Solving a Fractional-Order Differential Equation Using Rational Symmetric Contraction Mappings

Abstract

The intent of this manuscript is to present new rational symmetric $\varpi -\xi$-contractions and infer some fixed-points for such contractions in the setting of $\mathrm{\Theta }$-metric spaces. Furthermore, some related results such as Suzuki-type rational symmetric contractions, orbitally $\mathrm{{\rm Y}}$-complete, and orbitally continuous mappings in $\mathrm{\Theta }$-metric spaces are introduced. Ultimately, the theoretical results are shared to study the existence of the solution to a fractional-order differential equation with one boundary stipulation.

1. Introduction and Fundamental Facts

Recently, fixed-point (FP) theory has spread widely because of its entry into many vital disciplines, such as topology, game theory, artificial intelligence, dynamical systems (and chaos), logic programming, economics, and optimal control. Furthermore, it has become an essential pillar of nonlinear analysis, where it is used to study the existence and uniqueness of the solutions for many differential and nonlinear integral equations [1,2,3,4,5,6,7,8,9,10,11].

Research into fractional derivatives of the Atangana–Baleanu-type and the integral operator of the Atangana–Baleanu-type exploring improvements in engineering sciences has been largely interrelated, which has left open the question of whether a geometrical representation of Atangana–Baleanu fractional derivatives can be established.

To investigate the existence of unique solutions to Fredholm integral equations, many researchers have applied the “FP perspective” as the authoritative research model for different spaces in convergence analysis and compactness. Computer programming scientists study the logical programming semantics under ordinary metric spaces because it is easy to formulate and figure out and is used to prove the results.

Moreover, the applications of FP theory in fractional analysis cannot be denied. This trend is important in modeling many phenomena in many areas of science and construction. Fractional-order differential equations (FODEs) have numerous applications in electrochemistry, viscoelasticity, dynamic systems, porous media control, polymer functional science, electromagnetism, and so forth; for example, see [12,13,14,15,16,17,18,19].

In addition, the FP technique is heavily involved in the solution of many fractional differential and integral equations, as well as the boundary value problems (BVPs) resulting from the fractional input. For more details, we suggest the works of [20,21,22,23,24,25,26] to the reader.

The main advantage of using fractional differential equations is related to the fact that we can describe the dynamics of complex nonlocal systems with memory. Moreover, nonlinear analysis is used in the study of dynamical systems represented by nonlinear differential and integral equations. Since some of these equations representing a dynamic system have no analytic solution, studying the perturbation of these problems is very useful. There are different types of perturbed differential equations, and an important type here is called hybrid differential equations [27]. Since then, this method has become a way of life for researchers.

In the setting of partially metric spaces, interpolative-type contractions were introduced by Aydi et al. [28]. They called it interpolative contractions and obtained some recent FP results in the mentioned space.

Recently, the ordinary metric space was extended to a $\mathrm{\Theta }$-metric space ($\mathrm{\Theta }$-MS) by Jleli and Samet [29] as follows:

Definition 1.

[29] Assume that $\mathrm{\Theta }$ is a set function $\theta :(0,+\infty )\to (-\infty ,+\infty )$ that satisfies the hypotheses below

$\left({\mathrm{\Theta }}_1\right)$

For all $r<s$, we have $\theta \left(r\right)<\theta \left(s\right),$ that is θ is nondecreasing;

$\left({\mathrm{\Theta }}_2\right)$

For each sequence $\left\{u_i\right\}\subset (0,+\infty ),$ we obtain ${lim}_{i\to +\infty }{u}_i=0\iff {lim}_{i\to +\infty }\theta \left(u_i\right)=-\infty ,$

A $\mathrm{\Theta }$-MS is presented as follows:

Definition 2.

[29] Let B be a nonempty set and $\mathrm{\Omega }:B\times B\to [0,+\infty )$ be a given mapping. Suppose that there is $(\theta ,\nu )\in \mathrm{\Theta }\times [0,+\infty )$, such that for all $m,n\in B$

$\left({\mathrm{\Omega }}_1\right)$

$\mathrm{\Omega }(m,n)=0\iff m=n=0;$

$\left({\mathrm{\Omega }}_2\right)$

$\mathrm{\Omega }(m,n)=\mathrm{\Omega }(n,m);$

$\left({\mathrm{\Omega }}_3\right)$

For all $M\in \mathbb{N},$$M\ge 2$ and for each ${\left\{b_i\right\}}_{i=1}^M\subset B$ with $\left(b_1,b_N\right)=(m,n),$ we obtain

$$\mathrm{\Omega }(m,n)>0\Rightarrow \theta \left(\mathrm{\Omega }(m,n)\right)\le \theta \left(\sum _{i=1}^{M-1}\mathrm{\Omega }(b_i,b_{i+1})\right)+\nu ,$$

then the pair $(B,\mathrm{\Omega })$ is called a Θ-MS.

Definition 3.

[29] A sequence $\left\{m_i\right\}$ in $(B,\mathrm{\Omega })$ is Θ-Cauchy, if ${lim}_{i,j\to \infty }\mathrm{\Omega }(m_i,m_j)=0.$ Moreover, $(B,\mathrm{\Omega })$ is Θ-complete, if every Θ-Cauchy sequence is Θ-convergent in $B.$

The example below was shown by [29].

Example 1.

[29] Assume that $B=[0,3],$ then the pair $(B,\mathrm{\Omega })$ is a Θ-MS if we define Ω by

$$\mathrm{\Omega }(m,n)=\left\{\begin{array}{cc}\left|m-n\right|,& ifm,n\notin [0,3]\times [0,3],\\ {(m-n)}^2& ifm,n\in [0,3]\times [0,3],\end{array}\right.$$

for all $(m,n)\in B\times B,$$\theta \left(a\right)=ln\left(a\right)$ and $\nu =ln\left(3\right).$ Moreover, Ω does not form a metric, but it is a Θ-MS.

Based on the above result, a simple Banach fixed-point theorem was proved by Jleli and Samet [29] as follows:

Theorem 1.

[29] Let $(B,\mathrm{\Omega })$ be a Θ-MS and $L:B\to B$ be a give mapping, then L has a unique FP $m^{*}\in B$ provided that the assertions below are fulfilled

(1) 

$(B,\mathrm{\Omega })$ is Θ-complete;

(2) 

There is a constant $\varkappa \in (0,1)$ such that

$$\mathrm{\Omega }(Lm,Ln)\le \varkappa \mathrm{\Omega }\left(m,n\right),\left(m,n\right)\in B\times B.$$

The class of $\alpha$-admissible mappings was described in 2012 by Samet et al. [30] as follows:

Definition 4.

[30] Assume that $\xi :B\times B\to [0,+\infty ).$ A self-mapping L on B is called α-admissible, if $m,n\in B$ and $\alpha (m,n)\ge 1$ implies that $\alpha (Lm,Ln)\ge 1.$

After that, the notion of $\alpha$-admissible mappings was modified by Hussain and Salimi [31] as the definition below.

Definition 5.

[31] Suppose that $\alpha ,\xi :B\times B\to [0,+\infty )$ are two given functions. A self-mapping L on B is called an α-admissible mapping with respect to (w.r.t.) $\xi ,$ if $m,n\in B$ and $\alpha (m,n)\ge \xi (m,n)$ leads to $\alpha (Lm,Ln)\ge \xi (Lm,Ln).$

It should be noted that if we set $\xi =1,$ then Definition 5 reduces to Definition 4, and if we choose $\alpha =1,$ then L is said to be an $\xi$-subadmissible mapping.

Karapinar and Samet [32] presented the concept of $\alpha -\xi$-continuous mappings as follows.

Definition 6.

[32] Let $(B,\mathrm{\Omega })$ be a metric space and $\alpha ,\xi :B\times B\to [0,+\infty )$ be two functions. A self-mapping L is called an $\alpha -\xi$-continuous mapping, if there are $m\in B$ and the sequence $\left\{m_i\right\}\subset B$ such that ${lim}_{i\to \infty }{m}_i=m$ and for all $i\in \mathbb{N},$

$$\alpha (m_i,m_{i+1})\ge \xi (m_i,m_{i+1})\Rightarrow \underset{i\to \infty }{lim}L{m}_i=Lm.$$

Moreover, a mapping $L:B\to B$ is called orbitally continuous (OC) in $e\in B$ if

$$\underset{i\to \infty }{lim}{L}^{i}m=e\Rightarrow \underset{i\to \infty }{lim}{L}^{i}m=Le.$$

A mapping L is called OC on B if it is OC for all $e\in B.$

Based on what was previously mentioned in this paper, some new rational symmetric $\varpi -\xi$-contractions are shown. Furthermore, some fixed-point theorems were obtained under these conditions. In addition, some illustrative examples of the theoretical results are addressed. Finally, the theoretical results were involved in the treatment of the analytical solution to a fractional-order differential equation with one boundary condition.

2. Rational Symmetric $\mathbf{\varpi }-\mathbf{\xi }$-Contraction of Kind (I)

In this part, we introduce a novel rational symmetric $\varpi -\xi$-contraction of kind (I).

Definition 7.

Let $(B,\mathrm{\Omega })$ be a Θ-MS and $\varpi ,\xi :B\times B\to [0,+\infty )$ be two functions. A mapping $\mathrm{{\rm Y}}:B\to B$ is called a rational symmetric $\varpi -\xi$-contraction of kind (I) if for each $\zeta \in [0,1),$ there exist $j,k,l\in (0,1)$ such that, whenever $\varpi (m,n)\ge \xi (m,n),$ we obtain

$$\mathrm{\Omega }{(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)}^q\le \zeta \left(\widehat{A_1}(m,n)\right),$$

(1)

where

$$\begin{array}{ccc}\hfill \widehat{A_1}(m,n)& =& \mathrm{\Omega }{(m,n)}^q.\frac{\mathrm{\Omega }{(m,\mathrm{{\rm Y}}m)}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(n,\mathrm{{\rm Y}}n)}^{\frac{q}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m,n)}^q}.\hfill \\ & & \frac{{\left[\mathrm{\Omega }(m,\mathrm{{\rm Y}}m)+\mathrm{\Omega }(n,\mathrm{{\rm Y}}n)\right]}^{\frac{q}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m,n)}^q}.\frac{{\left[\mathrm{\Omega }(m,\mathrm{{\rm Y}}n)+\mathrm{\Omega }(n,\mathrm{{\rm Y}}m)\right]}^{\frac{q}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m,n)}^q},\hfill \end{array}$$

for all $q\in [1,\infty )$ and $m,n\in B\backslash F\left(\mathrm{{\rm Y}}\right)$ (where $F\left(\mathrm{{\rm Y}}\right)$ is the set of all FPs of the mapping $\mathrm{{\rm Y}}).$

Example 2.

