(J{v}_w,J{v}_{w+1}\right)$ in (4), we obtain that

$$m\left(J{v}_w,J{v}_{w+1}\right)\le {\mathit{\psi }}^w\left(F\left(m\left(J{v}_0,J{v}_1\right),\phi \left(J{v}_0\right),\phi \left(J{v}_1\right)\right)\right).$$

(5)

On the other hand, we obtain

$$\underset{w\to \infty }{lim}m\left(J{v}_w,J{v}_{w+1}\right)=0.$$

(6)

By assuming $min\left\{m\left(J{v}_w,J{v}_w\right),m\left(J{v}_{w+1},J{v}_{w+1}\right)\right\}=m\left(J{v}_w,J{v}_w\right),$ using (6) and the condition (m2) of definition (10), we have

$$\begin{array}{ccc}\hfill \underset{w\to \infty }{lim}m\left(J{v}_w,J{v}_w\right)& =& \underset{w\to \infty }{lim}min\left\{m\left(J{v}_w,J{v}_w\right),m\left(J{v}_{w+1},J{v}_{w+1}\right)\right\}\hfill \\ & =& \underset{w\to \infty }{lim}{m}_{J{v}_{w}J{v}_{w+1}}\hfill \\ & \le & \underset{w\to \infty }{lim}m\left(J{v}_w,J{v}_{w+1}\right)=0.\hfill \end{array}$$

Since $\underset{w\to \infty }{lim}m\left(J{v}_w,J{v}_w\right)=0,$ we have

$$\underset{w,m\to \infty }{lim}{m}_{J{v}_{w}J{v}_m}=0.$$

(7)

We now show that $\left\{J{v}_w\right\}$ is a $m-$Cauchy sequence. Consider $t,k\in \mathbb{N}$, such that $t>k.$ By using (5) and the condition (m4) of Definition (10), we obtain

$$\begin{array}{ccc}\hfill m(J{v}_t,J{v}_k)-m_{J{v}_{t}J{v}_k}& \le & m(J{v}_t,J{v}_{t+1})-m_{J{v}_{t}J{v}_{t+1}}+m(J{v}_{t+1},J{v}_k)\hfill \\ & & -m_{J{v}_{t+1}J{v}_k}\hfill \\ & \le & m(J{v}_t,J{v}_{t+1})-m_{J{v}_{t}J{v}_{t+1}}+m(J{v}_{t+1},J{v}_k)\hfill \\ & \le & m(J{v}_t,J{v}_{t+1})-m_{J{v}_{t}J{v}_{t+1}}+m(J{v}_{t+1},J{v}_{t+2})\hfill \\ & & -m_{J{v}_{t+1}J{v}_{t+2}}+m(J{v}_{t+2},J{v}_k)-m_{J{v}_{t+2}J{v}_k}\hfill \\ & \le & m(J{v}_t,J{v}_{t+1})-m_{J{v}_{t}J{v}_{t+1}}+m(J{v}_{t+1},J{v}_{t+2})\hfill \\ & & -m_{J{v}_{t+1}J{v}_{t+2}}+\dots +m(J{v}_{k-1},J{v}_k)-m_{J{v}_{k-1}J{v}_k}\hfill \\ & \le & m(J{v}_t,J{v}_{t+1})+m(J{v}_{t+1},J{v}_{t+2})+\dots +m(J{v}_{k-1},J{v}_k)\hfill \\ & \le & {\mathit{\psi }}^t\left(F\left(m\left(J{v}_0,J{v}_1\right),\phi \left(J{v}_0\right),\phi \left(J{v}_1\right)\right)\right)\hfill \\ & & +{\mathit{\psi }}^{t+1}\left(F\left(m\left(J{v}_0,J{v}_1\right),\phi \left(J{v}_0\right),\phi \left(J{v}_1\right)\right)\right)\hfill \\ & & +\dots +{\mathit{\psi }}^{k-1}\left(F\left(m\left(J{v}_0,J{v}_1\right),\phi \left(J{v}_0\right),\phi \left(J{v}_1\right)\right)\right)\hfill \\ & \le & \left(\begin{array}{c}\sum _{i=1}^{k-1}{\mathit{\psi }}^i\left(F\left(m\left(J{v}_0,J{v}_1\right),\phi \left(J{v}_0\right),\phi \left(J{v}_1\right)\right)\right)\\ -\sum _{j=1}^{t-1}{\mathit{\psi }}^j\left(F\left(m\left(J{v}_0,J{v}_1\right),\phi \left(J{v}_0\right),\phi \left(J{v}_1\right)\right)\right)\end{array}\right).\hfill \end{array}$$

(8)

It follows from Remark (9) and (8) that $m(J{v}_t,J{v}_k)-m_{J{v}_{t}J{v}_k}\to 0$ as $t\to \infty .$ Also, by (7), we obtain that

$$\underset{t,k\to \infty }{lim}\left(M_{J{v}_{t}J{v}_k}-m_{J{v}_{t}J{v}_k}\right)=0.$$

Thus, $\left\{J{v}_t\right\}$ is a m-Cauchy sequence in $W.$ Since W is m-complete, there exists $b\in A_0$, such that

$$\underset{t\to \infty }{lim}m(J{v}_t,b)-m_{J{v}_{t}b}=0\mathrm{and}\underset{t\to \infty }{lim}{M}_{J{v}_{t}b}-m_{J{v}_{t}b}=0.$$

Since $\underset{t\to \infty }{lim}m\left(J{v}_t,J{v}_t\right)=0,$ we have

$$\underset{t\to \infty }{lim}m(J{v}_t,b)=0\mathrm{and}\underset{t\to \infty }{lim}{M}_{J{v}_{t}b}=0.$$

(9)

Thus by Remark (8), we obtain

$$\underset{t\to \infty }{lim}m(b,b)=\underset{t\to \infty }{lim}\left[M_{J{v}_{t}b}+m_{J{v}_{t}b}-m\left(J{v}_t,J{v}_t\right)\right]=0.$$

This implies that

$$m(b,b)=0.$$

(10)

Suppose that W has a closed subspace $L\left(W\right).$ Next, we select $v\in W,$ such that

$$\underset{t\to \infty }{lim}J{v}_t=\underset{t\to \infty }{lim}L{v}_{t+1}=Lv=b.$$

Now we need to show that $\phi \left(b\right)=0.$ By assuming $max\left\{m\left(J{v}_t,J{v}_{t+1}\right),\phi \left(J{v}_t\right)\right\}=\phi \left(J{v}_t\right)$ in (4), we have

$$\phi \left(J{v}_t\right)\le {\mathit{\psi }}^t\left(F\left(m\left(J{v}_0,J{v}_1\right),\phi \left(J{v}_0\right),\phi \left(J{v}_1\right)\right)\right).$$

Assuming that the limit is $t\to \infty$ on both sides of the inequality above, we obtain

$$\underset{t\to \infty }{lim}\phi \left(J{v}_t\right)=0.$$

(11)

As $\phi$ is a lower semicontinuous, then from (9) and (11) we obtain

$$0\le \phi \left(b\right)\le \underset{t\to \infty }{lim}inf\phi \left(J{v}_t\right)=0.$$

Hence, $\phi \left(Lv\right)=\phi \left(b\right)=0.$ By taking $v=v$ and $w=v_t$ in the condition (1), we obtain

$$F\left(m\left(Jv,J{v}_t\right),\phi \left(Jv\right),\phi \left(J{v}_t\right)\right)\le \mathit{\psi }\left(Z\left(v,v_t\right)\right),$$

(12)

where

$$\begin{array}{ccc}\hfill Z\left(v,v_t\right)& =& max\left\{\begin{array}{c}F\left(m\left(Lv,L{v}_t\right),\phi \left(Lv\right),\phi \left(L{v}_t\right)\right),\\ F\left(m\left(Jv,J{v}_t\right),\phi \left(Jv\right),\phi \left(J{v}_t\right)\right),\\ F\left(m\left(L{v}_t,J{v}_t\right),\phi \left(L{v}_t\right),\phi \left(J{v}_t\right)\right)\\ {\displaystyle \frac{1}{2}}\left[\begin{array}{c}F\left(m\left(Lv,L{v}_t\right),\phi \left(Lv\right),\phi \left(L{v}_t\right)\right)\\ +F\left(m\left(Jv,J{v}_t\right),\phi \left(Jv\right),\phi \left(J{v}_t\right)\right)\end{array}\right]\end{array}\right\}\hfill \\ & \le & max\left\{\begin{array}{c}F\left(m\left(Lv,L{v}_t\right),\phi \left(Lv\right),\phi \left(L{v}_t\right)\right),\\ F\left(m\left(Jv,J{v}_t\right),\phi \left(Jv\right),\phi \left(J{v}_t\right)\right),\\ F\left(m\left(L{v}_t,J{v}_t\right),\phi \left(L{v}_t\right),\phi \left(J{v}_t\right)\right)\end{array}\right\}.\hfill \end{array}$$

