Abstract
We show that T-stability of Picard-Banach and Mann-Ishikawa iterations are equivalent.
Authors
Stefan M. Soltuz
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Keywords
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Paper coordinates
S.M. Şoltuz, The equivalence between the T-stabilities of Picard-Banach and Mann-Ishikawa iterations. Applied Math. E-Notes, 8 (2008), 109-114.
About this paper
Journal
Applied Mathematics E-Notes
Publisher Name
DOI
Print ISSN
1607-2510
Online ISSN
1687-0425
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[1] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1974), 147–150.
[2] A. M. Harder and T. Hicks, Stability results for fixed point iteration procedures, Math. Japonica, 33 (1988), 693–706.
[3] W. R. Mann, Mean value in iteration, Proc. Amer. Math. Soc., 4 (1953), 506–510.
[4] M. O. Osilike, Stability of the Mann and Ishikawa iteration procedures for φ-strong pseudocontractions and nonlinear equations of the φ-strongly accretive type, J.
Math. Anal. Appl., 227 (1998), 319–334.
[5] B. E. Rhoades and S.M. Soltuz, On the equivalence of Mann and Ishikawa iteration methods, Int. J. Math. Math. Sci., 2003(2003), 451–459.
[6] B. E. Rhoades and S. M. Soltuz, The equivalence between the T -stabilities of Mann and Ishikawa iterations, J. Math. Anal. Appl., 318(2006), 472-475.
Paper (preprint) in HTML form
The Equivalence Between The TT-Stabilities Of Picard-Banach And Mann-Ishikawa Iterations*
Received 5 February 2007
Abstract
We show that TT-stability of Picard-Banach and Mann-Ishikawa iterations are equivalent.
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 point 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. [2] If ((lim_(n rarr oo)epsilon_(n)=0)=>(lim_(n rarr oo)xi_(n)=p))\left(\left(\lim _{n \rightarrow \infty} \epsilon_{n}=0\right) \Rightarrow\left(\lim _{n \rightarrow \infty} \xi_{n}=p\right)\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] 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 xi_(2)\xi_{2}, the value f(T,xi_(1))f\left(T, \xi_{1}\right) is computed to yields 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\}.
Consider e_(0)=s_(0)=t_(0)=g_(0)=h_(0)e_{0}=s_{0}=t_{0}=g_{0}=h_{0}. The Picard-Banach iteration is given by
{:(1)b_(n+1)=Tb_(n):}\begin{equation*}
b_{n+1}=T b_{n} \tag{1}
\end{equation*}
The two most popular iteration procedures for obtaining fixed points of TT, when the Banach principle fails, are Mann iteration [3], defined by
{:(2)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{2}
\end{equation*}
and Ishikawa iteration [1], defined by
{:[(3)s_(n+1)=(1-alpha_(n))s_(n)+alpha_(n)Tt_(n)],[t_(n)=(1-beta_(n))s_(n)+beta_(n)Ts_(n)]:}\begin{align*}
s_{n+1} & =\left(1-\alpha_{n}\right) s_{n}+\alpha_{n} T t_{n} \tag{3}\\
t_{n} & =\left(1-\beta_{n}\right) s_{n}+\beta_{n} T s_{n}
\end{align*}
We have {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) and {alpha_(n)}\left\{\alpha_{n}\right\} usually satisfies
{:(4)lim_(n rarr oo)alpha_(n)=0","sum_(n=1)^(oo)alpha_(n)=oo:}\begin{equation*}
\lim _{n \rightarrow \infty} \alpha_{n}=0, \sum_{n=1}^{\infty} \alpha_{n}=\infty \tag{4}
\end{equation*}
Recently, the equivalence between the TT-stabilities of Mann and Ishikawa iterations was shown in [6]. In this note we shall prove the equivalence between TT-stabilities of (1), (2) and (3).
