# Semistability of iterations in cone spaces

- A Yadegarnegad
^{1}, - S Jahedi
^{2}Email author, - B Yousefi
^{1}and - SM Vaezpour
^{3}

**2011**:70

https://doi.org/10.1186/1687-1812-2011-70

© Yadegarnegad et al; licensee Springer. 2011

**Received: **8 April 2011

**Accepted: **28 October 2011

**Published: **28 October 2011

## Abstract

The aim of this work is to prove some iteration procedures in cone metric spaces. This extends some recent results of T-stability.

### Mathematics Subject Classification

47J25; 26A18.

## Keywords

## 1. Introduction

*E*be a real Banach space. A subset

*P*⊂

*E*is called a cone in

*E*if it satisfies in the following conditions:

- (i)
*P*is closed, nonempty and*P*≠ {0}. - (ii)
*a*,*b*∈*R*,*a*,*b*≥ 0 and*x*,*y*∈*P*imply that*ax*+*by*∈*P*. - (iii)
*x*∈*P*and -*x*∈*P*imply that*x*= 0.

The space *E* can be partially ordered by the cone *P* ⊂ *E*, by defining; *x* ≤ *y* if and only if *y* - *x* ∈ *P*, Also, we write *x* ≪ *y* if *y* - *x* ∈ int *P*, where int *P* denotes the interior of *P*. A cone *P* is called normal if there exists a constant *k* > 1 such that 0 ≤ *x* ≤ *y* implies ||*x*|| ≤ *k*||*y*||.

In the following we suppose that *E* is a real Banach space, *P* is a cone in *E* and ≤ is a partial ordering with respect to *P*.

**Definition 1.1**. ([1]) Let

*X*be a nonempty set. Assume that the mapping

*d*:

*X*×

*X*→

*E*satisfies in the following conditions:

- (i)
0 ≤

*d*(*x*,*y*) for all*x*,*y*∈*X*and*d*(*x*,*y*) = 0 if and only if*x*=*y*, - (ii)
*d*(*x*,*y*) =*d*(*y*,*x*) for all*x*,*y*∈*X*. - (iii)
*d*(*x*,*y*) ≤*d*(*x*,*z*) +*d*(*z*,*y*) for all*x*,*y*,*z*∈*X*.

Then *d* is called a cone metric on *X* and (*X*, *d*) is called a cone metric space.

If *T* is a self-map of *X*, then by *F*(*T*) we mean the set of fixed points of *T*. Also, **N**_{
0
} denotes the set of nonnegative integers, i.e., **N**_{
0
} = **N** ∪ {**0**}.

**Definition 1.2**. ([2]) If 0 < *α* < 1, 0 < *β*, $\gamma <\frac{1}{2}$ we say that a map *T*: *X* → *X* is Zamfirescu with respect to (*α*, *β*, *γ*), if for each pair *x*, *y* ∈ *X*, *T* satisfies at least one of the following conditions:

*Z*(1). *d*(*Tx, Ty*) ≤ *αd*(*x*, *y*),

*Z*(2). *d*(*Tx*, *Ty*) ≤ *β*(*d*(*x*, *Tx*) + *d*(*y*, *Ty*)),

*Z*(3). *d*(*Tx*, *Ty*) ≤ *γ* (*d*(*x, Ty*) + *d*(*y*, *Tx*)).

Usually for simplicity, *T* is called a Zamfirescu operator if *T* is Zamfirescu with respect to some (*α*, *β*, *γ*), for some scalars *α*, *β*, *γ* with above restrictions. Also, *T* is called a *f*-Zamfirescu operator if the relations *Z*(1), *Z*(2) and *Z*(3) hold for all *x* ∈ *X* and all *y* ∈ *F*(*T*).

**Definition 1.3**. ([3]) Let (*X*, *d*) be a cone metric space. A map *T: X* → *X* is called a quasi-contraction if for some constant *λ* ∈ (0, 1) and for every *x*, *y* ∈ *X*, there exists *u* ∈ *C*(*T*; *x*, *y*) ≡ {*d*(*x*, *y*), *d*(*x*, *Tx*), *d*(*y*, *Ty*), *d*(*y*, *Tx*), *d*(*x*, *Ty*)} such that *d*(*Tx*, *Ty*) ≤ *λu*. If this inequality holds for all *x* ∈ *X* and *y* ∈ *F*(*T*), we say that *T* is a *f*-quasi-contraction.

