Some New Observations and Results for Convex Contractions of Istratescu’s Type

Abstract

The purpose is to ensure that a continuous convex contraction mapping of order two in b-metric spaces has a unique fixed point. Moreover, this result is generalized for convex contractions of order n in b-metric spaces and also in almost and quasi b-metric spaces.

1. Introduction

In [1,2], the notion of a b-metric space was initiated and some usual fixed point results have been provided. Many new results in this space were obtained over the past ten years (see for example [3,4,5,6]). Istratescu [7] considered convex contraction mappings in metric spaces and showed that each convex contraction mapping of order two admits a unique fixed point. The Istratescu’s result has recently caused the attention and was the object of examination in b-metric spaces (see [8]). Our paper is a generalization of the Istratescu’s result for convex contractions of order n in b-metric spaces (and also in almost b-metric spaces and in quasi b-metric spaces).

2. Preliminaries

Definition 1.

Given a nonempty set $\mathsf{{\rm Y}} \mathrm{and} s\ge 1$. Let $\eta :\mathsf{{\rm Y}}\times \mathsf{{\rm Y}}\to [0,\infty )$ satisfy:

(1) 

$\eta (\omega ,\upsilon )=0$ if $\omega =\upsilon$;

(2) 

$\eta (\omega ,\upsilon )=\eta (\upsilon ,\omega )$;

(3) 

$\eta (\omega ,\varsigma )\le s\left[\eta \right(\omega ,\upsilon )+\eta (\upsilon ,\varsigma \left)\right]$,

for all $\omega ,\upsilon ,\varsigma \in \mathsf{{\rm Y}}$, then d is a b-metric. Here, $(\mathsf{{\rm Y}},\eta ,s)$ is called a b-metric space.

Definition 2.

Let $\left\{{\varsigma }_n\right\}$ be a sequence in a b-metric space $(\mathsf{{\rm Y}},\eta ,s)$. Take $\varsigma \in \mathsf{{\rm Y}}$.

(a) 

$\left\{{\varsigma }_n\right\}$ is convergent to ς, if for each $\epsilon >0$ there is $n_0\in \mathbb{N}$ so that $\eta ({\varsigma }_n,\varsigma )<\epsilon$ for all $n\ge n_0$;

(b) 

$\left\{{\varsigma }_n\right\}$ is Cauchy if for every $\epsilon >0$ there is $n_0\in \mathbb{N}$ so that $\eta ({\varsigma }_n,{\varsigma }_m)<\epsilon$ for all $n,m\ge n_0$;

(c) 

$(\mathsf{{\rm Y}},\eta ,s)$ is complete if every Cauchy sequence is convergent.

Miculescu and Mihail [9] (Lemma 2.2) and Suzuki [10] (Lemma 6) gave the following result (see also [11]).

Lemma 1.

Let $\left\{{\varsigma }_n\right\}$ be a sequence in the b-metric space $(\mathsf{{\rm Y}},\eta ,s)$ so that there is $\gamma \in [0,1)$ in order that for every $n\ge 1$, $\eta ({\varsigma }_{n+1},{\varsigma }_n)\le \gamma \eta ({\varsigma }_n,{\varsigma }_{n-1})$. Then $\left\{{\varsigma }_n\right\}$ is Cauchy.

The above lemma is an important tool to get variant results in b-metric spaces since it facilitates many proofs concerning various contraction conditions. The following is a consequence of the proof of Lemma 1.3 in [9]).

Lemma 2.

Let $\left\{{\varsigma }_n\right\}$ be a sequence in the b-metric space $(\mathsf{{\rm Y}},\eta ,s)$ so that there are $\gamma \in [0,1)$ and $C>0$ in order that for each $n\ge 0$,

$$\eta ({\varsigma }_{n+1},{\varsigma }_n)\le C{\gamma }^n.$$

Then $\left\{{\varsigma }_n\right\}$ is Cauchy.

Definition 3.

