The Results of Common Fixed Points in b-Metric Spaces

Abstract

In this paper, we present some results on the existence and uniqueness of common fixed points on $d^{*}$-complete topological spaces. Our results generalize and improve upon earlier results in the literature. Finally, we give some examples in $l_p$ spaces, $(p\in (0,1\left)\right)$, where we use the obtained results.

1. Introduction

In the past 50 years, non-metric generalizations of Banach’s fixed point theory and its applications have played an important role in nonlinear analysis; see [1,2,3,4,5]. There are many definitions of extended metric spaces (in these spaces, the distance need not satisfy the triangle inequality or need not be symmetric). Some examples of such spaces are d-complete L spaces or Kasahara spaces [6] (see also [7]). Hicks [8] first introduced the notion of d-complete topological spaces and obtained the topological properties of those spaces. In the paper [9], d-complete topological spaces were extended via the $d^{*}$-complete topological spaces. Fixed point theory for non-commutative mappings was introduced by Kannan [10] and further developed by Srivastava and Gupta [11], Wong [12] and Ćirić [13]. Classical results have been presented by Wong [12] and Ćirić [13], which were extended by George et al. [14] on b-metric spaces.

In this paper, we obtain some theorems that draw out significant results in [9], about the existence and uniqueness of fixed point obtained for $d^{*}$-complete topological spaces. The significance of our improvement is that we obtained results about common fixed points for two mappings that are not required to have a commutative property. Our results are shown at complete b-metric spaces $CbMS$ with an (SC) property.

Our results generalize previous results of Wong [12], Ćirić [13], Bianchini [15], Bryant [16], Caccioppoli [17], Marjanović [18], Reich [19], Tasković [20], Yen [21] and Zamfirescu [22] on b-metric spaces.

Moreover, we take into consideration some properties of b-spaces, a class of topological spaces which belong to E-spaces (spaces with regular écart) and include metric spaces. However, we draw the attention of the reader to the fact that we use $CbMS$ for a complete b-metric space and $bMS$ for a b-metric space.

2. Preliminary Notes

In this section, we list several well-known definitions, remarks and lemmas.

Definition 1.

Suppose that Ω and Γ are topological spaces. The mapping $f:\mathrm{\Omega }\to \mathrm{\Gamma }$ is sequentially continuous if for every sequence $\left({\omega }_n\right)\subseteq \mathrm{\Omega }$,

$$\underset{n\to +\infty }{lim}{\omega }_n=p implies\underset{n\to +\infty }{lim}f{\omega }_n=fp.$$

For ${\omega }_0\in \mathrm{\Omega }$, we affirm that the sequence $\left({\omega }_n\right)$ defined by ${\omega }_n=f^n{\omega }_0$ is a sequence of Picard iterates for f at point ${\omega }_0$ or that $\left({\omega }_n\right)$ is the orbit of f at point ${\omega }_0$.

Suppose that $\mathrm{\Omega }$ is a nonempty set, $f:\mathrm{\Omega }\to \mathrm{\Omega }$ is a mapping and $x\in \mathrm{\Omega }$ is a fixed point for f if $fx=x$.

The first part of the following statement was developed and proved by Adamović [23]. Its second part was presented in [24].

Lemma 1.

Let Ω be a nonempty set and $f:\mathrm{\Omega }\to \mathrm{\Omega }$ be a mapping. Let $l\in \mathbb{N}$, where $f^l$ has a unique fixed point $u_{*}$. Then,

(1)

$u_{*}$ is the unique fixed point for f;

(2)

If Ω is the topological space and a Picard-iterated sequence defined by $f^l$ converges to $u_{*}$, then the sequence of Picard iterates defined along f converges to $u_{*}$.

Definition 2

([8,9]). Suppose that Ω is a Hausdorff topological space and $d:\mathrm{\Omega }\times \mathrm{\Omega }\to [0,+\infty )$ is a mapping. We define the next three valuable properties:

(a)

For any $\omega ,\theta \in \mathrm{\Omega }$, $d(\omega ,\theta )=0$ if and only if $\omega =\theta$;

(b)

For every sequence $\left({\omega }_n\right)\subseteq \mathrm{\Omega }$,

$$\sum _{n=0}^{\infty }d({\omega }_n,{\omega }_{n+1})<\infty implies that\left({\omega }_n\right)converges in \mathrm{\Omega };$$

(1)

(c)

For each sequence of $\left({\omega }_n\right)\subseteq \Omega$, if there is $L>0$ and $\lambda \in [0,1)$, where

$$d({\omega }_n,{\omega }_{n+1})\le L{\lambda }^n,$$

(2)

for $n=0,1,2,\dots$, then $\left({\omega }_n\right)$ converges in Ω.

If $(\mathrm{\Omega },d)$ satisfies conditions (a) and (b) ((a) and (c)), then we say that $(\mathrm{\Omega },d)$ is a d-complete topological space ($d^{*}$-complete topological space).

Remark 1.

