Common fixed point and coincidence point results for generalized $\alpha$-$\varphi_{E}$-Geraghty contraction mappings in $b$-metric spaces

1.   Introduction

In 1973, Geraghty ^[1] generalized the Banach contraction principle in the setting of a complete metric space by considering an auxiliary function. This interesting result has attracted the attention of a great number of researchers. In 2012, Samet et al. ^[2] introduced the concept of $\alpha$-admissible and $\alpha$-$\psi$-contraction mappings and presented fixed point theorems for them. In 2013, Cho et al. ^[3] introduced the concept of $\alpha$-Geraghty contraction mappings in metric spaces and proved some fixed point results of such mappings. Karapinar et al. ^[4] gave the notion of an $\alpha$-$\zeta$-$E$-Pata contraction and proved the existence and uniqueness of a fixed point of such mappings in the setting of a complete metric space. In ^[5], authors established that the main result via $\psi$-Geraghty type contraction is equivalent to an existing related result in the literature. In the framework of a complete $b$-metric space, Karapinar et al. ^[6] investigated the existence of fixed points for $\alpha$-almost Istratesc contraction of type $E$ and of type $E^*$. Alghamdi et al. ^[7] considered a common fixed point theorem via extended $Z$-contraction with respect to $y$-simulation function over an auxiliary function $x$. In ^[8], Afshari et al. obtained a fixed point result of generalized $\alpha$-$\psi$-Geraghty contractive type mappings. Then, the notion of Geraghty contraction of type $E$ was introduced by Fulga and Proca ^[9]. Recently, some new set-valued Meir-Keeler, Geraghty and Edelstein type fixed point theorems were presented in ^[10]. In 2019, Aydi ^[11] introduced the notion of $\alpha$-$\beta_E$-Geraghty contraction mappings on $b$-metric spaces and proved the existence and uniqueness of fixed point for such mappings. In the following year, Alqahtani ^[12] proved the existence and uniqueness of a common fixed point for the Geraghty contraction of type $E_{S, T}$ on complete metric spaces. In ^[13], Debnath et al. extended the notions of orbitally continuous and asymptotically regular mappings in the set-valued context. They introduced two new contractive inequalities one of which is of Geraghty-type and the other is of Boyd and Wong-type and proved two new existence of fixed point results corresponding to those inequalities. Okeke et al. ^[14] proved some theorems on the existence and uniqueness of fixed point for Reich-type contraction mappings and Geraghty-type mappings satisfying rational inequalities in modular metric spaces. For recent development on fixed point theory of Geraghty type contraction, we refer to ^[15,16,17] and the related references therein.

Czerwik ^[18] introduced the concept of $b$-metric space in 1993, which is a generalization of metric space and proved some fixed point theorems of contractive mappings in this space. Afterwards, some authors had studied the fixed point theorems of a various new type of contractive conditions in $b$-metric spaces (see ^{[19,20,21,22,23,24,25,26]}). Recently, Abbas et al. ^[27] studied the existence of fixed points of $T$-Ciric type mappings in the setup of partially ordered spaces. Roshan et al. ^[28] proved a common fixed point theorem for three mappings in $G_b$-metric space which is not continuous. Mustafa et al. ^[29] introduced the class of extended rectangular $b$-metric spaces as a generalization of both rectangular metric and rectangular $b$-metric spaces. In addition, some fixed point results connected with certain contractions are obtained. In ^[30], Mustafa et al. presented some coincidence point results for six mappings satisfying the generalized $(\psi, \varphi)$-weakly contractive condition in the framework of partially ordered $G$-metric spaces.

Throughout this paper, we aim to obtain common fixed point of $\alpha_{i, j}$-$\varphi_{{E}_{M, N}}$-Geraghty contraction mappings and coincidence point of $\alpha_{i, j}$-$\varphi_{E_N}$-Geraghty contraction mapping in the framework of $b$-metric space. Furthermore, we provide two examples that elaborate on the usability of our results.

There are some basic definitions and theorems that need to be used later. We state them as follows.

Definition 1.1. ^[18] Let $X$ be a nonempty set and $s\geq 1$ be a given real number. A mapping $d:X\times X\rightarrow [0, +\infty)$ is said to be a $b$-metric if and only if, for all $a, b, c\in X,$ the following conditions are satisfied:

(i) $d(a, b) = 0$ if and only if $a = b$;

(ii) $d(a, b) = d(b, a)$;

(iii) $d(a, c)\leq s[d(a, b)+d(b, c)]$.

In general, $(X, d, s)$ is called a $b$-metric space with parameter $s\geq 1$.