2 The Equivalence between TT-Stabilities
Let XX be a normed space and T:X rarr XT: X \rightarrow X a map. Let {u_(n)},{p_(n)},{x_(n)},{y_(n)}sub X\left\{u_{n}\right\},\left\{p_{n}\right\},\left\{x_{n}\right\},\left\{y_{n}\right\} \subset X be such that u_(0)=p_(0)=x_(0)=y_(0)u_{0}=p_{0}=x_{0}=y_{0}, and consider
{:[epsi_(n):=||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||],[delta_(n):=||p_(n+1)-Tp_(n)||]:}\begin{aligned}
\varepsilon_{n}:=\left\|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right\| \\
\delta_{n}:=\left\|p_{n+1}-T p_{n}\right\|
\end{aligned}
For {beta_(n)}sub[0,1)\left\{\beta_{n}\right\} \subset[0,1), we consider y_(n)=(1-beta_(n))x_(n)+beta_(n)Tx_(n)y_{n}=\left(1-\beta_{n}\right) x_{n}+\beta_{n} T x_{n}, and
xi_(n):=||x_(n+1)-(1-alpha_(n))x_(n)-alpha_(n)Ty_(n)||\xi_{n}:=\left\|x_{n+1}-\left(1-\alpha_{n}\right) x_{n}-\alpha_{n} T y_{n}\right\|
DEFINITION 2. Definition 1 gives:
(i) The Ishikawa iteration (3), is said to be TT-stable if and only if for all {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1),AA{x_(n)}sub X\left\{\alpha_{n}\right\} \subset (0,1),\left\{\beta_{n}\right\} \subset[0,1), \forall\left\{x_{n}\right\} \subset X given, we have
(i) The Ishikawa iteration (3), is said to be TT-stable if and only if for all {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1),AA{x_(n)}sub X\left\{\alpha_{n}\right\} \subset (0,1),\left\{\beta_{n}\right\} \subset[0,1), \forall\left\{x_{n}\right\} \subset X given, we have
lim_(n rarr oo)xi_(n)=lim_(n rarr oo)||x_(n+1)-(1-alpha_(n))x_(n)-alpha_(n)Ty_(n)||=0=>lim_(n rarr oo)x_(n)=x^(**)\lim _{n \rightarrow \infty} \xi_{n}=\lim _{n \rightarrow \infty}\left\|x_{n+1}-\left(1-\alpha_{n}\right) x_{n}-\alpha_{n} T y_{n}\right\|=0 \Rightarrow \lim _{n \rightarrow \infty} x_{n}=x^{*}
The Mann iteration is said to be TT-stable if and only if for all {alpha_(n)}sub(0,1),AA{u_(n)}sub X\left\{\alpha_{n}\right\} \subset(0,1), \forall\left\{u_{n}\right\} \subset X given, we have
lim_(n rarr oo)epsi_(n)=lim_(n rarr oo)||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||=0=>lim_(n rarr oo)u_(n)=x^(**)\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 \Rightarrow \lim _{n \rightarrow \infty} u_{n}=x^{*}
(ii) The Picard iteration is said to be TT-stable if and only if for all {p_(n)}sub X\left\{p_{n}\right\} \subset X given, we have
lim_(n rarr oo)delta_(n)=lim_(n rarr oo)||p_(n+1)-Tp_(n)||=0=>lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty} \delta_{n}=\lim _{n \rightarrow \infty}\left\|p_{n+1}-T p_{n}\right\|=0 \Rightarrow \lim _{n \rightarrow \infty} p_{n}=x^{*}
It is obvious that for alpha_(n):=0,AA n inN,beta_(n):=0,AA n inN\alpha_{n}:=0, \forall n \in \mathbb{N}, \beta_{n}:=0, \forall n \in \mathbb{N}, one obtains xi_(n)=epsi_(n)=delta_(n)\xi_{n}=\varepsilon_{n}=\delta_{n}.