**Lemma 1.4**. ([4]) If *T* is a quasi-contraction with $0<\lambda <\frac{1}{2}$, then *T* is a Zamfirescu operator.

**Lemma 1.5**. ([4]) Let *P* be a normal cone, and let {*a*_{
n
} } and {*b*_{
n
} } be sequences in *E* satisfying the inequality *a*_{
n
}_{+1} ≤ *ha*_{
n
} + *b*_{
n
} , where *h* ∈ (0, 1) and *b*_{
n
} → 0 as *n* → ∞. Then lim _{
n
} *a*_{
n
} = 0.

**Definition 1.6**. A self-map *T* of a metric space (*X*, *d*) is called nonexpansive if *d*(*Tx*, *Ty*) ≤ *d*(*x*, *y*) for all *x*, *y* ∈ *X*.

**Definition 1.7**. A self-map *T* of (*X*, *d*) is called affine if *T*(*αx* + (1- *α*)*y*) = *αTx* + (1 -*α*)*Ty* for all *x*, *y* ∈ *X*, and *α* ∈ [0, 1].

**Definition 1.8**. A self-map *T* of (*X*, *d*) is called semi-compact if the convergence ||*x*_{
n
} - *Tx*_{
n
} ||→0 implies that there exist a subsequence $\left\{{x}_{{n}_{k}}\right\}$ of {*x*_{
n
} } and *x** ∈ *X* such that ${x}_{{n}_{k}}\to {x}^{*}$.

## 2. Main results

In this section we want to prove some iteration procedures in cone spaces. This extends some recent results of *T*-stability ([4]). Khamsi [5] has shown that any normal cone metric space can have a metric type defined on it. Consequently, our results are consistent for any metric spaces. Let (*X*, *d*) be a cone metric space and {*T*_{
n
} } _{
n
} be a sequence of self-maps of *x* with ∩ _{
n
}*F*(*T*_{
n
} ) ≠ ∅. Let *x*_{0} be a point of *X*, and assume that *x*_{
n
}_{+1} = *f*(*T*_{
n
} , *x*_{
n
} ) is an iteration procedure involving {*T*_{
n
} }_{
n
}, which yields a sequence {*x*_{
n
}} of points from *X*.

**Definition 2.1**. The iteration *x*_{
n
}_{+1} = *f*(*T*_{
n
} , *x*_{
n
} ) is said {*T*_{
n
} }-semistable (or semistable with respect to {*T*_{
n
} }) if {*x*_{
n
} } converges to a fixed point *q* in ∩ _{
n
}*F*(*T*_{
n
} ), and whenever {*y*_{
n
} } is a sequence in *X* with lim _{
n
} *d*(*y*_{
n
} , *f* (*T*_{
n
} , *y*_{
n
} )) = 0, and *d*(*y*_{
n
} , *f* (*T*_{
n
} , *y*_{
n
} )) = *o*(*t*_{
n
} ) for some sequence {*t*_{
n
} } ⊂ **R**^{+}, then we have *y*_{
n
} → *q*.

In practice, such a sequence {*y*_{
n
} } could arise in the following way. Let *x*_{0} be a point in *X*. Set *x*_{
n
}_{+1} = *f*(*T*_{
n
} , *x*_{
n
} ). Let *y*_{0} = *x*_{0}. Now *x*_{1} = *f*(*T*_{0}, *x*_{0}). Because of rounding or discretization in the function *T*_{0}, a new value *y*_{1} approximately equal to *x*_{1} might be obtained instead of the true value of *f*(*T*_{0}, *x*_{0}). Then to approximate *y*_{2}, the value *f*(*T*_{1}, *y*_{1}) is computed to yield *y*_{2}, approximation of *f*(*T*_{1}, *y*_{1}). This computation is continued to obtain {*y*_{
n
} } as an approximate sequence of {*x*_{
n
} }.

In the following we extend the definition of stability from a single self-map (see [6]) to a sequence of single-maps.

