Open Access

# Semistability of iterations in cone spaces

Fixed Point Theory and Applications20112011:70

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

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.

47J25; 26A18.

## Keywords

Cone metriccontractionstabilitynonexpansiveaffinesemi-compact

## 1. Introduction

Let E be a real Banach space. A subset P E is called a cone in E if it satisfies in the following conditions:
1. (i)

P is closed, nonempty and P ≠ {0}.

2. (ii)

a, b R, a, b ≥ 0 and x, y P imply that ax + by P.

3. (iii)

x P and -x P imply that x = 0.

The space E can be partially ordered by the cone P E, by defining; xy 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 ≤ xy 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 × XE satisfies in the following conditions:
1. (i)

0 ≤ d(x, y) for all x, y X and d(x, y) = 0 if and only if x = y,

2. (ii)

d(x, y) = d(y, x) for all x, y X.

3. (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: XX 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: XX 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 +1ha 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 x0 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 x0 be a point in X. Set x n +1 = f(T n , x n ). Let y0 = x0. Now x1 = f(T0, x0). Because of rounding or discretization in the function T0, a new value y1 approximately equal to x1 might be obtained instead of the true value of f(T0, x0). Then to approximate y2, the value f(T1, y1) is computed to yield y2, approximation of f(T1, y1). 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
$d\left({T}_{n}x,q\right)\le {a}_{n}d\left(x,{T}_{n}x\right)+{b}_{n}d\left(x,q\right)\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(*\right)$

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
$d\left(p,q\right)=d\left({T}_{n}p,q\right)\le {a}_{n}d\left(p,{T}_{n}p\right)+{b}_{n}d\left(p,q\right)\le \alpha d\left(p,q\right),$
where α = supnbn. This implies that p = q. So let ∩ n F(T n ) = {q0} 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 q0. For this by using the relation (*) we have:
$\begin{array}{cc}\hfill d\left({y}_{n+1},{q}_{0}\right)& \le d\left({y}_{n+1},{T}_{n}{y}_{n}\right)+d\left({T}_{n}{y}_{n},{q}_{0}\right)\hfill \\ \le d\left({y}_{n+1},{T}_{n}{y}_{n}\right)+{a}_{n}d\left({T}_{n}{y}_{n},{y}_{n}\right)+{b}_{n}d\left({y}_{n},{q}_{0}\right)\hfill \\ ={c}_{n}+\alpha d\left({y}_{n},{q}_{0}\right),\hfill \end{array}$

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 q0 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:
$d\left({T}_{n}x,q\right)\le {\gamma }_{n}\left(d\left(q,{T}_{n}x\right)+d\left(x,q\right)\right),$
where q n F(T n ). Thus we get
$d\left({T}_{n}x,q\right)\le {\gamma }_{n}d\left(x,{T}_{n}x\right)+2{\gamma }_{n}d\left(x,q\right).$

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
$d\left({T}_{n}x,q\right)\le {t}_{n}d\left(x,{T}_{n}x\right)+{s}_{n}d\left(x,q\right),$

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 -1y) ≤ λ n u n where u n = d(T n x, y) or u n = d(T n -1y, 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 -1yn-1). We have
${b}_{n}\le d\left({y}_{n},{T}_{n-1}{y}_{n-1}\right)+d\left({T}_{n}{y}_{n},{T}_{n-1}{y}_{n-1}\right)\le {c}_{n}+{s}_{n}{b}_{n-1},$

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: XE having the properties:
1. (a)

p(x) ≥ 0 for all x in X.

2. (b)

p(x + y) ≤ p(x) + p(y) for all x, y in X.

3. (c)

p(αx) = |α|p(x) for all α F and x X.

Then p is called a cone seminorm on X. A cone norm is a cone seminorm p such that
1. (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 +1t n + s n for all n ≥ 1. If ∑ n N s n converges, then lim n ||t n || exists.