It is obvious that the class of $b$-metric spaces is effectively larger than that of metric spaces since any metric space is a $b$-metric space with $s = 1$. The following examples show that, in general, a $b$-metric space need not necessarily be a metric space.

Example 1.2. ^[31] Let $X = \mathbb{R}$ and let the mapping $d:X\times X \to \mathbb{R}^+$ be defined by

for all $a, b \in X$. Then $(X, d, 2)$ is a $b$-metric space with parameter $s = 2$.

Example 1.3. ^[32] Let $X = \{0, 1, 2\}$ and $d(2, 0) = d(0, 2) = m\geq 2$, $d(0, 1) = d(1, 2) = d(1, 0) = d(2, 1) = 1$, $d(0, 0) = d(1, 1) = d(2, 2) = 0.$ Then

for all $a, b, c\in X$. If $m > 2,$ the ordinary triangle inequality does not hold.

In ^[33], the authors showed the generality of Example 1.2.

Example 1.4. ^[33] Let $(X, \rho)$ be a metric space, and $d(a, b) = (\rho(a, b))^p$, where $p > 1$ is a real number. Then $(X, d, s)$ is a $b$-metric space with $s = 2^{p-1}$.

In general, a $b$-metric function $d$ for $s > 1$ is not jointly continuous in all of its two variables. Following is an example of a $b$-metric which is not continuous.

Example 1.5. ^[34] Let $X = \mathbb{N}\cup\{\infty\}$ and $d:X \times X \to \mathbb{R}^+$ be defined by

Then it is easy to see that for all $m, n, p \in X$, we have

Thus, $(X, d, 3)$ is a $b$-metric space. If $a_n = 2n$, for each $n\in \mathbb{N}$, then

that is, $a_n \to \infty,$ but $d(x_{n}, 1) = 2\nrightarrow d(\infty, 1)$, as $n \to \infty$.

Definition 1.6. ^[35] Let $(X, d, s)$ be a $b$-metric space with parameter $s\geq 1$. Then a sequence $\{a_n\}$ in $X$ is said to be:

(i) $b$-convergent if and only if there exists $a\in X$ such that $d(a_n, a)\rightarrow 0$ as $n\rightarrow \infty$;

(ii) a Cauchy sequence if and only if $d(a_n, a_m)\rightarrow 0$ when $n, m\rightarrow \infty$.

As usual, a $b$-metric space is called complete if and only if each Cauchy sequence in this space is $b$-convergent.

As $b$-metric is not continuous in general, so we need the following simple lemma about the $b$-convergent sequences.

Lemma 1.7. ^[33] Let $(X, d, s)$ be a $b$-metric space with parameter $s\geq 1$. Assume that $\{a_n\}$ and $\{b_n\}$ are $b$-convergent to $a$ and $b$, respectively. Then we have

In particular, if $a = b$, then we have $\lim_{n\rightarrow \infty} d({a_n}, {b_n}) = 0$. Moreover, for each $c\in X$, we have

Definition 1.8. ^[36] Let $M$ and $N$ be two self-mappings on a nonempty set $X$. If $a^* = Ma = Na$, for some $a\in X,$ then $a$ is said to be the coincidence point of $M$ and $N$, where $a^*$ is called the point of coincidence of $M$ and $N$. Let $C(M, N)$ denote the set of all coincidence points of $M$ and $N$.

Definition 1.9. ^[36] Let $M$ and $N$ be two self-mappings defined on a nonempty set $X$. Then $M$ and $N$ are said to be weakly compatible if they commute at every coincidence point, that is, $Ma = Na\Rightarrow MNa = NMa$ for every $a\in C(M, N)$.

Inspired the concept $\alpha$ admissible mapping introduced by ^[2,32], Popescu ^[37] gave the following definition:

Definition 1.10. ^[37] Let $M:X\rightarrow X$ and $\alpha:X\times X\rightarrow \mathbb{R}$ be given mappings. We say that $M$ is $\alpha$-orbital admissible if for all $a\in X$, we have

Definition 1.11. ^[37] Let $M:X\rightarrow X$ and $\alpha:X\times X\rightarrow \mathbb{R}$ be given mappings. A mapping $M$ is called a triangular $\alpha$-orbital admissible mapping if

(i) $M$ is $\alpha$-orbital admissible;

(ii) $\alpha(a, b)\geq1 \text{ and }\alpha(b, Mb)\geq1\Rightarrow \alpha(a, Mb)\geq1, a, b\in X$.