Let $(\mathsf{{\rm Y}},\eta ,s)$ be a b-metric space. The mapping $\Omega :\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ is a convex Reich type contraction if

$$\begin{array}{ccc}\hfill \eta ({\mathsf{\Omega }}^2\varsigma ,{\mathsf{\Omega }}^2\omega )& \le & \alpha \eta (\varsigma ,\omega )+\beta \eta (\mathsf{\Omega }\varsigma ,\mathsf{\Omega }\omega )+\gamma \left[\eta \right(\varsigma ,\mathsf{\Omega }\varsigma )+\eta (\omega ,\mathsf{\Omega }\omega \left)\right]+\hfill \\ \hspace{1em}& \hspace{1em}& \delta [\eta (\mathsf{\Omega }\varsigma ,{\mathsf{\Omega }}^2\varsigma )+\eta (\mathsf{\Omega }\omega ,{\mathsf{\Omega }}^2\omega )],\hfill \end{array}$$

for all $\varsigma ,\omega \in \mathsf{{\rm Y}}$, where $\alpha ,\beta ,\gamma ,\delta \ge 0$ with $\alpha +\beta +2(\gamma +\delta )<1$.

Definition 4.

Let $(\mathsf{{\rm Y}},\eta ,s)$ be a b-metric space. A mapping $\mathsf{\Omega }:\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ is a convex contraction of Reich type of order $n(n\ge 2)$ if

$$\begin{array}{ccc}\hfill \eta ({\mathsf{\Omega }}^n\varsigma ,{\mathsf{\Omega }}^n\omega )& \le & a_0\eta (\varsigma ,\omega )+a_1\eta (\mathsf{\Omega }\varsigma ,\mathsf{\Omega }\omega )\hfill \\ \hspace{1em}& +& {\displaystyle \sum _{i=0}^{n-1}}{a}_{i+2}[\eta ({\mathsf{\Omega }}^i\varsigma ,{\mathsf{\Omega }}^{i+1}\varsigma )+\eta ({\mathsf{\Omega }}^i\omega ,{\mathsf{\Omega }}^{i+1}\omega )],\hfill \end{array}$$

for all $\varsigma ,\omega \in \mathsf{{\rm Y}}$, where $a_0,a_1,a_2,\dots ,a_{n+1}$ are nonnegative constants with $a_0+a_1+2{\displaystyle \sum _{i=0}^{n-1}}{a}_{i+2}<1$.

A slight modification of the definition of contraction mappings of order n was as follows:

Definition 5. 

([7]) Let $(\mathsf{{\rm Y}},\eta )$ be a complete metric space. A mapping $\mathsf{\Omega }:\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ is a convex contraction of order n if there are $a_0,\dots ,a_{n-1}$ in $(0,1)$ so that for all $\varsigma ,\omega \in \mathsf{{\rm Y}}$,

$$\eta ({\mathsf{\Omega }}^n\left(\varsigma \right),{\mathsf{\Omega }}^n\left(\omega \right))\le a_0\eta (\varsigma ,\omega )+a_1\eta (\mathsf{\Omega }\left(\varsigma \right),\mathsf{\Omega }\left(\omega \right))+\cdots +a_{n-1}\eta ({\mathsf{\Omega }}^{n-1}\left(\varsigma \right),{\mathsf{\Omega }}^{n-1}\left(\omega \right)),$$

(1)

where ${\mathsf{\Omega }}^n\left(\varsigma \right)=\mathsf{\Omega }\left({\mathsf{\Omega }}^{n-1}\left(\varsigma \right)\right)$ and $a_0+a_1+\cdots +a_{n-1}<1$.

If we exclude in Definition 1 the symmetry condition, $(\mathsf{{\rm Y}},\eta ,s)$ is said to be a quasi b-metric space. Now, we recall the definition of almost b-metric spaces, which relies on some symmetry-type limiting conditions of defined almost b-metrics.

Definition 6. 

([12]) Let Υ be a nonempty set and $s\ge 1$. Let ${\eta }_{ab}:\mathsf{{\rm Y}}\times \mathsf{{\rm Y}}\to [0,+\infty )$ and $\varsigma ,\omega ,\sigma ,{\varsigma }_n\in \mathsf{{\rm Y}}$ so that:

• (bM1) ${\eta }_{ab}(\varsigma ,\omega )=0$ if and only if $\varsigma =\omega$,
• (bM2l) ${\eta }_{ab}({\varsigma }_n,\varsigma )\to 0,n\to \infty$ implies ${\eta }_{ab}(\varsigma ,{\varsigma }_n)\to 0,n\to \infty$,
• (bM2r) ${\eta }_{ab}(\varsigma ,{\varsigma }_n)\to 0,n\to \infty$ implies ${\eta }_{ab}({\varsigma }_n,\varsigma )\to 0,n\to \infty$,
• (bM3) ${\eta }_{ab}(\varsigma ,\omega )\le s({\eta }_{ab}(\varsigma ,\sigma )+{\eta }_{ab}(\sigma ,\omega ))$.