Proof. Let t1 = 0 and P be normal with constant k. Since t n +1 - t n s n , thus ∑ n (t n +1 - t n ) ≤ ∑nsn. Hence ||∑ n (t n +1 - t n )|| ≤ k ||∑nsn|| < ∞. 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 ) ≠ , T0 = 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 = ∑nαn, γ0 = sup d c (x n , q) and b n = d c (x n , q) for each n. By taking α0 = 0, we get
$\begin{array}{cc}\hfill {b}_{n+1}& ={d}_{c}\left({x}_{n+1},q\right)\hfill \\ ={d}_{c}\left({\beta }_{n}{x}_{n}+\left(1-{\beta }_{n}\right){S}_{n}{x}_{n},q\right)\hfill \\ \le {\beta }_{n}{d}_{c}\left({x}_{n},q\right)+\left(1-{\beta }_{n}\right){d}_{c}\left({S}_{n}{x}_{n},q\right)\hfill \\ ={\beta }_{n}{b}_{n}+\left(1-{\beta }_{n}\right){d}_{c}\left({S}_{n}{x}_{n},q\right).\hfill \end{array}$
But,
$\begin{array}{cc}\hfill {d}_{c}\left({S}_{n}{x}_{n},q\right)& ={d}_{c}\left(\frac{1}{n}\left({x}_{n}+{T}_{1}{x}_{n}+\cdots +{T}_{n-1}{x}_{n}\right),q\right)\hfill \\ \le \frac{1}{n}\sum _{i=0}^{n-1}{d}_{c}\left({T}_{i}{x}_{n},q\right)\hfill \\ \le \frac{1}{n}\sum _{i=0}^{n-1}\left(1+{\alpha }_{i}\right){d}_{c}\left({x}_{n},q\right)\hfill \\ =\frac{1}{n}{b}_{n}\sum _{i=0}^{n-1}\left(1+{\alpha }_{i}\right)\hfill \\ ={b}_{n}+\frac{1}{n}\sum _{i=1}^{n-1}{b}_{n}{\alpha }_{i}.\hfill \end{array}$
Hence we get
$\begin{array}{cc}\hfill {b}_{n+1}& \le {\beta }_{n}{b}_{n}+\left(1-{\beta }_{n}\right)\left({b}_{n}+\frac{1}{n}{b}_{n}\sum _{i=1}^{n-1}{\alpha }_{i}\right)\hfill \\ ={b}_{n}+\frac{1}{n}\left(1-{\beta }_{n}\right)\sum _{i=1}^{n-1}{\alpha }_{i}{b}_{n}\hfill \\ \le {b}_{n}+\frac{1}{n}\left(1-{\beta }_{n}\right)\alpha {b}_{n}\hfill \\ \le {b}_{n}+\frac{1}{n}\left(1-{\beta }_{n}\right)\alpha {\gamma }_{0}.\hfill \end{array}$

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 → ∞.

Now for all m we have
${d}_{c}\left({T}_{m}q,q\right)\le {d}_{c}\left({T}_{m}q,{T}_{m}{x}_{{n}_{k}}\right)+{d}_{c}\left({T}_{m}{x}_{{n}_{k}},q\right)+{d}_{c}\left(q,{x}_{{n}_{k}}\right)$

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 T0 = I, ∩ n F(T n ) ≠ , and ||T m x - T m -1x || ≤ ||T m -1x - T m -2x|| 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
${d}_{c}\left(f\left({T}_{n},{y}_{n}\right),q\right)\le a{d}_{c}\left(f\left({T}_{n},{x}_{n}\right),{y}_{n}\right)+b{d}_{c}\left({y}_{n},q\right)\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(*\right)$

