Common Fixed Point Theorems in Complex-Valued Non-Negative Extended b-Metric Spaces

Abstract

We define modified F.P results for mappings, according to some rational contractions and present to C.C.N.V.b-M Spaces. The relevant F.P theorems in the context of extended b-metric b-metric and classical M.Spaces are improved and expanded by our proposal. There are presented nontrivial examples to back up the hypotheses and the value of primary conclusion achieved herein.

1. Introduction

F.P.Theory is one of the quite well-accepted fields in mathematics. Theory of Banach [1] fixed point plays the main rule in the execution of solution of nonlinear phenomena. This theory presents the concept of contractive mappings and C.V.M.Space to find out the fixed point of certain function. A different kind of contractive condition that explains the fixed point theorem was presented by Kannan [2] in 1969. The difference between the Banach theorem and the mapping theory of Kannan is that whereas Kannan maps do not always require continuity, Banach mappings do require continuity for contraction. Furthermore, Chaterjea [3] also demonstrated that type of contraction. Numerous academics have improved the Banach [1], F.P.Theorem in a number of different ways (see [4, 5, 6, 7]). For additional definitions of contractive sort of mappings, see Rhoades [7]. All of the modifications of the Banach F.P.Theorem are further divided into two categories: those that replace the contractive condition with a more comprehensive one and those that increase or weaken the axioms that define the ground set. In the latter case, a few of these type of M.Spaces have names such as semimetric, pseudometric, quasimetric, K-metric, and bmetric, to name a few. In agreement with all of this, by substituting a typical co-domain for a metric in place $\mathrm{ℝ}$ that is, the set of real numbers, Zhang and Huang [8] originated the cone metric space idea to the literature and enhanced it in different type of metric spaces. By continuing the setting of these ideas, the authors build up a numbers of contractive mappings in F.P. Theorems are proven here in (cone metric spaces). In the setting of cone metric spaces, other writers have been developed a number of significant fixed point results (see [9, 10]). For the in-depth research study, the interested scholar may go along the study on cone M.Spaces, which was introduced by mathematics scholars named Aleksic et al. [11].

The well-known property in cone M.Spaces, F.P.Theorems using rational con-tractions cannot be extended or rendered irrelevant. Azam et al. [4] developed the concept of C.V.M.Spaces to get around this issue and set up appropriate constraints for a pair of maps to the validity of contractive type inequalities including rational expressions in common fixed points [12, 13]. It is fascinating to note that a specific class of cone metric spaces includes C.V.M.Space. A division ring is not the basis for the definition of a cone metric, which is based on the ground set, that is, Banach space. As a result, many division-related clearly showing that we cannot extrapolated them into a cone metric spaces. The rational inequality was introduced in C.V.M.Spaces and thus new results are proved over there (see, [4, 14, 15, 16]). Along these lines, in 1993 Czerwik [17] proposed the notion of b-metric space and Branciari [18], made up a change in triangular inequality to create the idea of rectangle metric space. In addition to complex-valued metric spaces, Rao [19] also proposed the concept of fixed point results on complex-valued b-metric spaces. Every C.V.b-M.Space, however, is a cone b-metric space over Banach algebra C, where the cone is normal and the normalcy coefficient is $\kappa =1.$ and where the interior of the cone is not empty (i.e., solid cone). With references to C.V.b − M.Spaces, numerous authors have shown fixed point solutions for various maps that satisfy rational inequalities in the paragraphs that follow (see, for illustration, [19, 20, 21]).

Inspired by the concepts provided in [4, 17, 18], we define new F.P.Theorems for maps under particular rational constructive inequalities and introduce the notion of C.V.N.b-M.Spaces. The publications cited above, as well as a few others in the related literature, are improved and expanded upon by our idea. Our proposal enhances and expands upon the aforementioned papers as well as a few other sections of related literature.

2. Preliminaries

In this section, we review a few key ideas that are essential to present of our the primary conclusion. [4] Let $\mathbb{C}$ represent the number system of complex numbers and let $p_1,p_2\in \mathbb{C}.$ The definition of the partial order on $\mathbb{C}$ is:

$p_1\preccurlyeq p_2,\iff \mathit{Re}\left(p_1\right)\le \mathit{Re}\left(p_2\right)$ and $\mathit{Im}\left(p_1\right)\le \mathit{Im}\left(p_2\right).$ This implies that $p_1\preccurlyeq p_2$ if one of below follows:

• $\left[\left(C_i\right)\right]\mathit{Re}\left(p_1\right)=\mathit{Re}\left(p_2\right),\mathit{Im}\left(p_1\right)<\mathit{Im}\left(p_2\right),$

• $\left[\left(C_{\mathit{ii}}\right)\right]\mathit{Re}\left(p_1\right)<\mathit{Re}\left(p_2\right),\mathit{Im}\left(p_1\right)=\mathit{Im}\left(p_2\right),$

• $\left[\left(C_{\mathit{iii}}\right)\right]\mathit{Re}\left(p_1\right)<\mathit{Re}\left(p_2\right),\mathit{Im}\left(p_1\right)<\mathit{Im}\left(p_2\right),$

• $\left[\left(C_{\mathit{iv}}\right)\right]\mathit{Re}\left(p_1\right)=\mathit{Re}\left(p_2\right),\mathit{Im}\left(p_1\right)=\mathit{Im}\left(p_2\right).$

[4] Let $\mathbb{M}$ be a set with function ${\psi }_{\vartheta }:\mathbb{M}\times \mathbb{M}\to \mathbb{C}$_, which meet the below conditions:

$$\left(C_i\right)0\preccurlyeq {\psi }_{\vartheta }\left(u,v\right)$$

$$\left(C_{\mathit{ii}}\right){\psi }_{\vartheta }\left(u,v\right)=0\iff u=v;$$

$$\left(C_{\mathit{iii}}\right){\psi }_{\vartheta }\left(u,v\right)={\psi }_{\vartheta }\left(v,u\right);$$

$$\left(C_{\mathit{iv}}\right){\psi }_{\vartheta }\left(u,v\right)\preccurlyeq {\psi }_{\vartheta }\left(u,w\right)+{\psi }_{\vartheta }\left(w,v\right),\forall u,w,v\in \mathbb{M},$$

then ${\psi }_{\vartheta }$ is referred as a C.V.Metric on $\mathbb{M}$ and $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ is known as C.V.Mspace.