THEOREM 1. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map. If
THEOREM 1. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map. If
{:(5)lim_(n rarr oo)||p_(n)-Tp_(n)||=0" and "lim_(n rarr oo)||u_(n)-Tu_(n)||=0:}\begin{equation*}
\lim _{n \rightarrow \infty}\left\|p_{n}-T p_{n}\right\|=0 \text { and } \lim _{n \rightarrow \infty}\left\|u_{n}-T u_{n}\right\|=0 \tag{5}
\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.
PROOF. (i)=>(ii)(i) \Rightarrow(i i). Take lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0. Observe that
{:[epsi_(n)=||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||],[ <= ||u_(n+1)-Tu_(n)||+(1-alpha_(n))||u_(n+1)-u_(n)||+(1-alpha_(n))||u_(n+1)-Tu_(n)||],[ <= (2-alpha_(n))||u_(n+1)-Tu_(n)||+(1-alpha_(n))||u_(n+1)-u_(n)||],[ <= (2-alpha_(n))||u_(n+1)-Tu_(n)||+(1-alpha_(n))(||u_(n+1)-Tu_(n)||+||u_(n)-Tu_(n)||)],[=(3-2alpha_(n))||u_(n+1)-Tu_(n)||+(1-alpha_(n))||u_(n)-Tu_(n)||],[=(3-2alpha_(n))delta_(n)+(1-alpha_(n))||u_(n)-Tu_(n)||],[ rarr0]:}\begin{aligned}
\varepsilon_{n} & =\left\|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right\| \\
& \leq\left\|u_{n+1}-T u_{n}\right\|+\left(1-\alpha_{n}\right)\left\|u_{n+1}-u_{n}\right\|+\left(1-\alpha_{n}\right)\left\|u_{n+1}-T u_{n}\right\| \\
& \leq\left(2-\alpha_{n}\right)\left\|u_{n+1}-T u_{n}\right\|+\left(1-\alpha_{n}\right)\left\|u_{n+1}-u_{n}\right\| \\
& \leq\left(2-\alpha_{n}\right)\left\|u_{n+1}-T u_{n}\right\|+\left(1-\alpha_{n}\right)\left(\left\|u_{n+1}-T u_{n}\right\|+\left\|u_{n}-T u_{n}\right\|\right) \\
& =\left(3-2 \alpha_{n}\right)\left\|u_{n+1}-T u_{n}\right\|+\left(1-\alpha_{n}\right)\left\|u_{n}-T u_{n}\right\| \\
& =\left(3-2 \alpha_{n}\right) \delta_{n}+\left(1-\alpha_{n}\right)\left\|u_{n}-T u_{n}\right\| \\
& \rightarrow 0
\end{aligned}
as n rarr oon \rightarrow \infty. We know from ( ii ) that if lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0, then lim_(n rarr oo)u_(n)=x^(**)\lim _{n \rightarrow \infty} u_{n}=x^{*}, thus we have shown that if lim_(n rarr oo)delta_(n)=lim_(n rarr oo)||u_(n+1)-Tu_(n)||=0\lim _{n \rightarrow \infty} \delta_{n}=\lim _{n \rightarrow \infty}\left\|u_{n+1}-T u_{n}\right\|=0, then lim_(n rarr oo)u_(n)=x^(**)\lim _{n \rightarrow \infty} u_{n}=x^{*}.
For (ii)=>(i)(i i) \Rightarrow(i), take lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0. Observe that
{:[delta_(n)=||p_(n+1)-Tp_(n)||],[ <= ||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Tp_(n)||+(1-alpha_(n))||p_(n)-Tp_(n)||],[ <= epsi_(n)+(1-alpha_(n))||p_(n)-Tp_(n)||],[ rarr0]:}\begin{aligned}
\delta_{n} & =\left\|p_{n+1}-T p_{n}\right\| \\
& \leq\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T p_{n}\right\|+\left(1-\alpha_{n}\right)\left\|p_{n}-T p_{n}\right\| \\
& \leq \varepsilon_{n}+\left(1-\alpha_{n}\right)\left\|p_{n}-T p_{n}\right\| \\
& \rightarrow 0
\end{aligned}
as n rarr oon \rightarrow \infty. We know from ( iii i ) that if lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0, then lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty} p_{n}=x^{*}, thus we have shown that if lim_(n rarr oo)epsi_(n)=lim_(n rarr oo)||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Tp_(n)||=0\lim _{n \rightarrow \infty} \varepsilon_{n}=\lim _{n \rightarrow \infty}\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T p_{n}\right\|=0, then lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty} p_{n}=x^{*}.