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) = {q0} and {y n } X be such that lim n dc(y n +1, f(T n , y n ) = lim n ((1- α n )(n- 1)) d c (T1y n , y n ) = 0. Now we show that y n q0. To see this note that by using the relation (*) we have:
$\begin{array}{cc}\hfill {d}_{c}\left({y}_{n+1},{q}_{0}\right)& \le {d}_{c}\left({y}_{n+1},f\left({T}_{n},{y}_{n}\right)\right)+{d}_{c}\left(f\left({T}_{n},{y}_{n}\right),{q}_{0}\right)\hfill \\ \le {d}_{c}\left({y}_{n+1},f\left({T}_{n},{y}_{n}\right)\right)+a{d}_{c}\left(f\left({T}_{n},{y}_{n}\right),{y}_{n}\right)+b{d}_{c}\left({y}_{n},{q}_{0}\right)\hfill \\ ={c}_{n}+b{d}_{c}\left({y}_{n},{q}_{0}\right),\hfill \end{array}$
where 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
$\begin{array}{cc}\hfill {d}_{c}\left(f\left({T}_{n},{y}_{n}\right),{y}_{n}\right)& =\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\parallel f\left({T}_{n},{y}_{n}\right)-{y}_{n}{\parallel }_{c}\hfill \\ =\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}\parallel {\alpha }_{n}{y}_{n}+\left(1-{\alpha }_{n}\right){S}_{n}{y}_{n}-{y}_{n}{\parallel }_{c}\hfill \\ =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left(1-{\alpha }_{n}\right)\parallel {y}_{n}-{S}_{n}{y}_{n}{\parallel }_{c}\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\frac{1-{\alpha }_{n}}{n}\sum _{i=1}^{n-1}\parallel \left({T}_{i}{y}_{n}-{y}_{n}\right){\parallel }_{c}.\hfill \end{array}$
But for i ≥ 1, we have
$\begin{array}{l}||{T}_{i}{y}_{n}-{y}_{n}|{|}_{c}\phantom{\rule{0.5em}{0ex}}\le {d}_{c}\left({T}_{i}{y}_{n}-{T}_{i-1}{y}_{n}\right)+\cdots +{d}_{c}\left({T}_{1}{y}_{n}-{y}_{n}\right)\\ \phantom{\rule{1em}{0ex}}\le i{d}_{c}{\left(}_{1}T{y}_{n},{y}_{n}\right).\end{array}$
Therefore,
${d}_{c}\left(f\left({T}_{n},{y}_{n}\right),{y}_{n}\right)\le \frac{1-{\alpha }_{n}}{n}\sum _{i=1}^{n-1}i{d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)=\frac{\left(1-{\alpha }_{n}\right)\left(n-1\right)}{2}{d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right),$

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 q0 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 T0 = I, ∩ n F(T n ) ≠ , and ||T m x - T m -1x|| ≤ ||T m -1x- T m -2x|| 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
${d}_{c}\left({S}_{n}{y}_{n},q\right)\le {a}_{n}{d}_{c}\left({S}_{n}{y}_{n},{y}_{n}\right)+{b}_{n}{d}_{c}\left({y}_{n},q\right)$

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 T0 = I, ∩ n F(T n ) ≠ , and ||T m x - T m -1x|| ≤ ||T m -1x- T m -2x|| 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
${d}_{c}\left({S}_{n}{y}_{n},q\right)\le a{d}_{c}\left({S}_{n}{y}_{n},{y}_{n}\right)+b{d}_{c}\left({y}_{n},q\right)$

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 T0 = I, ∩ n F(T n ) ≠ , and d c (T m x - T m -1y) ≤ d c (T m -1x- T m -2y) 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
${d}_{c}\left(f\left({T}_{n},{y}_{n}\right),q\right)\le a{d}_{c}\left(f\left({T}_{n},{y}_{n}\right),{y}_{n}\right)+b{d}_{c}\left({y}_{n},q\right)\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\left(*\right)$

