Some generalized fixed point results in a b-metric space and application to matrix equations

Abstract

We have proved a generalized Presic-Hardy-Rogers contraction principle and Ciric-Presic type contraction principle for two mappings in a b-metric space. As an application, we derive some convergence results for a class of nonlinear matrix equations. Numerical experiments are also presented to illustrate the convergence algorithms.

1 Introduction

There appears in literature several generalizations of the famous Banach contraction principle. One such generalization was given by Presic [1, 2] as follows.

Theorem 1.1

[2]

Let $(X, d)$ be a metric space, k be a positive integer, $T : X^{k}\rightarrow X$ be a mapping satisfying the following condition:

$$\begin{aligned}& d\bigl(T(x_{1}, x_{2}, \ldots , x_{k}),T(x_{2}, x_{3}, \ldots , x_{k+1}) \bigr) \\ & \quad \le q_{1}\cdot d(x_{1}, x_{2}) + q_{2}\cdot d(x_{2}, x_{3}) + \cdots + q_{k}\cdot d(x_{k}, x_{k+1}), \end{aligned}$$

(1.1)

where $x_{1}, x_{2}, \ldots , x_{k+1}$ are arbitrary elements in X and $q_{1}, q_{2}, \ldots , q_{k}$ are nonnegative constants such that $q_{1} + q_{2} + \cdots + q_{k} < 1$. Then there exists some $x \in X$ such that $x = T(x, x, \ldots, x)$. Moreover, if $x_{1}, x_{2}, \ldots , x_{k}$ are arbitrary points in X and for $n \in N$, $x_{n+k} = T(x_{n}, x_{n+1}, \ldots, x_{n+k-1})$, then the sequence $\langle x_{n}\rangle$ is convergent and $\lim x_{n} = T(\lim x_{n}, \lim x_{n}, \ldots, \lim x_{n})$.

Note that for $k=1$ the above theorem reduces to the well-known Banach contraction principle. Ciric and Presic [3] generalizing the above theorem proved the following.

Theorem 1.2

[3]

Let $(X, d)$ be a metric space, k be a positive integer, $T : X^{k}\rightarrow X$ be a mapping satisfying the following condition:

$$\begin{aligned} \begin{aligned}[b] &d\bigl(T(x_{1}, x_{2}, \ldots , x_{k}),T(x_{2}, x_{3}, \ldots , x_{k+1}) \bigr) \\ &\quad \le\lambda\cdot \max\bigl\{ d(x_{1}, x_{2}), d(x_{2}, x_{3}),\ldots, d(x_{k}, x_{k+1}) \bigr\} , \end{aligned} \end{aligned}$$

(1.2)

where $x_{1}, x_{2}, \ldots , x_{k+1}$ are arbitrary elements in X and $\lambda\in(0, 1)$. Then there exists some $x \in X$ such that $x = T(x, x, \ldots, x)$. Moreover, if $x_{1}, x_{2}, \ldots , x_{k}$ are arbitrary points in X and for $n \in N$, $x_{n+k} = T(x_{n}, x_{n+1}, \ldots, x_{n+k-1})$, then the sequence $\langle x_{n}\rangle$ is convergent and $\lim x_{n} = T(\lim x_{n}, \lim x_{n}, \ldots, \lim x_{n})$. If in addition T satisfies $D(T(u, u, \ldots, u), T(v, v, \ldots, v)) < d(u, v)$ for all $u, v \in X$, then x is the unique point satisfying $x = T(x, x, \ldots, x)$.

In [4, 5] Pacurar gave a classic generalization of the above results. Later the above results were further extended and generalized by many authors (see [6–14]). Generalizing the concept of metric space, Bakhtin [15] introduced the concept of b-metric space which is not necessarily Hausdorff and proved the Banach contraction principle in the setting of a b-metric space. Since then several papers have dealt with fixed point theory or the variational principle for single-valued and multi-valued operators in b-metric spaces (see [16–23] and the references therein). In this paper we have proved common fixed point theorems for the generalized Presic-Hardy-Rogers contraction and Ciric-Presic contraction for two mappings in a b-metric space. Our results extend and generalize many well-known results. As an application, we have derived some convergence results for a class of nonlinear matrix equations. Numerical experiments are also presented to illustrate the convergence algorithms.

2 Preliminaries

Definition 2.1

[15]

Let X be a nonempty set and $d: X\times X\rightarrow[0,\infty )$ satisfy:

• (bM1) $d(x,y)=0$ if and only if $x=y$ for all $x,y\in X$;

• (bM2) $d(x,y)=d(y,x)$ for all $x,y\in X$;

• (bM3) there exists a real number $s \geq1$ such that $d(x,y)\leq s[d(x,z)+d(z,y)]$ for all $x,y,z\in X$.

Then d is called a b-metric on X and $(X,d)$ is called a b-metric space (in short bMS) with coefficient s.

Convergence, Cauchy sequence and completeness in b-metric space are defined as follows.

Definition 2.2

[15]

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

• (a) The sequence $\{x_{n}\}$ is said to be convergent in $(X,d)$, and it converges to x if for every $\varepsilon>0$ there exists $n_{0}\in\mathbb{N}$ such that $d(x_{n},x)<\varepsilon$ for all $n>n_{0}$, and this fact is represented by $\lim_{n\rightarrow \infty}x_{n}=x$ or $x_{n}\rightarrow x$ as $n\rightarrow\infty$.

• (b) The sequence $\{x_{n}\}$ is said to be Cauchy sequence in $(X,d)$ if for every $\varepsilon>0$ there exists $n_{0}\in\mathbb{N}$ such that $d(x_{n},x_{n+p})<\varepsilon$ for all $n>n_{0}$, $p>0$ or, equivalently, if $\lim_{n\rightarrow\infty }d(x_{n},x_{n+p})=0$ for all $p>0$.

• (c) $(X,d)$ is said to be a complete b-metric space if every Cauchy sequence in X converges to some $x\in X$.

Definition 2.3

[9]

Let $(X,d)$ be a metric space, k be a positive integer, $T:X^{k}\rightarrow X$ and $f:X\rightarrow X$ be mappings.

• (a) An element $x \in X$ is said to be a coincidence point of f and T if and only if $f(x)=T(x,x,\ldots,x)$. If $x = f(x)=T(x,x,\ldots,x)$, then we say that x is a common fixed point of f and T. If $w = f(x)=T(x,x,\ldots,x)$, then w is called a point of coincidence of f and T.

• (b) Mappings f and T are said to be commuting if and only if $f(T(x,x,\ldots, x))=T(fx,fx,\ldots, fx)$ for all $x\in X$.

• (c) Mappings f and T are said to be weakly compatible if and only if they commute at their coincidence points.

Remark 2.4

For $k=1$ the above definitions reduce to the usual definition of commuting and weakly compatible mappings in a metric space.

The set of coincidence points of f and T is denoted by $C(f,T)$.

Lemma 2.5

[24]

Let X be a nonempty set, k be a positive integer and $f:X^{k}\rightarrow X$, $g:X\rightarrow X$ be two weakly compatible mappings. If f and g have a unique point of coincidence $y=f(x,x,\ldots,x)=g(x)$, then y is the unique common fixed point of f and g.

Khan et al. [8] defined the set function $\theta:[0,\infty)^{4}\rightarrow[0,\infty)$ as follows:

• 1. θ is continuous,

• 2. for all $t_{1},t_{2},t_{3},t_{4}\in[0,\infty)$, $\theta(t_{1},t_{2},t_{3},t_{4})=0\Leftrightarrow t_{1}t_{2}t_{3}t_{4}=0$.

3 Main results

Throughout this paper we assume that the b-metric $d: X\times X\rightarrow[0,\infty)$ is continuous on $X^{2}$.

