# Some results on T-stability of Picard’s iteration

- Thokchom Chhatrajit
^{1}Email authorView ORCID ID profile and - Yumnam Rohen
^{1}

**Received: **8 January 2016

**Accepted: **25 February 2016

**Published: **5 March 2016

## Abstract

We prove the existence and uniqueness of fixed points of T-stability for an iteration on partial cone metric space of Zamfirescu contraction. As an application, we prove a theorem for integral equation. We also give illustrative examples to verify our results.

## Keywords

## Mathematics Subject Classification

## Background

In the year 1974, Rhoades (1974) showed that the iterative scheme converges to a fixed point of a self-mapping *f* for a particular space *X*. Rhoades (1991) provided a survey of iteration procedures that have been used to obtain fixed points for maps satisfying a variety of contractive conditions. Rhoades (1990) showed that several iteration procedures are *T*-stable for maps satisfying a fairly general contractive condition. Liu (1995) introduced the concept of the Ishikawa iteration process with errors and obtained a fixed point of the Lipschitzian local strictly pseudo-contractive mapping. Yousefi (2012) proved an iteration procedure in cone metric spaces. Rhoades and Soltuz (2006) showed that *T*-stability of Mann and Ishikawa iterations are equivalent. Qing and Rhoades (2008) established a general result for the stability of Picard’s iteration. Asadi et al. (2009) investigated the *T*-stability of Picard’s iteration procedures in cone metric spaces and gave an application. Saadati et al. (2009) showed that the variational iteration method for solving integral equations is T-stable. Recently, iteration scheme is extended to some other spaces. It is suitable for mathematician to consider *T*-stability for new iterations problems (see Saipriya et al. 2015; Kang et al. 2015; Yao et al. 2015; Haddadia 2014; Okeke and Olaleru 2014).

*X*,

*p*). Further, let \(F_{T}=\{x \in X: Tx=x\}\) is the set of fixed points of

*T*. In complete metric space, the Picard iteration process \(\{x_{n}\}\) is defined by

*Banach’s contraction condition*.

We shall state some of the iteration process generalising (1) as follows:

Kannan (1968) established an extension of the Banach’s fixed point theorem by using the following contractive definition:

*T*, there exists \(\beta \in (0,\frac{1}{2})\) such that

*T*, there exists \(\gamma \in (0,\frac{1}{2})\) such that

- 1.
\(d(Tx,Ty)\le \alpha d(x,y)\)

- 2.
\(d(Tx,Ty)\le \beta [d(x,Tx)+d(y,Ty)]\)

- 3.
\(d(Tx,Ty)\le \gamma [d(x,Ty)+d(y,Tx)]\)

Then the mapping \(T:E\rightarrow E\) satisfying all above three conditions is called a Zamfirescu operator. Any mapping satisfying the above condition 2 is called a Kannan mapping while the mapping satisfying the above condition 3 is called a Chatterjea operator. Results on stability and T-stability of Picard iteration using the contractive conditions can be found in Rhoades (1974, 1990, 1991), Liu (1995), Yousefi (2012), Rhoades and Soltuz (2006), Qing and Rhoades (2008), Asadi et al. (2009), Saadati et al. (2009), Olatinwo (2008) and references there in.

Huang and Zhang (2007) obtained a generalisation of metric space by introducing the concept of cone metric space. They used an ordered Banach space in place of set of real numbers in metric space. They also obtained some fixed point theorems in this space for mappings satisfying various types of contractive conditions. Some results on cone metric space can be found in Singh and Singh (2014), Singh and Singh (2015), Singh and Sing (2014), Singh (2014) and references there in.

*E*be a real Banach space. A subset

*P*of

*E*is called a cone if

- 1.
*P*is closed, non-empty and \(P \ne {0}\) - 2.
\(a,b \in {\mathbb {R}}, a,b\ge 0\) and \(x,y \in P\) imply \(ax+by \in P\)

- 3.
\(P\cap (-P)={0}\).

