 Review
 Open access
 Published:
Some results on Tstability of Picard’s iteration
SpringerPlus volume 5, Article number: 284 (2016)
Abstract
We prove the existence and uniqueness of fixed points of Tstability 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.
Background
In the year 1974, Rhoades (1974) showed that the iterative scheme converges to a fixed point of a selfmapping 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 Tstable 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 pseudocontractive mapping. Yousefi (2012) proved an iteration procedure in cone metric spaces. Rhoades and Soltuz (2006) showed that Tstability 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 Tstability 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 Tstable. Recently, iteration scheme is extended to some other spaces. It is suitable for mathematician to consider Tstability for new iterations problems (see Saipriya et al. 2015; Kang et al. 2015; Yao et al. 2015; Haddadia 2014; Okeke and Olaleru 2014).
Let us consider a self mapping \(T:X\rightarrow X\) in a complete partial cone metric space (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
It has been used to approximate the fixed points of mappings satisfying the contractive condition
over the years by many authors. The above contractive condition (2) is called Banach’s contraction condition.
We shall state some of the iteration process generalising (1) as follows:
For \(x_{0} \in E\), the sequence \(\{x_{n}\}_{n=0}^{\infty }\) defined by
where \(\{\alpha _{n}\}_{n=0}^{\infty }\subset [0,1]\), is called the Mann iteration process.
For \(x_{0} \in E\), the sequence \(\{x_{n}\}_{n=0}^{\infty }\) defined by
where \(\{\alpha _{n}\}_{n=0}^{\infty }\) and \(\{\beta _{n}\}_{n=0}^{\infty }\) are sequences in [0, 1] called the Ishikawa iteration process.
Kannan (1968) established an extension of the Banach’s fixed point theorem by using the following contractive definition:
For a self map T, there exists \(\beta \in (0,\frac{1}{2})\) such that
Chatterjea (1972) gave the following contractive condition:
For a selfmap T, there exists \(\gamma \in (0,\frac{1}{2})\) such that
Zamfirescu (1972) established the generalisation of the Banach’s fixed point theorem by combining (2), (5) and (6). For a mapping \(T:E\rightarrow E\), there exist real numbers \(\alpha ,\beta ,\gamma\) satisfying \(0\le \alpha <1, 0\le \beta <\frac{1}{2},0\le \gamma <\frac{1}{2}\) respectively such that for each \(x,y \in E\), at least one of the following is true:

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 Tstability 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.
Let E be a real Banach space. A subset P of E is called a cone if

1.
P is closed, nonempty 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 \(yx \in P\). We write \(x<y\) to denote that \(x\le y\) but \(x \ne y\), while \(x\ll y\) will stand for \(yx \in int.P\)(interior of P).
There are two kinds of cone. They are normal cone and nonnormal 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 Tstability 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
(Huang and Zhang 2007) Let 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 \(x,y,z \in X\).
Then d is called a cone metric on X and (X, d) is called a cone metric space.
Definition 2
(Sonmez 2011) A partial cone metric space on a nonempty set 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.
\(p(x,y)\le p(x,z)+p(z,y)p(z,z)\).
Then the pair (x, p) such that X is nonempty 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
(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 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
(Olatinwo 2008) Let (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
Then, the iteration procedure is said to be Tstable 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
(Olatinwo 2008) Since the metric space is induced by the norm, we have
in place of
in the definition of stability whenever we are working in normed linear space or Banach space.
Main results
In this section we establish iteration procedure in partial cone metric spaces. This is to stretch out some recent results of Tstability. 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 Tstability.
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 Tstable.
Proof
Let \(\{y_{n}\} \subseteq X, \epsilon _{n}=p(y_{n+1},Ty_{n})\) and \(\epsilon _{n} \rightarrow 0\) as \(n\rightarrow \infty\). Then for any \(n \in {\mathbb {N}}\), we have
Hence, \(p(y_{n+1},u) =0\). But since \(p(Ty_{n+1},Ty_{n+1})\le c p(y_{n+1},y_{n+1})=0\). We have that \(p(Ty_{n+1},Ty_{n+1})=p(Ty_{n+1},u)=p(u,u)=0\). This implies that \(Ty_{n}=u\). Therefore, \(Lim_{n\rightarrow \infty }y_{n}=q\).
For uniqueness: Let v be another fixed point of T, then
Since \(c <1\) we have \(p(u,v)=p(u,u)=p(v,v)\). Hence \(u=v\).
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 Tstable.
Proof
Let \(\{y_{n}\} \subseteq X, \epsilon _{n}=p(y_{n+1},Ty_{n})\) and \(\epsilon _{n} \rightarrow 0\) as \(n\rightarrow \infty\). Then for any \(n \in {\mathbb {N}}\), we have
Hence, \(p(Ty_{n+1},u)=0\). But since
We have that \(p(Ty_{n+1},Ty_{n+1})=p(Ty_{n+1},u)=p(u,u)=0\). This implies that \(Ty_{n+1}=u\).
For uniqueness: Let v be another fixed point of T, then
Hence \(p(u,v)=p(u,u)=p(v,v)=0\). We get \(u=v\).
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 Tstable.
Proof
Let \(\{y_{n}\} \subseteq X, \epsilon _{n}=p(y_{n+1},Ty_{n})\) and \(\epsilon _{n} \rightarrow 0\) as \(n\rightarrow \infty\). Then for any \(n \in {\mathbb {N}}\), we have
Hence, \(p(Ty_{n+1},u)=0\). But since
We have that \(p(Ty_{n+1},Ty_{n+1})=p(Ty_{n+1},u)=p(u,u)=0\). This implies that \(Ty_{n+1}=u\).
