The Equivalence between T-Stabilities of The Krasnoselskij and The Mann Iterations

Stefan Soltuz
(original), Fixed point theory, JCR/ISI, paper

Abstract

We prove the equivalence between the -stabilities of the Krasnoselskij and the Mann iterations; a consequence is the equivalence with the -stability of the Picard-Banach iteration.

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    S.M. Soltuz
    (Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

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    Ş.M. Şoltuz, The Equivalence between T-Stabilities of The Krasnoselskij and The Mann Iterations. Fixed Point Theory Appl 2007, 060732 (2007).
    doi: 10.1155/2007/60732

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    Fixed Point Theory Appl.

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    Springer

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    10.1155/2007/60732

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    0022-247X

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    60732

    Research Article
    The Equivalence between T T TTT-Stabilities of The Krasnoselskij and The Mann Iterations

    1. Introduction

    Let X X XXX be a normed space and T T TTT a selfmap of X X XXX. Let x 0 x 0 x_(0)x_{0}x0 be a point of X X XXX, and assume that x n + 1 = f ( T , x n ) x n + 1 = f T , x n x_(n+1)=f(T,x_(n))x_{n+1}=f\left(T, x_{n}\right)xn+1=f(T,xn) is an iteration procedure, involving T T TTT, which yields a sequence { x n } x n {x_(n)}\left\{x_{n}\right\}{xn} of points from X X XXX. Suppose { x n } x n {x_(n)}\left\{x_{n}\right\}{xn} converges to a fixed point x ∗ x ∗ x^(**)x^{*}x∗ of T T TTT. Let { ξ n } ξ n {xi_(n)}\left\{\xi_{n}\right\}{ξn} be an arbitrary sequence in X X XXX, and set ϵ n = ‖ ξ n + 1 − f ( T , ξ n ) ‖ ϵ n = ξ n + 1 − f T , ξ n epsilon_(n)=||xi_(n+1)-f(T,xi_(n))||\epsilon_{n}=\left\|\xi_{n+1}-f\left(T, \xi_{n}\right)\right\|ϵn=‖ξn+1−f(T,ξn)‖ for all n ∈ N n ∈ N n inNn \in \mathbb{N}n∈N.
    Definition 1.1 [1]. If ( lim n → ∞ ϵ n = 0 ) ⇒ ( lim n → ∞ ξ n = p ) lim n → ∞   ϵ n = 0 ⇒ lim n → ∞   ξ n = p (lim_(n rarr oo)epsilon_(n)=0)=>(lim_(n rarr oo)xi_(n)=p)\left(\lim _{n \rightarrow \infty} \epsilon_{n}=0\right) \Rightarrow\left(\lim _{n \rightarrow \infty} \xi_{n}=p\right)(limn→∞ϵn=0)⇒(limn→∞ξn=p), then the iteration procedure x n + 1 = f ( T , x n ) x n + 1 = f T , x n x_(n+1)=f(T,x_(n))x_{n+1}=f\left(T, x_{n}\right)xn+1=f(T,xn) is said to be T T TTT-stable with respect to T T TTT.
    Remark 1.2 [1]. In practice, such a sequence { ξ n } ξ n {xi_(n)}\left\{\xi_{n}\right\}{ξn} could arise in the following way. Let x 0 x 0 x_(0)x_{0}x0 be a point in X X XXX. Set x n + 1 = f ( T , x n ) x n + 1 = f T , x n x_(n+1)=f(T,x_(n))x_{n+1}=f\left(T, x_{n}\right)xn+1=f(T,xn). Let ξ 0 = x 0 ξ 0 = x 0 xi_(0)=x_(0)\xi_{0}=x_{0}ξ0=x0. Now x 1 = f ( T , x 0 ) x 1 = f T , x 0 x_(1)=f(T,x_(0))x_{1}=f\left(T, x_{0}\right)x1=f(T,x0). Because of rounding or discretization in the function T T TTT, a new value ξ 1 ξ 1 xi_(1)\xi_{1}ξ1 approximately equal to x 1 x 1 x_(1)x_{1}x1 might be obtained instead of the true value of f ( T , x 0 ) f T , x 0 f(T,x_(0))f\left(T, x_{0}\right)f(T,x0). Then to approximate x 2 x 2 x_(2)x_{2}x2, the value f ( T , ξ 1 ) f T , ξ 1 f(T,xi_(1))f\left(T, \xi_{1}\right)f(T,ξ1) is computed to yield ξ 2 ξ 2 xi_(2)\xi_{2}ξ2, an approximation of f ( T , ξ 1 ) f T , ξ 1 f(T,xi_(1))f\left(T, \xi_{1}\right)f(T,ξ1). This computation is continued to obtain { ξ n } ξ n {xi_(n)}\left\{\xi_{n}\right\}{ξn} an approximate sequence of { x n } x n {x_(n)}\left\{x_{n}\right\}{xn}.
    Let X X XXX be a normed space, D D DDD a nonempty, convex subset of X X XXX, and T T TTT a selfmap of D D DDD, let p 0 = e 0 ∈ D p 0 = e 0 ∈ D p_(0)=e_(0)in Dp_{0}=e_{0} \in Dp0=e0∈D. The Mann iteration (see [2]) is defined by
    (1.1) e n + 1 = ( 1 − α n ) e n + α n T e n , (1.1) e n + 1 = 1 − α n e n + α n T e n , {:(1.1)e_(n+1)=(1-alpha_(n))e_(n)+alpha_(n)Te_(n)",":}\begin{equation*} e_{n+1}=\left(1-\alpha_{n}\right) e_{n}+\alpha_{n} T e_{n}, \tag{1.1} \end{equation*}(1.1)en+1=(1−αn)en+αnTen,
    where { α n } ⊂ ( 0 , 1 ) α n ⊂ ( 0 , 1 ) {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1){αn}⊂(0,1). The Ishikawa iteration is defined (see [3]) by
    (1.2) x n + 1 = ( 1 − α n ) x n + α n T y n , y n = ( 1 − β n ) x n + β n T x n , (1.2) x n + 1 = 1 − α n x n + α n T y n , y n = 1 − β n x n + β n T x n , {:[(1.2)x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)Ty_(n)","],[y_(n)=(1-beta_(n))x_(n)+beta_(n)Tx_(n)","]:}\begin{align*} x_{n+1} & =\left(1-\alpha_{n}\right) x_{n}+\alpha_{n} T y_{n}, \tag{1.2}\\ y_{n} & =\left(1-\beta_{n}\right) x_{n}+\beta_{n} T x_{n}, \end{align*}(1.2)xn+1=(1−αn)xn+αnTyn,yn=(1−βn)xn+βnTxn,
    where { α n } ⊂ ( 0 , 1 ) , { β n } ⊂ [ 0 , 1 ) α n ⊂ ( 0 , 1 ) , β n ⊂ [ 0 , 1 ) {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1){αn}⊂(0,1),{βn}⊂[0,1). The Krasnoselskij iteration (see [4]) is defined by
    (1.3) p n + 1 = ( 1 − λ ) p n + λ T p n , (1.3) p n + 1 = ( 1 − λ ) p n + λ T p n , {:(1.3)p_(n+1)=(1-lambda)p_(n)+lambda Tp_(n)",":}\begin{equation*} p_{n+1}=(1-\lambda) p_{n}+\lambda T p_{n}, \tag{1.3} \end{equation*}(1.3)pn+1=(1−λ)pn+λTpn,
    where λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in(0,1)λ∈(0,1). Recently, the equivalence between the T T TTT-stabilities of Mann and Ishikawa iterations, respectively, for modified Mann-Ishikawa iterations was shown in [5]. In the present paper, we shall prove the equivalence between the T T TTT-stabilities of the Krasnoselskij and the Mann iterations. Next, { u n } , { v n } ⊂ X u n , v n ⊂ X {u_(n)},{v_(n)}sub X\left\{u_{n}\right\},\left\{v_{n}\right\} \subset X{un},{vn}⊂X are arbitrary.

