Some New Observations on Geraghty and Ćirić Type Results in b-Metric Spaces

Abstract

We discuss recent fixed point results in b-metric spaces given by Pant and Panicker (2016). Our results are with shorter proofs. In addition, for $\epsilon \in (1,3]$, our results are genuine generalizations of ones from Pant and Panicker.

1. Introduction

We start with the following.

Definition 1

([1,2]). Let f be a self-mapping on a metric space $(X,d)$. For $\mu \in X$, take

$$O\left(\mu ,n\right)=\left\{\mu ,f\mu ,\dots ,f^n\mu \right\}andO\left(\mu ,\infty \right)=\left\{\mu ,f\mu ,\dots ,f^n\mu ,\dots \right\},$$

where $n\in \mathbb{N}$. The set $O\left(\mu ,\infty \right)$ is called an orbit of f. Such $\left(X,d\right)$ is said to be f-orbitally complete if each Cauchy sequence in $O\left(\mu ,\infty \right)$ converges in $\left(X,d\right)$.

It is well known that every complete metric space is f-orbitally complete for each self-mapping f on X. Its converse does not hold (see [1,2]).

Two very known and important generalizations of the Banach contraction principle [3] obtained by Ćirić [1] and Geraghty [4] as follows:

Theorem 1

([1]). Let $\left(X,d\right)$ be an f-orbitally complete metric space and $f:X\to X$ be a quasi-contraction, i.e., there is $\lambda \in [0,1)$ so that

$$d\left(f\mu ,f\tau \right)\le \lambda ·max\left\{d\left(\mu ,\tau \right),d\left(\mu ,f\mu \right),d\left(\tau ,f\tau \right),d\left(\mu ,f\tau \right),d\left(f\mu ,\tau \right)\right\},$$

(1)

for all $\mu ,\tau \in X$. Then, f possesses a unique fixed point.

Theorem 2

([4]). Let $\left(X,d\right)$ be a complete metric space and $f:X\to X$ be such that

$$d\left(f\mu ,f\tau \right)\le \beta \left(d\left(\mu ,\tau \right)\right)d\left(\mu ,\tau \right),$$

(2)

for all $\mu ,\tau \in X,$ where $\beta :[0,\infty )\to [0,1)$ is such that $\beta \left(t_n\right)\to 1$ implies $t_n\to 0$ as $n\to \infty .$ Then, f has a unique fixed point.

The concept of quasi-contractions has been generalized by Kumam et al. [2].

Definition 2

([2]). A self-mapping f on a metric space $\left(X,d\right)$ is called a generalized quasi-contraction if there is $\lambda \in [0,1)$ so that

$$d\left(T\mu ,T\tau \right)\le \lambda ·M\left(\mu ,\tau \right),$$

(3)

for all $\mu ,\tau \in X,$ where

$$M\left(\mu ,\tau \right)=max\left\{d(\mu ,\tau ),d(\mu ,f\mu ),d(\tau ,f\tau ),d(\mu ,f\tau ),d(f\mu ,\tau ),d(f^2\mu ,\mu ),d(f^2\mu ,f\mu ),d(f^2\mu ,\tau ),d(f^2\mu ,f\tau )\right\}.$$

(4)

Theorem 3

([2]). Each generalized quasi-contraction self-mapping f on an f-orbitally complete metric space admits a unique fixed point.

Given $s\ge 1$. A function $d:X\times X\to [0,\infty )$ is called a b-metric on a non-empty set X if for all $\mu ,\tau ,\xi \in X$, $d\left(\mu ,\tau \right)=0$ iff $\mu =\tau ,d\left(\mu ,\tau \right)=d\left(\tau ,\mu \right)$ and $d\left(\mu ,\xi \right)\le s\left[d\left(\mu ,\tau \right)+d\left(\tau ,\xi \right)\right].$ The concept of b-convergence, b-completeness, b-Cauchyness in b-metric spaces can be found in [5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38].

Definition 3

([39]). Given $\mathsf{\Omega }:X\times X\to [0,\infty )$ and $f:X\to X$, such f is called Ω-admissible if, for all $\mu ,\tau \in X,$

$$\mathsf{\Omega }\left(\mu ,\tau \right)\ge 1implies\mathsf{\Omega }\left(f\mu ,f\tau \right)\ge 1.$$

Definition 4

([40]). The mapping $f:X\to X$ is called triangular Ω-admissible if for all $\mu ,\tau ,\xi \in X,$

(i)$\mathsf{\Omega }\left(\mu ,\tau \right)\ge 1$implies$\mathsf{\Omega }\left(f\mu ,f\tau \right)\ge 1;$

(ii)$\mathsf{\Omega }\left(\mu ,\xi \right)\ge 1$and$\mathsf{\Omega }\left(\xi ,\tau \right)\ge 1$implies$\mathsf{\Omega }\left(\mu ,\tau \right)\ge 1.$

Lemma 1

([40]). Let f be a triangular Ω-admissible mapping. Suppose there is ${\mu }_0\in X$ so that $\mathsf{\Omega }\left({\mu }_0,f{\mu }_0\right)\ge 1.$ Define $\left\{{\mu }_n\right\}$ by ${\mu }_n=f^n{\mu }_0$. Then, $\mathsf{\Omega }\left({\mu }_m,{\mu }_n\right)\ge 1$ for all $m,n\in N$ with $m<n$.