Let $\mathbb{M}=M_1\cup M_2,$ where

$$M_1=\left\{h\in \mathbb{C}:\mathit{Re}\left(h\right)\ge 0\text{and}\mathit{Im}\left(h\right)=0\right\}$$

and

$$M_2=\left\{h\in \mathbb{C}:\mathit{Re}\left(h\right)=0\text{and}\mathit{Im}\left(h\right)\ge 0\right\}.$$

Define $d:\mathbb{M}\times \mathbb{M}\to \mathbb{C}$ as follows:

$${\psi }_{\vartheta }\left(p_1,p_2\right)=\left\{\begin{array}{l}\frac{2}{3}\left|u_1-u_2\right|+\frac{i}{2}\left|u_1-u_2\right|,\text{if}{p}_1,p_2\in {\mathbb{M}}_1;\\ \frac{1}{2}\left|v_1-v_2\right|+\frac{i}{2}\left|v_1-v_2\right|,\text{if}{p}_1,p_2\in {\mathbb{M}}_2;\\ \left(\frac{1}{2}{v}_1+\frac{2}{3}{u}_2\right)+i\left(\frac{1}{3}{v}_1+\frac{1}{2}{u}_2\right),\text{if}{p}_1\in M_1,p_2\in M_2,\end{array}\right.$$

where $p_1=u_1+{\mathit{iv}}_1$ and $p_2=u_2+{\mathit{iv}}_2.$ Then, $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ is a C.V.M.Space. [4] Let $\mathbb{M}\ne \varphi$ be a set with mapping: ${\psi }_{\vartheta }:\mathbb{M}\times \mathbb{M}\to \mathbb{C}$ which meet the below requirements:

$$\left(C_i\right)0\preccurlyeq {\psi }_{\vartheta }\left(u,v\right)\left(C_{\mathit{ii}}\right){\psi }_{\vartheta }\left(u,v\right)=0\iff u=v;$$

$$\left(C_{\mathit{iii}}\right){\psi }_{\vartheta }\left(u,v\right)=\mathit{psi}\left(v,u\right);$$

$$\left(C_{\mathit{iii}}\right)d\left(u,v\right)\preccurlyeq \tau \left[d\left(u,w\right)+d\left(w,v\right)\right],\forall \text{where}\tau \ge 1u,w,v\in \mathbb{M},$$

then ${\psi }_{\vartheta }$ is known as a C.V.b-Metric on $\mathbb{M},$ and the $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ is said to be a C.V.b − M.Space [19]. Let $\mathbb{M}=\left[0,1\right].$ Define a function: ${\psi }_{\vartheta }:\mathbb{M}\times \mathbb{M}\to \mathbb{C}$ by

$${\psi }_{\vartheta }\left(u,v\right)={\left|u-v\right|}^2+i{\left|u-v\right|}^2$$

for all $u,v\in \mathbb{M}.$ Then $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ is a C.V.b − M.Space with $\mu =2.$ [4] Let $\mathbb{M}\ne \varphi$ a set with $\theta :\mathbb{M}\times \mathbb{M}\to \left[1,\infty \right)$ be a mapping and the function: ${\psi }_{\vartheta }:\mathbb{M}\times \mathbb{M}\to \mathbb{C}$ obeying the below conditions:

$$\left(C_i\right)0\preccurlyeq {\psi }_{\vartheta }\left(u,v\right)$$

$$\left(C_{\mathit{ii}}\right){\psi }_{\vartheta }\left(u,v\right)=0\iff u=v;$$

$$\left(C_{\mathit{iii}}\right){\psi }_{\vartheta }\left(u,v\right)={\psi }_{\vartheta }\left(v,u\right);$$

$$\left(C_{\mathit{iv}}\right){\psi }_{\vartheta }\left(u,v\right)\preccurlyeq \vartheta \left(u,v\right)\left[{\psi }_{\vartheta }\left(u,w\right)+{\psi }_{\vartheta }(w,v)\right],\forall u,w,v\in \mathbb{M},$$

then ${\psi }_{\vartheta }$ is called C.V.b-Metric on $\mathbb{M},$ and the $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ is known to be a C.V.b − M.Space [4]. Let $\mathbb{M}\ne C\left(\left[r,s\right],\mathrm{ℝ}\right),$ where$C(\left[r,s\right]$ represents a set of continuous mappings which are real-valued, defined on [r, s] and a map ${\psi }_{\vartheta }:\mathbb{M}\times \mathbb{M}\to \left[1,\infty \right)$ is

$$\vartheta \left(u,v\right)=\mid u\left(z\right)\mid +\mid v\left(z\right)\mid +2.$$

Also, define ${\psi }_{\vartheta }:\mathbb{M}\times \mathbb{M}\to \mathbb{C}$ by

$${\psi }_{\vartheta }\left(u,v\right)=\underset{z\in \left[r,s\right]}{max}{\left|u\left(z\right)-v\left(z\right)\right|}^2.$$