Given a cone \(P\subset E\) we define the partial ordering ≤ with respect to *P* by *x* ≤ *y* if and only if \(y-x \in P\). We write \(x<y\) to denote that \(x\le y\) but \(x \ne y\), while \(x\ll y\) will stand for \(y-x \in int.P\)(interior of *P*).

There are two kinds of cone. They are normal cone and non-normal cone. A cone \(P\subset E\) is normal if there is a number \(K> 0\) such that for all \(x,y \in P\), \(0\le x \le y \Rightarrow \parallel x\parallel \le K\parallel y\parallel\). In other words if \(x_{n}\le y_{n}\le z_{n}\) and \(lim_{n\rightarrow \infty }x_{n}=lim_{n\rightarrow \infty }z_{n}= x\) imply \(lim_{n\rightarrow \infty }y_{n}=x\). Also, a cone \(P\subset E\) is regular if every increasing sequence which is bounded above is convergent.

The aim of this paper is to show the existence and uniqueness of fixed points of T-stability for an iteration on partial cone metric space under Zamfirescu contraction. We give an application in integral equation. We also give illustrative examples that verifies our results.

We have the following basic definitions:

###
**Definition 1**

*X*be a nonempty set. Suppose the mapping \(d:X\times X\rightarrow E\) satisfies the following conditions:

- 1.
\(0< d(x,y)\) for all \(x,y \in X\) and \(d(x,y)=0\) iff \(x=y\).

- 2.
\(d(x,y)=d(y,x)\) for all \(x,y \in X\).

- 3.
\(d(x,y)\le d(x,z)+d(z,y)\) for all \(x,y,z \in X\).

*d*is called a cone metric on

*X*and (

*X*,

*d*) is called a cone metric space.

###
**Definition 2**

*X*is a function \(p:X \times X\rightarrow E\) such that for all \(x,y,z \in X\)

- 1.
\(x=y\) if and only if \(p(x,x)=p(x,y)=p(y,y)\),

- 2.
\(0\le p(x,x)\le p(x,y)\),

- 3.
\(p(x,y)=p(y,x)\),

- 4.
\(p(x,y)\le p(x,z)+p(z,y)-p(z,z)\).

*x*,

*p*) such that

*X*is non-empty set and

*p*is a partial cone metric on

*X*is called a partial cone metric space. We know that if

*p*(

*x*,

*y*) = 0, then

*x*=

*y*. But if

*x*=

*y*, then

*p*(

*x*,

*y*) may not be 0.

A cone metric space is a partial cone metric space. But there are partial cone metric space which are not cone metric space. The following example verifies the statement.

###
*Example 3*

(Sonmez 2011) Let \(E={\mathbb {R}}^{2},P=\{(x,y) \in E: x,y \ge 0\}\), \(X={\mathbb {R}}^{+}\) and \(p:X \times X\rightarrow E\) defined by \(p(x,y)=(max.\{x,y\},\alpha max.\{x,y\})\) where \(\alpha \ge 0\) is a constant. Then (*X*, *p*) is a partial cone metric space which is not a cone metric space.

###
**Definition 4**

*X*,

*p*) be a partial cone metric space. Let \(\{x_{n}\}\) be a sequence in

*X*and \(x \in X\). Then \(\{x_{n}\}\) is said to be convergent to

*x*and

*x*is called a limit of \(\{x_{n}\}\) if

###
**Definition 5**

(Sonmez 2011) Let (*X*, *p*) be a partial cone metric space. Let \(\{x_{n}\}\) be a sequence in *X* and \(x \in X\). Then \(\{x_{n}\}\) is said to be Cauchy sequence if there exists an \(a \in P\) such that for every \(\epsilon > 0\) there is *N* such that \(\parallel p(x_{n},x_{m})-a\parallel < \epsilon\) for all \(n,m > N\).

###
**Definition 6**

(Sonmez 2011) A partial cone metric space (*X*, *p*) is said to be complete if and only if every Cauchy sequence in *X* is convergent.

