Integral Equation via Fixed Point Theorems on a New Type of Convex Contraction in b-Metric and 2-Metric Spaces

Abstract

Our paper is devoted to describing a new way of generalized convex contraction of type-2 in the framework of b-metric spaces and 2-metric spaces. First, the concept of a new generalized convex contraction on b-metric spaces and 2-metric spaces is introduced, and fixed point theorem is extended to these spaces. Some examples supporting our main results are also presented. Finally, we apply our main result to approximating the solution of the Fredholm integral equation.

1. Introduction and Preliminaries

The theoretical framework of metric fixed point theory has been an active subject of study, and the contraction mapping principle is one of the most significant theorems in functional analysis. Banach [1] contraction mapping concept has several applications in fixed point theory. Istrǎţescu [2] established a class of convex contraction mappings in metric spaces and expanded the well-known Banach contraction concept. Following that, Boriceanu, Bota, and Petrusel [3,4] produced several concrete instances of b-metric spaces and examined the fixed point characteristics of set-valued operators in b-metric spaces. For examples of results in b-metric space, see [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19] and references therein. Following that, much research occurred on the generalization of such classes of mappings in the context of diverse spaces [20,21,22,23,24,25,26,27,28,29]. Singh et al. [30] presented the notions of a convex contraction via admissible mapping and its related fixed point theorems with an application in 2018. Chen et al. [7] also looked into different fixed point theorems and applications in convex b-metric spaces. In 2021, Gelete et al. [10] found fixed points of alpha convex contraction mapping in b-metric space. We will focus on the research works of Khan et al. [11,31] for our purposes. They developed the generalized convexity condition of type-2 in b-metric and 2-metric space and demonstrated fixed points of these maps in complete metric spaces. In this study, we present Khan et al.’s [11,31] main theorem about b-metric spaces and 2-metric spaces more generally.

Definition 1

([8]). Let ℧ be a nonempty set and $ħ\ge 1$ a given real number. A mapping $\pi :\mho \times \mho \to [0,\infty )$ is called a b-metric on ℧, if for all $\upsilon ,\varsigma ,\vartheta \in \mho$, it satisfies:

(i)

$\pi (\upsilon ,\varsigma )=0$, if and only if $\upsilon =\varsigma$;

(ii)

$\pi (\upsilon ,\varsigma )=\pi (\varsigma ,\upsilon )$;

(iii)

$\pi (\upsilon ,\zeta )\le ħ\left[\pi \right(\upsilon ,\vartheta )+\pi (\vartheta ,\varsigma \left)\right]$.

The pair $(\mho ,\pi )$ is called a b-metric space.

Example 1.

Let $\mho =\{0,1,2\}$, $\pi (2,0)=\pi (0,2)=\varpi >2,\pi (0,1)=\pi (1,0)=\pi (2,1)=\pi (1,2)=1$ and $\pi (0,0)=\pi (1,1)=\pi (2,2)=0$. Then,

$$\begin{array}{c}\hfill \pi (\upsilon ,\varsigma )\le \frac{\varpi }{2}[\pi (\upsilon ,\vartheta )+\pi (\vartheta ,\varsigma )],\mathrm{for}\mathrm{all}\upsilon ,\varsigma ,\vartheta \in \mho .\end{array}$$

The conventional triangle inequality does not hold if $\varpi >2$.

The papers of Boriceanu [3], Boriceanu and Bota [4] provide more information.

Definition 2

([32]). Let ℧ be a non-empty set and let $\pi :\mho \times \mho \times \mho \to \mathcal{R}$ be a map satisfying the following conditions:

(i)

For each $\upsilon ,\varsigma \in \mho$ with $\upsilon \ne \varsigma$, there exists a point $å\in \mho$ such that $\pi (\upsilon ,\varsigma ,å)\ne 0$;

(ii)

$\pi (\upsilon ,\varsigma ,\vartheta )=0$, when at least two of the points are equal;

(iii)

$\pi (\upsilon ,\varsigma ,\vartheta )=\pi (\varsigma ,\vartheta ,\upsilon )=\pi (\upsilon ,\vartheta ,\varsigma )$ for all $\upsilon ,\varsigma ,\vartheta \in \mho$;

(iv)

$\pi (\upsilon ,\varsigma ,\vartheta )\le \pi (\upsilon ,\varsigma ,å)+\pi (\upsilon ,å,\vartheta )+\pi (å,\varsigma ,\vartheta )$, for all $\upsilon ,\varsigma ,\vartheta ,å\in \mho$.

Then, we say that π is a 2-metric on ℧ and $(\mho ,\pi )$ is a 2-metric space.

Example 2.

Let $\mho ={\mathcal{R}}_{+}\times {\mathcal{R}}_{+}$, $\pi :\mho \times \mho \times \mho \to [0,\infty )$ be a function such that for all $\upsilon ,\varsigma ,\vartheta \in \mho$ and the area of a triangle be $\pi (\upsilon ,\varsigma ,\vartheta )$ with vertices $\upsilon =({\upsilon }_1,{\upsilon }_2),\varsigma =({\varsigma }_1,{\varsigma }_2),\vartheta =({\vartheta }_1,{\vartheta }_2)$. Then, π is a 2-metric space and $(\mho ,\pi )$ is a 2-metric space.

Furthermore, we refer to Gähler [32,33,34] for more information on 2-metric space. Iseki [35], Khan [36], Kumar and Poonam [21] and others have all written about fixed point theorems in 2-metric spaces.

Definition 3.

Let £ be a self-mapping on a metric space $(\mho ,\pi )$ with $\epsilon >0$; then, ${\upsilon }_0\in \mho$ is called an ε-fixed point of £ on ℧, whenever $\pi ({\upsilon }_0,\pounds {\upsilon }_0)<\epsilon$. It is denoted by ${\Theta }_{\epsilon >0}\left(\pounds \right)=\{\upsilon \in \mho |\pi (\pounds \upsilon ,\upsilon )<\epsilon \}$ and the set of all fixed points of £ is $Fix\left(\pounds \right)$. We say that £ has the approximate fixed point property (AFPP) if there exists an ε-fixed point of £, for all $\epsilon >0$, i.e., ${\Theta }_{\epsilon }\left(\pounds \right)\ne \varnothing$, or symbolically, ${inf}_{\upsilon \in \mho }\pi (\pounds \upsilon ,\upsilon )=0$.

Example 3.

On $\mho =[0,\infty )$, consider the mapping $\pounds :\mho \to \mho$ is given by the formula $\pounds \upsilon =\upsilon +\frac{1}{2\upsilon +1}$ for all $\upsilon \in \mho$. Define a mapping $\pi :\mho \times \mho \to [0,\infty )$ by $\pi (\pounds {\upsilon }_0,{\upsilon }_0)=|\pounds {\upsilon }_0-{\upsilon }_0|$, while letting $0<\epsilon <\frac{1}{2}$ and setting ${\upsilon }_0\in \mho$ such that ${\upsilon }_0>\frac{1-\epsilon }{2\epsilon }$. We obtain

$$\pi (\pounds {\upsilon }_0,{\upsilon }_0)=|\pounds {\upsilon }_0-{\upsilon }_0|=|\frac{1}{2{\upsilon }_0+1}|<\epsilon .$$

This proves that £ has an ε-fixed point, so ${\mathsf{\Theta }}_{\epsilon }\left(\pounds \right)\ne \varnothing$. Here £ has no fixed point in ℧.

Definition 4

([37]). Let $\pounds :\mho \to \mho$ be a self-mapping on a non-empty set ℧ and $\alpha :\mho \times \mho \to [0,\infty )$ be a mapping. We say that £ is α-admissible if $\upsilon ,\varsigma \in \mho ,\alpha (\upsilon ,\varsigma )\ge 1$ implies $\alpha (\pounds \upsilon ,\pounds \varsigma )\ge 1$.

Definition 5

([21]). Let $\pounds :\mho \to \mho$ be a self-mapping on a non-empty set ℧ and $\alpha :\mho \times \mho \times \mho \to [0,\infty )$ be a mapping. £ is said to be an α-admissible if $\upsilon ,\varsigma ,å\in \mho ,\alpha (\upsilon ,\varsigma ,å)\ge 1$ implies $\alpha (\pounds \upsilon ,\pounds \varsigma ,å)\ge 1$.

Definition 6

([31]). Suppose ℧ has the property (H) if for each $\upsilon ,\varsigma \in Fix\left(\pounds \right),$ there exists $\vartheta \in \mho$ such that $\alpha (\upsilon ,\vartheta )\ge 1$ and $\alpha (\varsigma ,\vartheta )\ge 1$.

In 2013, Miandaragh et al. [26] introduced the idea of generalized convex contraction of order-2 in a metric space scenario.

Definition 7

([26]). Let $\pounds :\mho \to \mho$ be a self-mapping on a non-empty set ℧. £ is said to be a generalized convex contraction if there exist a mapping $\alpha :\mho \times \mho \to [0,\infty )$ and $0\le {\stackrel{`}{\alpha }}_1,{\stackrel{`}{\alpha }}_2<1$ with ${\sum }_{\xi =1}^2{\stackrel{`}{\alpha }}_{\xi }<1$ such that

$$\begin{array}{c}\hfill \alpha (\upsilon ,\varsigma )\pi ({\pounds }^2\upsilon ,{\pounds }^2\varsigma )\le {\stackrel{`}{\alpha }}_1\pi (\upsilon ,\varsigma )+{\stackrel{`}{\alpha }}_2\pi (\pounds \upsilon ,\pounds \varsigma ),\end{array}$$

for all $\upsilon ,\varsigma \in \mho$, where $\stackrel{`}{\alpha }$ is the based mapping.

Definition 8

([26]). Let $\pounds :\mho \to \mho$ be a self-mapping on a non-empty set ℧. £ is said to be a generalized convex contraction of order-2 if there exist a mapping $\alpha :\mho \times \mho \to [0,\infty )$ and ${\stackrel{`}{\alpha }}_3,{\stackrel{`}{\alpha }}_4,{\stackrel{`}{\alpha }}_5,{\stackrel{`}{\alpha }}_6\in [0,1)$, with ${\sum }_{\xi =3}^6{\stackrel{`}{\alpha }}_{\xi }<1$ such that

$$\begin{array}{c}\hfill \alpha (\upsilon ,\varsigma )\pi ({\pounds }^2\upsilon ,{\pounds }^2\varsigma )\le {\stackrel{`}{\alpha }}_3\pi (\upsilon ,\pounds \upsilon )+{\stackrel{`}{\alpha }}_4\pi (\pounds \upsilon ,{\pounds }^2\upsilon )+{\stackrel{`}{\alpha }}_5\pi (\varsigma ,\pounds \varsigma )+{\stackrel{`}{\alpha }}_6\pi (\pounds \varsigma ,{\pounds }^2\varsigma ),\end{array}$$

for all $\upsilon ,\varsigma \in \mho$, where $\stackrel{`}{\alpha }$ is the based mapping.

