Common fixed point theorems for two weakly compatible self-mappings in cone b-metric spaces

Abstract

In this paper, we establish common fixed point theorems for two weakly compatible self-mappings satisfying the contractive condition or the quasi-contractive condition in the case of a quasi-contractive constant $\lambda \in (0,1/s)$ in cone b-metric spaces without the normal cone, where the coefficient s satisfies $s\ge 1$. The main results generalize, extend and unify several well-known comparable results in the literature.

1 Introduction and preliminaries

Huang and Zhang [1] introduced the concept of a cone metric space, proved the properties of sequences on cone metric spaces and obtained various fixed point theorems for contractive mappings. The existence of a common fixed point on cone metric spaces was considered recently in [2–5]. Also, Ilic and Rakocevic [6] introduced a quasi-contraction on a cone metric space when the underlying cone was normal. Later on, Kadelburg et al. obtained a few similar results without the normality of the underlying cone, but only in the case of a quasi-contractive constant $\lambda \in (0,1/2)$. However, Gajic [7] proved that result is true for $\lambda \in (0,1)$ on a cone metric space by a new way, which answered the open question whether the result is true for $\lambda \in (0,1)$. Recently, Hussain and Shah [8] introduced cone b-metric spaces, as a generalization of b-metric spaces and cone metric spaces, and established some important topological properties in such spaces. Following Hussain and Shah, Huang and Xu [9] obtained some interesting fixed point results for contractive mappings in cone b-metric spaces. Although Ion Marian [10] proved some common fixed point theorems in complete b-cone metric spaces, the main ways of the proof depend strongly on the nonlinear scalarization function ${\xi }_e:Y\to \mathbb{R}$. In the present paper, we will show common fixed point theorems for two weakly compatible self-mappings satisfying the contractive condition or quasi-contractive condition in the case of a quasi-contractive constant $\lambda \in (0,1/s)$ in cone b-metric spaces without the assumption of normality, where the coefficient s satisfies $s\ge 1$. As consequences, our results generalize, extend and unify several well-known comparable results (see, for example, [2–7, 9–13]).

Consistent with Huang and Zhang [1], the following definitions and results will be needed in the sequel.

Let E be a real Banach space and let P be a subset of E. By θ we denote the zero element of E and by intP the interior of P. The subset P is called a cone if and only if:

• (i) P is closed, nonempty, and $P\ne \{\theta \}$;

• (ii) $a,b\in \mathbb{R}$, $a,b\ge 0$, $x,y\in P\Rightarrow ax+by\in P$;

• (iii) $P\cap (-P)=\{\theta \}$.

On this basis, we define a partial ordering ⪯ with respect to P by $x⪯y$ if and only if $y-x\in P$. We write $x\prec y$ to indicate that $x⪯y$ but $x\ne y$, while $x\ll y$ stands for $y-x\in intP$. Write $\parallel \cdot \parallel$ as the norm on E. The cone P is called normal if there is a number $K>0$ such that for all $x,y\in E$, $\theta ⪯x⪯y$ implies $\parallel x\parallel \le K\parallel y\parallel$. The least positive number satisfying the above is called the normal constant of P. It is well known that $K\ge 1$.

In the following, we always suppose that E is a Banach space, P is a cone in E with $intP\ne \mathrm{\varnothing }$ and ⪯ is a partial ordering with respect to P.

Definition 1.1 [8]

Let X be a nonempty set and let $s\ge 1$ be a given real number. A mapping $d:X\times X\to E$ is said to be cone b-metric if and only if for all $x,y,z\in X$ the following conditions are satisfied:

• (i) $\theta \prec d(x,y)$ with $x\ne y$ and $d(x,y)=\theta$ if and only if $x=y$;

• (ii) $d(x,y)=d(y,x)$;

• (iii) $d(x,y)⪯s[d(x,z)+d(z,y)]$.

The pair $(X,d)$ is called a cone b-metric space.

Example 1.2 Consider the space $L_p$ ($0<p<1$) of all real function $x(t)$ ($t\in [0,1]$) such that ${\int }_0^1{|x(t)|}^{p}dt<\mathrm{\infty }$. Let $X=L_p$, $E={\mathbb{R}}^2$, $P=\{(x,y)\in E\mid x,y\ge 0\}\subset {\mathbb{R}}^2$ and $d:X\times X\to E$ such that

$$d(x,y)=(\alpha {\left\{{\int }_0^1{|x(t)-y(t)|}^{p}dt\right\}}^{\frac{1}{p}},\beta {\left\{{\int }_0^1{|x(t)-y(t)|}^{p}dt\right\}}^{\frac{1}{p}}),$$

where $\alpha ,\beta \ge 0$ are constants. Then $(X,d)$ is a cone b-metric space with the coefficient $s=2^{\frac{1}{p}-1}$.