Then, $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ is C.V.b − M.Space. Let $\mathbb{M}\ne \varphi$ be a set, ${\vartheta }_0,\vartheta :\mathbb{M}\times \mathbb{M}\to \left[0,\infty \right)$ be defined by

$$\vartheta \left(u,v\right)={\vartheta }_0\left(u,v\right)+\mu ,$$

for all $u,v\in \mathbb{M},$ and $\mu \ge 1.$ Define a mapping: ${\psi }_{\vartheta }:\mathbb{M}\times \mathbb{M}\to \mathbb{C},$ if for all $u,v,w\in \mathbb{M},$ the following assertion are valid.

$$\left(C_i\right)0\preccurlyeq {\psi }_{\vartheta }\left(u,v\right)$$

$$\left(C_{\mathit{ii}}\right){\psi }_{\vartheta }\left(u,v\right)=0\text{if\hspace{0.17em}\hspace{0.17em}and\hspace{0.17em}\hspace{0.17em}only\hspace{0.17em}\hspace{0.17em}if}u=v;$$

$$\left(C_{\mathit{iii}}\right){\psi }_{\vartheta }\left(u,v\right)={\psi }_{\vartheta }\left(v,u\right);$$

$$\left(C_{\mathit{iii}}\right){\psi }_{\vartheta }\left(u,v\right)\preccurlyeq \vartheta \left(u,v\right)\left[{\psi }_{\vartheta }\left(u,w\right)+d(w,v)\right].$$

Then ${\psi }_{\vartheta }$ is known as C.V.N.E.b-Metric on $\mathbb{M}$ and the pair $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ is known as a C.V.N.Eb-M.Space.

2.1 Ingredient of C.V.E.b − M.Space

If we setting ${\vartheta }_0\left(u,v\right)=u+v$ and $\mu =1,$ then we have $\vartheta \left(u,v\right)=1+u+v={\vartheta }^{\ast }\left(u,v\right).$.

Property $\left(C_{\mathit{iii}}\right)$ of 2 will be updated by

$${\psi }_{\vartheta }\left(u,v\right)\preccurlyeq {\vartheta }^{\ast }\left(u,v\right)\left[{\psi }_{\vartheta }\left(u,w\right)+{\psi }_{\vartheta }(w,v)\right].$$

Thus, the definition 2 with respect to that condition becomes C.V extended b-M.Space.

2.2 Ingredient of C.V b-M.Spaces

If we setting ${\vartheta }_0\left(u,v\right)=0,$ then $\theta \left(u,v\right)=0+\mu ,$ where $\mu \ge 1,$ then property $\left(C_{\mathit{iii}}\right)$ of 2 is

$${\psi }_{\vartheta }\left(u,v\right)\preccurlyeq \tau \left[{\psi }_{\vartheta }\left(u,w\right)+{\psi }_{\vartheta }(w,v)\right].$$

The notion of complex-valued b-metric spaces is thus defined by definition 2 with this aspect.

2.3 Ingredient of C.V.M.Spaces

Similar to that when we put $\mu =1,$ in the preceding definition, then its gives us the “C.V.M.Spaces.”

2.4 Development of ordinary M.Space

By replacing the ground set of the previous one’s $\mathbb{C}$ with $\mathrm{ℝ}.$ thus ordinary M.Spaces is defined. [19] Consider a set $\mathbb{M}=\left[0,1\right]$ and let a function ${\psi }_{\vartheta }:\mathbb{M}\times \mathbb{M}\to \mathbb{C}$ by

$${\psi }_{\vartheta }\left(u,v\right)={\left|u-v\right|}^2+i{\left|u-v\right|}^2$$

for all $u,v\in \mathbb{M}.$ Then, $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ is a C.V.N.E.b-M.Space and letting $\mu =2$ and ${\vartheta }_0\left(u,v\right)=0$.

[16] Let $\mathbb{M}\ne \varphi$ set with mapping ${\vartheta }_0,\vartheta :\mathbb{M}\times \mathbb{M}\to \left[0,\infty \right)$ be defined as:

$$\vartheta \left(u,v\right)=1+u+v,{\vartheta }_0\left(u,v\right)=u+v,\mu =1$$

Furthermore, let

• $\left[\left(C_i\right)\right]{\psi }_{\vartheta }\left(u,v\right)=\frac{i}{\mathit{uv}},\forall u,v\in \left(0,1\right];$

• $\left[\left(C_{\mathit{ii}}\right)\right]{\psi }_{\vartheta }\left(u,v\right)=0\iff u=v\text{for}\mathrm{all}u,v\in \left[0,1\right];$

• $\left[\left(C_{\mathit{iii}}\right)\right]{\psi }_{\vartheta }\left(u,0\right)={\psi }_{\vartheta }\left(0,u\right)=\frac{i}{u}\text{for}\mathrm{all}u\in \left(0,1\right].$

Then, the pair $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ is a C.V.N.E.b-M.Space. [16] Let $\mathbb{M}=\left[0,\infty \right).\vartheta :\mathbb{M}\times \mathbb{M}\to \left[0,\infty \right)$ be a mapping defined by $\vartheta \left(u,v\right)=1+u+v$ and ${\psi }_{\vartheta }:\mathbb{M}\times \mathbb{M}\to \mathbb{C}$ to be used as

$${\psi }_{\vartheta }\left(u,v\right)=\left\{\begin{array}{lll}0,& \text{whenever}& u=v\\ i,& \text{whenever}& u\ne v.\end{array}\right.$$

Then, $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ is a C.V non-negative extended b-M.Space. A non-empty set $\mathbb{M}$ with E and F be a self-maps then:

$\left(C_i\right)$ An element $u\in \mathbb{M}$ is known as F.Point of F if Fu = u.