Definition 9

([38]). Let $\pounds :\mho \to \mho$ be a self-mapping on a non-empty set ℧. £ is said to be asymptotically regular at a point $\upsilon \in \mho$ if $\pi ({\pounds }^{\omega }\upsilon ,{\pounds }^{\omega +1}\upsilon )\to 0$ as $\omega \to \infty$.

Definition 10

([11]). Let $(\mho ,\pi )$ be a b-metric space with $ħ\ge 1$. A self-mapping £ on ℧ is called a generalized convex contraction of type-2, if there exist a mapping $\alpha :\mho \times \mho \to [0,\infty )$ and ${\stackrel{`}{\alpha }}_{\xi }>0$ with ${\sum }_{\xi =1}^6{\stackrel{`}{\alpha }}_{\xi }<\frac{1}{{ħ}^2}$ such that

$$\begin{array}{cc}\hfill \alpha (\upsilon ,\varsigma )\pi ({\pounds }^2\upsilon ,{\pounds }^2\varsigma )& \le {\stackrel{`}{\alpha }}_1\pi (\upsilon ,\varsigma )+{\stackrel{`}{\alpha }}_2\pi (\pounds \upsilon ,\pounds \varsigma )+{\stackrel{`}{\alpha }}_3\pi (\upsilon ,\pounds \upsilon )\hfill \\ & +{\stackrel{`}{\alpha }}_4\pi (\pounds \upsilon ,{\pounds }^2\upsilon )+{\stackrel{`}{\alpha }}_5\pi (\varsigma ,\pounds \varsigma )+{\stackrel{`}{\alpha }}_6\pi (\pounds \varsigma ,{\pounds }^2\varsigma ),\hfill \end{array}$$

for all $\upsilon ,\varsigma \in \mho$.

We recall the following useful results.

Lemma 1

([39]). A self-mapping £ is asymptotically regular on a metric space $(\mho ,\pi )$; i.e., $\pi ({\pounds }^{\varpi }\upsilon ,{\pounds }^{\varpi +1}\upsilon )\to 0$, for all $\upsilon \in \mho$. Then, £ has the AFPP.

Lemma 2.

A self-mapping £ is asymptotically regular on a b-metric space $(\mho ,\pi )$ at a point ${\upsilon }_0\in \mho$—i.e., $\pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +1}{\upsilon }_0)\to 0$, and then £ has the AFPP.

Proof. 

The proof is exactly the same as in Lemma 2.5 in [22] for metric spaces. □

Remember that a self-mapping £ on a 2-metric space $(\mho ,\pi )$ is called asymptotically regular at a point $\upsilon \in \mho$, if $\pi ({\pounds }^{\omega }\upsilon ,{\pounds }^{\omega +1}\upsilon ,å)\to 0$ as $\omega \to \infty ,$ for all $å\in \mho$.

Lemma 3.

A self-mapping £ is asymptotically regular on a 2-metric space $(\mho ,\pi )$ at a point ${\upsilon }_0\in \mho$, i.e., $\pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +1}{\upsilon }_0,å)\to 0,$ for all $å\in \mho$. Then, £ has the AFPP.

Proof. 

For any positive integer $\omega$, we have

$$\begin{array}{c}\hfill \underset{\upsilon \in \mho }{inf}\pi ({\upsilon }_0,\pounds {\upsilon }_0,å)\le \pi ({\pounds }^{\omega }{\upsilon }_0,{\pounds }^{\omega +1}{\upsilon }_0,å),\mathrm{for}\mathrm{all}å\in \mho .\end{array}$$

Since £ is asymptotically regular at ${\upsilon }_0\in \mho$, $\pi ({\pounds }^{\omega }{\upsilon }_0,{\pounds }^{\omega +1}{\upsilon }_0,å)\to \infty$ as $\omega \to \infty$, which in turn gives ${inf}_{\upsilon \in \mho }\pi (\upsilon ,\pounds \upsilon ,å)=0$. Hence, £ has the AFPP. □

In this study, we present the notion of a novel new $\alpha$-convex contraction of type-2 in b-metric and 2-metric spaces, which was inspired by Khan et al. [11] publications. Using this style of mapping, we can find fixed points that are close to each other in b-metric and 2-metric spaces.

2. Main Results

In this section, we prove a fixed point result for new $\alpha$-convex contraction of type-2 mappings in b-metric spaces. Now we introduce our definition as follows.

Definition 11.

Let $(\mho ,\pi )$ be a b-metric space with $ħ\ge 1$. We say that a self-mapping £ on ℧ is a new α-convex contraction of type-2, if there exists a mapping $\alpha :\mho \times \mho \to [0,\infty )$ and ${\stackrel{`}{\alpha }}_i\ge 0$, for all $i\in \{1,\dots ,8\}$ with ${\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_6+2{\stackrel{`}{\alpha }}_7+2{\stackrel{`}{\alpha }}_8<\frac{1}{{ħ}^2}$ such that

$$\begin{array}{cc}\hfill \alpha (\upsilon ,\varsigma )\pi ({\pounds }^2\upsilon ,{\pounds }^2\varsigma )\le & {\stackrel{`}{\alpha }}_1\pi (\upsilon ,\varsigma )+{\stackrel{`}{\alpha }}_2\pi (\pounds \upsilon ,\pounds \varsigma )+{\stackrel{`}{\alpha }}_3\pi (\upsilon ,\pounds \upsilon )+{\stackrel{`}{\alpha }}_4\pi (\pounds \upsilon ,{\pounds }^2\upsilon )\hfill \\ & +{\stackrel{`}{\alpha }}_5\pi (\varsigma ,\pounds \varsigma )+{\stackrel{`}{\alpha }}_6\pi (\pounds \varsigma ,{\pounds }^2\varsigma )+{\stackrel{`}{\alpha }}_7\left(\frac{\pi (\upsilon ,\pounds \varsigma )+\pi (\varsigma ,\pounds \upsilon )}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi (\pounds \upsilon ,{\pounds }^2\varsigma )+\pi (\pounds \varsigma ,{\pounds }^2\upsilon )}{2}\right),\hfill \end{array}$$

(1)

for all $\upsilon ,\varsigma \in \mho$.

Theorem 1.

Let $(\mho ,\pi )$ be a complete b-metric space with coefficient $ħ\ge 1$ and let $\pounds :\mho \to \mho$ be a new α-convex contraction of type-2. Assume that £ is an α-admissible map and there exists ${\upsilon }_0\in \mho$ such that $\alpha (\pounds {\upsilon }_0,{\upsilon }_0)\ge 1$. Then, £ has an AFPP. Furthermore, £ has a fixed point if £ is continuous. Moreover, if for all $\upsilon ,\varsigma \in Fix\left(\pounds \right)$ we have $\alpha (\upsilon ,\varsigma )\ge 1$, then £ has a unique fixed point in ℧.

Proof. 

Let ${\upsilon }_0\in \mho$ be such that $\alpha (\pounds {\upsilon }_0,{\upsilon }_0)\ge 1$. Now, we define a sequence $\left\{{\upsilon }_{\omega }\right\}$ by ${\upsilon }_{\omega +1}={\pounds }^{\omega +1}{\upsilon }_0$, for all $\omega \ge 0$. If ${\upsilon }_{\omega }={\upsilon }_{\omega +1}$, i.e., ${\pounds }^{\omega }{\upsilon }_0=\pounds \left({\pounds }^{\omega }{\upsilon }_0\right)$ for each $\omega$, then we conclude that the theorem follows immediately. Suppose that ${\upsilon }_{\omega }\ne {\upsilon }_{\omega +1},$ for all $\omega \ge 0$. Since £ is $\alpha$-admissible, $\alpha (\pounds {\upsilon }_0,{\upsilon }_0)\ge 1$ implies $\alpha ({\pounds }^2{\upsilon }_0,\pounds {\upsilon }_0)\ge 1$. Therefore, we can obtain inductively that $\alpha ({\pounds }^{\omega +1}{\upsilon }_0,{\pounds }^{\omega }{\upsilon }_0)\ge 1$, for all $\omega \ge 0$. We put

$$\mathfrak{m}=max\{\pi ({\upsilon }_0,\pounds {\upsilon }_0),\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)\},$$

$$\nu ={\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+2{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8$$

and $♭=1-{\stackrel{`}{\alpha }}_6-{\stackrel{`}{\alpha }}_8.$ Since £ is $\alpha$-admissible, by (1), taking $\upsilon ={\upsilon }_0$, $\varsigma =\pounds {\upsilon }_0$, we obtain

$$\begin{array}{cc}\hfill \pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)& \le \alpha ({\upsilon }_0,\pounds {\upsilon }_0)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\hfill \\ & \le {\stackrel{`}{\alpha }}_1\pi ({\upsilon }_0,\pounds {\upsilon }_0)+{\stackrel{`}{\alpha }}_2\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+{\stackrel{`}{\alpha }}_3\pi ({\upsilon }_0,\pounds {\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_4\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+{\stackrel{`}{\alpha }}_5\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_7\left(\frac{\pi ({\upsilon }_0,{\pounds }^2{\upsilon }_0)+\pi (\pounds {\upsilon }_0,\pounds {\upsilon }_0)}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi (\pounds {\upsilon }_0,{\pounds }^3{\upsilon }_0)+\pi ({\pounds }^2{\upsilon }_0,{\pounds }^2{\upsilon }_0)}{2}\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\upsilon }_0,\pounds {\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_6\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+{\stackrel{`}{\alpha }}_7\left(\frac{\pi ({\upsilon }_0,{\pounds }^2{\upsilon }_0)}{2}\right)+{\stackrel{`}{\alpha }}_8\left(\frac{\pi (\pounds {\upsilon }_0,{\pounds }^3{\upsilon }_0)}{2}\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\upsilon }_0,\pounds {\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_6\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+\frac{{\stackrel{`}{\alpha }}_7}{2}\left(\pi ({\upsilon }_0,\pounds {\upsilon }_0)+\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)\right)\hfill \\ & +\frac{{\stackrel{`}{\alpha }}_8}{2}\left(\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le ({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\upsilon }_0,\pounds {\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_6\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+{\stackrel{`}{\alpha }}_7\left(\pi ({\upsilon }_0,\pounds {\upsilon }_0)+\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7)\pi ({\upsilon }_0,\pounds {\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)\hfill \\ & +({\stackrel{`}{\alpha }}_6+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0).\hfill \end{array}$$