Remark 1.3 It is obvious that any cone metric space must be a cone b-metric space. Moreover, cone b-metric spaces generalize cone metric spaces, b-metric spaces and metric spaces.

Definition 1.4 [8]

Let $(X,d)$ be a cone b-metric space, $x\in X$ and $\{x_n\}$ be a sequence in X. Then

• (i) $\{x_n\}$ converges to x whenever, for every $c\in E$ with $\theta \ll c$, there is a natural number N such that $d(x_n,x)\ll c$ for all $n\ge N$. We denote this by ${lim}_{n\to \mathrm{\infty }}{x}_n=x$ or $x_n\to x$ ($n\to \mathrm{\infty }$).

• (ii) $\{x_n\}$ is a Cauchy sequence whenever, for every $c\in E$ with $\theta \ll c$, there is a natural number N such that $d(x_n,x_m)\ll c$ for all $n,m\ge N$.

• (iii) $(X,d)$ is a complete cone b-metric space if every Cauchy sequence is convergent.

Lemma 1.5 [8]

Let $(X,d)$ be a cone b-metric space. The following properties are often used while dealing with cone b-metric spaces in which the cone is not necessarily normal.

• (1) If $u\ll v$ and $v⪯w$, then $u\ll w$;

• (2) If $\theta ⪯u\ll c$ for each $c\in intP$, then $u=\theta$;

• (3) If $a⪯b+c$ for each $c\in intP$, then $a⪯b$;

• (4) If $\theta ⪯d(x_n,x)⪯b_n$ and $b_n\to \theta$, then $x_n\to x$;

• (5) If $a⪯\lambda a$, where $a\in P$ and $0<\lambda <1$, then $a=\theta$;

• (6) If $c\in intP$, $\theta ⪯a_n$ and $a_n\to \theta$, then there exists $n_0\in \mathbb{N}$ such that $a_n\ll c$ for all $n>n_0$.

Lemma 1.6 [8]

The limit of a convergent sequence in a cone b-metric space is unique.

Definition 1.7 [2]

The mappings $f,g:X\to X$ are weakly compatible if for every $x\in X$, $fgx=gfx$ holds whenever $fx=gx$.

Definition 1.8 [3]

Let f and g be self-maps of a set X. If $w=fx=gx$ for some x in X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g.

Lemma 1.9 [3]

Let f and g be weakly compatible self-maps of a set X. If f and g have a unique point of coincidence $w=fx=gx$, then w is the unique common fixed point of f and g.

Definition 1.10 [13]

Let $(X,d)$ be a cone metric space. A mapping $f:X\to X$ is such that, for some constant $\lambda \in (0,1)$ and for every $x,y\in X$, there exists an element

$$u\in C(g,x,y)=\{d(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy),d(gy,fx)\}$$

for which $d(fx,fy)⪯\lambda u$ is called a g-quasi-contraction.

2 Main results

In this section, we give some common fixed point results for two weakly compatible self-mappings satisfying the contractive condition and quasi-contractive condition in the case of a contractive constant $\lambda \in (0,1/s)$ in cone b-metric spaces without the assumption of normality.

Theorem 2.1 Let $(X,d)$ be a cone b-metric space with the coefficient $s\ge 1$ and let $a_i\ge 0$ ($i=1,2,3,4,5$) be constants with $2s{a}_1+(s+1)(a_2+a_3)+(s^2+s)(a_4+a_5)<2$. Suppose that the mappings $f,g:X\to X$ satisfy the condition, for all $x,y\in X$,

$$d(fx,fy)⪯a_{1}d(gx,gy)+a_{2}d(gx,fx)+a_{3}d(gy,fy)+a_{4}d(gx,fy)+a_{5}d(gy,fx).$$

(2.1)

If the range of g contains the range of f and $g(X)$ or $f(X)$ is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in X.

Proof For an arbitrary $x_0\in X$, since $f(X)\subset g(X)$, there exists an $x_1\in X$ such that $f{x}_0=g{x}_1$. By induction, a sequence $\{x_n\}$ can be chosen such that $f{x}_n=g{x}_{n+1}$ ($n\ge 1$). If $g{x}_{n_0-1}=g{x}_{n_0}=f{x}_{n_0-1}$ for some natural number $n_0$, then $x_{n_0-1}$ is a coincidence point of f and g in X. Suppose that $g{x}_{n-1}\ne g{x}_n$ for all $n\ge 1$.