$\left(C_{\mathit{ii}}\right)$ An element $u\in \mathbb{M}$ is known as coincidence point of F and E if Fu = Eu and we will refer t = Tu = Su to a point of coincidence of F and E.

$\left(C_{\mathit{iii}}\right)$ An element $u\in \mathbb{M}$ is known as C.F.Point of F and E if u = Fu = Eu. [4] Let $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ be a C.V.R.E.b − M.Space and the $\left\{u_k\right\}$ sequence in $\mathbb{M}$ is a convergent sequence $\iff \left|{\psi }_{\vartheta }\left(u_k,u\right)\right|\to 0$ as $k\to \infty .$.

[4] Let $\left(\mathbb{M},{\psi }_{\vartheta }\right)$ be a C.V.N.E.b-M.Space and the $\left\{u_k\right\}$ a sequence in $\mathbb{M}.$ This implies that $\left\{u_k\right\}$ is a Cauchy sequence $\iff \left|{\psi }_{\vartheta }\left(u_k,u_l\right)\right|\to 0$ as $k,l\to \infty .$

3. Main results

Our primary results are presented below.

Theorem 1.1 A complete C.V non-negative extended b-M.Space $\left(\mathbb{M},{\psi }_{\vartheta }\right),$ with a mapping $\vartheta ,{\vartheta }_i:\mathbb{M}\times \mathbb{M}\to \left[0,\infty \right)\left(i=0,1,2\text{and}\vartheta ={\vartheta }_0+\mu ,\mu \ge 1\right)$ and $F,E:\mathit{\text{mathbbM}}\to \mathit{\text{mathbbM}}$ satisfying the assertions below:

1. ${\vartheta }_1<{\vartheta }_2;$

2. $\underset{k\to \infty }{lim}\vartheta \left(u_{k+1},u_l\right){\vartheta }_1\left(u_{k+1},u_{k+2}\right)+{\vartheta }_2\left(u_{k+1},u_{k+2}\right)<1;$

3. ${\psi }_{\vartheta }\left(\mathit{Fu},\mathit{Ev}\right)\preccurlyeq {\vartheta }_1\left(u,v\right){\psi }_{\vartheta }\left(u,v\right)+{\vartheta }_2\left(u,v\right)\frac{{\psi }_{\vartheta }\left(u,\mathit{Fu}\right){\psi }_{\vartheta }\left(v,\mathit{Ev}\right)}{1+{\psi }_{\vartheta }\left(u,v\right)}.$

Then F, E have a unique common fixed point in $\mathbb{M}.$.

Proof: Let $u_0\in \mathbb{M}$ be any element in $\mathbb{M}.$ Build up a sequence $\left\{u_k\right\}$ such that

$$u_{2n+1}={\mathit{Fu}}_{2n},u_{2n+2}={\mathit{Eu}}_{2n+1}$$E1

for all $n\ge 0.$ From hypothesis and 1, we get.

$$\begin{array}{l}{\psi }_{\vartheta }\left(u_{2k+1},u_{2k+2}\right)={\psi }_{\vartheta }\left({\mathit{Fu}}_{2k},{\mathit{Eu}}_{2k+1}\right)\preccurlyeq {\vartheta }_1\left(u_{2k},u_{2k+1}\right){\psi }_{\vartheta }\left(u_{2k},u_{2k+1}\right)\\ +\frac{{\vartheta }_2\left(u_{2k},u_{2k+1}\right){\psi }_{\vartheta }\left(u_{2k},{\mathit{Fu}}_{2k}\right){\psi }_{\vartheta }\left(u_{2k+1},{\mathit{Eu}}_{2k+1}\right)}{1+{\psi }_{\vartheta }\left(u_{2k},u_{2k+1}\right)},\\ \preccurlyeq {\vartheta }_1\left(u_{2k},u_{2k+1}\right){\psi }_{\vartheta }\left(u_{2k},u_{2k+1}\right)+{\vartheta }_2\left(u_{2k},u_{2k+1}\right){\psi }_{\vartheta }\left(u_{2k+1},{\mathit{Eu}}_{2k+1}\right)\frac{{\psi }_{\vartheta }\left(u_{2k},u_{2k+1}\right)}{1+{\psi }_{\vartheta }\left(u_{2k},u_{2k+1}\right)},\\ \preccurlyeq {\vartheta }_1\left(u_{2k},u_{2k+1}\right){\psi }_{\vartheta }\left(u_{2k},u_{2k+1}\right)+{\vartheta }_2\left(u_{2k},u_{2k+1}\right){\psi }_{\vartheta }\left(u_{2k+1},u_{2k+2}\right).\end{array}$$

This implies that

$$\left(1-{\vartheta }_2\left(u_{2k},u_{2k+1}\right)\right){\psi }_{\vartheta }\left(u_{2k+1},u_{2k+2}\right)\preccurlyeq {\vartheta }_1\left(u_{2k},u_{2k+1}\right){\psi }_{\vartheta }\left(u_{2k},u_{2k+1}\right).$$

That is,

$$\begin{array}{l}{\psi }_{\vartheta }\left(u_{2k+1},u_{2k+2}\right)\preccurlyeq \frac{{\vartheta }_1\left(u_{2k},u_{2k+1}\right)}{1-{\vartheta }_2\left(u_{2k},u_{2k+1}\right)}{\psi }_{\vartheta }\left(u_{2k},u_{2k+1}\right)\\ \preccurlyeq \frac{{\vartheta }_1\left(u_{2k},u_{2k+1}\right)}{1-{\vartheta }_2\left(u_{2k},u_{2k+1}\right)}\frac{{\vartheta }_1\left(u_{2k-1},u_{2k}\right)}{1-{\vartheta }_2\left(u_{2k-1},u_{2k}\right)}{\psi }_{\vartheta }\left(u_{2k-1},u_{2k}\right)\\ \preccurlyeq \frac{{\vartheta }_1\left(u_{2k},u_{2k+1}\right)}{1-{\vartheta }_2\left(u_{2k},u_{2k+1}\right)}\frac{{\vartheta }_1\left(u_{2k-1},u_{2k}\right)}{1-{\vartheta }_2\left(u_{2k-1},u_{2k}\right)}\frac{{\vartheta }_1\left(u_{2k-2},u_{2k-1}\right)}{1-{\vartheta }_2\left(u_{2k-2},u_{2k-1}\right)}{\psi }_{\vartheta }\left(u_{2k-2},u_{2k-1}\right).\end{array}$$