• If (bM1), (bM2l) and (bM3) are verified, then ${\eta }_{ab}$ is said to be an l-almost b-metric on Υ;
• If (bM1), (bM2r) and (bM3) are verified, then ${\eta }_{ab}$ is said to be an r-almost b-metric on Υ;
• If (bM1), (bM2l), (bM2r) and (bM3) are verified, then ${\eta }_{ab}$ is said to be an almost b-metric on Υ.

We refer readers to see more about those spaces in the papers [1,2,7,8,9,10,11,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]. For all contractive type conditions, see [34,35]. One of applications of contractive mappings was used for maximum likelihood estimation of the multiple linear regression parameters in the generalized Gauss–Laplace distribution assumption of the measurement’s errors [36].

3. Main Results

3.1. Some Lemmas

The first lemma is an auxiliary result. We use it to be ensured that in a convex contraction, the Cauchyness holds. The same result was obtained for metric and b-metric spaces.

Lemma 3.

Let $k\in \mathbb{N}$ and $\left\{{\varsigma }_n\right\}$ be a sequence in a b-metric space $(\mathsf{{\rm Y}},\eta ,s)$ so that

$$\eta ({\varsigma }_{n+k},{\varsigma }_{n+k-1})\le {\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta ({\varsigma }_{n+i},{\varsigma }_{n+i-1}),$$

(2)

for all $n\in \mathbb{N}$, where $a_i\ge 0$ such that ${\displaystyle \sum _{i=0}^{k-1}}{a}_i<1$. Then

$$\eta ({\varsigma }_{n+k},{\varsigma }_{n+k-1})\le c{\gamma }^n,$$

(3)

for all $n\in \mathbb{N}$, where $c=max\{\eta ({\varsigma }_1,{\varsigma }_0),\dots ,\eta ({\varsigma }_k,{\varsigma }_{k-1})\}$ and $\gamma ={\left({\displaystyle \sum _{i=0}^{k-1}}{a}_i\right)}^{1/k}.$

Proof. 

From (2), we obtain

$$\eta ({\varsigma }_{k+1},{\varsigma }_k)\le {\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta ({\varsigma }_{i+1},{\varsigma }_i)\le c{\displaystyle \sum _{i=0}^{k-1}}{a}_i=c{\gamma }^k\le c\gamma .$$

Similarly,

$$\eta ({\varsigma }_{k+2},{\varsigma }_{k+1})\le {\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta ({\varsigma }_{i+2},{\varsigma }_{i+1})\le c{\displaystyle \sum _{i=0}^{k-2}}{a}_i+c{a}_{k-1}\gamma \le c{\displaystyle \sum _{i=0}^{k-2}}{a}_i+c{a}_{k-1}=c{\gamma }^k\le c{\gamma }^2.$$

$$⋮$$

$$\eta ({\varsigma }_{2k},{\varsigma }_{2k-1})\le {\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta ({\varsigma }_{i+k},{\varsigma }_{i+k-1})\le c{\displaystyle \sum _{i=0}^{k-1}}{a}_i{\gamma }^i\le c{\displaystyle \sum _{i=0}^{k-1}}{a}_i=c{\gamma }^k.$$

So, by induction for $n\ge k$ we have that

$$\eta ({\varsigma }_{n+k},{\varsigma }_{n+k-1})\le {\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta ({\varsigma }_{n+i},{\varsigma }_{n+i-1})\le c{\displaystyle \sum _{i=0}^{k-1}}{a}_i{\gamma }^{n+i-k}\le c{\gamma }^{n-k}{\displaystyle \sum _{i=0}^{k-1}}{a}_i=c{\gamma }^n.$$

 □

Lemma 4.