Thus, by (2.1) for any $n\in \mathbb{N}$, we have

$$\begin{array}{rcl}d(g{x}_{n+1},g{x}_n)& =& d(f{x}_n,f{x}_{n-1})\\ ⪯& a_{1}d(g{x}_n,g{x}_{n-1})+a_{2}d(g{x}_n,f{x}_n)\\ +a_{3}d(g{x}_{n-1},f{x}_{n-1})+a_{4}d(g{x}_n,f{x}_{n-1})+a_{5}d(g{x}_{n-1},f{x}_n)\end{array}$$

and

$$\begin{array}{rcl}d(g{x}_n,g{x}_{n+1})& =& d(f{x}_{n-1},f{x}_n)\\ ⪯& a_{1}d(g{x}_{n-1},g{x}_n)+a_{2}d(g{x}_{n-1},f{x}_{n-1})\\ +a_{3}d(g{x}_n,f{x}_n)+a_{4}d(g{x}_{n-1},f{x}_n)+a_{5}d(g{x}_n,f{x}_{n-1}).\end{array}$$

Hence

$$\begin{array}{rcl}2d(g{x}_n,g{x}_{n+1})& =& d(g{x}_{n+1},g{x}_n)+d(g{x}_n,g{x}_{n+1})\\ ⪯& (2a_1+a_2+a_3+s{a}_4+s{a}_5)d(g{x}_n,g{x}_{n-1})\\ +(a_2+a_3+s{a}_4+s{a}_5)d(g{x}_{n+1},g{x}_n).\end{array}$$

Since $2s{a}_1+(s+1)(a_2+a_3)+(s^2+s)(a_4+a_5)<2$, we have

$$\begin{array}{rcl}d(g{x}_n,g{x}_{n+1})& ⪯& \frac{2a_1+a_2+a_3+s{a}_4+s{a}_5}{2-a_2-a_3-s{a}_4-s{a}_5}d(g{x}_n,g{x}_{n-1})\\ =& kd(g{x}_n,g{x}_{n-1})⪯k^{2}d(g{x}_{n-1},g{x}_{n-2})\\ ⪯& k^{3}d(g{x}_{n-2},g{x}_{n-3})⪯\cdots ⪯k^{n}d(g{x}_1,g{x}_0),\end{array}$$

where $k=\frac{2a_1+a_2+a_3+s{a}_4+s{a}_5}{2-a_2-a_3-s{a}_4-s{a}_5}$. Obviously, $k\in [0,\frac{1}{s})$.

Thus, setting any positive integers m and n, we have

$$\begin{array}{rcl}d(g{x}_n,g{x}_{n+m})& ⪯& sd(g{x}_n,g{x}_{n+1})+sd(g{x}_{n+1},g{x}_{n+m})\\ ⪯& sd(g{x}_n,g{x}_{n+1})+s^{2}d(g{x}_{n+1},g{x}_{n+2})+s^{2}d(g{x}_{n+2},g{x}_{n+m})\\ ⪯& sd(g{x}_n,g{x}_{n+1})+s^{2}d(g{x}_{n+1},g{x}_{n+2})+s^{3}d(g{x}_{n+2},g{x}_{n+3})\\ +\cdots +s^{m-1}d(g{x}_{n+m-2},g{x}_{n+m-1})+s^{m-1}d(g{x}_{n+m-1},g{x}_{n+m})\\ ⪯& sd(g{x}_n,g{x}_{n+1})+s^{2}d(g{x}_{n+1},g{x}_{n+2})+s^{3}d(g{x}_{n+2},g{x}_{n+3})\\ +\cdots +s^{m-1}d(g{x}_{n+m-2},g{x}_{n+m-1})+s^{m}d(g{x}_{n+m-1},g{x}_{n+m})\\ ⪯& (s{k}^n+s^2{k}^{n+1}+\cdots +s^m{k}^{n+m-1})d(g{x}_1,g{x}_0)\\ =& \frac{s{k}^n[1-{(sk)}^m]}{1-sk}d(g{x}_1,g{x}_0)\\ ⪯& \frac{s{k}^n}{1-sk}d(g{x}_1,g{x}_0).\end{array}$$

Since $k\in [0,1/s)$, we notice that $\frac{s{k}^n}{1-sk}d(g{x}_1,g{x}_0)\to \theta$ as $n\to \mathrm{\infty }$ for any $m\in {\mathbb{N}}_{+}$. By Lemma 1.5, for any $c\in intP$, we can choose $n_0\in \mathbb{N}$ such that $\frac{s{k}^n}{1-sk}d(g{x}_1,g{x}_0)\ll c$ for all $n>n_0$. Thus, for each $c\in intP$, $d(g{x}_{n+m},g{x}_n)\ll c$ for all $n>n_0$, $m\ge 1$. Therefore $\{g{x}_n\}$ is a Cauchy sequence in $g(X)$.