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 ) = {q0} and {y n } X be such that
$\underset{n}{lim}\phantom{\rule{0.3em}{0ex}}{d}_{c}\left({y}_{n+1},f\left({T}_{n},{y}_{n}\right)\right)=\underset{n}{lim}\phantom{\rule{0.3em}{0ex}}n{\alpha }_{n}{d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)=0.$
Now we show that y n q0. To see this note that by using the notation (*) we have:
$\begin{array}{cc}\hfill {d}_{c}\left({y}_{n+1},{q}_{0}\right)& \le \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{d}_{c}\left({y}_{n+1},f\left({T}_{n},{y}_{n}\right)\right)+{d}_{c}\left(f\left({T}_{n},{y}_{n}\right),{q}_{0}\right)\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{d}_{c}\left({y}_{n+1},f\left({T}_{n},{y}_{n}\right)\right)+a{d}_{c}\left(f\left({T}_{n},{y}_{n}\right),{y}_{n}\right)+b{d}_{c}\left({y}_{n},{q}_{0}\right)\hfill \\ =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{c}_{n}+b{d}_{c}\left({y}_{n},{q}_{0}\right),\hfill \end{array}$
where 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
$\begin{array}{cc}\hfill {d}_{c}\left(f\left({T}_{n},{y}_{n}\right),{y}_{n}\right)& =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\parallel f\left({T}_{n},{y}_{n}\right)-{y}_{n}{\parallel }_{c}\hfill \\ =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\parallel \left(1-{\alpha }_{n}\right){y}_{n}+{\alpha }_{n}{T}_{n}\left({z}_{n}\right)-{y}_{n}{\parallel }_{c}\hfill \\ =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{n}\parallel {T}_{n}{z}_{n}-{y}_{n}{\parallel }_{c}\hfill \\ =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{n}\parallel {T}_{n}\left(\left(1-{\beta }_{n}\right){y}_{n}+{\beta }_{n}{T}_{n}{y}_{n}\right)-{y}_{n}{\parallel }_{c}\hfill \\ =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{n}\parallel \left(\left(1-{\beta }_{n}\right){T}_{n}{y}_{n}+{\beta }_{n}{T}_{n}^{2}{y}_{n}\right)-{y}_{n}{\parallel }_{c}\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{n}\left(1-{\beta }_{n}\right){d}_{c}\left({T}_{n}{y}_{n},{y}_{n}\right)+{\alpha }_{n}{\beta }_{n}{d}_{c}\left({T}_{n}^{2}{y}_{n},{y}_{n}\right)\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{n}\left(1-{\beta }_{n}\right)\left[{d}_{c}\left({T}_{n}{y}_{n},{T}_{n-1}{y}_{n}\right)+\cdots +{d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)\right]\hfill \\ +\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{n}{\beta }_{n}\left[{d}_{c}\left({T}_{n}^{2}{y}_{n},{T}_{n-1}{T}_{n}{y}_{n}\right)+\cdots +{d}_{c}\left({T}_{1}{T}_{n}{y}_{n},{T}_{n}{y}_{n}\right)\right]\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{n}\left(1-{\beta }_{n}\right){d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)+n{\alpha }_{n}{\beta }_{n}{d}_{c}\left({T}_{1}{T}_{n}{y}_{n},{T}_{n}{y}_{n}\right)\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{n}\left(1-{\beta }_{n}\right){d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)+n{\alpha }_{n}{\beta }_{n}{d}_{c}\left({T}_{n}{T}_{1}{y}_{n},{T}_{n}{y}_{n}\right)\hfill \\ \le \phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}{\alpha }_{n}\left(1-{\beta }_{n}\right){d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)+n{\alpha }_{n}{\beta }_{n}{d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)\hfill \\ =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left[n{\alpha }_{n}\left(1-{\beta }_{n}\right)+n{\alpha }_{n}{\beta }_{n}\right]{d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)\hfill \\ =\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}n{\alpha }_{n}{d}_{c}\left({T}_{1}{y}_{n},{y}_{n}\right)\hfill \end{array}$

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 q0 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 T0 = I, ∩ n F(T n ) ≠ , and ||T m x - T m -1x|| ≤ ||T m -1x- T m -2x|| for all X, m ≥ 2. Consider the iteration procedure xn+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
${d}_{c}\left(f\left({T}_{n},{y}_{n}\right),q\right)\le a{d}_{c}\left(f\left({T}_{n},{x}_{n}\right),{y}_{n}\right)+b{d}_{c}\left({y}_{n},q\right)$

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.

## Authors’ Affiliations

(1)
Department of Mathematics, Payame Noor University, Tehran, Iran
(2)
Department of Mathematics, Shiraz University of Technology, Shiraz, Iran
(3)
Department of Mathematics, Amirkabir University of Technology, Tehran, Iran

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