By following the same step, we have

$$\begin{array}{l}\left(2\right){\psi }_{\vartheta }\left(u_{2n+1},u_{2n+2}\right)\preccurlyeq \frac{{\vartheta }_1\left(u_{2n},u_{2n+1}\right)}{1-{\vartheta }_2\left(u_{2n},u_{2n+1}\right)}\frac{{\vartheta }_1\left(u_{2n-1},u_{2n}\right)}{1-{\vartheta }_2\left(u_{2n-1},u_{2n}\right)}\frac{{\vartheta }_1\left(u_{2n-2},u_{2n-1}\right)}{1-{\vartheta }_2\left(u_{2n-2},u_{2n-1}\right)}\\ \left(3\right)⋮\\ \left(4\right)\times \frac{{\vartheta }_1\left(u_0,u_1\right)}{1-{\vartheta }_2\left(u_0,u_1\right)}{\psi }_{\vartheta }\left(u_0,u_1\right).\end{array}$$

Taking $l>k,$ we have

$$\begin{array}{l}{\psi }_{\vartheta }\left(u_k,u_l\right)\preccurlyeq \vartheta \left(u_k,u_l\right){\psi }_{\vartheta }\left(u_k,u_{k+1}\right)+\vartheta \left(u_k,u_l\right)\vartheta \left(u_{k+1},u_l\right){\psi }_{\vartheta }\left(u_{k+1},u_{k+2}\right)+.\dots \\ +\vartheta \left(u_k,u_l\right)\vartheta \left(u_{k+1},u_l\right)\vartheta \left(u_{k+2},u_l\right).\dots \vartheta \left(u_{l-1},u_l\right){\psi }_{\vartheta }\left(u_{l-1},u_l\right),\\ \preccurlyeq \vartheta \left(u_0,u_l\right)\vartheta \left(u_1,u_l\right)\vartheta \left(u_2,u_l\right).\dots \vartheta \left(u_{k-1},u_l\right)\vartheta \left(u_k,u_l\right)\\ \frac{{\vartheta }_1\left(u_0,u_1\right)}{1-{\vartheta }_2\left(u_0,u_1\right)}\frac{{\vartheta }_1\left(u_1,u_2\right)}{1-{\vartheta }_2\left(u_1,u_2\right)}.\dots \frac{{\vartheta }_1\left(u_{k-1},u_k\right)}{1-{\vartheta }_2\left(u_{k-1},u_k\right)}{\psi }_{\vartheta }\left(u_0,u_1\right)\\ +\vartheta \left(u_0,u_l\right)\vartheta \left(u_1,u_l\right)\vartheta \left(u_2,u_l\right).\dots \vartheta \left(u_k,u_l\right)\vartheta \left(u_{k+1},u_l\right)\\ \frac{{\vartheta }_1\left(u_0,u_1\right)}{1-{\vartheta }_2\left(u_0,u_1\right)}\frac{{\vartheta }_1\left(u_1,u_2\right)}{1-{\vartheta }_2\left(u_1,u_2\right)}.\dots \frac{{\vartheta }_1\left(u_{k+1},u_{k+2}\right)}{1-{\vartheta }_2\left(u_{k+1},u_{k+2}\right)}{\psi }_{\vartheta }\left(u_0,u_1\right)\\ .\\ .\\ .\\ +\vartheta \left(u_0,u_l\right)\vartheta \left(u_1,u_l\right)\vartheta \left(u_2,u_l\right).\dots \vartheta \left(u_{l-2},u_l\right)\vartheta \left(u_{l-1},u_l\right)\\ \frac{{\vartheta }_1\left(u_0,u_1\right)}{1-{\vartheta }_2\left(u_0,u_1\right)}\frac{{\vartheta }_1\left(u_1,u_2\right)}{1-{\vartheta }_2\left(u_1,u_2\right)}.\dots \frac{{\vartheta }_1\left(u_{l-2},u_{l-1}\right)}{1-{\vartheta }_2\left(u_{l-2},u_{l-1}\right)}{\psi }_{\vartheta }\left(u_0,u_1\right).\end{array}$$