Obviously, any d-complete topological space $(\mathrm{\Omega },d)$ is $d^{*}$-complete; however, the converse is not true (see [9]).

In the article [25], Fréchet established the classes of metric spaces E-spaces. The historical development of the theory of b-spaces was revisited in the paper by Berinde and Păcurar [26].

Definition 3.

The triplet $(B,\rho ,s)$, where B is a nonempty set, $\rho :B\times B\to [0,+\infty )$ and $s>0$, is a $bMS$ with constant s if the following conditions hold:

(B₁)

$\rho (u,v)=0$ if and only if $u=v$,

(B₂)

$\rho (u,v)=\rho (v,u)$,

(B₃)

$\rho (u,v)\le s\left[\rho \right(u,w)+\rho (w,v\left)\right]$,

for all $u,v,w\in B$.

Remark 2.

(i) It is clear that $(B,\rho ,1)$ is a metric space.

(ii) If $v=w$ is put into ($B_3$), we obtain $\rho (u,v)\le s\left[\rho \right(u,v)+\rho (v,v\left)\right]=s\rho (u,v)$. So, in a b-metric space $(B,\rho ,s)$, we have $s\ge 1$.

Here are some results that can be seen in [27,28].

In every b-metric space $(B,\rho ,s)$, one can propose the topology ${\tau }_{\rho }$ on behalf of defining the family of closed sets as follows:

A set $A\subseteq B$ is closed if and only if for every $u\in B$, $\rho (u,A)=0$ implies that $u\in A$, where

$$\rho (u,A)=inf\left\{\rho \right(u,a):a\in A\}.$$

The convergence of the sequence $\left(u_n\right)$ in the topology ${\tau }_{\rho }$ is not necessarily implied $\rho (u_n,u)\to 0$, although, the converse is true (see [24]).

Many notions in $bMS$ would be the same as those in metric spaces.

Definition 4.

A sequence $\left(u_n\right)\subseteq B$ is said to be a Cauchy sequence if for a given $\epsilon >0$, there is $N_{\epsilon }\in \mathbb{N}$ such that $d(u_m,u_n)<\epsilon$, for all $m,n\ge N_{\epsilon }.$ A $bMS$$(B,\rho ,s)$ is said to be complete if every Cauchy sequence converges to some $u\in \mathrm{\Omega }$.

Let $r>0$ and $u\in B$. By

$$B(u,r)=\{v\in \mathrm{\Omega }:\rho (u,v)<r\},$$

we denote an open ball with a center u and a radius r.

Many properties of $bMS$ would be the same as those in metric spaces (but, it is not all because there is no triangle inequality). For example, each $bMS$ is a Hausdorff space.

Further, An et al. [29] proved that every $bMS$ is a semi-metrizable space (for a definition of a semi-metrizable space, see [24]), but there exists $bMS$ in which open balls are not open sets. Additionally, every $bMS$ satisfies the first axiom of countability, which implies that continuity and sequential continuity in $bMS$ are equivalent notions.

In [30], Miculescu and Mihail proved the following result. Its simple and short proof was presented by Mitrović [31].

Lemma 2.

Suppose that $(B,\rho ,s)$ is a $bMS$ and sequence $\left\{u_n\right\}\subseteq B$. If there exists $\lambda \in [0,1)$ such that

$$\rho (u_{n+1},u_n)\le \lambda \rho (u_n,u_{n-1})$$

(3)

for all $n\in \mathbb{N},$ then $\left\{u_n\right\}$ is Cauchy.

Remark 3.

From Lemma 2, we have that $CbMS$ is a $d^{*}$ complete topological space.

Definition 5.

We say that b-metric space $(B,\rho ,s)$ has the property (SC) if

$$\underset{n\to +\infty }{lim}\rho (u_n,u)=0 implies{\overline{lim}}_{n\to +\infty }\rho (u_n,v)=\rho (u,v),$$

(4)

where $\left(u_n\right)\subset B,u,v\in B$.

Remark 4.

In [24], it was proved that $bMS$ has the property (SC) if all its open balls are open sets in topology ${\tau }_{\rho }$.

3. Main Results

In the following results, some properties of the finite product of $CbMS$ are given. We use the notation $\left[n\right]=\{1,\dots ,n\}$, where $n\in \mathbb{N}$.

Lemma 3.

Let $(B_i,{\rho }_i,s_i),i\in \left[n\right]$ be $bMS$, $B=B_1\times \cdots \times B_n$ and $\rho :B^2\to [0,+\infty )$ be defined by

$$\rho ((p_1,\dots ,p_n),(q_1,\dots ,q_n))=max\{{\rho }_i(p_i,q_i):i\in \left[n\right]\},$$

(5)

where $p_i,q_i\in B_i,i\in \left[n\right]$. Let $s=max\{s_i:i\in \left[n\right]\}$. Then,

(1)

$(B,\rho ,s)$ is a b-metric space;

(2)

$(B,\rho ,s)$ is complete if and only if $(B_i,{\rho }_i,s_i),i\in \left[n\right]$ is complete;

(3)

$(B,\rho ,s)$ has the property (SC) if and only if $(B_i,{\rho }_i,s_i),i\in \left[n\right]$ has the property (SC).