If $Z\left(v,v_t\right)=F\left(m\left(Jv,J{v}_t\right),\phi \left(Jv\right),\phi \left(J{v}_t\right)\right),$ then from (12) we obtain

$$\begin{array}{ccc}\hfill F\left(m\left(Jv,J{v}_t\right),\phi \left(Jv\right),\phi \left(J{v}_t\right)\right)& \le & \mathit{\psi }\left(F\left(m\left(Jv,J{v}_t\right),\phi \left(Jv\right),\phi \left(J{v}_t\right)\right)\right)\hfill \\ & <& F\left(m\left(Jv,J{v}_t\right),\phi \left(Jv\right),\phi \left(J{v}_t\right)\right),\hfill \end{array}$$

which is a contradiction. If $Z\left(v,v_t\right)=F\left(m\left(L{v}_t,J{v}_t\right),\phi \left(L{v}_t\right),\phi \left(J{v}_t\right)\right),$ then from (12) we obtain

$$\begin{array}{ccc}\hfill F\left(m\left(Jv,J{v}_t\right),\phi \left(Jv\right),\phi \left(J{v}_t\right)\right)& \le & \mathit{\psi }\left(F\left(m\left(L{v}_t,J{v}_t\right),\phi \left(L{v}_t\right),\phi \left(J{v}_t\right)\right)\right)\hfill \\ & \le & \mathit{\psi }\left(F\left(\begin{array}{c}m\left(L{v}_t,Lv\right)-m_{L{v}_t,Lv}+m\left(Lv,J{v}_t\right)\\ -m_{Lv,J{v}_t},\phi \left(L{v}_t\right),\phi \left(J{v}_t\right)\end{array}\right)\right)\hfill \\ & \le & \mathit{\psi }\left(F\left(m\left(L{v}_t,Lv\right)+m\left(Lv,J{v}_t\right),\phi \left(L{v}_t\right),\phi \left(J{v}_t\right)\right)\right)\hfill \\ & =& \mathit{\psi }\left(F\left(m\left(J{v}_{t-1},Lv\right)+m\left(Lv,J{v}_t\right),\phi \left(J{v}_{t-1}\right),\phi \left(J{v}_t\right)\right)\right)\hfill \\ & <& F\left(m\left(J{v}_{t-1},Lv\right)+m\left(Lv,J{v}_t\right),\phi \left(J{v}_{t-1}\right),\phi \left(J{v}_t\right)\right).\hfill \end{array}$$

(13)

It follows from the condition (F1) of definition (13) and (13) that

$$\begin{array}{ccc}\hfill m\left(Jv,J{v}_t\right)& \le & max\left\{m\left(Jv,J{v}_t\right),\phi \left(Jv\right)\right\}\hfill \\ & <& F\left(m\left(J{v}_{t-1},Lv\right)+m\left(Lv,J{v}_t\right),\phi \left(J{v}_{t-1}\right),\phi \left(J{v}_t\right)\right).\hfill \end{array}$$

Assuming that the limit is $t\to \infty$ on both sides of the above inequality, we obtain

$$\underset{t\to \infty }{lim}m\left(Jv,J{v}_t\right)=0.$$

If $Z\left(v,v_t\right)=F\left(m\left(Lv,L{v}_t\right),\phi \left(Lv\right),\phi \left(L{v}_t\right)\right),$ then from (12) we obtain

$$\begin{array}{ccc}\hfill \psi \left(F\left(m\left(Jv,J{v}_t\right),\phi \left(Jv\right),\phi \left(J{v}_t\right)\right)\right)& \le & \mathit{\psi }\left(F\left(m\left(Lv,L{v}_t\right),\phi \left(Lv\right),\phi \left(L{v}_t\right)\right)\right)\hfill \\ & <& F\left(m\left(Lv,L{v}_t\right),\phi \left(Lv\right),\phi \left(L{v}_t\right)\right)\hfill \\ & =& F\left(m\left(Lv,L{v}_t\right),0,\phi \left(J{v}_{t-1}\right)\right).\hfill \end{array}$$

(14)

It follows from the condition (F1) of definition (13) and (14) that

$$m\left(Jv,J{v}_t\right)\le max\left\{m\left(Jv,J{v}_t\right),\phi \left(Jv\right)\right\}=F\left(m\left(Lv,L{v}_t\right),0,\phi \left(J{v}_{t-1}\right)\right).$$

Then, by taking the limit as $t\to \infty$ on the both sides of the above inequality, we obtain

$$\underset{t\to \infty }{lim}m\left(Jv,J{v}_t\right)=0.$$

Hence, $\underset{t\to \infty }{lim}m\left(Jv,J{v}_t\right)=0.$ By using the condition (m4) of definition (10), we have

$$\begin{array}{ccc}\hfill m(Jv,Lv)-m_{JvLv}& \le & m(Jv,J{v}_t)-m_{JvJ{v}_t}+m(J{v}_t,Lv)-m_{J{v}_{t}Lv}\hfill \\ \hfill m(Jv,Lv)-m_{JvLv}& \le & m(Jv,J{v}_t)+m(J{v}_t,Lv)\hfill \\ \hfill \underset{t\to \infty }{lim}m(Jv,Lv)-m_{JvLv}& \le & \underset{t\to \infty }{lim}m(Jv,J{v}_t)+\underset{t\to \infty }{lim}m(J{v}_t,Lv)\hfill \\ \hfill \underset{t\to \infty }{lim}m(Jv,Lv)-m_{JvLv}& =& 0.\hfill \end{array}$$

Thus by using (10), we see that

$$m(Lv,Lv)=m(b,b)=0.$$

Therefore, $Jv=Lv.$ That is, b is a $\phi$-Poc of the pair $(J,L).$ Moreover, if s is another $\phi$-Poc of the pair $(J,L)$ then $Js=Ls=s$ and $\phi \left(s\right)=0.$ Since L is a $(F,\phi ,\psi ,Z)$-contraction, we have

$$\begin{array}{ccc}\hfill F\left(m\left(s,b\right),0,0\right)& =& F\left(m\left(Js,Js\right),0,0\right)\hfill \\ & \le & \mathit{\psi }\left(Z\left(s,s\right)\right)\hfill \\ & <& Z\left(s,s\right),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill Z\left(s,s\right)& =& max\left\{\begin{array}{c}F\left(m\left(Ls,Ls\right),\phi \left(Ls\right),\phi \left(Ls\right)\right),\\ F\left(m\left(Js,Js\right),\phi \left(Js\right),\phi \left(Js\right)\right),\\ F\left(m\left(Ls,Js\right),\phi \left(Ls\right),\phi \left(Js\right)\right),\\ {\displaystyle \frac{1}{2}}\left[\begin{array}{c}F\left(m\left(Ls,Ls\right),\phi \left(Ls\right),\phi \left(Ls\right)\right)\\ +F\left(m\left(Js,Js\right),\phi \left(Js\right),\phi \left(Js\right)\right)\end{array}\right]\end{array}\right\}\hfill \\ & \le & max\left\{\begin{array}{c}F\left(m\left(Ls,Ls\right),\phi \left(Ls\right),\phi \left(Ls\right)\right),\\ F\left(m\left(Js,Js\right),\phi \left(Js\right),\phi \left(Js\right)\right),\\ F\left(m\left(Ls,Js\right),\phi \left(Ls\right),\phi \left(Js\right)\right)\end{array}\right\}\hfill \\ & =& max\left\{\begin{array}{c}F\left(m\left(s,b\right),\phi \left(s\right),\phi \left(b\right)\right),\\ F\left(m\left(s,b\right),\phi \left(s\right),\phi \left(b\right)\right),\\ F\left(m\left(b,b\right),\phi \left(b\right),\phi \left(b\right)\right)\end{array}\right\}\hfill \\ & =& max\left\{F\left(m\left(s,b\right),0,0\right),0\right\}\hfill \\ & =& F\left(m\left(s,b\right),0,0\right),\hfill \end{array}$$

it is a contradiction. Hence, b is a unique $\phi$-Poc of the pair $(J,L).$ Hence, $D\left(Poc\right(J,L\left)\right)=0.$ As $(J,L)$ is a S-Op, then

$$m(b,Lb)\le D\left(Poc\right(J,L\left)\right)=0.$$

Suppose that $Jb\ne b,$ then by putting $v=b$ and $w=v$ in (1) we obtain

$$F\left(m\left(Jb,b\right),\phi \left(Jb\right),\phi \left(b\right)\right)=F\left(m\left(Jb,Jv\right),\phi \left(Jb\right),\phi \left(Jv\right)\right)\le \mathit{\psi }\left(Z\left(b,v\right)\right),$$

where

$$\begin{array}{ccc}\hfill Z\left(b,v\right)& =& max\left\{\begin{array}{c}F\left(m\left(Lb,Lv\right),\phi \left(Lb\right),\phi \left(Lv\right)\right),\\ F\left(m\left(Jb,Jv\right),\phi \left(Jb\right),\phi \left(Jv\right)\right),\\ F\left(m\left(Lv,Jv\right),\phi \left(Lv\right),\phi \left(Jv\right)\right),\\ {\displaystyle \frac{1}{2}}\left[\begin{array}{c}F\left(m\left(Lb,Lv\right),\phi \left(Lb\right),\phi \left(Lv\right)\right)\\ +F\left(m\left(Jb,Jv\right),\phi \left(Jb\right),\phi \left(Jv\right)\right)\end{array}\right]\end{array}\right\}\hfill \\ & \le & max\left\{\begin{array}{c}F\left(m\left(Lb,Lv\right),\phi \left(Lb\right),\phi \left(Lv\right)\right),\\ F\left(m\left(Jb,Jv\right),\phi \left(Jb\right),\phi \left(Jv\right)\right),\\ F\left(m\left(Lv,Jv\right),\phi \left(Lv\right),\phi \left(Jv\right)\right)\end{array}\right\}\hfill \\ & =& max\left\{\begin{array}{c}F\left(m\left(b,b\right),\phi \left(b\right),\phi \left(b\right)\right),\\ F\left(m\left(Jb,b\right),\phi \left(Jb\right),\phi \left(b\right)\right)\end{array}\right\}\hfill \\ & =& max\left\{F\left(0,0,0\right),F\left(m\left(Jb,b\right),\phi \left(Jb\right),\phi \left(b\right)\right)\right\}\hfill \\ & =& max\left\{0,F\left(m\left(Jb,b\right),\phi \left(Jb\right),\phi \left(b\right)\right)\right\}\hfill \\ & =& F\left(m\left(Jb,b\right),\phi \left(Jb\right),\phi \left(b\right)\right).\hfill \end{array}$$