Therefore, we get

$$\begin{array}{cc}\hfill (1-{\stackrel{`}{\alpha }}_6-{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)& \le ({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7)\mathfrak{m}+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\mathfrak{m}\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+2{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\mathfrak{m},\hfill \end{array}$$

that is, $\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\le \left(\frac{\nu }{♭}\right)\mathfrak{m}$, where $\frac{\nu }{♭}<1$ as ${\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_6+2{\stackrel{`}{\alpha }}_7+2{\stackrel{`}{\alpha }}_8<\frac{1}{{ħ}^2}.$ Since £ is $\alpha$-admissible, by (11), taking $\upsilon =\pounds {\upsilon }_0,\varsigma ={\pounds }^2{\upsilon }_0$, we obtain

$$\begin{array}{cc}\hfill \pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)& \le \alpha (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\hfill \\ & \le {\stackrel{`}{\alpha }}_1\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+{\stackrel{`}{\alpha }}_2\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+{\stackrel{`}{\alpha }}_3\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+{\stackrel{`}{\alpha }}_4\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_5\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_7\left(\frac{\pi (\pounds {\upsilon }_0,{\pounds }^3{\upsilon }_0)+\pi ({\pounds }^2{\upsilon }_0,{\pounds }^2{\upsilon }_0)}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi ({\pounds }^2{\upsilon }_0,{\pounds }^4{\upsilon }_0)+\pi ({\pounds }^3{\upsilon }_0,{\pounds }^3{\upsilon }_0)}{2}\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_6\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+{\stackrel{`}{\alpha }}_7\left(\frac{\pi (\pounds {\upsilon }_0,{\pounds }^3{\upsilon }_0)}{2}\right)+{\stackrel{`}{\alpha }}_8\left(\frac{\pi ({\pounds }^2{\upsilon }_0,{\pounds }^4{\upsilon }_0)}{2}\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\hfill \\ & +\frac{{\stackrel{`}{\alpha }}_7}{2}\left(\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\right)+\frac{{\stackrel{`}{\alpha }}_8}{2}\left(\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\right)\hfill \\ & \le ({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_7\left(\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\right)+{\stackrel{`}{\alpha }}_8\left(\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\hfill \\ & +({\stackrel{`}{\alpha }}_6+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0).\hfill \end{array}$$

Therefore, we get

$$\begin{array}{cc}\hfill (1-{\stackrel{`}{\alpha }}_6-{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)& \le ({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7)\mathfrak{m}+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\mathfrak{m}\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+2{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\mathfrak{m},\hfill \end{array}$$

that is, $\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\le \left(\frac{\nu }{♭}\right)\mathfrak{m}$ and

$$\begin{array}{cc}\hfill \pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)& \le \alpha ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)\hfill \\ & \le {\stackrel{`}{\alpha }}_1\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+{\stackrel{`}{\alpha }}_2\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+{\stackrel{`}{\alpha }}_3\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+{\stackrel{`}{\alpha }}_4\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_5\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)+{\stackrel{`}{\alpha }}_7\left(\frac{\pi ({\pounds }^2{\upsilon }_0,{\pounds }^4{\upsilon }_0)+\pi ({\pounds }^3{\upsilon }_0,{\pounds }^3{\upsilon }_0)}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi ({\pounds }^3{\upsilon }_0,{\pounds }^5{\upsilon }_0)+\pi ({\pounds }^4{\upsilon }_0,{\pounds }^4{\upsilon }_0)}{2}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_7\left(\frac{\pi ({\pounds }^2{\upsilon }_0,{\pounds }^4{\upsilon }_0)}{2}\right)+{\stackrel{`}{\alpha }}_8\left(\frac{\pi ({\pounds }^3{\upsilon }_0,{\pounds }^5{\upsilon }_0)}{2}\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_6\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)+\frac{{\stackrel{`}{\alpha }}_7}{2}\left(\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\right)\hfill \\ & +\frac{{\stackrel{`}{\alpha }}_8}{2}\left(\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)\right)\hfill \\ & \le ({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_6\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)+{\stackrel{`}{\alpha }}_7\left(\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\hfill \\ & +({\stackrel{`}{\alpha }}_6+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7)\left(\frac{\nu }{♭}\right)\mathfrak{m}\hfill \\ & +({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\left(\frac{\nu }{♭}\right)\mathfrak{m}+({\stackrel{`}{\alpha }}_6+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+2{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\left(\frac{\nu }{♭}\right)\mathfrak{m}+({\stackrel{`}{\alpha }}_6+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0).\hfill \end{array}$$

It follows that

$$\begin{array}{cc}\hfill \pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)& \le \frac{{\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+2{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8}{1-{\stackrel{`}{\alpha }}_6-{\stackrel{`}{\alpha }}_8}\left(\frac{\nu }{♭}\right)\mathfrak{m}\hfill \\ & ={\left(\frac{\nu }{♭}\right)}^2\mathfrak{m}.\hfill \end{array}$$

Additionally, we obtain

$$\begin{array}{cc}\hfill \pi ({\pounds }^5{\upsilon }_0,{\pounds }^6{\upsilon }_0)& \le \alpha ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)\pi ({\pounds }^5{\upsilon }_0,{\pounds }^6{\upsilon }_0)\hfill \\ & \le {\stackrel{`}{\alpha }}_1\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+{\stackrel{`}{\alpha }}_2\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)+{\stackrel{`}{\alpha }}_3\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+{\stackrel{`}{\alpha }}_4\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_5\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^5{\upsilon }_0,{\pounds }^6{\upsilon }_0)+{\stackrel{`}{\alpha }}_7\left(\frac{\pi ({\pounds }^3{\upsilon }_0,{\pounds }^5{\upsilon }_0)+\pi ({\pounds }^4{\upsilon }_0,{\pounds }^4{\upsilon }_0)}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi ({\pounds }^4{\upsilon }_0,{\pounds }^6{\upsilon }_0)+\pi ({\pounds }^5{\upsilon }_0,{\pounds }^5{\upsilon }_0)}{2}\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^4{\upsilon }_0,{\pounds }^6{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_7\left(\frac{\pi ({\pounds }^3{\upsilon }_0,{\pounds }^5{\upsilon }_0)}{2}\right)+{\stackrel{`}{\alpha }}_8\left(\frac{\pi ({\pounds }^4{\upsilon }_0,{\pounds }^6{\upsilon }_0)}{2}\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^4{\upsilon }_0,{\pounds }^6{\upsilon }_0)\hfill \\ & +\frac{{\stackrel{`}{\alpha }}_7}{2}\left(\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)\right)+\frac{{\stackrel{`}{\alpha }}_8}{2}\left(\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)+\pi ({\pounds }^5{\upsilon }_0,{\pounds }^6{\upsilon }_0)\right)\hfill \\ & \le ({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^4{\upsilon }_0,{\pounds }^6{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_7\left(\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)\right)+{\stackrel{`}{\alpha }}_8\left(\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)+\pi ({\pounds }^5{\upsilon }_0,{\pounds }^6{\upsilon }_0)\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7)\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0)\hfill \\ & +({\stackrel{`}{\alpha }}_6+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^5{\upsilon }_0,{\pounds }^6{\upsilon }_0)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7)\left(\frac{\nu }{♭}\right)\mathfrak{m}+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\left(\frac{\nu }{♭}\right)\mathfrak{m}\hfill \\ & +({\stackrel{`}{\alpha }}_6+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^5{\upsilon }_0,{\pounds }^6{\upsilon }_0)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+2{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\left(\frac{\nu }{♭}\right)\mathfrak{m}+({\stackrel{`}{\alpha }}_6+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^5{\upsilon }_0,{\pounds }^6{\upsilon }_0),\hfill \end{array}$$

that is,

$$\begin{array}{cc}\hfill \pi ({\pounds }^5{\upsilon }_0,{\pounds }^6{\upsilon }_0)& \le \frac{{\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+2{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8}{1-{\stackrel{`}{\alpha }}_6-{\stackrel{`}{\alpha }}_8}\left(\frac{\nu }{♭}\right)\mathfrak{m}\hfill \\ & ={\left(\frac{\nu }{♭}\right)}^2\mathfrak{m}.\hfill \end{array}$$

Continuing in this way, we get $\pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +1}{\upsilon }_0)\le {\left(\frac{\nu }{♭}\right)}^{ȷ}\mathfrak{m}$, whenever $\varpi =2ȷ$ or $\varpi =2ȷ+1$, for $ȷ>1$ or $\pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +1}{\upsilon }_0)\le {\left(\frac{\nu }{♭}\right)}^{ȷ-1}\mathfrak{m}$, whenever $\varpi =2ȷ$ or $\varpi =2ȷ-1$, for $ȷ>2$. Therefore, $\pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +1}{\upsilon }_0)\to 0$ as $\varpi \to \infty$, so we get that £ is asymptotically regular at ${\upsilon }_0$. By Lemma 2, conclude that £ has an AFPP.

Now, assume that $(\mho ,\pi )$ is a complete b-metric space and £ is continuous. To prove that $\left\{{\upsilon }_{\omega }\right\}$ is a Cauchy sequence in ℧. Let $\varpi$ and $\omega$ be two distinct non-zero positive integers such that $\varpi <\omega$, which arises the two cases.