**Definition 2.2**. The iteration *x*_{
n
}_{+1} = *f*(*T*_{
n
} , *x*_{
n
} ) is said {*T*_{
n
} }-stable (or stable with respect to ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$) if {*x*_{
n
} } converges to a fixed point *q* in ∩ _{
n
}*F*(*T*_{
n
} ), and whenever {*y*_{
n
} } is a sequence in *X* with lim _{
n
} *d*(*y*_{
n
}_{+1} , *f*(*T*_{
n
} , *y*_{
n
} )) = 0, we have *y*_{
n
} → *q*.

Note that if *T*_{
n
} = *T* for all *n*, then Definition 2.2. gives the definition of T-stability ([6]).

**Definition 2.3**. For a sequence of self-maps ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$, the iteration *x*_{
n
}_{+1} = *T*_{
n
}*x*_{
n
} is called the Picard's S-iteration.

The stability of some iterations have been studied in metric spaces in [7, 8]. Here we want to investigate the semistability and stability of Picard's S-iteration.

**Theorem 2.4**. Let (

*X*,

*d*) be a cone metric space,

*P*a normal cone and {

*T*

_{ n }}

_{ n }=

**N**

_{ 0 }be a sequence of self-maps of

*X*with ∩

_{ n }

*F*(

*T*

_{ n }) ≠ ∅. Suppose that there exist nonnegative bounded sequences {

*a*

_{ n }}, {

*b*

_{ n }} with sup

_{ n }

*b*

_{ n }< 1, such that

for each *n* ∈ **N**_{
0
}, **x** ∈ **X** and *q* ∈ ∩ _{
n
}*F*(*T*_{
n
} ). Then the Picard's S-iteration is semistable with respect to {*T*_{
n
} }_{
n
}.

*Proof*. First we note that relation (*) implies that ∩

_{ n }

*F*(

*T*

_{ n }) is a singleton. Indeed, if

*p*and

*q*belong to ∩

_{ n }

*F*(

*T*

_{ n }), then by (*) we get

_{n}b

_{n}. This implies that

*p*=

*q*. So let ∩

_{ n }

*F*(

*T*

_{ n }) = {

*q*

_{0}} and {

*y*

_{ n }} ⊂

*X*be such that lim

_{ n }

*d*(

*y*

_{ n }

_{+1},

*T*

_{ n }

*y*

_{ n }) = lim

_{ n }

*d*(

*T*

_{ n }

*y*

_{ n },

*y*

_{ n }) = 0. Now we show that

*y*

_{ n }→

*q*

_{0}. For this by using the relation (*) we have:

where *c*_{
n
} = *d*(*y*_{
n
}_{+1} , *T*_{
n
}*y*_{
n
} ) + *a*_{
n
} *d*(*T*_{
n
}*y*_{
n
} , *y*_{
n
} ) tends to 0 as *n* → ∞, and 0 ≤ *α* < 1. Now by Lemma 1.5, *y*_{
n
} → *q*_{0} and so the Picard's S-iteration is {*T*_{
n
} }_{
n
}-semistable. This completes the proof.□

**Corollary 2.5**. Let (*X*, *d*) be a cone metric space, *P* a normal cone and ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$ be a sequence of self-maps of *X* with ∩ _{
n
}*F*(*T*_{
n
} ) ≠ ∅. If there exists a nonnegative sequence {*λ*_{
n
} } with sup _{
n
}*λ*_{
n
} < 1 such that *d*(*T*_{
n
}*x*, *T*_{
n
}*y*) ≤ *λ*_{
n
} *d*(*x*, *y*) for each *x*, *y* ∈ *X* and *n* ∈ **N**_{
0
}, then the Picard's S-iteration is semistable with respect to {*T*_{
n
} }_{
n
}.

**Corollary 2.6**. Let (*X*, *d*) be a cone metric space, *P* a normal cone and ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$ be a sequence of self-maps of *X* with ∩ _{
n
}*F*(*T*_{
n
} ) ≠ ∅. If for all *n* ∈ **N**_{
0
}, *T*_{
n
} is a *f*-Zamfirescu operator with respect to (*α*_{
n
} , *β*_{
n
} , *γ*_{
n
} ) with sup _{
n
}*γ*_{
n
} < 1/2, then the Picard's S-iteration is semistable with respect to {*T*_{
n
} }_{
n
}.