    Definition 1.3.

    (i) The Mann iteration (1.1) is said to be T T TTT-stable if and only if for all { α n } ⊂ ( 0 , 1 ) α n ⊂ ( 0 , 1 ) {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1){αn}⊂(0,1) and for every sequence { u n } ⊂ X u n ⊂ X {u_(n)}sub X\left\{u_{n}\right\} \subset X{un}⊂X,
    (1.4) lim n → ∞ ε n = 0 ⟹ lim n → ∞ u n = x ∗ (1.4) lim n → ∞   ε n = 0 ⟹ lim n → ∞   u n = x ∗ {:(1.4)lim_(n rarr oo)epsi_(n)=0Longrightarrowlim_(n rarr oo)u_(n)=x^(**):}\begin{equation*} \lim _{n \rightarrow \infty} \varepsilon_{n}=0 \Longrightarrow \lim _{n \rightarrow \infty} u_{n}=x^{*} \tag{1.4} \end{equation*}(1.4)limn→∞εn=0⟹limn→∞un=x∗
    where ε n := ‖ u n + 1 − ( 1 − α n ) u n − α n T u n ‖ ε n := u n + 1 − 1 − α n u n − α n T u n epsi_(n):=||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||\varepsilon_{n}:=\left\|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right\|εn:=‖un+1−(1−αn)un−αnTun‖.
    (ii) The Krasnoselskij iteration (1.3) is said to be T T TTT-stable if and only if for all λ ∈ ( 0 , 1 ) λ ∈ ( 0 , 1 ) lambda in(0,1)\lambda \in (0,1)λ∈(0,1), and for every sequence { v n } ⊂ X v n ⊂ X {v_(n)}sub X\left\{v_{n}\right\} \subset X{vn}⊂X,
    (1.5) lim n → ∞ δ n = 0 ⟹ lim n → ∞ v n = x ∗ (1.5) lim n → ∞   δ n = 0 ⟹ lim n → ∞   v n = x ∗ {:(1.5)lim_(n rarr oo)delta_(n)=0Longrightarrowlim_(n rarr oo)v_(n)=x^(**):}\begin{equation*} \lim _{n \rightarrow \infty} \delta_{n}=0 \Longrightarrow \lim _{n \rightarrow \infty} v_{n}=x^{*} \tag{1.5} \end{equation*}(1.5)limn→∞δn=0⟹limn→∞vn=x∗
    where δ n := ‖ v n + 1 − ( 1 − λ ) v n − λ T v n ‖ δ n := v n + 1 − ( 1 − λ ) v n − λ T v n delta_(n):=||v_(n+1)-(1-lambda)v_(n)-lambda Tv_(n)||\delta_{n}:=\left\|v_{n+1}-(1-\lambda) v_{n}-\lambda T v_{n}\right\|δn:=‖vn+1−(1−λ)vn−λTvn‖.