###
**Definition 7**

*X*,

*d*) be a complete metric space, \(T:X\rightarrow X\) a selfmap of

*X*. Suppose that \(F_{T}=\{p \in E:Tp=p\}\) is the set of fixed points of

*T*. Let \(\{x_{n}\}_{n=0}^{\infty }\subset E\) be the sequence generated by an iteration procedure involving

*T*which is defined by \(x_{n+1}=f(T,x_{n}),n=0,1,2\dots\) where \(x_{0}\in X\) is the initial approximation and

*f*is some function. Suppose \(\{x_{n}\}_{n=0}^{\infty }\) converges to fixed point

*p*of

*T*. Let \(\{y_{n}\}_{n=0}^{\infty }\subset X\) and set

*T*-stable or stable with respect to

*T*if and only if \(\lim _{n\rightarrow \infty }\epsilon _{n}=0\) implies \(\lim _{n\rightarrow \infty }y_{n}=p\).

###
*Remark 8*

## Main results

In this section we establish iteration procedure in partial cone metric spaces. This is to stretch out some recent results of T-stability. Let (*X*, *p*) be a partial cone metric space. Let \(\{T_{n}\}_{n}\) be a sequence of self maps of *X* with \(\bigcap _{n}F(T_{n})\ne \phi\). Let \(x_{0}\) be a point of *X* and posit that \(y_{n+1}=F(T_{n},y_{n})\) is an iteration procedure involving \(\{T_{n}\}_{n}\), which gives a sequence \(\{y_{n}\}\) of points from X.

In general, such a sequence \(\{z_{n}\}\) can be acquired in the following way. Let \(y_{0}\) be a point in *X*. Put \(y_{n+1}=f(T_{n},y_{n})\). Let \(y_{0}=z_{0}\). Now, \(y_{1}=f(T_{0},y_{0})\). Because of rounding or in the function \(T_{0}\), a new value \(z_{1}\) approximately equal to \(y_{0}\) might be procured in place of \(f(T_{0},y_{0})\). Then to approximate \(z_{1}\), the value \(f(T_{1},y_{1})\) is determined to furnish \(z_{2}\), approximation of \(f(T_{1},z_{1})\). This computation is persisted to obtain \(\{z_{n}\}\) as an approximate sequence of \(\{y_{n}\}\).

###
**Definition 9**

The iteration \(y_{n+1}=F(T_{n},y_{n})\) is said to be \(\{T_{n}\}\)-semistable (or semistable) with respect to \(\{T_{n}\}\) if \(\{y_{n}\}\) converges to a fixed point *q* in \(\bigcap _{n}F(T_{n})\ne \phi\) and whenever \(\{z_{n}\}\) is a sequence in *X* with \(lim_{n\rightarrow \infty }p(y_{n},f(T_{n},y_{n}))=0\) and \(p(y_{n},f(T_{n},z_{n}))=o(t_{n})\) for some sequence \(t_{n}\subset {\mathbb {R}}^{+}\), then we have \(z_{n}\rightarrow z\).

###
**Definition 10**

The iteration \(y_{n+1}=F(T_{n},y_{n})\) is said to be \(\{T_{n}\}\) stable(or stable) with respect to \(\{T_{n}\}\) if \(\{z_{n}\}\) converges to a fixed point *q* in \(\bigcap _{n}F(T_{n})\ne \phi\) and whenever \(\{z_{n}\}\) is a sequence in *X* with \(lim_{n\rightarrow \infty }p(y_{n},f(T_{n},z_{n}))=0\), then we have \(z_{n}\rightarrow z\).

###
*Remark 11*

\(T_{n} = T\) for all n that gives the definition of T-stability.

###
**Theorem 12**

*Let (X, p) be a complete partial cone metric space. Let P be a normal cone with normal constant K and*
\(T:X\rightarrow X\)
*with*
\(F(T)\ne \phi\)
*. If there exists*
\(c \in (0,\frac{1}{2})\)
*such that*
\(p(Tx,Ty)\le c p(x,y)\)
*for all*
\(x,y \in X\)
*and*
\(u \in F(T)\)
*and in addition, whenever*
\(\{y_{n}\}\)
*is a sequence with*
\(p(y_{n},Ty_{n})\rightarrow 0\)
*as*
\(n\rightarrow \infty\)
*, then Picard iteration is T-stable.*