Let $k\in \mathbb{N}$ and $\left\{{\varsigma }_n\right\}$ be a sequence in a b-metric space $(\mathsf{{\rm Y}},\eta ,s)$ so that

$$\eta ({\varsigma }_{n+k},{\varsigma }_{n+k-1})\le {\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta ({\varsigma }_{n+i},{\varsigma }_{n+i-1}),$$

(4)

for all $n\in \mathbb{N}$, where $a_i\ge 0$ such that ${\displaystyle \sum _{i=0}^{k-1}}{a}_i<1$. Then $\left\{{\varsigma }_n\right\}$ is Cauchy.

Proof. 

It follows directly from Lemma 2 and Lemma 3. □

Remark 1.

Lemma 4 with $k=1$ corresponds to Lemma 1, while for $k>1$, Lemma 4 is more general.

3.2. On Convex Contractions of Order k in b-Metric Spaces

Our next theorem is a generalization of Istratescu’s result about convex contractions. We generalize the result of [7] in two directions by proving it for any $k\in \mathbb{N}$ and by considering the class of b-metric spaces. What distinguishes our obtained result is the fact that it is the same as in usual metric spaces and in b-metric spaces. There are two reasons for our new result: The first is due to Lemma 4 and the other is adding the assumption that the considered mapping is continuous.

Theorem 1.

Let $\mathsf{\Omega }:\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ be a continuous convex contraction of order k, on a complete b-metric space $(\mathsf{{\rm Y}},\eta )$, so that

$$\eta ({\mathsf{\Omega }}^k\varsigma ,{\mathsf{\Omega }}^k\omega )\le {\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta ({\mathsf{\Omega }}^i\varsigma ,{\mathsf{\Omega }}^i\omega ),$$

(5)

for all $\varsigma ,\omega \in \mathsf{{\rm Y}}$, where $a_i\ge 0$ such that ${\displaystyle \sum _{i=0}^{k-1}}{a}_i<1$. Then there is a unique fixed point of Ω.

Proof. 

Let $t_0$ be in $\mathsf{{\rm Y}}$. Consider $t_n={\mathsf{\Omega }}^n\left(t_0\right)$.

$$\begin{array}{cc}\hfill \eta (t_{n+k},t_{n+k-1})& =\eta ({\mathsf{\Omega }}^k\left(t_n\right),{\mathsf{\Omega }}^k\left(t_{n-1}\right))\le {\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta ({\mathsf{\Omega }}^i\left(t_n\right),{\mathsf{\Omega }}^i\left(t_{n-1}\right))\\ \hspace{1em}& ={\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta (t_{n+i},t_{n+i-1})\end{array}$$

and directly from Lemma 4, we conclude that $\left\{t_n\right\}$ is a Cauchy sequence in $(\mathsf{{\rm Y}},\eta )$ (which is complete). Hence, there is $t\in \mathsf{{\rm Y}}$ so that $\underset{n\to \infty }{lim}{t}_n=t$. Since $\mathsf{\Omega }$ is continuous, we obtain that

$$\mathsf{\Omega }\left(t\right)=\underset{n\to \infty }{lim}\mathsf{\Omega }\left(t_n\right)=\underset{n\to \infty }{lim}{t}_{n+1}=t,$$

i.e., t is a fixed point of $\mathsf{\Omega }$. Its uniqueness follows from (5). □

Example 1.

The space $\mathsf{{\rm Y}}=l^p=\{\left(x_n\right)\subset \mathbb{R}:{\displaystyle \sum _{n=1}^{+\infty }}|x_n{|}^p<+\infty \},p\in (0,1)$, together with the function $d:l^p\times l^p\to \mathbb{R},$

$$d(x,y)={\left({\displaystyle \sum _{n=1}^{+\infty }}{|x_n-y_n|}^p\right)}^{\frac{1}{p}},$$

where $x=\left(x_n\right),y=\left(y_n\right)\in l^p$, is a b-metric space with $s=2^{\frac{1}{p}}$, [1]. Indeed, by an elementary calculation, we obtain

$$d(x,z)\le 2^{\frac{1}{p}}[d(x,y)+d(y,z)].$$

Let $\mathsf{\Omega }:l^p\to l^p$ be a mapping defined by

$$\mathsf{\Omega }(x_1,x_2,x_3,x_4,\dots )=(0,x_1,\frac{x_2}{2},\frac{x_3}{2},\frac{x_4}{2},\dots ).$$

