 Research
 Open Access
Iterative approximation of fixed points of quasicontraction mappings in cone metric spaces
 Petko D Proinov^{1}Email author and
 Ivanka A Nikolova^{1}
https://doi.org/10.1186/1029242X2014226
© Proinov and Nikolova; licensee Springer. 2014
 Received: 19 February 2014
 Accepted: 23 May 2014
 Published: 3 June 2014
Abstract
In this paper, we establish a full statement of Ćirić’s fixed point theorem in the setting of cone metric spaces. More exactly, we obtain a priori and a posteriori error estimates for approximating fixed points of quasicontractions in a cone metric space. Our result complements recent results of Zhang (Comput. Math. Appl. 62:16271633, 2011), Ding et al. (J. Comput. Anal. Appl. 15:463470, 2013) and others.
MSC: Primary 54H25; secondary 47H10, 46A19.
Keywords
 Picard iteration
 cone metric space
 solid vector space
 fixed points
 quasicontractions
 error estimates
1 Introduction
In this paper, we study fixed points of quasicontraction mappings in a cone metric space $(X,d)$ over a solid vector space $(Y,\u2aaf)$. Cone metric spaces have a long history (see Collatz [1], Zabrejko [2], Janković et al. [3], Proinov [4] and references therein). A unified theory of cone metric spaces over a solid vector space was developed in a recent paper of Proinov [4]. Recall that an ordered vector space with convergence structure $(Y,\u2aaf)$ is called:

a solid vector space if it can be endowed with a strict vector ordering $(\prec )$;

a normal vector space if the convergence of Y has the sandwich property.
Every metric space $(X,d)$ is a cone metric space over ℝ (with usual ordering and usual convergence). On the other hand, every cone metric space over a solid vector space is a metrizable topological space (see Proinov [4] and references therein). It is well known that a lot of fixed point results in cone metric setting can be directly obtained from their metric versions (see Du [5], AminiHarandi and Fakhar [6], Feng and Mao [7], Kadelburg et al. [8], Asadi et al. [9], Proinov [4], and Ercan [10]).
For instance, for this purpose we can use the following theorem. This theorem shows that every cone metric is equivalent to a metric which preserves the completeness as well as some inequalities.
Theorem 1.1 ([[4], Theorem 9.3])
 (i)
The topology of $(X,d)$ coincides with the topology of $(X,\rho )$.
 (ii)
$(X,d)$ is complete if and only if $(X,\rho )$ is complete.
 (iii)For $x,{x}_{1},\dots ,{x}_{n}\in X$, $y,{y}_{1},\dots ,{y}_{n}\in X$ and ${\lambda}_{1},\dots ,{\lambda}_{n}\in \mathbb{R}$,$d(x,y)\u2aaf\sum _{i=1}^{n}{\lambda}_{i}d({x}_{i},{y}_{i})\phantom{\rule{1em}{0ex}}\mathit{\text{implies}}\phantom{\rule{1em}{0ex}}\rho (x,y)\le \sum _{i=1}^{n}{\lambda}_{i}\rho ({x}_{i},{y}_{i}).$
In 1922, Banach [11] proved his famous fixed point theorem for contraction mappings. Banach’s contraction principle is one of the most useful theorems in the fixed point theory. It has two versions: a short version and a full version. In a metric space setting its full statement can be seen, for example, in the monograph of Berinde [[12], Theorem 2.1]. Recently, full statements of Banach’s fixed point theorem in a cone metric spaces over a solid vector space were given by Radenović and Kadelburg [[13], Theorem 3.3] and Proinov [[4], Theorem 11.1].
Definition 1.2 ([14])
for all $x,y\in X$.
There are a large number of generalizations of Banach’s contraction principle (see, for example, [14–18] and references therein). In 1974, Ćirić [14] introduced contraction mappings and proved the following well known generalization of Banach’s fixed point theorem.
 (i)
Existence and uniqueness. T has a unique fixed point ξ in X.
 (ii)
Convergence of Picard iteration. For every starting point $x\in X$ the Picard iteration sequence $({T}^{n}x)$ converges to ξ.
 (iii)A priori error estimate. For every point $x\in X$ the following a priori error estimate holds:$d({T}^{n}x,\xi )\le \frac{{\lambda}^{n}}{1\lambda}d(x,Tx)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}n\ge 0.$(2)
Following Zhang [19], in the next definition, we define a useful binary relation between an ordered vector space Y and the set of all subsets of Y. It plays a very important role in this paper as it is used to prove our main result.
