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.
Authors
S.M. Soltuz
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
Keywords
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Paper coordinates
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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Research Article
The Equivalence between TT-Stabilities of The Krasnoselskij and The Mann Iterations
1. Introduction
Let XX be a normed space and TT a selfmap of XX. Let x_(0)x_{0} be a point of XX, and assume that x_(n+1)=f(T,x_(n))x_{n+1}=f\left(T, x_{n}\right) is an iteration procedure, involving TT, which yields a sequence {x_(n)}\left\{x_{n}\right\} of points from XX. Suppose {x_(n)}\left\{x_{n}\right\} converges to a fixed point x^(**)x^{*} of TT. Let {xi_(n)}\left\{\xi_{n}\right\} be an arbitrary sequence in XX, and set epsilon_(n)=||xi_(n+1)-f(T,xi_(n))||\epsilon_{n}=\left\|\xi_{n+1}-f\left(T, \xi_{n}\right)\right\| for all n inNn \in \mathbb{N}.
Definition 1.1 [1]. If (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), then the iteration procedure x_(n+1)=f(T,x_(n))x_{n+1}=f\left(T, x_{n}\right) is said to be TT-stable with respect to TT.
Remark 1.2 [1]. In practice, such a sequence {xi_(n)}\left\{\xi_{n}\right\} could arise in the following way. Let x_(0)x_{0} be a point in XX. Set x_(n+1)=f(T,x_(n))x_{n+1}=f\left(T, x_{n}\right). Let xi_(0)=x_(0)\xi_{0}=x_{0}. Now x_(1)=f(T,x_(0))x_{1}=f\left(T, x_{0}\right). Because of rounding or discretization in the function TT, a new value xi_(1)\xi_{1} approximately equal to x_(1)x_{1} might be obtained instead of the true value of f(T,x_(0))f\left(T, x_{0}\right). Then to approximate x_(2)x_{2}, the value f(T,xi_(1))f\left(T, \xi_{1}\right) is computed to yield xi_(2)\xi_{2}, an approximation of f(T,xi_(1))f\left(T, \xi_{1}\right). This computation is continued to obtain {xi_(n)}\left\{\xi_{n}\right\} an approximate sequence of {x_(n)}\left\{x_{n}\right\}.
Let XX be a normed space, DD a nonempty, convex subset of XX, and TT a selfmap of DD, let p_(0)=e_(0)in Dp_{0}=e_{0} \in D. The Mann iteration (see [2]) is defined by
{:(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*}
where {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1). The Ishikawa iteration is defined (see [3]) by
{:[(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*}
where {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1). The Krasnoselskij iteration (see [4]) is defined by
{:(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*}
where lambda in(0,1)\lambda \in(0,1). Recently, the equivalence between the TT-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 TT-stabilities of the Krasnoselskij and the Mann iterations. Next, {u_(n)},{v_(n)}sub X\left\{u_{n}\right\},\left\{v_{n}\right\} \subset X are arbitrary.
Definition 1.3.
(i) The Mann iteration (1.1) is said to be TT-stable if and only if for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) and for every sequence {u_(n)}sub X\left\{u_{n}\right\} \subset 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*}
where 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\|.
(ii) The Krasnoselskij iteration (1.3) is said to be TT-stable if and only if for all lambda in(0,1)\lambda \in (0,1), and for every sequence {v_(n)}sub X\left\{v_{n}\right\} \subset X,
(ii) The Krasnoselskij iteration (1.3) is said to be TT-stable if and only if for all lambda in(0,1)\lambda \in (0,1), and for every sequence {v_(n)}sub X\left\{v_{n}\right\} \subset 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*}
where 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\|.
2. Main results
Theorem 2.1. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map with bounded range and {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfy lim_(n rarr oo)alpha_(n)=lambda,lambda in(0,1)\lim _{n \rightarrow \infty} \alpha_{n}=\lambda, \lambda \in(0,1). Then the following are equivalent:
(i) the Mann iteration is TT-stable,
(ii) the Krasnoselskij iteration is TT-stable.
(i) the Mann iteration is TT-stable,
(ii) the Krasnoselskij iteration is TT-stable.
Proof. We prove that (i) ⇒ (ii). If lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0, then {v_(n)}\left\{v_{n}\right\} is bounded. Set
{:(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*}
Observe that ||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}. Set M:=M_(1)+1//lambdaM:=M_{1}+1 / \lambda. Suppose that ||v_(n)|| <= M\left\|v_{n}\right\| \leq M to prove that ||v_(n+1)|| <= M\left\|v_{n+1}\right\| \leq M. Remark that
{:[||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*}
Suppose that lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0 to note that
{:[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*}
Condition (i) assures that if lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0, then lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}. Thus, for a {v_(n)}\left\{v_{n}\right\} satisfying
{:(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*}
we have shown that lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}.