Theorem 3.1

Let $(X,d)$ be a b-metric space with coefficient $s \ge1$. For any positive integer k, let $f:X^{k} \rightarrow X$ and $g:X\rightarrow X$ be mappings satisfying the following conditions:

$$\begin{aligned}& f\bigl(X^{k}\bigr) \subseteq g(X), \end{aligned}$$

(3.1)

$$\begin{aligned}& d\bigl(f(x_{1},x_{2},\ldots, x_{k}), f(x_{2},x_{3},\ldots, x_{k+1})\bigr) \\& \quad \leq\sum_{i=1}^{k} \alpha_{i}d(gx_{i},gx_{i+1})+\sum _{i=1}^{k+1}\sum_{j=1}^{k+1} \beta_{i,j}d\bigl(gx_{i},f(x_{j},x_{j}, \ldots, x_{j})\bigr) \\& \qquad {}+ L\cdot\theta \bigl(d\bigl(gx_{1},f(x_{k+1},x_{k+1},x_{k+1}, \ldots,x_{k+1})\bigr),d\bigl(gx_{k+1},f(x_{1},x_{1},x_{1}, \ldots,x_{1})\bigr), \\& \qquad d\bigl(gx_{1},f(x_{1},x_{1}, \ldots,x_{1})\bigr),d\bigl(gx_{k+1},f(x_{k+1},x_{k+1}, \ldots,x_{k+1})\bigr)\bigr), \end{aligned}$$

(3.2)

where $x_{1},x_{2},\ldots, x_{k+1}$ are arbitrary elements in X and $\alpha _{i}$, $\beta_{ij}$, L are nonnegative constants such that $\sum_{n=1}^{k}s^{k+3-n}[\alpha_{n}+\sum_{i=1}^{k+1}\sum_{j=1}^{k+1}\beta _{i,j}]<1$ and

$$ g(X) \textit{ is complete}. $$

(3.3)

Then f and g have a unique coincidence point, i.e., $C(f, g) \neq \emptyset$. In addition, if f and g are weakly compatible, then f and g have a unique common fixed point. Moreover, for any $x_{1}\in X$, the sequence $\{y_{n}\}$ defined by $y_{n} = g(x_{n}) = f(x_{n-1}, x_{n-1}, \ldots, x_{n-1})=Fx_{n-1}$ converges to the common fixed point of f and g.

Proof

Let $x_{1}\in X$, then $f(x_{1},x_{1},\ldots,x_{1})\in f(X^{k})\subset g(X)$. So there exists $x_{2}\in X$ such that $f(x_{1},x_{1},\ldots,x_{1})=g(x_{2})$. Now $f(x_{2},x_{2},\ldots,x_{2})\in f(X^{k})\subset g(X)$ and so there exists $x_{3}\in X$ such that $f(x_{2},x_{2},\ldots,x_{2})=g(x_{3})$. Continuing this process we define the sequence $\langle y_{n}\rangle$ in $g(X)$ as $y_{n} = g(x_{n}) = f(x_{n-1}, x_{n-1}, \ldots, x_{n-1})=Fx_{n-1}$, $n = 1,2,\ldots,k+1$, where F is the associate operator for f. Let $d_{n} = d(y_{n}, y_{n+1})=d(gx_{n}, gx_{n+1})$ and $D_{ij}=d(gx_{i},f(x_{j},x_{j},\ldots,x_{j}))$.

Then we have

$$\begin{aligned} d_{n+1} =&d\bigl(g(x_{n+1}),g(x_{n+2})\bigr) \\ =& d(Fx_{n},Fx_{n+1}) \\ =& d\bigl(f(x_{n},x_{n},\ldots,x_{n}),f(x_{n+1},x_{n+1}, \ldots,x_{n+1})\bigr) \\ \leq& sd\bigl(f(x_{n},x_{n},\ldots,x_{n}),f(x_{n},x_{n}, \ldots,x_{n+1})\bigr) \\ &{}+s^{2}d\bigl(f(x_{n},x_{n}, \ldots,x_{n+1}),f(x_{n},x_{n},\ldots, x_{n+1},x_{n+1})\bigr) \\ &{}+s^{3}d\bigl(f(x_{n},x_{n},\ldots, x_{n+1},x_{n+1}), f(x_{n},\ldots,x_{n+1},x_{n+1},x_{n+1}) \bigr)+\cdots \\ &{}+s^{k}d\bigl(f(x_{n},x_{n+1}, \ldots,x_{n+1},x_{n+1}), f(x_{n+1}, \ldots,x_{n+1},x_{n+1},x_{n+1})\bigr). \end{aligned}$$

Using (3.2) we get

$$\begin{aligned} d_{n+1} \leq&s\Biggl\{ \alpha_{k} d_{n}+\Biggl[\sum _{j=1}^{k}\beta_{1,j}+\sum _{j=1}^{k}\beta_{2,j}+\cdots+\sum _{j=1}^{k}\beta_{kj}\Biggr]D_{n,n} \\ &{}+\Biggl[\sum_{i=1}^{k} \beta_{i,k+1}\Biggr]D_{n,n+1}+\Biggl[\sum _{j=1}^{k}\beta _{k+1,j}\Biggr]D_{n+1,n}+ \beta_{k+1,k+1}D_{n+1,n+1}\Biggr\} \\ &{}+s^{2}\Biggl\{ \alpha_{k-1} d_{n}+\Biggl[\sum _{j=1}^{k-1}\beta_{1,j}+\sum _{j=1}^{k-1}\beta_{2,j}+\cdots+\sum _{j=1}^{k-1}\beta_{k-1,j}\Biggr]D_{n,n} \\ &{}+\Biggl[\sum_{i=1}^{k-1} \beta_{i,k}+\sum_{i=1}^{k-1}\beta _{i,k+1}\Biggr]D_{n,n+1}+\Biggl[\sum_{j=1}^{k-1} \beta_{k,j}+\sum_{j=1}^{k-1} \beta_{k+1,j}\Biggr]D_{n+1,n} \\ &{}+\Biggl[\sum_{j=k}^{k+1} \beta_{k,j}+\sum_{j=k}^{k+1}\beta _{k+1,j}\Biggr]D_{n+1,n+1}\Biggr\} \\ &{}+\cdots+s^{k}\Biggl\{ \alpha_{1} d_{n}+\beta _{1,1}D_{n,n}+\Biggl[\sum_{j=2}^{k+1} \beta_{1,j}\Biggr]D_{n,n+1}+\Biggl[\sum _{i=2}^{k+1}\beta_{i,1}\Biggr]D_{n+1,n} \\ &{}+\Biggl[\sum_{j=2}^{k+1}\beta _{2,j}+\sum_{j=2}^{k+1} \beta_{3,j}+\cdots+\sum_{j=2}^{k+1} \beta _{k+1,j}\Biggr]D_{n+1,n+1}\Biggr\} \\ &{}+ L\cdot \theta \bigl(d\bigl(gx_{n},(fx_{n+1},x_{n+1},x_{n+1}, \ldots,x_{n+1})\bigr),d\bigl(gx_{n+1},f(x_{n},x_{n},x_{n}, \ldots,x_{n})\bigr), \\ &d\bigl(gx_{n},f(x_{n},x_{n}, \ldots,x_{n})\bigr),d\bigl(gx_{n+1},f(x_{n+1},x_{n+1}, \ldots,x_{n+1})\bigr)\bigr), \end{aligned}$$

i.e.,

$$\begin{aligned} d_{n+1} \leq&\bigl[s\alpha_{k}+s^{2} \alpha_{k-1}+s^{3}\alpha _{k-2}+\cdots+s^{k} \alpha_{1}\bigr]d_{n}+s\Biggl\{ \Biggl[\sum _{i=1}^{k}\sum_{j=1}^{k} \beta _{i,j}\Biggr]D_{n,n} \\ &{}+\Biggl[\sum_{i=1}^{k} \beta_{i,k+1}\Biggr]D_{n,n+1}+\Biggl[\sum _{j=1}^{k}\beta _{k+1,j}\Biggr]D_{n+1,n}+ \beta_{k+1,k+1}D_{n+1,n+1}\Biggr\} \\ &{}+s^{2}\Biggl\{ \Biggl[\sum_{i=1}^{k-1} \sum_{j=1}^{k-1}\beta_{i,j} \Biggr]D_{n,n}+\Biggl[\sum_{i=1}^{k-1} \sum_{j=k}^{k+1}\beta_{i,j} \Biggr]D_{n,n+1}+\Biggl[\sum_{i=k}^{k+1} \sum_{j=1}^{k-1}\beta_{i,j} \Biggr]D_{n+1,n} \\ &{}+\Biggl[\sum_{i=k}^{k+1}\sum _{j=k}^{k+1}\beta_{i,j}\Biggr]D_{n+1,n+1} \Biggr\} +\cdots+s^{k}\Biggl\{ \beta_{1,1}D_{n,n}+ \Biggl[\sum_{j=2}^{k+1}\beta_{1,j} \Biggr]D_{n,n+1} \\ &{}+\Biggl[\sum_{i=2}^{k+1} \beta_{i,1}\Biggr]D_{n+1,n}+\Biggl[\sum _{i=2}^{k+1}\sum_{j=2}^{k+1} \beta_{i,j}\Biggr]D_{n+1,n+1}\Biggr\} +L\cdot 0, \end{aligned}$$