Definition 1.4 ([19])
Let $(Y,\u2aaf)$ be an ordered vector space, $x\in Y$ and $A\subset Y$. We say that $x\u2aafA$ if there exists at least one vector $y\in A$ such that $x\u2aafy$.
for all $x,y\in X$. They proved the following result [[20], Theorem 2.1]: Let $(X,d)$ be a cone metric space over a normal solid Banach space $(Y,\u2aaf)$; then every quasicontraction T of the type (3) has a unique fixed point in X, and for all $x\in X$ the Picard iterative sequence $({T}^{n}x)$ converges to this fixed point. Kadelburg et al. [[21], Theorem 2.2] improved this result by omitting the assumption of normality provided that $\lambda \in [0,1/2)$. Gajić and Rakočević [[22], Theorem 3] proved this result for any contraction constant $\lambda \in [0,1)$. Rezapour et al. [[23], Theorem 2.1] proved this result in the case when Y is a solid topological vector space and $\lambda \in [0,1)$. Furthermore, Kadelburg et al. [[8], Theorem 3.5(b)] proved that this result is equivalent to the short version of Ćirić’s fixed point theorem.
In 2011, Zhang [19] presented the following new definition for quasicontractions in cone metric spaces.
Definition 1.5 ([19])
for all $x,y\in X$.
By applying Theorem 1.1 to the first two conclusions of Theorem 1.3, we obtain the following fixed point theorem in a cone metric setting.
 (i)
Existence and uniqueness. T has a unique fixed point ξ in X.
 (ii)
Convergence of Picard iteration. For every starting point $x\in X$ the Picard iteration sequence $({T}^{n}x)$ converges to ξ.
In 2011, Zhang [19] proved Theorem 1.6 in the case when Y is a normal solid Banach space. In 2013, Ding et al. [24] proved this theorem in the case when Y is a solid topological vector space.
In this paper, we establish a full statement of Ćirić’s fixed point theorem in the setting of cone metric spaces. Our result complements Theorem 1.6. Thus it extends and complements the corresponding results of Zhang [19], Ding et al. [24], and others.
For some recent results on the topic, we refer the reader to [25–38]. In the papers [37, 38] one can find some applications of cone metric spaces to iterative methods for finding all zeros of polynomial simultaneously.
2 Preliminaries
In this section, we introduce some basic definitions and theorems of cone metric spaces over a solid vector space.
Definition 2.1 ([4])
Let Y be a real vector space and S be the set of all infinite sequences in Y. A binary relation → between S and Y is called a convergence on Y if it satisfies the following axioms:
(C1) If ${x}_{n}\to x$ and ${y}_{n}\to y$, then ${x}_{n}+{y}_{n}\to x+y$.
(C2) If ${x}_{n}\to x$ and $\lambda \in \mathbb{R}$, then $\lambda {x}_{n}\to \lambda x$.
(C3) If ${\lambda}_{n}\to \lambda $ in ℝ and $x\in Y$, then ${\lambda}_{n}x\to \lambda x$.
A real vector space Y endowed with convergence is said to be a vector space with convergence. If ${x}_{n}\to x$, then $({x}_{n})$ is said to be a convergent sequence in Y, and the vector x is said to be a limit of $({x}_{n})$.
Definition 2.2 ([4])
Let $(Y,\to )$ be a vector space with convergence. An ordering ⪯ on Y is said to be a vector ordering if it is compatible with the algebraic and convergence structures on Y in the sense that the following are true:
(V1) If $x\u2aafy$, then $x+z\u2aafy+z$.
(V2) If $\lambda \ge 0$ and $x\u2aafy$, then $\lambda x\u2aaf\lambda y$.
(V3) If ${x}_{n}\to x$, ${y}_{n}\to y$, ${x}_{n}\u2aaf{y}_{n}$ for all n, then $x\u2aafy$.
A vector space with convergence Y endowed with vector ordering is called an ordered vector space with convergence.
Definition 2.3 ([4])
Let $(Y,\u2aaf,\to )$ be an ordered vector space with convergence. A strict ordering ≺ on Y is said to be a strict vector ordering if it is compatible with the vector ordering, the algebraic structure and the convergence structure on Y in the sense that the following are true:
(S1) If $x\prec y$, then $x\u2aafy$.