Hence

$$\begin{array}{ccc}\hfill F\left(m\left(Jb,b\right),\phi \left(Jb\right),\phi \left(b\right)\right)& \le & \mathit{\psi }\left(F\left(m\left(Jb,b\right),\phi \left(Jb\right),\phi \left(b\right)\right)\right)\hfill \\ & <& F\left(m\left(Jb,b\right),\phi \left(Jb\right),\phi \left(b\right)\right),\hfill \end{array}$$

which is a contradiction. Hence, $b=Lb=Jb.$ Then there exists a $\phi$-Cfp $b=Jb=Lb.$

Uniqueness: suppose that b and n are two $\phi$-Cfp of J and L with $n\ne$b and

$$\phi \left(b\right)=\phi \left(n\right)=0.$$

Since L is a $(F,\phi ,\psi ,Z)-$ contraction, we have

$$\begin{array}{ccc}\hfill F\left(m\left(b,n\right),0,0\right)& =& F\left(m\left(b,n\right),\phi \left(b\right),\phi \left(n\right)\right)\hfill \\ & =& F\left(m\left(Jb,Jn\right),\phi \left(Jb\right),\phi \left(Jn\right)\right)\hfill \\ & \le & \mathit{\psi }\left(Z\left(b,n\right)\right),\hfill \end{array}$$

(15)

where

$$\begin{array}{ccc}\hfill Z\left(b,n\right)& =& max\left\{\begin{array}{c}F\left(m\left(Lb,Ln\right),\phi \left(Lb\right),\phi \left(Ln\right)\right),\\ F\left(m\left(Jb,Jn\right),\phi \left(Jb\right),\phi \left(Jn\right)\right),\\ F\left(m\left(Ln,Jn\right),\phi \left(Ln\right),\phi \left(Jn\right)\right),\\ {\displaystyle \frac{1}{2}}\left[\begin{array}{c}F\left(m\left(Lb,Ln\right),\phi \left(Lb\right),\phi \left(Ln\right)\right)\\ +F\left(m\left(Jb,Jn\right),\phi \left(Jb\right),\phi \left(Jn\right)\right)\end{array}\right]\end{array}\right\}\hfill \\ & =& max\left\{\begin{array}{c}F\left(m\left(b,n\right),\phi \left(b\right),\phi \left(n\right)\right),\\ F\left(m\left(b,n\right),\phi \left(b\right),\phi \left(n\right)\right),\\ F\left(m\left(n,n\right),\phi \left(n\right),\phi \left(n\right)\right),\\ {\displaystyle \frac{1}{2}}\left[\begin{array}{c}F\left(m\left(b,n\right),\phi \left(b\right),\phi \left(n\right)\right)+\\ F\left(m\left(b,n\right),\phi \left(b\right),\phi \left(n\right)\right)\end{array}\right]\end{array}\right\}\hfill \\ & =& max\left\{F\left(m\left(b,n\right),0,0\right),0\right\}\hfill \\ & =& F\left(m\left(b,n\right),0,0\right),\hfill \end{array}$$

(16)

then by using (15) and (16), we obtain

$$\begin{array}{ccc}\hfill F\left(m\left(b,n\right),0,0\right)& \le & \mathit{\psi }\left(F\left(m\left(b,n\right),0,0\right)\right)\hfill \\ & <& F\left(m\left(b,n\right),0,0\right),\hfill \end{array}$$

which is a contradiction. Hence b is a unique $\phi$-Cfp of mappings J and L. □

Now, we obtain the following common $\phi$-fixed point result of S-Op $(J,L)$ with a $(F,\phi ,\psi )$-contraction.

Corollary 1.

Let $(W,m)$ be a complete m-MS, $J,L:W\to W$ and $F\in \mathcal{F}$ with $J\left(W\right)\subseteq L\left(W\right)$. Let us assume that one of the subspaces $JW$ and $LW$ is closed in $W,$ where $(J,L)$ is a S-Op and φ is a lower semicontinuous function. If J is a $(F,\phi ,\psi )$-contraction with respect to $L,$ then J and L has a unique φ-Cfp.

Proof. 

Theorem (1) yields the result that follows by choosing $Z(v,w)=F\left(m\right(Lv,Lw),\phi (Lv)$, $\phi \left(Lw\right))$ and the remaining proof follows under the same lines. □

Also, we obtain the following common $\phi$-fixed point result of WComp-map $(J,L)$ with $(F,\phi ,\psi )$-contraction.

Corollary 2.

Let $(W,m)$ be a complete m-MS, $J,L:W\to W$ and $F\in \mathcal{F}$ with $J\left(W\right)\subseteq L\left(W\right)$. Let us assume that one of the subspaces $JW$ and $LW$ is closed in $W,$ where $(J,L)$ is a WComp-map and φ is a lower semicontinuous function. If J is a $(F,\phi ,\psi )$-contraction with respect to $L,$ then J and L has a unique φ-Cfp.

To support Corollary (1), we provide the following example.

Example 6.

Let $W=\left[0,1\right]$ and $m:W\times W\to \left[0,\infty \right)$ be defined by

$$m(v,w)={\displaystyle \frac{v+w}{2}}.$$

Then $(W,m)$ is an m-MS. Define a mapping $J,L:W\to W$ as:

$$Jv=\left\{\begin{array}{c}3-2v,\mathit{if}v\in \left\{0,1\right\},\\ 0,\mathit{if}0<v<1,\end{array}\right.$$

and

$$Lv=v,\mathit{for}\mathit{all}v\in W.$$

Note that $J\left(W\right)\subseteq L\left(W\right).$ Define functions $\psi :[0,\infty )\to [0,\infty )$, $\phi :W\to [0,\infty )$ and $F:{[0,\infty )}^3\to [0,\infty )$ by

$$\begin{array}{ccc}\hfill \mathit{\psi }\left(p\right)& =& {\displaystyle \frac{p}{2}},\mathit{for}\mathit{all}p\in [0,\infty )\hfill \\ \hfill \phi \left(v\right)& =& v,\mathit{for}\mathit{all}v\in W\hfill \\ \hfill \mathit{and}F(u,x,i)& =& u+x+i,\mathit{for}\mathit{all}u,x,i\in [0,\infty ).\hfill \end{array}$$

Now we will show that J is a $(F,\phi ,\psi )$-contraction with respect to $L.$ Let $v,w\in W,$ then

$$F\left(m\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right)=9-3v-3w\le {\displaystyle \frac{3v}{4}}+{\displaystyle \frac{3w}{4}}=\mathit{\psi }\left(F\left(m\left(Lv,Lw\right),\phi \left(Lv\right),\phi \left(Lw\right)\right)\right).$$

Note that $Cp(J,L)=\left\{1\right\},$ $Poc(J,L)=\left\{1\right\}$ and $D\left(Poc\right(J,L\left)\right)=0.$ Then $(J,L)$ is a S-Op. Therefore, all the requirements of Corollary (1) are satisfied. Furthermore, $w=1$ is a unique φ-Cfp.

Since an m-MS is a p-MS, from Theorem (1) we immediately conclude the following result. It should be noted that the notions in Definitions (17) and (18) are considered in the framework of p-MS in the following result.

Corollary 3.

Let $(W,\rho )$ be a complete p-MS, $J,L:W\to W$ and $F\in \mathcal{F}$ with $J\left(W\right)\subseteq L\left(W\right)$. Let us assume that one of the subspaces $JW$ and $LW$ is closed in $W,$ where $(J,L)$ is a S-Op and φ is a lower semicontinuous function. If J is a $(F,\phi ,\psi ,Z)$-contraction with respect to $L,$ then J and L has a unique φ-Cfp.

Proof. 

We deduce the result from Theorem (1) since a m-MS is a generalization of p-MS. □

Corollary 4.