REMARK 2. Note that no boundedness condition is needed in the above result. Note that lim_(n rarr oo)||u_(n)-Tu_(n)||=0\lim _{n \rightarrow \infty}\left\|u_{n}-T u_{n}\right\|=0 is used in order to prove that lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0, hence can not be avoided. Analogously, lim_(n rarr oo)||p_(n)-Tp_(n)||=0\lim _{n \rightarrow \infty}\left\|p_{n}-T p_{n}\right\|=0 is used in order to prove that lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0, hence can not be avoided.
THEOREM 2. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map with bounded range. If
lim_(n rarr oo)||p_(n)-Tp_(n)||=0" and "lim_(n rarr oo)||x_(n)-Tx_(n)||=0\lim _{n \rightarrow \infty}\left\|p_{n}-T p_{n}\right\|=0 \text { and } \lim _{n \rightarrow \infty}\left\|x_{n}-T x_{n}\right\|=0
then the following are equivalent:
(i) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) and {beta_(n)}sub[0,1)\left\{\beta_{n}\right\} \subset[0,1), satisfying (4), the Ishikawa 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) and {beta_(n)}sub[0,1)\left\{\beta_{n}\right\} \subset[0,1), satisfying (4), the Ishikawa iteration is TT-stable,
(ii) the Picard iteration is TT-stable.
PROOF. Let
M:=max{s u p_(x in X){||T(x)||},||x_(0)||}.M:=\max \left\{\sup _{x \in X}\{\|T(x)\|\},\left\|x_{0}\right\|\right\} .
Since TT has bounded range, we have M < ooM<\infty.
We shall prove that (i)=>(ii)(i) \Rightarrow(i i). Take lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0. Observe that
{:[xi_(n)=||x_(n+1)-(1-alpha_(n))x_(n)-alpha_(n)Ty_(n)||],[ <= ||x_(n+1)-Tx_(n)||+||(1-alpha_(n))x_(n)-alpha_(n)Ty_(n)+Tx_(n)||],[=||x_(n+1)-Tx_(n)||+||(1-alpha_(n))x_(n)-alpha_(n)Ty_(n)+Tx_(n)-alpha_(n)Tx_(n)+alpha_(n)Tx_(n)||],[ <= ||x_(n+1)-Tx_(n)||+(1-alpha_(n))||x_(n)-Tx_(n)||+alpha_(n)||Tx_(n)-Ty_(n)||],[=delta_(n)+(1-alpha_(n))||x_(n)-Tx_(n)||+2alpha_(n)M],[ rarr0]:}\begin{aligned}
\xi_{n} & =\left\|x_{n+1}-\left(1-\alpha_{n}\right) x_{n}-\alpha_{n} T y_{n}\right\| \\
& \leq\left\|x_{n+1}-T x_{n}\right\|+\left\|\left(1-\alpha_{n}\right) x_{n}-\alpha_{n} T y_{n}+T x_{n}\right\| \\
& =\left\|x_{n+1}-T x_{n}\right\|+\left\|\left(1-\alpha_{n}\right) x_{n}-\alpha_{n} T y_{n}+T x_{n}-\alpha_{n} T x_{n}+\alpha_{n} T x_{n}\right\| \\
& \leq\left\|x_{n+1}-T x_{n}\right\|+\left(1-\alpha_{n}\right)\left\|x_{n}-T x_{n}\right\|+\alpha_{n}\left\|T x_{n}-T y_{n}\right\| \\
& =\delta_{n}+\left(1-\alpha_{n}\right)\left\|x_{n}-T x_{n}\right\|+2 \alpha_{n} M \\
& \rightarrow 0
\end{aligned}
as n rarr oon \rightarrow \infty. Condition (i) assures that lim_(n rarr oo)xi_(n)=0=>lim_(n rarr oo)x_(n)=x^(**)\lim _{n \rightarrow \infty} \xi_{n}=0 \Rightarrow \lim _{n \rightarrow \infty} x_{n}=x^{*}. Thus, for a {x_(n)}\left\{x_{n}\right\} satisfying
lim_(n rarr oo)delta_(n)=lim_(n rarr oo)||x_(n+1)-Tx_(n)||=0\lim _{n \rightarrow \infty} \delta_{n}=\lim _{n \rightarrow \infty}\left\|x_{n+1}-T x_{n}\right\|=0
we have shown that lim_(n rarr oo)x_(n)=x^(**)\lim _{n \rightarrow \infty} x_{n}=x^{*}.