*Proof*. It is sufficient to show that condition (*) in Theorem 2.4 is consistent. Clearly the conditions

*Z*(1) and

*Z*(2) imply that (*) holds. Also, note that by using condition

*Z*(3) for

*T*

_{ n }we have:

*q*∈ ∩

_{ n }

*F*(

*T*

_{ n }). Thus we get

Since sup _{
n
}*γ*_{
n
} < 1/2, so clearly (*) holds.□

**Corollary 2.7**. Under the conditions of Corollary 2.6 if *T*_{
n
} is a Zamfirescu operator for all *n*, then the Picard's S-iteration is semistable with respect to {*T*_{
n
} }_{
n
}.

**Corollary 2.8**. Let (*X*, *d*) be a cone metric space, *P* a normal cone and ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$ be a sequence of self-maps of *X* with ∩ _{
n
}*F*(*T*_{
n
} ) ≠ ∅. If for all *n* ∈ **N**_{
0
}, *T*_{
n
} is a *f*-quasi-contraction with *λ*_{
n
} such that sup _{
n
}*λ*_{
n
} < 1, then the Picard's S-iteration is semistable with respect to {*T*_{
n
} }_{
n
}.

*Proof*. It is sufficient to show that condition (*) holds. For every

*x*∈

*X*and

*q*∈ ∩

_{ n }

*F*(

*T*

_{ n }) we have

*d*(

*T*

_{ n }

*x*,

*q*) ≤

*γ*

_{ n }

*u*

_{ n }for some

*u*

_{ n }∈

*C*(

*T*

_{ n };

*x*,

*q*). Hence

where *s*_{
n
} , *t*_{
n
} ∈ {0, *λ*_{
n
} }. This completes the proof. □

**Theorem 2.9**. Under the conditions of Theorem 2.4, suppose that there exists a sequence of nonnegative scalars ${\left\{{\lambda}_{n}\right\}}_{n\in {\mathbf{N}}_{\mathbf{0}}}$ with sup _{
n
}*λ*_{
n
} < 1/2, such that for all *x*, *y* ∈ *X*, *n* ≥ 1 we have *d*(*T*_{
n
}*x*, *T*_{
n
}_{-1}*y*) ≤ *λ*_{
n
}*u*_{
n
} where *u*_{
n
} = *d*(*T*_{
n
}*x*, *y*) or *u*_{
n
} = *d*(*T*_{
n
}_{-1}*y*, *y*). Then the Picard's S-iteration is semistable with respect to {*T*_{
n
} }_{
n
}.

*Proof*. It is sufficient to show that

*d*(

*y*

_{ n },

*T*

_{ n }

*y*

_{ n }) → 0 whenever

*d*(

*y*

_{ n }

_{+1},

*T*

_{ n }

*y*

_{ n }) → 0. Put

*b*

_{ n }=

*d*(

*y*

_{ n },

*T*

_{ n }

*y*

_{ n }) and

*c*

_{ n }=

*d*(

*y*

_{ n },

*T*

_{ n }

_{-1}

*y*

_{n-1}). We have

where *s*_{
n
} = *λ*_{
n
} or ${s}_{n}=\frac{{\lambda}_{n}}{1-{\lambda}_{n}}$. Hence by Lemma 1.5, *b*_{
n
} → 0, and so by the proof of Theorem 2.4, the proof is complete.□

Now we want to investigate the semistability in the cone normed spaces.

**Definition 2.10**. Let

*X*be a vector space over the field

*F*. Assume that the function

*p*:

*X*→

*E*having the properties:

- (a)
*p*(*x*) ≥ 0 for all*x*in*X*. - (b)
*p*(*x*+*y*) ≤*p*(*x*) +*p*(*y*) for all*x*,*y*in*X*. - (c)
*p*(*αx*) = |*α*|*p*(*x*) for all*α*∈*F*and*x*∈*X*.

*p*is called a cone seminorm on

*X*. A cone norm is a cone seminorm

*p*such that

- (d)
*x*= 0 if*p*(*x*) = 0.

We will denote a cone norm by ||·||_{
c
}and (*X*, ||·||_{
c
}) is called a cone normed space. Also, *d*_{
c
} (*x*, *y*) = ||*x* - *y*|| _{
c
} defines a cone metric on *X*.