Definition 1.12. ^[1] Let $(X, d)$ be a complete metric space. An operator $M:X\rightarrow X$ is called a Geraghty contraction if there exists a function $\varphi:[0, \infty)\rightarrow [0, 1)$ which satisfies the following condition

such that

In the following, we denote the class of functions $\varphi:[0, \infty)\rightarrow [0, 1)$ which satisfies the condition (1.1) by $\Phi$.

Definition 1.13. ^[12] Suppose that $M$ and $N$ are two self-mappings on a metric space $(X, d)$. Suppose that there is a $\varphi\in\Phi$ such that the inequality

is satisfied, where

Then, we say that the mappings $M$ and $N$ satisfy the Geraghty contraction of type $E_{M, N}$.

Now, for $s\geq1$, let $\Phi_{s}$ denotes the family of functions $\varphi:[0, \infty)\rightarrow [0, \frac{1}{s})$ satisfying the condition:

Definition 1.14. ^[11] Let $(X, d)$ be a metric space and $\alpha:X\times X\rightarrow R$ be a function. A mapping $M:X\rightarrow X$ is said to be an $\alpha$-$\varphi_E$-Geraghty contraction mapping if there exists $\varphi\in\Phi_s$ such that

for all $a, b\in X$, where

2.   Main results

In this section, we introduce some new definitions and concepts and prove some new common fixed point theorems and coincidence point theorems in a $b$-metric space which is not required the continuity of $b$-metric. Meanwhile, we provide two examples to support our results.

In the following, we assume that $i, j$ are two arbitrary positive integers unless otherwise state.

Definition 2.1. Let $(X, d, s)$ be a $b$-metric space with parameter $s\geq1$ and $\alpha_{i, j}:X\times X\rightarrow [0, \infty)$ be a function. Two mappings $M, N:X\rightarrow X$ are called $\alpha_{i, j}$-$\varphi_{{E}_{M, N}}$-Geraghty contraction mappings if there exists a function $\varphi\in\Phi_s$ such that

where

and $p\geq2$ is a constant.

Remark 2.2.

(i) If $s = 1, \alpha_{i, j}(a, b) = 1\text{ and }i = j = 1$, we can get the Geraghty contraction of type $E_{M, N}$.

(ii) If $M = N\text{ and }i = j = 1$, we can get an $\alpha$-$\varphi_E$-Geraghty contraction.

Definition 2.3. Let $(X, d, s)$ be a $b$-metric space with parameter $s\geq1$ and $\alpha_{i, j}:X\times X\rightarrow [0, \infty)$ be a function. The self-mappings $M, N:X\rightarrow X$ are said to be $\alpha_{i, j}$-orbital admissible, if the following conditions hold:

for $a\in X$, where $p\geq2$ is a constant.

Definition 2.4. Let $(X, d, s)$ be a $b$-metric space with parameter $s\geq1$ and $\alpha_{i, j}:X\times X\rightarrow [0, \infty)$ be a function. Let $M, N:X\rightarrow X$ be two given self-mappings. The pair $(M, N)$ is said to be triangular $\alpha_{i, j}$-orbital admissible, if

(i) $M, N$ are $\alpha_{i, j}$-orbital admissible;

(ii) $\alpha_{i, j}(a, b)\geq s^p, \alpha_{i, j}(b, M^{i}b)\geq s^p$ and $\alpha_{i, j}(b, N^{j}b)\geq s^p$ implies $\alpha_{i, j}(a, M^{i}b)\geq s^p$ and $\alpha_{i, j}(a, N^{j}b)\geq s^p$, where $p\geq2$ is a constant.

Lemma 2.5. Let $(X, d, s)$ be a complete $b$-metric space with parameter $s\geq1$. Let $M, N:X\rightarrow X$ be two self-mappings such that the pair $(M, N)$ is triangular $\alpha_{i, j}$-orbital admissible. Assume that there exists $a_0 \in X$ such that $\alpha_{i, j}(a_0, M^{i}a_0)\geq s^p$. Define a sequence $\{a_n\}$ in $X$ by $a_{2n} = N^{j}a_{2n-1}, a_{2n+1} = M^{i}a_{2n}$ where $n = 0, 1, 2, \cdots$. Then for $n, m \in \mathbb{N}\cup \{0\}$ with $m > n$, we have $\alpha_{i, j}(a_n, a_m)\geq s^p$.