Case (i). For $\varpi =2ȷ$ with $ȷ,\ell >1$, then

$$\begin{array}{cc}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +\ell }{\upsilon }_0)& =\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+\ell }{\upsilon }_0)\hfill \\ & \le ħ\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0)+{ħ}^2\pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0)\hfill \\ & +{ħ}^3\pi ({\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0)+{ħ}^4\pi ({\pounds }^{2ȷ+3}{\upsilon }_0,{\pounds }^{2ȷ+4}{\upsilon }_0)+....\hfill \\ & \le ħ{\left(\frac{\nu }{♭}\right)}^{ȷ}\mathfrak{m}+{ħ}^2{\left(\frac{\nu }{♭}\right)}^{ȷ}\mathfrak{m}+{ħ}^3{\left(\frac{\nu }{♭}\right)}^{ȷ+1}\mathfrak{m}+{ħ}^4{\left(\frac{\nu }{♭}\right)}^{ȷ+1}\mathfrak{m}+\cdots \hfill \\ & \le ħ{\left(\frac{\nu }{♭}\right)}^{ȷ}(1+{ħ}^2\left(\frac{\nu }{♭}\right)+\cdots )\mathfrak{m}+{ħ}^2{\left(\frac{\nu }{♭}\right)}^{ȷ}\{1+{ħ}^2\left(\frac{\nu }{♭}\right)+\cdots \}\mathfrak{m}\hfill \\ & =(ħ+{ħ}^2){\left(\frac{\nu }{♭}\right)}^{ȷ}(1+{ħ}^2\left(\frac{\nu }{♭}\right)+\cdots )\mathfrak{m}\hfill \\ & \le (ħ+{ħ}^2){\left(\frac{\nu }{♭}\right)}^{ȷ}\frac{1}{1-{ħ}^2\left(\frac{\nu }{♭}\right)}\mathfrak{m}.\hfill \end{array}$$

Case (ii). For $\varpi =2ȷ+1$ with $ȷ,\ell >1$, we have

$$\begin{array}{cc}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +\ell }{\upsilon }_0)& =\pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+\ell +1}{\upsilon }_0)\hfill \\ & \le ħ\pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0)+{ħ}^2\pi ({\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0)\hfill \\ & +{ħ}^3\pi ({\pounds }^{2ȷ+3}{\upsilon }_0,{\pounds }^{2ȷ+4}{\upsilon }_0)+{ħ}^4\pi ({\pounds }^{2ȷ+4}{\upsilon }_0,{\pounds }^{2ȷ+5}{\upsilon }_0)+\cdots \hfill \\ & \le ħ{\left(\frac{\nu }{♭}\right)}^{ȷ}\mathfrak{m}+{ħ}^2{\left(\frac{\nu }{♭}\right)}^{ȷ+1}\mathfrak{m}+{ħ}^3{\left(\frac{\nu }{♭}\right)}^{ȷ+1}\mathfrak{m}+{ħ}^4{\left(\frac{\nu }{♭}\right)}^{ȷ+2}\mathfrak{m}+\cdots \hfill \\ & \le ħ{\left(\frac{\nu }{♭}\right)}^{ȷ}(1+{ħ}^2\left(\frac{\nu }{♭}\right)+\cdots )\mathfrak{m}+{ħ}^2{\left(\frac{\nu }{♭}\right)}^{ȷ}(1+{ħ}^2\left(\frac{\nu }{♭}\right)+\cdots )\mathfrak{m}\hfill \\ & =(ħ+{ħ}^2){\left(\frac{\nu }{♭}\right)}^{ȷ}(1+{ħ}^2\left(\frac{\nu }{♭}\right)+\cdots )\mathfrak{m}\hfill \\ & \le (ħ+{ħ}^2){\left(\frac{\nu }{♭}\right)}^{ȷ}\frac{1}{1-{ħ}^2\left(\frac{\nu }{♭}\right)}\mathfrak{m}.\hfill \end{array}$$

When taking $ȷ\to \infty$ in all cases, since ${ħ}^2\frac{\nu }{♭}<1$, we obtain $\pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\omega }{\upsilon }_0)\to 0$. Therefore, $\left\{{\upsilon }_{\omega }\right\}$ is a Cauchy sequence in ℧. Since ℧ is complete, there exists a point $\vartheta \in \mho$ such that ${\upsilon }_{\omega }={\pounds }^{\omega }{\upsilon }_0\to \vartheta \in \mho$ as $\omega \to \infty$. By the continuity of £, we obtain $\vartheta =\underset{\omega \to \infty }{lim}\pounds \left({\pounds }^{\omega }{\upsilon }_0\right)=\pounds \vartheta$. Therefore, we obtain that $\vartheta$ is a fixed point of £.

Now, we show the uniqueness of the fixed point.

Let $\vartheta ,{\vartheta }^{*}\in Fix\left(\pounds \right)$ with $\vartheta \ne {\vartheta }^{*}$. By being $\alpha$-admissible, we have $\alpha (\vartheta ,{\vartheta }^{*})>1$ and from (1), taking $\upsilon =\vartheta$ and $\varsigma ={\vartheta }^{*}$, we obtain

$$\begin{array}{cc}\hfill \pi (\vartheta ,{\vartheta }^{*})& =\pi ({\pounds }^2\vartheta ,{\pounds }^2{\vartheta }^{*})\hfill \\ & \le \alpha (\vartheta ,{\vartheta }^{*})\pi ({\pounds }^2\vartheta ,{\pounds }^2{\vartheta }^{*})\hfill \\ & \le {\stackrel{`}{\alpha }}_1\pi (\vartheta ,{\vartheta }^{*})+{\stackrel{`}{\alpha }}_2\pi (\pounds \vartheta ,\pounds {\vartheta }^{*})+{\stackrel{`}{\alpha }}_3\pi (\vartheta ,\pounds \vartheta )+{\stackrel{`}{\alpha }}_4\pi (\pounds \vartheta ,{\pounds }^2\vartheta )\hfill \\ & +{\stackrel{`}{\alpha }}_5\pi ({\vartheta }^{*},\pounds {\vartheta }^{*})+{\stackrel{`}{\alpha }}_6\pi (\pounds {\vartheta }^{*},{\pounds }^2{\vartheta }^{*})+{\stackrel{`}{\alpha }}_7\left(\frac{\pi (\vartheta ,\pounds {\vartheta }^{*})+\pi ({\vartheta }^{*},\pounds \vartheta )}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi (\pounds \vartheta ,{\pounds }^2{\vartheta }^{*})+\pi (\pounds {\vartheta }^{*},{\pounds }^2\vartheta )}{2}\right)\hfill \\ & \le {\stackrel{`}{\alpha }}_1\pi (\vartheta ,{\vartheta }^{*})+{\stackrel{`}{\alpha }}_2\pi (\vartheta ,{\vartheta }^{*})+{\stackrel{`}{\alpha }}_7\pi (\vartheta ,{\vartheta }^{*})+{\stackrel{`}{\alpha }}_8\pi (\vartheta ,{\vartheta }^{*})\hfill \\ & \le ({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\pi (\vartheta ,{\vartheta }^{*}).\hfill \end{array}$$

It follows that $(1-{\stackrel{`}{\alpha }}_1-{\stackrel{`}{\alpha }}_2-{\stackrel{`}{\alpha }}_7-{\stackrel{`}{\alpha }}_8)\pi (\vartheta ,{\vartheta }^{*})\le 0$, which is a contradiction. Therefore, $\pi (\vartheta ,{\vartheta }^{*})=0$. Hence, £ has a unique fixed point in ℧. □

In Equation (1), put ${\sum }_{\xi =3}^8{\stackrel{`}{\alpha }}_{\xi }=0$ (resp. ${\sum }_{\xi =1}^2{\stackrel{`}{\alpha }}_{\xi }=0$) with $ħ=1$. We have new $\alpha$-convex contraction of order-2.

Corollary 1.

Let $(\mho ,\pi )$ be a complete b-metric space with coefficient $ħ\ge 1$ and let $\pounds :\mho \to \mho$ be a new generalized convex contraction. Assume that £ is an α-admissible and there exists ${\upsilon }_0\in \mho$ such that $\alpha (\pounds {\upsilon }_0,{\upsilon }_0)\ge 1$. Then, £ has an AFPP. Furthermore, £ has a fixed point if £ is continuous. Moreover, if for all $\upsilon ,\varsigma \in Fix\left(\pounds \right)$, we get $\alpha (\upsilon ,\varsigma )\ge 1$. Then, £ has a unique fixed point in ℧.

Proof. 

By generalized convex contraction, we have

$$\begin{array}{cc}\hfill \alpha (\upsilon ,\varsigma )\pi ({\pounds }^2\upsilon ,{\pounds }^2\varsigma )& \le {\stackrel{`}{\alpha }}_1\pi (\upsilon ,\varsigma )+{\stackrel{`}{\alpha }}_2\pi (\pounds \upsilon ,\pounds \varsigma )\hfill \\ & \le {\stackrel{`}{\alpha }}_1\pi (\upsilon ,\varsigma )+{\stackrel{`}{\alpha }}_2\pi (\pounds \upsilon ,\pounds \varsigma )+{\stackrel{`}{\alpha }}_3\pi (\upsilon ,\pounds \upsilon )+{\stackrel{`}{\alpha }}_4\pi (\pounds \upsilon ,{\pounds }^2\upsilon )\hfill \\ & +{\stackrel{`}{\alpha }}_5\pi (\varsigma ,\pounds \varsigma )+{\stackrel{`}{\alpha }}_6\pi (\pounds \varsigma ,{\pounds }^2\varsigma )+{\stackrel{`}{\alpha }}_7\left(\frac{\pi (\upsilon ,\pounds \varsigma )+\pi (\varsigma ,\pounds \upsilon )}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi (\pounds \upsilon ,{\pounds }^2\varsigma )+\pi (\pounds \varsigma ,{\pounds }^2\upsilon )}{2}\right).\hfill \end{array}$$

We conclude that £ is new $\alpha$-convex contraction of type-2. Thus, all the conditions of Theorem 1 are satisfied. □

Example 4.