    2. Main results

    Theorem 2.1. Let X X XXX be a normed space and T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X a map with bounded range and { α n } ⊂ ( 0 , 1 ) α n ⊂ ( 0 , 1 ) {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1){αn}⊂(0,1) satisfy lim n → ∞ α n = λ , λ ∈ ( 0 , 1 ) lim n → ∞   α n = λ , λ ∈ ( 0 , 1 ) lim_(n rarr oo)alpha_(n)=lambda,lambda in(0,1)\lim _{n \rightarrow \infty} \alpha_{n}=\lambda, \lambda \in(0,1)limn→∞αn=λ,λ∈(0,1). Then the following are equivalent:
    (i) the Mann iteration is T T TTT-stable,
    (ii) the Krasnoselskij iteration is T T TTT-stable.
    Proof. We prove that (i) ⇒ (ii). If lim n → ∞ δ n = 0 lim n → ∞   δ n = 0 lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0limn→∞δn=0, then { v n } v n {v_(n)}\left\{v_{n}\right\}{vn} is bounded. Set
    (2.1) M 1 := max { sup x ∈ X { ‖ T ( x ) ‖ } , ‖ v 0 ‖ , ‖ u 0 ‖ } . (2.1) M 1 := max sup x ∈ X   { ‖ T ( x ) ‖ } , v 0 , u 0 . {:(2.1)M_(1):=max{s u p_(x in X){||T(x)||},||v_(0)||,||u_(0)||}.:}\begin{equation*} M_{1}:=\max \left\{\sup _{x \in X}\{\|T(x)\|\},\left\|v_{0}\right\|,\left\|u_{0}\right\|\right\} . \tag{2.1} \end{equation*}(2.1)M1:=max{supx∈X{‖T(x)‖},‖v0‖,‖u0‖}.
    Observe that ‖ v 1 ‖ ≤ δ 0 + ( 1 − λ ) ‖ v 0 ‖ + λ ‖ T v 0 ‖ ≤ δ 0 + M 1 v 1 ≤ δ 0 + ( 1 − λ ) v 0 + λ T v 0 ≤ δ 0 + M 1 ||v_(1)|| <= delta_(0)+(1-lambda)||v_(0)||+lambda||Tv_(0)|| <= delta_(0)+M_(1)\left\|v_{1}\right\| \leq \delta_{0}+(1-\lambda)\left\|v_{0}\right\|+\lambda\left\|T v_{0}\right\| \leq \delta_{0}+M_{1}‖v1‖≤δ0+(1−λ)‖v0‖+λ‖Tv0‖≤δ0+M1. Set M := M 1 + 1 / λ M := M 1 + 1 / λ M:=M_(1)+1//lambdaM:=M_{1}+1 / \lambdaM:=M1+1/λ. Suppose that ‖ v n ‖ ≤ M v n ≤ M ||v_(n)|| <= M\left\|v_{n}\right\| \leq M‖vn‖≤M to prove that ‖ v n + 1 ‖ ≤ M v n + 1 ≤ M ||v_(n+1)|| <= M\left\|v_{n+1}\right\| \leq M‖vn+1‖≤M. Remark that
    ‖ v n + 1 ‖ ≤ δ n + ( 1 − λ ) δ n − 1 + ⋯ + ( 1 − λ ) n δ 0 + M 1 (2.2) ≤ 1 + ( 1 − λ ) + ⋯ + ( 1 − λ ) n + M 1 ≤ 1 1 − ( 1 − λ ) + M 1 = M . v n + 1 ≤ δ n + ( 1 − λ ) δ n − 1 + ⋯ + ( 1 − λ ) n δ 0 + M 1 (2.2) ≤ 1 + ( 1 − λ ) + ⋯ + ( 1 − λ ) n + M 1 ≤ 1 1 − ( 1 − λ ) + M 1 = M . {:[||v_(n+1)|| <= delta_(n)+(1-lambda)delta_(n-1)+cdots+(1-lambda)^(n)delta_(0)+M_(1)],[(2.2) <= 1+(1-lambda)+cdots+(1-lambda)^(n)+M_(1)],[ <= (1)/(1-(1-lambda))+M_(1)=M.]:}\begin{align*} \left\|v_{n+1}\right\| & \leq \delta_{n}+(1-\lambda) \delta_{n-1}+\cdots+(1-\lambda)^{n} \delta_{0}+M_{1} \\ & \leq 1+(1-\lambda)+\cdots+(1-\lambda)^{n}+M_{1} \tag{2.2}\\ & \leq \frac{1}{1-(1-\lambda)}+M_{1}=M . \end{align*}‖vn+1‖≤δn+(1−λ)δn−1+⋯+(1−λ)nδ0+M1(2.2)≤1+(1−λ)+⋯+(1−λ)n+M1≤11−(1−λ)+M1=M.
    Suppose that lim n → ∞ δ n = 0 lim n → ∞   δ n = 0 lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0limn→∞δn=0 to note that
    ε n = ‖ v n + 1 − ( 1 − α n ) v n − α n T v n ‖ = ‖ v n + 1 − v n + λ v n − λ v n + α n v n − λ T v n + λ T v n − α n T v n ‖ (2.3) ≤ ‖ v n + 1 − ( 1 − λ ) v n − λ T v n ‖ + | λ − α n | ‖ v n − T v n ‖ ≤ ‖ v n + 1 − ( 1 − λ ) v n − λ T v n ‖ + 2 M | λ − α n | = δ n + 2 M | λ − α n | ⟶ 0 as n ⟶ ∞ . ε n = v n + 1 − 1 − α n v n − α n T v n = v n + 1 − v n + λ v n − λ v n + α n v n − λ T v n + λ T v n − α n T v n (2.3) ≤ v n + 1 − ( 1 − λ ) v n − λ T v n + λ − α n v n − T v n ≤ v n + 1 − ( 1 − λ ) v n − λ T v n + 2 M λ − α n = δ n + 2 M λ − α n ⟶ 0  as  n ⟶ ∞ . {:[epsi_(n)=||v_(n+1)-(1-alpha_(n))v_(n)-alpha_(n)Tv_(n)||],[=||v_(n+1)-v_(n)+lambdav_(n)-lambdav_(n)+alpha_(n)v_(n)-lambda Tv_(n)+lambda Tv_(n)-alpha_(n)Tv_(n)||],[(2.3) <= ||v_(n+1)-(1-lambda)v_(n)-lambda Tv_(n)||+|lambda-alpha_(n)|||v_(n)-Tv_(n)||],[ <= ||v_(n+1)-(1-lambda)v_(n)-lambda Tv_(n)||+2M|lambda-alpha_(n)|],[=delta_(n)+2M|lambda-alpha_(n)|longrightarrow0quad" as "n longrightarrow oo.]:}\begin{align*} \varepsilon_{n} & =\left\|v_{n+1}-\left(1-\alpha_{n}\right) v_{n}-\alpha_{n} T v_{n}\right\| \\ & =\left\|v_{n+1}-v_{n}+\lambda v_{n}-\lambda v_{n}+\alpha_{n} v_{n}-\lambda T v_{n}+\lambda T v_{n}-\alpha_{n} T v_{n}\right\| \\ & \leq\left\|v_{n+1}-(1-\lambda) v_{n}-\lambda T v_{n}\right\|+\left|\lambda-\alpha_{n}\right|\left\|v_{n}-T v_{n}\right\| \tag{2.3}\\ & \leq\left\|v_{n+1}-(1-\lambda) v_{n}-\lambda T v_{n}\right\|+2 M\left|\lambda-\alpha_{n}\right| \\ & =\delta_{n}+2 M\left|\lambda-\alpha_{n}\right| \longrightarrow 0 \quad \text { as } n \longrightarrow \infty . \end{align*}εn=‖vn+1−(1−αn)vn−αnTvn‖=‖vn+1−vn+λvn−λvn+αnvn−λTvn+λTvn−αnTvn‖(2.3)≤‖vn+1−(1−λ)vn−λTvn‖+|λ−αn|‖vn−Tvn‖≤‖vn+1−(1−λ)vn−λTvn‖+2M|λ−αn|=δn+2M|λ−αn|⟶0 as n⟶∞.
    Condition (i) assures that if lim n → ∞ ε n = 0 lim n → ∞   ε n = 0 lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0limn→∞εn=0, then lim n → ∞ v n = x ∗ lim n → ∞   v n = x ∗ lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}limn→∞vn=x∗. Thus, for a { v n } v n {v_(n)}\left\{v_{n}\right\}{vn} satisfying