Conversely, we prove (ii) ⇒ (i). First, we prove that {u_(n)}\left\{u_{n}\right\} is bounded. Since lim_(n rarr oo)alpha_(n)=lambda\lim _{n \rightarrow \infty} \alpha_{n}= \lambda, for beta in(0,1)\beta \in(0,1) given, there exists n_(0)in Nn_{0} \in N, such that 1-alpha_(n) <= beta1-\alpha_{n} \leq \beta, for all n >= n_(0)n \geq n_{0}. Set 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\} and M:=n_(0)+1+beta//(1-beta)+M_(1)M:=n_{0}+1+\beta /(1-\beta)+M_{1} to obtain
Conversely, we prove (ii) ⇒ (i). First, we prove that {u_(n)}\left\{u_{n}\right\} is bounded. Since lim_(n rarr oo)alpha_(n)=lambda\lim _{n \rightarrow \infty} \alpha_{n}= \lambda, for beta in(0,1)\beta \in(0,1) given, there exists n_(0)in Nn_{0} \in N, such that 1-alpha_(n) <= beta1-\alpha_{n} \leq \beta, for all n >= n_(0)n \geq n_{0}. Set 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\} and M:=n_(0)+1+beta//(1-beta)+M_(1)M:=n_{0}+1+\beta /(1-\beta)+M_{1} to obtain
{:[||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*}
Suppose lim_(n rarr oo)E_(n)=0\lim _{n \rightarrow \infty} \mathcal{E}_{n}=0. Observe that
{:[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*}
Condition (ii) assures that if lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0, then lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}. Thus, for a {u_(n)}\left\{u_{n}\right\} satisfying
{:(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*}
we have shown that lim_(n rarr oo)u_(n)=x^(**)\lim _{n \rightarrow \infty} u_{n}=x^{*}.
Remark 2.2. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map with bounded range and {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfy lim_(n rarr oo)alpha_(n)=lambda,lambda in(0,1)\lim _{n \rightarrow \infty} \alpha_{n}=\lambda, \lambda \in(0,1). If the Mann iteration is not TT-stable, then the Krasnoselskij iteration is not TT-stable, and conversely.
Remark 2.2. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map with bounded range and {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfy lim_(n rarr oo)alpha_(n)=lambda,lambda in(0,1)\lim _{n \rightarrow \infty} \alpha_{n}=\lambda, \lambda \in(0,1). If the Mann iteration is not TT-stable, then the Krasnoselskij iteration is not TT-stable, and conversely.
Example 2.3. Let T:[0,1)rarr[0,1)T:[0,1) \rightarrow[0,1) be given by Tx=x^(2)T x=x^{2}, and lambda=1//2\lambda=1 / 2. Then the Krasnoselskij iteration converges to the unique fixed point x^(**)=0x^{*}=0, and it is not TT-stable.
The Krasnoselskij iteration converges because, supposing F:=s u p_(n)p_(n) < 1F:=\sup _{n} p_{n}<1, the sequence p_(n)rarr0p_{n} \rightarrow 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)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*}
set v_(n)=n//(n+1)v_{n}=n /(n+1) and note that v_(n)v_{n} does not converge to zero, while delta_(n)\delta_{n} does:
{:(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*}
The Mann iteration also converges because (supposing E:=s u p_(n)e_(n) < 1E:=\sup _{n} e_{n}<1 ) one has
{:[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*}
the last inequality is true because 1-x <= exp(-x),AA x >= 01-x \leq \exp (-x), \forall x \geq 0, and sumalpha_(n)=+oo\sum \alpha_{n}=+\infty.
Take u_(n)=n//(n+1)rarr1u_{n}=n /(n+1) \rightarrow 1, and note that epsi_(n)rarr0\varepsilon_{n} \rightarrow 0 because
Take u_(n)=n//(n+1)rarr1u_{n}=n /(n+1) \rightarrow 1, and note that epsi_(n)rarr0\varepsilon_{n} \rightarrow 0 because
{:(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*}
So the Mann iteration is not TT-stable. Actually, by use of Theorem 2.1, one can easily obtain the non- TT-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 {alpha_(n)}\left\{\alpha_{n}\right\} are imposed.
Theorem 2.4. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map, and {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1). If
{:(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*}
then the following are equivalent:
(i) the Mann iteration is TT-stable,
(ii) the Krasnoselskij iteration is TT-stable.
(i) the Mann iteration is TT-stable,
(ii) the Krasnoselskij iteration is TT-stable.