Note that Ω has a unique fixed point, which is $(0,0,0,\dots )$ (see also [37]). We have

$${\mathsf{\Omega }}^2(x_1,x_2,x_3,x_4,\dots )=(0,0,\frac{x_1}{2},\frac{x_2}{2^2},\frac{x_3}{2^2},\frac{x_4}{2^2},\dots ),$$

$${\mathsf{\Omega }}^3(x_1,x_2,x_3,x_4,\dots )=(0,0,0,\frac{x_1}{2^2},\frac{x_2}{2^3},\frac{x_3}{2^3},\frac{x_4}{2^3},\dots ),$$

$$⋮$$

$${\mathsf{\Omega }}^n(x_1,x_2,x_3,x_4,\dots )=(\underset{n}{\underbrace{0,\dots ,0}},\frac{x_1}{2^{n-1}},\frac{x_2}{2^n},\frac{x_3}{2^n},\frac{x_4}{2^n},\dots ).$$

Further, for $x,y\in l^p$, we have

$$\begin{array}{ccc}\hfill d({\mathsf{\Omega }}^{n}x,{\mathsf{\Omega }}^{n}y)& =& {\left(\frac{|x_1-y_1{|}^p}{2^{p(n-1)}}+\frac{|x_2-y_2{|}^p}{2^{pn}}+\frac{|x_3-y_3{|}^p}{2^{pn}}+\cdots \right)}^{\frac{1}{p}}\hfill \\ \hspace{1em}& \le & {\left[\frac{1}{2^{p(n-1)}}\left(|x_1-y_1{|}^p+|x_2-y_2{|}^p+{|x_3-y_3|}^p+\cdots \right)\right]}^{\frac{1}{p}}\hfill \\ \hspace{1em}& \le & \frac{1}{2^{n-1}}d(x,y).\hfill \end{array}$$

So, the mapping $\mathsf{\Omega }:l^p\to l^p$ is a convex contraction of order n. On the other hand, Ω is not a contraction. Indeed, for $x=(1,0,0,\dots )$ and $y=(2,0,0,\dots )$, we have $\mathsf{\Omega }x=(0,1,0,0,\dots )$, $\mathsf{\Omega }y=(0,2,0,0,\dots )$, $d(x,y)=1$ and $d(\mathsf{\Omega }x,\mathsf{\Omega }y)=1$. Consequently,

$$d(\mathsf{\Omega }x,\mathsf{\Omega }y)>\lambda d(x,y),$$

for each $\lambda \in (0,1)$.

Example 2. 

(a) Let $\mathsf{{\rm Y}}=[1,4]$ be endowed with $\eta (\varsigma ,\omega )=|\varsigma -\omega |$. Consider $\mathsf{\Omega }:\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ defined by $\mathsf{\Omega }\varsigma =2\sqrt{\varsigma }$. Here, Ω is not a contraction on the metric space $(\mathsf{{\rm Y}},\eta )$. Also, Ω is a convex contraction mapping of order 2. Namely, we have

$$|{\mathsf{\Omega }}^2\varsigma -{\mathsf{\Omega }}^2\omega |\le \frac{\sqrt{2}}{2}|\varsigma -\omega |,$$

for all $\varsigma ,\omega \in \mathsf{\Omega }$ and $\varsigma =4$ is the unique fixed point of Ω.

(b) Let $\mathsf{{\rm Y}}=[1,4]$. Consider $\mathsf{\Omega }\varsigma =2\sqrt{\varsigma }$ and $\eta (\varsigma ,\omega )={(\varsigma -\omega )}^2$. Here, $(\mathsf{{\rm Y}},\eta ,s=2)$ is a b-metric space. Note that Ω is not contraction, but since

$$\eta ({\mathsf{\Omega }}^2\varsigma ,{\mathsf{\Omega }}^2,\omega )=8{(\sqrt[4]{\varsigma }-\sqrt[4]{\omega })}^2\le \frac{1}{4}\left(\eta (\mathsf{\Omega }\varsigma ,\mathsf{\Omega }\omega )+\eta (\varsigma ,\omega )\right)$$

Ω is also a convex contraction of order 2.