Let $(W,\rho )$ be a complete p-MS, $J,L:W\to W$ and $F\in \mathcal{F}$ with $J\left(W\right)\subseteq L\left(W\right)$. Let us assume that one of the subspaces $JW$ and $LW$ is closed in $W,$ where $(J,L)$ is a S-Op and φ is a lower semicontinuous function. If J is a $(F,\phi ,\psi )$-contraction with respect to $L,$ then J and L has a unique φ-Cfp.

Proof. 

We deduce the result from Corollary (1) since a m-MS is a generalization of p-MS. □

Corollary 5.

Let $(W,\rho )$ be a complete p-MS, $J,L:W\to W$ and $F\in \mathcal{F}$ with $J\left(W\right)\subseteq L\left(W\right)$. Let us assume that one of the subspaces $JW$ and $LW$ is closed in $W,$ where $(J,L)$ is a WComp-map and φ is a lower semicontinuous function. If J is a $(F,\phi ,\psi )$-contraction with respect to $L,$ then J and L has a unique φ-Cfp.

We present the following example in order to bolster Corollary (4).

Example 7.

Let $W=\left[0,1\right]$. Define the mapping $\rho :W^2\to \mathbb{R}$ by

$$\rho (v,w)=max\left\{v,w\right\},\mathit{for}\mathit{all}v,w\in W.$$

Then $(W,\rho )$ is a p-MS. Define a mapping $J,L:W\to W$ as:

$$Jv=\left\{\begin{array}{c}{\displaystyle \frac{v}{2}}\mathit{if}v\in \left\{0,1\right\},\\ 0,\mathit{otherwise},\end{array}\right.$$

and

$$Lv=\left\{\begin{array}{c}2v,\mathit{if}v\in \left\{0,{\displaystyle \frac{1}{2}}\right\},\\ {\displaystyle \frac{1}{2}},\mathit{if}v={\displaystyle \frac{1}{3}},\\ 0,\mathit{otherwise}.\end{array}\right.$$

Note that $J\left(W\right)\subseteq L\left(W\right).$ Define functions $\psi :[0,\infty )\to [0,\infty )$, $\phi :W\to [0,\infty )$ and $F:{[0,\infty )}^3\to [0,\infty )$ by

$$\begin{array}{ccc}\hfill \mathit{\psi }\left(p\right)& =& {\displaystyle \frac{p}{2}},\mathit{for}\mathit{all}p\in [0,\infty ),\hfill \\ \hfill \phi \left(v\right)& =& v,\mathit{for}\mathit{all}v\in W\hfill \\ \hfill \mathit{and}F(u,x,i)& =& u+x+i,\mathit{for}\mathit{all}u,x,i\in [0,\infty ).\hfill \end{array}$$

Now we will show that J is a $(F,\phi ,\psi )$-contraction with respect to $L.$ Let $v,w\in W,$ then

$$F\left(\rho \left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right)\le \mathit{\psi }\left(F\left(\rho \left(Lv,Lw\right),\phi \left(Lv\right),\phi \left(Lw\right)\right)\right).$$

Note that $Cp(J,L)=\left\{0\right\}$, $Poc(J,L)=\left\{0\right\}$ and $D\left(Poc\right(J,L\left)\right)=0$. Then the pair $(J,L)$ is a $S-$Op. Therefore, all the requirements of Corollary (4) are satisfied. Furthermore, $w=0$ is a unique φ-Cfp.

Since m-MS and p-MS are an extension of MS, from Theorem (1) we deduce immediately the following results. Note that in the following results we consider the notions in Definitions (17) and (18) in the framework of MS.

Corollary 6.

Let $(W,l)$ be a complete MS, $J,L:W\to W$ and $F\in \mathcal{F}$ with $J\left(W\right)\subseteq L\left(W\right)$. Let us assume that one of the subspaces $JW$ and $LW$ is closed in $W,$ where $(J,L)$ is a S-Op and φ is a lower semicontinuous function. If J is a $(F,\phi ,\psi ,Z)$-contraction with respect to $L,$ then J and L has a unique φ-Cfp.

Proof. 

We deduce the result from Theorem (1) and Corollary (3) since MS is a generalization of p-MS and m-MS. □

Corollary 7.

Let $(W,l)$ be a complete MS, $J,L:W\to W$ and $F\in \mathcal{F}$ with $J\left(W\right)\subseteq L\left(W\right)$. Let us assume that one of the subspaces $JW$ and $LW$ is closed in $W,$ where $(J,L)$ is a S-Op and φ is a lower semicontinuous function. If J is a $(F,\phi ,\psi )$-contraction with respect to $L,$ then J and L has a unique φ-Cfp.

Proof. 

We deduce the result from Corollaries (1) and (4) since MS is a generalization of p-MS and m-MS. □

Corollary 8.

Let $(W,l)$ be a complete MS, $J,L:W\to W$ and $F\in \mathcal{F}$ with $J\left(W\right)\subseteq L\left(W\right)$. Let us assume that one of the subspaces $JW$ and $LW$ is closed in $W,$ where $(J,L)$ is a WComp-map and φ is a lower semicontinuous function. If J is a $(F,\phi ,\psi )$-contraction with respect to $L,$ then J and L has a unique φ-Cfp.

Remark 12.

It should be noted that Corollary (8) is a result deduced by Karapinar et al. [27].

4. $\phi$-Fixed Point Results

In this section, we derive some corollaries as generalizations of various results in the literature, as a consequence of our new proved results. Putting $L=I_W$, the identity mapping on W, in Theorem (1), we conclude the following definitions and corollaries, which seem to be new to the existing literature.

Definition 19.

Let $(W,m)$ be a m-MS. A mapping J is known as a $(F,\phi ,\psi ,Z)$-contraction if there is a lower semicontinuous function $\phi :W\to \left[0,\infty \right)$, such that

$$F\left(m\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right)\le \mathit{\psi }\left(Z(v,w)\right),$$

where

$$Z(v,w)=max\left\{\begin{array}{c}F\left(m\left(v,w\right),\phi \left(v\right),\phi \left(w\right)\right),\\ F\left(m\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right),\\ F\left(m\left(w,Jw\right),\phi \left(w\right),\phi \left(Jw\right)\right),\\ {\displaystyle \frac{1}{2}}\left[\begin{array}{c}F\left(m\left(v,w\right),\phi \left(v\right),\phi \left(w\right)\right)\\ +F\left(m\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right)\end{array}\right]\end{array}\right\},$$

for all $v,w\in W$, $\psi \in \mathsf{\Psi }$ and $F\in \mathcal{F}$.

Definition 20.

Let $(W,m)$ be a m-MS. A mapping J is known as a $(F,\phi ,\psi )$-contraction if there is a lower semicontinuous function $\phi :W\to \left[0,\infty \right)$, such that

$$F\left(m\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right)\le \mathit{\psi }\left(F\left(m\left(v,w\right),\phi \left(v\right),\phi \left(w\right)\right)\right),$$

for all $v,w\in W$, $\psi \in \mathsf{\Psi }$ and $F\in \mathcal{F}$.

Corollary 9.

Let $(W,m)$ be a m-complete m-MS and $J:W\to W$, $F\in \mathcal{F}$. Then J has a unique φ-fp provided that J is a $(F,\phi ,\psi ,Z)$-contraction.

Proof. 

By putting $L=I_W$ and using Definition (19) in Theorem (1), we immediately deduce the result. □

Corollary 10.

Let $(W,m)$ be a m-complete m-MS and $J:W\to W$, $F\in \mathcal{F}$. Then J has a unique φ-fp provided that J is a $(F,\phi ,\psi )$-contraction.

Proof. 

By taking $Z(v,w)=F$$\left(m\left(v,w\right),\phi \left(v\right),\phi \left(w\right)\right)$ in Corollary (9), we immediately deduce the result. □

Since an m-MS is a p-MS, from Corollaries (9) and (10) we immediately conclude the following results. It should be noted that the notions in Definitions (19) and (20) are considered in the framework of p-MS in the following results.

Corollary 11.

Let $(W,p)$ be a complete p-MS and $J:W\to W$, $F\in \mathcal{F}$. Then J has a unique φ-fp provided that J is a $(F,\phi ,\psi ,Z)$-contraction.

Proof. 

We deduce the result from Corollary (9) since an m-MS is a generalization of p-MS. □

Corollary 12.

Let $(W,p)$ be a complete p-MS and $J:W\to W$, $F\in \mathcal{F}$. Then J has a unique φ-fp provided that J is a $(F,\phi ,\psi )$-contraction.

Proof. 

We deduce the result from Corollary (10) since an m-MS is a generalization of p-MS. □

Since m-MS and p-MS are an extension of MS, then we deduce the following two Corollaries in the setting of MS. Note that in the following Corollaries we consider the notions in Definitions (17) and (18) in the framework of MS.

Corollary 13.

Let $(W,l)$ be a complete MS and $J:W\to W$, $F\in \mathcal{F}$. Then J has a unique φ-fp provided that J is a $(F,\phi ,\psi ,Z)$-contraction.

Proof. 

We deduce the result from Corollary (9) and Corollary (11) since m-MS and p-MS are a generalization of MS. □

Corollary 14.

Let $(W,l)$ be a complete MS and $J:W\to W$, $F\in \mathcal{F}$. Then J has a unique φ-fp provided that J is a $(F,\phi ,\psi )$-contraction.