Proof. Since $\alpha_{i, j}(a_0, M^{i}a_0) = \alpha_{i, j}(a_0, a_1)\geq s^p$ and $(M, N)$ is triangular $\alpha_{i, j}$-orbital admissible, we obtain

Applying the above argument repeatedly, it follows that $\alpha_{i, j}(a_n, a_{n+1})\geq s^p$ for all $n \in \mathbb{N}\cup \{0\}$. Since $(M, N)$ is triangular $\alpha_{i, j}$-orbital admissible, $\alpha_{i, j}(a_n, a_m)\geq s^p$ for all $n, m \in \mathbb{N}\cup \{0\}$ with $m > n$.

Theorem 2.6. Let $(X, d, s)$ be a complete $b$-metric space with parameter $s\geq 1$, $\alpha_{i, j}:X\times X\rightarrow [0, \infty)$ be a symmetrical function and $M, N:X\rightarrow X$ be two given mappings. Suppose that the following conditions are satisfied:

(i) The pair $(M, N)$ is triangular $\alpha_{i, j}$-orbital admissible;

(ii) $M, N$ are $\alpha_{i, j}$-$\varphi_{{E}_{M, N}}$-Geraghty contraction mappings;

(iii) there is $a_0\in X$ satisfying $\alpha_{i, j}(a_0, M^{i}a_0)\geq s^p$;

(iv) if $\{a_{n}\}$ is a sequence in $X$ such that $\alpha_{i, j}(a_n, a_{n+1})\geq s^p$ for all $n\in\mathbb{N}$ and $a_{n}\rightarrow a$ as $n\rightarrow \infty$, then there exists a subsequence $\{a_{n_{k}}\}$ of $\{a_{n}\}$ such that $\alpha_{i, j}(a_{n_{k}}, a)\geq s^p$ for all $k\in\mathbb{N}$;

(v) for all $a, b\in Fix(M^i)$ or $Fix(N^j)$, we have $\alpha_{i, j}(a, b)\geq s^p$, where $Fix(M^i)$ denotes the set of fixed points of $M^i$.

Then $M$ and $N$ have a unique common fixed point.

Proof. Let $a_0 \in X$ satisfy $\alpha_{i, j}(a_0, M^{i}a_0)\geq s^p$. Define a sequence $\{a_n\}$ in $X$ by $a_{2n} = N^ja_{2n-1},$ $a_{2n+1} = M^ia_{2n}$ for $n = 0, 1, 2, \cdots$. Suppose there exists a $n_0\in \mathbb{N}$ such that $a_{n_0} = a_{n_0+1}$. We consider two cases:

(i) $n_0$ is odd. We have $a_{n_0+1} = N^ja_{n_0} = a_{n_0}$, that is, $a_{n_0}$ is a fixed point of $N^j$. Next, we will prove that $a_{n_0} = a_{n_0+1} = N^ja_{n_0} = M^{i}a_{n_0+1}$. Considering $N^ja_{n_0}\neq M^{i}a_{n_0+1}$ and according to Lemma 2.5, we get $\alpha_{i, j}(a_{n_0}, a_{n_0+1}) = \alpha_{i, j}(a_{n_0+1}, a_{n_0})\geq s^p$ and

where

It follows that

which is a contradiction. Hence, $d(N^ja_{n_0}, M^{i}a_{n_0+1}) = 0$ and $a_{n_0} = a_{n_0+1} = N^ja_{n_0} = M^{i}a_{n_0+1}$. Then $a_{n_0}$ is a fixed point of $M^{i}$, that is, $a_{n_0}$ is a common fixed point of $M^{i}$ and $N^j$.

(ii) $n_0$ is even. We have $a_{n_0+1} = M^ia_{n_0} = a_{n_0}$, that is, $a_{n_0}$ is a fixed point of $M^i$. By the same way, we obtain that $a_{n_0}$ is a common fixed point of $M^{i}$ and $N^j$.

Consequently, throughout the proof, we assume $a_{n}\neq a_{n+1}$ for all $n\geq0$. We consider the following cases:

Case$(1)$. In $(2.1)$, let $a = a_{2n}$ and $b = a_{2n-1}$. Then we have $\alpha_{i, j}(a_{2n}, a_{2n-1})\geq s^p$. Hence, we get

where

If $d(a_{2n}, a_{2n+1})\geq d(a_{2n-1}, a_{2n})$, we have $E(a_{2n}, a_{2n-1}) = d(a_{2n}, a_{2n+1})$ and

which is a contradiction. Therefore, $d(a_{2n}, a_{2n+1}) < d(a_{2n-1}, a_{2n})$.