Consider $B=\{0,1,2,3\}$ with the Θ-metric Ω described by

$$\mathrm{\Omega }(m,n)=\left\{\begin{array}{cc}\left|m-n\right|,& ifm,n\notin B\times B,\\ {(m-n)}^2& ifm,n\in B\times B.\end{array}\right.$$

Let $\theta \left(a\right)=ln\left(a\right)$ and $\nu =ln\left(3\right).$ Define the mapping $\mathrm{{\rm Y}}:B\to B$ and functions $\varpi ,\xi :B\times B\to [0,\infty )$ as follows:

$$\mathrm{{\rm Y}}1=\mathrm{{\rm Y}}2=0,\mathrm{{\rm Y}}0=0and\mathrm{{\rm Y}}3=3,$$

$$\varpi (m,n)=\left\{\begin{array}{cc}2,& ifm,n\in B,\\ 0,& \mathrm{otherwise},\end{array}\right.$$

and

$$\xi (m,n)=\left\{\begin{array}{cc}1,& ifm,n\in B,\\ 0,& \mathrm{otherwise},\end{array}\right.$$

respectively. We find that, for each $m,n\in B\backslash F\left(\mathrm{{\rm Y}}\right),$$\varpi (m,n)\in B\ge \xi (m,n)$ such that

$$\begin{array}{ccc}\hfill \mathrm{\Omega }{(\mathrm{{\rm Y}}1,\mathrm{{\rm Y}}2)}^q& =& 0\le \zeta \left(\begin{array}{c}\mathrm{\Omega }{(1,2)}^q.\frac{\mathrm{\Omega }{(1,\mathrm{{\rm Y}}1)}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(2,\mathrm{{\rm Y}}2)}^{\frac{q}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(1,2)}^q}\\ .\frac{{\left[\mathrm{\Omega }(1,\mathrm{{\rm Y}}1)+\mathrm{\Omega }(2,\mathrm{{\rm Y}}2)\right]}^{\frac{q}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(1,2)}^q}.\frac{{\left[\mathrm{\Omega }(1,\mathrm{{\rm Y}}2)+\mathrm{\Omega }(2,\mathrm{{\rm Y}}1)\right]}^{\frac{q}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(1,2)}^q}\end{array}\right)\hfill \\ & =& \zeta \left(\begin{array}{c}1.\mathrm{\Omega }{(1,0)}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(2,0)}^{\frac{q}{(j-k)(j-l)}}.{\left[\mathrm{\Omega }(1,0)+\mathrm{\Omega }(2,0)\right]}^{\frac{q}{(k-j)(k-l)}}\\ .{\left[\mathrm{\Omega }(1,0)+\mathrm{\Omega }(2,0)\right]}^{\frac{q}{(l-j)(l-k)}}\end{array}\right)\hfill \\ & =& \zeta \left({\left(1\right)}^{\frac{q}{(j-k)(j-l)}}.{\left(4\right)}^{\frac{q}{(j-k)(j-l)}}.{\left(1+4\right)}^{\frac{q}{(k-j)(k-l)}}.{\left(1+4\right)}^{\frac{q}{(l-j)(l-k)}}\right)\hfill \\ & \le & \zeta \left({\left(1\right)}^{\frac{q}{(j-k)(j-l)}}.{\left(5\right)}^{\frac{q}{(j-k)(j-l)}}.{\left(1.5\right)}^{\frac{q}{(k-j)(k-l)}}.{\left(1.5\right)}^{\frac{q}{(l-j)(l-k)}}\right)\hfill \\ & =& \zeta \left({(\left(1\right).\left(5\right))}^{\frac{q}{(j-k)(j-l)}+\frac{q}{(k-j)(k-l)}+\frac{q}{(l-j)(l-k)}}\right)=\zeta .\hfill \end{array}$$

It is clear that the inequality (1) holds for any value of $\zeta \in [0,1)$ and $j,k,l\in (0,1).$ Moreover, B has two FPs of 0 and $3.$

Theorem 2.

Let $(B,\mathrm{\Omega })$ be a Θ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a rational symmetric $\varpi -\xi$-contraction of kind (I) verifying the hypotheses below:

(1) 

Υ is $\varpi -\xi$-continuous;

(2) 

Υ is a ϖ-admissible mapping concerning $\xi ;$

(3) 

There is $m_0\in B$ such that $\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge \xi (m_0,\mathrm{{\rm Y}}{m}_0).$

Then, Υ has an FP in $B.$

Proof. 

Let $m_0\in B$ such that $\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge \xi (m_0,\mathrm{{\rm Y}}{m}_0).$ Extract the sequence ${\left\{m_i\right\}}_{i=1}^{\infty }$ in the following way: $m_1=\mathrm{{\rm Y}}{m}_0$ and $m_2=\mathrm{{\rm Y}}{m}_1={\mathrm{{\rm Y}}}^2{m}_0.$ Continue with the same scenario, $m_{i+1}=\mathrm{{\rm Y}}{m}_i={\mathrm{{\rm Y}}}^{i+1}{m}_0,$ for $i\in \mathbb{N}.$ It follows from Stipulation $\left(2\right)$ that

$$\varpi (m_0,m_1)=\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge \xi (m_0,\mathrm{{\rm Y}}{m}_0)=\xi (m_0,m_1).$$

Proceeding in this manner, we obtain

$$\varpi (m_{i-1},m_i)\ge \xi (m_{i-1},\mathrm{{\rm Y}}{m}_i)=\xi (m_{i-1},m_i),\forall i\in \mathbb{N}.$$

If $m_{i+1}=m_i$ for some $i\in \mathbb{N},$ then $m_i=\tilde{m}$, and there is nothing to prove. Therefore, assume that $m_{i+1}\ne m_i$ accompanied by

$$\mathrm{\Omega }(\mathrm{{\rm Y}}{m}_{i-1},\mathrm{{\rm Y}}{m}_i)=\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)>0,\forall i\in \mathbb{N}.$$

Because $\mathrm{{\rm Y}}:B\to B$ is a rational symmetric $\varpi -\xi$-contraction of kind (I), then for some $i\in \mathbb{N},$ we obtain

$$\begin{array}{ccc}\hfill \mathrm{\Omega }{(m_i,m_{i+1})}^q& =& \mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_{i-1},\mathrm{{\rm Y}}{m}_i)}^q\hfill \\ & \le & \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^q.\frac{\mathrm{\Omega }{(m_{i-1},\mathrm{{\rm Y}}{m}_{i-1})}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,\mathrm{{\rm Y}}{m}_i)}^{\frac{q}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\right.\hfill \\ & & \frac{{\left[\mathrm{\Omega }(m_{i-1},\mathrm{{\rm Y}}{m}_{i-1})+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)\right]}^{\frac{q}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\hfill \\ & & \left.\frac{{\left[\mathrm{\Omega }(m_{i-1},\mathrm{{\rm Y}}{m}_i)+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_{i-1})\right]}^{\frac{q}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}\right)\hfill \\ & =& \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^q.\frac{\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{q}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\right.\hfill \\ & & \frac{{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\hfill \\ & & \left.\frac{{\left[\mathrm{\Omega }(m_{i-1},m_{i+1})+\mathrm{\Omega }(m_i,m_i)\right]}^{\frac{q}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}\right)\hfill \\ & \le & \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^q.\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{q}{(j-k)(j-l)}}\right..\hfill \\ & & \left.{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q}{(k-j)(k-l)}}.{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q}{(l-j)(l-k)}}\right)\hfill \\ & =& \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^q.\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{q}{(j-k)(j-l)}}\right..\hfill \\ & & \left.{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q}{(k-j)(k-l)}+\frac{q}{(l-j)(l-k)}}\right)\hfill \\ & \le & \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^q.\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{q}{(j-k)(j-l)}}\right..\hfill \\ & & \left.{\left[\mathrm{\Omega }(m_{i-1},m_i).\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q}{(k-j)(k-l)}+\frac{q}{(l-j)(l-k)}}\right)\hfill \\ & =& \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^{q+\left(\frac{q}{(j-k)(j-l)}+\frac{q}{(k-j)(k-l)}+\frac{q}{(l-j)(l-k)}\right)}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\left(\frac{q}{(j-k)(j-l)}+\frac{q}{(k-j)(k-l)}+\frac{q}{(l-j)(l-k)}\right)}\right)\hfill \\ & =& \zeta \mathrm{\Omega }{(m_{i-1},m_i)}^q,\hfill \end{array}$$

which leads to

$$\mathrm{\Omega }{(m_i,m_{i+1})}^q\le \zeta \mathrm{\Omega }{(m_{i-1},m_i)}^q,$$

and we deduce that

$$\mathrm{\Omega }(m_i,m_{i+1})\le \zeta \mathrm{\Omega }(m_{i-1},m_i).$$

(2)

It follows that $\left\{\mathrm{\Omega }(m_{i-1},m_i)\right\}$ is a nonincreasing sequence with non-negative terms. Hence, there exists a positive constant $\vartheta \ge 0$ such that ${lim}_{i\to \infty }\mathrm{\Omega }(m_{i-1},m_i)=\vartheta .$ Based on (2), one can write

$$\mathrm{\Omega }(m_i,m_{i+1})\le \zeta \mathrm{\Omega }(m_{i-1},m_i)\le {\zeta }^i\mathrm{\Omega }(m_0,m_1).$$

Using the triangle inequality, for $h>i,$ we obtain

$$\sum _{z=i}^{h-1}\mathrm{\Omega }(m_z,m_{z+1})\le \frac{{\zeta }^i}{1-\zeta }\mathrm{\Omega }(m_0,m_1).$$

Considering

$$\underset{i\to +\infty }{lim}\frac{{\zeta }^i}{1-\zeta }=0,$$

there exists some $M\in \mathbb{N}$ analogous to

$$0<\frac{{\zeta }^i}{1-\zeta }<\delta ,i\ge M.$$

Suppose that $ϵ>0$ is fixed and the stipulation $\left({\mathrm{\Omega }}_3\right)$ is verified, then there is $\delta >0$ and $(\theta ,\nu )\in \mathrm{\Theta }\times [0,+\infty )$ such that:

$$0<s<\delta \mathrm{implies}\theta \left(s\right)+\nu <\theta \left(ϵ\right).$$

Thus, by $\left({\theta }_1\right),$ we have

$$\theta \left(\sum _{z=i}^{h-1}\mathrm{\Omega }(m_z,m_{z+1})\right)\le \theta \left(\frac{{\zeta }^i}{1-\zeta }\mathrm{\Omega }(m_0,m_1)\right)<\theta \left(ϵ\right)-\nu ,$$

(3)

for each $i,h\in \mathbb{N}$ with the aim that $h>i\ge M$ together $\mathrm{\Omega }(m_i,m_h)>0.$ Therefore, from $\left({\mathrm{\Omega }}_3\right)$ and (3), we obtain

$$\theta \left(\sum _{z=i}^{h-1}\mathrm{\Omega }(m_i,m_h)\right)\le \theta \left(\sum _{z=i}^{h-1}\mathrm{\Omega }(m_z,m_{z+1})\right)+\nu <\theta \left(ϵ\right),$$

which implies by $\left({\theta }_1\right)$ that

$$\mathrm{\Omega }(m_i,m_h)<ϵ,\mathrm{for}h>i\ge M.$$

As a result, the sequence $\left\{m_i\right\}$ is $\mathrm{\Theta }$-Cauchy. Because $(B,\mathrm{\Omega })$ is $\mathrm{\Theta }$-complete $\mathrm{\Theta }$-MS, then there is $\tilde{m}\in B$ such that $\left\{m_i\right\}$ is $\mathrm{\Theta }$-convergent to $\tilde{m},$ i.e.,

$$\underset{i\to \infty }{lim}\mathrm{\Omega }(m_i,\tilde{m})=0.$$

(4)