(S2) If $x\u2aafy$ and $y\prec z$, then $x\prec z$.
(S3) If $x\prec y$, then $x+z\prec y+z$.
(S4) If $\lambda >0$ and $x\prec y$, then $\lambda x\prec \lambda y$.
(S5) If ${x}_{n}\to x$, ${y}_{n}\to y$ and $x\prec y$, then ${x}_{n}\prec {y}_{n}$ for all but finitely many n.
It turns out that an ordered vector space can be endowed with at most one strict vector ordering (see Proinov [[4], Theorem 5.1]).
Definition 2.4 (Solid vector space)
An ordered vector space with convergence endowed with a strict vector ordering is said to be a solid vector space.
Let us consider an important example of a solid vector space.
Then Y is a solid vector space called a solid topological vector space.
Now let us recall the definition of a cone metric space known also as ‘Kmetric spaces’ (see Zabrejko [2], Proinov [4] and references therein).
Definition 2.6 (Cone metric space)
 (i)
$d(x,y)\u2ab00$ for all $x,y\in X$ and $d(x,y)=0$ if and only if $x=y$;
 (ii)
$d(x,y)=d(y,x)$ for all $x,y\in X$;
 (iii)
$d(x,y)\u2aafd(x,z)+d(z,y)$ for all $x,y,z\in X$.
The pair $(X,d)$ is called a cone metric space over Y.
Let $(X,d)$ be a cone metric space over a solid vector space $(Y,\u2aaf,\prec )$, ${x}_{0}\in X$ and $r\in Y$ with $r\succ 0$. Then the set $U({x}_{0},r)=\{x\in X:d(x,{x}_{0})\prec r\}$ is called an open ball with center ${x}_{0}$ and radius r.
Every cone metric space X over a solid vector space Y is a Hausdorff topological space with topology generated by the basis of all open balls. Then a sequence $({x}_{n})$ of points in X converges to $x\in X$ if and only if for every vector $c\in Y$ with $c\succ 0$, $d({x}_{n},x)\prec c$ for all but finitely many n.
Recall also that a sequence $({x}_{n})$ in X is called a Cauchy sequence if for every $c\in Y$ with $c\succ 0$ there is $N\in \mathbb{N}$ such that $d({x}_{n},{x}_{m})\prec c$ for all $n,m>N$. A cone metric space X is called complete if each Cauchy sequence in X is convergent.
In order to prove our main result we need the following two theorems.
Theorem 2.7 ([4])
Theorem 2.8 ([4])
Then ξ is a fixed point of T.
3 Auxiliary results
Let A be a subset of a real vector space Y. Recall that the convex hull of A, denoted coA, is the smallest convex set including A. Suppose $x,{x}_{1},\dots ,{x}_{n}\in Y$. It is well known that $x\in co\{{x}_{1},\dots ,{x}_{n}\}$ if and only if there exist nonnegative numbers ${\alpha}_{1},\dots ,{\alpha}_{n}$ such that ${\sum}_{i=1}^{n}{\alpha}_{i}=1$ and $x={\sum}_{i=1}^{n}{\alpha}_{i}{x}_{i}$.
Lemma 3.1 Let $(Y,\u2aaf)$ be an ordered vector space. Suppose that x, y, ${x}_{1},\dots ,{x}_{n}$, ${y}_{1},\dots ,{y}_{m}$ are vectors in Y and λ is a real number. Then:
(P1) $x\u2aafco\{{x}_{1},\dots ,{x}_{n}\}\Rightarrow x\u2aafco\{{x}_{1},\dots ,{x}_{n},y\}$;
(P2) $x\u2aafco\{{x}_{1},\dots ,{x}_{n}\}$ and ${x}_{i}\u2aaf{y}_{i}$ for all $i\Rightarrow x\u2aafco\{{y}_{1},\dots ,{y}_{n}\}$;
(P3) $x\u2aafco\{{x}_{1},\dots ,{x}_{n},y\}$ and $y\u2aafco\{{y}_{1},\dots ,{y}_{m}\}\Rightarrow x\u2aafco\{{x}_{1},\dots ,{x}_{n},{y}_{1},\dots ,{y}_{m}\}$;
(P4) $x\u2aafco\{0,{x}_{1},\dots ,{x}_{n}\}\iff x\u2aafco\{{x}_{1},\dots ,{x}_{n}\}$ if ${x}_{i}\u2ab00$ for some i;
(P5) $x\u2aafco\{\lambda x,{x}_{1},\dots ,{x}_{n}\}\iff x\u2aafco\{{x}_{1},\dots ,{x}_{n}\}$ if $\lambda <1$ and ${x}_{i}\u2ab00$ for some i;
(P6) $x\u2aafco\{{x}_{1},\dots ,{x}_{n},y\}\iff x\u2aafco\{{x}_{1},\dots ,{x}_{n}\}$ if $y={x}_{i}$ for some i.