Very recently, Pant and Panciker [35] initiated the concept of generalized Ω-quasi-contraction in b-metric spaces. Namely, they defined and proved the following:

Definition 5

([35]). Let $\left(X,d\right)$ be a b-metric space with constant $s\ge 1.$ The self-mapping f on X is called a generalized Ω-quasi-contraction if there are Ω: $X\times X\to [0,\infty )$ and a real number q with $0<q<\frac{1}{s^2}$ such that

$$\mathsf{\Omega }\left(\mu ,\tau \right)d\left(f\mu ,f\tau \right)\le qM\left(\mu ,\tau \right),$$

(5)

where $M\left(\mu ,\tau \right)$ is given by (4).

Lemma 2

([35]). Let $\left(X,d\right)$ be a b-metric space with $s\ge 1$ and $f:X\to X$ be a generalized Ω-quasi contraction such that

(A): f is triangular Ω-admissible;

(B): there is${\mu }_0\in X$so that $\mathsf{\Omega }\left({\mu }_0,f{\mu }_0\right)\ge 1$.

Then, for all$p,k\in \left\{1,2,\dots ,n\right\}$with$\left(p<k\right)$, we have

$$d\left(f^p{\mu }_0,f^k{\mu }_0\right)\le q\delta \left[O\left({\mu }_0,n\right)\right].$$

(6)

Theorem 4

([35]). Let $\left(X,d\right)$ be a f-orbitally complete b-metric space with $s\ge 1$ and $f:X\to X$ be a generalized Ω-quasi-contraction so that (A) and (B) of Lemma 2 hold. Then, f admits a fixed point.

The proofs of Lemma 2 and Theorem 4 are based on Lemma 1.

Motivated by Ćirić [1], Geraghty [4], Kumam et al. [2], Samet et al. [39] as well as Pant and Panciker [35], we improve some related fixed point theorems in b-metric spaces. Our proofs are much shorter and nicer than the ones in [35].

Corollary 1.

Let$\left(X,d\right)$be a b-complete b-metric space with a constant$s\ge 1$. Given$\mathsf{\Omega }:X\times X\to [0,\infty )$a functional, let$f:X\to X$be an Ω-quasi-contraction, i.e.,

$$\mathsf{\Omega }\left(\mu ,\tau \right)d\left(f\mu ,f\tau \right)\le q·m\left(\mu ,\tau \right),$$

(7)

for all $\mu ,\tau \in X,$ where $0\le q<1$ and $m\left(\mu ,\tau \right)=max\left\{d\left(\mu ,\tau \right),d\left(\mu ,f\mu \right),d\left(\tau ,f\tau \right),d\left(\mu ,f\tau \right),d\left(f\mu ,\tau \right)\right\}$.

Suppose that

(i) f is Ω-admissible;

(ii) there is${\mu }_0\in X$so that$\mathsf{\Omega }\left({\mu }_0,f{\mu }_0\right)\ge 1.$

If$q<\frac{1}{s^2+s}$, then f admits a fixed point.

Chandok [41] defined the following.

Definition 6

([41]). Let $\left(X,d\right)$ be a b-metric space with constant $s\ge 1$, $f:X\to X$ and $\mathsf{\Omega },\omega :X\times X\to [0,\infty ).$ The mapping f is said to be $\left(\mathsf{\Omega },\omega \right)$-admissible if $\mathsf{\Omega }\left(\mu ,\tau \right)\ge 1$ and $\omega \left(\mu ,\tau \right)\ge 1$ implies $\mathsf{\Omega }\left(f\mu ,f\tau \right)\ge 1$ and $\omega \left(f\mu ,f\tau \right)\ge 1$ for all $\mu ,\tau \in X$.