###
*Proof*

*v*be another fixed point of

*T*, then

Thus the fixed point of *T* is unique. \(\square\)

###
**Theorem 13**

*Let (X, p) be a complete partial cone metric space. Let P be a normal cone with normal constant K and*
\(T:X\rightarrow X\)
*with*
\(F(T)\ne \phi .\)
*If there exists*
\(c \in (0,\frac{1}{2})\)
*such that*
\(p(Tx,Ty)\le c(p(Tx,x)+p(Ty,y))\)
*for all*
\(x,y \in X\)
*and*
\(u \in F(T)\)
*and in addition, whenever*
\(\{y_{n}\}\)
*is a sequence with*
\(p(y_{n},Ty_{n})\rightarrow 0\)
*as*
\(n\rightarrow \infty ,\)
*then Picard iteration is T-stable.*

###
*Proof*

*v*be another fixed point of

*T*, then

Thus the fixed point of *T* is unique. \(\square\)

###
**Theorem 14**

*Let (X, p) be a complete partial cone metric space. Let P be a normal cone with normal constant K and*
\(T:X\rightarrow X\)
*with*
\(F(T)\ne \phi\)
*. If there exists*
\(c \in (0,\frac{1}{2})\)
*such that*
\(p(Tx,Ty)\le c(p(x,Ty)+p(y,Tx))\)
*for all*
\(x,y \in X\)
*and*
\(u \in F(T)\)
*and in addition, whenever*
\(\{y_{n}\}\)
*is a sequence with*
\(p(y_{n},Ty_{n})\rightarrow 0\)
*as*
\(n\rightarrow \infty\)
*, then Picard iteration is T-stable.*

###
*Proof*

*v*be another fixed point of

*T*, then

We get \(u=v\). Thus the fixed point of *T* is unique. \(\square\)

###
*Example 15*

*p*be the partial cone metric on

*X*defined by \(p(x,y)=\,\mid x-y\mid\). Let \(T:X\rightarrow X\) such that

## An application

###
**Theorem 16**

*Let X*=

*C*[0, 1], \({\mathbb {R}}\)

*with*\(\parallel f \parallel _{\infty }\,=Sup_{0\le x\le 1} \mid f(x)\mid\)

*for*\(f \in X\)

*and let*\(T:X\rightarrow X\)

*defined by*\(Tf(x)=\int ^{1}_{0}F(x,f(t))dt\)

*where*

- 1.
\(F:[0,1]\times {\mathbb {R}}\rightarrow {\mathbb {R}}\)

*is a continuous function.* - 2.
*The partial derivative*\(F_{y}\) of*F**with respect to y exists and*\(\mid F_{y}(x,y)\mid \le c\)*for some*\(c \in [0,1)\) - 3.
*For every real number*\(0\le a< 1\)*one has*\(ax\le F(x, ay)\)*for every*\(x, y \in [0,1]\)

*Let*
\(P= \{(x,y)\in {\mathbb {R}}^{2}: x,y \ge 0\}\)
*be a normal cone and (X, p) the complete partial cone metric space defined*
\(p(f,g)=(\parallel f- g\parallel _{\infty },\alpha \parallel f-g\parallel _{\infty })\)
*where*
\(\alpha \ge 0\)
*. Then Picard’s iteration is T-stable if*
\(0\le c \le \frac{1}{2}\).

###
*Example 17*

Let \(F(x,y)=\frac{x+y}{4}\). Then *F* satisfies of Theorem 16 if \(0\le c <1\). Let \(T:X \rightarrow X\) be a self-map defined by \(Tf(x)=x+(\frac{1}{4})+\int ^{1}_{0}f(t)dt\). Then *T* has unique fixed point and Picard’s iteration is *T*-stable.

## Conclusion

We extend and prove the T-stability of Picard’s iteration satisfying Zamfirescu contraction in partial cone metric space. Our results are more general than that of the results of metric and cone metric spaces. This result can be extended to other spaces.

## Declarations

### Authors’ contributions

Both authors contributed equally in preparation of the final manuscript. Both authors read and approved the final manuscript.

### Acknowledgements

Authors would like to thank the referees for their suggestions.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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