i.e.,

$$\begin{aligned} d_{n+1} \leq& \bigl[s\alpha_{k}+s^{2} \alpha_{k-1}+s^{3}\alpha _{k-2}+\cdots+s^{k} \alpha_{1}\bigr]d_{n} \\ &{}+\Biggl[s\sum_{i=1}^{k}\sum _{j=1}^{k}\beta _{i,j}+s^{2}\sum _{i=1}^{k-1}\sum_{j=1}^{k-1} \beta _{i,j}+\cdots+s^{k-1}\sum_{i=1}^{2} \sum_{j=1}^{2}\beta_{i,j}+s^{k} \beta _{1,1}\Biggr]D_{n,n} \\ &{}+\Biggl[s\sum_{i=1}^{k} \beta_{i,k+1}+s^{2}\sum_{i=1}^{k-1} \sum_{j=k}^{k+1}\beta_{i,j}+ \cdots+s^{k-1}\sum_{i=1}^{2}\sum _{j=3}^{k+1}\beta_{i,j}+s^{k} \sum_{j=2}^{k+1}\beta_{1,j} \Biggr]D_{n,n+1} \\ &{}+\Biggl[s\sum_{j=1}^{k} \beta_{k+1,j}+s^{2}\sum_{i=k}^{k+1} \sum_{j=1}^{k-1}\beta_{i,j}+ \cdots+s^{k-1}\sum_{i=3}^{k+1}\sum _{j=1}^{2}\beta_{i,j}+s^{k} \sum_{i=2}^{k+1}\beta_{i,1} \Biggr]D_{n+1,n} \\ &{}+\Biggl[s^{k}\sum_{i=2}^{k+1} \sum_{j=2}^{k+1}\beta_{i,j}+s^{k-1} \sum_{i=3}^{k+1}\sum _{j=3}^{k+1}\beta_{i,j}+\cdots+s^{2} \sum_{i=k}^{k+1}\sum _{j=k}^{k+1}\beta_{i,j}+s\beta_{k+1,k+1} \Biggr]D_{n+1,n+1} \\ =&Ad_{n}+BD_{n,n}+CD_{n,n+1}+ED_{n+1,n}+FD_{n+1,n+1}, \end{aligned}$$

where A, B, C, E and F are the coefficients of $d_{n}$, $D_{n,n}$, $D_{n,n+1}$, $D_{n+1,n}$ and $D_{n+1,n+1}$ respectively in the above inequality. By the definition, $D_{n,n}=d(gx_{n},f(x_{n},x_{n},\ldots,x_{n}))= d(gx_{n},gx_{n+1})=d_{n}$, $D_{n,n+1}=d(gx_{n},f(x_{n+1},x_{n+1},\ldots,x_{n+1}))=d(gx_{n},gx_{n+2})$, $D_{n+1,n}=d(gx_{n+1},f(x_{n},x_{n},\ldots,x_{n}))=d(gx_{n+1},gx_{n+1})=0$, $D_{n+1,n+1}=d(gx_{n+1},f(x_{n+1},x_{n+1},\ldots,x_{n+1}))=d(gx_{n+1},gx_{n+2})=d_{n+1}$; therefore,

$$\begin{aligned} d_{n+1} \leq& Ad_{n}+Bd_{n}+Cd(gx_{n},gx_{n+2})+Fd_{n+1} \\ \leq&Ad_{n}+Bd_{n}+Csd(gx_{n},gx_{n+1})+Csd(gx_{n+1},gx_{n+2})+Fd_{n+1} \\ =&(A+B+Cs)d_{n}+(Cs+F)d_{n+1}, \end{aligned}$$

i.e., $(1-Cs-F)d_{n+1}\leq(A+B+Cs)d_{n}$. Again, interchanging the role of $x_{n}$ and $x_{n+1}$ and repeating the above process, we obtain $(1-Es-B)d_{n+1}\leq(A+F+Es)d_{n}$. It follows that

$$\begin{aligned} \bigl(2-(C+E)s-F-B\bigr)d_{n+1} \leq& \bigl(2A+B+F+s(C+E) \bigr)d_{n} \\ d_{n+1} \leq&\frac{2A+B+F+s(C+E)}{2-B-F-(C+E)s}d_{n} \\ d_{n+1} \leq&\lambda d_{n}, \end{aligned}$$

where $\lambda=\frac{2A+B+F+s(C+E)}{2-B-F-(C+E)s}$. Thus we have

$$ d_{n+1}\leq\lambda^{n+1}d_{0} \quad \mbox{for all } n\geq0 . $$

(3.4)

We will show that $\lambda< 1$ and $s\lambda< 1$. We have

$$\begin{aligned}& A+B+F+s(C+E) \\& \quad \le s[A+B+C+E+F] \\& \quad = s\bigl[s\alpha_{k}+s^{2}\alpha_{k-1}+s^{3} \alpha_{k-2}+\cdots+s^{k}\alpha_{1}\bigr] \\& \qquad {} + s\Biggl[s\sum_{i=1}^{k}\sum _{j=1}^{k}\beta_{i,j}+s^{2} \sum_{i=1}^{k-1}\sum _{j=1}^{k-1}\beta_{i,j}+\cdots+s^{k-1} \sum_{i=1}^{2}\sum _{j=1}^{2}\beta_{i,j}+s^{k} \beta_{1,1}\Biggr] \\& \qquad {} + s\Biggl[s\sum_{i=1}^{k} \beta_{i,k+1}+s^{2}\sum_{i=1}^{k-1} \sum_{j=k}^{k+1}\beta_{i,j}+ \cdots+s^{k-1}\sum_{i=1}^{2}\sum _{j=3}^{k+1}\beta_{i,j}+s^{k} \sum_{j=2}^{k+1}\beta_{1,j}\Biggr] \\& \qquad {} + s\Biggl[s\sum_{j=1}^{k} \beta_{k+1,j}+s^{2}\sum_{i=k}^{k+1} \sum_{j=1}^{k-1}\beta_{i,j}+ \cdots+s^{k-1}\sum_{i=3}^{k+1}\sum _{j=1}^{2}\beta_{i,j}+s^{k} \sum_{i=2}^{k+1}\beta_{i,1}\Biggr] \\& \qquad {} + s\Biggl[s\beta_{k+1,k+1}+s^{2}\sum _{i=k}^{k+1}\sum_{j=k}^{k+1} \beta _{i,j}+\cdots+s^{k-1}\sum_{i=3}^{k+1} \sum_{j=3}^{k+1}\beta_{i,j}+ s^{k}\sum_{i=2}^{k+1}\sum _{j=2}^{k+1}\beta_{i,j}\Biggr] \\& \quad = \bigl[s^{2}\alpha_{k}+s^{3} \alpha_{k-1}+s^{4}\alpha_{k-2}+\cdots+s^{k+1} \alpha _{1}\bigr]+s^{2}\sum_{i=1}^{k+1} \sum_{j=1}^{k+1}\beta_{i,j}+s^{3} \sum_{i=1}^{k+1}\sum _{j=1}^{k+1}\beta_{i,j} \\& \qquad {} + s^{4}\sum_{i=1}^{k+1} \sum_{j=1}^{k+1}\beta_{i,j}+\cdots +s^{k+1}\sum_{i=1}^{k+1}\sum _{j=1}^{k+1}\beta_{i,j} \\& \quad = \bigl[s^{2}\alpha_{k}+s^{3} \alpha_{k-1}+s^{4}\alpha_{k-2}+\cdots+s^{k+1} \alpha _{1}\bigr] \\ & \qquad {}+ \bigl[s^{2}+s^{3}+s^{4}+ \cdots+s^{k+1}\bigr]\sum_{i=1}^{k+1} \sum_{j=1}^{k+1}\beta _{i,j} \\ & \quad \leq \bigl[s^{3}\alpha_{k}+s^{4} \alpha_{k-1}+s^{5}\alpha_{k-2}+\cdots +s^{k+2}\alpha_{1}\bigr] \\ & \qquad {}+ \bigl[s^{3}+s^{4}+s^{5}+ \cdots+s^{k+2}\bigr]\sum_{i=1}^{k+1} \sum_{j=1}^{k+2}\beta _{i,j} \\ & \quad = \sum_{n=1}^{k}s^{k+3-n} \Biggl[\alpha_{n}+\sum_{i=1}^{k+1} \sum_{j=1}^{k+1}\beta_{i,j}\Biggr]< 1, \end{aligned}$$