If $g(X)\subset X$ is complete, there exist $q\in g(X)$ and $p\in X$ such that $g{x}_n\to q$ as $n\to \mathrm{\infty }$ and $gp=q$. (If $f(X)$ is complete, there exists $q\in f(X)$ such that $f{x}_n\to q$ as $n\to \mathrm{\infty }$. Since $f(X)\subset g(X)$, we can find $p\in X$ such that $gp=q$.)

Now, from (2.1) we show that $fp=q$,

$$\begin{array}{rcl}d(g{x}_{n+2},fp)& =& d(f{x}_{n+1},fp)\\ ⪯& a_{1}d(g{x}_{n+1},q)+a_{2}d(g{x}_{n+1},g{x}_{n+2})\\ +a_{3}d(q,fp)+a_{4}d(g{x}_{n+1},fp)+a_{5}d(q,g{x}_{n+2}).\end{array}$$

Similarly,

$$\begin{array}{rcl}d(fp,g{x}_{n+2})& =& d(fp,f{x}_{n+1})\\ ⪯& a_{1}d(q,g{x}_{n+1})+a_{2}d(q,fp)\\ +a_{3}d(g{x}_{n+1},g{x}_{n+2})+a_{4}d(q,g{x}_{n+2})+a_{5}d(g{x}_{n+1},fp),\end{array}$$

thus, we have

$$\begin{array}{rcl}2d(g{x}_{n+2},fp)& ⪯& 2a_{1}d(g{x}_{n+1},q)+(a_2+a_3)d(g{x}_{n+1},g{x}_{n+2})+(a_2+a_3)d(q,fp)\\ +(a_4+a_5)d(g{x}_{n+1},fp)+(a_4+a_5)d(q,g{x}_{n+2})\\ ⪯& (2s{a}_1+s{a}_2+s{a}_3+a_4+a_5)d(g{x}_{n+2},q)\\ +(s{a}_2+s{a}_3+s{a}_4+s{a}_5)d(g{x}_{n+2},fp)\\ +(2s{a}_1+a_2+a_3+s{a}_4+s{a}_5)d(g{x}_{n+1},g{x}_{n+2}).\end{array}$$

Since $0\le a_2+a_3+a_4+a_5<2/s$, by the triangular inequality, it follows that

$$\begin{array}{rcl}d(g{x}_{n+2},fp)& ⪯& \frac{2s{a}_1+s{a}_2+s{a}_3+a_4+a_5}{2-s{a}_2-s{a}_3-s{a}_4-s{a}_5}d(g{x}_{n+2},q)\\ +\frac{2s{a}_1+a_2+a_3+s{a}_4+s{a}_5}{2-s{a}_2-s{a}_3-s{a}_4-s{a}_5}d(g{x}_{n+1},g{x}_{n+2}).\end{array}$$

Since $\{g{x}_n\}$ is a Cauchy sequence and $g{x}_n\to q$ ($n\to \mathrm{\infty }$), for any $c\in intP$, we can choose $n_1\in \mathbb{N}$ such that for all $n\ge n_1$,

$$d(g{x}_{n+1},g{x}_{n+2})\ll \frac{(2-s{a}_2-s{a}_3-s{a}_4-s{a}_5)c}{2(2s{a}_1+a_2+a_3+s{a}_4+s{a}_5)}$$

and

$$d(g{x}_{n+2},q)\ll \frac{(2-s{a}_2-s{a}_3-s{a}_4-s{a}_5)c}{2(2s{a}_1+s{a}_2+s{a}_3+a_4+a_5)}.$$

Thus, for any $c\in intP$, $d(g{x}_{n+2},fp)\ll c$ for all $n\ge n_1$. Therefore, by Lemma 1.5, we have $fp=q=gp$.

Assume that there exist u, w in X such that $fu=gu=w$.

$$\begin{array}{rcl}d(gu,gp)& =& d(fu,fp)\\ ⪯& a_{1}d(gu,gp)+a_{2}d(fu,gu)+a_{3}d(fp,gp)+a_{4}d(fp,gu)+a_{5}d(fu,gp)\\ =& (a_1+a_4+a_5)d(gu,gp).\end{array}$$

Since $0\le a_1+a_4+a_5<1$, by Lemma 1.5, we can obtain that $d(gu,gp)=\theta$, i.e., $w=gu=gp=q$. Moreover, the mappings f and g are weakly compatible, by Lemma 1.9, we know that q is the unique common fixed point of f and g. □