Definition 7

([41]). Let $\mathsf{\Omega },\omega :X\times X\to [0,\infty )$. A b-metric space $\left(X,d\right)$ with a constant $s\ge 1$ is $\left(\mathsf{\Omega },\omega \right)$-regular if $\left\{{\mu }_n\right\}$ is a sequence in X such that ${\mu }_n\to x\in X,\mathsf{\Omega }\left({\mu }_n,{\mu }_{n+1}\right)\ge 1,\omega \left({\mu }_n,{\mu }_{n+1}\right)\ge 1$, for all n; then, there is a subsequence $\left\{{\mu }_{n_k}\right\}$ of $\left\{{\mu }_n\right\}$ such that $\mathsf{\Omega }\left({\mu }_{n_k},{\mu }_{n_k+1}\right)\ge 1,\omega \left({\mu }_{n_k},{\mu }_{n_k+1}\right)\ge 1$, for all $k\in \mathbb{N}$ and $\mathsf{\Omega }\left(\mu ,f\mu \right)\ge 1$, $\omega \left(\mu ,f\mu \right)\ge 1$.

The two following classes of functions are defined in [35].

(1) $\mathsf{\Theta }$ denotes the family of functions $\theta :[0,\infty )\to [0,1)$ so that, for any bounded sequence $\left\{t_n\right\}$ of positive reals, $\theta \left(t_n\right)\to 1$ implies $t_n\to 0;$

(2) $\mathsf{\Psi }$ denotes the set of functions $\psi :[0,\infty )\to [0,\infty )$ so that $\psi$ is continuous, strictly increasing and $\psi \left(0\right)=0$.

Definition 8

([35]). Let $\left(X,d\right)$ be a b-metric space with constant $s\ge 1$. The mapping $f:X\to X$ is called an $\left(\mathsf{\Omega },\omega \right)$-Geraghty type contraction if there are $\theta \in \mathbb{\Theta }$, $\psi \in \mathsf{\Psi }$ and $\mathsf{\Omega },\omega :X\times X\to [0,\infty )$ such that

$$\mathsf{\Omega }\left(\mu ,f\mu \right)\omega \left(\tau ,f\tau \right)\psi \left(s^{3}d\left(f\mu ,f\tau \right)\right)\le \theta \left(\psi \left(N\left(\mu ,\tau \right)\right)\right)\psi \left(N\left(\mu ,\tau \right)\right),$$

(8)

for all $\mu ,\tau \in X,$ where $N\left(\mu ,\tau \right)=max\left\{d\left(\mu ,\tau \right),d\left(\mu ,f\mu \right),d\left(\tau ,f\tau \right),\frac{d\left(\mu ,f\tau \right)+d\left(\tau ,f\mu \right)}{2s}\right\}$.

Theorem 5

([35]). Let $\left(X,d\right)$ be a b-complete b-metric space with a constant $s\ge 1$ and $f:X\to X$ be a self-mapping. Suppose that the following assertions hold:

(A): f is $\left(\mathsf{\Omega },\omega \right)$-admissible;

(B): f is an $\left(\mathsf{\Omega },\omega \right)$-Geraghty type contraction;

(C): there is ${\mu }_0\in X$ so that $\mathsf{\Omega }\left({\mu }_0,f{\mu }_0\right)\ge 1$ and$\omega \left({\mu }_0,f{\mu }_0\right)\ge 1;$

(D): either f is continuous, or $\left(X,d\right)$ is $\left(\mathsf{\Omega },\omega \right)$-regular.

Then, f has a unique fixed point.

Corollary 2

([35]). Let $\left(X,d\right)$ be a b-complete b-metric space with $s\ge 1$ and $f:X\to X$ be a self-mapping. Suppose that the following assertions hold:

(A): f is $\mathsf{\Omega }$-admissible;

(B): f is an $\mathsf{\Omega }$-Geraghty type contraction;

(C): there is ${\mu }_0\in X$ so that$\mathsf{\Omega }\left({\mu }_0,f{\mu }_0\right)\ge 1;$

(D): either f is continuous or $\left(X,d\right)$ is $\mathsf{\Omega }$-regular.

Then, f admits a unique fixed point.

The proofs of Theorem 5 (and Corollary 2) are based on the following crucial lemma.

Lemma 3

([8]). Let $\left(X,d\right)$ be a b-metric space with $s\ge 1$. Let $\left\{{\mu }_n\right\}$ and $\left\{{\tau }_n\right\}$ b-converge to $\mu ,\tau \in X$, respectively. We have

$$\frac{1}{s^2}d\left(\mu ,\tau \right)\le \underset{n\to \infty }{\lim \inf }d\left({\mu }_n,{\tau }_n\right)\le \underset{n\to \infty }{\lim \inf }d\left({\mu }_n,{\tau }_n\right)\le s^{2}d\left(\mu ,\tau \right).$$

(9)

In particular, if $\mu =\tau$, then ${\displaystyle \underset{n\to \infty }{lim}d\left({\mu }_n,{\tau }_n\right)=0}$. In addition, for any $\xi \in X$,

$$\frac{1}{s}d\left(\mu ,\xi \right)\le \underset{n\to \infty }{\lim \inf }d\left({\mu }_n,\xi \right)\le \underset{n\to \infty }{\lim \inf }d\left({\mu }_n,\xi \right)\le sd\left(\mu ,\xi \right).$$

(10)

2. Main Results

Instead of Lemma 3, we will use in this paper the next result to establish our main results.