and so $\lambda<1$. We also have $sA+sB+sF+s(C+E) = s(A+B+F+C+E) < 1$ (proved above) and

$$\begin{aligned}& sA + B + F +s^{2}(C+E) \\& \quad \le s^{2}[A+B+C+E+F] \\& \quad = s^{2}\bigl[s\alpha_{k}+s^{2} \alpha_{k-1}+s^{3}\alpha_{k-2}+\cdots+s^{k} \alpha_{1}\bigr] \\& \qquad {} + s^{2}\Biggl[s\sum_{i=1}^{k} \sum_{j=1}^{k}\beta_{i,j}+s^{2} \sum_{i=1}^{k-1}\sum _{j=1}^{k-1}\beta_{i,j}+\cdots+s^{k-1} \sum_{i=1}^{2}\sum _{j=1}^{2}\beta_{i,j}+s^{k} \beta_{1,1}\Biggr] \\& \qquad {} + s^{2}\Biggl[s\sum_{i=1}^{k} \beta_{i,k+1}+s^{2}\sum_{i=1}^{k-1} \sum_{j=k}^{k+1}\beta_{i,j}+ \cdots+s^{k-1}\sum_{i=1}^{2}\sum _{j=3}^{k+1}\beta_{i,j}+s^{k} \sum_{j=2}^{k+1}\beta_{1,j}\Biggr] \\& \qquad {} + s^{2}\Biggl[s\sum_{j=1}^{k} \beta_{k+1,j}+s^{2}\sum_{i=k}^{k+1} \sum_{j=1}^{k-1}\beta_{i,j}+ \cdots+s^{k-1}\sum_{i=3}^{k+1}\sum _{j=1}^{2}\beta_{i,j}+s^{k} \sum_{i=2}^{k+1}\beta_{i,1}\Biggr] \\& \qquad {} + s^{2}\Biggl[s\beta_{k+1,k+1}+s^{2}\sum _{i=k}^{k+1}\sum_{j=k}^{k+1} \beta _{i,j}+\cdots+s^{k-1}\sum_{i=3}^{k+1} \sum_{j=3}^{k+1}\beta_{i,j}+ s^{k}\sum_{i=2}^{k+1}\sum _{j=2}^{k+1}\beta_{i,j}\Biggr] \\& \quad = \bigl[s^{3}\alpha_{k}+s^{4} \alpha_{k-1}+s^{5}\alpha_{k-2}+\cdots+s^{k+2} \alpha _{1}\bigr]+s^{3}\sum_{i=1}^{k+1} \sum_{j=1}^{k+1}\beta_{i,j}+s^{4} \sum_{i=1}^{k+1}\sum _{j=1}^{k+1}\beta_{i,j} \\& \qquad {} + s^{5}\sum_{i=1}^{k+1} \sum_{j=1}^{k+1}\beta_{i,j}+\cdots +s^{k+2}\sum_{i=1}^{k+1}\sum _{j=1}^{k+1}\beta_{i,j} \\& \quad = \bigl[s^{3}\alpha_{k}+s^{4} \alpha_{k-1}+s^{5}\alpha_{k-2}+\cdots+s^{k+2} \alpha _{1}\bigr] \\& \qquad {} + \bigl[s^{3}+s^{4}+s^{5}+ \cdots+s^{k+2}\bigr]\sum_{i=1}^{k+1} \sum_{j=1}^{k+1}\beta _{i,j} \\& \quad = \sum_{n=1}^{k}s^{k+3-n} \Biggl[\alpha_{n}+\sum_{i=1}^{k+1} \sum_{j=1}^{k+1}\beta_{i,j}\Biggr]< 1, \end{aligned}$$

and so $s\lambda<1$.

Thus, for all $n,p\in N$,

$$\begin{aligned} d(gx_{n},gx_{n+p}) \leq& sd(gx_{n},gx_{n+1})+s^{2}d(gx_{n+1},gx_{n+2})+ \cdots+s^{p-1}d(gx_{n+(p-1)},gx_{n+p}) \\ =&sd_{n}+s^{2}d_{n+1}+\cdots+s^{p-1}d_{n+(p-1)} \\ \leq&s\lambda^{n}d_{0}+s^{2} \lambda^{n+1}d_{0}+\cdots+s^{p-1}\lambda ^{n+(p-1)}d_{0} \\ \leq& \frac{s\lambda^{n}}{1-s\lambda}d_{0} \rightarrow0\quad \mbox{as } n \rightarrow\infty. \end{aligned}$$

Thus $\{gx_{n}\}$ is a Cauchy sequence. By completeness of $g(X)$, there exists $u\in X$ such that

$$ \lim_{n\rightarrow\infty}gx_{n}=u \mbox{ and there exists } p\in X \mbox{ such that } g(p)=u. $$

(3.5)

We shall show that u is the fixed point of f and g. Using a similar process as the one used in the calculation of $d_{n+1}$, we obtain

$$\begin{aligned} d\bigl(g(p),f(p,\ldots,p)\bigr) \leq&s\bigl[d\bigl(g(p),y_{n+1}\bigr)+d \bigl(y_{n+1},f(p,p,\ldots, p)\bigr)\bigr] \\ \leq&s\bigl[d\bigl(g(p),y_{n+1}\bigr)+d(Fx_{n},Fp)\bigr] \\ \leq&s\bigl[d\bigl(g(p),y_{n+1}\bigr)+Ad(gx_{n},gp)+Bd \bigl(gx_{n},f(x_{n},x_{n},\ldots,x_{n}) \bigr) \\ &{}+Cd\bigl(gx_{n},f(p,p,\ldots,p)\bigr) \\ &{}+Ed\bigl(gp,f(x_{n},x_{n},\ldots,x_{n}) \bigr)+Fd\bigl(gp,f(p,p,\ldots,p)\bigr)\bigr]. \end{aligned}$$

It follows from (3.5) that

$$ d\bigl(g(p),f(p,\ldots,p)\bigr)\leq s(C+F)d\bigl(gp,f(p,p,\ldots,p)\bigr). $$

(3.6)

As $s(C+F)<1$, we obtain $F(p)=g(p)=f(p,p,\ldots,p)=u$ Thus, u is a point of coincidence of f and g. If $u'$ is another point of coincidence of f and g, then there exists $p'\in X$ such that $F(p')=g(p')=f(p',p',\ldots,p')=u'$.

Then we have

$$\begin{aligned} d\bigl(u,u'\bigr) =&d\bigl(Fp,Fp'\bigr) \\ \leq&Ad\bigl(gp,gp'\bigr)+Bd\bigl(gp,f(p,p,\ldots,p)\bigr) \\ &{}+Cd\bigl(gp,f\bigl(p',p',\ldots,p' \bigr)\bigr) \\ &{}+Ed\bigl(gp',f(p,p,\ldots,p)\bigr)+Fd\bigl(gp',f \bigl(p',p',\ldots,p'\bigr)\bigr) \\ =&Ad\bigl(u,u'\bigr)+Bd(u,u)+Cd\bigl(u,u'\bigr)+Ed \bigl(u',u\bigr)+Fd\bigl(u',u\bigr) \\ =&(A+C+E+F)d\bigl(u,u'\bigr). \end{aligned}$$

As $A+C+E+F<1$, we obtain from the above inequality that $d(u,u')=0$, that is, $u=u'$. Thus the point of coincidence u is unique. Further, if f and g are weakly compatible, then by Lemma 2.5, u is the unique common fixed point of f and g. □

Remark 3.2

Taking $s=1$, $g=I$ and $\theta(t_{1},t_{2},t_{3},t_{4})=0$ in Theorem 3.1, we get Theorem 4 of Shukla et al. [13].

Remark 3.3

For $s=1$, $g=I$, $i=j$, $\beta_{ij}=\delta_{k+1}$, ∀i, $L=1$, we obtain Theorem 2.1 of Khan et al. [8].

Remark 3.4

For $s=1$, $g=I$, $\beta_{ij}=0$, $\forall i,j \in\{1,2,\ldots,k+1\}$ and $\theta(t_{1},t_{2},t_{3},t_{4})=\min\{(t_{1},t_{2}, t_{3},t_{4})\}$, we obtain the result of Pacurar [5].

Remark 3.5

For $s=1$, $g=I$, $\alpha_{i}=0$, $i=j$, $\beta_{ij}=a$, $L=0$, we obtain the result of Pacurar [4].