This implies $x\u2aafco\{{x}_{1},\dots ,{x}_{n}\}$ since ${\sum}_{i=1}^{n}{\gamma}_{i}=1$. □
Remark 3.2 Note that Lemma 3.1 remains true if we omit the expression ‘co’ from its formulation.
The following lemma was given by Zhang [[19], Lemma 6] in a slightly different form. We give a simple proof of this lemma.
We divide the proof of (8) into three steps.
From the last four inequalities and properties (P3) and (P6), we obtain the desired inequality.
From (10), (11), and properties (P3) and (P6), we obtain (9).
Finally, by the claim of Step 2 and the inequality (12), taking into account the properties (P3) and (P6), we obtain (8). This completes the proof of the lemma. □
In the following lemma, we show that if T is a quasicontraction of a cone metric space X, then for every starting point $x\in X$, the Picard iteration sequence $({T}^{n}x)$ is bounded in the space X.
Multiplying both sides of this inequality by $1/(1\lambda +\alpha \lambda )$, we obtain (14). This completes the proof of the lemma. □
where $\alpha =\lambda /(1\lambda )$ and $\beta =(1+\lambda )/(1\lambda )$.
Multiplying both sides of this inequality by $1/(1\lambda )$, we get (18). □
Lemma 3.6 Let $(X,d)$ be a cone metric space over an ordered vector space $(Y,\u2aaf)$. Then every quasicontraction $T:X\to X$ has at most one fixed point in X.
Proof Suppose that x and y are two fixed points of T. It follows from the inequality (4) and properties (P4) and (P6) that $d(x,y)\u2aaf\lambda d(x,y)$ which implies $d(x,y)\u2aaf0$. On the other hand, $d(x,y)\u2ab00$. Hence, $d(x,y)=0$, which yields $x=y$. □
4 Main result
Now we are ready to state the main result of this paper. Let $(X,d)$ be a complete cone metric space over an ordered vector space Y. Recall that for a point ${x}_{0}\in X$ and a vector $r\in Y$ with $r\u2ab00$, the set $\overline{U}({x}_{0},r)=\{x\in X:d(x,{x}_{0})\u2aafr\}$ is called a closed ball with center ${x}_{0}$ and radius r.
 (i)Existence, uniqueness and localization. T has a unique fixed point ξ which belongs to the closed ball $\overline{U}(x,r)$ with radius$r=\frac{1}{1\lambda}d(x,Tx),$
 (ii)
Convergence of Picard iteration. Starting from any point $x\in X$ the Picard sequence $({T}^{n}x)$ remains in the closed ball $\overline{U}(x,r)$ and converges to ξ.
 (iii)A priori error estimate. For every point $x\in X$ the following a priori estimate holds:$d({T}^{n}x,\xi )\u2aaf\frac{{\lambda}^{n}}{1\lambda}d(x,Tx)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}n\ge 0.$(20)
 (iv)A posteriori error estimates. For every point $x\in X$ the following a posteriori estimate holds:$d({T}^{n}x,\xi )\u2aaf\frac{1}{1\lambda}d({T}^{n}x,{T}^{n+1}x)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}n\ge 0,$(21)$d({T}^{n}x,\xi )\u2aaf\frac{\lambda}{1\lambda}d({T}^{n}x,{T}^{n1}x)\phantom{\rule{1em}{0ex}}\mathit{\text{for all}}n\ge 1.$(22)
Applying (25) to the point ${T}^{n1}x$, we get (22). Setting $n=0$ in (23), we obtain $d(x,{T}^{m}x)\u2aaf{b}_{0}$ for every $m\ge 0$. Hence, the sequence $({T}^{n}x)$ lies in the ball $\overline{U}(x,r)$ since $r={b}_{0}$.