Lemma 4.

([28], Lemma 3.1) Let$\left\{{\mu }_n\right\}$be a sequence in a b-metric space$\left(X,d,s\ge 1\right)$such that

$$d\left({\mu }_{n+1},{\mu }_{n+2}\right)\le \lambda d\left({\mu }_n,{\mu }_{n+1}\right),n\ge 0,$$

for some$\lambda \in [0,\frac{1}{s})$. Then,$\left\{{\mu }_n\right\}$is a b-Cauchy sequence in$X$.

By Lemma 4, it would be good to note that each Picard sequence is b-Cauchy, and so the proofs of some results (such as, the ones in the sequel) become shorter.

Remark 1.

In many results based on b-metric spaces with a constant$s\ge 1$, people often suppose that$\lambda \in [0,\frac{1}{s})$instead of$\lambda \in [0,1)$, which is clearly a stronger condition due to the fact that$[0,\frac{1}{s})\subseteq [0,1)$. To ensure this fact, the following inequality is utilized:

$$d\left({\mu }_m,{\mu }_n\right)\le sd\left({\mu }_m,{\mu }_{m+1}\right)+s^{2}d\left({\mu }_{m+1},{\mu }_{m+2}\right)+···+s^{n-m-1}d\left({\mu }_{n-2},{\mu }_{n-1}\right)+s^{n-m-1}d\left({\mu }_{n-1},{\mu }_n\right),$$

(11)

for$n,m\in N$and$n>m$.

Since there is a doubt in the proof of Theorem 3.5 of [35] (see [35], page 6, line 13: about the inequality: $d\left(Tu,T{T}^n{\mu }_0\right)\le \mathsf{\Omega }\left(u,T^n{\mu }_0\right)d\left(Tu,T{T}^n{\mu }_0\right)$), we give the following new result.

Theorem 6.

Let$\left(X,d\right)$be an f-orbitally b-complete b-metric space (with$s>1$) and let$f:X\to X$be a generalized Ω-quasi-contraction where (A) and (B) of Lemma 2 both hold. If either f is continuous, or $\left(X,d\right)$ is Ω-regular, then f possesses a fixed point.

Proof. 

By assumption, there is ${\mu }_0\in X$ so that $\mathsf{\Omega }\left({\mu }_0,f{\mu }_0\right)\ge 1$. We will show that the sequence ${\{{\mu }_n=f^n{\mu }_0\}}_{n\in \mathbb{N}}$ is b-Cauchy in the b-metric space $\left(X,d\right)$. Indeed, let ${\mu }_n\ne {\mu }_{n-1}$ for all $n\in \mathbb{N}$. According to (A) from Lemma 2, it follows that $\mathsf{\Omega }\left({\mu }_n,{\mu }_{n+1}\right)\ge 1$ for each $n\ge 0$. Therefore,

$$d\left({\mu }_{n+1},{\mu }_{n+2}\right)\le \mathsf{\Omega }\left({\mu }_n,{\mu }_{n+1}\right)d\left(f{\mu }_n,f{\mu }_{n+1}\right)\le qM\left({\mu }_n,{\mu }_{n+1}\right),$$

(12)

where

$$\begin{array}{ccc}\hfill M\left({\mu }_n,{\mu }_{n+1}\right)& =& max\left\{d\left({\mu }_n,{\mu }_{n+1}\right),d\left({\mu }_{n+1},{\mu }_{n+2}\right),d\left({\mu }_n,{\mu }_{n+2}\right)\right\}\hfill \\ & \le & max\left\{d\left({\mu }_n,{\mu }_{n+1}\right),d\left({\mu }_{n+1},{\mu }_{n+2}\right),s\left(d\left({\mu }_n,{\mu }_{n+1}\right)+d\left({\mu }_{n+1},{\mu }_{n+2}\right)\right)\right\}\hfill \\ & =& s\left(d\left({\mu }_n,{\mu }_{n+1}\right)+d\left({\mu }_{n+1},{\mu }_{n+2}\right)\right).\hfill \end{array}$$

Hence,

$$d\left({\mu }_{n+1},{\mu }_{n+2}\right)\le \frac{qs}{1-qs}d\left({\mu }_n,{\mu }_{n+1}\right)=\lambda d\left({\mu }_n,{\mu }_{n+1}\right),$$

(13)

where $\lambda =\frac{qs}{1-qs}<\frac{1}{s}$ because $q<\frac{1}{s^2}$.