Proof. 

(1) Conditions (1), (2) and (3) are trivial-satisfied.

(2) Suppose that $(B,\rho ,s)$ is $CbMS$. Suppose that $k\in \left[n\right]$, $\left(u_i^k\right)\subset B_k$ is Cauchy sequence in $(B_k,{\rho }_k,s_k)$ and $p^j\in B_j$ such that $j\in \left[n\right]\setminus \left\{k\right\}$. Then, sequence $\left(u_i\right)=(u_i^1,\dots ,u_i^n)\in B$ defined by

$$u_i^j=\left\{\begin{array}{c}{u}_i^k,\mathrm{if} j=k,\hfill \\ p^j,\mathrm{if} j\ne k,\hfill \end{array}\right.$$

is a Cauchy sequence in $(B,\rho )$. So, $x_i$ converges. Hence, we have that $\left(u_i^k\right)$ converges.

Now, assume that $(B_k,{\rho }_k,s_k)$ is a $CbMS$ for each $k\in \left[n\right]$, and that sequence $\left(u_i\right)\subseteq B$ defined by $u_i=(u_i^1,\dots ,u_i^n)$ is a Cauchy sequence in $(B,\rho ,s)$. We can see that then, $\left(u_i^k\right)\subset B_k$ is a Cauchy sequence in $(B_k,{\rho }_k)$. Let ${lim}_{i\to +\infty }{x}_i^k=p^k$, $k\in \left[n\right]$. Then, for each $\epsilon >0$ and every $k\in \left[n\right]$, there exists $m_0^k\in \mathbb{N}$, where $i\ge m_0^k$ implies ${\rho }_{(}{u}_i^k,p^k)<\epsilon$. Let $p=(p^1,\dots ,p^n)$. So, $i\ge m_0=max\{m_0^1,\dots ,m_0^n\}$ implies $\rho (u_i,p)\le \epsilon$. Hence, ${lim}_{i\to +\infty }{x}_i=p$.

(3) Let $U_i\subset B_i,i\in \left[n\right]$ be open balls and $U=U_1\times \cdots \times U_n$. Then, U is an open set in topology ${\tau }_{\rho }$ if each $U_i,i\in \left[n\right]$ is an open set in topology ${\tau }_{{\rho }_i}$. So, if U is open, then all $U_i$ are open.

Further, if $B(p,r)\subseteq B$, where $p=(p_1,\dots ,p_n)$ is an open set, then $B(p_i,r)\subseteq B_i$ is an open set as a projection of $B(p,r)$ to $B_i$. □

Next a common fixed point theorem extends previous fixed point results presented by Mitrović et al. [9] and Tasković [20].

Theorem 1.

Let $(B_i,{\rho }_i,s_i),i\in \left[n\right]$ be a $CbMS$ with the (SC) property. Let $B=B_1\times \cdots \times B_n$ and $\rho :B^2\to [0,+\infty )$ be a mapping defined as

$$\rho (({\omega }_1,\dots ,{\omega }_n),(v_1,\dots ,v_n))=max\{{\rho }_i({\omega }_i,y_i):i\in \left[n\right]\}.$$

(6)

Let $s=max\{s_i:i\in \left[n\right]\}$, $f_i,g_i:B\to B_i,i\in \left[n\right]$ and $F,G:B\to B$ be defined by $F=(f_1,\dots ,f_n)$ and $G=(g_1,\dots ,g_n)$. Suppose that

$$\rho (F\omega ,G\theta )\le \lambda \rho (\omega ,\theta )$$

(7)

for all$\omega ,\theta \in \mathrm{\Omega }$ and some $\lambda \in [0,1)$. Then, F and G have a unique common fixed point $\mathrm{\Theta }\in \mathrm{\Omega }$. Also, Θ is a unique limit of all Picard sequences defined by F and a unique limit of all Picard sequences defined by G.

Proof. 

Suppose that ${\mathrm{\Theta }}_0=({\omega }_1^0,\dots ,{\omega }_n^0)\in B$ and $\left({\mathrm{\Theta }}_i\right)$ is a sequence defined by ${\mathrm{\Theta }}_{2i+1}=F{\mathrm{\Theta }}_{2i}$ and ${\mathrm{\Theta }}_{2i+2}=G{\mathrm{\Theta }}_{2i+1}$. We have that

$$\rho ({\mathrm{\Theta }}_{i+1},{\mathrm{\Theta }}_{i+2})\le \lambda \rho ({\mathrm{\Theta }}_i,{\mathrm{\Theta }}_{i+1})$$