Example 2.2 Let $E=C_{\mathbb{R}}^1([0,1])$, $P=\{\phi \in E:\phi \ge 0\}\subset E$, $X=[1,\mathrm{\infty })$ and $d(x,y)={|x-y|}^2{e}^t$. Then $(X,d)$ is a cone b-metric space with the coefficient $s=2$, but it is not a cone metric space. We consider the functions $f,g:X\to X$ defined by $fx=\frac{1}{6}lnx+1$, $gx=lnx+1$. Hence

$$\begin{array}{rcl}d(fx,fy)& =& {|\frac{1}{6}lnx+1-\frac{1}{6}lny-1|}^2{e}^t\\ ⪯& {|\frac{1}{6}lnx+\frac{1}{6}lny|}^2{e}^t\\ =& {|\frac{1}{5}(lnx-\frac{1}{6}lny)+\frac{1}{5}(lny-\frac{1}{6}lnx)|}^2{e}^t\\ ⪯& \frac{2}{25}{|lnx-\frac{1}{6}lny|}^2{e}^t+\frac{2}{25}{|lny-\frac{1}{6}lnx|}^2{e}^t\\ =& \frac{2}{25}d(gx,fy)+\frac{2}{25}d(gy,fx).\end{array}$$

Here $1\in X$ is the unique common fixed point of f and g.

Example 2.3 Let X be the set of Lebesgue measurable functions on $[0,1]$ such that ${\int }_0^1{|u(x)|}^{2}dx<\mathrm{\infty }$, $E=C_{\mathbb{R}}([0,1])$, $P=\{\phi \in E:\phi \ge 0\}\subset E$. We define $d:X\times X\to E$ as

$$d(u(t),v(t))=e^t{\int }_0^1{|u(s)-v(s)|}^{2}ds,$$

for all $x,y\in X$. Then $(X,d)$ is a cone b-metric space with the coefficient $s=2$, but it is not a cone metric space. Considering the functions $fu=\frac{1}{4}u(t)$ and $gu=\frac{1}{2}u(t)$ ($t\in [0,1]$), we have

$$\begin{array}{rcl}d(fu,fv)& =& e^t{\int }_0^1{|\frac{1}{4}u(s)-\frac{1}{4}v(s)|}^{2}ds\\ =& \frac{e^t}{4}{\int }_0^1{|\frac{1}{2}u(s)-\frac{1}{2}v(s)|}^{2}ds\\ =& \frac{1}{4}d(gu,gv).\end{array}$$

Clearly, $0\in X$ is the unique common fixed point of f and g.

Remark 2.4 Compared with the common fixed point results on cone metric spaces in [2, 3, 5], the common fixed point theorems in complete b-cone metric spaces in [10] and the fixed point results in cone b-metric spaces in [9], Theorem 2.1 is shown to be a proper generalization by Examples 2.2 and 2.3. Furthermore, Theorem 2.1 generalizes and unifies [[9], Theorem 2.1 and 2.3].

Definition 2.5 Let $(X,d)$ be a cone b-metric space with the coefficient $s\ge 1$. A mapping $f:X\to X$ is such that, for some constant $\lambda \in (0,1/s)$ and for every $x,y\in X$, there exists an element

$$\nu \in C(g,x,y)=\{d(gx,gy),d(gx,fx),d(gy,fy),d(gx,fy),d(gy,fx)\}$$

(2.2)

for which $d(fx,fy)⪯\lambda u$ is called a g-quasi-contraction.

Theorem 2.6 Let $(X,d)$ be a cone b-metric space with the coefficient $s\ge 1$ and let the mapping $f:X\to X$ be a g-quasi-contraction. If the range of g contains the range of f and $g(X)$ or $f(X)$ is a complete subspace of X, then f and g have a unique point of coincidence in X. Moreover, if f and g are weakly compatible, then f and g have a unique common fixed point in X.

Proof For each $x_0\in X$, set $g{x}_1=f{x}_0$ and $g{x}_{n+1}=f{x}_n$ ($n\in \mathbb{N}$). If $g{x}_{n_0-1}=g{x}_{n_0}=f{x}_{n_0-1}$ for some natural number $n_0$, then $x_{n_0-1}$ is a coincidence point of f and g in X.