Proof. 

We deduce the result from Corollaries (10) and (12) since m-MS and p-MS are a generalization of MS. □

Remark 13.

It should be noted that Corollary (14) is a result deduced by Imdad et al. [26].

5. Application to System of Integral Equations

We investigated in the existence and uniqueness of a common solution for the following system of integral equations in this part to demonstrate the applicability of Corollary (1):

$$\begin{array}{ccc}\hfill h\left(j\right)& =& q\left(j\right)+{\int }_0^{j}K(j,\mathfrak{o},h\left(\mathfrak{o}\right))d\mathfrak{o},j\in \left[0,1\right],\hfill \end{array}$$

(17)

$$\begin{array}{ccc}\hfill h\left(j\right)& =& f\left(j\right)+{\int }_0^{j}P(j,\mathfrak{o},h\left(\mathfrak{o}\right))d\mathfrak{o},j\in \left[0,1\right],\hfill \end{array}$$

(18)

where $K,P:\left[0,1\right]\times \left[0,1\right]\times \mathbb{R}\to \mathbb{R}$ and $q,f:\left[0,1\right]\to \mathbb{R}$ are given functions. Let $W=C\left(\left[0,1\right],\mathbb{R}\right)$ denote the set of all real valued continuous functions defined on $\left[0,1\right].$ For an arbitrary $h\in W,$ define $m:W\times W\to \mathbb{R},$ that is

$$m(h\left(j\right),v\left(j\right))=\underset{j\in \left[\mathfrak{a},b\right]}{sup}\left({\displaystyle \frac{\left|h\left(j\right)\right|+\left|v\left(j\right)\right|}{2}}\right),\mathrm{for}\mathrm{all}v,w\in W.$$

Then $(W,m)$ is an m-complete $m-$MS.

Now we are equipped to state and prove our result as follows:

Theorem 2.

Take into consideration the system of Equations (17) and (18). Assume that the following conditions are satisfied:

(1)

$K,P,q$ and f are continuous functions.

(2)

$L,J:W\to W$ are two mappings defined by

$$\begin{array}{ccc}\hfill Jh\left(j\right)& =& q\left(j\right)+{\int }_0^{j}K(j,\mathfrak{o},h\left(\mathfrak{o}\right))d\mathfrak{o},j\in \left[0,1\right],\hfill \\ \hfill Lh\left(j\right)& =& f\left(j\right)+{\int }_0^{j}P(j,\mathfrak{o},h\left(\mathfrak{o}\right))d\mathfrak{o},j\in \left[0,1\right],\hfill \end{array}$$

with property that $l(v,Lv)\le D\left(Poc\right(J,L\left)\right),$ for some $v\in Poc(J,L).$

(3)

For all $h,v\in W$ and $j,o\in \left[0,1\right],$ we have

$$\left|K(j,\mathfrak{o},h(\mathfrak{o}\left)\right)+K(j,\mathfrak{o},v(\mathfrak{o}\left)\right)\right|\le {\displaystyle \frac{1}{2}}\left|Lh+Lv\right|.$$

Then there exists a unique common solution for the system of integral Equations (17) and (18).

Proof. 

For all $h,v\in W$ and $j,o\in \left[0,1\right],$ we have

$$\begin{array}{ccc}\hfill m\left(Jh\right(j),Jv(j\left)\right)& =& \left|{\displaystyle \frac{J\left(h\left(j\right)\right)+J\left(v\left(j\right)\right)}{2}}\right|\hfill \\ & =& \left|{\int }_0^j\left({\displaystyle \frac{K(j,\mathfrak{o},h(\mathfrak{o}\left)\right)+K(j,\mathfrak{o},v(\mathfrak{o}\left)\right)}{2}}\right)d\mathfrak{o}\right|\hfill \\ & \le & {\int }_0^j\left|{\displaystyle \frac{K(j,\mathfrak{o},h(\mathfrak{o}\left)\right)+K(j,\mathfrak{o},v(\mathfrak{o}\left)\right)}{2}}\right|d\mathfrak{o}\hfill \\ & \le & {\int }_0^j\left|{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{Lh+Lv}{2}}\right)\right|d\mathfrak{o}\hfill \\ & \le & {\displaystyle \frac{1}{2}}\left({\int }_0^j{\displaystyle \frac{\left|Lh\right|+\left|Lv\right|}{2}}d\mathfrak{o}\right)\hfill \\ & \le & {\displaystyle \frac{1}{2}}\left(\underset{j\in \left[\mathfrak{a},b\right]}{sup}{\displaystyle \frac{\left|Lh\right|+\left|Lv\right|}{2}}j\right)\hfill \\ & =& {\displaystyle \frac{1}{2}}\left(m\left(Lh,Lv\right)j\right),\hfill \end{array}$$

which on taking supremum leads to

$$m(Jh\left(j\right),Jv\left(j\right))\le {\displaystyle \frac{1}{2}}\left(m\left(Lh,Lv\right)\right).$$

Now, we write the three essential functions $\psi ,F$ and $\phi$ as

$$F(u,x,i)=u+x+i,\mathrm{for}\mathrm{all}u,x,i\in [0,\infty ),\mathit{\psi }\left(p\right)={\displaystyle \frac{p}{2}},$$

for all $p\in [0,\infty )$ where $\psi \in \mathsf{\Psi }$ and

$$\phi \left(v\right)=0\mathrm{for}\mathrm{all}v\in W.$$

Hence, the above inequality can be written as

$$F\left(m\left(Jh\left(j\right),Jv\left(j\right)\right),\phi \left(Jh\left(j\right)\right),\phi \left(Jv\left(j\right)\right)\right)\le \mathit{\psi }\left(F\left(m\left(Lh\left(j\right),Lv\left(j\right)\right),\phi \left(Lh\left(j\right)\right),\phi \left(Lv\left(j\right)\right)\right)\right).$$

By Corollary (1), there exists a unique common solution for the system of integral Equations (17) and (18). □

6. Existence of a Solution for Nonlinear Fractional Differential Equation

This section presents an application of Corollary (1) to a nonlinear fractional differential equation in the context of m-MS, where we can use $(F,\phi ,\psi ,Z)$-contractive mapping to achieve a common solution. Here, we look into the nonlinear fractional differential equation’s Caputo derivative using the fractional order. For a continuous function $q:\left[0,\infty \right)\to \mathbb{R}$, this fractional derivative form is provided as

$$\left({}^{c}D_r^{\gamma }\right)q\left(r\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(w-\gamma \right)}}{\int }_a^r{\left(r-\mathfrak{o}\right)}^{w-\gamma -1}{q}^w\left(\mathfrak{o}\right)d\mathfrak{o},\left(w-1<\gamma ,w=\left[\gamma \right]+1\right),$$

(19)

where $\left[\gamma \right]$ stands for the real number’s integer part $\gamma$ (see [28,29]). Furthermore, the order $\gamma$ fractional integral of Riemann–Liouville is denoted by

$$\left(I_{\mathfrak{o}}^{\gamma }\right)q\left(r\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left(\gamma \right)-1}}{\int }_a^r{\left(r-\mathfrak{o}\right)}^{\gamma }q\left(\mathfrak{o}\right)d\mathfrak{o},\left(\gamma >0\right),$$

(20)

The Caputo fractional differential equation finds wide-ranging mathematical applications in fields such as probability theory, physics, acoustics, digital data processing, electrical signals, image processing and physics (see [6]). Inspired by Kilbas et al. [30], Budhia et al. [31], Kanwal et al. [29] and Baleanu et al. [28], the following nonlinear fractional differential equation is

$$\left\{\begin{array}{c}{}^{c}D_{\mathfrak{o}}h\left(r\right)=q\left(r,h\left(r\right)\right),r\in \left(0,1\right),1<\gamma \le 2,\\ h\left(0\right)=0,h\left(1\right)={\int }_0^{v}h\left(\mathfrak{o}\right)d\mathfrak{o},\left(0<v<1\right),\end{array}\right.$$

(21)

The Caputo fractional derivative of order $\gamma$ is represented as ${}^{c}D_{\mathfrak{o}}$ and $q:\left[0,1\right]\to W$ is a continuous function. Let the set of all real valued continuous functions defined on $\left[0,1\right]$ be represented by the symbol $W=C\left(\left[0,1\right],\mathbb{R}\right)$. Define $m:W\times W\to \mathbb{R}$ for an arbitrary $h\in W$, that is

$$m(h\left(r\right),v\left(r\right))=\underset{r\in \left[\mathfrak{a},b\right]}{sup}\left({\displaystyle \frac{\left|h\left(r\right)\right|+\left|v\left(r\right)\right|}{2}}\right),\mathrm{for}\mathrm{all}h,v\in W.$$

(22)

Then $(W,m)$ is an m-complete m-MS. The expression for the nonlinear fractional Equation (21) is

$$\begin{array}{ccc}\hfill h\left(r\right)& =& {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^r{\left(r-\mathfrak{o}\right)}^{\gamma -1}q\left(\mathfrak{o},h\left(\mathfrak{o}\right)\right)d\mathfrak{o}-{\displaystyle \frac{2\iota }{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^1{\left(1-\mathfrak{o}\right)}^{\gamma -1}q\left(\mathfrak{o},h\left(\mathfrak{o}\right)\right)d\mathfrak{o}\hfill \\ & & +{\displaystyle \frac{2\iota }{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^v\left({\int }_0^{\mathfrak{o}}{\left(\mathfrak{o}-w\right)}^{\gamma -1}q\left(w,h\left(w\right)\right)dw\right)d\mathfrak{o}.\hfill \end{array}$$

(23)

A function $h\in C\left(\left[0,1\right],\mathbb{R}\right)$ is a solution of the fractional differential integral Equation (23) if and only if a solution of the nonlinear fractional differential Equation (21) is h. Now, we prove the following theorem.