    (2.4) lim n → ∞ δ n = lim n → ∞ ‖ v n + 1 − ( 1 − λ ) v n − λ T v n ‖ = 0 , (2.4) lim n → ∞   δ n = lim n → ∞   v n + 1 − ( 1 − λ ) v n − λ T v n = 0 , {:(2.4)lim_(n rarr oo)delta_(n)=lim_(n rarr oo)||v_(n+1)-(1-lambda)v_(n)-lambda Tv_(n)||=0",":}\begin{equation*} \lim _{n \rightarrow \infty} \delta_{n}=\lim _{n \rightarrow \infty}\left\|v_{n+1}-(1-\lambda) v_{n}-\lambda T v_{n}\right\|=0, \tag{2.4} \end{equation*}(2.4)limn→∞δn=limn→∞‖vn+1−(1−λ)vn−λTvn‖=0,
    we have shown that lim n → ∞ v n = x ∗ lim n → ∞   v n = x ∗ lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}limn→∞vn=x∗.
    Conversely, we prove (ii) ⇒ (i). First, we prove that { u n } u n {u_(n)}\left\{u_{n}\right\}{un} is bounded. Since lim n → ∞ α n = λ lim n → ∞   α n = λ lim_(n rarr oo)alpha_(n)=lambda\lim _{n \rightarrow \infty} \alpha_{n}= \lambdalimn→∞αn=λ, for β ∈ ( 0 , 1 ) β ∈ ( 0 , 1 ) beta in(0,1)\beta \in(0,1)β∈(0,1) given, there exists n 0 ∈ N n 0 ∈ N n_(0)in Nn_{0} \in Nn0∈N, such that 1 − α n ≤ β 1 − α n ≤ β 1-alpha_(n) <= beta1-\alpha_{n} \leq \beta1−αn≤β, for all n ≥ n 0 n ≥ n 0 n >= n_(0)n \geq n_{0}n≥n0. Set M 1 := max { sup x ∈ X ‖ T x ‖ , ‖ u 0 ‖ } M 1 := max sup x ∈ X   ‖ T x ‖ , u 0 M_(1):=max{s u p_(x in X)||Tx||,||u_(0)||}M_{1}:= \max \left\{\sup _{x \in X}\|T x\|,\left\|u_{0}\right\|\right\}M1:=max{supx∈X‖Tx‖,‖u0‖} and M := n 0 + 1 + β / ( 1 − β ) + M 1 M := n 0 + 1 + β / ( 1 − β ) + M 1 M:=n_(0)+1+beta//(1-beta)+M_(1)M:=n_{0}+1+\beta /(1-\beta)+M_{1}M:=n0+1+β/(1−β)+M1 to obtain
    ‖ u n + 1 ‖ ≤ [ ε n + ( 1 − α 1 ) ε n − 1 + ( 1 − α 1 ) ( 1 − α 2 ) ε n − 2 + ⋯ + ( 1 − α 1 ) ( 1 − α 2 ) ⋯ ( 1 − α n 0 ) ε n − n 0 ] + ( 1 − α 1 ) ( 1 − α 2 ) ⋯ ( 1 − α n 0 ) ( 1 − α n 0 + 1 ) ε n − n 0 − 1 (2.5) + ⋯ + ( 1 − α 1 ) ( 1 − α 2 ) ⋯ ( 1 − α n ) ε 0 + M 1 ≤ ( n 0 + 1 ) + ( 1 − α n 0 + 1 ) + ( 1 − α n 0 + 1 ) ( 1 − α n 0 + 2 ) ⋯ + ( 1 − α n 0 + 1 ) ⋯ ( 1 − α n ) ε 0 + M 1 ≤ n 0 + 1 + β + β 2 + ⋯ + β n − n 0 + M 1 < M . u n + 1 ≤ ε n + 1 − α 1 ε n − 1 + 1 − α 1 1 − α 2 ε n − 2 + ⋯ + 1 − α 1 1 − α 2 ⋯ 1 − α n 0 ε n − n 0 + 1 − α 1 1 − α 2 ⋯ 1 − α n 0 1 − α n 0 + 1 ε n − n 0 − 1 (2.5) + ⋯ + 1 − α 1 1 − α 2 ⋯ 1 − α n ε 0 + M 1 ≤ n 0 + 1 + 1 − α n 0 + 1 + 1 − α n 0 + 1 1 − α n 0 + 2 ⋯ + 1 − α n 0 + 1 ⋯ 1 − α n ε 0 + M 1 ≤ n 0 + 1 + β + β 2 + ⋯ + β n − n 0 + M 1 < M . {:[||u_(n+1)|| <= [epsi_(n)+(1-alpha_(1))epsi_(n-1)+(1-alpha_(1))(1-alpha_(2))epsi_(n-2):}],[{:+cdots+(1-alpha_(1))(1-alpha_(2))cdots(1-alpha_(n_(0)))epsi_(n-n_(0))]],[+(1-alpha_(1))(1-alpha_(2))cdots(1-alpha_(n_(0)))(1-alpha_(n_(0)+1))epsi_(n-n_(0)-1)],[(2.5)+cdots+(1-alpha_(1))(1-alpha_(2))cdots(1-alpha_(n))epsi_(0)+M_(1)],[ <= (n_(0)+1)+(1-alpha_(n_(0)+1))+(1-alpha_(n_(0)+1))(1-alpha_(n_(0)+2))cdots],[+(1-alpha_(n_(0)+1))cdots(1-alpha_(n))epsi_(0)+M_(1)],[ <= n_(0)+1+beta+beta^(2)+cdots+beta^(n-n_(0))+M_(1) < M.]:}\begin{align*} \left\|u_{n+1}\right\| \leq & {\left[\varepsilon_{n}+\left(1-\alpha_{1}\right) \varepsilon_{n-1}+\left(1-\alpha_{1}\right)\left(1-\alpha_{2}\right) \varepsilon_{n-2}\right.} \\ & \left.+\cdots+\left(1-\alpha_{1}\right)\left(1-\alpha_{2}\right) \cdots\left(1-\alpha_{n_{0}}\right) \varepsilon_{n-n_{0}}\right] \\ & +\left(1-\alpha_{1}\right)\left(1-\alpha_{2}\right) \cdots\left(1-\alpha_{n_{0}}\right)\left(1-\alpha_{n_{0}+1}\right) \varepsilon_{n-n_{0}-1} \\ & +\cdots+\left(1-\alpha_{1}\right)\left(1-\alpha_{2}\right) \cdots\left(1-\alpha_{n}\right) \varepsilon_{0}+M_{1} \tag{2.5}\\ \leq & \left(n_{0}+1\right)+\left(1-\alpha_{n_{0}+1}\right)+\left(1-\alpha_{n_{0}+1}\right)\left(1-\alpha_{n_{0}+2}\right) \cdots \\ & +\left(1-\alpha_{n_{0}+1}\right) \cdots\left(1-\alpha_{n}\right) \varepsilon_{0}+M_{1} \\ \leq & n_{0}+1+\beta+\beta^{2}+\cdots+\beta^{n-n_{0}}+M_{1}<M . \end{align*}‖un+1‖≤[εn+(1−α1)εn−1+(1−α1)(1−α2)εn−2+⋯+(1−α1)(1−α2)⋯(1−αn0)εn−n0]+(1−α1)(1−α2)⋯(1−αn0)(1−αn0+1)εn−n0−1(2.5)+⋯+(1−α1)(1−α2)⋯(1−αn)ε0+M1≤(n0+1)+(1−αn0+1)+(1−αn0+1)(1−αn0+2)⋯+(1−αn0+1)⋯(1−αn)ε0+M1≤n0+1+β+β2+⋯+βn−n0+M1<M.
    Suppose lim n → ∞ E n = 0 lim n → ∞   E n = 0 lim_(n rarr oo)E_(n)=0\lim _{n \rightarrow \infty} \mathcal{E}_{n}=0limn→∞En=0. Observe that
    δ n = ‖ u n + 1 − ( 1 − λ ) u n − λ T u n ‖ = ‖ u n + 1 − u n + λ u n − λ T u n + α n u n − α n u n − α n T u n + α n T u n ‖ (2.6) ≤ ‖ u n + 1 − ( 1 − α n ) u n − α n T u n ‖ + | λ − α n | ‖ u n − T u n ‖ ≤ ‖ u n + 1 − ( 1 − α n ) u n − α n T u n ‖ + 2 M | λ − α n | = ε n + 2 M | λ − α n | ⟶ 0 as n ⟶ ∞ . δ n = u n + 1 − ( 1 − λ ) u n − λ T u n = u n + 1 − u n + λ u n − λ T u n + α n u n − α n u n − α n T u n + α n T u n (2.6) ≤ u n + 1 − 1 − α n u n − α n T u n + λ − α n u n − T u n ≤ u n + 1 − 1 − α n u n − α n T u n + 2 M λ − α n = ε n + 2 M λ − α n ⟶ 0  as  n ⟶ ∞ . {:[delta_(n)=||u_(n+1)-(1-lambda)u_(n)-lambda Tu_(n)||],[=||u_(n+1)-u_(n)+lambdau_(n)-lambda Tu_(n)+alpha_(n)u_(n)-alpha_(n)u_(n)-alpha_(n)Tu_(n)+alpha_(n)Tu_(n)||],[(2.6) <= ||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||+|lambda-alpha_(n)|||u_(n)-Tu_(n)||],[ <= ||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||+2M|lambda-alpha_(n)|],[=epsi_(n)+2M|lambda-alpha_(n)|longrightarrow0" as "n longrightarrow