Now, by ([28], Lemma 3.1), the sequence $\{{\mu }_n=f^n{\mu }_0\}$ is b-Cauchy. Since $\left(X,d\right)$ is f-orbitally b–complete, there is $u\in X$ so that ${\displaystyle \underset{n\to \infty }{lim}{f}^n{\mu }_0=u}$. If f is continuous, then obviously $fu=u$. When $\left(X,d\right)$ is Ω-regular, we have (because $\mathsf{\Omega }\left(u,{\mu }_n\right)\ge 1$)

$$\begin{array}{ccc}\hfill \frac{1}{s}d\left(u,fu\right)& \le & d\left(u,{\mu }_{n+1}\right)+d\left(f{\mu }_n,fu\right)\hfill \\ & \le & d\left(u,{\mu }_{n+1}\right)+\mathsf{\Omega }\left(u,{\mu }_n\right)d\left(f{\mu }_n,fu\right)\hfill \\ & \le & d\left(u,{\mu }_{n+1}\right)+qM\left({\mu }_n,u\right),\hfill \end{array}$$

where

$$\begin{array}{ccc}\hfill M\left({\mu }_n,u\right)& =& max\left\{d\left({\mu }_n,u\right),d\left({\mu }_n,{\mu }_{n+1}\right),d\left(u,fu\right),d\left({\mu }_n,fu\right),d\left(u,{\mu }_{n+1}\right),d\left({\mu }_{n+2},u\right),\right.\hfill \\ & & d\left.\left({\mu }_{n+2},{\mu }_{n+1}\right),d\left({\mu }_{n+2},u\right),d\left({\mu }_{n+2},fu\right)\right\}.\hfill \end{array}$$

Furthermore,

$$\begin{array}{ccc}\hfill M\left({\mu }_n,u\right)& \le & max\left\{d\left({\mu }_n,u\right),d\left({\mu }_n,{\mu }_{n+1}\right),d\left(u,fu\right),s\left(d\left({\mu }_n,u\right)+d\left(u,fu\right)\right),d\left(u,{\mu }_{n+1}\right),d\left({\mu }_{n+2},u\right),\right.\hfill \\ & & d\left.\left({\mu }_{n+2},{\mu }_{n+1}\right),d\left({\mu }_{n+2},u\right),s\left(d\left({\mu }_{n+2},u\right)+d\left(u,fu\right)\right)\right\}\hfill \\ & \to & max\left\{0,0,d\left(u,fu\right),sd\left(u,fu\right),0,0,0,0,sd\left(u,fu\right)\right\}\hfill \\ & =& sd\left(u,fu\right),\hfill \end{array}$$

as $n\to \infty$.

Hence,

$$\frac{1}{s}d\left(u,fu\right)\le sqd\left(u,fu\right)\mathrm{or}\mathrm{equivalently}\frac{1}{s^2}d\left(u,fu\right)\le qd\left(u,fu\right),$$

which is possible only if $fu=u$.  □

Remark 2.

If$q<\frac{1}{s^2}$, then our approach gives a short and nice proof that a generalized Ω-quasi-contraction $f:X\to X$ has a fixed point. However, the proof of the corresponding result in ([35], page 6, line 13+) is not correct without the assumption that the b–metric space $\left(X,d\right)$ is Ω-regular. This new compliment of the proof of Theorem 3.5 in [35] is correct if $q<\frac{1}{s^2}$.

We introduce the following.

Definition 9.

Let$\left(X,d,s>1\right)$be a b-metric space. The mapping$f:X\to X$is said to be an$\left(\mathsf{\Omega },\omega \right)$-type contraction if there are$\mathsf{\Omega },\omega :X\times X\to [0,\infty )$,$\epsilon >1$and$\psi \in \mathsf{\Psi }$such that

$$\mathsf{\Omega }\left(\mu ,f\mu \right)\omega \left(\tau ,f\tau \right)\psi \left(s^{\epsilon }d\left(f\mu ,f\tau \right)\right)\le \psi \left(N\left(\mu ,\tau \right)\right),$$

(14)

for all$\mu ,\tau \in X$, where$N\left(\mu ,\tau \right)=max\left\{d\left(\mu ,\tau \right),d\left(\mu ,f\mu \right),d\left(\tau ,f\tau \right),\frac{d\left(\mu ,f\tau \right)+d\left(\tau ,f\mu \right)}{2s}\right\}$.

Remark 3.

The contraction (14) generalizes the corresponding ones from ([35], Definition 4.3) in several directions.

Now, we can prove the next result.

Theorem 7.