Let $\mho =[0,1]$ be endowed with $\pi (\upsilon ,\varsigma )={|\upsilon -\varsigma |}^2$. Then, $(\mho ,\pi ,ħ=2)$ is a complete b-metric on ℧. As in [24], we define a self-mapping $\pounds :\mho \to \mho$ by $\pounds \upsilon =\frac{{\upsilon }^2}{2}+\frac{1}{4}$. By setting $\alpha (\upsilon ,\varsigma )=1$, then $\alpha (\pounds \upsilon ,\pounds \varsigma )=1,$ for all $\upsilon ,\varsigma \in \mho$. Therefore, a map £ is continuous and α-admissible. Now, we obtain

$$\begin{array}{cc}\hfill \alpha (\upsilon ,\varsigma )\pi ({\pounds }^2\upsilon ,{\pounds }^2\varsigma )& =|{\pounds }^2\upsilon -{\pounds }^2\varsigma {|}^2\hfill \\ & =|\frac{{\upsilon }^4+{\upsilon }^2+\frac{9}{4}}{8}-\frac{{\varsigma }^4+{\varsigma }^2+\frac{9}{4}}{8}{|}^2\hfill \\ & =\frac{1}{64}|({\upsilon }^4-{\varsigma }^4)+({\upsilon }^2-{\varsigma }^2){|}^2\hfill \\ & \le \frac{ħ}{64}\left\{\right|{\upsilon }^4-{\varsigma }^4{|}^2+|{\upsilon }^2-{\varsigma }^2{|}^2\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill \alpha (\upsilon ,\varsigma )\pi ({\pounds }^2\upsilon ,{\pounds }^2\varsigma )& =\frac{1}{32}\left\{\right|{\upsilon }^4-{\varsigma }^4{|}^2+|{\upsilon }^2-{\varsigma }^2{|}^2\}\hfill \\ & =\frac{1}{2}\left\{\right|\frac{({\upsilon }^2+{\varsigma }^2)}{2}\frac{({\upsilon }^2-{\varsigma }^2)}{2}{|}^2+\left|\frac{(\upsilon +\varsigma )}{2}\frac{(\upsilon -\varsigma )}{2}{|}^2\right\}\hfill \\ & =\frac{1}{2}\left\{\right|\frac{{\upsilon }^2+{\varsigma }^2}{2}{|}^2|\frac{{\upsilon }^2-{\varsigma }^2}{2}{|}^2+|\frac{\upsilon +\varsigma }{2}{|}^2\left|\frac{\upsilon -\varsigma }{2}{|}^2\right\}\hfill \\ & \le \frac{1}{2}|\frac{{\upsilon }^2-{\varsigma }^2}{2}{|}^2+\frac{1}{8}|\upsilon -\varsigma {|}^2\hfill \\ \hfill & \le \frac{1}{2}\pi (\pounds \upsilon ,\pounds \varsigma )+\frac{1}{8}\pi (\upsilon ,\varsigma ),\hfill \end{array}$$

where $|\frac{{\upsilon }^2+{\varsigma }^2}{2}{|}^2\le 1,|\frac{\upsilon +\varsigma }{2}{|}^2\le 1$, which shows that £ is a new generalized convex contraction with ${\stackrel{`}{\alpha }}_1=\frac{1}{8}$ and ${\stackrel{`}{\alpha }}_2=\frac{1}{2}$. We construct a sequence $\left\{{\upsilon }_{\omega }\right\}$ by ${\upsilon }_{\omega }=\frac{\omega }{\omega +1}-\frac{1}{\sqrt{2}}$. Then, ${\upsilon }_{\omega }\to 1-\frac{1}{\sqrt{2}}$, as $\omega \to \infty$. Therefore, ${\upsilon }_{\omega +1}=\pounds {\upsilon }_{\omega }=[\frac{1}{2}{(\frac{\omega }{\omega +1}-\frac{1}{\sqrt{2}})}^2+\frac{1}{4}]\to (1-\frac{1}{\sqrt{2}})$, as $\omega \to \infty$. Thus, all the conditions of Corollary 1 are satisfied and $\upsilon =1-\frac{1}{\sqrt{2}}$ is the unique fixed point of £ in ℧.

Now, in the context of 2-metric space, we create a new $\alpha$-convex contraction of type-2.

Definition 12.

Let $(\mho ,\pi )$ be a 2-metric space and £ be a self-mapping on ℧. Then, £ is called a new α-convex contraction of type-2, if there exists a mapping $\alpha :\mho \times \mho \times \mho \to [0,\infty )$ and ${\stackrel{`}{\alpha }}_i\in [0,1)$, for all $i\in \{1,\dots ,8\}$ with ${\sum }_{\xi =1}^8{\stackrel{`}{\alpha }}_{\xi }<1$ such that

$$\begin{array}{cc}\hfill \alpha (\upsilon ,\varsigma ,å)\pi ({\pounds }^2\upsilon ,{\pounds }^2\varsigma ,å)& \le {\stackrel{`}{\alpha }}_1\pi (\upsilon ,\varsigma ,å)+{\stackrel{`}{\alpha }}_2\pi (\pounds \upsilon ,\pounds \varsigma ,å)+{\stackrel{`}{\alpha }}_3\pi (\upsilon ,\pounds \upsilon ,å)\hfill \\ & +{\stackrel{`}{\alpha }}_4\pi (\pounds \upsilon ,{\pounds }^2\upsilon ,å)+{\stackrel{`}{\alpha }}_5\pi (\varsigma ,\pounds \varsigma ,å)+{\stackrel{`}{\alpha }}_6\pi (\pounds \varsigma ,{\pounds }^2\varsigma ,å)\hfill \\ & +{\stackrel{`}{\alpha }}_7\left(\frac{\pi (\upsilon ,\pounds \varsigma ,å)+\pi (\varsigma ,\pounds \upsilon ,å)}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi (\pounds \upsilon ,{\pounds }^2\varsigma ,å)+\pi (\pounds \varsigma ,{\pounds }^2\upsilon ,å)}{2}\right),\hfill \end{array}$$

(2)

for all $\upsilon ,\varsigma ,å\in \mho$.

Theorem 2.

Let $(\mho ,\pi )$ be a complete 2-metric space and $\pounds :\mho \to \mho$ be a new α-convex contraction of type-2. Assume that £ is α-admissible and there exists ${\upsilon }_0\in \mho$ such that $\alpha (\pounds {\upsilon }_0,{\upsilon }_0,å)\ge 1$ for all $å\in \mho$. Then, £ has an AFPP. Furthermore, £ has a fixed point if £ is continuous. Moreover, if $\upsilon ,\varsigma \in Fix\left(\pounds \right)$, where we have $\alpha (\upsilon ,\varsigma ,å)\ge 1$ for all $å\in \mho$, then £ has a unique fixed point in ℧.

Proof. 

Let ${\upsilon }_0\in \mho$ be such that $\alpha (\pounds {\upsilon }_0,{\upsilon }_0,å)\ge 1,$ for all $å\in \mho$. We construct a sequence $\left\{{\upsilon }_{\omega }\right\}$ by ${\upsilon }_{\omega +1}={\pounds }^{\omega +1}{\upsilon }_0,$ for all $\omega \ge 0$, as in Theorem 1. If ${\upsilon }_{\omega }={\upsilon }_{\omega +1}$, i.e., ${\pounds }^{\omega }{\upsilon }_0=\pounds \left({\pounds }^{\omega }{\upsilon }_0\right)$ for each $\omega$, then we conclude that it follows immediately. Suppose that ${\upsilon }_{\omega }\ne {\upsilon }_{\omega +1}$ for all $\omega \ge 0$. Since £ is $\alpha$-admissible, we obtain that $\alpha (\pounds {\upsilon }_0,{\upsilon }_0,å)\ge 1,forallå\in \mho implies\alpha ({\pounds }^2{\upsilon }_0,\pounds {\upsilon }_0)\ge 1.$ Therefore, we obtain inductively that $\alpha ({\pounds }^{\omega +1}{\upsilon }_0,{\pounds }^{\omega }{\upsilon }_0,å)\ge 1$ for all $\omega \ge 0$. Denote

$$\mathfrak{m}=max\{\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å),\pi ({\upsilon }_0,\pounds {\upsilon }_0,å)\},\mathrm{for} \mathrm{all}å\in \mho ,$$

$$\nu ={\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+2{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_{8}and♭=(1-{\stackrel{`}{\alpha }}_6-{\stackrel{`}{\alpha }}_8).$$

Since £ is $\alpha$-admissible, from (2), by taking $\upsilon ={\upsilon }_0,\varsigma =\pounds {\upsilon }_0$, we obtain

$$\begin{array}{cc}\hfill \pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å)& \le \alpha ({\upsilon }_0,\pounds {\upsilon }_0,å)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å)\hfill \\ & \le {\stackrel{`}{\alpha }}_1\pi ({\upsilon }_0,\pounds {\upsilon }_0,å)+{\stackrel{`}{\alpha }}_2\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)+{\stackrel{`}{\alpha }}_3\pi ({\upsilon }_0,\pounds {\upsilon }_0,å)\hfill \\ & +{\stackrel{`}{\alpha }}_4\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)+{\stackrel{`}{\alpha }}_5\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å)\hfill \\ & +{\stackrel{`}{\alpha }}_7\left(\frac{\pi ({\upsilon }_0,{\pounds }^2{\upsilon }_0,å)+\pi (\pounds {\upsilon }_0,\pounds {\upsilon }_0,å)}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi (\pounds {\upsilon }_0,{\pounds }^3{\upsilon }_0,å)+\pi ({\pounds }^2{\upsilon }_0,{\pounds }^2{\upsilon }_0,å)}{2}\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\upsilon }_0,\pounds {\upsilon }_0,å)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)\hfill \\ & +{\stackrel{`}{\alpha }}_6\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å)+{\stackrel{`}{\alpha }}_7\left(\frac{\pi ({\upsilon }_0,{\pounds }^2{\upsilon }_0,å)}{2}\right)+{\stackrel{`}{\alpha }}_8\left(\frac{\pi (\pounds {\upsilon }_0,{\pounds }^3{\upsilon }_0,å)}{2}\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\upsilon }_0,\pounds {\upsilon }_0,å)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å)\hfill \\ & +\frac{{\stackrel{`}{\alpha }}_7}{2}\left(\pi ({\upsilon }_0,\pounds {\upsilon }_0,å)+\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)\right)+\frac{{\stackrel{`}{\alpha }}_8}{2}\left(\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)+\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å)\right)\hfill \\ & \le ({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3)\pi ({\upsilon }_0,\pounds {\upsilon }_0,å)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å)\hfill \\ & +{\stackrel{`}{\alpha }}_7\left(\pi ({\upsilon }_0,\pounds {\upsilon }_0,å)+\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)\right)+{\stackrel{`}{\alpha }}_8\left(\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)+\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å)\right)\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7)\pi ({\upsilon }_0,\pounds {\upsilon }_0,å)+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\pi (\pounds {\upsilon }_0,{\pounds }^2{\upsilon }_0,å)\hfill \\ & +({\stackrel{`}{\alpha }}_6+{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å).\hfill \end{array}$$

Therefore, we get

$$\begin{array}{cc}\hfill (1-{\stackrel{`}{\alpha }}_6-{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å)& \le ({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7)\mathfrak{m}+({\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\mathfrak{m}\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\mathfrak{m}\hfill \\ & =({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_3+{\stackrel{`}{\alpha }}_4+{\stackrel{`}{\alpha }}_5+2{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\mathfrak{m};\hfill \end{array}$$

that is $\pi ({\pounds }^2{\upsilon }_0,{\pounds }^3{\upsilon }_0,å)\le \left(\frac{\nu }{♭}\right)\mathfrak{m}$.