Example 3.

Let $\mathsf{{\rm Y}}=\mathbb{R}$. Take $\mathsf{\Omega }\left(\zeta \right)=\frac{\zeta +3}{4}$ and $\eta \left(\omega ,v\right)={\left|\omega -v\right|}^2.$ Then $\left(\mathsf{{\rm Y}},\eta \right)$ is a complete $b$—metric space with the coefficient $s=2.$ Given $\mathsf{\Omega }:\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ as $\mathsf{\Omega }\left(\zeta \right)=\frac{\zeta +3}{4}$ for all $\zeta \in \mathsf{{\rm Y}}.$ We obtain that

$$d\left({\mathsf{\Omega }}^2\omega ,{\mathsf{\Omega }}^{2}v\right)={\left|\frac{\omega +15}{16}-\frac{v+15}{16}\right|}^2=\frac{{\left|\omega -v\right|}^2}{256}andd\left(\mathsf{\Omega }\omega ,\mathsf{\Omega }v\right)=\frac{{\left|\omega -v\right|}^2}{16},$$

that is,

$$d\left({\mathsf{\Omega }}^2\omega ,{\mathsf{\Omega }}^{2}v\right)\le a·d\left(\mathsf{\Omega }\omega ,\mathsf{\Omega }v\right)+b·d\left(\omega ,v\right),$$

i.e.,

$$\frac{{\left|\omega -v\right|}^2}{256}\le a·\frac{{\left|\omega -v\right|}^2}{16}+b·{\left|\omega -v\right|}^2.$$

Or, equivalently $\frac{1}{256}\le \frac{a}{16}+b$ where $a,b\ge 0$ and $a+b<1.$ Putting, for example $a=b$ we get that for $a=b\ge \frac{1}{272}$, all the conditions of Theorem 1 are satisfied, i.e., Ω is a convex contraction of order 2 and therefore has a unique fixed point (which is, $\zeta =2).$

Remark 2.

Theorem 2.1 of [7] and Theorem 4 of [8] follow from Theorem 1. Also Theorem 1 improves the contraction condition (in b-metric spaces) stated for convex contraction mappings of order 2 in [8].

In the previous lemmas and in Theorem 1, we did not use the symmetry of the b-metric $\eta$. Thus, with minor changes in the contraction conditions, the main results are similarly valid in the setting of almost b-metric spaces (and also in quasi b-metric spaces). Notice that previous formulations can also applied in l-almost b-metric spaces. Next, we state results for r-almost b-metric spaces.

Remark 3.

Let $k\in \mathbb{N}$ and $\left\{{\varsigma }_n\right\}$ be a sequence in the r-almost b-metric space $(\mathsf{{\rm Y}},\eta ,s)$ so that

$$\eta ({\varsigma }_{n+k-1},{\varsigma }_{n+k})\le {\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta ({\varsigma }_{n+i-1},{\varsigma }_{n+i}),$$

(6)

for all $n\in \mathbb{N}$, where $a_i\ge 0$ such that ${\displaystyle \sum _{i=0}^{k-1}}{a}_i<1$. Then

$$\eta ({\varsigma }_{n+k},{\varsigma }_{n+k-1})\le c{\gamma }^n,$$

(7)

for all $n\in \mathbb{N}$, where $c=max\{\eta ({\varsigma }_0,{\varsigma }_1),\dots ,\eta ({\varsigma }_{k-1},{\varsigma }_k)\}$ and $\gamma ={\left({\displaystyle \sum _{i=0}^{k-1}}{a}_i\right)}^{1/k}.$

Since the result is similar to Lemma 2 and is valid in r-almost b-metric spaces (see [12]), the r-almost version of Lemma 4 is as follows: let $k\in \mathbb{N}$ and $\left\{{\varsigma }_n\right\}$ be a sequence in the r-almost b-metric space $(\mathsf{{\rm Y}},\eta ,s)$ so that

$$\eta ({\varsigma }_{n+k-1},{\varsigma }_{n+k})\le {\displaystyle \sum _{i=0}^{k-1}}{a}_i\eta ({\varsigma }_{n+i-1},{\varsigma }_{n+i}),$$

for all $n\in \mathbb{N}$, where $a_i\ge 0$ such that ${\displaystyle \sum _{i=0}^{k-1}}{a}_i<1$. Then $\left\{{\varsigma }_n\right\}$ is right-Cauchy sequence.