Let$\left(X,d,s>1\right)$be a b-complete b-metric space,$f:X\to X$, and$\mathsf{\Omega },\omega :X\times X\to [0,\infty )$. Suppose that the following assertions hold:

(A) f is$\left(\mathsf{\Omega },\omega \right)$-admissible;

(B) f is an$\left(\mathsf{\Omega },\omega \right)$-contraction;

(C) there is${\mu }_0\in X$so that$\mathsf{\Omega }\left({\mu }_0,f{\mu }_0\right)\ge 1$and$\omega \left({\mu }_0,f{\mu }_0\right)\ge 1$;

(D) either f is continuous or$\left(X,d,s>1\right)$is$\left(\mathsf{\Omega },\omega \right)$-regular.

Then, f has a unique fixed point.

Proof. 

As in [35], page 5748, we get

$$\begin{array}{ccc}\hfill \psi \left(d\left({\mu }_{n+1},{\mu }_{n+2}\right)\right)& =& \psi \left(d\left(f{\mu }_n,f{\mu }_{n+1}\right)\right)\hfill \\ & \le & \psi \left(s^{\epsilon }d\left(f{\mu }_n,f{\mu }_{n+1}\right)\right)\hfill \\ & \le & \mathsf{\Omega }\left({\mu }_n,f{\mu }_n\right)\omega \left({\mu }_{n+1},f{\mu }_{n+1}\right)\psi \left(s^{\epsilon }d\left(f{\mu }_n,f{\mu }_{n+1}\right)\right)\hfill \\ & \le & \psi \left(N\left({\mu }_n,{\mu }_{n+1}\right)\right),\hfill \end{array}$$

or equivalently, $s^{\epsilon }d\left(f{\mu }_n,f{\mu }_{n+1}\right)\le N\left({\mu }_n,{\mu }_{n+1}\right)$, where

$$\begin{array}{ccc}\hfill N\left({\mu }_n,{\mu }_{n+1}\right)& =& max\left\{d\left({\mu }_n,{\mu }_{n+1}\right),d\left({\mu }_{n+1},{\mu }_{n+2}\right),\frac{d\left({\mu }_n,{\mu }_{n+2}\right)}{2s}\right\}\hfill \\ & \le & max\left\{d\left({\mu }_n,{\mu }_{n+1}\right),d\left({\mu }_{n+1},{\mu }_{n+2}\right),\frac{d\left({\mu }_n,{\mu }_{n+1}\right)+d\left({\mu }_{n+1},{\mu }_{n+2}\right)}{2}\right\}\hfill \\ & \le & max\left\{d\left({\mu }_n,{\mu }_{n+1}\right),d\left({\mu }_{n+1},{\mu }_{n+2}\right)\right\}\hfill \\ & \le & N\left({\mu }_n,{\mu }_{n+1}\right).\hfill \end{array}$$

Hence, $s^{\epsilon }d\left(f{\mu }_n,f{\mu }_{n+1}\right)\le max\left\{d\left({\mu }_n,{\mu }_{n+1}\right),d\left({\mu }_{n+1},{\mu }_{n+2}\right)\right\}$.  □

It is not hard to see that $s^{\epsilon }d\left(f{\mu }_n,f{\mu }_{n+1}\right)\le d\left({\mu }_n,{\mu }_{n+1}\right)$. That is,

$$d\left({\mu }_{n+1},{\mu }_{n+2}\right)\le \frac{1}{s^{\epsilon }}d\left({\mu }_n,{\mu }_{n+1}\right)=\lambda d\left({\mu }_n,{\mu }_{n+1}\right),$$

where $\lambda =\frac{1}{s^{\epsilon }}<\frac{1}{s}$.

As in the proof of Theorem 6, the sequence $\{{\mu }_n=f^n{\mu }_0\}$ is b-Cauchy in b-complete b-metric space, so there is $u\in X$ so that ${\mu }_n\to u$ as $n\to \infty$. In the case that f is continuous, one writes

$$u=\underset{n\to \infty }{lim}{\mu }_{n+1}=\underset{n\to \infty }{lim}f{\mu }_n=f\left(\underset{n\to \infty }{lim}{\mu }_n\right)=fu.$$

In the case that $\left(X,d\right)$ is $\left(\mathsf{\Omega },\omega \right)$-regular, there is $\left\{{\mu }_{n_k}\right\}$ of $\left\{{\mu }_n\right\}$ so that $\mathsf{\Omega }\left({\mu }_{n_k+1},{\mu }_{n_k}\right)\ge 1$ and $\omega \left({\mu }_{n_k+1},{\mu }_{n_k}\right)\ge 1$ for all $k\in N$ and $\mathsf{\Omega }\left(u,fu\right)\ge 1$ and $\omega \left(u,fu\right)\ge 1.$ Using Equation (14) with $\mu ={\mu }_{n_k}$ and $\tau =u$, we have

$$\begin{array}{ccc}\hfill \psi \left(s^{\epsilon }d\left(f{\mu }_{n_k},fu\right)\right)& \le & \mathsf{\Omega }\left({\mu }_{n_k}f{\mu }_{n_k}\right)\omega \left(u,fu\right)\psi \left(s^{\epsilon }d\left(f{\mu }_{n_k},fu\right)\right)\hfill \\ & \le & \psi \left(N\left({\mu }_{n_k},u\right)\right).\hfill \end{array}$$