Similarly, we obtain

$$\pi ({\pounds }^3{\upsilon }_0,{\pounds }^4{\upsilon }_0,å)\le \left(\frac{\nu }{♭}\right)\mathfrak{m},and\pi ({\pounds }^4{\upsilon }_0,{\pounds }^5{\upsilon }_0,å)\le {\left(\frac{\nu }{♭}\right)}^2\mathfrak{m}.$$

Additionally, $\pi ({\pounds }^5{\upsilon }_0,{\pounds }^6{\upsilon }_0,å)\le {\left(\frac{\nu }{♭}\right)}^2\mathfrak{m}.$ By continuing this process, we obtain

$$\begin{array}{c}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +1}{\upsilon }_0,å)\le {\left(\frac{\nu }{♭}\right)}^{ȷ}\mathfrak{m},\mathrm{whenever}\varpi =2ȷ\mathrm{or}\varpi =2ȷ+1,\mathrm{for}ȷ>1\end{array}$$

or

$$\begin{array}{c}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +1}{\upsilon }_0,å)\le {\left(\frac{\nu }{♭}\right)}^{ȷ-1}\mathfrak{m},\mathrm{whenever}\varpi =2ȷ\mathrm{or}\varpi =2ȷ-1,\mathrm{for}ȷ>2.\end{array}$$

Therefore, $\pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +1}{\upsilon }_0,å)\to 0$ as $\varpi \to \infty$; we get that £ is asymptotically regular at ${\upsilon }_0\in \mho ,\forall å\in \mho$. By Lemma 3, £ has an AFPP.

Now, prove a sequence $\left\{{\upsilon }_{\omega }\right\}$ is a Cauchy sequence in ℧. Let $\varpi ,\omega$ be two distinct non-zero positive integers such that $\varpi <\omega$, which arises the two cases.

Case (i). For $\varpi =2ȷ$ with $ȷ>1$, then

$$\begin{array}{c}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +1}{\upsilon }_0,å)=\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0,å)\le {\left(\frac{\nu }{♭}\right)}^{ȷ}\mathfrak{m}.\end{array}$$

Now, we have

$$\begin{array}{cc}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +2}{\upsilon }_0,å)& =\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,å)\hfill \\ & \le \pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0)+\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0,å)+\pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,å).\hfill \end{array}$$

Therefore, we obtain

$$\begin{array}{c}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +2}{\upsilon }_0,å)\le \pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0)+\sum _{\mathfrak{r}=0}^{\mathfrak{r}=1}\pi ({\pounds }^{2ȷ+\mathfrak{r}}{\upsilon }_0,{\pounds }^{2ȷ+\mathfrak{r}+1}{\upsilon }_0,å).\end{array}$$

(3)

Since £ is $\alpha$-admissible, from (2), we obtain

$$\begin{array}{cc}\hfill \pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)& =\pi ({\pounds }^2\left({\pounds }^{2ȷ-1}\right){\upsilon }_0,{\pounds }^2\left({\pounds }^{2ȷ}\right){\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)\hfill \\ & \le \alpha ({\pounds }^{2ȷ-1}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)\pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\stackrel{`}{\alpha }}_1\pi ({\pounds }^{2ȷ-1}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)+{\stackrel{`}{\alpha }}_2\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_3\pi ({\pounds }^{2ȷ-1}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)+{\stackrel{`}{\alpha }}_4\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_5\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)+{\stackrel{`}{\alpha }}_6\pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)\hfill \\ & +{\stackrel{`}{\alpha }}_7\left(\frac{\pi ({\pounds }^{2ȷ-1}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)+\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)+\pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ}{\upsilon }_0)}{2}\right).\hfill \end{array}$$

Therefore,

$$(1-{\stackrel{`}{\alpha }}_6-{\stackrel{`}{\alpha }}_8)\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0)\le 0,$$

which in turn gives $\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0)=0$. Thus, inequality (3) reduces to

$$\begin{array}{c}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +2}{\upsilon }_0,å)\le \sum _{\mathfrak{r}=0}^{\mathfrak{r}=1}\pi ({\pounds }^{2ȷ+\mathfrak{r}}{\upsilon }_0,{\pounds }^{2ȷ+\mathfrak{r}+1}{\upsilon }_0,å).\end{array}$$

Again, we have

$$\begin{array}{cc}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +3}{\upsilon }_0,å)& =\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0,å)\hfill \\ & \le \pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0)+\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,å)+\pi ({\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0,å)\hfill \\ & =\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0)+\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,å)+\pi ({\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0,å),\hfill \end{array}$$

which implies that

$$\begin{array}{cc}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +3}{\upsilon }_0,å)& \le \pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0)+\sum _{\mathfrak{r}=0}^{\mathfrak{r}=2}\pi ({\pounds }^{2ȷ+\mathfrak{r}}{\upsilon }_0,{\pounds }^{2ȷ+\mathfrak{r}+1}{\upsilon }_0,å).\hfill \end{array}$$

One can show that $\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0)=0$. Similarly, we obtain

$$\begin{array}{c}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +4}{\upsilon }_0,å)\le \sum _{\mathfrak{r}=0}^{\mathfrak{r}=3}\pi ({\pounds }^{2ȷ+\mathfrak{r}}{\upsilon }_0,{\pounds }^{2ȷ+\mathfrak{r}+1}{\upsilon }_0,å).\end{array}$$

Continuing in this way, we have

$$\begin{array}{cc}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +\ell }{\upsilon }_0,å)& \le \sum _{\mathfrak{r}=0}^{\mathfrak{r}=\ell -1}\pi ({\pounds }^{2ȷ+\mathfrak{r}}{\upsilon }_0,{\pounds }^{2ȷ+\mathfrak{r}+1}{\upsilon }_0,å)\hfill \\ & =\pi ({\pounds }^{2ȷ}{\upsilon }_0,{\pounds }^{2ȷ+1}{\upsilon }_0,å)+\pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+2}{\upsilon }_0,å)+\pi ({\pounds }^{2ȷ+2}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0,å)\hfill \\ & +\cdots +\pi ({\pounds }^{2ȷ+\ell -1}{\upsilon }_0,{\pounds }^{2ȷ+\ell }{\upsilon }_0,å)\hfill \\ & \le {\left(\frac{\nu }{♭}\right)}^{ȷ}\mathfrak{m}+{\left(\frac{\nu }{♭}\right)}^{ȷ}\mathfrak{m}+{\left(\frac{\nu }{♭}\right)}^{ȷ+1}\mathfrak{m}+{\left(\frac{\nu }{♭}\right)}^{ȷ+1}\mathfrak{m}+\cdots \hfill \\ & \le 2{\left(\frac{\nu }{♭}\right)}^{ȷ}\frac{1}{1-\frac{\nu }{♭}}\mathfrak{m}.\hfill \end{array}$$

Case (ii). For $\varpi =2ȷ+1$ with $ȷ>1$,

$$\pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +1}{\upsilon }_0,å)=\pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0,å)\le {\left(\frac{\nu }{♭}\right)}^{ȷ}\mathfrak{m}.$$

Similarly as in Case (i), we obtain

$$\begin{array}{c}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +3}{\upsilon }_0,å)\le \sum _{\mathfrak{r}=0}^{\mathfrak{r}=2}\pi ({\pounds }^{2ȷ+\mathfrak{r}+1}{\upsilon }_0,{\pounds }^{2ȷ+\mathfrak{r}+2}{\upsilon }_0,å),\end{array}$$

when $\pi ({\pounds }^{2ȷ+1}{\upsilon }_0,{\pounds }^{2ȷ+3}{\upsilon }_0,{\pounds }^{2ȷ+4}{\upsilon }_0)=0$. By continuing this way, we get

$$\begin{array}{cc}\hfill \pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\varpi +\ell }{\upsilon }_0,å)& \le \sum _{\mathfrak{r}=0}^{\mathfrak{r}=\ell -1}\pi ({\pounds }^{2ȷ+\mathfrak{r}}{\upsilon }_0,{\pounds }^{2ȷ+\mathfrak{r}+1}{\upsilon }_0,å)\hfill \\ & \le 2{\left(\frac{\nu }{♭}\right)}^{ȷ}\frac{1}{1-\frac{\nu }{♭}}\mathfrak{m}.\hfill \end{array}$$

Taking $ȷ\to \infty$ in all cases, $\frac{\nu }{♭}<1$, we obtain $\pi ({\pounds }^{\varpi }{\upsilon }_0,{\pounds }^{\omega }{\upsilon }_0,å)\to 0$. Therefore, the sequence $\left\{{\upsilon }_{\omega }\right\}$ in ℧ is a Cauchy sequence.

Since completeness of 2-metric space, there exists a point $\vartheta \in \mho$ such that ${\upsilon }_{\omega }={\pounds }^{\omega }{\upsilon }_0\to \vartheta \in \mho$ as $\omega \to \infty$.

Since £ is continuous, we have $\vartheta =\underset{\omega \to \infty }{lim}\pounds \left({\pounds }^{\omega }{\upsilon }_0\right)=\pounds \vartheta$. Therefore, £ has a fixed point $\vartheta$ in £.

Now, we show the uniqueness of the fixed point.