(8)

for $i=0,1,2,\dots$. Now, from (8), Lemmas 2 and 3, we conclude that there exists $\mathrm{\Theta }\in B$ where $\mathrm{\Theta }={lim}_{i\to +\infty }{\mathrm{\Theta }}_i$. Also, we have

$$\mathrm{\Theta }=\underset{i\to +\infty }{lim}{\mathrm{\Theta }}_i=\underset{i\to +\infty }{lim}{\mathrm{\Theta }}_{2i+1}=\underset{i\to +\infty }{lim}F{\mathrm{\Theta }}_{2i}$$

and

$$\mathrm{\Theta }=\underset{i\to +\infty }{lim}{\mathrm{\Theta }}_i=\underset{i\to +\infty }{lim}{\mathrm{\Theta }}_{2i+2}=\underset{i\to +\infty }{lim}G{\mathrm{\Theta }}_{2i+1}.$$

From

$$\rho (F\mathrm{\Theta },{\mathrm{\Theta }}_{2i+2})\le \lambda \rho (\mathrm{\Theta },{\mathrm{\Theta }}_{2i+1}),$$

using the (SC) property, we obtain

$${\overline{lim}}_{i\to +\infty }\rho (F\mathrm{\Theta },{\mathrm{\Theta }}_{2i+2})\le \lambda {\overline{lim}}_{i\to +\infty }\rho (\mathrm{\Theta },{\mathrm{\Theta }}_{2i+1})=0.$$

Therefore, $F\mathrm{\Theta }={lim}_{i\to +\infty }{\mathrm{\Theta }}_{2i+2}=\mathrm{\Theta }$. Further, from

$$\rho (G\mathrm{\Theta },{\mathrm{\Theta }}_{2i+1})\le \lambda \rho (\mathrm{\Theta },{\mathrm{\Theta }}_{2i}),$$

again, using the (SC) property, we have

$${\overline{lim}}_{i\to +\infty }\rho (G\mathrm{\Theta },{\mathrm{\Theta }}_{2i+1})\le \lambda {\overline{lim}}_{i\to +\infty }\rho (\mathrm{\Theta },{\mathrm{\Theta }}_{2i})=0.$$

So, $G\mathrm{\Theta }={lim}_{i\to +\infty }{\mathrm{\Theta }}_{2i+1}=\mathrm{\Theta }$. Let $\omega \in B$, where $F\omega =\omega$ and $\omega \ne \mathrm{\Theta }$. Then, we deduce that

$$\rho (\omega ,\mathrm{\Theta })=\rho (F\omega ,G\mathrm{\Theta })\le \lambda \rho (\vartheta ,\mathrm{\Theta }).$$

This is a contradiction.

Similarly, let $\omega \in B$ where $G\omega =\omega$ and $\omega \ne \mathrm{\Theta }$. Thereafter, we have

$$\rho (\omega ,\mathrm{\Theta })=\rho (G\omega ,F\mathrm{\Theta })\le \lambda \rho (\omega ,\mathrm{\Theta }).$$

This is a contradiction.

So, $\mathrm{\Theta }$ is a unique common fixed point for both F and G.

We obtain the convergence of Picard sequences defined by F and Picard sequences defined by G from

$$\rho (F^{n+1}\omega ,G\mathrm{\Theta })\le \lambda \rho (F^n\omega ,\mathrm{\Theta })\le \dots \le {\lambda }^{n+1}\rho (\vartheta ,\mathrm{\Theta }),$$

and

$$\rho (G^{n+1}\omega ,F\mathrm{\Theta })\le \lambda \rho (G^n\omega ,\mathrm{\Theta })\le \dots \le {\lambda }^{n+1}\rho (\omega ,\mathrm{\Theta }).$$

□

Remark 5.

In [32], the authors used additive metrics instead of max and obtained similar results.

From Theorem 1, we obtain the next corollary which generalizes the well-known results initiated by Bryant [16].

Corollary 1.

Let $(B,\rho ,s)$ be a $CbMS$, $\lambda \in [0,1)$, $f,g:B\to B$ and $n\in \mathbb{N}$, where

$$\rho (f^n\omega ,g^n\theta )\le \lambda \rho (\omega ,\theta ),$$

(9)

for all $u,v\in B$. Then, both f and g have a unique common fixed point $q\in B$. Also, q is a unique limit of all Picard sequences defined along f and a unique limit of all Picard sequences defined along g.

By Corollary 1, we arrive at the following common fixed point result that provides the theorem for Yen [21].

Corollary 2.

Let $(B,\rho ,s)$ be a $CbMS$, $\lambda \in [0,1)$, $f,g:B\to B$ and $m,n\in \mathbb{N}$, where

$$\rho (f^m\omega ,g^n\omega )\le \lambda \rho (\omega ,\theta ),$$

(10)

for all $\omega ,\theta \in \mathrm{\Omega }$. Then, f and g have a unique common fixed point $q\in B$. Also, q is a unique limit of all Picard sequences defined by f and a unique limit of all Picard sequences defined by g.

Proof. 