Remark 3.6

For $s=1$, $g=I$, $\beta_{ij}=0$, $\forall i,j \in\{1,2,\ldots ,k+1\}$, $L=0$, we obtain the result of Presic [2].

Next we prove a generalized Ciric-Presic type fixed point theorem in a b-metric space. Consider a function $\phi: R^{k}\rightarrow R$ such that

• 1. ϕ is an increasing function, i.e., $x_{1}< y_{1},x_{2}< y_{2},\ldots,x_{k}< y_{k}$ implies $\phi(x_{1},x_{2},\ldots,x_{k})< \phi (y_{1},y_{2},\ldots,y_{k})$;

• 2. $\phi(t,t,t,\ldots)\leq t$ for all $t\in X$;

• 3. ϕ is continuous in all variables.

Theorem 3.7

Let $(X,d)$ be a b-metric space with $s \ge1$. For any positive integer k, let $f:X^{k} \rightarrow X$ and $g:X\rightarrow X$ be mappings satisfying the following conditions:

$$\begin{aligned}& f\bigl(X^{k}\bigr) \subseteq g(X), \end{aligned}$$

(3.7)

$$\begin{aligned}& d\bigl(f(x_{1},x_{2},\ldots, x_{k}), f(x_{2},x_{3},\ldots, x_{k+1})\bigr) \\& \quad \leq\lambda\phi\bigl(d(gx_{1},gx_{2}),d(gx_{2},gx_{3}),d(gx_{3},gx_{4}), \ldots ,d(gx_{k},gx_{k+1})\bigr), \end{aligned}$$

(3.8)

where $x_{1},x_{2},\ldots,x_{k+1}$ are arbitrary elements in X, $\lambda \in(0,\frac{1}{s^{k}})$,

$$ g(X)\textit{ is complete} $$

(3.9)

and

$$ d\bigl(f(u,u,\ldots,u),f(v,v,\ldots,v)\bigr)< d(gu,gv) $$

(3.10)

for all $u,v\in X$. Then f and g have a coincidence point, i.e., $C(f, g) \neq\emptyset$. In addition, if f and g are weakly compatible, then f and g have a unique common fixed point. Moreover, for any $x_{1}\in X$, the sequence $\{y_{n}\}$ defined by $y_{n} = g(x_{n}) = f(x_{n}, x_{n+1}, \ldots, x_{n+k-1})$ converges to the common fixed point of f and g.

Proof

For arbitrary $x_{1},x_{2},\ldots,x_{k}$ in X, let

$$ R=\max\biggl(\frac{d(gx_{1},gx_{2})}{\theta },\frac{d(gx_{2},gx_{3})}{\theta^{2}},\ldots, \frac {d(gx_{k},f(x_{1},x_{2},\ldots,x_{k}))}{\theta^{k}}\biggr), $$

(3.11)

where $\theta=\lambda^{\frac{1}{k}}$. By (3.7) we define the sequence $\langle y_{n}\rangle$ in $g(X)$ as $y_{n}=gx_{n}$ for $n = 1,2,\ldots, k$ and $y_{n+k} = g(x_{n+k}) = f(x_{n}, x_{n+1}, \ldots, x_{n+k-1})$, $n = 1,2,\ldots$ .

Let $\alpha_{n} = d(y_{n}, y_{n+1})$. By the method of mathematical induction, we will prove that

$$ \alpha_{n}\leq R\theta^{n} \quad \mbox{for all }n. $$

(3.12)

Clearly, by the definition of R, (3.12) is true for $n = 1,2, \ldots, k$. Let the k inequalities $\alpha_{n}\leq R\theta^{n} ,\alpha_{n+1}\leq R\theta^{n+1},\ldots,\alpha _{n+k-1}\leq R\theta^{n+k-1}$ be the induction hypothesis. Then we have

$$\begin{aligned} \alpha_{n+k} =& d(y_{n+k}, y_{n+k+1}) \\ =&d\bigl(f(x_{n}, x_{n+1}, \ldots, x_{n+k-1}), f(x_{n+1}, x_{n+2}, \ldots, x_{n+k})\bigr) \\ \leq&\lambda\phi \bigl(d(gx_{n},gx_{n+1}),d(gx_{n+1},gx_{n+2}), \ldots,d(gx_{n+k-1},gx_{n+k}), \\ &d\bigl(gx_{n},f(x_{n},x_{n}, \ldots,x_{n})\bigr),d\bigl(gx_{n+k},f(x_{n+k},x_{n+k}, \ldots,x_{n+k})\bigr)\bigr) \\ =&\lambda\phi({ \alpha_{n}, \alpha_{n+1},\ldots, \alpha_{n+k-1}}) \\ \leq&\lambda\phi\bigl({R\theta^{n}, R\theta^{n+1},\ldots,R \theta ^{n+k-1}}\bigr) \\ \leq&\lambda\phi\bigl({R\theta^{n}, R\theta^{n},\ldots,R \theta^{n}}\bigr) \\ \leq&\lambda R\theta^{n} \\ =&R\theta^{n+k}. \end{aligned}$$

Thus the inductive proof of (3.12) is complete. Now, for $n,p \in N$, we have

$$\begin{aligned} \begin{aligned} d(y_{n}, y_{n+p}) &\leq sd(y_{n}, y_{n+1})+s^{2}d(y_{n+1},y_{n+2})+ \cdots+s^{p-1}d(y_{n+p-1},y_{n+p}), \\ &\leq sR\theta^{n}+s^{2}R\theta^{n+1}+ \cdots+s^{p-1}R\theta^{n+p-1} \\ &\leq sR\theta^{n}\bigl(1+s\theta+s^{2} \theta^{2}+\cdots\bigr) \\ &= {sR\theta^{n} \over 1-s\theta}. \end{aligned} \end{aligned}$$

Hence the sequence $\langle y_{n}\rangle$ is a Cauchy sequence in $g(X)$ and since $g(X)$ is complete, there exist $v, u\in X$ such that $\lim_{n\rightarrow\infty}y_{n} = v = g(u)$,

$$\begin{aligned} d\bigl(gu, f(u,u,\ldots,u)\bigr) \leq& s\bigl[d(gu, y_{n+k}) + d \bigl(y_{n+k}, f(u,u,\ldots ,u)\bigr)\bigr] \\ =&s\bigl[d(gu,y_{n+k})+d\bigl(f(x_{n}, x_{n+1}, \ldots, x_{n+k-1}),f(u,u,\ldots,u)\bigr)\bigr] \\ =& sd(gu, y_{n+k}) + sd\bigl(f(x_{n}, x_{n+1}, \ldots,x_{n+k-1}), f(u,u,\ldots,u)\bigr) \\ \leq& sd(gu,y_{n+k})+s^{2}d\bigl(f(u,u,\ldots,u),f(u,u, \ldots,x_{n})\bigr) \\ &{}+s^{3}d\bigl(f(u,u,\ldots,x_{n}),f(u,u, \ldots,x_{n},x_{n+1})\bigr) \\ &{}+\cdots+s^{k-1}d\bigl(f(u,x_{n},\ldots ,x_{n+k-2}),f(x_{n},x_{n+1},\ldots,x_{n+k-1}) \bigr) \\ \leq& sd(gu,y_{n+k})+s^{2}\lambda\phi\bigl\{ d(gu,gu),d(gu,gu),\ldots ,d(gu,gx_{n})\bigr\} \\ &{}+s^{3}\lambda\phi\bigl\{ d(gu, gu),d(gu, gu),\ldots,d(gu, gx_{n}),d(gx_{n}, gx_{n+1})\bigr\} +\cdots \\ &{}+s^{k-1}\lambda\phi\bigl\{ d(gu, gx_{n}),d(gx_{n}, gx_{n+1}),\ldots ,d(gx_{n+k-2}, gx_{n+k-1})\bigr\} \\ =&sd(gu,y_{n+k})+ s^{2}\lambda\phi\bigl(0,0, \ldots,d(gu,gx_{n})\bigr) \\ &{}+s^{3} \lambda\phi\bigl(0,0,\ldots,d(gu, gx_{n}),d(gx_{n}, gx_{n+1})\bigr)+\cdots \\ &{}+ s^{k-1}\lambda\phi\bigl(d(gu, gx_{n}),d(gx_{n}, gx_{n+1}),\ldots ,d(gx_{n+k-2}, gx_{n+k-1})\bigr). \end{aligned}$$