**Lemma 2.11**. Let *P* be a normal cone, and the sequences {*t*_{
n
} } and (*s*_{
n
} } be such that 0 ≤ *t*_{
n
}_{+1} ≤ *t*_{
n
} + *s*_{
n
} for all *n* ≥ 1. If ∑_{
n
}_{∈}_{
N
} *s*_{
n
} converges, then lim _{
n
} ||*t*_{
n
} || exists.

*Proof*. Let *t*_{1} = 0 and *P* be normal with constant *k*. Since *t*_{
n
}_{+1} - *t*_{
n
} ≤ *s*_{
n
} , thus ∑ _{
n
} (*t*_{
n
}_{+1} - *t*_{
n
} ) ≤ ∑_{n}s_{n}. Hence ||∑ _{
n
} (*t*_{
n
}_{+1} - *t*_{
n
} )|| ≤ *k* ||∑_{n}s_{n}|| < ∞. So ${lim}_{k}\parallel {\sum}_{n=1}^{k}\left({t}_{n+1}-{t}_{n}\right)\parallel $ exists. But ${\sum}_{n=1}^{k}\left({t}_{n+1}-{t}_{n}\right)={t}_{k+1}-{t}_{1}$. Thus indeed lim _{
n
} ||*t*_{
n
} || exists.□

**Theorem 2.12**. Let (*X*, ||·||_{
c
}) be a cone normed space with respect to a normal cone *P* in the real Banach space *E*, and ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$ be a sequence of self-maps of *X* with ∩ _{
n
}*F*(*T*_{
n
} ) ≠ ∅, *T*_{0} = *I* and *d*_{
c
} (*T*_{
n
}*x*, *q*) ≤ (1 + *α*_{
n
} ) *d*_{
c
} (*x*, *q*) for all *n* ∈ **N**_{
0
}, *x* ∈ *X* and *q* ∈ ∩ _{
n
}*F*(*T*_{
n
} ) where ${\sum}_{n\in {\mathbf{N}}_{0}}{\alpha}_{n}<\infty $. Suppose that there exists a sequence {*β*_{
n
} } ⊂ (0, 1] such that ${\sum}_{n}\frac{1-{\beta}_{n}}{n}<\infty $ and the sequence {*x*_{
n
} } _{
n
} obtained by the iteration procedure *x*_{
n
}_{+1} = *β*_{
n
}*x*_{
n
} + (1 - *β*_{
n
} )*S*_{
n
}*x*_{
n
} be bounded where ${S}_{n}=\frac{1}{n}\left({T}_{0}+{T}_{1}+\cdots +{T}_{n-1}\right)$. Then lim *d*_{
c
} (*x*_{
n
} , *q*) exists for all *q* ∈ ∩ _{
n
}*F*(*T*_{
n
} ). Moreover, if for all *m*, *T*_{
m
} is a continuous semi-compact mapping and *d*_{
c
} (*T*_{
m
}*x*_{
n
} , *x*_{
n
} ) → 0 as *n* → ∞, then {*x*_{
n
} } converges to a point in ∩ _{
n
}*F*(*T*_{
n
} ).

*Proof*. Let

*q*∈ ∩

_{ n }

*F*(

*T*

_{ n }) and put

*α*= ∑

_{n}α

_{n},

*γ*

_{0}= sup

*d*

_{ c }(

*x*

_{ n },

*q*) and

*b*

_{ n }=

*d*

_{ c }(

*x*

_{ n },

*q*) for each

*n*. By taking

*α*

_{0}= 0, we get

But ${\sum}_{n}\frac{1-{\beta}_{n}}{n}<\infty $, so by lemma 2.11 we conclude that lim _{
n
} *b*_{
n
} exits and so the proof of the first part is complete. Now let *T*_{
m
} 's be continuous semi-compact and for all *m*, *d*_{
c
} (*T*_{
m
}*x*_{
n
} , *x*_{
n
} ) → 0 as *n* → ∞. Since *T*_{
m
} is semi-compact, there exists a subsequence $\left\{{x}_{{n}_{k}}\right\}$ of {*x*_{
n
} } and *q* ∈ *X* such that ${d}_{c}\left({x}_{{n}_{k}},q\right)\to 0$. But *T*_{
m
} is continuous, thus for all $m,{d}_{c}\left({T}_{m}{x}_{{n}_{k}},{T}_{m}q\right)\to 0$ as *k* → ∞.