Since $\mathrm{{\rm Y}}$ is $\varpi -\xi$-continuous and $\varpi (m_{i-1},m_i)\ge \xi (m_{i-1},m_i)$ for each $i\in \mathbb{N},$ then we have

$$m_{i+1}=\mathrm{{\rm Y}}{m}_i\to \mathrm{{\rm Y}}\tilde{m},\mathrm{as}i\to \infty .$$

(5)

On the other hand, we shall prove that $\tilde{m}$ is an FP of $\mathrm{{\rm Y}}$ using the contradiction method. Let $\mathrm{\Omega }(\mathrm{{\rm Y}}\tilde{m},\tilde{m})>0,$ then by $\left({\mathrm{\Omega }}_3\right),$ we obtain

$$\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\le \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\mathrm{{\rm Y}}{m}_i)}^q+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu ,n\in \mathbb{N}.$$

Using $\left({\theta }_1\right)$ and (1), one sees that

$$\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\le \theta \left(\zeta \left[\begin{array}{c}\mathrm{\Omega }{(\tilde{m},m_i)}^q.\frac{\mathrm{\Omega }{(\tilde{m},\mathrm{{\rm Y}}\tilde{m})}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,\mathrm{{\rm Y}}{m}_i)}^{\frac{q}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}\\ \frac{{\left[\mathrm{\Omega }(\tilde{m},\mathrm{{\rm Y}}\tilde{m})+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)\right]}^{\frac{q}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}.\frac{{\left[\mathrm{\Omega }(\tilde{m},\mathrm{{\rm Y}}{m}_i)+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}\tilde{m})\right]}^{\frac{q}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}\end{array}\right]+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu .$$

It follows from (4) and (5) that

$$\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\le \underset{i\to \infty }{lim}\theta \left(\mathrm{\Omega }{(m_{i+1},\tilde{m})}^q\right)+\nu =\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)+\nu ,$$

and making use of $\left({\theta }_1\right)$, we obtain

$$\underset{i\to \infty }{lim}\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)+\nu =-\infty ,$$

which is a contradiction. Therefore, $\mathrm{\Omega }(\mathrm{{\rm Y}}\tilde{m},\tilde{m})=0$; hence, $\mathrm{{\rm Y}}$ has an FP. □

If we omit the continuity condition, we have the following theorem:

Theorem 3.

Let $(B,\mathrm{\Omega })$ be a Θ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a rational symmetric $\varpi -\xi$-contraction of kind (I) such that the assertions below hold

(a) 

Υ is a ϖ-admissible mapping concerning $\xi ;$

(b) 

There is $m_0\in B$ such that $\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge \xi (m_0,\mathrm{{\rm Y}}{m}_0);$

(c) 

For each $i\in \mathbb{N},$ there is an iteration $\left\{m_i\right\}\in B$ such that $\varpi (m_i,m_{i+1})\ge \xi (m_i,m_{i+1})$ with $m_i\to \tilde{m}$ as $i\to \infty .$ Moreover, the inequality $\varpi (m_i,\tilde{m})\ge \xi (m_i,\tilde{m})$ for every $i\in \mathbb{N}.$

Then, there is an FP of Υ in $B.$

Proof. 

In the final lines of the proof of Theorem 2, we acquire $\varpi (m_i,\tilde{m})\ge \xi (m_i,\tilde{m}),$ for all $i\in \mathbb{N}.$ From $\left({\mathrm{\Omega }}_3\right),$ we can obtain

$$\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\le \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\mathrm{{\rm Y}}{m}_i)}^q+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu .$$

Using (1) and $\left({\theta }_1\right)$, we can write

$$\begin{array}{ccc}& & \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\hfill \\ & \le & \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\mathrm{{\rm Y}}{m}_i)}^q+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu \hfill \\ & \le & \theta \left(\zeta \left[\begin{array}{c}\mathrm{\Omega }{(\tilde{m},m_i)}^q.\frac{\mathrm{\Omega }{(\tilde{m},\mathrm{{\rm Y}}\tilde{m})}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,\mathrm{{\rm Y}}{m}_i)}^{\frac{q}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}.\\ \frac{{\left[\mathrm{\Omega }(\tilde{m},\mathrm{{\rm Y}}\tilde{m})+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)\right]}^{\frac{q}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}.\frac{{\left[\mathrm{\Omega }(\tilde{m},\mathrm{{\rm Y}}{m}_i)+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}\tilde{m})\right]}^{\frac{q}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}\end{array}\right]+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu ,\hfill \end{array}$$

From (4) together with the result ${lim}_{i\to \infty }\mathrm{\Omega }(m_{i+1},\tilde{m})=0,$ we have

$$\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\le \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)+\nu ,$$

and making use of $\left({\theta }_1\right)$, we obtain

$$\underset{i\to \infty }{lim}\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)+\nu =-\infty ,$$

which is a contradiction. Therefore, $\mathrm{\Omega }(\mathrm{{\rm Y}}\tilde{m},\tilde{m})=0$; hence; $\mathrm{{\rm Y}}$ has an FP. □

The following example supports the above results.

Example 3.

Consider $B=(0,1]\subset \mathbb{R}$ with Θ-metric $\mathrm{\Omega }:(0,1]\times (0,1]\to [0,\infty )$ given by

$$\mathrm{\Omega }(m,n)=\left\{\begin{array}{cc}\left|m-n\right|,& ifm,n\notin B\times B,\\ {(m-n)}^2& ifm,n\in B\times B,\end{array}\right.$$

accompanied by $\theta \left(a\right)=ln\left(a\right)$ with $\nu =ln\left(1\right),$$q=1.$ Define the mapping $\mathrm{{\rm Y}}:B\to B$ and functions $\varpi ,\xi :B\times B\to [0,\infty )$ by

$$\mathrm{{\rm Y}}\left(m\right)=\left\{\begin{array}{cc}\frac{m^2}{e},& ifm\in B,\\ 0,& ifm\notin B,\end{array}\right.$$

$$\varpi (m,n)=\left\{\begin{array}{cc}2,& ifm,n\in B,\\ 0,& \mathrm{otherwise},\end{array}\right.$$

and

$$\xi (m,n)=\left\{\begin{array}{cc}1,& ifm,n\in B,\\ 0,& \mathrm{otherwise},\end{array}\right.$$

respectively. Clearly $(B,\mathrm{\Omega })$ is a Θ-MS. In order to realize that Υ is a rational symmetric $\varpi -\xi$-contraction of kind (I), we examine the following cases:

(i) 

If $m=n,$ obviously $\mathrm{\Omega }(m,n)=0$; hence; each stipulation of Theorem 2 is fulfilled;

(ii) 

If $m,n\in B$, clearly, Υ is a ϖ-admissible mapping w.r.t. $\xi ,$ as long as $\varpi (m,n)\ge \xi (m,n),$ such that

$$\begin{array}{ccc}\hfill \mathrm{\Omega }(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)& =& \mathrm{\Omega }(\frac{m^2}{e},\frac{n^2}{e})=\frac{1}{e^2}{\left(m^2-n^2\right)}^2\hfill \\ & \le & \zeta \left(\begin{array}{c}{(m^2-n^2)}^2.\frac{{\left[{(m-\frac{m^2}{e})}^2\right]}^{\frac{1}{(j-k)(j-l)}}.{\left[{(n-\frac{n^2}{e})}^2\right]}^{\frac{1}{(j-k)(j-l)}}}{1+\frac{1}{e^2}{(m^2-n^2)}^2}\\ .\frac{{\left[{\left((m-\frac{m^2}{e}\right)}^2+{(n-\frac{n^2}{e})}^2\right]}^{\frac{1}{(k-j)(k-l)}}}{1+\frac{1}{e^2}{(m^2-n^2)}^2}.\frac{{\left[{(m-\frac{n^2}{e})}^2+{(n-\frac{m^2}{e})}^2\right]}^{\frac{1}{(l-j)(l-k)}}}{1+\frac{1}{e^2}{(m^2-n^2)}^2}\end{array}\right)\hfill \end{array}$$

with $\zeta =\frac{1}{e^2}<1,$ and $j,k,l\in (0,1),$ for all $m,n\in B\backslash F\left(\mathrm{{\rm Y}}\right).$

(iii) 

If $m,n\notin B,$ then the state of contraction is directly realized. Therefore, all the constraints of Theorem 2 are fulfilled, and Υ is a rational symmetric $\varpi -\xi$-contraction of kind (I). Moreover, e is an FP of Υ in $B.$ (Note that 0 is also an FP of Υ, but does not belong to B.)

Definition 8.

Let $(B,\mathrm{\Omega })$ be a Θ-MS and $\varpi ,\xi :B\times B\to [0,\infty )$ be functions. Then, a Θ-MS on B is called $\varpi -\xi$-complete iff every Θ-Cauchy sequence $\left\{m_i\right\}$ with $\varpi (m_i,m_{i+1})\ge \xi (m_i,m_{i+1})$ is Θ-convergent in B for each $i\in \mathbb{N}.$

Remark 1.

The constraints of Theorems 2 and 3 hold for a $\varpi -\xi$-complete Θ-MS in lieu of a Θ-complete Θ-MS; for more clarification, see [33].

3. Rational Symmetric $\mathbf{\varpi }-\mathbf{\xi }$-Contraction of Kind (II)

This part is devoted to presenting a rational symmetric $\varpi -\xi$-contraction of kind (II) in the setting of a $\mathrm{\Theta }$-complete $\mathrm{\Theta }$-MS. Moreover, some FP consequences are obtained via this notion.

Definition 9.

Let $(B,\mathrm{\Omega })$ be a Θ-MS and $\varpi ,\xi :B\times B\to [0,\infty )$ be two functions. A mapping $\mathrm{{\rm Y}}:B\to B$ is called a rational symmetric $\varpi -\xi$-contraction of kind (II) if for each $\zeta \in [0,1),$ there exist $j,k,l\in (0,1)$ such that, whenever $\varpi (m,n)\ge \xi (m,n),$ we have

$$\mathrm{\Omega }{(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)}^q\le \zeta \left(\widehat{A_2}(m,n)\right),$$

(6)

where

$$\begin{array}{ccc}\hfill \widehat{A_2}(m,n)& =& \mathrm{\Omega }{(m,n)}^q.\frac{\mathrm{\Omega }{(m,\mathrm{{\rm Y}}m)}^{\frac{qj}{(j-k)(j-l)}}.\mathrm{\Omega }{(n,\mathrm{{\rm Y}}n)}^{\frac{qj}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m,n)}^q}.\hfill \\ & & \frac{{\left[\mathrm{\Omega }(m,\mathrm{{\rm Y}}m)+\mathrm{\Omega }(n,\mathrm{{\rm Y}}n)\right]}^{\frac{qk}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m,n)}^q}.\frac{{\left[\mathrm{\Omega }(m,\mathrm{{\rm Y}}n)+\mathrm{\Omega }(n,\mathrm{{\rm Y}}m)\right]}^{\frac{ql}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m,n)}^q},\hfill \end{array}$$

for all $q\in [1,\infty )$ and $m,n\in B\backslash F\left(\mathrm{{\rm Y}}\right).$

Now, the main theorem of this part is valid for presentation.