Put $\omega =f^n\mu$ and $\theta =g^m\nu$, where $\mu ,\nu \in B$. We have that $f^{m+n}$ and $g^{m+n}$ hold all conditions of Corollary 1. □

By Corollary 1, we obtain the next common fixed point, which expands upon the well-known theorem of R. Caccioppoli [17].

Corollary 3.

Let $(B,\rho ,s)$ be a $CbMS$, $\lambda \in [0,1)$, $f,g:B\to B$, $\left(c_n\right)$ be a sequence such that ${\sum }_{n=1}^{+\infty }{c}_n<+\infty ,c_n\ge 0,n\in \mathbb{N}$ and

$$\rho (f^n\omega ,g^n\theta )\le c_n\rho (\omega ,\theta )$$

(11)

for all $\omega ,\theta \in B,n\in \mathbb{N}$. Then, f and g have a unique common fixed point $q\in B$. Also, q is a unique limit for all Picard sequences defined along f and a unique limit for all Picard sequences defined along g.

Proof. 

For some positive integer n, we have $c_n<1$. Now, the statement follows from Corollary 1. □

Lemma 4.

Let $(B,\rho ,s)$ be a $CbMS$ with an (SC) property, $\lambda \in [0,1)$, $f,g:B\to B$ and ${\rho }_{*}:B^2\to [0,+\infty )$ be defined by ${\rho }_{*}(\omega ,\theta )=0 for \omega =\theta$ and

$${\rho }_{*}(\omega ,\theta )=max\{\rho (\omega ,\theta ),\rho (f\omega ,g\theta ),\dots ,\rho (f^n\omega ,g^n\theta )\}$$

(12)

for $\omega \ne \theta$. Then, space $(B,{\rho }_{*},s)$ is a $CbMS$ with an (SC) property.

Proof. 

The space $(B,{\rho }_{*},s)$ is $bMS$ because conditions (B1), (B2) and (B3) are trivial-satisfied. Also, we have $\rho (\omega ,\theta )\le {\rho }_{*}(\omega ,\theta )$ for any $\omega ,\theta \in X$. Further, if $\left({\theta }_j\right)\subseteq B$ is an arbitrary Cauchy sequence in $(B,{\rho }_{*})$, $\left({\theta }_j\right)$ is a Cauchy sequence in $(B,\rho )$, which implies that $(B,{\rho }_{*})$ is $CbMS$ because $(B,\rho ,s)$ is complete. Further, for every $k\in \left[n\right]$, we have ${{\displaystyle \overline{lim}}}_{i\to +\infty }\rho (f^k\omega ,g^k{\theta }_i)\le \rho (f^k\omega ,g^k\theta )$, which implies that ${\displaystyle \overline{lim}}{\rho }_{*}(\omega ,{\theta }_n)\le {\rho }_{*}(\omega ,\theta )$. Hence, $(B,\rho ,s)$ has the property (SC). □

The following theorem extends the previous results presented by M. Marjanović [18], from $CMS$ to $CbMS$.

Theorem 2.

Suppose that $(B,\rho ,s)$ is a $CbMS$ that satisfies the (SC) property, and $\lambda \in [0,1)$ and $f,g:B\to B$ are two mappings such that

$$\rho (f^{n+1}\omega ,g^{n+1}\theta )\le \lambda \underset{0\le i\le n}{max}\left\{\rho (f^i\omega ,g^i\theta )\right\},$$

(13)

for all $\omega ,\theta \in \mathrm{\Omega }$. Then, f and g have a unique common fixed point $z\in B$, which is a unique limit for all Picard sequences defined along f and a unique limit for all Picard sequences defined along g.

Proof. 

By Lemma 4, space $(B,{\rho }_{*},s)$ is a $CbMS$ that has the (SC) property. Further, we have that

$${\rho }_{*}(f^{n+1}\omega ,g^{n+1}\theta )\le \lambda {\rho }_{*}(\omega ,\theta ).$$

Based on the one-dimensional case of Theorem 1, it follows that $f^{n+1}$ and $g^{n+1}$ have a unique common fixed point, say q, which is a unique limit for all Picard sequences defined by $f^{n+1}$ and a unique common fixed point that is a unique limit for all Picard sequences defined by $f^{n+1}$. By Lemma 1 we obtain that q is a unique fixed point for f and a unique limit for all Picard sequences defined along f, and q is a unique fixed point for g and a unique limit for all Picard sequences defined along g. Hence, q is a unique common fixed point for f and g. □

Next, the common fixed point theorem extends the results from George et al. [14].

Theorem 3.