*m*we have

which tends to 0 as *k* → ∞. Hence *T*_{
m
}*q* = *q* for all *m*. So *q* ∈ ∩ _{
m
}*F*(*T*_{
m
} ) and ${d}_{c}\left({x}_{{n}_{k}},q\right)\to 0$. Also, we saw by the first part of the proof, ${lim}_{n}{d}_{c}\left({x}_{{n}_{k}},q\right)$exists. This implies that ${d}_{c}\left({x}_{{n}_{k}},q\right)\to 0$ and so the proof is complete.□

**Theorem 2.13**. Let (

*X*, ||·||

_{ c }) be a cone normed space with respect to a normal cone

*P*in the real Banach space

*E*, and ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$ be a sequence of self-maps of

*X*with

*T*

_{0}=

*I*, ∩

_{ n }

*F*(

*T*

_{ n }) ≠ ∅, and ||

*T*

_{ m }

*x*-

*T*

_{ m }

_{-1}

*x*|| ≤ ||

*T*

_{ m }

_{-1}

*x*-

*T*

_{ m }

_{-2}

*x*|| for all

*x*∈

*X*,

*m*≥ 2. Consider the iteration procedure

*x*

_{ n }

_{+1}=

*f*(

*T*

_{ n },

*x*

_{ n }) =

*α*

_{ n }

*x*

_{ n }+ (1 -

*α*

_{ n })

*S*

_{ n }

*x*

_{ n }where ${S}_{n}=\frac{1}{n}\left({T}_{0}+{T}_{1}+\cdots +{T}_{n-1}\right)$ and

*α*

_{ n }∈ [0, 1). If there exist

*a*≥ 0 and

*b*∈ (0, 1) such that

for all sequences {*y*_{
n
} } with ${d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)=o\left(\frac{1}{\left(1-{\alpha}_{n}\right)\left(n-1\right)}\right)$, and all *q* ∈ ∩ _{
n
}*F*(*T*_{
n
} ), then the given iteration is {*T*_{
n
} }-semistable.

*Proof*. First note that the relation (*) implies that ∩

_{ n }

*F*(

*T*

_{ n }) is a singleton. Indeed, if

*p*and

*q*belong to

*F*(

*T*), then by setting

*y*

_{ n }=

*p*in (*) for all

*n*, we get

*d*

_{ c }(

*p*,

*q*) ≤

*bd*

_{ c }(

*p*,

*q*). This implies that

*p*=

*q*. Now let

*F*(

*T*) = {

*q*

_{0}} and {

*y*

_{ n }} ⊂

*X*be such that lim

_{ n }d

_{c}(

*y*

_{ n }

_{+1},

*f*(

*T*

_{ n },

*y*

_{ n }) = lim

_{ n }((1-

*α*

_{ n })(

*n*- 1))

*d*

_{ c }(

*T*

_{1}

*y*

_{ n },

*y*

_{ n }) = 0. Now we show that

*y*

_{ n }→

*q*

_{0}. To see this note that by using the relation (*) we have:

*c*

_{ n }=

*d*

_{ c }(

*y*

_{ n }

_{+1},

*f*(

*T*

_{ n },

*y*

_{ n })) +

*a d*

_{ c }(

*f*(

*T*

_{ n },

*y*

_{ n }),

*y*

_{ n }). By Lemma 1.5, it suffices to show that

*c*

_{ n }→ 0. For this we show that

*d*

_{ c }(

*f*(

*T*

_{ n },

*y*

_{ n }),

*y*

_{ n }) → 0 as

*n*→ ∞. We have

*i*≥ 1, we have

which tends to 0 since ${d}_{c}\left({T}_{1}{y}_{n}{y}_{n}\right)=o\left(\frac{1}{\left(1-{\alpha}_{n}\right)\left(n-1\right)}\right)$. Thus *y*_{
n
} → *q*_{0} and so the iteration *x*_{
n
}_{+1} = *f*(*T*_{
n
} , *x*_{
n
} ) is {*T*_{
n
} }-semistable. This completes the proof.□

**Corollary 2.14**. Let (

*X*, ||·||

_{c}) be a cone normed space with respect to a normal cone

*P*in the real Banach space

*E*, and ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$ be a sequence of self-maps of