Theorem 4.

Let $(B,\mathrm{\Omega })$ be a Θ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a rational symmetric $\varpi -\xi$-contraction of kind (II) satisfying the same assertions of Theorem 2. Then, Υ possesses an FP in $B.$

Proof. 

Consider $m_0\in B$ such that $\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge \xi (m_0,\mathrm{{\rm Y}}{m}_0).$ For $i\in \mathbb{N},$ we extract the sequence ${\left\{m_i\right\}}_{i=1}^{\infty }$ in the following manner: $m_1=\mathrm{{\rm Y}}{m}_0$ and $m_2=\mathrm{{\rm Y}}{m}_1={\mathrm{{\rm Y}}}^2{m}_0.$ Proceeding with the same method, $m_{i+1}=\mathrm{{\rm Y}}{m}_i={\mathrm{{\rm Y}}}^{i+1}{m}_0.$ According to the condition $\left(2\right)$, we obtain

$$\varpi (m_0,m_1)=\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge \xi (m_0,\mathrm{{\rm Y}}{m}_0)=\xi (m_0,m_1).$$

Proceeding with this exercise, we can write

$$\varpi (m_{i-1},m_i)\ge \xi (m_{i-1},\mathrm{{\rm Y}}{m}_i)=\xi (m_{i-1},m_i),\forall i\in \mathbb{N}.$$

If $m_{i+1}=m_i$ for some $i\in \mathbb{N},$ then $m_i=\tilde{m}$, and there is nothing to prove. Therefore, we suppose that $m_{i+1}\ne m_i$ accompanied by

$$\mathrm{\Omega }(\mathrm{{\rm Y}}{m}_{i-1},\mathrm{{\rm Y}}{m}_i)=\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)>0,\forall i\in \mathbb{N}.$$

By the definition of $\mathrm{{\rm Y}}$, for some $i\in \mathbb{N},$ we have

$$\begin{array}{ccc}\hfill \mathrm{\Omega }{(m_i,m_{i+1})}^q& =& \mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_{i-1},\mathrm{{\rm Y}}{m}_i)}^q\hfill \\ & \le & \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^q.\frac{\mathrm{\Omega }{(m_{i-1},\mathrm{{\rm Y}}{m}_{i-1})}^{\frac{qj}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,\mathrm{{\rm Y}}{m}_i)}^{\frac{qj}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\right.\hfill \\ & & \frac{{\left[\mathrm{\Omega }(m_{i-1},\mathrm{{\rm Y}}{m}_{i-1})+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)\right]}^{\frac{qk}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\hfill \\ & & \left.\frac{{\left[\mathrm{\Omega }(m_{i-1},\mathrm{{\rm Y}}{m}_i)+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_{i-1})\right]}^{\frac{ql}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}\right)\hfill \\ & =& \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^q.\frac{\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{qj}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{qj}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}\right..\hfill \\ & & \left.\frac{{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{qk}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\frac{{\left[\mathrm{\Omega }(m_{i-1},m_{i+1})+\mathrm{\Omega }(m_i,m_i)\right]}^{\frac{ql}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}\right)\hfill \\ & \le & \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^q.\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{qj}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{qj}{(j-k)(j-l)}}\right..\hfill \\ & & \left.{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{qk}{(k-j)(k-l)}}.{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{ql}{(l-j)(l-k)}}\right)\hfill \\ & =& \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^q.\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{q}{(j-k)(j-l)}}\right..\hfill \\ & & \left.{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q}{(k-j)(k-l)}+\frac{q}{(l-j)(l-k)}}\right)2\hfill \\ & \le & \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^q.\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{qj}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{qj}{(j-k)(j-l)}}\right..\hfill \\ & & \left.{\left[\mathrm{\Omega }(m_{i-1},m_i).\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{qk}{(k-j)(k-l)}+\frac{qk}{(l-j)(l-k)}}\right)\hfill \\ & =& \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^{q+\left(\frac{qj}{(j-k)(j-l)}+\frac{qk}{(k-j)(k-l)}+\frac{ql}{(l-j)(l-k)}\right)}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\left(\frac{qj}{(j-k)(j-l)}+\frac{qk}{(k-j)(k-l)}+\frac{ql}{(l-j)(l-k)}\right)}\right)\hfill \\ & =& \zeta \mathrm{\Omega }{(m_{i-1},m_i)}^q,\hfill \end{array}$$

which implies that

$$\mathrm{\Omega }{(m_i,m_{i+1})}^q\le \zeta \mathrm{\Omega }{(m_{i-1},m_i)}^q,$$

and one can deduce that

$$\mathrm{\Omega }(m_i,m_{i+1})\le \zeta \mathrm{\Omega }(m_{i-1},m_i).$$

The rest of the proof is the same as the proof lines of Theorem 2. □

Theorem 5.

Let $(B,\mathrm{\Omega })$ be a Θ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a rational symmetric $\varpi -\xi$-contraction of kind (II) meeting the same assumptions of Theorem 3. Then, there is an FP of Υ in $B.$

Proof. 

In a similar way to proving Theorem 3, since, by Stipulation $\left(c\right)$, $\varpi (m_i,\tilde{m})\ge \xi (m_i,\tilde{m})$ holds for all $i\in \mathbb{N}.$ From $\left({\mathrm{\Omega }}_3\right),$ we have

$$\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\le \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\mathrm{{\rm Y}}{m}_i)}^q+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu .$$

Using (6) and $\left({\theta }_1\right)$, we obtain

$$\begin{array}{ccc}& & \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\hfill \\ & \le & \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\mathrm{{\rm Y}}{m}_i)}^q+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu \hfill \\ & \le & \theta \left(\zeta \left[\begin{array}{c}\mathrm{\Omega }{(\tilde{m},m_i)}^q.\frac{\mathrm{\Omega }{(\tilde{m},\mathrm{{\rm Y}}\tilde{m})}^{\frac{qj}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,\mathrm{{\rm Y}}{m}_i)}^{\frac{qj}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}\\ \frac{{\left[\mathrm{\Omega }(\tilde{m},\mathrm{{\rm Y}}\tilde{m})+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)\right]}^{\frac{qk}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}.\frac{{\left[\mathrm{\Omega }(\tilde{m},\mathrm{{\rm Y}}{m}_i)+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}\tilde{m})\right]}^{\frac{ql}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}\end{array}\right]+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu .\hfill \end{array}$$

Making use of (4), we have

$$\underset{i\to \infty }{lim}\mathrm{\Omega }{(m_i,\tilde{m})}^q=0\mathrm{together}\underset{i\to \infty }{lim}\mathrm{\Omega }{(m_{i+1},\tilde{m})}^q=0,$$

and we procure

$$\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\le \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)+\nu ,$$

then making use of $\left({\theta }_1\right)$, we obtain

$$\underset{i\to \infty }{lim}\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)+\nu =-\infty ,$$

which leads to a logical contradiction. Therefore, $\mathrm{\Omega }(\mathrm{{\rm Y}}\tilde{m},\tilde{m})=0;$ hence, $\mathrm{{\rm Y}}$ has an FP in B. □

4. Rational Symmetric $\mathbf{\varpi }-\mathbf{\xi }$-Contraction of Kind (III)

This part is devoted to presenting a rational symmetric $\varpi -\xi$-contraction of kind (III) in the framework of a $\mathrm{\Theta }$-complete $\mathrm{\Theta }$-MS. Moreover, some FP consequences are obtained under this concept.

Definition 10.

Let $(B,\mathrm{\Omega })$ be a Θ-MS and $\varpi ,\xi :B\times B\to [0,\infty )$ be given functions. We say that a mapping $\mathrm{{\rm Y}}:B\to B$ is a rational symmetric $\varpi -\xi$-contraction of kind (III) if for each $\zeta \in [0,1),$ there exist $j,k,l\in (0,1)$ such that, whenever $\varpi (m,n)\ge \xi (m,n),$ we have

$$\mathrm{\Omega }{(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)}^q\le \zeta \left(\widehat{A_3}(m,n)\right),$$

(7)

where

$$\widehat{A_3}(m,n)=max\left\{\begin{array}{c}\mathrm{\Omega }{(m,n)}^q,\frac{\mathrm{\Omega }{(m,\mathrm{{\rm Y}}m)}^{\frac{q{j}^2}{(j-k)(j-l)}}.\mathrm{\Omega }{(n,\mathrm{{\rm Y}}n)}^{\frac{q{j}^2}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m,n)}^q}.\\ \frac{{\left[\mathrm{\Omega }(m,\mathrm{{\rm Y}}m)+\mathrm{\Omega }(n,\mathrm{{\rm Y}}n)\right]}^{\frac{q{k}^2}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m,n)}^q}.\frac{{\left[\mathrm{\Omega }(m,\mathrm{{\rm Y}}n)+\mathrm{\Omega }(n,\mathrm{{\rm Y}}m)\right]}^{\frac{q{l}^2}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m,n)}^q}\end{array}\right\},$$

for all $q\in [1,\infty )$ and $m,n\in B\backslash F\left(\mathrm{{\rm Y}}\right).$

Now, we demonstrate our next theorem.

Theorem 6.

Let $(B,\mathrm{\Omega })$ be a Θ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a rational symmetric $\varpi -\xi$-contraction of kind (III) meeting the same hypotheses of Theorem 3. Then, Υ possesses an FP in $B.$

Proof. 

