Semistability of iterations in cone spaces

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.

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:

• (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 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

$$d\left(T_{n}x,q\right)\le a_{n}d\left(x,T_{n}x\right)+b_{n}d\left(x,q\right)\left(*\right)$$

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 = sup_nb_n. 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 -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

$$b_n\le d\left(y_n,T_{n-1}{y}_{n-1}\right)+d\left(T_n{y}_n,T_{n-1}{y}_{n-1}\right)\le c_n+s_n{b}_{n-1},$$

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: 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.

Then 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 t₁ = 0 and P be normal with constant k. Since t _{n +1} - t _n ≤ s _n , thus ∑ _n (t _{n +1} - t _n ) ≤ ∑_ns_n. Hence ||∑ _n (t _{n +1} - t _n )|| ≤ k ||∑_ns_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 ) ≠ ∅, 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 = ∑_nα_n, 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 -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

$$d_c\left(f\left(T_n,y_n\right),q\right)\le a{d}_c\left(f\left(T_n,x_n\right),y_n\right)+b{d}_c\left(y_n,q\right)\left(*\right)$$

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 d_c(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

$$\begin{array}{cc}\hfill d_c\left(f\left(T_n,y_n\right),y_n\right)& =\parallel f\left(T_n,y_n\right)-y_n{\parallel }_c\hfill \\ =\parallel {\alpha }_n{y}_n+\left(1-{\alpha }_n\right)S_n{y}_n-y_n{\parallel }_c\hfill \\ =\left(1-{\alpha }_n\right)\parallel y_n-S_n{y}_n{\parallel }_c\hfill \\ \le \frac{1-{\alpha }_n}{n}\sum _{i=1}^{n-1}\parallel \left(T_i{y}_n-y_n\right){\parallel }_c.\hfill \end{array}$$

But for i ≥ 1, we have

$$\begin{array}{l}\left|\right|T_i{y}_n-y_n|{|}_c\le d_c(T_i{y}_n-T_{i-1}{y}_n)+\cdots +d_c(T_1{y}_n-y_n)\\ \le i{d}_c{(}_{1}T{y}_n,y_n).\end{array}$$

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 -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

$$d_c\left(S_n{y}_n,q\right)\le a_n{d}_c\left(S_n{y}_n,y_n\right)+b_n{d}_c\left(y_n,q\right)$$

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 -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

$$d_c\left(S_n{y}_n,q\right)\le a{d}_c\left(S_n{y}_n,y_n\right)+b{d}_c\left(y_n,q\right)$$

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 -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

$$d_c\left(f\left(T_n,y_n\right),q\right)\le a{d}_c\left(f\left(T_n,y_n\right),y_n\right)+b{d}_c\left(y_n,q\right)\left(*\right)$$

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

$$\underset{n}{lim}{d}_c\left(y_{n+1},f\left(T_n,y_n\right)\right)=\underset{n}{lim}n{\alpha }_n{d}_c\left(T_1{y}_n,y_n\right)=0.$$

Now we show that y _n → q₀. To see this note that by using the notation (*) we have:

$$\begin{array}{cc}\hfill d_c\left(y_{n+1},q_0\right)& \le d_c\left(y_{n+1},f\left(T_n,y_n\right)\right)+d_c\left(f\left(T_n,y_n\right),q_0\right)\hfill \\ \le d_c\left(y_{n+1},f\left(T_n,y_n\right)\right)+a{d}_c\left(f\left(T_n,y_n\right),y_n\right)+b{d}_c\left(y_n,q_0\right)\hfill \\ =c_n+b{d}_c\left(y_n,q_0\right),\hfill \end{array}$$

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 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

$$\begin{array}{cc}\hfill d_c\left(f\left(T_n,y_n\right),y_n\right)& =\parallel f\left(T_n,y_n\right)-y_n{\parallel }_c\hfill \\ =\parallel \left(1-{\alpha }_n\right)y_n+{\alpha }_n{T}_n\left(z_n\right)-y_n{\parallel }_c\hfill \\ ={\alpha }_n\parallel T_n{z}_n-y_n{\parallel }_c\hfill \\ ={\alpha }_n\parallel T_n\left(\left(1-{\beta }_n\right)y_n+{\beta }_n{T}_n{y}_n\right)-y_n{\parallel }_c\hfill \\ ={\alpha }_n\parallel \left(\left(1-{\beta }_n\right)T_n{y}_n+{\beta }_n{T}_n^2{y}_n\right)-y_n{\parallel }_c\hfill \\ \le {\alpha }_n\left(1-{\beta }_n\right)d_c\left(T_n{y}_n,y_n\right)+{\alpha }_n{\beta }_n{d}_c\left(T_n^2{y}_n,y_n\right)\hfill \\ \le {\alpha }_n\left(1-{\beta }_n\right)\left[d_c\left(T_n{y}_n,T_{n-1}{y}_n\right)+\cdots +d_c\left(T_1{y}_n,y_n\right)\right]\hfill \\ +{\alpha }_n{\beta }_n\left[d_c\left(T_n^2{y}_n,T_{n-1}{T}_n{y}_n\right)+\cdots +d_c\left(T_1{T}_n{y}_n,T_n{y}_n\right)\right]\hfill \\ \le {\alpha }_n\left(1-{\beta }_n\right)d_c\left(T_1{y}_n,y_n\right)+n{\alpha }_n{\beta }_n{d}_c\left(T_1{T}_n{y}_n,T_n{y}_n\right)\hfill \\ \le {\alpha }_n\left(1-{\beta }_n\right)d_c\left(T_1{y}_n,y_n\right)+n{\alpha }_n{\beta }_n{d}_c\left(T_n{T}_1{y}_n,T_n{y}_n\right)\hfill \\ \le {\alpha }_n\left(1-{\beta }_n\right)d_c\left(T_1{y}_n,y_n\right)+n{\alpha }_n{\beta }_n{d}_c\left(T_1{y}_n,y_n\right)\hfill \\ =\left[n{\alpha }_n\left(1-{\beta }_n\right)+n{\alpha }_n{\beta }_n\right]d_c\left(T_1{y}_n,y_n\right)\hfill \\ =n{\alpha }_n{d}_c\left(T_1{y}_n,y_n\right)\hfill \end{array}$$

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 -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

$$d_c\left(f\left(T_n,y_n\right),q\right)\le a{d}_c\left(f\left(T_n,x_n\right),y_n\right)+b{d}_c\left(y_n,q\right)$$

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.