Consequently, $s^{\epsilon }d\left(f{\mu }_{n_k},fu\right)\le N\left({\mu }_{n_k},u\right)$, where

$$\begin{array}{ccc}\hfill N\left({\mu }_{n_k},u\right)& =& max\left\{d\left({\mu }_{n_k},u\right),d\left({\mu }_{n_k},f{\mu }_{n_k}\right),d\left(u,fu\right),\frac{d\left({\mu }_{n_k},fu\right)+d\left(f{\mu }_{n_k},u\right)}{2s}\right\}\hfill \\ & =& max\left\{d\left({\mu }_{n_k},u\right),d\left({\mu }_{n_k},{\mu }_{n_k+1}\right),d\left(u,fu\right),\frac{d\left({\mu }_{n_k},fu\right)+d\left({\mu }_{n_k+1},u\right)}{2s}\right\}.\hfill \end{array}$$

Since $\frac{1}{s}d\left({\mu }_{n_k},fu\right)\le d\left({\mu }_{n_k},u\right)+d\left(u,fu\right)$, we get $N\left({\mu }_{n_k},u\right)\to d\left(u,fu\right)$ as $k\to \infty$

On the other hand,

$$\frac{1}{s}d\left(u,fu\right)\le d\left(u,{\mu }_{n_k+1}\right)+d\left(f{\mu }_{n_k},fu\right),$$

that is,

$$\frac{1}{s}d\left(u,fu\right)\le d\left(u,{\mu }_{n_k+1}\right)+\frac{1}{s^{\epsilon }}max\left\{d\left({\mu }_{n_k},u\right),d\left({\mu }_{n_k},{\mu }_{n_k+1}\right),d\left(u,fu\right),\frac{d\left({\mu }_{n_k},fu\right)+d\left({\mu }_{n_k+1},u\right)}{2s}\right\},$$

i.e., $\frac{1}{s}d\left(u,fu\right)\le \frac{1}{s^{\epsilon }}d\left(u,fu\right).$ Since $\epsilon >1$, the last inequality holds unless $u=fu$.

Now, suppose v is so that $fu=u\ne v=fv$. Putting $\mu =u$ and $\tau =v$ in (14),

$$\psi \left(s^{\epsilon }d\left(u,v\right)\right)\le \mathsf{\Omega }\left(u,fu\right)\omega \left(v,fv\right)\psi \left(s^{\epsilon }d\left(u,v\right)\right)\le \psi \left(N\left(u,v\right)\right),$$

where

$$\begin{array}{ccc}\hfill N\left(u,v\right)& =& max\left\{d\left(u,v\right),d\left(u,fu\right),d\left(v,fv\right),\frac{d\left(u,fv\right)+d\left(v,fu\right)}{2s}\right\}\hfill \\ & =& max\left\{d\left(u,v\right),0,0,\frac{d\left(u,v\right)}{s}\right\}\hfill \\ & =& d\left(u,v\right).\hfill \end{array}$$

Hence, $\psi \left(s^{\epsilon }d\left(u,v\right)\right)\le \psi \left(d\left(u,v\right)\right)$ is possible only if $u=v$. The proof of the result is finished.  □

Remark 4.

It is not hard to check that Example 4.6 from [35] satisfies all conditions of Theorem 7 for$\epsilon \in (1,3]$. Indeed, since for all$x,y\in X$and for all$\epsilon \in (1,3]$, it follows that

$$\begin{array}{ccc}\hfill \alpha \left(x,y\right)\beta \left(x,y\right)\psi \left(s^{\epsilon }d\left(Tx,Ty\right)\right)& \le & \alpha \left(x,y\right)\beta \left(x,y\right)\psi \left(s^{3}d\left(Tx,Ty\right)\right)\hfill \\ & \le & \theta \left(\psi \left(N\left(x,y\right)\right)\right)\psi \left(N\left(x,y\right)\right).\hfill \end{array}$$

That is,

$$\alpha \left(x,y\right)\beta \left(x,y\right)\psi \left(s^{\epsilon }d\left(Tx,Ty\right)\right)\le \theta \left(\psi \left(N\left(x,y\right)\right)\right)\psi \left(N\left(x,y\right)\right),$$

for all$x,y\in X$and for all$\epsilon \in (1,3]$. This means that Example 4.6. from [35] supports Theorem 7. On the other hand, Theorem 7 extends the main result from [35] of$\left\{\epsilon =3\right\}$to$\epsilon \in (1,3]$. Thus, our results are genuine generalizations of ones from [35].