Consider $m_0\in B$ with the goal that $\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge \xi (m_0,\mathrm{{\rm Y}}{m}_0),$ and select any $m_0\in B;$ we erect a recapitulated ${\left\{m_i\right\}}_{i=1}^{\infty }$ as follows: $m_1=\mathrm{{\rm Y}}{m}_0$ and $m_2=\mathrm{{\rm Y}}{m}_1={\mathrm{{\rm Y}}}^2{m}_0.$ Proceeding with this practice, $m_{i+1}=\mathrm{{\rm Y}}{m}_i={\mathrm{{\rm Y}}}^{i+1}{m}_0.$ Based on Stipulation $\left(2\right)$ of Theorem 2, we have

$$\varpi (m_0,m_1)=\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge \xi (m_0,\mathrm{{\rm Y}}{m}_0)=\xi (m_0,m_1),$$

Proceeding with this exercise, we can write

$$\varpi (m_{i-1},m_i)\ge \xi (m_{i-1},\mathrm{{\rm Y}}{m}_i)=\xi (m_{i-1},m_i),\forall i\in \mathbb{N}.$$

When $m_{i+1}=m_i$ for some $i\in \mathbb{N},$ then $m_i=\tilde{m}$, and there is nothing to prove. Therefore, we assume that $m_{i+1}\ne m_i$ accompanied by

$$\mathrm{\Omega }(\mathrm{{\rm Y}}{m}_{i-1},\mathrm{{\rm Y}}{m}_i)=\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)>0,\forall i\in \mathbb{N}.$$

By the definition of $\mathrm{{\rm Y}}$, for some $i\in \mathbb{N},$ we can obtain

$$\begin{array}{ccc}\hfill \mathrm{\Omega }{(m_i,m_{i+1})}^q& =& \mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_{i-1},\mathrm{{\rm Y}}{m}_i)}^q\hfill \\ & \le & \zeta max\left\{\begin{array}{c}\mathrm{\Omega }{(m_{i-1},m_i)}^q,\frac{\mathrm{\Omega }{(m_{i-1},\mathrm{{\rm Y}}{m}_{i-1})}^{\frac{q{j}^2}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,\mathrm{{\rm Y}}{m}_i)}^{\frac{q{j}^2}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\\ \frac{{\left[\mathrm{\Omega }(m_{i-1},\mathrm{{\rm Y}}{m}_{i-1})+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)\right]}^{\frac{q{k}^2}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\frac{{\left[\mathrm{\Omega }(m_{i-1},\mathrm{{\rm Y}}{m}_i)+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_{i-1})\right]}^{\frac{q{l}^2}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}\end{array}\right\}\hfill \\ & =& \zeta max\left\{\begin{array}{c}\mathrm{\Omega }{(m_{i-1},m_i)}^q,\frac{\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q{j}^2}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{q{j}^2}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\\ \frac{{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q{k}^2}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\frac{{\left[\mathrm{\Omega }(m_{i-1},m_{i+1})+\mathrm{\Omega }(m_i,m_i)\right]}^{\frac{q{l}^2}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}\end{array}\right\}\hfill \\ & \le & \zeta max\left\{\begin{array}{c}\mathrm{\Omega }{(m_{i-1},m_i)}^q,\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q{j}^2}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{q{j}^2}{(j-k)(j-l)}}.\\ {\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q{k}^2}{(k-j)(k-l)}}.\\ {\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q{l}^2}{(l-j)(l-k)}}\end{array}\right\}\hfill \\ & =& \zeta max\left\{\begin{array}{c}\mathrm{\Omega }{(m_{i-1},m_i)}^q,\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q{j}^2}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{q{j}^2}{(j-k)(j-l)}}.\\ {\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q{k}^2}{(k-j)(k-l)}+\frac{q{l}^2}{(l-j)(l-k)}}\end{array}\right\}\hfill \\ & \le & \zeta max\left\{\begin{array}{c}\mathrm{\Omega }{(m_{i-1},m_i)}^q,\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q{j}^2}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{q{j}^2}{(j-k)(j-l)}}.\\ {\left[\mathrm{\Omega }(m_{i-1},m_i).\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q{k}^2}{(k-j)(k-l)}+\frac{q{l}^2}{(l-j)(l-k)}}\end{array}\right\}\hfill \\ & =& \zeta max\left\{\begin{array}{c}\mathrm{\Omega }{(m_{i-1},m_i)}^q,\mathrm{\Omega }{(m_{i-1},m_i)}^{\left(\frac{q{j}^2}{(j-k)(j-l)}+\frac{q{k}^2}{(k-j)(k-l)}+\frac{q{l}^2}{(l-j)(l-k)}\right)}.\\ \mathrm{\Omega }{(m_i,m_{i+1})}^{\left(\frac{q{j}^2}{(j-k)(j-l)}+\frac{q{k}^2}{(k-j)(k-l)}+\frac{q{l}^2}{(l-j)(l-k)}\right)}\end{array}\right\}\hfill \\ & =& \zeta max\{\mathrm{\Omega }{(m_{i-1},m_i)}^q,\mathrm{\Omega }{(m_i,m_{i+1})}^q\}.\hfill \end{array}$$

On the condition that

$$max\{\mathrm{\Omega }(m_{i-1},m_i),\mathrm{\Omega }(m_i,m_{i+1})\}=\mathrm{\Omega }(m_{i-1},m_i),$$

at that time

$$\mathrm{\Omega }(m_i,m_{i+1})\le \zeta \mathrm{\Omega }(m_{i+1},m_i),$$

which is a contradiction, so we must write

$$\mathrm{\Omega }(m_i,m_{i+1})\le \zeta \mathrm{\Omega }(m_i,m_{i-1}).$$

which implies that

$$\mathrm{\Omega }{(m_i,m_{i+1})}^q\le \zeta \mathrm{\Omega }{(m_{i-1},m_i)}^q,$$

and one can deduce that

$$\mathrm{\Omega }(m_i,m_{i+1})\le \zeta \mathrm{\Omega }(m_{i-1},m_i).$$

(8)

This leads to $\left\{\mathrm{\Omega }(m_{i-1},m_i)\right\}$ being a nonincreasing sequence with non-negative terms. Therefore, there is a non-negative constant $\vartheta$ such that ${lim}_{i\to \infty }\mathrm{\Omega }(m_{i-1},m_i)=\vartheta .$ We shall indicate that $\vartheta >0.$ Indeed, by (8), we can write

$$\mathrm{\Omega }(m_i,m_{i+1})\le \zeta \mathrm{\Omega }(m_{i-1},m_i)\le {\zeta }^n\mathrm{\Omega }(m_0,m_1).$$

Stop the proof, and go over the closing lines of Theorem 2. □

Theorem 7.

Let $(B,\mathrm{\Omega })$ be a Θ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a rational symmetric $\varpi -\xi$-contraction of kind (III) satisfying the same assumptions of Theorem 3. Then, Υ has an FP in $B.$

Proof. 

Similar to the proof of Theorem 3, considering $\left(c\right)$, $\varpi (m_i,\tilde{m})\ge \xi (m_i,\tilde{m})$ for all $i\in \mathbb{N}.$ Using $\left({\mathrm{\Omega }}_3\right),$ we obtain

$$\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\le \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\mathrm{{\rm Y}}{m}_i)}^q+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu .$$

Using (6) and $\left({\theta }_1\right)$, we obtain

$$\begin{array}{ccc}& & \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\hfill \\ & \le & \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\mathrm{{\rm Y}}{m}_i)}^q+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu \hfill \\ & \le & \theta \left(\zeta max\left\{\begin{array}{c}\mathrm{\Omega }{(\tilde{m},m_i)}^q,\frac{\mathrm{\Omega }{(\tilde{m},\mathrm{{\rm Y}}\tilde{m})}^{\frac{q{j}^2}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,\mathrm{{\rm Y}}{m}_i)}^{\frac{q{j}^2}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}\\ \frac{{\left[\mathrm{\Omega }(\tilde{m},\mathrm{{\rm Y}}\tilde{m})+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)\right]}^{\frac{q{k}^2}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}.\frac{{\left[\mathrm{\Omega }(\tilde{m},\mathrm{{\rm Y}}{m}_i)+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}\tilde{m})\right]}^{\frac{q{l}^2}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(\tilde{m},m_i)}^q}\end{array}\right\}+\mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_i,\tilde{m})}^q\right)+\nu .\hfill \end{array}$$

Making use of (4), we have

$$\underset{i\to \infty }{lim}\mathrm{\Omega }{(m_i,\tilde{m})}^q=0\mathrm{as}\mathrm{long}\mathrm{as}\underset{i\to \infty }{lim}\mathrm{\Omega }{(m_{i+1},\tilde{m})}^q=0,$$

and we obtain

$$\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)\le \theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)+\nu .$$

Utilizing $\left({\theta }_1\right)$, we obtain

$$\underset{i\to \infty }{lim}\theta \left(\mathrm{\Omega }{(\mathrm{{\rm Y}}\tilde{m},\tilde{m})}^q\right)+\nu =-\infty ,$$

which leads to a logical inconsistency. Therefore, $\mathrm{\Omega }(\mathrm{{\rm Y}}\tilde{m},\tilde{m})=0;$ hence, $\mathrm{{\rm Y}}$ has an FP in B. □

5. Rational Symmetric $\mathbf{\varpi }-\mathbf{\xi }$-Contraction of Kind (IV)

In this section, we present the notion of a rational symmetric $\varpi -\xi$-contraction of kind (IV) in the setting of a $\mathrm{\Theta }$-complete $\mathrm{\Theta }$-MS. Furthermore, some FP results are discussed under this idea.

Definition 11.

Let $(B,\mathrm{\Omega })$ be a Θ-MS and $\varpi ,\xi :B\times B\to [0,\infty )$ be given functions. We say that a mapping $\mathrm{{\rm Y}}:B\to B$ is a rational symmetric $\varpi -\xi$-contraction of kind (IV) if for each $\zeta \in [0,1),$ there exist $j,k,l\in (0,1)$ with $j+k+l<1$ such that, whenever $\varpi (m,n)\ge \xi (m,n),$ we obtain

$$\mathrm{\Omega }{(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)}^q\le \zeta \left(\widehat{A_4}(m,n)\right),$$

where

$$\widehat{A_4}(m,n)=\left(\begin{array}{c}\mathrm{\Omega }{(m,n)}^{\frac{q{j}^3}{(j-k)(j-l)}}.\frac{\mathrm{\Omega }{(n,\mathrm{{\rm Y}}n)}^{\frac{q{j}^3}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m,n)}^q}.\\ \frac{{\left[\mathrm{\Omega }(m,\mathrm{{\rm Y}}m)+\mathrm{\Omega }(n,\mathrm{{\rm Y}}n)\right]}^{\frac{q{k}^3}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m,n)}^q}.\frac{{\left[\mathrm{\Omega }(m,\mathrm{{\rm Y}}n)+\mathrm{\Omega }(n,\mathrm{{\rm Y}}m)\right]}^{\frac{q{l}^3}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m,n)}^q}\end{array}\right),$$

for all $q\in [1,\infty )$ and $m,n\in B\backslash F\left(\mathrm{{\rm Y}}\right).$

Now, we present our main theorem.

Theorem 8.

Let $(B,\mathrm{\Omega })$ be a Θ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a rational symmetric $\varpi -\xi$-contraction of kind (IV) justifying the same hypotheses of Theorem 2. Then, Υ has an FP in $B.$

Proof. 