This implies

$$\begin{array}{l}{\psi }_{\vartheta }\left(u_k,u_l\right)\preccurlyeq \vartheta \left(u_k,u_l\right){\psi }_{\vartheta }\left(u_k,u_{k+1}\right)+\vartheta \left(u_k,u_l\right)\vartheta \left(u_{k+1},u_l\right){\psi }_{\vartheta }\left(u_{k+1},u_{k+2}\right)+.\dots \\ +\vartheta \left(u_k,u_l\right)\vartheta \left(u_{k+1},u_l\right)\vartheta \left(u_{k+2},u_l\right).\dots \vartheta \left(u_{l-1},u_l\right){\psi }_{\vartheta }\left(u_{l-1},u_l\right),\\ \preccurlyeq {\psi }_{\vartheta }\left(u_0,u_1\right)\left[\vartheta \left(u_0,u_l\right)\vartheta \left(u_1,u_l\right)\vartheta \left(u_2,u_l\right).\dots \vartheta \left(u_{k-1},u_l\right)\vartheta \left(u_k,u_l\right)\right.\\ \frac{{\vartheta }_1\left(u_0,u_1\right)}{1-{\vartheta }_2\left(u_0,u_1\right)}\frac{{\vartheta }_1\left(u_1,u_2\right)}{1-{\vartheta }_2\left(u_1,u_2\right)}.\dots \frac{{\vartheta }_1\left(u_{k-1},u_k\right)}{1-{\vartheta }_2\left(u_{k-1},u_k\right)}\\ +\vartheta \left(u_0,u_l\right)\vartheta \left(u_1,u_l\right)\vartheta \left(u_2,u_l\right).\dots \vartheta \left(u_k,u_l\right)\vartheta \left(u_{k+1},u_l\right)\\ \frac{{\vartheta }_1\left(u_0,u_1\right)}{1-{\vartheta }_2\left(u_0,u_1\right)}\frac{{\vartheta }_1\left(u_1,u_2\right)}{1-{\vartheta }_2\left(u_1,u_2\right)}.\dots \frac{{\vartheta }_1\left(u_{k+1},u_{k+2}\right)}{1-{\vartheta }_2\left(u_{k+1},u_{k+2}\right)}\\ .\\ .\\ .\\ +\begin{array}{c}\vartheta \left(u_0,u_l\right)\vartheta \left(u_1,u_l\right)\vartheta \left(u_2,u_l\right).\dots \vartheta \left(u_{l-2},u_l\right)\vartheta \left(u_{l-1},u_l\right)\end{array}\\ \left.\frac{{\vartheta }_1\left(u_0,u_1\right)}{1-{\vartheta }_2\left(u_0,u_1\right)}\frac{{\vartheta }_1\left(u_1,u_2\right)}{1-{\vartheta }_2\left(u_1,u_2\right)}.\dots \frac{{\vartheta }_1\left(u_{l-2},u_{l-1}\right)}{1-{\vartheta }_2\left(u_{l-2},u_{l-1}\right)}\right].\end{array}$$

Since, ${lim}_{k\to \infty }\vartheta \left(u_{k+1},u_l\right){\vartheta }_1\left(u_{k+1},u_{k+2}\right)+{\vartheta }_2\left(u_{k+1},u_{k+2}\right)<1,$ thus the series.

${\sum }_{k=0}^{\infty }{\prod }_{i=0}^k\frac{\vartheta \left(u_i,u_l\right){\vartheta }_1\left(u_i,u_{i+1}\right)}{1-{\vartheta }_1\left(u_i,u_{i+1}\right)}$ converges under the ratio test. For each $l\in \mathrm{\mathbb{N}},$ let

$$S_{l-1}=\sum _{j=0}^{l-1}\prod _{i=0}^j\frac{\vartheta \left(u_i,u_l\right){\vartheta }_1\left(u_i,u_{i+1}\right)}{1-{\vartheta }_1\left(u_i,u_{i+1}\right)},S_n=\sum _{j=0}^n\prod _{i=0}^j\frac{\vartheta \left(u_i,u_l\right){\vartheta }_1\left(u_i,u_{i+1}\right)}{1-{\vartheta }_1\left(u_i,u_{i+1}\right)}.$$

Conceder $l>k,$ for which the above inequality becomes.

$${\psi }_{\vartheta }\left(u_k,u_l\right)\preccurlyeq {\psi }_{\vartheta }\left(u_0,u_1\right)\left[S_{l-1}-S_k\right].$$

Which means that,

$$\mid {\psi }_{\vartheta }\left(u_k,u_l\right)\mid \le \mid {\psi }_{\vartheta }\left(u_0,u_1\right)\mid \left[S_{l-1}-S_k\right].$$E2

Taking $k\to \infty$ in (5), we observe that $\left\{u_k\right\}$ becomes a Cauchy sequence. Since $\mathbb{M}$ is complete, thus we find out $u\in \mathbb{M}$ such that $u_k\to u\left(k\to \infty \right).$.

To verify that Fu = u, take.

$\begin{array}{l}{\psi }_{\vartheta }\left(u,\mathit{Fu}\right)\preccurlyeq {\psi }_{\vartheta }\left(u,u_{2k+2}\right)+{\psi }_{\vartheta }\left(u_{2k+2},\mathit{Fu}\right)={\psi }_{\vartheta }\left(u,u_{2k+2}\right)+{\psi }_{\vartheta }\left({\mathit{Eu}}_{2k+1},\mathit{Fu}\right)=\\ {\psi }_{\vartheta }\left(u,u_{2k+2}\right)+{\psi }_{\vartheta }\left(\mathit{Fu},{\mathit{Eu}}_{2k+1}\right)\preccurlyeq {\psi }_{\vartheta }\left(u,u_{2k+2}\right)+{\vartheta }_1\left(u,u_{2k+1}\right){\psi }_{\vartheta }\left(u,u_{2k+1}\right)+\frac{{\vartheta }_2\left(u,u_{2k+1}\right){\psi }_{\vartheta }\left(u,\mathit{Fu}\right){\psi }_{\vartheta }\left(u_{2k+1},{\mathit{Eu}}_{2k+1}\right)}{1+{\psi }_{\vartheta }\left(u,u_{2k+1}\right)}\end{array}$. Then, it becomes:

$\left|{\psi }_{\vartheta }\left(u,\mathit{Su}\right)\right|\le \left|{\psi }_{\vartheta }\left(u,u_{2k+2}\right)\right|+\left|{\vartheta }_1\left(u,u_{2k+1}\right){\psi }_{\vartheta }\left(u,u_{2k+1}\right)\right|+\left|\frac{{\vartheta }_2\left(u,u_{2k+1}\right){\psi }_{\vartheta }\left(u,\mathit{Fu}\right){\psi }_{\vartheta }\left(u_{2k+1},{\mathit{Eu}}_{2k+1}\right)}{1+{\psi }_{\vartheta }\left(u,u_{2k+1}\right)}\right|.$ By taking $k\to \infty$ implies that, ${\psi }_{\vartheta }\left(u,\mathit{Fu}\right)\mid \le 0.$ Then, we conclude that, u = Fu. repeating the same way, we can show that u = Eu. Hence, F and E have C.F.Point.