Next, we extend Lemma 3 to quasi b-metric spaces.

Remark 4.

Let $(\mathsf{{\rm Y}},\eta ,s)$ be a quasi b-metric space and let $\left\{{\varsigma }_n\right\}$ be a sequence in Υ and $k\in \mathbb{N}$ such that (4) is satisfied for all $n\in \mathbb{N}$, where $a_i\ge 0$ such that ${\displaystyle \sum _{i=0}^{k-1}}{a}_i<1$. Then

$$\eta ({\varsigma }_{n+k},{\varsigma }_{n+k-1})\le c{\gamma }^n,$$

(8)

for all $n\in \mathbb{N}$, where $c=max\{\eta ({\varsigma }_1,{\varsigma }_0),\eta ({\varsigma }_0,{\varsigma }_1)\dots ,\eta ({\varsigma }_k,{\varsigma }_{k-1}),\eta ({\varsigma }_{k-1},{\varsigma }_k)\}$ and $\gamma ={\left({\displaystyle \sum _{i=0}^{k-1}}{a}_i\right)}^{1/k}.$

3.3. On Convex Reich Type Contractions of Order k in b-Metric Spaces

Our next result is about convex Reich type contractions of order k ($k\ge 2$), (see [18,35,38,39]).

Theorem 2.

Let $(\mathsf{{\rm Y}},\eta ,s)$ be a complete b-metric space and $\mathsf{\Omega }:\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ be a continuous convex Reich type contraction mapping of order k $(k\ge 2)$ so that

$$\begin{array}{ccc}\hfill \eta ({\mathsf{\Omega }}^k\varsigma ,{\mathsf{\Omega }}^k\omega )& \le & a_0\eta (\varsigma ,\omega )+a_1\eta (\mathsf{\Omega }\varsigma ,\mathsf{\Omega }\omega )\hfill \\ \hspace{1em}& +& {\displaystyle \sum _{i=0}^{k-1}}{a}_{i+2}[\eta ({\mathsf{\Omega }}^i\varsigma ,{\mathsf{\Omega }}^{i+1}\varsigma )+\eta ({\mathsf{\Omega }}^i\omega ,{\mathsf{\Omega }}^{i+1}\omega )],\hfill \end{array}$$

for all $\varsigma ,\omega \in \mathsf{{\rm Y}}$, where $a_0,a_1,a_2,\dots ,a_{k+1}\ge 0$ with $a_0+a_1+2{\displaystyle \sum _{i=0}^{k-1}}{a}_{i+2}<1$. Then there is a unique fixed point of Υ.

Proof. 

Let ${\varsigma }_0$ be $\mathsf{{\rm Y}}$. Take ${\varsigma }_n={\mathsf{\Omega }}^n\left({\varsigma }_0\right)$ for all $n\in \mathbb{N}.$ Now, since $\mathsf{\Omega }$ is a convex Reich type contraction of order k we obtain

$$\begin{array}{ccc}\hfill \eta ({\varsigma }_{n+k},{\varsigma }_{n+k-1})& \le & a_0\eta ({\varsigma }_n,{\varsigma }_{n-1})+a_1\eta ({\varsigma }_{n+1},{\varsigma }_n)\hfill \\ \hspace{1em}& +& {\displaystyle \sum _{i=0}^{k-1}}{a}_{i+2}[\eta ({\varsigma }_{n+i},{\varsigma }_{n+i+1})+\eta ({\varsigma }_{n+i-1},{\varsigma }_{n+i})].\hfill \end{array}$$

Since

$${\displaystyle \sum _{i=0}^{k-1}}{a}_{i+2}\eta ({\varsigma }_{n+i-1},{\varsigma }_{n+i})=a_2\eta ({\varsigma }_{n-1},{\varsigma }_n)+{\displaystyle \sum _{i=0}^{k-2}}{a}_{i+3}\eta ({\varsigma }_{n+i},{\varsigma }_{n+i+1}),$$

we obtain

$$\begin{array}{ccc}\hfill \eta ({\varsigma }_{n+k},{\varsigma }_{n+k-1})& \le & (a_0+a_2)\eta ({\varsigma }_n,{\varsigma }_{n-1})+a_1\eta ({\varsigma }_{n+1},{\varsigma }_n)\hfill \\ \hspace{1em}& +& {\displaystyle \sum _{i=0}^{k-2}}(a_{i+2}+a_{i+3})\eta ({\varsigma }_{n+i},{\varsigma }_{n+i+1})+a_{k+1}\eta ({\varsigma }_{n+k-1},{\varsigma }_{n+k}).\hfill \end{array}$$