Let $m_0\in B$ with the aim that $\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge \xi (m_0,\mathrm{{\rm Y}}{m}_0),$ and choose any $m_0\in B;$ we erect a chain ${\left\{m_i\right\}}_{i=1}^{\infty }$ as follows: $m_1=\mathrm{{\rm Y}}{m}_0$ and $m_2=\mathrm{{\rm Y}}{m}_1={\mathrm{{\rm Y}}}^2{m}_0.$ Proceeding with this practice, $m_{i+1}=\mathrm{{\rm Y}}{m}_i={\mathrm{{\rm Y}}}^{i+1}{m}_0.$ According to Condition $\left(2\right)$ of Theorem 2, we obtain

$$\varpi (m_0,m_1)=\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge \xi (m_0,\mathrm{{\rm Y}}{m}_0)=\xi (m_0,m_1).$$

Proceeding with this scenario, we have

$$\varpi (m_{i-1},m_i)\ge \xi (m_{i-1},\mathrm{{\rm Y}}{m}_i)=\xi (m_{i-1},m_i),\forall i\in \mathbb{N}.$$

When $m_{i+1}=m_i$ for some $i\in \mathbb{N},$ then $m_i=\tilde{m}$, and the proof is complete. Therefore, we assume that $m_{i+1}\ne m_i$ accompanied by

$$\mathrm{\Omega }(\mathrm{{\rm Y}}{m}_{i-1},\mathrm{{\rm Y}}{m}_i)=\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)>0,\forall i\in \mathbb{N}.$$

Because $\mathrm{{\rm Y}}$ is a rational symmetric $\varpi -\xi$-contraction of kind (IV), for some $i\in \mathbb{N},$ we have

$$\begin{array}{ccc}\hfill \mathrm{\Omega }{(m_i,m_{i+1})}^q& =& \mathrm{\Omega }{(\mathrm{{\rm Y}}{m}_{i-1},\mathrm{{\rm Y}}{m}_i)}^q\hfill \\ & \le & \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q{j}^3}{(j-k)(j-l)}}.\frac{\mathrm{\Omega }{(m_i,\mathrm{{\rm Y}}{m}_i)}^{\frac{q{j}^3}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\right.\hfill \\ & & \left.\frac{{\left[\mathrm{\Omega }(m_{i-1},\mathrm{{\rm Y}}{m}_{i-1})+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_i)\right]}^{\frac{q{k}^3}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\frac{{\left[\mathrm{\Omega }(m_{i-1},\mathrm{{\rm Y}}{m}_i)+\mathrm{\Omega }(m_i,\mathrm{{\rm Y}}{m}_{i-1})\right]}^{\frac{q{l}^3}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}\right)\hfill \\ & =& \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q{j}^3}{(j-k)(j-l)}}.\frac{\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{qj3}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\right.\hfill \\ & & \left.\frac{{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q{k}^3}{(k-j)(k-l)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}.\frac{{\left[\mathrm{\Omega }(m_{i-1},m_{i+1})+\mathrm{\Omega }(m_i,m_i)\right]}^{\frac{q{l}^3}{(l-j)(l-k)}}}{1+\mathrm{\Omega }{(m_{i-1},m_i)}^q}\right)\hfill \\ & \le & \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q{j}^3}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{qj3}{(j-k)(j-l)}}.{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q{k}^3}{(k-j)(k-l)}}\right.\hfill \\ & & \left..{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q{l}^3}{(l-j)(l-k)}}\right)\hfill \\ & =& \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q{j}^3}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{qj3}{(j-k)(j-l)}}.{\left[\mathrm{\Omega }(m_{i-1},m_i)+\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q{k}^3}{(k-j)(k-l)}+\frac{q{l}^3}{(l-j)(l-k)}}\right)\hfill \\ & \le & \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^{\frac{q{j}^3}{(j-k)(j-l)}}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\frac{qj3}{(j-k)(j-l)}}.{\left[\mathrm{\Omega }(m_{i-1},m_i).\mathrm{\Omega }(m_i,m_{i+1})\right]}^{\frac{q{k}^3}{(k-j)(k-l)}+\frac{q{l}^3}{(l-j)(l-k)}}\right)\hfill \\ & =& \zeta \left(\mathrm{\Omega }{(m_{i-1},m_i)}^{\left(\frac{q{j}^3}{(j-k)(j-l)}+\frac{q{k}^3}{(k-j)(k-l)}+\frac{q{l}^3}{(l-j)(l-k)}\right)}.\mathrm{\Omega }{(m_i,m_{i+1})}^{\left(\frac{q{j}^3}{(j-k)(j-l)}+\frac{q{k}^3}{(k-j)(k-l)}+\frac{q{l}^3}{(l-j)(l-k)}\right)}\right)\hfill \\ & =& \zeta {\left(\mathrm{\Omega }(m_{i-1},m_i).\mathrm{\Omega }(m_i,m_{i+1})\right)}^{q\left(j+k+l\right)}\hfill \\ & \le & \zeta max\{\mathrm{\Omega }{(m_{i-1},m_i)}^q,\mathrm{\Omega }{(m_i,m_{i+1})}^q\}.\hfill \end{array}$$

On the condition that

$$max\{\mathrm{\Omega }(m_{i-1},m_i),\mathrm{\Omega }(m_i,m_{i+1})\}=\mathrm{\Omega }(m_{i-1},m_i),$$

at that time

$$\mathrm{\Omega }(m_i,m_{i+1})\le \zeta \mathrm{\Omega }(m_{i+1},m_i),$$

which is a contradiction, so we must write

$$\mathrm{\Omega }(m_i,m_{i+1})\le \zeta \mathrm{\Omega }(m_i,m_{i-1}).$$

Go over the closing lines of Theorem 2. □

Theorem 9.

Let $(B,\mathrm{\Omega })$ be a Θ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a rational symmetric $\varpi -\xi$-contraction of kind (IV) fulfilling the same assumptions of Theorem 3. Then, Υ has an FP in $B.$

Proof. 

The proof is performed in a similar way to the proof of Theorem 3 under the concept of a rational symmetric $\varpi -\xi$-contraction of kind (VI). □

If we take $\xi (m,n)=1$ in Theorems 2 and 3, we have the following results:

Corollary 1.

Let $(B,\mathrm{\Omega })$ be a Θ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a rational symmetric $\varpi -1$-contraction of kind (I) fulfilling the accompanying affirmations

• Υ is ϖ-continuous;
• Υ is a ϖ-admissible mapping;
• There is an $m_0\in B$ such that $\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge 1.$

Then, Υ has an FP in $B.$

Corollary 2.

Assume that $(B,\mathrm{\Omega })$ is a Θ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ is a rational symmetric $\varpi -1$-contraction of kind (I) so that the assertions below hold

• Υ is a ϖ-admissible mapping;
• There is $m_0\in B$ such that $\varpi (m_0,\mathrm{{\rm Y}}{m}_0)\ge 1;$
• For each $i\in \mathbb{N},$ there is an iteration $\left\{m_i\right\}\in B$ such that $\varpi (m_i,m_{i+1})\ge 1$ with $m_i\to \tilde{m}$ as $i\to \infty .$ Moreover, the inequality $\varpi (m_i,\tilde{m})\ge 1$ for every $i\in \mathbb{N}.$

Then, there is an FP of Υ in $B.$

Remark 2.

It is clear that

• For each $j,k,l\in (0,1),$

  $$\begin{array}{ccc}\hfill \frac{1}{(j-k)(j-l)}+\frac{1}{(k-j)(k-l)}+\frac{1}{(l-j)(l-k)}& =& 0,\hfill \\ \hfill \frac{j}{(j-k)(j-l)}+\frac{k}{(k-j)(k-l)}+\frac{l}{(l-j)(l-k)}& =& 0,\hfill \\ \hfill \frac{j^2}{(j-k)(j-l)}+\frac{k^2}{(k-j)(k-l)}+\frac{l^2}{(l-j)(l-k)}& =& 1,\hfill \\ \hfill \frac{j^2}{(j-k)(j-l)}+\frac{k^2}{(k-j)(k-l)}+\frac{l^2}{(l-j)(l-k)}& =& j+k+l;\hfill \end{array}$$
• Corollaries 1 and 2 can be derived easily for a rational symmetric $\varpi -1$-contraction of kind (II), kind (III), and kind (IV), respectively.

6. Some Related Results

This part was prepared to study Suzuki-type rational symmetric contractions and orbitally $\mathrm{{\rm Y}}$-complete and OC mappings in $\mathrm{{\rm Y}}$-MSs, as a consequence of our results presented in the previous parts.

Theorem 10.

Let $(B,\mathrm{\Omega })$ be a Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a continuous mapping on B. Assume that $\ell \in [0,1)$ and $j,k,l\in (0,1)$ such that

$$\mathrm{\Omega }(m,\mathrm{{\rm Y}}m)\le \mathrm{\Omega }(m,n)⟹\mathrm{\Omega }{(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)}^q\le \ell \left(\widehat{A_1}(m,n)\right),$$

where $\widehat{A_1}(m,n)$ is described as Definition 7 and $q\in [1,\infty ),$ for all $m,n\in B\backslash F\left(\mathrm{{\rm Y}}\right).$ Then, Υ has an FP in $B.$

Proof. 

Define $\varpi ,\xi :B\times B\to [0,+\infty )$ by

$$\varpi (m,n)=\mathrm{\Omega }(m,n)and\xi (m,n)=\mathrm{\Omega }(m,\mathrm{{\rm Y}}m),\forall m,n\in B,$$

with $j,k,l\in (0,1)$ and $\ell \in [0,1)$. Clearly, $\varpi (m,n)\ge \xi (m,n),$ for all $m,n\in B.$ Therefore, in turn, Conditions $\left(1\right)-\left(3\right)$ of Theorem 2 are fulfilled. Consider

$$\xi (m,n)\le \varpi (m,n)⟹\mathrm{\Omega }(m,\mathrm{{\rm Y}}m)\le \mathrm{\Omega }(m,n),$$

which leads to the contractive condition

$$\mathrm{\Omega }{(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)}^q\le \ell \left(\widehat{A_1}(m,n)\right).$$

Ultimately, every stipulation of Theorem 2 holds true. Thus, $\mathrm{{\rm Y}}$ has an FP in $B.$ □

Theorem 11.

Let $(B,\mathrm{\Omega })$ be an orbitally Υ-complete Θ-MS and $\mathrm{{\rm Y}}:B\to B$ be a mapping of B. Suppose that $\ell \in [0,1)$ and $j,k,l\in (0,1)$ such that

$$\mathrm{\Omega }{(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)}^q\le \ell \left(\widehat{A_1}(m,n)\right),$$

(9)

where $\widehat{A_1}(m,n)$ is given in Definition 7 and $q\in [1,\infty ),$ for all $m,n\in O\left(\omega \right),$ for some $\omega \in B.$ Then, Υ has an FP in $B,$ provided that the following assumption holds

(A) 

If $\left\{n_i\right\}$ is a sequence such that $\left\{n_i\right\}\subseteq O\left(\omega \right)$ with ${lim}_{i\to \infty }{n}_i=n,$ then $n\in O\left(\omega \right),$ where $O\left(\omega \right)$ is an orbit of $\omega .$

Proof. 

Define $\varpi ,\xi :B\times B\to [0,+\infty )$ as

$$\varpi (m,n)=\left\{\begin{array}{cc}3,& ifm,n\in O\left(\omega \right)\times O\left(\omega \right),\\ 0,& \mathrm{otherwise},\end{array}\right.$$

and $\xi (m,n)=1,$ for all $m,n\in B$ (Remark 6 [34]). Hence, $(B,\mathrm{\Omega })$ is a $\varpi -\xi$-complete $\mathrm{\Theta }$-metric and $\mathrm{{\rm Y}}$ is a $\varpi$-admissible mapping concerning $\xi .$ Consider $\xi (m,n)\le \varpi (m,n),$ then $m,n\in O\left(\omega \right),$ and hence, from (9), we find that

$$\mathrm{\Omega }{(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)}^q\le \ell \left(\widehat{A_1}(m,n)\right),$$

where $\widehat{A_1}(m,n)$ is defined in Definition 7 and $q\in [1,\infty )$. This implies that $\mathrm{{\rm Y}}$ is a rational symmetric $\varpi -\xi$-contraction of kind (I). Suppose that $\left\{n_i\right\}$ is a sequence commensurate with $\varpi (n_i,n_{i+1})\ge \xi (n_i,n_{i+1})$ and ${lim}_{i\to \infty }{n}_i=n.$ From Assertion (A), $n\in O\left(\omega \right).$ This means $\varpi (n_i,n)\ge \xi (n_i,n).$ Based on the foregoing, we concluded that all the assumptions of Theorem 2 are fulfilled. Thus, $\mathrm{{\rm Y}}$ has an FP. □

Theorem 12.