Theorem 3.

Assume that the following conditions are satisfied:

(i)

$q\in C\left(I\times W,W\right)$ is sequentially continuous.

(ii)

There exists a continuous function $q:\left[0,1\right]\times \mathbb{R}\to {\mathbb{R}}^{+},$ such that

$$\left|q\left(r,h\left(\mathfrak{o}\right)\right)-q\left(r,v\left(\mathfrak{o}\right)\right)\right|\le {\displaystyle \frac{P}{2}}\left|Lh\left(\mathfrak{o}\right)+Lv\left(\mathfrak{o}\right)\right|,$$

for all $r\in \left[0,1\right]$ and for all $h,v\in W$, such that $l(h\left(r\right),v\left(r\right)\ge 0)$ and a constant P with a constant K relate with $P,$ such that

$$\begin{array}{ccc}\hfill PK& \le & 1,\hfill \\ \hfill K& =& \left({\displaystyle \frac{r^{\gamma }}{\alpha \mathsf{\Gamma }\left(\gamma \right)}}+{\displaystyle \frac{2\iota }{\left(2-v^2\right)\alpha \mathsf{\Gamma }\left(\gamma \right)}}+{\displaystyle \frac{2\iota v^{\gamma +1}}{\left(2-v^2\right)\gamma \left(\gamma +1\right)\mathsf{\Gamma }\left(\gamma \right)}}\right).\hfill \end{array}$$

Then, nonlinear fractional differential Equation (21) has a common solution as a φ-Cfp $w\in C\left(I,W\right).$

Proof. 

Let us define $J:C\left(I\right)\to C\left(I\right)$ by

$$\begin{array}{ccc}\hfill Jh\left(r\right)& =& {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^r{\left(r-\mathfrak{o}\right)}^{\gamma -1}q\left(\mathfrak{o},h\left(\mathfrak{o}\right)\right)d\mathfrak{o}-{\displaystyle \frac{2\iota }{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^1{\left(1-\mathfrak{o}\right)}^{\gamma -1}q\left(\mathfrak{o},h\left(\mathfrak{o}\right)\right)d\mathfrak{o}\hfill \\ & & +{\displaystyle \frac{2\iota }{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^v\left({\int }_0^{\mathfrak{o}}{\left(\mathfrak{o}-w\right)}^{\gamma -1}q\left(w,h\left(w\right)\right)dw\right)d\mathfrak{o}.\hfill \end{array}$$

Now we will show that J is a $(F,\phi ,\psi ,Z)$-contraction with respect to $L.$ By condition (ii), we have

$$\begin{array}{ccc}\hfill m\left(Jh\right(r),Jv(r\left)\right)& =& \underset{r\in \left[0,1\right]}{sup}\left|{\displaystyle \frac{J\left(h\left(r\right)\right)+J\left(v\left(r\right)\right)}{2}}\right|\hfill \\ & =& {\displaystyle \frac{1}{2}}\underset{r\in \left[0,1\right]}{sup}\left|\begin{array}{c}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^r{\left(r-\mathfrak{o}\right)}^{\gamma -1}q\left(\mathfrak{o},h\left(\mathfrak{o}\right)\right)d\mathfrak{o}-\\ {\displaystyle \frac{2\iota }{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^1{\left(1-\mathfrak{o}\right)}^{\gamma -1}q\left(\mathfrak{o},h\left(\mathfrak{o}\right)\right)d\mathfrak{o}+\\ {\displaystyle \frac{2\iota }{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^v\left({\int }_0^{\mathfrak{o}}{\left(\mathfrak{o}-w\right)}^{\gamma -1}q\left(w,h\left(w\right)\right)dw\right)d\mathfrak{o}+\\ {\displaystyle \frac{1}{\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^r{\left(r-\mathfrak{o}\right)}^{\gamma -1}q\left(\mathfrak{o},v\left(\mathfrak{o}\right)\right)d\mathfrak{o}-\\ {\displaystyle \frac{2\iota }{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^1{\left(1-\mathfrak{o}\right)}^{\gamma -1}q\left(\mathfrak{o},v\left(\mathfrak{o}\right)\right)d\mathfrak{o}+\\ {\displaystyle \frac{2\iota }{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^v\left({\int }_0^{\mathfrak{o}}{\left(\mathfrak{o}-w\right)}^{\gamma -1}q\left(w,v\left(w\right)\right)dw\right)d\mathfrak{o}\end{array}\right|\hfill \\ & \le & {\displaystyle \frac{1}{2}}\underset{r\in \left[0,1\right]}{sup}\left(\begin{array}{c}{\displaystyle \frac{1}{\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^r{\left(r-\mathfrak{o}\right)}^{\gamma -1}\left|q\left(\mathfrak{o},h\left(\mathfrak{o}\right)\right)+q\left(\mathfrak{o},v\left(\mathfrak{o}\right)\right)\right|d\mathfrak{o}\\ +{\displaystyle \frac{2\iota }{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^1{\left(1-\mathfrak{o}\right)}^{\gamma -1}\left|q\left(\mathfrak{o},h\left(\mathfrak{o}\right)\right)+q\left(\mathfrak{o},v\left(\mathfrak{o}\right)\right)\right|d\mathfrak{o}\\ +{\displaystyle \frac{2\iota }{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^v\left({\int }_0^{\mathfrak{o}}{\left(\mathfrak{o}-w\right)}^{\gamma -1}\left|q\left(w,h\left(w\right)\right)+q\left(w,v\left(w\right)\right)\right|dw\right)d\mathfrak{o}\end{array}\right)\hfill \\ & \le & {\displaystyle \frac{1}{2}}\underset{r\in \left[0,1\right]}{sup}\left(\begin{array}{c}{\displaystyle \frac{L}{2\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^r{\left(r-\mathfrak{o}\right)}^{\gamma -1}\left|Lh\left(\mathfrak{o}\right)+Lv\left(\mathfrak{o}\right)\right|d\mathfrak{o}\\ +{\displaystyle \frac{2\iota L}{2\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^1{\left(1-\mathfrak{o}\right)}^{\gamma -1}\left|Lh\left(\mathfrak{o}\right)+Lv\left(\mathfrak{o}\right)\right|d\mathfrak{o}\\ +{\displaystyle \frac{2\iota L}{2\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}{\int }_0^v\left({\int }_0^{\mathfrak{o}}{\left(\mathfrak{o}-w\right)}^{\gamma -1}\left|Lh\left(w\right)+Lv\left(w\right)\right|dw\right)d\mathfrak{o}\end{array}\right)\hfill \end{array}$$

So, we obtain

$$\begin{array}{ccc}\hfill m\left(Jh\right(r),Jv(r\left)\right)& \le & {\displaystyle \frac{1}{2}}\left(\begin{array}{c}{\displaystyle \frac{L}{\mathsf{\Gamma }\left(\gamma \right)}}\underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{\left|Lh\left(\mathfrak{o}\right)|+|Lv\left(\mathfrak{o}\right)\right|}{2}}{\int }_0^r{\left(r-\mathfrak{o}\right)}^{\gamma -1}d\mathfrak{o}\\ +{\displaystyle \frac{2\iota L}{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}\underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{\left|Lh\left(\mathfrak{o}\right)|+|Lv\left(\mathfrak{o}\right)\right|}{2}}{\int }_0^1{\left(1-\mathfrak{o}\right)}^{\gamma -1}d\mathfrak{o}\\ +{\displaystyle \frac{2\iota L}{\left(2-v^2\right)\mathsf{\Gamma }\left(\gamma \right)}}\underset{w\in \left[0,1\right]}{sup}{\displaystyle \frac{\left|Lh\left(w\right)|+|Lv\left(w\right)\right|}{2}}{\int }_0^v\left({\int }_0^{\mathfrak{o}}{\left(\mathfrak{o}-w\right)}^{\gamma -1}\right)d\mathfrak{o}\end{array}\right)\hfill \\ & \le & {\displaystyle \frac{1}{2}}\left({\displaystyle \frac{L{\iota }^{\gamma }}{\alpha \mathsf{\Gamma }\left(\gamma \right)}}+{\displaystyle \frac{2\iota L}{\left(2-v^2\right)\alpha \mathsf{\Gamma }\left(\gamma \right)}}+{\displaystyle \frac{2\iota L{v}^{\gamma +1}}{\left(2-v^2\right)\gamma \left(\gamma +1\right)\mathsf{\Gamma }\left(\gamma \right)}}\right)m\left(Lh,Lv\right)\hfill \\ & =& {\displaystyle \frac{L}{2}}\left({\displaystyle \frac{r^{\gamma }}{\alpha \mathsf{\Gamma }\left(\gamma \right)}}+{\displaystyle \frac{2\iota }{\left(2-v^2\right)\alpha \mathsf{\Gamma }\left(\gamma \right)}}+{\displaystyle \frac{2\iota v^{\gamma +1}}{\left(2-v^2\right)\gamma \left(\gamma +1\right)\mathsf{\Gamma }\left(\gamma \right)}}\right)m\left(Lh,Lv\right)\hfill \\ & \le & {\displaystyle \frac{LK}{2}}m\left(Lh,Lv\right),\hfill \end{array}$$

which on taking supremum leads to

$$m(Jh\left(r\right),Jv\left(r\right))\le {\displaystyle \frac{LK}{2}}m\left(Lh,Lv\right).$$