That is,

$$\begin{array}{ccc}\hfill (1-a_{k+1})\eta ({\varsigma }_{n+k},{\varsigma }_{n+k-1})& \le & (a_0+a_2)\eta ({\varsigma }_n,{\varsigma }_{n-1})+a_1\eta ({\varsigma }_{n+1},{\varsigma }_n)\hfill \\ \hspace{1em}& +& {\displaystyle \sum _{i=0}^{k-2}}(a_{i+2}+a_{i+3})\eta ({\varsigma }_{n+i},{\varsigma }_{n+i+1}).\hfill \end{array}$$

Thus,

$$\eta ({\varsigma }_{n+k},{\varsigma }_{n+k-1})\le {\displaystyle \sum _{i=0}^{k-1}}{b}_i\eta ({\varsigma }_{n+i},{\varsigma }_{n+i-1}).$$

for all $n\in \mathbb{N}$, where $b_0=\frac{a_0+a_2}{1-a_{k+1}}$, $b_1=\frac{a_1+a_2+a_3}{1-a_{k+1}}$, $b_i=\frac{a_{i+1}+a_{i+2}}{1-a_{k+1}}$, $i=2,\dots ,k-1.$ Since $a_0+a_1+2{\displaystyle \sum _{i=0}^{k-1}}{a}_{i+2}<1$, we obtain that ${\displaystyle \sum _{i=0}^{k-1}}{b}_i<1$, therefore using Lemma 4 we conclude that ${\varsigma }_n$ is a Cauchy sequence, and so there is $\varsigma \in \mathsf{{\rm Y}}$ such that $\underset{n\to \infty }{lim}{\varsigma }_n=\varsigma$. It further shows that $\varsigma$ is the unique fixed point of $\mathsf{\Omega }$ (as in Theorem 1). □

Remark 5.

From Theorem 2, we obtain Theorem 2.3 of [7] (case $k=2$).

Finally, as an open problem, we may ask the following question:

Let $(\mathsf{{\rm Y}},\eta ,s)$ be a b-metric space. Find the conditions for the constants $a_0,a_1,b_i,c_i$ such that the mapping $\mathsf{\Omega }:\mathsf{{\rm Y}}\to \mathsf{{\rm Y}}$ has a unique fixed point, if the next condition (Hardy–Rogers type contraction order of k) is fulfilled:

$$\begin{array}{ccc}\hfill \eta ({\mathsf{\Omega }}^k\varsigma ,{\mathsf{\Omega }}^k\omega )& \le & a_0\eta (\varsigma ,\omega )+a_1\eta (\mathsf{\Omega }\varsigma ,\mathsf{\Omega }\omega )\hfill \\ \hspace{1em}& +& {\displaystyle \sum _{i=0}^{k-1}}{b}_i[\eta ({\mathsf{\Omega }}^i\varsigma ,{\mathsf{\Omega }}^{i+1}\varsigma )+\eta ({\mathsf{\Omega }}^i\omega ,{\mathsf{\Omega }}^{i+1}\omega )]\hfill \\ \hspace{1em}& +& {\displaystyle \sum _{i=0}^{k-1}}{c}_i[\eta ({\mathsf{\Omega }}^i\varsigma ,{\mathsf{\Omega }}^{i+1}\omega )+\eta ({\mathsf{\Omega }}^i\omega ,{\mathsf{\Omega }}^{i+1}\varsigma )],\hfill \end{array}$$

for all $\varsigma ,\omega \in \mathsf{{\rm Y}}$.

4. Conclusions

We generalized the Istratescu’s result for convex contractions. Particulary, we considered convex contractions of order k in the setting of b-metric spaces. Some related observations have been made for the classes of almost and quasi b-metric spaces. The presented results have been supported by some examples.