Let $(B,\mathrm{\Omega })$ be a Θ-MS and Υ be a self-mapping of B. Assume that $\ell \in [0,1)$ and $j,k,l\in (0,1)$ such that

$$\mathrm{\Omega }{(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)}^q\le \ell \left(\widehat{A_1}(m,n)\right),$$

(10)

where $\widehat{A_1}(m,n)$ is defined in Definition 7 and $q\in [1,\infty ),$ for all $m,n\in O\left(\omega \right),$ for some $\omega \in B.$ Then, Υ has an FP in $B,$ provided that the operator Υ is OC.

Proof. 

Describe $\varpi ,\xi :B\times B\to [0,+\infty )$ as

$$\varpi (m,n)=\left\{\begin{array}{cc}3,& ifm,n\in O\left(\omega \right)\times O\left(\omega \right),\\ 0,& \mathrm{otherwise},\end{array}\right.$$

and $\xi (m,n)=1,$ for all $m,n\in B$ (Remark 1.1 [35]); we know that $\mathrm{{\rm Y}}$ is a $\varpi$-admissible mapping. Suppose that $\xi (m,n)\le \varpi (m,n),$ then $m,n\in O\left(\omega \right).$ Therefore, $\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n\in O\left(\omega \right),$ i.e., $\xi (\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)\le \varpi (\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n).$ This implies that $\mathrm{{\rm Y}}$ is a $\varpi$-admissible mapping w.r.t. $\xi .$ Using (10), we obtain

$$\mathrm{\Omega }{(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)}^q\le \ell \left(\widehat{A_1}(m,n)\right),$$

where $\widehat{A_1}(m,n)$ is defined in Definition 7 and $q\in [1,\infty )$. This implies that $\mathrm{{\rm Y}}$ is a rational symmetric $\varpi -\xi$-contraction of kind (I). Therefore, all hypotheses of Theorem 2 are fulfilled. Hence, $\mathrm{{\rm Y}}$ has an FP. □

Remark 3.

Theorems 10–12 can be presented easily for a rational symmetric $\varpi -\xi$-contraction of kind (I), kind (II), kind (III), and kind (IV).

7. Solve Fractional-Order Differential Equations

We discuss in this part the existence and uniqueness of the bounded solution to an FODE by using some of the theoretical results presented at the beginning of the paper.

Let $Z:(0,\infty )\to \mathbb{R}$ be a given function. The conformable derivative of order $\tau$ of Z at $\varrho >0$ is described as [36]

$$D^{\tau }Z\left(\varrho \right)=\underset{ı\to 0}{lim}\frac{Z(\varrho +ı{\varrho }^{1-\tau })-Z\left(\varrho \right)}{ı}.$$

Abdeljawad [37] presented the notion of the conformable fractional integral of the function Z as follows:

$$I_0^{\tau }Z\left(\varrho \right)=\underset{0}{\overset{\varrho }{\int }}{\wp }^{\tau -1}Z(\wp )d\wp .$$

Now, we consider the following BVP of a conformable FODE:

$$\left\{\begin{array}{cc}{D}^{\tau }m\left(\varrho \right)=\delta Z(\varrho ,m\left(\varrho \right)),& 0<\varrho <1,\tau \in (1,2)\\ m\left(0\right)=0,m\left(1\right)=\underset{0}{\overset{1}{\int }}m(\wp )d\wp .\end{array}\right.$$

(11)

The BVP (11) can be expressed as the integral equation as follows:

$$m\left(\varrho \right)=\delta \underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )Z(\wp ,m(\wp ))d\wp ,$$

(12)

where $Q(\delta ,\wp )$ is Green’s function, described as

$$Q(\varrho ,\wp )=\left\{\begin{array}{cc}-2\varrho {\wp }^{\tau }+{\wp }^{\tau -1},& 0\le \wp \le \varrho \le 1,\\ -2\varrho {\wp }^{\tau },& 0\le \varrho \le \wp \le 1,\end{array}\right.$$

and $\underset{0}{\overset{1}{\int }}m(\wp )d\wp$ refers to the Riemann integrable of m w.r.t. ℘ and $Z:[0,1]\times \mathbb{R}\to \mathbb{R}$ a continuous function.

Consider

$$m\left(\varrho \right)={\mathrm{\Xi }}_0+{\mathrm{\Xi }}_1\varrho +\delta \underset{0}{\overset{\varrho }{\int }}{\wp }^{\tau -1}Z(\wp ,m(\wp ))d\wp ,$$

(13)

when $m\left(0\right)=0,$ then ${\mathrm{\Xi }}_0=0$ and from the condition $m\left(1\right)=\underset{0}{\overset{1}{\int }}m(\wp )d\wp ,$ we have:

$$\begin{array}{ccc}\hfill \underset{0}{\overset{1}{\int }}m(\wp )d\wp & =& {\mathrm{\Xi }}_1\underset{0}{\overset{1}{\int }}\wp d\wp +\delta \underset{0}{\overset{1}{\int }}\underset{0}{\overset{\wp }{\int }}{\vartheta }^{\tau -1}Z(\vartheta ,m\left(\vartheta \right))d\vartheta d\wp \hfill \\ & =& \frac{1}{2}{\mathrm{\Xi }}_1+\delta \underset{0}{\overset{1}{\int }}\underset{\vartheta }{\overset{1}{\int }}{\vartheta }^{\tau -1}Z(\vartheta ,m\left(\vartheta \right))d\wp d\vartheta \hfill \\ & =& \frac{1}{2}{\mathrm{\Xi }}_1+\delta \underset{\vartheta }{\overset{1}{\int }}(1-{\vartheta }^{\tau }){\vartheta }^{\tau -1}Z(\vartheta ,m\left(\vartheta \right))d\vartheta \hfill \\ & =& \frac{1}{2}{\mathrm{\Xi }}_1+\delta \underset{\vartheta }{\overset{1}{\int }}({\wp }^{\tau -1}-{\wp }^{\tau })Z(\wp ,m(\wp ))d\wp ;\hfill \end{array}$$

this implies that

$$\begin{array}{ccc}\hfill \frac{1}{2}{\mathrm{\Xi }}_1& =& -\delta \underset{0}{\overset{1}{\int }}{\wp }^{\tau -1}Z(\wp ,m(\wp ))d\wp +\delta \underset{\vartheta }{\overset{1}{\int }}({\wp }^{\tau -1}-{\wp }^{\tau })Z(\wp ,m(\wp ))d\wp \hfill \\ & =& -\delta \underset{0}{\overset{1}{\int }}{\wp }^{\tau }Z(\wp ,m(\wp ))d\wp ;\hfill \end{array}$$

hence, ${\mathrm{\Xi }}_1=-2\delta \underset{0}{\overset{1}{\int }}{\wp }^{\tau }Z(\wp ,m(\wp ))d\wp .$ It follows from (13), ${\mathrm{\Xi }}_0$, and ${\mathrm{\Xi }}_1$ that

$$\begin{array}{ccc}\hfill m\left(\varrho \right)& =& -2\delta \varrho \underset{0}{\overset{1}{\int }}{\wp }^{\tau }Z(\wp ,m(\wp ))d\wp +\delta \underset{0}{\overset{\varrho }{\int }}{\wp }^{\tau -1}Z(\wp ,m(\wp ))d\wp \hfill \\ & =& -2\delta \varrho \underset{0}{\overset{\varrho }{\int }}{\wp }^{\tau }Z(\wp ,m(\wp ))d\wp -2\delta \varrho \underset{\varrho }{\overset{1}{\int }}{\wp }^{\tau }Z(\wp ,m(\wp ))d\wp +\delta \underset{0}{\overset{\varrho }{\int }}{\wp }^{\tau -1}Z(\wp ,m(\wp ))d\wp \hfill \\ & =& \delta \underset{0}{\overset{\varrho }{\int }}\left(-2\varrho {\wp }^{\tau }+{\wp }^{\tau -1}\right)Z(\wp ,m(\wp ))d\wp +\delta \underset{\varrho }{\overset{1}{\int }}(-2\varrho {\wp }^{\tau })Z(\wp ,m(\wp ))d\wp \hfill \\ & =& \delta \underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )Z(\wp ,m(\wp ))d\wp .\hfill \end{array}$$

Assume that $C\left(I\right)$ is the space of all continuous functions defined on $I=[0,1]$, and consider

$$\mathrm{\Omega }(m,n)={∥m-n∥}_{\infty }=\underset{\varrho \in I}{sup}\left|m\left(\varrho \right)-n\left(\varrho \right)\right|,\forall m,n\in C\left(I\right).$$

Hence, the pair $\left(C\right(I),\mathrm{\Omega })$ is a $\mathrm{\Theta }$-complete MS.

Now, the BVP (11) is considered under the stipulations below:

$\left({♡}_1\right)$

There is a $\zeta \in (0,1)$ and $\beth ,\daleth :{\mathbb{R}}^2\to \mathbb{R}$ that are functions for each $f,g\in \mathbb{R}$ with $\beth (f,g)\ge \daleth (f,g)$ such that

$$\begin{array}{ccc}\hfill \left|Z(\wp ,m(\wp \left)\right)-Z(\wp ,n(\wp \left)\right)\right|& \le & \zeta \left|m(\wp )-n(\wp )\right|.\hfill \\ & & \frac{{\left|m(\wp )-\mathrm{{\rm Y}}m(\wp )\right|}^{\frac{j}{(j-k)(j-l)}}.{\left|n(\wp )-\mathrm{{\rm Y}}n(\wp )\right|}^{\frac{j}{(j-k)(j-l)}}}{1+\left|m(\wp )-n(\wp )\right|}.\hfill \\ & & \frac{{\left[\left|m(\wp )-\mathrm{{\rm Y}}m(\wp )\right|+\left|n(\wp )-\mathrm{{\rm Y}}n(\wp )\right|\right]}^{\frac{k}{(k-j)(k-l)}}}{1+\left|m(\wp )-n(\wp )\right|}.\hfill \\ & & \frac{{\left[\left|m(\wp )-\mathrm{{\rm Y}}n(\wp )\right|+\left|n(\wp )-\mathrm{{\rm Y}}m(\wp )\right|\right]}^{\frac{l}{(l-j)(l-k)}}}{1+\left|m(\wp )-n(\wp )\right|},\hfill \end{array}$$

where $j,k,l\in (0,1);$

$\left({♡}_2\right)$

For all $\varrho \in I,$ there is an $m_1\in C\left(I\right)$ such that

$$\beth \left(m_1\left(\varrho \right),\underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )Z(\wp ,m_1(\wp ))d\wp \right)\ge \daleth \left(m_1\left(\varrho \right),\underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )Z(\wp ,m_1(\wp ))d\wp \right);$$

$\left({♡}_3\right)$

For all $\varrho \in I$ and for each $m,n\in C\left(I\right)$, there are $m_1,n_1\in C\left(I\right)$ such that

$$\begin{array}{ccc}\hfill \beth \left(m\left(\varrho \right),n\left(\varrho \right)\right)& \ge & \daleth \left(m\left(\varrho \right),n\left(\varrho \right)\right)\mathrm{implies}\hfill \\ & & \beth \left(\underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )Z(\wp ,m_1(\wp ))d\wp ,\underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )Z(\wp ,n_1(\wp ))d\wp \right)\hfill \\ & \ge & \daleth \left(\underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )Z(\wp ,m_1(\wp ))d\wp ,\underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )Z(\wp ,n_1(\wp ))d\wp \right);\hfill \end{array}$$

$\left({♡}_4\right)$

For any sequence $\left\{m_i\right\}$ in $C\left(I\right),$ there is a cluster point m of $\left\{m_i\right\}$ with $\beth \left(m_i,m_{i+1}\right)\ge \daleth \left(m_i,m_{i+1}\right)$ such that

$$\underset{i\to \infty }{lim}inf\beth \left(m_i,m\right)\ge \underset{i\to \infty }{lim}inf\daleth \left(m_i,m\right).$$

Now, our main theorem is valid for presentation.