Suppose that $(B,\rho ,s)$ is a $CbMS$, $f,g:B\to B$ and $\alpha ,\beta \in [0,1)$, where $\alpha +2\beta <1$ and

$$\begin{array}{cc}\hfill \rho (fx,gy)& \le max\left\{\rho (u,v),\rho (u,fu),\rho (v,gv),\frac{\rho (u,gv)+\rho (fu,v)}{2s}\right\}\hfill \\ \hfill & +\frac{\beta \left[\rho \right(u,gv)+\rho (fu,v\left)\right]}{s},\hfill \end{array}$$

(14)

for all $u,v\in B$. If one of the next conditions are satisfied,

(1)

f and g are sequentially continuous or ρ is sequentially continuous;

(2)

$(B,\rho ,s)$ has the (SC) property and $s(\alpha +\beta )<1$,

then f and g have a unique common fixed point $z\in B$. Also, a sequence $\left(u_n\right)$ defined by $u_{2n+1}=f{x}_{2n},u_{2n+2}=g{u}_{2n+1},n=0,1,2,\dots$, where $u_0\in B$, converges to w.

Proof. 

Let $u_0\in B$ be arbitrary and a $\left(u_n\right)$ sequence defined by $u_{2n+1}=f{u}_{2n}$ and $u_{2n+2}=g{u}_{2n+1}$. Then, there exists $w\in \mathrm{\Omega }$ such that $w=lim{u}_n$; the proof is the same as that in [14] (Theorem 13).

Case (1): If f and g are sequentially continuous or $\rho$ is sequentially continuous, then by Theorem 13 in [14], we obtain that f and g have a unique common fixed point $w\in \mathrm{\Omega }$. The rest of the proof is like that in Case (2).

Case (2): Let $(B,\rho ,s)$ satisfy the (SC). From (14), we have

$$\begin{array}{cc}\hfill \rho (fw,g{u}_{2n+1})& \le \alpha max\{\rho (w,u_{2n+1}),\rho (w,fw),\rho (u_{2n+1},g{u}_{2n+1}),\hfill \\ \hfill & \frac{\rho (u_{2n+1},fw)+\rho (g{u}_{2n+1},w)}{2s}\}+\frac{\beta [\rho (u_{2n+1},fw)+\rho (g{u}_{2n+1},w)]}{s},\hfill \end{array}$$

which implies

$$\begin{array}{ccc}& & \overline{lim}\rho (fw,g{u}_{2n+1})\le \overline{lim}[\alpha max\{\rho (w,u_{2n+1}),\rho (w,fw),\rho (u_{2n+1},u_{2n+2}),\hfill \\ & & \frac{\rho (u_{2n+1},fw)+\rho (g{u}_{2n+1},w)}{2s}\}+\beta \frac{\rho (u_{2n+1},fw)}{s}+\beta \frac{\rho (g{u}_{2n+1},w)}{s}].\hfill \end{array}$$

So, we have that

$$\begin{array}{ccc}& & \overline{lim}\rho (g{u}_{2n+1},fw)\le \alpha max\{\rho (w,w),\rho (w,fw),\rho (w,w),\hfill \\ & & \frac{\rho (w,fw)+\rho (w,w)}{2s}\}+\beta \frac{\rho (w,fw)}{s}+\beta \frac{\rho (w,w)}{s}].\hfill \end{array}$$

Therefore,

$$\begin{array}{cc}& \overline{lim}\rho (g{u}_{2n+1},fw)\le (\alpha +\beta )\rho (w,fw).\end{array}$$

Now, we have

$$\rho (w,fw)\le s[\overline{lim}\rho (g{u}_{2n+1},fw)+\overline{lim}\rho (g{u}_{2n+1},w)]\le s(\alpha +\beta )\rho (w,fw).$$

(15)

Since $s(\alpha +\beta )\in [0,1)$ from (15), we have $w=f\left(w\right)$. Further, we have

$$\begin{array}{ccc}& & \rho (f{u}_{2n},gw)\le \alpha max\{\rho (u_{2n},w),),\rho (u_{2n},f{u}_{2n}),\rho (w,gw)\hfill \\ & & \frac{\rho (u_{2n},gw)+\rho (f{u}_{2n},w)}{2s}\}+\beta \frac{\rho (u_{2n},gw}{s}+\beta \frac{\rho (f{u}_{2n},w)}{s},\hfill \end{array}$$

which implies

$$\begin{array}{ccc}& & \overline{lim}\rho (f{u}_{2n},gw)\le \overline{lim}[\alpha max\{\rho (u_{2n},w),\rho (u_{2n},u_{2n+1}),\rho (w,gw),\hfill \\ & & \frac{\rho ((u_{2n},gz)+\rho (u_{2n+1},w))}{2s}\}+\beta \frac{\rho (u_{2n},gw)}{s}+\beta \frac{\rho (f{u}_{2n},w)}{s}].\hfill \end{array}$$

So, we have that

$$\begin{array}{ccc}& & \overline{lim}\rho (f{u}_{2n},gw)\le \alpha max\{\rho (w,w),\rho (w,fw),\rho (w,w),\hfill \\ & & \frac{\rho (w,fw)+\rho (w,w)}{2s}\}+\beta \frac{\rho (w,fw)}{s}+\beta \frac{\rho (w,w)}{s}].\hfill \end{array}$$

Therefore,

$$\overline{lim}\rho (f{u}_{2n},gw)\le (\alpha +\beta )\rho (w,gw).$$

Now,

$$\rho (w,gw)\le s[\overline{lim}\rho (f{u}_{2n},gw)+\overline{lim}\rho (f{u}_{2n},w)]\le s(\alpha +\beta )\rho (w,gw),$$

so $w=g\left(w\right)$.