Since $LK<1,$ we have

$$m(Jh\left(r\right),Jv\left(r\right))\le {\displaystyle \frac{1}{2}}m\left(Lh,Lv\right).$$

Now, we write the three essential functions $\psi ,F$ and $\phi$ as

$$F(u,x,i)=u+x+i,\mathrm{for}\mathrm{all}u,x,i\in [0,\infty ),\mathit{\psi }\left(p\right)={\displaystyle \frac{p}{2}},$$

for all $p\in [0,\infty )$, where $\psi \in \mathsf{\Psi }$ and

$$\phi \left(v\right)=0,\mathrm{for}\mathrm{all}v\in W.$$

Hence, the above inequality can be written as

$$F\left(m\left(Jh\left(r\right),Jv\left(r\right)\right),\phi \left(Jh\left(r\right)\right),\phi \left(Jv\left(r\right)\right)\right)\le \mathit{\psi }\left(F\left(m\left(Lh\left(r\right),Lv\left(r\right)\right),\phi \left(Lh\left(r\right)\right),\phi \left(Lv\left(r\right)\right)\right)\right).$$

Hence, J is a $(F,\phi ,\psi )$-contraction with respect to $L.$ Since all the criteria of Corollary (1) are satisfied, there exists $w\in C\left(I\right)$, a $\phi$-Cfp of J and $L,$ that is, w is a solution to the nonlinear fractional differential Equation (21).

□

7. Existence of Common Solution of Ordinary Differential Equations for Damped Forced Oscillations

In this section, the damped forced oscillation differential equations problem is examined in the context of m-MS. In 2020, the damped forced oscillation differential problem of an object of mass m moving to and fro on the v-axis around an equilibrium position $v=0$ was examined by Shoib et al. [32]. At time r, the object’s position is $h\left(r\right).$ It experiences a force due to spring

$$F_s=ku,$$

Moreover, a damping force that resists the object’s movement is shown:

$$F_f=b{\displaystyle \frac{dh}{dr}}.$$

Now, by the second law of motion

$$F_{net}=m{\displaystyle \frac{d^{2}h}{d{r}^2}}.$$

where $m,$ b and k are positive constants. Up to that feature, the system is simply the damped harmonic oscillator. Now, suppose the additional time-dependent force $q\left(r\right)$ is applied to the object. Then, by the second law given by Newton,

$$m{\displaystyle \frac{d^{2}h}{d{r}^2}}+b{\displaystyle \frac{dh}{dr}}+kh=q\left(r\right),$$

(24)

If it is assumed that the initial conditions are

$$h\left(0\right)=0,h^{\prime }\left(0\right)=0,$$

the problem (24) can be expressed as

$$h\left(r\right)={\int }_0^{\varkappa }M(r,\mathfrak{o})K\left(r,o,h\left(\mathfrak{o}\right)\right)d\mathfrak{o},\mathrm{for}\mathrm{all}r,\mathfrak{o}\in \left[0,\varkappa \right],$$

(25)

where $\varkappa >0.$ Let $W=C\left(I\right),$ $I=\left[0,\varkappa \right].$ For forced damped oscillation, the green function is defined as

$$M(r,\mathfrak{o})=\left\{\begin{array}{c}-\mathfrak{o}{e}^{\tau (\varkappa +\mathfrak{o}-r)},0\le \mathfrak{o}\le r\le \varkappa ,\\ -r{e}^{\tau (r-\mathfrak{o})},0\le r\le \mathfrak{o}\le \varkappa .\end{array}\right.$$

where $\tau$ can be expressed using $b,$ k and m. Drawing inspiration from the work of Shoaib et al. [32], we apply the fixed point method to determine the common solution of a damped forced oscillation differential equation.

Theorem 4.

Assume that the following conditions are satisfied:

(i)

There exists a continuous function $K:{\left[0,\varkappa \right]}^2\to \left[0,\infty \right),$ such that

$$\left|K\left(r,\mathfrak{o},h\left(o\right)\right)-K\left(r,\mathfrak{o},v\left(o\right)\right)\right|\le {\displaystyle \frac{1}{2}}\left|Lh+Lv\right|,$$

for $r,o\in \left[0,\varkappa \right]$, $\gamma >0$ and $h,v\in \mathbb{R}.$

(ii)

There exists a function $L:I\times I\to {\mathbb{R}}_{+},$ such that

$$\underset{r\in \left[0,\varkappa \right]}{sup}{\int }_0^{\varkappa }L(r,\mathfrak{o})d\mathfrak{o}\le 1.$$

Then, problem (25) has a φ-Cfp $w\in W,$ which is a solution of (24).

Proof. 

Let $J:C^1\left[0,1\right]\to C^1\left[0,1\right]$ be an operator defined by

$$Jh\left(r\right)={\int }_0^{\varkappa }L\left(r,\mathfrak{o}\right)K(r,\mathfrak{o},h\left(\mathfrak{o}\right))d\mathfrak{o}.$$

Consider $h>v$, for all $h,v\in C\left(I\right)$ and (25), we have

$$\begin{array}{ccc}\hfill \underset{r\in \left[0,1\right]}{sup}\left|{\displaystyle \frac{Jh\left(r\right)+Jv\left(r\right)}{2}}\right|& =& \underset{r\in \left[0,1\right]}{sup}{\int }_0^{\varkappa }\left({\displaystyle \frac{L\left(r,\mathfrak{o}\right)\left[K(r,\mathfrak{o},h\left(\mathfrak{o}\right)-K(r,\mathfrak{o},v\left(\mathfrak{o}\right)\right]}{2}}\right)d\mathfrak{o}\hfill \\ & \le & \underset{r\in \left[0,1\right]}{sup}{\int }_0^{\varkappa }L\left(r,\mathfrak{o}\right){\displaystyle \frac{1}{2}}\left|{\displaystyle \frac{Lh\left(\mathfrak{o}\right)+Lv\left(\mathfrak{o}\right)}{2}}\right|d\mathfrak{o}\hfill \\ & \le & {\displaystyle \frac{1}{2}}{\displaystyle \frac{\left|Lh\left(\mathfrak{o}\right)\right|+\left|Lv\left(\mathfrak{o}\right)\right|}{2}}\underset{r\in \left[0,1\right]}{sup}{\int }_0^{\varkappa }L\left(r,\mathfrak{o}\right)d\mathfrak{o}\hfill \\ & \le & {\displaystyle \frac{1}{2}}{\displaystyle \frac{\left|Lh\left(\mathfrak{o}\right)\right|+\left|Lv\left(\mathfrak{o}\right)\right|}{2}}\hfill \\ & \le & {\displaystyle \frac{1}{2}}\underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{\left|Lh\left(\mathfrak{o}\right)\right|+\left|Lv\left(\mathfrak{o}\right)\right|}{2}}\hfill \\ & =& {\displaystyle \frac{1}{2}}\left(m\left(Lh,Lv\right)\right),\hfill \end{array}$$

hence, we obtain

$$m(Jh\left(r\right),Jv\left(r\right))\le {\displaystyle \frac{1}{2}}\left(m\left(Lh,Lv\right)\right).$$

Now, we write the three essential functions $\psi ,F$ and $\phi$ as

$$F(u,x,i)=u+x+i,\mathrm{for}\mathrm{all}u,x,i\in [0,\infty ),\mathit{\psi }\left(p\right)={\displaystyle \frac{p}{2}},$$

for all $p\in [0,\infty )$, where $\psi \in \mathsf{\Psi }$ and

$$\phi \left(v\right)=0\mathrm{for}\mathrm{all}v\in W.$$

Hence, the above inequality can be written as

$$F\left(m\left(Jh\left(r\right),Jv\left(r\right)\right),\phi \left(Jh\left(r\right)\right),\phi \left(Jv\left(r\right)\right)\right)\le \mathit{\psi }\left(F\left(m\left(Lh\left(r\right),Lv\left(r\right)\right),\phi \left(Lh\left(r\right)\right),\phi \left(Lv\left(r\right)\right)\right)\right).$$

Hence, J is a $(F,\phi ,\psi )$-contraction with respect to $L.$ Since all the criteria of Corollary (1) are satisfied, there exists $w\in C\left(I\right)$, a $\phi$-Cfp of J and $L,$ that is, w is a solution to the integral Equation (25).