Taking the limit when n tends to infinity, we obtain $d(gu, f(u,u,\ldots, u))\leq0$. Thus $gu = f(u,u,u,\ldots,u)$, i.e., $C(g,f)\neq\emptyset$. Thus there exist $v, u\in X$ such that $\lim_{n\rightarrow \infty}y_{n} = v = g(u)=f(u,u,u,\ldots,u)$. Since g and f are weakly compatible, $g(f(u,u,\ldots,u)) = f(gu,gu,gu,\ldots,gu)$. By (3.10) we have that

$$\begin{aligned} d(ggu,gu) =&d\bigl(gf(u,u,\ldots,u),f(u,u,\ldots,u)\bigr) \\ =&d\bigl(f(gu,gu,gu,\ldots,gu),f(u,u,\ldots,u)\bigr) \\ < &d(ggu,gu) \end{aligned}$$

implies $d(ggu,gu)=0$ and so $ggu=gu$. Hence we have $gu=ggu=g(f(u,u,\ldots,u))=f(gu,gu, gu,\ldots,gu)$, i.e., gu is a common fixed point of g and f, and $\lim_{n\rightarrow\infty} y_{n}=g(u)$. Now suppose that x, y are two fixed points of g and f. Then

$$\begin{aligned} d(x,y) =& d\bigl(f(x,x,x,\ldots,x),f(y,y,y,\ldots,y)\bigr) \\ < &d(gx,gy) \\ =&d(x,y). \end{aligned}$$

This implies $x=y$. Hence the common fixed point is unique. □

Remark 3.8

Taking $s=1$, $g=I$ and $\phi(x_{1},x_{2}, \ldots, x_{k}) = \max\{ x_{1},x_{2}, \ldots, x_{k}\}$ in Theorem 3.7, we obtain Theorem 1.2, i.e., the result of Ciric and Presic [3].

Remark 3.9

For $\lambda\in(0,\frac{1}{s^{k+1}})$, we can drop the condition (3.10) of Theorem 3.7. In fact we have the following.

Theorem 3.10

Let $(X,d)$ be a b-metric space with $s\ge 2$. For any positive integer k, let $f:X^{k} \rightarrow X$ and $g:X\rightarrow X$ be mappings satisfying conditions (3.7), (3.8) and (3.9) with $\lambda\in(0,\frac{1}{s^{k+1}})$. Then all conclusions of Theorem  3.7 hold.

Proof

As proved in Theorem 3.7, there exist $v, u\in X$ such that $\lim_{n\rightarrow\infty}y_{n} = v = g(u)=f(u,u,\ldots,u)$, i.e., $C(g,f)\neq\emptyset$. Since g and f are weakly compatible, $g(f(u,u,\ldots,u)) = f(gu,gu,gu,\ldots,gu)$. By (3.8) we have

$$\begin{aligned} d(ggu,gu) =&d\bigl(gf(u,u,\ldots,u),f(u,u,\ldots,u)\bigr) \\ =&d\bigl(f(gu,gu,gu,\ldots,gu),f(u,u,\ldots,u)\bigr) \\ \leq& sd\bigl(f(gu,gu,gu,\ldots,gu),f(gu,gu,\ldots,gu,u)\bigr) \\ &{}+s^{2}d\bigl(f(gu,gu,\ldots,gu,u),f(gu,gu,\ldots,u,u)\bigr) \\ &{}+\cdots+s^{k-1}d\bigl(f(gu,gu,\ldots,u,u),f(u,u,\ldots,u)\bigr) \\ &{}+s^{k-1}d\bigl(f(gu,u,\ldots,u,u),f(u,u,\ldots,u)\bigr) \\ \leq& s\lambda\phi\bigl(d(ggu,ggu),\ldots,d(ggu,ggu),d(ggu,gu)\bigr) \\ &{}+s^{2}\lambda\phi\bigl(d(ggu,ggu),\ldots,d(ggu,gu),d(gu,gu)\bigr) \\ &{}+ \cdots +s^{k-1}\lambda\phi\bigl(d(ggu,gu),\ldots,d(gu,gu),d(gu,gu) \bigr) \\ =&s\lambda\phi\bigl(0,0,0,\ldots,d(ggu,gu)\bigr)+s^{2}\lambda\phi \bigl(0,0,\ldots,0,d(ggu,gu),0\bigr) \\ &{}+\cdots+s^{k-1}\lambda\phi\bigl(d(ggu,gu),0,0,\ldots,0\bigr) \\ =&s\lambda\bigl[1+s+s^{2}+s^{3}+\cdots+s^{k-2}+s^{k-2} \bigr]d(ggu,gu) \\ \leq&s\lambda\bigl[1+s+s^{2}+s^{3}+\cdots+s^{k-2}+s^{k-1} \bigr]d(ggu,gu) \\ =&s\lambda\frac{s^{k}-1}{s-1}d(ggu,gu). \end{aligned}$$

$s\lambda\frac{s^{k}-1}{s-1}<1$ implies $d(ggu,gu)=0$ and so $ggu=gu$. Hence we have $gu=ggu=g(f(u,u, \ldots,u))=f(gu,gu,gu,\ldots,gu)$, i.e., gu is a common fixed point of g and f, and $\lim_{n\rightarrow\infty} y_{n}=g(u)$. Now suppose that x, y are two fixed points of g and f. Then

$$\begin{aligned} d(x,y) =& d\bigl(f(x,x,x,\ldots,x),f(y,y,y,\ldots,y)\bigr) \\ \leq& sd\bigl(f(x,x,\ldots,x),f(x,x,\ldots,x,y)\bigr)+s^{2} d \bigl(f(x,x,\ldots,x,y), \\ &f(x,x,x,\ldots,x,y,y)\bigr) +\cdots+s^{k-1}d\bigl(f(x,x,y, \ldots,y),f(y,y,\ldots,y)\bigr) \\ &{}+s^{k-1}d\bigl(f(x,y,y,\ldots,y),f(y,y,\ldots,y)\bigr) \\ \leq&s \lambda\phi\bigl\{ d(fx,fx),d(fx,fx),\ldots,d(fx,fy)\bigr\} +s^{2}\lambda \phi\bigl\{ d(fx, fx), \\ &d(fx, fx),\ldots,d(fx,fy),d(fy, fy)\bigr\} \\ &{}+\cdots+s^{k-1}\lambda\phi\bigl\{ d(fx,fy),d(fy,fy),\ldots,d(fy,fy) \bigr\} \\ =& s\lambda\phi\bigl(0,0,\ldots,d(fx,fy)\bigr)+s^{2} \lambda\phi \bigl(0,0,\ldots,d(fx, fy),0\bigr)+\cdots \\ &{}+s^{k-1} \lambda\phi\bigl(d(fx, fy),0,0,0,\ldots,0\bigr) \\ =&\lambda\bigl[s+s^{2}+s^{3}+\cdots+s^{k-1}+s^{k-1} \bigr] d(fx, fy) \\ =&s\lambda\bigl[1+s+s^{2}+s^{3}+\cdots+s^{k-2}+s^{k-2} \bigr] d(fx, fy) \\ \leq&s\lambda\bigl[1+s+s^{2}+s^{3}+\cdots+s^{k-2}+s^{k-1} \bigr] d(fx, fy) \\ =&s\lambda\frac{s^{k}-1}{s-1}d(fx, fy). \\ =&s\lambda\frac{s^{k}-1}{s-1}d(x, y). \end{aligned}$$

This implies $x=y$. Hence the common fixed point is unique. □

Example 3.11

Let $X=R$ and $d:X\times X\rightarrow X$ such that $d(x,y)=| x-y|^{3}$. Then d is a b-metric on X with $s=4$. Let $f:X^{2} \rightarrow X$ and $g:X \rightarrow X$ be defined as follows:

$$\begin{aligned}& f(x,y) = \frac{x^{2}+y^{2}}{13}+\frac{18}{13} \quad \mbox{if }(x,y) \in R, \\& gx = x^{2}-2\quad \mbox{if }x \in R. \end{aligned}$$