oo.]:}\begin{align*} \delta_{n} & =\left\|u_{n+1}-(1-\lambda) u_{n}-\lambda T u_{n}\right\| \\ & =\left\|u_{n+1}-u_{n}+\lambda u_{n}-\lambda T u_{n}+\alpha_{n} u_{n}-\alpha_{n} u_{n}-\alpha_{n} T u_{n}+\alpha_{n} T u_{n}\right\| \\ & \leq\left\|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right\|+\left|\lambda-\alpha_{n}\right|\left\|u_{n}-T u_{n}\right\| \tag{2.6}\\ & \leq\left\|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right\|+2 M\left|\lambda-\alpha_{n}\right| \\ & =\varepsilon_{n}+2 M\left|\lambda-\alpha_{n}\right| \longrightarrow 0 \text { as } n \longrightarrow \infty . \end{align*}δn=‖un+1−(1−λ)un−λTun‖=‖un+1−un+λun−λTun+αnun−αnun−αnTun+αnTun‖(2.6)≤‖un+1−(1−αn)un−αnTun‖+|λ−αn|‖un−Tun‖≤‖un+1−(1−αn)un−αnTun‖+2M|λ−αn|=εn+2M|λ−αn|⟶0 as n⟶∞.
    Condition (ii) assures that if lim n → ∞ δ n = 0 lim n → ∞   δ n = 0 lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0limn→∞δn=0, then lim n → ∞ v n = x ∗ lim n → ∞   v n = x ∗ lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}limn→∞vn=x∗. Thus, for a { u n } u n {u_(n)}\left\{u_{n}\right\}{un} satisfying
    (2.7) lim n → ∞ ε n = lim n → ∞ ‖ u n + 1 − ( 1 − α n ) u n − α n T u n ‖ = 0 (2.7) lim n → ∞   ε n = lim n → ∞   u n + 1 − 1 − α n u n − α n T u n = 0 {:(2.7)lim_(n rarr oo)epsi_(n)=lim_(n rarr oo)||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||=0:}\begin{equation*} \lim _{n \rightarrow \infty} \varepsilon_{n}=\lim _{n \rightarrow \infty}\left\|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right\|=0 \tag{2.7} \end{equation*}(2.7)limn→∞εn=limn→∞‖un+1−(1−αn)un−αnTun‖=0
    we have shown that lim n → ∞ u n = x ∗ lim n → ∞   u n = x ∗ lim_(n rarr oo)u_(n)=x^(**)\lim _{n \rightarrow \infty} u_{n}=x^{*}limn→∞un=x∗.
    Remark 2.2. Let X X XXX be a normed space and T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X a map with bounded range and { α n } ⊂ ( 0 , 1 ) α n ⊂ ( 0 , 1 ) {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1){αn}⊂(0,1) satisfy lim n → ∞ α n = λ , λ ∈ ( 0 , 1 ) lim n → ∞   α n = λ , λ ∈ ( 0 , 1 ) lim_(n rarr oo)alpha_(n)=lambda,lambda in(0,1)\lim _{n \rightarrow \infty} \alpha_{n}=\lambda, \lambda \in(0,1)limn→∞αn=λ,λ∈(0,1). If the Mann iteration is not T T TTT-stable, then the Krasnoselskij iteration is not T T TTT-stable, and conversely.
    Example 2.3. Let T : [ 0 , 1 ) → [ 0 , 1 ) T : [ 0 , 1 ) → [ 0 , 1 ) T:[0,1)rarr[0,1)T:[0,1) \rightarrow[0,1)T:[0,1)→[0,1) be given by T x = x 2 T x = x 2 Tx=x^(2)T x=x^{2}Tx=x2, and λ = 1 / 2 λ = 1 / 2 lambda=1//2\lambda=1 / 2λ=1/2. Then the Krasnoselskij iteration converges to the unique fixed point x ∗ = 0 x ∗ = 0 x^(**)=0x^{*}=0x∗=0, and it is not T T TTT-stable.
    The Krasnoselskij iteration converges because, supposing F := sup n p n < 1 F := sup n   p n < 1 F:=s u p_(n)p_(n) < 1F:=\sup _{n} p_{n}<1F:=supnpn<1, the sequence p n → 0 p n → 0 p_(n)rarr0p_{n} \rightarrow 0pn→0, as we can see from
    p n + 1 = ( 1 − 1 2 ) p n + 1 2 p n 2 = 1 2 p n + 1 2 p n 2 (2.8) = 1 2 p n ( 1 + p n ) ≤ 1 + F 2 p n = ( 1 + F 2 ) n p 0 ⟶ 0 p n + 1 = 1 − 1 2 p n + 1 2 p n 2 = 1 2 p n + 1 2 p n 2 (2.8) = 1 2 p n 1 + p n ≤ 1 + F 2 p n = 1 + F 2 n p 0 ⟶ 0 {:[p_(n+1)=(1-(1)/(2))p_(n)+(1)/(2)p_(n)^(2)=(1)/(2)p_(n)+(1)/(2)p_(n)^(2)],[(2.8)=(1)/(2)p_(n)(1+p_(n)) <= (1+F)/(2)p_(n)=((1+F)/(2))^(n)p_(0)longrightarrow0]:}\begin{align*} p_{n+1} & =\left(1-\frac{1}{2}\right) p_{n}+\frac{1}{2} p_{n}^{2}=\frac{1}{2} p_{n}+\frac{1}{2} p_{n}^{2} \\ & =\frac{1}{2} p_{n}\left(1+p_{n}\right) \leq \frac{1+F}{2} p_{n}=\left(\frac{1+F}{2}\right)^{n} p_{0} \longrightarrow 0 \tag{2.8} \end{align*}pn+1=(1−12)pn+12pn2=12pn+12pn2(2.8)=12pn(1+pn)≤1+F2pn=(1+F2)np0⟶0
    set v n = n / ( n + 1 ) v n = n / ( n + 1 ) v_(n)=n//(n+1)v_{n}=n /(n+1)vn=n/(n+1) and note that v n v n v_(n)v_{n}vn does not converge to zero, while δ n δ n delta_(n)\delta_{n}δn does:
    (2.9) δ n = | n + 1 n + 2 − 1 2 n n + 1 − 1 2 n 2 ( n + 1 ) 2 | = n 2 + 4 n + 2 2 ( n + 1 ) 2 ( n + 2 ) ⟶ 0 (2.9) δ n = n + 1 n + 2 − 1 2 n n + 1 − 1 2 n 2 ( n + 1 ) 2 = n 2 + 4 n + 2 2 ( n + 1 ) 2 ( n + 2 ) ⟶ 0 {:(2.9)delta_(n)=|(n+1)/(n+2)-(1)/(2)(n)/(n+1)-(1)/(2)(n^(2))/((n+1)^(2))|=(n^(2)+4n+2)/(2(n+1)^(2)(n+2))longrightarrow0:}\begin{equation*} \delta_{n}=\left|\frac{n+1}{n+2}-\frac{1}{2} \frac{n}{n+1}-\frac{1}{2} \frac{n^{2}}{(n+1)^{2}}\right|=\frac{n^{2}+4 n+2}{2(n+1)^{2}(n+2)} \longrightarrow 0 \tag{2.9} \end{equation*}(2.9)δn=|n+1n+2−12nn+1−12n2(n+1)2|=n2+4n+22(n+1)2(n+2)⟶0
    The Mann iteration also converges because (supposing E := sup n e n < 1 E := sup n   e n < 1 E:=s u p_(n)e_(n) < 1E:=\sup _{n} e_{n}<1E:=supnen<1 ) one has