Further, we prove that the fixed point is unique. Suppose that there are two common fixed points w and $w^{\prime }$, i.e. $gw=fw=w$ and $g{w}^{\prime }=f{w}^{\prime }=w^{\prime }$. Then, we obtain

$$\begin{array}{ccc}\hfill d(w,w^{\prime })=d(fw,g{w}^{\prime })& \le & \alpha \{d(w,w^{\prime }),d(w,w)d(w^{\prime },w^{\prime }),\frac{d(w,w^{\prime })+d(w,w^{\prime })}{2s}\}\hfill \\ & +& \beta \frac{d(w,w^{\prime })}{2s}+\beta d(w,w^{\prime })\le (\alpha +2\beta )d(w,w^{\prime }),\hfill \end{array}$$

which implies that $w=w^{\prime }$.

Finally, we prove the convergence of sequences of a corresponding Picard iteration:

$$\rho (f^{n+1}u,gw)\le \lambda \rho (f^{n}u,w))\le \dots \le {\lambda }^{n+1}\rho (u,w),$$

and

$$\rho (g^{n+1}v,fw)\le \lambda \rho (g^{n}v,w))\le \dots \le {\lambda }^{n+1}\rho (v,w).$$

□

The next Corollary extends the known results presented by Reich [19], Bianchini [15], Singh [33], Srivastava, and Gupta [11] and Ray [34] to b-metric spaces.

Corollary 4.

Suppose that $(B,\rho ,s)$ is a $CbMS$, $\lambda \in [0,1)$ and $f,g:B\to B$. Suppose that

$$d(f\omega ,g\theta )\le \lambda max\left\{\rho \right(\omega ,\theta ),\rho (\omega ,f\omega ),\rho (\theta ,g\theta \left)\right\},$$

(16)

for all $\omega ,\theta \in B$ and one of the following conditions is satisfied:

(1)

f and g are sequentially continuous or ρ is sequentially continuous;

(2)

$(B,\rho ,s)$ satisfies the (SC) property and $s\lambda <1$.

Then, f and g have a unique common fixed point $w\in \mathrm{\Omega }$. Also, w is a unique limit of all Picard sequences defined by f and a unique limit of all Picard sequences defined by g.

Finally, we prove the following statement, which extends earlier results presented by Marjanović [18] and Zamfirescu [22].

Corollary 5.

Let $(B,\rho ,s)$ be a $CbMS$ with the (SC) property, $\lambda \in [0,1)$ and $f,g:\mathrm{\Omega }\to \mathrm{\Omega }$. If there exist positive integers i, j and k such that $m=max\{i,j,k\}$

$$\begin{array}{ccc}& & \rho (f^{m+1}\omega ,g^{m+1}\theta )\le \lambda max\{\underset{0\le n\le i}{max}\left\{\rho (f^n\omega ,g^n\theta )\right\},\hfill \\ & & \frac{\underset{0\le n\le j-1}{max}\{\rho (f^n\omega ,g^{n+1}\theta \}+\underset{0\le n\le k-1}{max}\left\{\rho (g^n\omega ,f^{n+1}\theta )\right\}}{2}\},\hfill \end{array}$$

for all $\omega ,\theta \in \mathrm{\Omega }$, then f and g have a unique common fixed point $w\in X$. Also, w is a unique limit of all Picard sequences defined by f and a unique limit of all Picard sequences defined by g.

Proof. 

By Lemma 4, the space $(B,{\rho }_{*},s)$ is a $CbMS$ with the (SC) property. Further, we have that

$${\rho }_{*}(f^{m+1}\omega ,g^{m+1}\theta )\le \lambda max\{{\rho }_{*}(\omega ,\theta ),{\rho }_{*}(\omega ,f\theta ),{\rho }_{*}(g\omega ,\theta ))\}.$$

By Theorem 3, we obtain that $f^{m+1}$ and $g^{m+1}$ have a unique common fixed point q, and q is a unique limit of all Picard sequences defined by $f^{n+1}$ and a unique limit of all Picard sequences defined by $f^{n+1}$. By Lemma 1, we obtain that q is a unique fixed point for f and a unique limit of all Picard sequences defined by f, and q is a unique fixed point for g and a unique limit of all Picard sequences defined by g. Hence, q is a unique common fixed point for f and g. □

Remark 6.

Note that Theorem 3 generalizes the classical results presented by Ćirić [13] and Wong [12] obtained in complete metric spaces.

4. Some Examples

Example 1.