It follows from Lemma 3.5 that ξ satisfies condition (5). Hence, by Theorem 2.8, we conclude that ξ is a fixed point of T. The uniqueness of the fixed point follows from Lemma 3.6. □
Theorem 4.1 extends and complements the recent results of Ding et al. [[24], Theorem 3.1] and Zhang [[19], Theorem 3] as well as previous results due to Ilić and Rakočević [[20], Theorem 2.1], Kadelburg et al. [[21], Theorem 2.2], Rezapour et al. [[23], Theorem 2.1] and Kadelburg et al. [[8], Theorem 3.5(b)] who have studied quasicontraction mappings of the type (3).
for all $x,y\in X$, where α, β, γ, μ and ν are five nonnegative constants such that $\alpha +\beta +\gamma +\mu +\nu <1$. In this case Theorem 4.1 holds with $\lambda =\alpha +\beta +\gamma +\mu +\nu $ since condition (26) implies condition (4) with this λ. Let us note that in this special case Theorem 4.1 holds even with $\lambda =(\alpha +\delta )/(1\delta )$, where $\delta =(\beta +\gamma +\mu +\nu )/2$.
5 Examples
Zhang [[19], Example 1] gives an example showing that the set of all quasicontractions of the type (3) is a proper subset of the set of all quasicontractions defined by (4). In order to prove this, he considers a selfmapping of a cone metric space X over a normal solid vector space Y. Ding et al. [[24], Example 4.1] provide a similar example, but for the case of a nonnormal solid vector space Y.
Now we shall construct a family of examples which show that ℬ is a proper subset of . In particular, this family contains both the example of Zhang [19] and the example of Ding et al. [24].
Definition 5.1 Let $(Y,\u2aaf)$ be an ordered vector space, and let $a,b,c\u2ab00$ be three vectors in Y. We say that the triple $(a,b,c)$ satisfies property (C) if the following two statements hold:

$a\u2aaf\lambda co\{b,c\}$ for some $\lambda \in [0,1)$, $b\u2aafa+c$ and $c\u2aafa+b$.

$a\u2aafk\{b,c\}$ is wrong for every $k\in [0,1)$.
Proposition 5.2 Let $Y=\mathbb{R}$ be endowed with the usual ordering ≤. Then there are no triples $(a,b,c)$ in Y satisfying property (C).
Proof Assume that there is a triple $(a,b,c)$ in Y with property (C). Then $a\u2aaf\lambda max\{b,c\}$ for some $\lambda \in [0,1)$. On the other hand, $a\u2aafkmax\{b,c\}$ is wrong for every $k\in [0,1)$. This is a contradiction which proves the proposition. □
Proposition 5.3 Let $Y={\mathbb{R}}^{n}$ ($n\ge 2$) be endowed with coordinatewise ordering ⪯. Then in Y there exist infinitely many triples $(a,b,c)$ satisfying property (C).
Then the vectors $a=(\alpha ,\dots ,\alpha )$, $b=(\beta ,\dots ,\beta ,\gamma )$ and $c=(\gamma ,\dots ,\gamma ,\beta )$ satisfy property (C) with $\lambda =\frac{2\alpha}{\beta +\gamma}$. □
Proposition 5.4 Let $Y={C}^{n}[0,1]$ ($n\ge 2$) be endowed with pointwise ordering ⪯. Then in Y there exist infinitely many triples $(a,b,c)$ satisfying property (C).
Then the functions $a(t)=\alpha $, $b(t)=\beta +\delta t$ and $c(t)=\gamma \delta t$ satisfy property (C) with $\lambda =\frac{2\alpha}{\beta +\gamma}$. □
Using Lemma 3.1, it is easy to prove that $T\in \mathcal{C}$ if and only if $a\u2aaf\lambda co\{b,c\}$ for some $\lambda \in [0,1)$. Analogously, taking into account Remark 3.2, one can easily prove that $T\in \mathcal{B}$ if and only if $a\u2aafk\{b,c\}$ for some $k\in [0,1)$. Now taking into account that the triple $(a,b,c)$ satisfies property (C), we conclude that $T\in \mathcal{C}$ and $T\notin \mathcal{B}$. Hence, ℬ is a proper subset of .
Declarations
Acknowledgements
The research is supported by Project NI13 FMI002 of Plovdiv University.
Authors’ Affiliations
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