Proof. We prove that (i) =>\Rightarrow (ii). Suppose lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0, to note that,
{:[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*}
Condition (i) assures that if lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0, then lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}. Thus, for a {v_(n)}\left\{v_{n}\right\} satisfying
{:(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*}
we have shown that lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}.
Conversely, we prove (ii) ⇒ (i). Suppose lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0. Observe that
Conversely, we prove (ii) ⇒ (i). Suppose lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0. Observe that
{:[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*}
Condition (ii) assures that if lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0, then lim_(n rarr oo)v_(n)=x^(**)\lim _{n \rightarrow \infty} v_{n}=x^{*}. Thus, for a {u_(n)}\left\{u_{n}\right\} satisfying
{:(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*}
we have shown that lim_(n rarr oo)u_(n)=x^(**)\lim _{n \rightarrow \infty} u_{n}=x^{*}.
Remark 2.5. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map, {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) and 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} \|=0. If the Mann iteration is not TT-stable, then the Krasnoselskij iteration is not TT-stable, and conversely.
Remark 2.5. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map, {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) and 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} \|=0. If the Mann iteration is not TT-stable, then the Krasnoselskij iteration is not TT-stable, and conversely.
Note that one can consider the usual conditions lambda=1//2,limalpha_(n)=0\lambda=1 / 2, \lim \alpha_{n}=0, and sumalpha_(n)=oo\sum \alpha_{n}=\infty in Theorem 2.4 and Remark 2.5.
Example 2.6. Again, let T:[0,1)rarr[0,1)T:[0,1) \rightarrow[0,1) be given by Tx=x^(2)T x=x^{2}, and lambda=1//2,alpha_(n)=1//n\lambda=1 / 2, \alpha_{n}=1 / n. Set v_(n)=u_(n)=n//(n+1)v_{n}=u_{n}=n /(n+1), to note that lim_(n rarr oo)u_(n)=1\lim _{n \rightarrow \infty} u_{n}=1, and
{:(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*}
Hence, neither the Mann nor the Krasnoselskij iteration is TT-stable, as we can see from Example 2.3.
3. Further results
Let q_(0)in Xq_{0} \in X be fixed, and let q_(n+1)=Tq_(n)q_{n+1}=T q_{n} be the Picard-Banach iteration.
Definition 3.1. The Picard iteration is said to be TT-stable if and only if for every sequence {q_(n)}sub X\left\{q_{n}\right\} \subset X given,
Definition 3.1. The Picard iteration is said to be TT-stable if and only if for every sequence {q_(n)}sub X\left\{q_{n}\right\} \subset X given,
{:(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*}
where Delta_(n):=||q_(n+1)-Tq_(n)||\Delta_{n}:=\left\|q_{n+1}-T q_{n}\right\|.
In [6], the equivalence between the TT-stabilities of Picard-Banach iteration and Mann iteration is given, that is, the following holds.
In [6], the equivalence between the TT-stabilities of Picard-Banach iteration and Mann iteration is given, that is, the following holds.
Theorem 3.2 [6]. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map. If
{:(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*}
then the following are equivalent:
(i) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1), the Mann iteration is TT - stable,
(ii) the Picard iteration is TT-stable.
(i) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1), the Mann iteration is TT - stable,
(ii) the Picard iteration is TT-stable.
Theorems 2.4 and 3.2 lead to the following conclusion.
Corollary 3.3. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map. If
Corollary 3.3. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map. If
{:(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*}
then the following are equivalent:
(i) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1), the Mann iteration is TT-stable,
(ii) the Picard-Banach iteration is TT-stable,
(iii) the Krasnoselskij iteration is TT-stable.
(i) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1), the Mann iteration is TT-stable,
(ii) the Picard-Banach iteration is TT-stable,
(iii) the Krasnoselskij iteration is TT-stable.
Remark 3.4. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map, {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) and 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\|=0. If the Mann or Krasnoselskij iteration is not TT-stable, then the Picard-Banach iteration is not TT-stable, and conversely.
Example 3.5. To see that the Picard-Banach iteration is also not TT-stable, consider TT : [0,1)rarr[0,1)[0,1) \rightarrow[0,1), by Tx=x^(2)T x=x^{2}.
Indeed, setting q_(n)=n//(n+1)q_{n}=n /(n+1), we have
{:[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*}
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 TT-stabilities of Picard-Banach and Mann-Ishikawa iterations," to appear in Applied Mathematics E-Notes.
[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 TT-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
Current address: Tiberiu Popoviciu Institute of Numerical Analysis, 400110 Cluj-Napoca, Romania
Email address: smsoltuz@gmail.com