We will prove that f and g satisfy condition (3.8):

$$\begin{aligned} d\bigl(f(x,y),f(y,z)\bigr) =& \bigl\vert {f(x,y)-f(y,z)}\bigr\vert ^{3} \\ =&\biggl\vert \frac{x^{2}-z^{2}}{13}\biggr\vert ^{3}= \biggl\vert \frac {x^{2}-y^{2}+y^{2}-z^{2}}{13}\biggr\vert ^{3} \\ \leq&4\biggl(\biggl\vert \frac{x^{2}-y^{2}}{13}\biggr\vert ^{3}+ \biggl\vert \frac{y^{2}-z^{2}}{13}\biggr\vert ^{3}\biggr) \\ =& \frac{4}{13^{3}}\bigl[\bigl\vert x^{2}-y^{2}\bigr\vert ^{3}+ \bigl\vert y^{2}-z^{2}\bigr\vert ^{3}\bigr] \\ =& \frac{8}{13^{3}}\frac{1}{2}\bigl[\bigl\vert x^{2}-y^{2} \bigr\vert ^{3}+ \bigl\vert y^{2}-z^{2}\bigr\vert ^{3}\bigr] \\ \leq& \frac{8}{13^{3}}\max \bigl\{ \bigl\vert x^{2}-y^{2}\bigr\vert ^{3}, \bigl\vert y^{2}-z^{2}\bigr\vert ^{3}\bigr\} \\ =& \frac{8}{13^{3}}\max \bigl\{ d(gx,gy),d(gy,gz)\bigr\} . \end{aligned}$$

Thus, f and g satisfy condition (3.8) with $\lambda= \frac{8}{13^{3}}\in(0,\frac{1}{4^{3}})$. Clearly $C(f, g) =2$, f and g commute at 2. Finally, 2 is the unique common fixed point of f and g. But f and g do not satisfy condition (3.10) as at $x=-1$ and $y=1$, $d(f(x,x),f(y,y))=d(f(-1,-1),f(1,1))=d(\frac{2}{13}+\frac {18}{13},\frac{2}{13}+\frac{18}{13})=0=d(-1,-1)=d(g(-1),g(1))=d(gx,gy)$.

4 Application to matrix equation

In this section we have applied Theorem 3.7 to study the existence of solutions of the nonlinear matrix equation

$$ X=Q+\sum_{i=1}^{m}A_{i}X^{\delta _{i}}A_{i}^{*}, \quad 0< |\delta_{i}|< 1, $$

(4.1)

where Q is an $n\times n$ positive semidefinite matrix and $A_{i}$’s are nonsingular $n\times n$ matrices, or Q is an $n\times n$ positive definite matrix and $A_{i}$’s are arbitrary $n\times n$ matrices, and a positive definite solution X is sought. Here $A_{i}^{*}$ denotes the conjugate transpose of the matrix $A_{i}$. The existence and uniqueness of positive definite solutions and numerical methods for finding a solution of (4.1) have recently been studied by many authors (see [25–30]). The Thompson metric on the open convex cone $P(N)$ ($N\geq2$), the set of all $N\times N$ Hermitian positive definite matrices, is defined by

$$ d(A,B)=\max\bigl\{ \log M(A/B),\log M(B/A)\bigr\} , $$

(4.2)

where $M(A/B)=\inf\{\lambda>0:A\leq\lambda B\}=\lambda _{\mathrm{max}}(B^{-1/2}AB^{-1/2})$, the maximal eigenvalue of $B^{-1/2}AB^{-1/2}$. Here $X\leq Y$ means that $Y-X$ is positive semidefinite and $X< Y$ means that $Y-X$ is positive definite. Thompson [31] has proved that $P(N)$ is a complete metric space with respect to the Thompson metric d and $d(A,B)=\| \log(A^{-1/2}BA^{-1/2})\|$, where $\|\cdot\|$ stands for the spectral norm. The Thompson metric exists on any open normal convex cone of real Banach spaces [31, 32]; in particular, the open convex cone of positive definite operators of a Hilbert space. It is invariant under the matrix inversion and congruence transformations:

$$ d(A,B)=d\bigl(A^{-1},B^{-1}\bigr)=d \bigl(MAM^{*},MBM^{*}\bigr) $$

(4.3)

for any nonsingular matrix M. One remarkable and useful result is the nonpositive curvature property of the Thompson metric:

$$ d\bigl(X^{r},Y^{r}\bigr)\leq rd(X,Y), \quad r \in[0,1]. $$

(4.4)

By the invariant properties of the metric, we then have

$$ d\bigl(MX^{r}M^{*},MY^{r}M^{*}\bigr)=\vert r \vert d(X,Y),\quad r\in[-1,1] $$

(4.5)

for any $X,Y\in P(N)$ and a nonsingular matrix M. Proceeding as in [30] we prove the following lemma.

Lemma 4.1

For any $A_{1},A_{2},\ldots,A_{k}\in P(N)$, $B_{1},B_{2},\ldots ,B_{k}\in P(N)$, $d(A_{1}+A_{2}+\cdots+A_{k},B_{1}+B_{2}+\cdots+B_{k})\leq \max\{ d(A_{1},B_{1}),d(A_{2},B_{2}),\ldots,d(A_{k},B_{k})\}$.

Proof

Without loss of generality we can assume that $d(A_{1},B_{1})\leq d(A_{2},B_{2})\leq\cdots\leq d(A_{k},B_{k})=\log r$. Then $B_{1}\leq rA_{1},B_{2}\leq rA_{2},\ldots,B_{k}\leq rA_{k}$ and $A_{1}\leq rB_{1},A_{2}\leq rB_{2},\ldots,A_{k}\leq rB_{k}$, and thus $B_{1}+A_{1}\leq r(A_{1}+B_{1}),B_{2}+A_{2}\leq r(A_{2}+B_{2}),\ldots,B_{k}+A_{k}\leq r(A_{k}+B_{k})$. Hence $A_{1}+A_{2}+\cdots+A_{k}\leq r[B_{1}+B_{2}+\cdots+B_{k}]$ and $B_{1}+B_{2}+\cdots+B_{k}\leq r[A_{1}+A_{2}+\cdots+A_{k}]$. Hence $d(A_{1}+A_{2}+\cdots+A_{k},B_{1}+B_{2}+\cdots+B_{k})\leq \log r=d(A_{k},B_{k})=\max\{ d(A_{1},B_{1}),d(A_{2},B_{2}),\ldots,d(A_{k},B_{k})\}$. □

For arbitrarily chosen positive definite matrices $X_{n-r}, X_{n-(r-1)},\ldots, X_{n}$, consider the iterative sequence of matrices, given by

$$ X_{n+1}=Q+A_{1}^{*}X_{n-r}^{\alpha _{1}}A_{1}+A_{2}^{*}X_{n-(r-1)}^{\alpha_{2}}A_{2}+ \cdots+A_{r+1}^{*}X_{n}^{\alpha _{r+1}}A_{r+1}, $$

(4.6)

$\alpha_{1}, \alpha_{2},\ldots,\alpha_{r+1}$ are real numbers.

Theorem 4.2

Suppose that $\lambda=\max\{\vert\alpha _{1}\vert,\vert\alpha_{2}\vert,\ldots,\vert\alpha_{r+1}\vert\} \in(0,1)$.

• (i) Equation (4.6) has a unique equilibrium point in $P(N)$, that is, there exists unique $U\in P(N)$ such that

  $$ U=Q+A_{1}^{*}U^{\alpha_{1}}A_{1}+A_{2}^{*}U^{\alpha _{2}}A_{2}+ \cdots+A_{r+1}^{*}U^{\alpha_{r+1}}A_{r+1}. $$

  (4.7)

• (ii) The iterative sequence $\{X_{n}\}$ defined by (4.6) converges to a unique solution of (4.1).

Proof

Define the mapping $f:P(N)\times P(N)\times P(N)\times \cdots\times P(N)\rightarrow P(N)$ by

$$ f(X_{1},X_{2},X_{n-(r-2)},\ldots, X_{k})=Q+A_{1}^{*}X_{1}^{\alpha_{1}}A_{1}+A_{2}^{*}X_{2}^{\alpha_{2}}A_{2}+ \cdots +A_{r+1}^{*}X_{k}^{\alpha_{r+1}}A_{r+1}, $$

(4.8)

where $X_{1},X_{2},\ldots, X_{k}\in P(N)$.