3.1 Uniqueness

To verify the uniqueness of u, let us assume that F and E have another F. Point $u^{\ast }$ with $u\ne u^{\ast }.$ Then,

${\psi }_{\vartheta }\left(u,u^{\ast }\right)={\psi }_{\vartheta }\left(\mathit{Fu},{\mathit{Eu}}^{\ast }\right)\preccurlyeq {\vartheta }_1\left(u,u^{\ast }\right){\psi }_{\vartheta }\left(u,u^{\ast }\right)+\frac{{\vartheta }_2\left(u,u^{\ast }\right){\psi }_{\vartheta }\left(u,\mathit{Fu}\right){\psi }_{\vartheta }\left(u^{\ast },{\mathit{Tu}}^{\ast }\right)}{1+{\psi }_{\vartheta }\left(u,u^{\ast }\right)},$which gives

$$\left|{\psi }_{\vartheta }\left(u,u^{\ast }\right)\right|\le {\vartheta }_1\left(u,u^{\ast }\right)\left|{\psi }_{\vartheta }\left(u,u^{\ast }\right)\right|+\left|\frac{{\vartheta }_2\left(u,u^{\ast }\right){\psi }_{\vartheta }\left(u,\mathit{Fu}\right){\psi }_{\vartheta }\left(u^{\ast },{\mathit{Eu}}^{\ast }\right)}{1+{\psi }_{\vartheta }\left(u,u^{\ast }\right)}\right|$$

and

$$\left(1-{\vartheta }_1\left(u,u^{\ast }\right)\right){\psi }_{\vartheta }\left(u,u^{\ast }\right)\le 0,$$

which verified that $u=u^{\ast }.$ Hence, F and E have a unique F.Point.

A complete C.V.N.E.b-M.Space $\left(\mathbb{M},{\psi }_{\vartheta }\right),$ with $\vartheta ,{\vartheta }_i:\mathbb{M}\times \mathbb{M}\to \left[0,\infty \right)\left(i=0,1,2\text{and}\vartheta ={\vartheta }_0+\mu ,\mu \ge 1\right)$ and $F:\mathbb{M}\to \mathbb{M}$ be a function with the below conditions:

1. ${\vartheta }_1<{\vartheta }_2;$

2. $\underset{k\to \infty }{lim}\vartheta \left(u_{k+1},u_m\right){\vartheta }_1\left(u_{k+1},u_{k+2}\right)+{\vartheta }_2\left(u_{k+1},u_{k+2}\right)<1;$

3. ${\psi }_{\vartheta }\left(\mathit{Fu},\mathit{Fv}\right)\preccurlyeq {\vartheta }_1\left(u,v\right){\psi }_{\vartheta }\left(u,v\right)+{\vartheta }_2\left(u,v\right)\frac{{\psi }_{\vartheta }\left(u,\mathit{Fu}\right){\psi }_{\vartheta }\left(v,\mathit{Fv}\right)}{1+{\psi }_{\vartheta }\left(u,v\right)}.$

Hence, this leads us to the unique F.Point of F.

Proof: Letting F = E in theorem 1.1.

Let us take a result from ([4], Theorem 4) (Author A. Azam et al.) and consider $\mathbb{M}$ as a complete C.V.M.Space and $F,E:\mathbb{M}\to \mathbb{M}.$ If F and E following the below inequality:

$${\psi }_{\vartheta }\left(\mathit{Fu},\mathit{Ev}\right)\preccurlyeq {\mathit{\alpha \psi }}_{\vartheta }\left(u,v\right)+\frac{{\mathit{\kappa \psi }}_{\vartheta }\left(u,\mathit{Fu}\right){\psi }_{\vartheta }\left(v,\mathit{Ev}\right)}{1+{\psi }_{\vartheta }\left(u,v\right)}$$

for all $u,v\in \mathbb{M},$ where non-negative real numbers $\alpha ,\kappa ,$ with $\alpha +\kappa <1.$ Then F and E have a C.F.Point in $\mathbb{M}.$.

Proof: Put ${\vartheta }_1\left(u,v\right)=\alpha$ and ${\vartheta }_2\left(u,v\right)=\kappa$ in 3.

Theorem 1.2 Let $\mathbb{M}$ be a complete C.V.N.E.b-M.Space, $\vartheta ,{\vartheta }_i:\mathbb{M}\times \mathbb{M}\to \left[0,\infty \right)\left(i=0,1,2\text{and}\vartheta ={\vartheta }_0+\mu ,\mathit{mu}\ge 1\right)$ and $E:\mathbb{M}\to \mathbb{M}$ be mappings verifying the following assertions:

1. ${\vartheta }_1<{\vartheta }_2;$

2. $\underset{k\to \infty }{lim}\vartheta \left(u_{k+1},u_l\right){\vartheta }_1\left(u_{k+1},u_{k+2}\right)+{\vartheta }_2\left(u_{k+1},u_{k+2}\right)<1;$

3. ${\psi }_{\vartheta }\left(E^{k}u,E^{k}v\right)\preccurlyeq {\vartheta }_1\left(u,v\right){\psi }_{\vartheta }\left(u,v\right)+{\vartheta }_2\left(u,v\right)\frac{{\psi }_{\vartheta }\left(u,E^{k}u\right){\psi }_{\vartheta }\left(v,E^{k}v\right)}{1+{\psi }_{\vartheta }\left(u,v\right)}.$

Then E has a unique F.Point in $\mathbb{M}.$.