    e n + 1 = ( 1 − α n ) e n + α n e n 2 = ( 1 − ( 1 − E ) α n ) e n (2.10) ≤ ∏ k = 1 n ( 1 − ( 1 − E ) α k ) e 0 ≤ exp ⁡ ( − ( 1 − E ) ∑ k = 1 n α k ) e 0 ⟶ 0 e n + 1 = 1 − α n e n + α n e n 2 = 1 − ( 1 − E ) α n e n (2.10) ≤ ∏ k = 1 n   1 − ( 1 − E ) α k e 0 ≤ exp ⁡ − ( 1 − E ) ∑ k = 1 n   α k e 0 ⟶ 0 {:[e_(n+1)=(1-alpha_(n))e_(n)+alpha_(n)e_(n)^(2)=(1-(1-E)alpha_(n))e_(n)],[(2.10) <= prod_(k=1)^(n)(1-(1-E)alpha_(k))e_(0) <= exp(-(1-E)sum_(k=1)^(n)alpha_(k))e_(0)longrightarrow0]:}\begin{align*} e_{n+1} & =\left(1-\alpha_{n}\right) e_{n}+\alpha_{n} e_{n}^{2}=\left(1-(1-E) \alpha_{n}\right) e_{n} \\ & \leq \prod_{k=1}^{n}\left(1-(1-E) \alpha_{k}\right) e_{0} \leq \exp \left(-(1-E) \sum_{k=1}^{n} \alpha_{k}\right) e_{0} \longrightarrow 0 \tag{2.10} \end{align*}en+1=(1−αn)en+αnen2=(1−(1−E)αn)en(2.10)≤∏k=1n(1−(1−E)αk)e0≤exp⁡(−(1−E)∑k=1nαk)e0⟶0
    the last inequality is true because 1 − x ≤ exp ⁡ ( − x ) , ∀ x ≥ 0 1 − x ≤ exp ⁡ ( − x ) , ∀ x ≥ 0 1-x <= exp(-x),AA x >= 01-x \leq \exp (-x), \forall x \geq 01−x≤exp⁡(−x),∀x≥0, and ∑ α n = + ∞ ∑ α n = + ∞ sumalpha_(n)=+oo\sum \alpha_{n}=+\infty∑αn=+∞.
    Take u n = n / ( n + 1 ) → 1 u n = n / ( n + 1 ) → 1 u_(n)=n//(n+1)rarr1u_{n}=n /(n+1) \rightarrow 1un=n/(n+1)→1, and note that ε n → 0 ε n → 0 epsi_(n)rarr0\varepsilon_{n} \rightarrow 0εn→0 because
    (2.11) ε n = | n + 1 n + 2 − ( 1 − α n ) n n + 1 − α n n 2 ( n + 1 ) 2 | = α n n 2 + ( 2 α n + 1 ) n + 1 ( n + 1 ) 2 ( n + 2 ) (2.11) ε n = n + 1 n + 2 − 1 − α n n n + 1 − α n n 2 ( n + 1 ) 2 = α n n 2 + 2 α n + 1 n + 1 ( n + 1 ) 2 ( n + 2 ) {:(2.11)epsi_(n)=|(n+1)/(n+2)-(1-alpha_(n))(n)/(n+1)-alpha_(n)(n^(2))/((n+1)^(2))|=(alpha_(n)n^(2)+(2alpha_(n)+1)n+1)/((n+1)^(2)(n+2)):}\begin{equation*} \varepsilon_{n}=\left|\frac{n+1}{n+2}-\left(1-\alpha_{n}\right) \frac{n}{n+1}-\alpha_{n} \frac{n^{2}}{(n+1)^{2}}\right|=\frac{\alpha_{n} n^{2}+\left(2 \alpha_{n}+1\right) n+1}{(n+1)^{2}(n+2)} \tag{2.11} \end{equation*}(2.11)εn=|n+1n+2−(1−αn)nn+1−αnn2(n+1)2|=αnn2+(2αn+1)n+1(n+1)2(n+2)
    So the Mann iteration is not T T TTT-stable. Actually, by use of Theorem 2.1, one can easily obtain the non- T T TTT-stability of the other iteration, provided that the previous one is not stable.
    The following result takes in consideration the case in which no condition on { α n } α n {alpha_(n)}\left\{\alpha_{n}\right\}{αn} are imposed.
    Theorem 2.4. Let X X XXX be a normed space and T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X a map, and { α n } ⊂ ( 0 , 1 ) α n ⊂ ( 0 , 1 ) {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1){αn}⊂(0,1). If
    (2.12) lim n → ∞ ‖ v n − T v n ‖ = 0 , lim n → ∞ ‖ u n − T u n ‖ = 0 (2.12) lim n → ∞   v n − T v n = 0 , lim n → ∞   u n − T u n = 0 {:(2.12)lim_(n rarr oo)||v_(n)-Tv_(n)||=0","quadlim_(n rarr oo)||u_(n)-Tu_(n)||=0:}\begin{equation*} \lim _{n \rightarrow \infty}\left\|v_{n}-T v_{n}\right\|=0, \quad \lim _{n \rightarrow \infty}\left\|u_{n}-T u_{n}\right\|=0 \tag{2.12} \end{equation*}(2.12)limn→∞‖vn−Tvn‖=0,limn→∞‖un−Tun‖=0
    then the following are equivalent:
    (i) the Mann iteration is T T TTT-stable,
    (ii) the Krasnoselskij iteration is T T TTT-stable.
    Proof. We prove that (i) ⇒ ⇒ =>\Rightarrow⇒ (ii). Suppose lim n → ∞ δ n = 0 lim n → ∞   δ n = 0 lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0limn→∞δn=0, to note that,
    ε n = ‖ v n + 1 − ( 1 − α n ) v n − α n T v n ‖ = ‖ v n + 1 − v n + λ v n − λ v n + α n v n − λ T v n + λ T v n − α n T v n ‖ (2.13) ≤ ‖ v n + 1 − ( 1 − λ ) v n − λ T v n ‖ + | λ − α n | ‖ v n − T v n ‖ ≤ δ n + 2 ‖ v n − T v n ‖ ⟶ 0 as n → ∞ . ε n = v n + 1 − 1 − α n v n − α n T v n = v n + 1 − v n + λ v n − λ v n + α n v n − λ T v n + λ T v n − α n T v n (2.13) ≤ v n + 1 − ( 1 − λ ) v n − λ T v n + λ − α n v n − T v n ≤ δ n + 2 v n − T v n ⟶ 0  as  n → ∞ . {:[epsi_(n)=||v_(n+1)-(1-alpha_(n))v_(n)-alpha_(n)Tv_(n)||],[=||v_(n+1)-v_(n)+lambdav_(n)-lambdav_(n)+alpha_(n)v_(n)-lambda Tv_(n)+lambda Tv_(n)-alpha_(n)Tv_(n)||],[(2.13) <= ||v_(n+1)-(1-lambda)v_(n)-lambda Tv_(n)||+|lambda-alpha_(n)|||v_(n)-Tv_(n)||],[ <= delta_(n)+2||v_(n)-Tv_(n)||longrightarrow0" as "n rarr oo.]:}\begin{align*} \varepsilon_{n} & =\left\|v_{n+1}-\left(1-\alpha_{n}\right) v_{n}-\alpha_{n} T v_{n}\right\| \\ & =\left\|v_{n+1}-v_{n}+\lambda v_{n}-\lambda v_{n}+\alpha_{n} v_{n}-\lambda T v_{n}+\lambda T v_{n}-\alpha_{n} T v_{n}\right\| \\ & \leq\left\|v_{n+1}-(1-\lambda) v_{n}-\lambda T v_{n}\right\|+\left|\lambda-\alpha_{n}\right|\left\|v_{n}-T v_{n}\right\| \tag{2.13}\\ & \leq \delta_{n}+2\left\|v_{n}-T v_{n}\right\| \longrightarrow 0 \text { as } n \rightarrow \infty . \end{align*}εn=‖vn+1−(1−αn)vn−αnTvn‖=‖vn+1−vn+λvn−λvn+αnvn−λTvn+λTvn−αnTvn‖(2.13)≤‖vn+1−(1−λ)vn−λTvn‖+|λ−αn|‖vn−Tvn‖≤δn+2‖vn−Tvn‖⟶0 as n→∞.
    Condition (i) assures that if lim n → ∞ ε n = 0 lim n → ∞   ε n = 0 lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0limn→∞εn=0, then lim n → ∞ v n = x ∗ lim n → ∞   v n = x ∗ lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}limn→∞vn=x∗. Thus, for a { v n } v n {v_(n)}\left\{v_{n}\right\}{vn} satisfying
    (2.14) lim n → ∞ δ n = lim n → ∞ ‖ v n + 1 − ( 1 − λ ) v n − λ T v n ‖ = 0 , (2.14) lim n → ∞   δ n = lim n → ∞   v n + 1 − ( 1 − λ ) v n − λ T v n = 0 , {:(2.14)lim_(n rarr oo)delta_(n)=lim_(n rarr oo)||v_(n+1)-(1-lambda)v_(n)-lambda Tv_(n)||=0",":}\begin{equation*} \lim _{n \rightarrow \infty} \delta_{n}=\lim _{n \rightarrow \infty}\left\|v_{n+1}-(1-\lambda) v_{n}-\lambda T v_{n}\right\|=0, \tag{2.14} \end{equation*}(2.14)limn→∞δn=limn→∞‖vn+1−(1−λ)vn−λTvn‖=0,