In [30], the authors introduced so-called Geraghty type functions. Denote by $\mathsf{\Psi }$ the set of continuous increasing nonnegative functions $\psi$ defined on $[0,\infty )$ so that ${\psi }^{-1}\left(0\right)=\left\{0\right\}$. Take $s\ge 1$. Let $\mathcal{F}$ be the family of all nondecreasing functions $\beta :[0,\infty )\to [0,\frac{1}{s})$ so that

$$\underset{k\infty }{lim}\beta \left(l_k\right)=\frac{1}{s}\mathrm{implies}\underset{k\to \infty }{lim}{l}_k=0.$$

Definition 10.

Let T be a self-mapping on a b-metric space$\left(M,d\right)$. T is a generalized$\mathsf{\Omega }-\psi$-Geraghty contractive mapping if there are$\mathsf{\Omega }:M\times M\to [0,\infty )$,$\beta \in \mathcal{F}$,$\psi ,\varphi \in \mathsf{\Psi }$and$L\ge 0$so that for

$$E\left(\mu ,\tau \right)=max\left\{d\left(\mu ,\tau \right),d\left(\mu ,T\mu \right),d\left(\tau ,T\tau \right),\frac{d\left(\mu ,T\tau \right)+d\left(\tau ,T\mu \right)}{2s}\right\}$$

and

$$N\left(\mu ,\tau \right)=min\left\{d\left(\mu ,T\mu \right),d\left(\tau ,T\tau \right)\right\},$$

we have

$$\mathsf{\Omega }\left(\mu ,\tau \right)\psi \left(s^{3}d\left(T\mu ,T\tau \right)\right)\le \beta \left(E\left(\mu ,\tau \right)\right)\psi \left(E\left(\mu ,\tau \right)\right)+L\varphi \left(N\left(\mu ,\tau \right)\right),$$

(15)

for all$\mu ,\tau \in M$.

Theorem 8.

Let$\left(M,d,s\ge 1\right)$be a b-complete b-metric space and$T:M\to M$be a generalized$\mathsf{\Omega }-\psi$-Geraghty contraction so that

(i) 

T is triangular $\mathsf{\Omega }$-orbital admissible;

(ii) 

there is ${\mu }_0\in M$ so that $\mathsf{\Omega }\left({\mu }_0,T{\mu }_0\right)\ge 1$;

(iii) 

T is continuous.

Then, T has a fixed point.

Note that for the proof of the announced result in [30], the authors used Lemma 3. However, our approach does not require this lemma and the proof is much shorter. Namely, we consider the following:

$$\mathsf{\Omega }\left(\mu ,\tau \right)\psi \left(s^{\epsilon }d\left(T\mu ,T\tau \right)\right)\le \beta \left(E\left(\mu ,\tau \right)\right)\psi \left(E\left(\mu ,\tau \right)\right)+L\varphi \left(N\left(\mu ,\tau \right)\right),$$

where $\epsilon >1$, instead Equation (15). On the other hand,

$$d\left({\mu }_{n+1},{\mu }_n\right)\le \frac{1}{s^{\epsilon }}d\left({\mu }_n,{\mu }_{n-1}\right),n\ge 1.$$

This further implies that the sequence $\left\{{\mu }_n\right\}$ is b-Cauchy. The proof is now similar to its corresponding one in [30].

Remark 5.

Since$\beta \left([0,\infty )\right)\subseteq [0,1)$, it is not hard to see that Equation (14) becomes

$$\mathsf{\Omega }(\mu ,\tau )\psi \left(s^{3}d(T\mu ,T\tau )\right)\le \psi \left(E(\mu ,\tau )\right)+L\varphi (N(\mu ,\tau ),$$

that is, the Geraghty type case in b-metric spaces is superfluous.

It is worth mentioning the following:

Theorem 3 is a consequence of an old theorem of Hegedus [26]. In addition, Ćirić’s Definition of quasi-contractions and Definition 2 are special cases of the following Definition of Hegedus [26].

Definition 11

([26]). A self-mapping T on a metric space X is called a generalized Banach contraction if, for all $x,y\in X,\delta \left(x,y\right)<\infty$ and $d\left(Tx,Ty\right)\le \lambda \delta \left(x,y\right)$ for some $\lambda <1$, where $\delta \left(x,y\right)=diam[O\left(x,\infty \right)\cup O\left(y,\infty \right)]$.

Furthermore, Theorem 1 and Theorem 3 are special cases of the following theorem of Hegedus [26] (by omitting the approximation part of the theorem).

Theorem 9

([26]). Every generalized Banach contraction on a T-orbitally complete metric space has a unique fixed point.

3. Conclusions

This paper contains much shorter and more elementary proof of ones given in existing literature for some mappings in the context of b-metric spaces.