□

8. Application to a Satellite Web Coupling Problem

We solved a satellite web coupling boundary value problem using Corollary (10), inspired by the use of fixed point techniques in real-world problems [33]. A satellite web connection can be imagined as a thin sheet joining two cylindrical satellites. The problem of radiation from the web coupling between two satellites leads to the following nonlinear boundary value problem defined as:

$$-{\displaystyle \frac{d^{2}h}{d{r}^2}}=\mu h^4,0<r<1,h\left(0\right)=h\left(1\right)=0,$$

(26)

where $h\left(r\right)$ denotes the temperature of radiation at any point $r\in \left[0,1\right],$ $\mu ={\displaystyle \frac{2a{l}^2{K}^3}{\zeta \hslash }}>0$ is a non-dimensional positive constant, K is the constant absolute temperature of both satellites, while heat is radiated from the surface of the web into space at 0 absolute temperature, I represents the distance between two satellites, a is a positive constant that describes the radiation qualities of the web’s surface, factor 2 is needed because there is radiation from both top and bottom surfaces, and $\zeta$ is thermal conductivity, where ℏ is the thickness. The green function is defined as

$$G\left(r,\xi \right)=\left\{\begin{array}{c}r\left(1-\xi \right),0<r<\xi \\ \xi \left(1-r\right),\xi <r<1\end{array}\right.$$

Problem (26) is equivalent to

$$h\left(r\right)=1-\mu {\int }_0^{1}G\left(r,\xi \right)h^4\left(\xi \right)d\xi .$$

Let $W=\mathcal{R}\left[0,1\right]$ be a set of Riemann integrable functions on $\left[0,1\right]$. Define $m:W\times W\to \mathbb{R}$ for an arbitrary $h\in W$, that is

$$m(h,v)=\underset{r\in \left[\mathfrak{a},b\right]}{sup}\left({\displaystyle \frac{\left|h\right|+\left|v\right|}{2}}\right),\mathrm{for}\mathrm{all}h,v\in W.$$

Clearly, $\left(W,m\right)$ is a complete m-metric space.

Theorem 5.

Let $J:W\to W$ be a self mapping in a complete m-metric space, satisfying

$$\left|h^2\left(\xi \right)+v^2\left(\xi \right)\right|\left|h\left(\xi \right)-v\left(\xi \right)\right|\le {\displaystyle \frac{K}{\mu }},$$

(27)

and

$$2+K\left|h\left(\xi \right)+v\left(\xi \right)\right|-8\left|h\left(\xi \right)+v\left(\xi \right)\right|\le 0.$$

(28)

Then, the satellite web coupling boundary value problem (26) has a unique φ-fp.

Proof. 

Define a self mapping $J:W\to W$ by

$$Jh\left(r\right)=1-\mu {\int }_0^{1}G\left(r,\xi \right)h^4\left(\xi \right)d\xi ,\xi \in \left[0,1\right].$$

A solution to the satellite web coupling problem (26) is clearly a fixed point of a self map A. Consider $h>v$, for all $h,v\in C\left(I\right)$ and (25), we have

$$\begin{array}{ccc}\hfill \underset{r\in \left[0,1\right]}{sup}\left|{\displaystyle \frac{Jh\left(r\right)+Jv\left(r\right)}{2}}\right|& =& \underset{r\in \left[0,1\right]}{sup}\left|{\displaystyle \frac{1-\mu {\int }_0^{1}G\left(r,\xi \right)h^4\left(\xi \right)d\xi +1-\mu {\int }_0^{1}G\left(r,\xi \right)v^4\left(\xi \right)d\xi }{2}}\right|\hfill \\ & =& \underset{r\in \left[0,1\right]}{sup}\left|{\displaystyle \frac{2-\mu {\int }_0^{1}G\left(r,\xi \right)\left(h^4\left(\xi \right)-v^4\left(\xi \right)\right)d\xi }{2}}\right|\hfill \\ & =& \underset{r\in \left[0,1\right]}{sup}\left|{\displaystyle \frac{2-\mu {\int }_0^{1}G\left(r,\xi \right)\left(h^2\left(\xi \right)+v^2\left(\xi \right)\right)\left(h\left(\xi \right)-v\left(\xi \right)\right)\left(h\left(\xi \right)+v\left(\xi \right)\right)d\xi }{2}}\right|\hfill \\ & =& \underset{r\in \left[0,1\right]}{sup}\left|{\displaystyle \frac{2-\mu \left(h^2\left(\xi \right)+v^2\left(\xi \right)\right)\left(h\left(\xi \right)-v\left(\xi \right)\right)\left(h\left(\xi \right)+v\left(\xi \right)\right){\int }_0^{1}G\left(r,\xi \right)d\xi }{2}}\right|\hfill \\ & \le & \underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{\left|2\right|+\left|\mu \left(h^2\left(\xi \right)+v^2\left(\xi \right)\right)\left(h\left(\xi \right)-v\left(\xi \right)\right)\left(h\left(\xi \right)+v\left(\xi \right)\right){\int }_0^{1}G\left(r,\xi \right)d\xi \right|}{2}}\hfill \\ & \le & \underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{2+\left|\begin{array}{c}\mu \left(h^2\left(\xi \right)+v^2\left(\xi \right)\right)\left(h\left(\xi \right)-v\left(\xi \right)\right)\left(h\left(\xi \right)+v\left(\xi \right)\right)\\ \left[{\int }_0^t\xi (1-r)d\xi +{\int }_t^{1}r\left(1-\xi \right)d\xi \right]\end{array}\right|}{2}}\hfill \\ & =& {\displaystyle \frac{1}{16}}\underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{2+\left|\mu \left(h^2\left(\xi \right)+v^2\left(\xi \right)\right)\left(h\left(\xi \right)-v\left(\xi \right)\right)\left(h\left(\xi \right)+v\left(\xi \right)\right)\right|}{2}}\hfill \\ & =& {\displaystyle \frac{1}{16}}\underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{2+\mu \left|h^2\left(\xi \right)+v^2\left(\xi \right)\right|\left|h\left(\xi \right)-v\left(\xi \right)\right|\left|h\left(\xi \right)+v\left(\xi \right)\right|}{2}}.\hfill \end{array}$$

Then, by using Conditions (27) and (28), we obtain

$$\begin{array}{ccc}\hfill \underset{r\in \left[0,1\right]}{sup}\left|{\displaystyle \frac{Jh\left(r\right)+Jv\left(r\right)}{2}}\right|& =& {\displaystyle \frac{1}{16}}\underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{2+K\left|h\left(\xi \right)+v\left(\xi \right)\right|}{2}}\hfill \\ & \le & {\displaystyle \frac{1}{16}}\underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{8\left|h\left(\xi \right)+v\left(\xi \right)\right|}{2}}\hfill \\ & =& {\displaystyle \frac{1}{2}}\underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{\left|h\left(\xi \right)+v\left(\xi \right)\right|}{2}}\hfill \\ & \le & {\displaystyle \frac{1}{2}}\underset{r\in \left[0,1\right]}{sup}{\displaystyle \frac{\left|h\left(\xi \right)\right|+\left|v\left(\xi \right)\right|}{2}}\hfill \\ & =& {\displaystyle \frac{1}{2}}\left(m\left(h,v\right)\right),\hfill \end{array}$$

hence, we obtain

$$m(Jh\left(r\right),Jv\left(r\right))\le {\displaystyle \frac{1}{2}}\left(m\left(h,v\right)\right).$$

Now, we write the three essential functions $\psi ,F$ and $\phi$ as

$$F(u,x,i)=u+x+i,\mathrm{for}\mathrm{all}u,x,i\in [0,\infty ),\mathit{\psi }\left(p\right)={\displaystyle \frac{p}{2}},$$

for all $p\in [0,\infty )$, where $\psi \in \mathsf{\Psi }$ and

$$\phi \left(v\right)=0\mathrm{for}\mathrm{all}v\in W.$$

Hence, the above inequality can be written as

$$F\left(m\left(Jv,Jw\right),\phi \left(Jv\right),\phi \left(Jw\right)\right)\le \mathit{\psi }\left(F\left(m\left(v,w\right),\phi \left(v\right),\phi \left(w\right)\right)\right).$$

Hence, J is a $(F,\phi ,\psi )$-contraction. Hence, all the postulates of Corollary (10) are validated. As a result, A has a unique $\phi$-fp, and a satellite web coupling problem (26) has a unique solution. □

9. Conclusions

In this article, we have proposed a new class of noncommuting self mappings known as S-Op. Further, we have proved some common $\phi$-fixed point results satisfying $(F,\phi ,\psi ,Z)$-contraction, which generalized the main results in [16,17,23,34] in the sense of m-MS. We have also proved some common $\phi$-fixed point results in the setting of p-MS. The supportive examples of obtained results to illustrate the usability are also provided. Also, we have presented some corollaries and related $\phi$-fixed point results to prove the validity of our results. Furthermore, we also provided the common $\phi$-fixed point theorems for $(F,\phi ,\psi )$-contraction to illustrate the existence of a solution to a system of integral equations, a nonlinear fractional differential equation, and an ordinary differential equation for damped forced oscillations. Additionally, the satellite web coupling problem is proved. The obtained results extend from those in [5] and also generalize the main theorem of M. Imdad et al. [26].