    we have shown that lim n → ∞ v n = x ∗ lim n → ∞   v n = x ∗ lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}limn→∞vn=x∗.
    Conversely, we prove (ii) ⇒ (i). Suppose lim n → ∞ ε n = 0 lim n → ∞   ε n = 0 lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0limn→∞εn=0. Observe that
    δ n = ‖ u n + 1 − ( 1 − λ ) u n − λ T u n ‖ = ‖ u n + 1 − u n + λ u n − λ T u n + α n u n − α n u n − α n T u n + α n T u n ‖ (2.15) ≤ ‖ u n + 1 − ( 1 − α n ) u n − α n T u n ‖ + | λ − α n | ‖ u n − T u n ‖ ≤ ε n + 2 ‖ u n − T u n ‖ ⟶ 0 as n → ∞ . δ n = u n + 1 − ( 1 − λ ) u n − λ T u n = u n + 1 − u n + λ u n − λ T u n + α n u n − α n u n − α n T u n + α n T u n (2.15) ≤ u n + 1 − 1 − α n u n − α n T u n + λ − α n u n − T u n ≤ ε n + 2 u n − T u n ⟶ 0  as  n → ∞ . {:[delta_(n)=||u_(n+1)-(1-lambda)u_(n)-lambda Tu_(n)||],[=||u_(n+1)-u_(n)+lambdau_(n)-lambda Tu_(n)+alpha_(n)u_(n)-alpha_(n)u_(n)-alpha_(n)Tu_(n)+alpha_(n)Tu_(n)||],[(2.15) <= ||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||+|lambda-alpha_(n)|||u_(n)-Tu_(n)||],[ <= epsi_(n)+2||u_(n)-Tu_(n)||longrightarrow0quad" as "n rarr oo.]:}\begin{align*} \delta_{n} & =\left\|u_{n+1}-(1-\lambda) u_{n}-\lambda T u_{n}\right\| \\ & =\left\|u_{n+1}-u_{n}+\lambda u_{n}-\lambda T u_{n}+\alpha_{n} u_{n}-\alpha_{n} u_{n}-\alpha_{n} T u_{n}+\alpha_{n} T u_{n}\right\| \\ & \leq\left\|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right\|+\left|\lambda-\alpha_{n}\right|\left\|u_{n}-T u_{n}\right\| \tag{2.15}\\ & \leq \varepsilon_{n}+2\left\|u_{n}-T u_{n}\right\| \longrightarrow 0 \quad \text { as } n \rightarrow \infty . \end{align*}δn=‖un+1−(1−λ)un−λTun‖=‖un+1−un+λun−λTun+αnun−αnun−αnTun+αnTun‖(2.15)≤‖un+1−(1−αn)un−αnTun‖+|λ−αn|‖un−Tun‖≤εn+2‖un−Tun‖⟶0 as n→∞.
    Condition (ii) assures that if lim n → ∞ δ n = 0 lim n → ∞   δ n = 0 lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0limn→∞δn=0, then lim n → ∞ v n = x ∗ lim n → ∞   v n = x ∗ lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}limn→∞vn=x∗. Thus, for a { u n } u n {u_(n)}\left\{u_{n}\right\}{un} satisfying
    (2.16) lim n → ∞ ε n = lim n → ∞ ‖ u n + 1 − ( 1 − α n ) u n − α n T u n ‖ = 0 , (2.16) lim n → ∞   ε n = lim n → ∞   u n + 1 − 1 − α n u n − α n T u n = 0 , {:(2.16)lim_(n rarr oo)epsi_(n)=lim_(n rarr oo)||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||=0",":}\begin{equation*} \lim _{n \rightarrow \infty} \varepsilon_{n}=\lim _{n \rightarrow \infty}\left\|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right\|=0, \tag{2.16} \end{equation*}(2.16)limn→∞εn=limn→∞‖un+1−(1−αn)un−αnTun‖=0,
    we have shown that lim n → ∞ u n = x ∗ lim n → ∞   u n = x ∗ lim_(n rarr oo)u_(n)=x^(**)\lim _{n \rightarrow \infty} u_{n}=x^{*}limn→∞un=x∗.
    Remark 2.5. Let X X XXX be a normed space and T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X a map, { α n } ⊂ ( 0 , 1 ) α n ⊂ ( 0 , 1 ) {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1){αn}⊂(0,1) and lim n → ∞ ‖ v n − T v n ‖ = 0 , lim n → ∞ ‖ u n − T u n ‖ = 0 lim n → ∞   ‖ v n − T v n = 0 , lim n → ∞   u n − T u n ‖ = 0 lim_(n rarr oo)||v_(n)-Tv_(n)||=0,lim_(n rarr oo)||u_(n)-Tu_(n)||=0\lim _{n \rightarrow \infty} \| v_{n}- T v_{n}\left\|=0, \lim _{n \rightarrow \infty}\right\| u_{n}-T u_{n} \|=0limn→∞‖vn−Tvn‖=0,limn→∞‖un−Tun‖=0. If the Mann iteration is not T T TTT-stable, then the Krasnoselskij iteration is not T T TTT-stable, and conversely.
    Note that one can consider the usual conditions λ = 1 / 2 , lim α n = 0 λ = 1 / 2 , lim α n = 0 lambda=1//2,limalpha_(n)=0\lambda=1 / 2, \lim \alpha_{n}=0λ=1/2,limαn=0, and ∑ α n = ∞ ∑ α n = ∞ sumalpha_(n)=oo\sum \alpha_{n}=\infty∑αn=∞ in Theorem 2.4 and Remark 2.5.
    Example 2.6. Again, let T : [ 0 , 1 ) → [ 0 , 1 ) T : [ 0 , 1 ) → [ 0 , 1 ) T:[0,1)rarr[0,1)T:[0,1) \rightarrow[0,1)T:[0,1)→[0,1) be given by T x = x 2 T x = x 2 Tx=x^(2)T x=x^{2}Tx=x2, and λ = 1 / 2 , α n = 1 / n λ = 1 / 2 , α n = 1 / n lambda=1//2,alpha_(n)=1//n\lambda=1 / 2, \alpha_{n}=1 / nλ=1/2,αn=1/n. Set v n = u n = n / ( n + 1 ) v n = u n = n / ( n + 1 ) v_(n)=u_(n)=n//(n+1)v_{n}=u_{n}=n /(n+1)vn=un=n/(n+1), to note that lim n → ∞ u n = 1 lim n → ∞   u n = 1 lim_(n rarr oo)u_(n)=1\lim _{n \rightarrow \infty} u_{n}=1limn→∞un=1, and
    (2.17) lim n → ∞ ‖ v n − T v n ‖ = lim n → ∞ n ( n + 1 ) 2 = 0 (2.17) lim n → ∞   v n − T v n = lim n → ∞   n ( n + 1 ) 2 = 0 {:(2.17)lim_(n rarr oo)||v_(n)-Tv_(n)||=lim_(n rarr oo)(n)/((n+1)^(2))=0:}\begin{equation*} \lim _{n \rightarrow \infty}\left\|v_{n}-T v_{n}\right\|=\lim _{n \rightarrow \infty} \frac{n}{(n+1)^{2}}=0 \tag{2.17} \end{equation*}(2.17)limn→∞‖vn−Tvn‖=limn→∞n(n+1)2=0
    Hence, neither the Mann nor the Krasnoselskij iteration is T T TTT-stable, as we can see from Example 2.3.