For all $X_{n-r},X_{n-(r-1)},X_{n-(r-2)},\ldots, X_{n+1}\in P(N)$, we have

$$\begin{aligned}& d\bigl(f(X_{n-r},X_{n-(r-1)},X_{n-(r-2)}, \ldots, X_{n}),f(X_{n-(r-1)},X_{n-(r-2)},X_{n-(r-2)}, \ldots, X_{n+1})\bigr) \\& \quad =d\bigl(Q+A_{1}^{*}X_{n-r}^{\alpha_{1}}A_{1}+A_{2}^{*}X_{n-(r-1)}^{\alpha _{2}}A_{2}+ \cdots+A_{r+1}^{*}X_{n}^{\alpha_{r+1}}A_{r+1}, \\& \qquad Q+A_{2}^{*}X_{n-(r-1)}^{\alpha_{1}}A_{2}+A_{3}^{*}X_{n-(r-2)}^{\alpha _{3}}A_{3}+ \cdots+A_{r+2}^{*}X_{n+1}^{\alpha_{r+2}}A_{r+2}\bigr) \\& \quad \leq d\bigl(A_{1}^{*}X_{n-r}^{\alpha_{1}}A_{1}+A_{2}^{*}X_{n-(r-1)}^{\alpha _{2}}A_{2}+ \cdots+A_{r+1}^{*}X_{n}^{\alpha_{r+1}}A_{r+1}, \\& \qquad A_{2}^{*}X_{n-(r-1)}^{\alpha_{1}}A_{2}+A_{3}^{*}X_{n-(r-2)}^{\alpha _{3}}A_{3}+ \cdots+A_{r+2}^{*}X_{n+1}^{\alpha_{r+2}}A_{r+2}\bigr) \\& \quad \leq\max\bigl\{ d\bigl(A_{1}^{*}X_{n-r}^{\alpha_{1}}A_{1},A_{2}^{*}X_{n-(r-1)}^{\alpha _{1}}A_{2} \bigr),d\bigl(A_{2}^{*}X_{n-(r-1)}^{\alpha_{2}}A_{2},A_{3}^{*}X_{n-(r-2)}^{\alpha _{3}}A_{3} \bigr), \\& \qquad \ldots, d\bigl(A_{r+1}^{*}X_{n}^{\alpha _{r+1}}A_{r+1},A_{r+2}^{*}X_{n+1}^{\alpha_{r+2}}A_{r+2} \bigr)\bigr\} \\& \quad \leq \max\bigl\{ \vert\alpha_{1}\vert d(X_{n-r},X_{n-(r-1)}), \vert\alpha _{2}\vert d(X_{n-(r-1)},X_{n-(r-2)}), \\& \qquad \ldots,\vert\alpha_{r+1}\vert d(X_{n},X_{n+1}) \bigr\} \\& \quad \leq \max\bigl\{ \vert \alpha_{1}\vert ,\vert \alpha_{2}\vert,\ldots,\vert \alpha_{r+1}\vert\bigr\} \max \bigl\{ d(X_{n-r},X_{n-(r-1)}),d(X_{n-(r-1)},X_{n-(r-2)}), \\& \qquad \ldots, d(X_{n},X_{n+1})\bigr\} \\& \quad \leq\lambda \max \bigl\{ d(X_{n-r},X_{n-(r-1)}),d(X_{n-(r-1)},X_{n-(r-2)}), \ldots, d(X_{n},X_{n+1})\bigr\} \end{aligned}$$

(4.9)

for all $X_{n-r},X_{n-(r-1)},X_{n-(r-2)},\ldots, X_{n+1}\in P(N)$. $X,Y\in P(N)$, we have

$$\begin{aligned}& d\bigl(f(X,X,\ldots, X),f(Y,Y,\ldots, Y)\bigr) \\& \quad = d\bigl(Q+A_{1}^{*}X^{\alpha_{1}}A_{1}+A_{2}^{*}X^{\alpha_{2}}A_{2}+ \cdots +A_{r+1}^{*}X^{\alpha_{r+1}}A_{r+1}, \\& \qquad Q+A_{2}^{*}Y^{\alpha_{1}}A_{2}+A_{3}^{*}Y^{\alpha_{3}}A_{3}+ \cdots +A_{r+2}^{*}Y^{\alpha_{r+2}}A_{r+2}\bigr) \\& \quad \leq d\bigl(A_{1}^{*}X^{\alpha_{1}}A_{1}+A_{2}^{*}X^{\alpha_{2}}A_{2}+ \cdots +A_{r+1}^{*}X^{\alpha_{r+1}}A_{r+1}, \\& \qquad A_{2}^{*}Y^{\alpha_{1}}A_{2}+A_{3}^{*}Y^{\alpha_{3}}A_{3}+ \cdots +A_{r+2}^{*}Y^{\alpha_{r+2}}A_{r+2}\bigr) \\& \quad \leq \max\bigl\{ d\bigl(A_{1}^{*}X^{\alpha_{1}}A_{1},A_{2}^{*}Y^{\alpha _{1}}A_{2} \bigr),d\bigl(A_{2}^{*}X^{\alpha_{2}}A_{2},A_{3}^{*}Y^{\alpha_{3}}A_{3} \bigr), \\& \qquad \ldots, d\bigl(A_{r+1}^{*}X^{\alpha_{r+1}}A_{r+1},A_{r+2}^{*}Y^{\alpha_{r+2}}A_{r+2} \bigr)\bigr\} \\& \quad \leq \max\bigl\{ \vert\alpha_{1}\vert d(X,Y),\vert \alpha_{2}\vert d(X,Y), \ldots,\vert\alpha_{r+1}\vert d(X,Y)\bigr\} \\& \quad \leq \max\bigl\{ \vert \alpha_{1}\vert ,\vert \alpha_{2}\vert,\ldots,\vert \alpha_{r+1}\vert\bigr\} \max \bigl\{ d(X,Y),d(X,Y), \ldots, d(X,Y)\bigr\} \\& \quad \leq \lambda \max \bigl\{ d(X,Y),d(X,Y),\ldots, d(X,Y)\bigr\} \\& \quad < d(X,Y). \end{aligned}$$

Since $\lambda\in(0,1)$, (i) and (ii) follow immediately from Theorem 3.7 with $s=1$ and $g=I$. □

Numerical experiment illustrating the above convergence algorithm

Consider the nonlinear matrix equation

$$ X=Q+A^{*}X^{\frac{1}{2}}A+B^{*}X^{\frac {1}{3}}B+C^{*}X^{\frac{1}{4}}C, $$

(4.10)

where

$$\begin{aligned}& A=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 14/3 &1/3& 1/4 \\ 2/15 &1/12 &1/23 \\ 3/10 &9/20& 11/4 \end{array}\displaystyle \right ),\qquad B=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 2/5& 3/2& 4/6 \\ 10/4& 6/13& 7/46 \\ 5/2& 4/7& 6/13 \end{array}\displaystyle \right ) , \\& C=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1/3& 19/24 &22/55 \\ 17/10& 27/15 &45/17 \\ 13/8& 1/3& 1/4 \end{array}\displaystyle \right ), \qquad Q=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 &2& 3 \\ 2 &6 &4 \\ 1 &2 &7 \end{array}\displaystyle \right ). \end{aligned}$$

We define the iterative sequence $\{X_{n}\}$ by

$$ X_{n+1}=Q+A^{*}X_{n-2}^{\frac {1}{2}}A+B^{*}X_{n-1}^{\frac{1}{3}}B+C^{*}X_{n}^{\frac{1}{4}}C. $$

(4.11)

Let $R_{m}$ ($m\geq2$) be the residual error at the iteration m, that is, $R_{m}=\| X_{m+1}-(Q+A^{*}X_{m+1}^{\frac {1}{2}}A+B^{*}X_{m+1}^{\frac{1}{3}}B+C^{*}X_{m+1}^{\frac{1}{4}}C)\|$, where $\|\cdot\|$ is the spectral norm. For initial values

$$\begin{aligned}& X_{0}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 &0& 0 \\ 0 &1 &0 \\ 0 &0 &1 \end{array}\displaystyle \right ),\qquad X_{1}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 &1& 0 \\ 1 &1 &0 \\ 1 &0 &1 \end{array}\displaystyle \right ), \\& X_{2}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 1 &1&-1 \\ -1 &1 &1 \\ -1 &1 &1 \end{array}\displaystyle \right ), \end{aligned}$$

we computed the successive iterations and the error $R_{m}$ using MATLAB and found that after thirty five iterations the sequence given by (4.11) converges to

$$U=X_{35}=\left ( \textstyle\begin{array}{@{}c@{\quad}c@{\quad}c@{}} 639.1810& 54.1681 &107.3574 \\ 54.1285& 44.7768 & 44.1469 \\ 104.3977& 42.1095& 112.5509 \end{array}\displaystyle \right ), $$

which is clearly a solution of (4.10). The convergence history of algorithm (4.11) is given in Figure 1.

Figure 1

Convergence history for Equation ( 4.11 ).