We suppose that £ is $\alpha$-admissible. Since $Fix\left(\pounds \right)\ne \varnothing$, let $\vartheta ,{\vartheta }^{*}\in Fix\left(\pounds \right)$. By $\alpha$-admissible of £, we have $\stackrel{`}{\alpha }(\vartheta ,{\vartheta }^{*},\mathfrak{u})\ge 1,$ for all $\mathfrak{u}\in \mho$, and by Equation (2) with $\upsilon =\vartheta$ and $\varsigma ={\vartheta }^{*}$, we have

$$\begin{array}{cc}\hfill \pi (\vartheta ,{\vartheta }^{*},\mathfrak{u})& =\pi ({\pounds }^2\vartheta ,{\pounds }^2{\vartheta }^{*},\mathfrak{u})\hfill \\ & \le \alpha (\vartheta ,{\vartheta }^{*},\mathfrak{u})\pi ({\pounds }^2\vartheta ,{\pounds }^2{\vartheta }^{*},\mathfrak{u})\hfill \\ & \le {\stackrel{`}{\alpha }}_1\pi (\vartheta ,{\vartheta }^{*},\mathfrak{u})+{\stackrel{`}{\alpha }}_2\pi (\pounds \vartheta ,\pounds {\vartheta }^{*},\mathfrak{u})+{\stackrel{`}{\alpha }}_3\pi (\vartheta ,\pounds \vartheta ,\mathfrak{u})+{\stackrel{`}{\alpha }}_4\pi (\pounds \vartheta ,{\pounds }^2\vartheta ,\mathfrak{u})\hfill \\ & +{\stackrel{`}{\alpha }}_5\pi ({\vartheta }^{*},\pounds {\vartheta }^{*},\mathfrak{u})+{\stackrel{`}{\alpha }}_6\pi (\pounds {\vartheta }^{*},{\pounds }^2{\vartheta }^{*},\mathfrak{u})+{\stackrel{`}{\alpha }}_7\left(\frac{\pi (\vartheta ,\pounds {\vartheta }^{*},\mathfrak{u})+\pi ({\vartheta }^{*},\pounds \vartheta ,\mathfrak{u})}{2}\right)\hfill \\ & +{\stackrel{`}{\alpha }}_8\left(\frac{\pi (\pounds \vartheta ,{\pounds }^2{\vartheta }^{*},\mathfrak{u})+\pi (\pounds {\vartheta }^{*},{\pounds }^2\vartheta ,\mathfrak{u})}{2}\right)\hfill \\ & \le ({\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\pi (\vartheta ,{\vartheta }^{*},\mathfrak{u}),\hfill \end{array}$$

implying that $(1-{\stackrel{`}{\alpha }}_1+{\stackrel{`}{\alpha }}_2+{\stackrel{`}{\alpha }}_7+{\stackrel{`}{\alpha }}_8)\pi (\vartheta ,{\vartheta }^{*},\mathfrak{u})\le 0$, which is a contradiction. Hence, we get $\pi (\vartheta ,{\vartheta }^{*},\mathfrak{u})=0$. Therefore, £ has a unique fixed point in ℧. □

Remark 1.

By putting ${\sum }_{\xi =1}^2{\stackrel{`}{\alpha }}_{\xi }<1$, and ${\sum }_{\xi =3}^8{\stackrel{`}{\alpha }}_{\xi }=0$, inequality (2) becomes

$$\begin{array}{c}\hfill \alpha (\upsilon ,\varsigma ,å)\pi ({\pounds }^2\upsilon ,{\pounds }^2\varsigma ,å)\le {\stackrel{`}{\alpha }}_1\pi (\upsilon ,\varsigma ,å)+{\stackrel{`}{\alpha }}_2\pi (\pounds \upsilon ,\pounds \varsigma ,å).\end{array}$$

(4)

Again, by taking ${\sum }_{\xi =3}^8{\stackrel{`}{\alpha }}_{\xi }<1$, where ${\sum }_{\xi =1}^2{\stackrel{`}{\alpha }}_{\xi }=0$, inequality (2) becomes

$$\begin{array}{cc}\hfill \alpha (\upsilon ,\varsigma ,å)\pi ({\pounds }^2\upsilon ,{\pounds }^2\varsigma ,å)& \le {\stackrel{`}{\alpha }}_3\pi (\upsilon ,\pounds \upsilon ,å)+{\stackrel{`}{\alpha }}_4\pi (\pounds \upsilon ,{\pounds }^2\upsilon ,å)\hfill \\ & +{\stackrel{`}{\alpha }}_5\pi (\varsigma ,\pounds \varsigma ,å)+{\stackrel{`}{\alpha }}_6\pi (\pounds \varsigma ,{\pounds }^2\varsigma ,å)+{\stackrel{`}{\alpha }}_7\left(\frac{\pi (\upsilon ,\pounds \varsigma ,å)+\pi (\varsigma ,\pounds \upsilon ,å)}{2}\right)\hfill \end{array}$$

$$\begin{array}{c}\hfill +{\stackrel{`}{\alpha }}_8\left(\frac{\pi (\pounds \upsilon ,{\pounds }^2\varsigma ,å)+\pi (\pounds \varsigma ,{\pounds }^2\upsilon ,å)}{2}\right).\end{array}$$

(5)

Now, in the context of 2-metric space, note that (4) (resp. (5)) gives new α-convex contraction of type-2.

3. Application

In this section, we study the existence of a solution for an integral equation using the results proved in Section 2. Let $\mho =\mathcal{C}[\mathfrak{o},\mathfrak{e}]$ be a set of all real valued continuous functions on $[\mathfrak{o},\mathfrak{e}]$, where $[\mathfrak{o},\mathfrak{e}]$ is a closed and bounded interval in $\mathcal{R}$. For a real number $\wp >1$, define $\mathit{d}:\mho \times \mho \to [0,\infty )$ by

$$\begin{array}{c}\hfill d(\varphi ,\psi )=\underset{\eta \in [\mathfrak{o},\mathfrak{e}]}{max}{|\varphi \left(\eta \right)-\psi \left(\eta \right)|}^{\wp },\mathrm{for}\mathrm{all}\varphi ,\psi \in \mathcal{C}[\mathfrak{o},\mathfrak{e}].\end{array}$$

Therefore, $(\mho ,d)$ is a complete b-metric space with $ħ=2^{\wp -1}$. We apply Theorem 1 to establish the existence of the solution of Fredholm type defined by

$$\begin{array}{c}\hfill \upsilon \left(\eta \right)=\aleph \left(\eta \right)+\lambda {\int }_{\mathfrak{o}}^{\mathfrak{e}}\mathcal{K}(\eta ,\varrho ,\upsilon \left(\varrho \right))\mathit{d}\varrho ,\end{array}$$

(6)

where $\eta ,\varrho \in [\mathfrak{o},\mathfrak{e}]\subset [0,\infty )$. A solution of the Equation (6) is a function $\upsilon \in [\mathfrak{o},\mathfrak{e}]$.

Assume that

(i)

$\mathcal{K}:[\mathfrak{o},\mathfrak{e}]\times [\mathfrak{o},\mathfrak{e}]\times \mathcal{R}\to \mathcal{R}$ and $\aleph :[\mathfrak{o},\mathfrak{e}]\to \mathcal{R}$ are continuous functions on $[\mathfrak{o},\mathfrak{e}]$;

(ii)

$\left|\lambda \right|\le 1$;

(iii)

for every $\upsilon ,\varsigma \in \mho$ with $\upsilon \ne \varsigma$ and $\varrho ,\eta \in [\mathfrak{o},\mathfrak{e}]$ satisfying the following inequality

$$\begin{array}{cc}\hfill {|\mathcal{K}(\eta ,\varrho ,\pounds \upsilon \left(\varrho \right))-\mathcal{K}(\eta ,\varrho ,\pounds \varsigma \left(\varrho \right))|}^{\wp }& \le \mathcal{D}(\eta ,\varrho )max\left\{\right|\upsilon \left(\varrho \right)-\varsigma \left(\varrho \right)|,|\pounds \upsilon \left(\varrho \right)-\pounds \varsigma \left(\varrho \right)|,\hfill \\ & |\upsilon \left(\varrho \right)-\pounds \upsilon \left(\varrho \right)|,|\pounds \upsilon \left(\varrho \right)-{\pounds }^2\upsilon \left(\varrho \right)|,|\varsigma \left(\varrho \right)-\pounds \varsigma \left(\varrho \right)|,\hfill \\ & |\pounds \varsigma \left(\varrho \right)-{\pounds }^2\varsigma \left(\varrho \right)|,\left(\frac{\left|\upsilon \right(\varrho )-\pounds \varsigma (\varrho \left)\right|+\left|\varsigma \right(\varrho )-\pounds \upsilon (\varrho \left)\right|}{2}\right),\hfill \\ & \left(\frac{|\pounds \upsilon \left(\varrho \right)-{\pounds }^2\varsigma \left(\varrho \right)|+|\pounds \varsigma \left(\varrho \right)-{\pounds }^2\upsilon \left(\varrho \right)|}{2}\right){\}}^{\wp };\hfill \end{array}$$

(7)

(iii)

${max}_{\eta \in [\mathfrak{o},\mathfrak{e}]}{\int }_{\mathfrak{o}}^{\mathfrak{e}}\mathcal{D}(\eta ,\varrho )d\varrho \le \frac{1}{{(\mathfrak{e}-\mathfrak{o})}^{\wp -1}}$, where $ħ=2^{\wp -1}$.

Now, we are equipped to state and prove our main result in this section.

Theorem 3.

Under the assumptions (i)–(iv), the integral Equation (6) has a solution in ℧.

Proof. 

Define $\pounds :\mho \to \mho$ by

$$\begin{array}{c}\hfill \pounds \upsilon \left(\eta \right)=\aleph \left(\eta \right)+\lambda {\int }_{\mathfrak{o}}^{\mathfrak{e}}\mathcal{K}(\eta ,\varrho ,\upsilon \left(\varrho \right))d\varrho .\end{array}$$

(8)