*X*with

*T*

_{0}=

*I*, ∩

_{ n }

*F*(

*T*

_{ n }) ≠ ∅, and ||

*T*

_{ m }

*x*-

*T*

_{ m }

_{-1}

*x*|| ≤ ||

*T*

_{ m }

_{-1}

*x*-

*T*

_{ m }

_{-2}

*x*|| for all

*x*∈

*X*,

*m*≥ 2. Consider the iteration procedure

*x*

_{ n }

_{+1}=

*S*

_{ n }

*x*

_{ n }where ${S}_{n}=\frac{1}{n}\left({T}_{0}+{T}_{1}+\cdots +{T}_{n-1}\right)$. If there exist nonnegative bounded sequences {

*a*

_{ n }} and {

*b*

_{ n }} with sup

_{ n }

*b*

_{ n }< 1, such that

for all sequences {*y*_{
n
} } with ${d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)=o\left(\frac{1}{n-1}\right)$, and for all *q* ∈ ∩ _{
n
}*F*(*T*_{
n
} ), then the given iteration is {*T*_{
n
} }-semistable.

**Corollary 2.15**. Let (

*X*, ||·||

_{ c }) be a cone normed space with respect to a normal cone

*P*in the real Banach space

*E*, and ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$ be a sequence of self-maps of

*X*with

*T*

_{0}=

*I*, ∩

_{ n }

*F*(

*T*

_{ n }) ≠ ∅, and ||

*T*

_{ m }

*x*-

*T*

_{ m }

_{-1}

*x*|| ≤ ||

*T*

_{ m }

_{-1}

*x*-

*T*

_{ m }

_{-2}

*x*|| for all

*x*∈

*X*,

*m*≥ 2. Consider the iteration procedure

*x*

_{ n }

_{+1}=

*S*

_{ n }

*x*

_{ n }where ${S}_{n}=\frac{1}{n}\left({T}_{0}+{T}_{1}+\cdots +{T}_{n-1}\right)$. If there exist

*a*≥ 0 and

*b*∈ (0, 1) such that

for all sequences {*y*_{
n
} } with ${d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)=o\left(\frac{1}{n-1}\right)$, and for all *q* ∈∩ _{
n
}*F*(*T*_{
n
} ), then the given iteration is {*T*_{
n
} }-semistable.

**Theorem 2.16**. Let (

*X*, ||·||

_{ c }) be a cone normed space with respect to a normal cone

*P*in the real Banach space

*E*, and ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$ be a sequence of affine self-maps of

*X*with

*T*

_{0}=

*I*, ∩

_{ n }

*F*(

*T*

_{ n }) ≠ ∅, and

*d*

_{ c }(

*T*

_{ m }

*x*-

*T*

_{ m }

_{-1}

*y*) ≤

*d*

_{ c }(

*T*

_{ m }

_{-1}

*x*-

*T*

_{ m }

_{-2}

*y*) for all

*x*∈

*X*,

*m*≥ 2. Consider the iteration procedure

*x*

_{ n }

_{+1}=

*f*(

*T*

_{ n },

*x*

_{ n }) = (1 -

*α*

_{ n })

*x*

_{ n }+

*α*

_{ n }

*T*

_{ n }

*z*

_{ n }where

*z*

_{ n }= (1 -

*β*

_{ n })

*x*

_{ n }+

*β*

_{ n }

*T*

_{ n }

*x*

_{ n }and

*α*

_{ n },

*β*

_{ n }∈ [0, 1]. Suppose that there exist

*a*≥ 0 and

*b*∈ (0, 1) such that

for all sequences {*y*_{
n
} } with ${d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)=o\left(\frac{1}{n{\alpha}_{n}}\right),$ and all *q* ∈ ∩ _{
n
}*F*(*T*_{
n
} ). Then the given iteration is {*T*_{
n
} }-semistable.