Case$(2)$. In $(2.1)$, let $a = a_{2n}$ and $b = a_{2n+1}$. Then we get $\alpha_{i, j}(a_{2n}, a_{2n+1})\geq s^p$. It follows that

where

If $d(a_{2n+1}, a_{2n+2})\geq d(a_{2n}, a_{2n+1})$, we have $E(a_{2n}, a_{2n+1}) = d(a_{2n+1}, a_{2n+2})$ and

which is a contradiction. Therefore, $d(a_{2n+1}, a_{2n+2}) < d(a_{2n}, a_{2n+1})$.

To sum up, $\{d(a_n, a_{n+1})\}$ is non-increasing. Thus, there exists a $\gamma$ such that

We claim that $\gamma = 0$. In fact, by taking limits as $n\rightarrow \infty$ in (2.3), we get

Letting $n\rightarrow \infty$ in (2.2) and combining the above equality, we obtain

As a result, we get

It follows that

which implies $\lim_{n\rightarrow \infty}E(a_{2n}, a_{2n-1}) = \gamma = 0$, that is,

Next, we shall prove that $\{a_n\}$ is a Cauchy sequence in $X.$ To verify this, it is sufficient to prove that $\{a_{2n}\}$ is Cauchy. Instead, we assume that there exists $\ell > 0$ for which one can find subsequences $\{a_{2m_{k}}\}$ and $\{a_{2n_{k}}\}$ of $\{a_{2n}\}$ satisfying $m_k$ is the smallest index for which $m_k > n_k > k$, and

In view of the triangle inequality, one can deduce that

So, we have

From Lemma 2.5, we have $\alpha_{i, j}(a_{2m_{k}-1}, a_{2n_{k}})\geq s^p$ and

where

So there is

Taking $k\rightarrow \infty$ in inequality (2.5) and combining (2.4), we obtain

At the same time, we get

Therefore

Similarly, we have

and

It follows that

Combining (2.6) and (2.7), we get

which is a contradiction. Hence, $\{a_n\}$ is a Cauchy sequence. The completeness of $(X, d, s)$ ensures that there exists an $a^*$ in $X$ such that

Then from condition (iv), we can deduce that there exists a subsequence $\{a_{2n_k}\}$ of $\{a_n\}$ such that $\alpha_{i, j}(a_{2n_k}, a^*)\geq s^p$ and

where

and

In view of the triangle inequality, we have

Combining (2.8) and (2.9) and letting $k\rightarrow \infty$, we conclude

Thus, we obtain

that is,

In the same method, we get $d(a^*, M^ia^*) = 0$. Therefore, $a^*$ is a common fixed point of $M^i$ and $N^j$.

Next, we prove the uniqueness of the common fixed point. Suppose there exists another $b^*\in X$ such that $b^* = M^ib^*$. We have $\alpha_{i, j}(b^*, a^*)\geq s^p$ by condition (v). Consequently, we get

where

Hence, we have $d(b^*, a^*) < \frac{1}{s}d(b^*, a^*)$ which is contradictory. Then, we can obtain $a^*$ which is a unique fixed point of $M^i$. By the similar method, one can obtain $a^*$ is a unique fixed point of $N^j$. Since

we find $a^*$ is a common fixed point of $M$ and $N$ because of the uniqueness. It's easy to prove that $a^*$ is a unique common fixed point of $M$ and $N$.

Example 2.7. Let $X = [0, 1]$ and $d:X\times X\rightarrow \mathbb{R}^+$ be defined as $d(a, b) = |a-b|^2$. It is clear that $(X, d, s)$ forms a $b$-metric space with $s = 2$. Put $p = 2, i = 4, j = 2$ and

Let $M, N:X\rightarrow X$ be defined by $M(a) = \frac{a}{2}, N(a) = \frac{a}{4}$ and take $\varphi(t) = \frac{1}{64}$ for all $t > 0$. According to the definition of $M$ and $N$, it is easy to get that the pair $(M, N)$ is triangular $\alpha_{i, j}$-orbital admissible. Next, we prove that $M, N$ are $\alpha_{i, j}$-$\varphi_{{E}_{M, N}}$-Geraghty contraction mappings. Indeed, we have

Hence, we obtain

which satisfies $(2.1)$. In conclusion, for any $a, b\in X$, all the presumptions of Theorem 2.6 are satisfied. It follows that $M$ and $N$ have exactly one common fixed point in $X$. It is obvious that $0$ is the unique common fixed point of $M$ and $N$.

If $(X, d)$ is a metric space and let $p = 1, i = j = 1,$ and $\alpha_{i, j} = s^p = 1$ in Theorem 2.6, we obtain Theorem 5 in ^[12] immediately:

Corollary 2.8. Let $(X, d)$ be a complete metric space, and $M, N:X\rightarrow X$ be two given mappings. If the pair $(M, N)$ forms a Geraghty contraction of type $E_{M, N}$, then the pair of mappings $M, N$ has a unique common fixed point.