For uniqueness: Let v be another fixed point of T, then
Hence \(p(u,v)=p(u,u)=p(v,v)=0\).
We get \(u=v\). Thus the fixed point of T is unique. \(\square\)
Example 15
Let \(X = [0,\infty )\) and let p be the partial cone metric on X defined by \(p(x,y)=\,\mid xy\mid\). Let \(T:X\rightarrow X\) such that
Then \(p(Tx,1)\le p(x, Tx)\) for each \(x \in [0,\infty )\). If \(Tx=1\), then the inequality of the Theorem 12 is true. If \(x \in [\frac{1}{2n},\frac{1}{2n1}),n\ge 1\), then \(Tx=n\) and
If \(x \in (n1,n]), n\ge 2\), then \(Tx=\frac{1}{n}\) and
for each \(x \in X\), where \(q=1 \in F(T)\) and It is easy to see that Picard iteration \(x_{n+1} = Tx_{n}\) converges to 1 for every \(x_{0} \in X\). Let \(y_{2n}=\frac{1}{2n}, y_{2n+1}=\frac{1}{4n+4}, n\ge 1\). Then
and
so \(p(y_{n+1}, Ty_{n}) \rightarrow 0\).
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 \(p(f,g)=(\parallel f g\parallel _{\infty },\alpha \parallel fg\parallel _{\infty })\) where \(\alpha \ge 0\) . Then Picard’s iteration is Tstable 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 selfmap 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 Tstable.
Conclusion
We extend and prove the Tstability 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.
References
Asadi M, Soleimani H, Vaezpour SM, Rhoades BE (2009) On Tstability of Picard’s iteration in cone metric space. Fixed Point Theory Appl. doi:10.1155/2009/751090
Chatterjea SK (1972) Fixed point theorems. CR Acad Bulg Sci 10:727–730
Haddadia MR (2014) Best proximity point iteration for nonexpensive mapping in Banach spaces. J Nonlinear Sci Appl 7:126–130
Huang LG, Zhang X (2007) Cone metric spaces and fixed point theorems of contractive mapping. J Math Anal Appl 332:1468–1476
Kang SM, Alsulami HH, Rafiq A, Shahid AA (2015) Siteration scheme and polynomiography. J Nonlinear Sci Appl 8:617–627
Kannan R (1968) Some results on fixed points. Bull Calcutta Math Soc 10:71–76
Liu L (1995) Fixed points of local strickly pseudo contarctive mappings using Mann and Ishikawa iteration with errors. Indian J Pure Appl Math 26:649–659
Okeke GA, Olaleru JO (2014) Modified Noor iterations with errors for nonlinear equations in Banach spaces. J Nonlinear Sci Appl 7:180–187
Olatinwo MO (2008) Some stability results for two hybrid fixed point iterative algorithm of Kirk–Ishikawa and Kirk–Mann type. J Adv Math Stud 1:87–96
Qing Y, Rhoades BE (2008) Tstability of Picard iteration in metric space. Fixed Point Theory Appl. doi:10.1155/2008/418971
Rhoades BE (1974) Fixed point iteration using infinite matices. Trans Am Math Soc 196:161–176
Rhoades BE (1990) Fixed point theorems and stability results for fixed point iteration procedures. Indian J Pure Appl Math 21:1–9
Rhoades BE (1991) Research expository and survey article; some fixed point iteration procedures. Int J Math Math Sci 40:1–16
Rhoades BE, Soltuz SM (2006) The equivalence between the Tstablilities of Mann and Ishikawa iteration. J Math Anal Appl 318:472–475
Saadati R, Vaezpour SM, Rhoades BE (2009) Tstability approach to variational iteration method for solving integral equations. Fixed Point Theory Appl. doi:10.1155/2009/393245
Saipriya P, Chaipunya P, Cho YJ, Kumam P (2015) On strong and \(\Delta\)convergence of modified Siteration for uniformly continuous total asymptotically nonexpansive mappings in CAT κ spaces. J Nonlinear Sci Appl 8:965–975
Singh TC, Singh YR (2014) Some remarks on Dquasi contraction on cone symmetric space. Int J Math Arch 5:75–78
Singh TC, Singh YR (2014) Triple fixed points theorems on cone Banach space. J Glob Res Math Arch 2:43–49
Singh YR, Singh TC (2014) Semicompatible mappings and fixed point theorems in cone metric space. Wulfenia 21:216–224
Singh TC, Singh YR (2015) A comparative study of relationship among various types of spaces. Int J Appl Math 25:29–36
Sonmez A (2011) Fixed point theorems in partial cone metric spaces. Xiv:1101.2741v1 [math.GN]
Yao Z, Zhu LJ, Liou YC (2015) Strong covergence of Halperntype iteration algorithm for fixed point problems in Banach spaces. J Nonlinear Sci Appl 8:489–495
Yousefi B (2012) Stability of an iteration in cone metric space. Int J Pure Appl Math 76:9–13
Zamfirescu T (1972) Fixed point theorem in metric spaces. Arch Math 23:292–298
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.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
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.
About this article
Cite this article
Chhatrajit, T., Rohen, Y. Some results on Tstability of Picard’s iteration. SpringerPlus 5, 284 (2016). https://doi.org/10.1186/s4006401619395
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s4006401619395