Suppose that $g{x}_{n-1}\ne g{x}_n$ for all $n\ge 1$. Now we prove that $\{g{x}_n\}$ is a Cauchy sequence. First, we show that

$$d(g{x}_n,g{x}_1)=d(f{x}_{n-1},f{x}_0)⪯\frac{s\lambda }{1-s\lambda }d(g{x}_1,g{x}_0)\text{for all }n\in {\mathbb{N}}_{+}.$$

(2.3)

Clearly, we note (2.3) holds when $n=1$. We assume that (2.3) holds for some $n\le N-1$ ($N\in {\mathbb{N}}_{+}$), then we prove that (2.3) holds for all $n=N$. Because f is a g-quasi-contractive mapping, there exists a real number $k\le N$ such that

$$d(g{x}_N,g{x}_1)⪯\lambda d(g{x}_k,g{x}_0).$$

(2.4)

In order to prove that (2.4) holds, we show that for all $1\le i,j\le N$, there exists $1\le k\le N$ such that

$$d(g{x}_i,g{x}_j)⪯\lambda d(g{x}_k,g{x}_0).$$

(2.5)

Clearly, (2.5) is true for $N=1$. Suppose that (2.5) is true for each $N=P\in \mathbb{N}$, that is, for all $1\le i,j\le P$, there exists $1\le k\le P$ such that

$$d(g{x}_i,g{x}_j)⪯\lambda d(g{x}_k,g{x}_0).$$

(2.6)

Let us prove (2.5) holds for $N=P+1$.

By (2.6), we only show that for any $1\le i_0\le P+1$, there exists $1\le k\le P+1$ such that

$$d(g{x}_{P+1},g{x}_{i_0})⪯\lambda d(g{x}_k,g{x}_0).$$

Since f is a g-quasi-contractive mapping, there exists

$$\begin{array}{rcl}{\nu }_{i_0}\in C(g,x_P,x_{i_0-1})& =& \{d(g{x}_P,g{x}_{i_0-1}),d(g{x}_P,g{x}_{P+1}),\\ d(g{x}_{i_0-1},g{x}_{i_0}),d(g{x}_P,g{x}_{i_0}),d(g{x}_{i_0-1},g{x}_{P+1})\}\end{array}$$

such that $d(g{x}_{P+1},g{x}_{i_0})⪯\lambda {\nu }_{i_0}$.

By (2.6), we discuss that there exists an element

$$\begin{array}{rcl}d(g{x}_{P+1},g{x}_{i_1})& \in & \{d(g{x}_P,g{x}_{i_0-1}),d(g{x}_P,g{x}_{P+1}),d(g{x}_{i_0-1},g{x}_{i_0}),\\ d(g{x}_P,g{x}_{i_0}),d(g{x}_{i_0-1},g{x}_{P+1})\}\end{array}$$

such that $d(g{x}_{P+1},g{x}_{i_0})⪯\lambda d(g{x}_{P+1},g{x}_{i_1})$ ($1\le i_1\le P+1$).

If the above inequality does not hold for $1\le i_1\le P+1$, then (2.5) is true for $N=P+1$ by (2.6).

We continue in the same way, and after $P+1$ steps, we get $1\le i_j\le P+1$ ($0\le j\le P+1$) such that

$$d(g{x}_{P+1},g{x}_{i_j})⪯\lambda d(g{x}_{P+1},g{x}_{i_{j+1}})(0\le j\le P).$$

Notice that there exist $0\le r<s\le P+1$ such that $i_r=i_s$. That is,

$$d(g{x}_{P+1},g{x}_{i_r})⪯{\lambda }^{s-r}d(g{x}_{P+1},g{x}_{i_s})={\lambda }^{s-r}d(g{x}_{P+1},g{x}_{i_r})(0\le r<s\le P+1).$$

As $\lambda \in (0,1)$, by Lemma 1.5(5), we get a contradiction. From (2.6), (2.5) is true for $N=P+1$.

Hence, (2.5) is true for all $N\in \mathbb{N}$, which implies that (2.4) holds for $N\in \mathbb{N}$.

Next, let us prove that for all $n\in {\mathbb{N}}_{+}$,

$$d(g{x}_n,g{x}_0)⪯\frac{s}{1-s\lambda }d(g{x}_0,g{x}_1).$$

(2.7)

Using the triangular inequality, from (2.3) we obtain

$$\begin{array}{rcl}d(g{x}_n,g{x}_0)& ⪯& s[d(g{x}_n,g{x}_1)+d(g{x}_1,g{x}_0)]\\ ⪯& \frac{s^2\lambda }{1-s\lambda }d(g{x}_0,g{x}_1)+sd(g{x}_1,g{x}_0)\\ =& \frac{s}{1-s\lambda }d(g{x}_1,g{x}_0).\end{array}$$

Now, we show that $\{g{x}_n\}$ is a Cauchy sequence. For all $n>m$, there exists

$$\begin{array}{rcl}{\nu }_1\in C(g,x_{m-1},x_{n-1})& =& \{d(g{x}_{m-1},g{x}_{n-1}),d(g{x}_{m-1},g{x}_m),\\ d(g{x}_{n-1},g{x}_n),d(g{x}_{m-1},g{x}_n),d(g{x}_m,g{x}_{n-1})\}\end{array}$$

(2.8)

such that $d(g{x}_m,g{x}_n)=d(f{x}_{m-1},f{x}_{n-1})⪯\lambda {\nu }_1$.