If $p = 1, M = N$ and $i = j = 1$ in Theorem 2.6, we obtain Theorem 2.2 in ^[11] immediately :

Corollary 2.9. Let $(X, d, s)$ be a complete $b$-metric space with parameter $s\geq 1$, $\alpha:X\times X\rightarrow [0, \infty)$ be a function and $M:X\rightarrow X$ be a given mapping. Suppose that the following conditions are satisfied:

(i) $M$ is triangular $\alpha-$orbital admissible;

(ii) $M$ is an $\alpha-\varphi_{E}-$Geraghty contraction mapping;

(iii) there is $a_0\in X$ satisfying $\alpha(a_0, Ma_0)\geq 1$;

(iv) if $\{a_{n}\}$ is a sequence in $X$ such that $\alpha(a_n, a_{n+1})\geq 1$ for all $n\in\mathbb{N}$ and $a_{n}\rightarrow a$ as $n\rightarrow \infty$, then there exists a subsequence $\{a_{n_{k}}\}$ of $\{a_{n}\}$ such that $\alpha(a_{n_{k}}, a)\geq 1$ for all $k\in\mathbb{N}$.

Then $M$ has a fixed point $a^*\in X$.

Definition 2.10. Let $M$ and $N$ be two self-mappings defined on a nonempty set. Then, $M$ and $N$ are said to be $(i, j)$-weakly compatible if $M^ia = N^ja\Rightarrow MN^ja = NM^ia$ for every $a\in C(M, N)$.

Remark 2.11. For $i = j = 1$, the definition reduces to the definition of weakly compatible.

Definition 2.12. Let $(X, d, s)$ be a $b-$metric space with parameter $s\geq1$, and let $M, N:X\rightarrow X$ and $\alpha_{i, j}:X\times X\rightarrow [0, \infty)$ be given mappings satisfying

for all $a, b, c\in X$, where $p\geq1$ is an arbitrary constant. The mapping $M$ is said to be $N$-$\alpha_{i, j}$-admissible if, for all $a, b\in X$,

Remark 2.13.

(i) For $i = j = 1$, the definition reduces to an $N$-$\alpha_{s^p}$-admissible mapping in a $b$-metric space.

(ii) For $i = j = 1$ and $N = I$, the definition reduces to an $\alpha_{s^p}$-admissible mapping in a $b$-metric space.

(iii) For $s = p = i = j = 1$ and $N = I$, the definition reduces to the definition of an $\alpha$-admissible mapping in a metric space.

Lemma 2.14. Let $(X, d, s)$ be a $b$-metric space with parameter $s\geq1$. Let $M:X\rightarrow X$ be a self-mapping such that $M$ is $N$-$\alpha_{i.j}$-admissible. Assume that there exists $a_0 \in X$ such that $\alpha_{i, j}(N^ja_0, M^{i}a_0)\geq s^p$. Define sequences $\{a_n\}$ and $\{b_n\}$ in $X$ by $b_{n} = M^ia_{n} = N^ja_{n+1}$ where $n = 0, 1, 2, \cdots$. Then for $n, m \in \mathbb{N}\cup \{0\}$ with $m > n$, we have $\alpha_{i, j}(b_n, b_m)\geq s^p$.

Proof. Since $M$ is $N$-$\alpha_{i, j}$-admissible, then we have

Eventually, we get

Hence, we get $\alpha_{i, j}(b_{n}, b_{n+1})\geq s^p$ for all $n\in \mathbb{N}$. Since $\alpha_{i, j}(b_{n}, b_{n+1})\geq s^p$ and $\alpha_{i, j}(b_{n+1}, b_{n+2})\geq s^p$, we deduce $\alpha_{i, j}(b_{n}, b_{n+2})\geq s^p$. It follows that one can get $\alpha_{i, j}(b_n, b_m)\geq s^p$ for all $n, m \in \mathbb{N}\cup \{0\}$ with $m > n$.