    3. Further results

    Let q 0 ∈ X q 0 ∈ X q_(0)in Xq_{0} \in Xq0∈X be fixed, and let q n + 1 = T q n q n + 1 = T q n q_(n+1)=Tq_(n)q_{n+1}=T q_{n}qn+1=Tqn be the Picard-Banach iteration.
    Definition 3.1. The Picard iteration is said to be T T TTT-stable if and only if for every sequence { q n } ⊂ X q n ⊂ X {q_(n)}sub X\left\{q_{n}\right\} \subset X{qn}⊂X given,
    (3.1) lim n → ∞ Δ n = 0 ⟹ lim n → ∞ q n = x ∗ (3.1) lim n → ∞   Δ n = 0 ⟹ lim n → ∞   q n = x ∗ {:(3.1)lim_(n rarr oo)Delta_(n)=0Longrightarrowlim_(n rarr oo)q_(n)=x^(**):}\begin{equation*} \lim _{n \rightarrow \infty} \Delta_{n}=0 \Longrightarrow \lim _{n \rightarrow \infty} q_{n}=x^{*} \tag{3.1} \end{equation*}(3.1)limn→∞Δn=0⟹limn→∞qn=x∗
    where Δ n := ‖ q n + 1 − T q n ‖ Δ n := q n + 1 − T q n Delta_(n):=||q_(n+1)-Tq_(n)||\Delta_{n}:=\left\|q_{n+1}-T q_{n}\right\|Δn:=‖qn+1−Tqn‖.
    In [6], the equivalence between the T T TTT-stabilities of Picard-Banach iteration and Mann iteration is given, that is, the following holds.
    Theorem 3.2 [6]. Let X X XXX be a normed space and T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X a map. If
    (3.2) lim n → ∞ ‖ q n − T q n ‖ = 0 , lim n → ∞ ‖ u n − T u n ‖ = 0 (3.2) lim n → ∞   q n − T q n = 0 , lim n → ∞   u n − T u n = 0 {:(3.2)lim_(n rarr oo)||q_(n)-Tq_(n)||=0","quadlim_(n rarr oo)||u_(n)-Tu_(n)||=0:}\begin{equation*} \lim _{n \rightarrow \infty}\left\|q_{n}-T q_{n}\right\|=0, \quad \lim _{n \rightarrow \infty}\left\|u_{n}-T u_{n}\right\|=0 \tag{3.2} \end{equation*}(3.2)limn→∞‖qn−Tqn‖=0,limn→∞‖un−Tun‖=0
    then the following are equivalent:
    (i) for all { α n } ⊂ ( 0 , 1 ) α n ⊂ ( 0 , 1 ) {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1){αn}⊂(0,1), the Mann iteration is T T TTT - stable,
    (ii) the Picard iteration is T T TTT-stable.
    Theorems 2.4 and 3.2 lead to the following conclusion.
    Corollary 3.3. Let X X XXX be a normed space and T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X a map. If
    (3.3) lim n → ∞ ‖ q n − T q n ‖ = 0 , lim n → ∞ ‖ v n − T v n ‖ = 0 , lim n → ∞ ‖ u n − T u n ‖ = 0 , (3.3) lim n → ∞   q n − T q n = 0 , lim n → ∞   v n − T v n = 0 , lim n → ∞   u n − T u n = 0 , {:(3.3)lim_(n rarr oo)||q_(n)-Tq_(n)||=0","quadlim_(n rarr oo)||v_(n)-Tv_(n)||=0","quadlim_(n rarr oo)||u_(n)-Tu_(n)||=0",":}\begin{equation*} \lim _{n \rightarrow \infty}\left\|q_{n}-T q_{n}\right\|=0, \quad \lim _{n \rightarrow \infty}\left\|v_{n}-T v_{n}\right\|=0, \quad \lim _{n \rightarrow \infty}\left\|u_{n}-T u_{n}\right\|=0, \tag{3.3} \end{equation*}(3.3)limn→∞‖qn−Tqn‖=0,limn→∞‖vn−Tvn‖=0,limn→∞‖un−Tun‖=0,
    then the following are equivalent:
    (i) for all { α n } ⊂ ( 0 , 1 ) α n ⊂ ( 0 , 1 ) {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1){αn}⊂(0,1), the Mann iteration is T T TTT-stable,
    (ii) the Picard-Banach iteration is T T TTT-stable,
    (iii) the Krasnoselskij iteration is T T TTT-stable.
    Remark 3.4. Let X X XXX be a normed space and T : X → X T : X → X T:X rarr XT: X \rightarrow XT:X→X a map, { α n } ⊂ ( 0 , 1 ) α n ⊂ ( 0 , 1 ) {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1){αn}⊂(0,1) and lim n → ∞ ‖ q n − T q n ‖ = 0 , lim n → ∞ ‖ v n − T v n ‖ = 0 , lim n → ∞ ‖ u n − T u n ‖ = 0 lim n → ∞   q n − T q n = 0 , lim n → ∞   v n − T v n = 0 , lim n → ∞   u n − T u n = 0 lim_(n rarr oo)||q_(n)-Tq_(n)||=0,lim_(n rarr oo)||v_(n)-Tv_(n)||=0,lim_(n rarr oo)||u_(n)-Tu_(n)||=0\lim _{n \rightarrow \infty}\left\|q_{n}-T q_{n}\right\|=0, \lim _{n \rightarrow \infty}\left\|v_{n}-T v_{n}\right\|=0, \lim _{n \rightarrow \infty}\left\|u_{n}-T u_{n}\right\|=0limn→∞‖qn−Tqn‖=0,limn→∞‖vn−Tvn‖=0,limn→∞‖un−Tun‖=0. If the Mann or Krasnoselskij iteration is not T T TTT-stable, then the Picard-Banach iteration is not T T TTT-stable, and conversely.
    Example 3.5. To see that the Picard-Banach iteration is also not T T TTT-stable, consider T T TTT : [ 0 , 1 ) → [ 0 , 1 ) [ 0 , 1 ) → [ 0 , 1 ) [0,1)rarr[0,1)[0,1) \rightarrow[0,1)[0,1)→[0,1), by T x = x 2 T x = x 2 Tx=x^(2)T x=x^{2}Tx=x2.
    Indeed, setting q n = n / ( n + 1 ) q n = n / ( n + 1 ) q_(n)=n//(n+1)q_{n}=n /(n+1)qn=n/(n+1), we have
    lim n → ∞ q n = lim n → ∞ n n + 1 = 1 (3.4) lim n → ∞ | n n + 1 − ( n n + 1 ) 2 | = n ( n + 1 ) 2 = 0 lim n → ∞   q n = lim n → ∞   n n + 1 = 1 (3.4) lim n → ∞   n n + 1 − n n + 1 2 = n ( n + 1 ) 2 = 0 {:[lim_(n rarr oo)q_(n)=lim_(n rarr oo)(n)/(n+1)=1],[(3.4)lim_(n rarr oo)|(n)/(n+1)-((n)/(n+1))^(2)|=(n)/((n+1)^(2))=0]:}\begin{gather*} \lim _{n \rightarrow \infty} q_{n}=\lim _{n \rightarrow \infty} \frac{n}{n+1}=1 \\ \lim _{n \rightarrow \infty}\left|\frac{n}{n+1}-\left(\frac{n}{n+1}\right)^{2}\right|=\frac{n}{(n+1)^{2}}=0 \tag{3.4} \end{gather*}limn→∞qn=limn→∞nn+1=1(3.4)limn→∞|nn+1−(nn+1)2|=n(n+1)2=0

    Acknowledgment

    The author is indebted to referee for carefully reading the paper and for making useful suggestions.

    References

    [1] A. M. Harder and T. L. Hicks, "Stability results for fixed point iteration procedures," Mathematica Japonica, vol. 33, no. 5, pp. 693-706, 1988.
    [2] W. R. Mann, "Mean value methods in iteration," Proceedings of the American Mathematical Society, vol. 4, no. 3, pp. 506-510, 1953.
    [3] S. Ishikawa, "Fixed points by a new iteration method," Proceedings of the American Mathematical Society, vol. 44, no. 1, pp. 147-150, 1974.
    [4] M. A. Krasnosel'skiĭ, "Two remarks on the method of successive approximations," Uspekhi Matematicheskikh Nauk, vol. 10, no. 1(63), pp. 123-127, 1955.
    [5] B. E. Rhoades and Ş. M. Şoltuz, "The equivalence between the T-stabilities of Mann and Ishikawa iterations," Journal of Mathematical Analysis and Applications, vol. 318, no. 2, pp. 472475, 2006.
    [6] Ş. M. Şoltuz, "The equivalence between the T T TTT-stabilities of Picard-Banach and Mann-Ishikawa iterations," to appear in Applied Mathematics E-Notes.
    Ştefan M. Şoltuz: Departamento de Matematicas, Universidad de Los Andes, Carrera 1 no. 18A-10, Bogota, Colombia
    Current address: Tiberiu Popoviciu Institute of Numerical Analysis, 400110 Cluj-Napoca, Romania
    Email address: smsoltuz@gmail.com

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