Theorem 13.

BVP (11) has at least one solution $\tilde{m}\in C\left(I\right),$ provided that the stipulations $\left({♡}_1\right)-\left({♡}_4\right)$ hold.

Proof. 

It is obvious that $m\in C\left(I\right)$ is a solution of (11) iff $m\in C\left(I\right)$ is a solution of the FODE

$$m\left(\varrho \right)=\delta \underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )Z(\wp ,m(\wp ))d\wp ,\forall \delta ,\varrho \in I.$$

Define an operator $\mathrm{{\rm Y}}:C\left(I\right)\to C\left(I\right)$ by

$$\mathrm{{\rm Y}}m\left(\varrho \right)=\delta \underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )Z(\wp ,m(\wp ))d\wp ,\forall \varrho \in I.$$

Therefore, if it is possible to find the FP $\tilde{m}\in C\left(I\right)$ of the mapping $\mathrm{{\rm Y}}$, then Problem (11) has at least one solution, which is the same FP.

Assume that $m,n\in C\left(I\right)$ such that $\beth \left(m\left(\varrho \right),n\left(\varrho \right)\right)\ge 0,$ for all $\varrho \in I.$ From $\left({♡}_1\right),$ we have

$$\begin{array}{ccc}\hfill \left|\mathrm{{\rm Y}}m(\wp )-\mathrm{{\rm Y}}n(\wp )\right|& =& \left|\delta \right|\left|\underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )\left[Z(\wp ,m(\wp \left)\right)-Z(\wp ,n(\wp \left)\right)\right]d\wp \right|\hfill \\ & \le & \left|\delta \right|\underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )\left|Z(\wp ,m(\wp \left)\right)-Z(\wp ,n(\wp \left)\right)\right|d\wp \hfill \\ & \le & \left|\delta \right|\underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )\zeta d\wp .\left|m(\wp )-n(\wp )\right|.\hfill \\ & & \frac{{\left|m(\wp )-\mathrm{{\rm Y}}m(\wp )\right|}^{\frac{j}{(j-k)(j-l)}}.{\left|n(\wp )-\mathrm{{\rm Y}}n(\wp )\right|}^{\frac{j}{(j-k)(j-l)}}}{1+\left|m(\wp )-n(\wp )\right|}.\hfill \\ & & \frac{{\left[\left|m(\wp )-\mathrm{{\rm Y}}m(\wp )\right|+\left|n(\wp )-\mathrm{{\rm Y}}n(\wp )\right|\right]}^{\frac{k}{(k-j)(k-l)}}}{1+\left|m(\wp )-n(\wp )\right|}.\hfill \\ & & \frac{{\left[\left|m(\wp )-\mathrm{{\rm Y}}n(\wp )\right|+\left|n(\wp )-\mathrm{{\rm Y}}m(\wp )\right|\right]}^{\frac{l}{(l-j)(l-k)}}}{1+\left|m(\wp )-n(\wp )\right|}\hfill \\ & \le & \underset{\wp \in I}{sup}\underset{0}{\overset{1}{\int }}Q(\varrho ,\wp )d\wp .\zeta {∥m(\wp )-n(\wp )∥}_{\infty }.\hfill \\ & & \frac{{∥m(\wp )-\mathrm{{\rm Y}}m(\wp )∥}_{\infty }^{\frac{j}{(j-k)(j-l)}}.{∥n(\wp )-\mathrm{{\rm Y}}n(\wp )∥}_{\infty }^{\frac{j}{(j-k)(j-l)}}}{1+{∥m(\wp )-n(\wp )∥}_{\infty }}.\hfill \\ & & \frac{{\left[{∥m(\wp )-\mathrm{{\rm Y}}m(\wp )∥}_{\infty }+{∥n(\wp )-\mathrm{{\rm Y}}n(\wp )∥}_{\infty }\right]}^{\frac{k}{(k-j)(k-l)}}}{1+{∥m(\wp )-n(\wp )∥}_{\infty }}.\hfill \\ & & \frac{{\left[{∥m(\wp )-\mathrm{{\rm Y}}n(\wp )∥}_{\infty }+{∥n(\wp )-\mathrm{{\rm Y}}m(\wp )∥}_{\infty }\right]}^{\frac{l}{(l-j)(l-k)}}}{1+{∥m(\wp )-n(\wp )∥}_{\infty }}\hfill \\ & \le & \zeta {∥m(\wp )-n(\wp )∥}_{\infty }.\frac{{∥m(\wp )-\mathrm{{\rm Y}}m(\wp )∥}_{\infty }^{\frac{j}{(j-k)(j-l)}}.{∥n(\wp )-\mathrm{{\rm Y}}n(\wp )∥}_{\infty }^{\frac{j}{(j-k)(j-l)}}}{1+{∥m(\wp )-n(\wp )∥}_{\infty }}.\hfill \\ & & \frac{{\left[{∥m(\wp )-\mathrm{{\rm Y}}m(\wp )∥}_{\infty }+{∥n(\wp )-\mathrm{{\rm Y}}n(\wp )∥}_{\infty }\right]}^{\frac{k}{(k-j)(k-l)}}}{1+{∥m(\wp )-n(\wp )∥}_{\infty }}.\hfill \\ & & \frac{{\left[{∥m(\wp )-\mathrm{{\rm Y}}n(\wp )∥}_{\infty }+{∥n(\wp )-\mathrm{{\rm Y}}m(\wp )∥}_{\infty }\right]}^{\frac{l}{(l-j)(l-k)}}}{1+{∥m(\wp )-n(\wp )∥}_{\infty }}.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill \mathrm{\Omega }(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n)& \le & \zeta \left(\mathrm{\Omega }(m,n).\frac{\mathrm{\Omega }{(m,\mathrm{{\rm Y}}m)}^{\frac{j}{(j-k)(j-l)}}.\mathrm{\Omega }{(n,\mathrm{{\rm Y}}n)}^{\frac{j}{(j-k)(j-l)}}}{1+\mathrm{\Omega }{(m,n)}^q}.\right)\hfill \\ & & \left.\frac{{\left[\mathrm{\Omega }(m,\mathrm{{\rm Y}}m)+\mathrm{\Omega }(n,\mathrm{{\rm Y}}n)\right]}^{\frac{k}{(k-j)(k-l)}}}{1+\mathrm{\Omega }(m,n)}.\frac{{\left[\mathrm{\Omega }(m,\mathrm{{\rm Y}}n)+\mathrm{\Omega }(n,\mathrm{{\rm Y}}m)\right]}^{\frac{l}{(l-j)(l-k)}}}{1+\mathrm{\Omega }(m,n)}\right).\hfill \end{array}$$

for each $m,n\in C\left(I\right)$ such that $\beth \left(m\left(\varrho \right),n\left(\varrho \right)\right)\ge \daleth \left(m\left(\varrho \right),n\left(\varrho \right)\right),$ for all $\varrho \in I.$

Now, describe $\varpi ,\xi :C\left(I\right)\times C\left(I\right)\to [0,\infty )$ as

$$\varpi (m,n)=\left\{\begin{array}{cc}1,& if\beth \left(m\left(\varrho \right),n\left(\varrho \right)\right)\ge 0,\varrho \in I.\\ 0,& \mathrm{otherwise},\end{array}\right.$$

$$\xi (m,n)=\left\{\begin{array}{cc}\frac{1}{3},& if\daleth \left(m\left(\varrho \right),n\left(\varrho \right)\right)\ge 0,\varrho \in I.\\ 0,& \mathrm{otherwise}.\end{array}\right.$$

Clearly, $\varpi (m,n)\ge \xi (m,n)$ for all $m,n\in C\left(i\right).$ Using $\left({♡}_3\right)$, we find that

$$\beth \left(\mathrm{{\rm Y}}m\left(\varrho \right),\mathrm{{\rm Y}}n\left(\varrho \right)\right)\ge \daleth \left(\mathrm{{\rm Y}}m\left(\varrho \right),\mathrm{{\rm Y}}n\left(\varrho \right)\right),$$

and so

$$\varpi \left(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n\right)\ge \xi \left(\mathrm{{\rm Y}}m,\mathrm{{\rm Y}}n\right);$$

therefore, $\mathrm{{\rm Y}}$ is a $\varpi$-admissible map concerning $\xi .$ Based on the stipulation $\left({♡}_2\right),$ there exists $m_1\in C\left(I\right)$ parallel to $\varpi \left(m_1,\mathrm{{\rm Y}}{m}_1\right)\ge \xi \left(m_1,\mathrm{{\rm Y}}{m}_1\right)$. By $\left({♡}_4\right),$ we obtain a cluster point m of the sequence $\left\{m_i\right\}$ in $C\left(I\right)$ with $\beth \left(m_i,m_{i+1}\right)\ge \daleth \left(m_i,m_{i+1}\right)$ implying $\varpi \left(m_i,m_{i+1}\right)\ge \xi \left(m_i,m_{i+1}\right)$ and ${lim}_{i\to \infty }inf\varpi \left(m_i,m\right)\ge {lim}_{i\to \infty }inf\xi \left(m_i,m\right).$ Thus, all constraints of Theorem 4 are fulfilled with $q=1$. Then, the mapping $\mathrm{{\rm Y}}$ has an FP $\tilde{m}\in C\left(I\right)$, which is a solution to the BVP (11). □

8. Conclusions

The objective of this manuscript was to produce four new classes of type contractions. This research focused on new concepts of symmetric $\varpi -\xi$-contractions of type (I), type (II), type (III), and type (IV) and inferred some fixed-points for such contractions in the setting of $\mathrm{\Theta }$-metric spaces. This research will open a new avenue of fractional fixed-point theory. We developed Suzuki-type fixed-point results in orbitally complete F-metric space. These, new investigations and applications will enhance the impact of the new setup.