By the contractive condition, there exist but not all

$${\nu }_k\in \{d(g{x}_i,g{x}_j)|0\le i<j\le n\}(k=1,2,3,\dots ,m)$$

such that

$${\nu }_k⪯\lambda {\nu }_{k+1}(k=1,2,3,\dots ,m-1).$$

(2.9)

In fact, from (2.8) we have

$$\begin{array}{rcl}{\nu }_1& \in & C(g,x_{m-1},x_{n-1})\\ =& \{d(g{x}_{m-1},g{x}_{n-1}),d(g{x}_{m-1},g{x}_m),d(g{x}_{n-1},g{x}_n),d(g{x}_{m-1},g{x}_n),d(g{x}_m,g{x}_{n-1})\}\\ \subset & A_{m-1,n-1}=\{d(g{x}_i,g{x}_j){|}_{j=m,n,n-1,}^{i=m,m-1,n-1;}i<j\}.\end{array}$$

Let ${\nu }_1=d(g{x}_i,g{x}_j)=d(f{x}_{i-1},f{x}_{j-1})⪯\lambda {\nu }_2$, where

$$\begin{array}{rcl}{\nu }_2& \in & C(g,x_{i-1},x_{j-1})\subset A_{i-1,j-1}=\{d(g{x}_r,g{x}_s){|}_{s=i,j,j-1,}^{r=i,i-1,j-1;}r<s\}\\ =& \{d(g{x}_r,g{x}_s){|}_{s=m,m-1,n,n-1,n-2,}^{r=m,m-1,m-2,n-1,n-2;}r<s\}.\end{array}$$

In general, if there exists

$${\nu }_k\in \{d(g{x}_i,g{x}_j){|}_{j=m,m-1,m-2,\dots ,m-k+1,n,n-1,n-2,\dots ,n-k,}^{i=m,m-1,m-2,\dots ,m-k,n-1,n-2,\dots ,n-k;}i<j\}(1\le k\le m),$$

then we have

$${\nu }_{k+1}\in C(g,x_{i-1},x_{j-1})\subset A_{i-1,j-1}=\{d(g{x}_r,g{x}_s){|}_{s=i,j,j-1,}^{r=i,i-1,j-1;}r<s\}(1\le k\le m-1)$$

such that ${\nu }_k=d(g{x}_i,g{x}_j)=d(f{x}_{i-1},f{x}_{j-1})⪯\lambda {\nu }_{k+1}$ ($1\le k\le m-1$).

As

we can obtain (2.9).

Using the triangular inequality, we get

$$d(g{x}_i,g{x}_j)⪯sd(g{x}_i,g{x}_0)+sd(g{x}_0,g{x}_j)(0\le i,j\le n),$$

so we obtain

$$\begin{array}{rcl}d(g{x}_n,g{x}_m)& =& d(f{x}_{n-1},f{x}_{m-1})⪯\lambda {\nu }_1⪯{\lambda }^2{\nu }_2⪯\cdots ⪯{\lambda }^m{\nu }_m\\ ⪯& {\lambda }^{m}sd(g{x}_i,g{x}_0)+{\lambda }^{m}sd(g{x}_0,g{x}_j)\\ ⪯& \frac{2s^2{\lambda }^m}{1-s\lambda }d(g{x}_1,g{x}_0).\end{array}$$

Since $\frac{2s^2{\lambda }^m}{1-s\lambda }d(g{x}_1,g{x}_0)\to \theta$ as $m\to \mathrm{\infty }$, by Lemma 1.5, it is easy to see that for any $c\in intP$, there exists $n_0\in \mathbb{N}$ such that for all $n>m>n_0$,

$$d(g{x}_n,g{x}_m)⪯\frac{2s^2{\lambda }^m}{1-s\lambda }d(g{x}_1,g{x}_0)\ll c.$$

So, $\{g{x}_n\}$ is a Cauchy sequence in $g(X)$. If $g(X)\subset X$ is complete, there exist $q\in g(X)$ and $p\in X$ such that $g{x}_n\to q$ as $n\to \mathrm{\infty }$ and $g(p)=q$.

Now, from (2.2) we get

$$\nu \in C(g,x_n,p)=\{d(g{x}_n,gp),d(g{x}_n,f{x}_n),d(gp,fp),d(g{x}_n,fp),d(f{x}_n,gp)\}$$

such that $d(f{x}_n,fp)⪯\lambda \nu$.