Proof: From 3, we observe that $E^k$ has one and only one F.Point u, such that, $E^{k}u=u.$.

Thus, the result is follow below:

$$\begin{array}{l}{\psi }_{\vartheta }\left(\mathit{Eu},u\right)={\psi }_{\vartheta }\left({\mathit{EE}}^{k}u,E^{k}u\right)={\psi }_{\vartheta }\left(E^k\mathit{Eu},E^{k}u\right)\\ \preccurlyeq {\vartheta }_1\left(\mathit{Eu},u\right){\psi }_{\vartheta }\left(\mathit{Eu},u\right)+\frac{{\vartheta }_2\left(\mathit{Eu},u\right){\psi }_{\vartheta }\left(\mathit{Eu},E^n\mathit{Eu}\right){\psi }_{\vartheta }\left(u,E^{n}u\right)}{1+{\psi }_{\vartheta }\left(\mathit{Eu},u\right)},\\ \preccurlyeq {\vartheta }_1\left(\mathit{Eu},u\right){\psi }_{\vartheta }\left(\mathit{Eu},u\right)+\frac{{\vartheta }_2\left(\mathit{Eu},u\right){\psi }_{\vartheta }\left(\mathit{Eu},E^n\mathit{Eu}\right){\psi }_{\vartheta }\left(u,u\right)}{1+{\psi }_{\vartheta }\left(\mathit{Eu},u\right)}\\ ={\vartheta }_1\left(\mathit{Eu},u\right){\psi }_{\vartheta }\left(\mathit{Eu},u\right),\end{array}$$

these give us, $\left(1-{\vartheta }_1\left(\mathit{Eu},u\right)\right){\psi }_{\vartheta }\left(\mathit{Eu},u\right)\preccurlyeq 0.$ As a result, E has a unique F.Point.

Let $\mathbb{M}=C\left(\left[1,3\right],\mathrm{ℝ}\right),d>0$ and for any $u,y\in \mathbb{M},$ take

$${\psi }_{\vartheta }\left(u,y\right)=\underset{z\in \left[1,3\right]}{max}\mid u\left(z\right)-v\left(z\right)\mid \sqrt{1+d^2}{e}^{\mathit{\text{itan}}-1}d.$$

Define $E:\mathbb{M}\to \mathbb{M}$ by

$$E\left(r\left(z\right)\right)=4+{\int }_1^z\left(u\left(r\right)+u^2\right)e^{r-1}\mathit{dr},z\in \left[1,3\right].$$

Then, for every $u,v\in \mathbb{M},$

$$\begin{array}{l}{\psi }_{\vartheta }\left(\mathit{Eu},\mathit{Ev}\right)=\underset{t\in \left[1,3\right]}{max}\mid E\left(u\left(z\right)\right)-E\left(v\left(z\right)\right)\mid \sqrt{1+d^2}{e}^{{\mathit{\text{itan}}}^{-1}d},\\ \preccurlyeq {\vartheta }_1\left(u,v\right){\int }_1^3\underset{z\in \left[1,3\right]}{max}\mid u\left(r\right)-v\left(r\right)\mid e^2\sqrt{1+d^2}{e}^{{\mathit{\text{itan}}}^{-1}d}\mathit{dr},\\ \preccurlyeq 2{\vartheta }_1\left(u,v\right)e^2{\psi }_{\vartheta }\left(u,v\right).\end{array}$$

$$e^{2k}{\vartheta }_1^k\left(u,v\right)\frac{2^k}{k!}=\left\{\begin{array}{ll}109,& \text{if}k=2\\ 1987,& \text{if}k=4\\ 1.31,& \text{if}k=37\\ 0.53,& \text{if}k=38.\end{array}\right.$$

By using a typical calculation, we have

$${\psi }_{\vartheta }\left(E^{k}u,E^{k}v\right)\preccurlyeq e^{2k}{\vartheta }_1^k\left(u,v\right)\frac{2^k}{k!}{\psi }_{\vartheta }\left(u,v\right).$$

Thus, for ${\vartheta }_1\left(u,v\right)=0.53,{\vartheta }_2\left(u,v\right)=0,k=38,$ all the assertions of 1.2 are verified and thus, E has a unique F.Point, this implies that the integral equation has unique solution:

$$u\left(z\right)=4+{\int }_1^z\left(u\left(r\right)+r^2\right)e^{r-1}\mathit{dr},z\in \left[1,3\right],$$

or the differential equation:

$$\frac{\mathit{du}}{\mathit{dz}}=\left(u+z^2\right)e^{z-1},z\in \left[1,3\right],u\left(1\right)=4.$$

4. Conclusions

Inspired by the concepts provided in [4, 17, 18], we define new F.P.Theorems for maps under particular rational constructive inequalities and introduce the notion of C.V.N.b-M.Spaces. The publications cited above, as well as a few others in the related literature, are improved and expanded upon by our idea. Our proposal enhances and expands upon the aforementioned papers as well as a few other sections of related literature.

Conflict of interest

The authors declare that they have no conflict of interests.

Abbreviations

C.C.V.R.b-M.S	Complete Complex Valued Rectangular Extended b-Metric
F.P.Theorem	Fixed Point Theorem
C.C.V.b-M.S	Complete Complex Valued Extended b-Metric Spaces
C.F.Point	Common Fixed Point
M.Spaces	Metric Spaces
F.P	Fixed Point