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

$$d_p(u,v)={\left(\sum _{n=1}^{+\infty }{|u_n-v_n|}^p\right)}^{\frac{1}{p}}$$

where $u=\left\{u_n\right\},v=\left\{v_n\right\}\in l^p$ is not a metric space (the function $d_p$ does not satisfy the triangle inequality), but $(l^p,d_p,s)$ is a b-metric space with $s=2^{\frac{1}{p}-1}$ [35,36]. Let $f,g:l^p\to l^p,i=1,2$ be defined by

$$f\left(u\right)=\left\{\begin{array}{cc}(0,\frac{u_1}{4},\frac{u_2}{4},\frac{u_3}{4},\dots ),\hfill & ifx\ne (1,0,0,0,\dots ),\hfill \\ (\frac{1}{4},0,0,0,\dots ),\hfill & ifu=(1,0,0,0,\dots ),\hfill \end{array}\right.$$

$$g\left(v\right)=(0,\frac{v_1}{4},\frac{v_2}{4},\frac{v_3}{4},\dots ).$$

Then, we have

(1)

If $u,v\in l^p\setminus \left\{(1,0,0,0,\dots )\right\}$, then it is $d_p(fu,gv)\le \frac{1}{4}{d}_p(u,v)$;

(2)

If $u=(1,0,0,0,\dots ),v\in l^p\setminus \left\{(1,0,0,0,\dots )\right\}$, then it is

$$\begin{array}{ccc}\hfill d_p(fu,gv)& =& d_p((\frac{1}{4},0,0,0,\dots ),(0,\frac{v_1}{4},\frac{v_2}{4},\frac{v_3}{4},\dots ))\hfill \\ & =& \frac{1}{4}(1+\sum _{i=1}^{+\infty }|y_1{{|}^p)}^{\frac{1}{p}}\hfill \\ & =& \frac{1}{4}{d}_p(u,gv);\hfill \end{array}$$

(3)

If $u\in l^p\setminus \left\{(1,0,0,0,\dots )\right\},v=(1,0,0,0,\dots )$, then we obtain

$$\begin{array}{ccc}\hfill d_p(fu,gv)& =& d_p((0,\frac{u_1}{4},\frac{u_2}{4},\frac{u_3}{4},\dots ),(0,\frac{1}{4},0,0,\dots ))\hfill \\ & =& {\left({\left|\frac{u_1-1}{4}\right|}^p+{\left|\frac{u_2}{4}\right|}^p+{\left|\frac{u_3}{4}\right|}^p+{\left|\frac{u_3}{4}\right|}^p+\cdots \right)}^{\frac{1}{p}}\hfill \\ & =& \frac{1}{4}{d}_p(u,v);\hfill \end{array}$$

(4)

If $u=v=(1,0,0,0,\dots )$, we obtain

$$\begin{array}{ccc}\hfill d_p(fu,gv)& =& (\frac{1}{4},0,0,0,\dots ),(0,\frac{1}{4},0,0,\dots ))\hfill \\ & =& {\left(\frac{1}{4^p}+\frac{1}{4^p}\right)}^{\frac{1}{p}}\hfill \\ & =& \frac{2^{\frac{1}{p}}}{4}.\hfill \end{array}$$

On the other hand,

$$\begin{array}{ccc}\hfill d_p(u,fu)& =& d_p((1,0,0,0,\dots ),(\frac{1}{4},0,0,0,\dots ))\hfill \\ & =& \frac{3}{4}.\hfill \end{array}$$

So, we conclude that the conditions of Corollary 4 are satisfied for $\lambda =\frac{2^{\frac{1}{p}}}{3}<1$ and $p\in ({log}_{3}2,1)$.

Example 2.

Let $B=[0,1]$ and $\rho (a,b)={(a-b)}^2$ for any $a,b\in B$. Then, $(B,\rho ,2)$ is $CbMS$ and ρ is sequentially continuous. Let

$$f\left(u\right)=\left\{\begin{array}{c}\frac{u}{2},if u\in [0,1);\hfill \\ \frac{9}{10},if u=1,\hfill \end{array}\right.$$

and $g\left(v\right)=\frac{v}{2}$ for every $v\in [0,1]$. We have that $\rho (u,v)=\frac{1}{4}{(u-v)}^2$ for $u,v\in [0,1)$. Further, we have

$$\rho (f1,g0)=\frac{81}{100}<1=\rho (1,0)$$

and

$$\rho (f1,g1)=\frac{16}{100}<\frac{25}{100}=\rho (1,g0).$$

Since the conditions of Corollary 4 are satisfied for $\lambda =\frac{9}{10}$, we conclude that f and g have a unique common fixed point $w\in \mathrm{\Omega }$, which is a unique limit of all Picard sequences defined by f and a unique limit of all Picard sequences defined by g.

5. Conclusions

We present some theorems that extended some results on the existence and uniqueness of a fixed point obtained for $d^{*}$-complete topological spaces in [9]. The significance of our improvement is that we obtained results about common fixed points for two mappings that do not possess the property of commutativity. Our results are given on $CbMS$ with the (SC) property. Our results generalize previous results in the literature. Also, we considered some properties of b-spaces, a class of topological spaces that belong to E-spaces and includes metric spaces.