We have the following five cases:

• (1) $d(f{x}_n,fp)⪯\lambda d(g{x}_n,gp)⪯s\lambda d(g{x}_{n+1},gp)+s\lambda d(g{x}_{n+1},g{x}_n)$;

• (2) $d(f{x}_n,fp)⪯\lambda d(g{x}_n,f{x}_n)=\lambda d(g{x}_n,g{x}_{n+1})$;

• (3) $d(f{x}_n,fp)⪯\lambda d(gp,fp)⪯s\lambda d(g{x}_{n+1},gp)+s\lambda d(g{x}_{n+1},fp)$, that is, $d(f{x}_n,fp)⪯\frac{s\lambda }{1-s\lambda }d(g{x}_{n+1},gp)$;

• (4) $d(f{x}_n,fp)⪯\lambda d(g{x}_n,fp)⪯s\lambda d(g{x}_{n+1},fp)+s\lambda d(g{x}_{n+1},g{x}_n)$, that is, $d(f{x}_n,fp)⪯\frac{s\lambda }{1-s\lambda }d(g{x}_{n+1},g{x}_n)$;

• (5) $d(f{x}_n,fp)⪯\lambda d(f{x}_n,gp)=\lambda d(g{x}_{n+1},gp)$.

As $\frac{s\lambda }{1-s\lambda }>s\lambda$, then we obtain that

$$d(g{x}_{n+1},fp)⪯\frac{s\lambda }{1-s\lambda }[d(g{x}_{n+1},g{x}_n)+d(g{x}_{n+1},q)].$$

Since $g{x}_n\to q$ as $n\to \mathrm{\infty }$, for any $c\in intP$, there exists $n_1\in \mathbb{N}$ such that for all $n>n_1$,

$$d(g{x}_{n+1},g{x}_n)\ll \frac{(1-s\lambda )c}{2s\lambda }\text{and}d(g{x}_{n+1},q)\ll \frac{(1-s\lambda )c}{2s\lambda }.$$

By Lemmas 1.5 and 1.6, we have $g{x}_n\to fp$ as $n\to \mathrm{\infty }$ and $q=fp$.

Now, if w is another point such that $gu=fu=w$, then

$$d(w,q)=d(fu,fp)⪯\lambda \nu ,$$

where $\lambda \in (0,\frac{1}{s})$ and

$$\nu \in C(f;u,p)=\{d(gu,gp),d(gu,fu),d(gp,fp),d(gu,fp),d(fu,gp)\}.$$

It is obvious that $d(w,q)=\theta$, i.e., $w=q$. Therefore, q is the unique point of coincidence of f, g in X. Moreover, the mappings f and g are weakly compatible, by Lemma 1.9 we know that q is the unique common fixed point of f and g.

Similarly, if $f(X)$ is complete, the above conclusion is also established. □

Example 2.7 Let $X=\mathbb{R}$, $E=C_{\mathbb{R}}^1[0,1]$ and $P=\{f\in E:f\ge 0\}$. Define $d:X\times X\to E$ by $d(x,y)={|x-y|}^{\frac{3}{2}}\phi$ where $\phi :[0,1]\to \mathbb{R}$ such that $\phi (t)=e^t$. It is easy to see that $(X,d)$ is a cone b-metric space with the coefficient $s=2^{\frac{1}{2}}$, but it is not a cone metric space. The mappings $f,g:X\to X$ are defined by $fx=\alpha x$ and $gx=\sqrt{\alpha }x$ ($\alpha \in [\frac{1}{\sqrt[3]{8}},\frac{1}{\sqrt[3]{4}})$). The mapping f is a g-quasi-contraction with the constant $\lambda ={\alpha }^{\frac{3}{4}}\in [\frac{1}{2},\frac{\sqrt{2}}{2})$. Moreover, $0\in X$ is the unique common fixed point of f and g.

Remark 2.8 Kadelburg and Radenovi [11] obtained a fixed point result without the normality of the underlying cone, but only in the case of a quasi-contractive constant $\lambda \in (0,1/2)$ (see [[11], Theorem 2.2]). However, Ljiljana [7] proved the result is true for $\lambda \in (0,1)$ on a cone metric space by a new way. Referring to this way, Theorem 2.6 presents a similar common fixed point result in the case of the contractive constant $\lambda \in (0,1/s)$ in cone b-metric spaces without the assumption of normality. Moreover, it is obvious that Example 2.7 given above shows that Theorem 2.6 not only improves and generalizes [[11], Theorem 2.2], but also generalizes and unifies [[7], Theorem 3].