*Proof*. If

*p*and

*q*belong to ∩

_{ n }

*F*(

*T*

_{ n }), then by setting

*y*

_{ n }=

*p*in (*) for all

*n*, we get

*d*

_{ c }(

*p*,

*q*) ≤

*bd*

_{ c }(

*p*,

*q*). This implies that

*p*=

*q*. Now let ∩

_{ n }

*F*(

*T*

_{ n }) = {

*q*

_{0}} and {

*y*

_{ n }} ⊆

*X*be such that

*y*

_{ n }→

*q*

_{0}. To see this note that by using the notation (*) we have:

*c*

_{ n }=

*d*

_{ c }(

*y*

_{ n }+

_{1},

*f*(

*T*

_{ n },

*y*

_{ n })) +

*a d*

_{ c }(

*f*(

*T*

_{ n },

*y*

_{ n }),

*y*

_{ n }). By Lemma 1.5, it is sufficient to show that

*c*

_{ n }→ 0. For this we show that

*d*

_{ c }(

*f*(

*T*

_{ n },

*y*

_{ n }),

*y*

_{ n }) → 0 as

*n*→ ∞. Note that

which tends to 0 since ${d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)=o\left(\frac{1}{n{\alpha}_{n}}\right),$. Thus *y*_{
n
} → *q*_{0} and so the iteration *x*_{
n
}_{+1} = *f*(*T*_{
n
} , *x*_{
n
} ) is {*T*_{
n
} }-semistable. This completes the proof.□

**Corollary 2.17**. Let (

*X*, ||·||

_{ c }) be a cone normed space with respect to a normal cone

*P*in the real Banach space

*E*, and ${\left\{{T}_{n}\right\}}_{n\in {N}_{0}}$ be a sequence of self-maps of

*X*with

*T*

_{0}=

*I*, ∩

_{ n }

*F*(

*T*

_{ n }) ≠ ∅, and ||

*T*

_{ m }

*x*-

*T*

_{ m }

_{-1}

*x*|| ≤ ||

*T*

_{ m }

_{-1}

*x*-

*T*

_{ m }

_{-2}

*x*|| for all ∈

*X*,

*m*≥ 2. Consider the iteration procedure

*x*

_{n+1}=

*f*(

*T*

_{ n },

*x*

_{ n }) =

*α*

_{ n }

*x*

_{ n }+ (1 -

*α*

_{ n })

*T*

_{ n }

*x*

_{ n }where ${S}_{n}=\frac{1}{n}\left({T}_{0}+{T}_{1}+\cdots +{T}_{n-1}\right)$ and

*α*

_{ n }∈ [0, 1). If there exist

*a*≥ 0 and

*b*∈ (0, 1) such that

for all sequences {*y*_{
n
} } with ${d}_{c}\left(T{y}_{n},{y}_{n}\right)=o\left(\frac{n+{n}^{2}}{1-{\alpha}_{n}}\right)$, and all *q* ∈ ∩ _{
n
}*F*(*T*_{
n
} ), then the given iteration is {*T*_{
n
} }-semistable.

## Declarations

## Authors’ Affiliations

## References

- Huang LG, Zheng X:
**Cone metric space and fixed point theorems of contractive mapping.***J Math Anal Appl*2007,**332**(2):1468–1476. 10.1016/j.jmaa.2005.03.087MathSciNetView ArticleGoogle Scholar - Zamfirescu T:
**Fixed point theorem in metric spaces.***Arch Math (Basel)*1972,**23:**91–101.MathSciNetView ArticleGoogle Scholar - Ilic D, Rakocevic V:
**Quasi-contraction on a cone metric space.***Appl Math Lett*2009,**22**(5):728–7310. 10.1016/j.aml.2008.08.011MathSciNetView ArticleGoogle Scholar - Asadi M, Soleimani H, Vaezpour SM, Rhoades BE:
**On T-stability of picard iteration in cone metric spaces.***Fixed Point Theory Appl*2009,**2009:**6. Article ID 751090MathSciNetGoogle Scholar - Khamsi MA:
**Remarks on cone metric spaces and fixed point theorems of contractive mappings.***Fixed Point Theory Appl*2010,**2010:**7. Article ID 315398Google Scholar - Harder AM, Hicks TL:
**Stability results for fixed point iteration procedures.***Math Japanica*1988,**33**(5):693–706.MathSciNetGoogle Scholar - Qing Y, Rhoades BE:
**T-stability of picard iteration in metric spaces.***Fixed Point Theory Appl*2008,**2008:**4. Article ID 418971Google Scholar - Rhoades BE, Soltus SM:
**The equivalence between the T-stabilities of Mann and Ishikawa iterations.***J Math Anal*2006,**318:**472–475. 10.1016/j.jmaa.2005.05.066View ArticleGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.