Definition 2.15. Let $(X, d, s)$ be a $b$-metric space with parameter $s\geq1$, and let $M, N:X\rightarrow X$ be two self-mappings and $\alpha_{i, j}:X\times X\rightarrow [0, \infty)$ be a function. A mapping $M$ is called a $\alpha_{i, j}$-$\varphi_{E_N}$-Geraghty contraction mapping, if there exists a function $\varphi\in \Phi_{s}$ satisfying the following:

where

Theorem 2.16. Let $(X, d, s)$ be a complete $b$-metric space with parameter $s\geq 1$ and $\alpha_{i, j}:X\times X\rightarrow [0, \infty)$ be a function. Let $M, N:X\rightarrow X$ be two given self-mappings satisfying $M^i(X)\subset N^j(X)$ and $N^j(X)$ is closed. Suppose that the following conditions are satisfied:

(i) $M$ is $N$-$\alpha_{i, j}$-admissible;

(ii) $M$ is a $\alpha_{i, j}$-$\varphi_{E_N}$-Geraghty contraction mapping;

(iii) there is $a_0\in X$ satisfying $\alpha_{i, j}(N^ja_0, M^{i}a_0)\geq s^p$;

(iv) if $\{a_n\}$ is a sequence in $X$ such that $N^ja_n\rightarrow N^ja$ as $n\rightarrow \infty$, then there exists a subsequence $\{N^ja_{n_k}\}$ of $\{N^ja_{n}\}$ such that $\alpha_{i, j}(N^ja_{n_k}, N^ja)\geq s^p$ for all $k\in \mathbb{N}$;

(v) $M$ and $N$ are $(i, j)$-weakly compatible.

Then $M$ and $N$ have a coincidence point in $X$.

Proof. According to condition $(iii)$, there exists an $a_0\in X$ such that $\alpha_{i, j}(N^ja_0, M^{i}a_0)\geq s^p$. Now, we define sequences $\{a_n\}$ and $\{b_n\}$ in $X$ by $b_n = M^ia_n = N^ja_{n+1}$ for all $n\in\mathbb{ N}$. If $b_n = b_{n+1}$ for some $n\in\mathbb{ N}$, then we deduce $N^ja_{n+1} = b_n = b_{n+1} = M^ia_{n+1}$ and $M^i$ and $N^j$ have a coincidence point $a_{n+1}$. Since $M$ and $N$ are $(i, j)$-weakly compatible mappings, then we get

Therefore,

that is, $b_n$ is a coincidence point of $M$ and $N$.

Without loss of generality, we assume that $b_n\neq b_{n+1}$ for all $n\in\mathbb{ N}$. It follows from Lemma 2.14 that $\alpha_{i, j}(b_{n-1}, b_{n}) = \alpha_{i, j}(N^ja_{n}, N^ja_{n+1})\geq s^p$. In light of condition $(ii)$, we obtain

where

If $d(b_{n+1}, b_{n})\geq d(b_n, b_{n-1})$, we have $E(a_n, a_{n+1}) = d(b_n, b_{n+1}) > 0$, which means

a contradiction. Hence, $d(b_{n+1}, b_{n}) < d(b_n, b_{n-1})$, that is, $\{d(b_{n}, b_{n+1})\}$ is a non-increasing sequence and so there exists $\gamma\geq0$ such that

In view of (2.12), we obtain

Substituting (2.13) into (2.11) and taking the limit as $n\rightarrow \infty$, we get

The above formula means

So there is

Next, we aim to prove that $\{b_n\}$ is a Cauchy sequence. Suppose on the contrary, there exists $\ell > 0$ for which one can find sequences $\{b_{m_k}\}$ and $\{b_{n_k}\}$ of $\{b_n\}$ satisfying $n_k$ is the smallest index for which $n_k > m_k > k$, and

Since $\alpha_{i, j}(b_{n_{k}-1}, b_{m_{k}-1}) = \alpha_{i, j}(N^ja_{n_k}, N^ja_{m_k})\geq s^p$, we have

where

Passing to the limit as $k\rightarrow \infty$ in the above inequality and (2.14), we obtain

and

We deduce

Therefore,

which is a contradiction. Hence, $\{b_n\}$ is a Cauchy sequence. The completeness of $(X, d, s)$ ensures that there exists a $b^*\in X$ such that $b_n\rightarrow b^*$ as $n\rightarrow \infty$, that is,

Since $N^j(A)$ is closed, we have $b^*\in N^j(A)$. It follows that one can choose an $a^*\in X$ such that $b^* = N^ja^*$, and we can write

By using the condition $(iv)$, we get that there exists a subsequence $\{b_{n_k}\}$ of $\{b_n\}$ so that $\alpha_{i, j}(b_{n_{k}-1}, N^ja^*) = \alpha_{i, j}(N^ja_{n_k}, N^ja^*)\geq s^p$. Applying contractive condition (2.10), we have

where

and

Since $d(M^ia^*, N^ja^*)\leq sd(M^ia^*, M^ia_{n_k})+sd(M^ia_{n_k}, N^ja^*)$, then

Taking the limit as $k\rightarrow \infty$ in the above inequality, we obtain

As a result, we get

Thus, $b^* = M^ia^* = N^ja^*$ is a point of coincidence for $M^i$ and $N^j$. Since $M$ and $N$ are $(i, j)$-weakly compatible mappings, then we get

Therefore

that is, $b^*$ is a coincidence point of $M$ and $N$.