Observe that $\upsilon$ is a solution for (6) if and only if $\upsilon$ is a fixed point of £. Let $\mathrm{q}\in \mathcal{R}$ such that $\frac{1}{\wp }+\frac{1}{\mathrm{q}}=1$ using the Holder inequality and conditions (i)–(iv). We prove that £ is a new generalized convex contraction on $\mathcal{C}[\mathfrak{o},\mathfrak{e}]$. By Equations (7) and (8), we obtain

$$\begin{array}{cc}\hfill d({\pounds }^2\upsilon ,{\pounds }^2\varsigma )& =\underset{\eta \in [\mathfrak{o},\mathfrak{e}]}{max}{|{\pounds }^2\upsilon \left(\eta \right)-{\pounds }^2\varsigma \left(\eta \right)|}^{\wp }\hfill \\ & =\underset{\eta \in [\mathfrak{o},\mathfrak{e}]}{max}{\left|\lambda {\int }_{\mathfrak{o}}^{\mathfrak{e}}\mathcal{K}(\eta ,\varrho ,\pounds \upsilon \left(\varrho \right))d\varrho -\mathcal{K}(\eta ,\varrho ,\pounds \varsigma \left(\varrho \right))d\varrho \right|}^{\wp }\hfill \\ & \le \underset{\eta \in [\mathfrak{o},\mathfrak{e}]}{max}{\left|\lambda \right|}^{\wp }{\int }_{\mathfrak{o}}^{\mathfrak{e}}{\left|\mathcal{K}(\eta ,\varrho ,\pounds \upsilon (\varrho \left)\right)-\mathcal{K}(\eta ,\varrho ,\pounds \varsigma (\varrho \left)\right)\right|}^{\wp }d\varrho \hfill \\ & \le [\underset{\eta \in [\mathfrak{o},\mathfrak{e}]}{max}{\left({\int }_{\mathfrak{o}}^{\mathfrak{e}}{1}^{\wp }d\varrho \right)}^{\frac{1}{\mathrm{q}}}{\left({\int }_{\mathfrak{o}}^{\mathfrak{e}}{\left|\mathcal{K}(\eta ,\varrho ,\pounds \upsilon (\varrho \left)\right)-\mathcal{K}(\eta ,\varrho ,\pounds \varsigma (\varrho \left)\right)\right|}^{\wp }d\varrho \right)}^{\frac{1}{\wp }}{]}^{\wp }\hfill \\ & \le {(\mathfrak{e}-\mathfrak{o})}^{\frac{\wp }{\mathrm{q}}}\underset{\eta \in [\mathfrak{o},\mathfrak{e}]}{max}{\int }_{\mathfrak{o}}^{\mathfrak{e}}\mathcal{D}(\eta ,\varrho )max\left\{\right|\upsilon \left(\varrho \right)-\varsigma \left(\varrho \right)|,|\pounds \upsilon \left(\varrho \right)-\pounds \varsigma \left(\varrho \right)|,\hfill \\ & |\upsilon \left(\varrho \right)-\pounds \upsilon \left(\varrho \right)|,|\pounds \upsilon \left(\varrho \right)-{\pounds }^2\upsilon \left(\varrho \right)|,|\varsigma \left(\varrho \right)-\pounds \varsigma \left(\varrho \right)|,|\pounds \varsigma \left(\varrho \right)-{\pounds }^2\varsigma \left(\varrho \right)|,\hfill \\ & \left(\frac{\left|\upsilon \right(\varrho )-\pounds \varsigma (\varrho \left)\right|+\left|\varsigma \right(\varrho )-\pounds \upsilon (\varrho \left)\right|}{2}\right),\left(\frac{|\pounds \upsilon \left(\varrho \right)-{\pounds }^2\varsigma \left(\varrho \right)|+|\pounds \varsigma \left(\varrho \right)-{\pounds }^2\upsilon \left(\varrho \right)|}{2}\right){\}}^{\wp }d\varrho \hfill \\ & \le {(\mathfrak{e}-\mathfrak{o})}^{\wp -1}max[\underset{\mathfrak{r}\in [\mathfrak{o},\mathfrak{e}]}{max}\left\{\right|\upsilon \left(\mathfrak{r}\right)-\varsigma \left(\mathfrak{r}\right)|,|\pounds \upsilon \left(\mathfrak{r}\right)-\pounds \varsigma \left(\mathfrak{r}\right)|,|\upsilon \left(\mathfrak{r}\right)-\pounds \upsilon \left(\mathfrak{r}\right)|,\hfill \\ & |\pounds \upsilon \left(\mathfrak{r}\right)-{\pounds }^2\upsilon \left(\mathfrak{r}\right)|,|\varsigma \left(\mathfrak{r}\right)-\pounds \varsigma \left(\mathfrak{r}\right)|,|\pounds \varsigma \left(\mathfrak{r}\right)-{\pounds }^2\varsigma \left(\mathfrak{r}\right)|,\left(\frac{\left|\upsilon \right(\mathfrak{r})-\pounds \varsigma (\mathfrak{r}\left)\right|+\left|\varsigma \right(\mathfrak{r})-\pounds \upsilon (\mathfrak{r}\left)\right|}{2}\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \left(\frac{|\pounds \upsilon \left(\mathfrak{r}\right)-{\pounds }^2\varsigma \left(\mathfrak{r}\right)|+|\pounds \varsigma \left(\mathfrak{r}\right)-{\pounds }^2\upsilon \left(\mathfrak{r}\right)|}{2}\right){\}}^{\wp }]{\int }_{\mathfrak{o}}^{\mathfrak{e}}\mathcal{D}(\eta ,\varrho )d\varrho \hfill \\ & ={(\mathfrak{e}-\mathfrak{o})}^{\wp -1}.\frac{1}{{(\mathfrak{e}-\mathfrak{o})}^{\wp -1}}max\{d(\upsilon ,\varsigma ),d(\pounds \upsilon ,\pounds \varsigma ),d(\upsilon ,\pounds \upsilon ),d(\pounds \upsilon ,{\pounds }^2\upsilon ),d(\varsigma ,\pounds \varsigma ),\hfill \\ & d(\pounds \varsigma ,{\pounds }^2\varsigma ),\left(\frac{d(\upsilon ,\pounds \varsigma )+d(\varsigma ,\pounds \upsilon )}{2}\right),\left(\frac{d(\pounds \upsilon ,{\pounds }^2\varsigma )+d(\pounds \varsigma ,{\pounds }^2\upsilon )}{2}\right)\}\hfill \\ & =\mathfrak{R}(\upsilon ,\varsigma ).\hfill \end{array}$$

Therefore,

$$\begin{array}{c}\hfill \alpha (\upsilon ,\varsigma )d(\pounds \upsilon ,\pounds \varsigma )\le \mathfrak{R}(\upsilon ,\varsigma ).\end{array}$$

Define a mapping $\stackrel{`}{\alpha }:\mho \times \mho \to {\mathcal{R}}^{+}$ by $\alpha (\upsilon ,\varsigma )=1$ for all $\upsilon ,\varsigma \in \mho$.

Therefore, £ is $\alpha$-admissible.

Letting $\Theta \in \Im$ such that $\Theta \left(\gamma \right)=ln\gamma ,\gamma >0$.

Let ${\upsilon }_0\in \mho$ and a sequence $\left\{{\upsilon }_{\omega }\right\}$ in ℧ defined by ${\upsilon }_{\omega +1}=\pounds {\upsilon }_{\omega }={\pounds }^{\omega +1}{\upsilon }_0$ for all $\omega \ge 0$. By Equation (8), we have

$$\begin{array}{c}\hfill {\upsilon }_{\omega +1}=\pounds {\upsilon }_{\omega }\left(\eta \right)=\aleph \left(\eta \right)+\frac{1}{\mathfrak{e}-\mathfrak{o}}{\int }_{\mathfrak{o}}^{\mathfrak{e}}\mathcal{K}(\eta ,\varrho ,{\upsilon }_{\omega }\left(\varrho \right))d\varrho .\end{array}$$

(9)

Thus, all the assertions of Theorem 1 are satisfied, and hence, £ has a unique fixed point solution $\vartheta \in \mho$. □

The following example shows the existence of unique solution of an integral operator satisfying all the hypothesis in Theorem 3.

Example 5.

Let us consider the following nonlinear integral equation:

$$\begin{array}{c}\hfill \upsilon \left(\eta \right)={\int }_0^{\eta }[{\left(\eta (1-\varrho )\right)}^{\tau -1}-{(\eta -\varrho )}^{\tau -1}]cos\left(\upsilon \left(\varrho \right)\right)d\varrho ,\end{array}$$

(10)

with $0\le \varrho \le \eta \le 1$. Define $\pounds :\mathcal{C}\left(\right[0,\mathfrak{e}],\mathcal{R})\to \mathcal{C}\left(\right[0,\mathfrak{e}],\mathcal{R})$ by

$$\begin{array}{c}\hfill \pounds \upsilon \left(\eta \right)={\int }_0^{\eta }[{\left(\eta (1-\varrho )\right)}^{\tau -1}-{(\eta -\varrho )}^{\tau -1}]cos\left(\upsilon \left(\varrho \right)\right)d\varrho .\end{array}$$

Given the conditions of Theorem (3), it is simple to demonstrate that Equation (10) has a unique solution. Additionally, we will emphasize the viability of our strategies using the iteration process.

$$\begin{array}{c}\hfill {\upsilon }_{\vartheta +1}\left(\eta \right)={\int }_0^1[{\left(\eta (1-\varrho )\right)}^{\tau -1}-{(\eta -\varrho )}^{\tau -1}]cos\left({\upsilon }_{\vartheta }\left(\varrho \right)\right)d\varrho .\end{array}$$

Let $\tau \in (1,2)$. Let us take $\tau =1.5$ and initial point ${\upsilon }_0\left(\eta \right)=0$. The sequence ${\upsilon }_{\vartheta +1}\left(\eta \right)={\int }_0^1[{\left(\eta (1-\varrho )\right)}^{\tau -1}-{(\eta -\varrho )}^{\tau -1}]cos\left({\upsilon }_{\vartheta }\left(\varrho \right)\right)d\varrho$ is shown in

Table 1. For $\eta =0.1$, the exact solution is $\upsilon \left(0.1\right)=0.033$.

$\mathit{\vartheta }$	${\mathit{\upsilon }}_{\mathit{\vartheta }+1}\left(0.1\right)$	Approximate Solution	Absolute Error
0	${\upsilon }_1\left(0.1\right)$	0.0308	$2.5\times {10}^{-3}$
1	${\upsilon }_2\left(0.1\right)$	0.0307	$2.6\times {10}^{-3}$
2	${\upsilon }_3\left(0.1\right)$	0.0307	$2.6\times {10}^{-3}$

for $\eta =0.1$ converging to the exact solution $\upsilon \left(0.1\right)=\pounds \left(\upsilon \right(0.1\left)\right)=0.033$.

We obtain the interpolated graphs of nonlinear integral equation for $\eta =0.1$. We get the following interpolated graphs in Figure 1.

4. Conclusions

Examining the existence and uniqueness of fixed points of self-mappings defined on generalized metric spaces has played a significant role in mathematics in the last ten years. In this paper, the notion of generalized convex contraction condition in b-metric space and 2-metric space with eight suitable combination was presented. We have obtained some results on fixed points in those spaces, which were illustrated by appropriate examples. At the end of the paper, we gave applications of our results for solving nonlinear Fredholm integral equations.

We believe that the way is open for investigating the existence and uniqueness of fixed points in m-metric spaces and 2-metric spaces while taking into account the work in this study. Recently, Gordji et al. [40,41] introduced the orthogonal concept of b-metric space and obtained a generalization of the Banach fixed point theorem. It is an interesting open problem to study the orthogonal $b_2$-metric space and orthogonal complete metric space with a w-distance [42] instead of b-metric space and 2-metric space.