Example 2.17. Let $X = \{0, 1, 2\}$ and defined $d:X\times X\rightarrow \mathbb{R}$ by $d(a, b) = |a-b|^2$ for all $a, b\in X$. Therefore, $(X, d, s)$ is a $b$-metric space with $s = 2$. Put $p = 2, i = 2, j = 3$ and

Let $M, N:X\rightarrow X$ be two mappings defined by $M(0) = 1, M(1) = 1, M(2) = 0$, $N(0) = 2, N(1) = 1,$ and $N(2) = 0$. Take $\varphi(t) = \frac{1}{2s}$ for all $t\geq0$.

It is obvious that $M$ is $N$-$\alpha_{s^p}$-admissible. Meanwhile, $M$ is a $\alpha_{i, j}$-$\varphi_{E_N}$-Geraghty contraction mapping. Indeed, for $a = b$, we get $\alpha_{i, j}(N^ja, N^jb)d(M^{i}a, M^{i}b)\leq\varphi(E(a, b))E(a, b)$. On the other hand, we have

So, we obtain the following cases:

(a) $a = 0$ and $b = 1$. Then,

(b) $a = 0$ and $b = 2$. Then,

(c) $a = 1$ and $b = 2$. Then,

Considering the symmetry of $d$, we have (3.10) is true for all $a, b\in X$ and

which means $M$ and $N$ are $(i, j)$-weakly compatible. All hypotheses of Theorem 2.16 are satisfied. So, $M$ and $N$ have a coincidence point in $X$. Obviously, $1$ is the coincidence point of $M$ and $N$.

If $(X, d)$ is a metric space and let $i = j = 1, p = 1, N = I$ in Theorem 2.16, we obtain the following result immediately:

Corollary 2.18. Let $(X, d)$ be a complete metric space and $\alpha:X\times X\rightarrow [0, \infty)$ be a function. Let $M:X\rightarrow X$ be a given self-mapping. Suppose that the following conditions are satisfied:

(i) $M$ is $\alpha$-admissible;

(ii) $M$ is a $\alpha$-$\varphi_{E_N}$-Geraghty contraction mapping;

(iii) there is $a_0\in X$ with satisfying $\alpha(Na_0, Ma_0)\geq 1$;

(iv) if $\{a_n\}$ is a sequence in $X$ such that $a_n\rightarrow a$ as $n\rightarrow \infty$, then there exists a subsequence $\{a_{n_k}\}$ of $\{a_{n}\}$ such that $\alpha(a_{n_k}, a)\geq 1$ for all $k\in \mathbb{N}$;

Then $M$ has a fixed point in $X$.

If $i = j = 1$ in Theorem 2.16, we have the following result:

Corollary 2.19. Let $(X, d, s)$ be a complete $b$-metric space with parameter $s\geq 1$ and $\alpha:X\times X\rightarrow [0, \infty)$ be a function. Let $M, N:X\rightarrow X$ be two given self-mappings satisfying $M(X)\subset N(X)$ and $N(X)$ is closed. Suppose that the following conditions are satisfied:

(i) $M$ is $N$-$\alpha_{s^p}$-admissible;

(ii) $M$ is a $\alpha$-$\varphi_{E_N}$-Geraghty contraction mapping;

(iii) there is $a_0\in X$ satisfying $\alpha(Na_0, Ma_0)\geq s^p$;

(iv) if $\{a_n\}$ is a sequence in $X$ such that $Na_n\rightarrow Na$ as $n\rightarrow \infty$, then there exists a subsequence $\{Na_{n_k}\}$ of $\{Na_{n}\}$ such that $\alpha(Na_{n_k}, Na)\geq s^p$ for all $k\in \mathbb{N}$;

(v) $M$ and $N$ are weakly compatible.

Then $M$ and $N$ have a coincidence point in $X$.

3.   Conclusions

In this manuscript, we introduced two new classes of Geraghty contraction mappings and established common fixed point and coincidence point results involving these new classes of mappings in the framework of $b$-metric spaces. Further, we provided two examples that elaborated on the usability of our results.

Acknowledgments

This work was financially supported by the Science and Research Project Foundation of Liaoning Province Education Department (No:LJC202003).

Conflict of interest

The authors declare that they have no conflicts of interest regarding the publication of this paper.