Conversely, we prove (ii)=>(i)(i i) \Rightarrow(i). Take lim_(n rarr oo)xi_(n)=0\lim _{n \rightarrow \infty} \xi_{n}=0. Observe that
Conversely, we prove (ii)=>(i)(i i) \Rightarrow(i). Take lim_(n rarr oo)xi_(n)=0\lim _{n \rightarrow \infty} \xi_{n}=0. Observe that
{:[delta_(n)=||p_(n+1)-Tp_(n)||],[ <= ||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Ty_(n)||+||(1-alpha_(n))p_(n)+alpha_(n)Ty_(n)-Tp_(n)||],[ <= ||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Ty_(n)||+alpha_(n)(||p_(n)||+||Ty_(n)||)+||p_(n)-Tp_(n)||],[ <= epsi_(n)+alpha_(n)(||p_(n)||+M)+||p_(n)-Tp_(n)||],[ rarr0]:}\begin{aligned}
& \delta_{n}=\left\|p_{n+1}-T p_{n}\right\| \\
& \leq\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T y_{n}\right\|+\left\|\left(1-\alpha_{n}\right) p_{n}+\alpha_{n} T y_{n}-T p_{n}\right\| \\
& \leq\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T y_{n}\right\|+\alpha_{n}\left(\left\|p_{n}\right\|+\left\|T y_{n}\right\|\right)+\left\|p_{n}-T p_{n}\right\| \\
& \leq \varepsilon_{n}+\alpha_{n}\left(\left\|p_{n}\right\|+M\right)+\left\|p_{n}-T p_{n}\right\| \\
& \rightarrow 0
\end{aligned}
as n rarr oon \rightarrow \infty. Note that lim_(n rarr oo)||p_(n)-Tp_(n)||=0\lim _{n \rightarrow \infty}\left\|p_{n}-T p_{n}\right\|=0 and using the boundedness of {Tp_(n)}\left\{T p_{n}\right\} we obtain the boundedness of {p_(n)}\left\{p_{n}\right\}. Condition (ii) assures that
lim_(n rarr oo)delta_(n)=0=>lim_(n rarr oo)x_(n)=x^(**)\lim _{n \rightarrow \infty} \delta_{n}=0 \Rightarrow \lim _{n \rightarrow \infty} x_{n}=x^{*}
Thus, for a {p_(n)}\left\{p_{n}\right\} satisfying lim_(n rarr oo)xi_(n)=lim_(n rarr oo)||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Ty_(n)||=0\lim _{n \rightarrow \infty} \xi_{n}=\lim _{n \rightarrow \infty}\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T y_{n}\right\|=0, we have shown that lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty} p_{n}=x^{*}.
Theorems 1 and 2 lead to the following result.
COROLLARY 1. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map with bounded range. If
COROLLARY 1. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map with bounded range. If
lim_(n rarr oo)||p_(n)-Tp_(n)||=0,lim_(n rarr oo)||x_(n)-Tx_(n)||=0" and "lim_(n rarr oo)||u_(n)-Tu_(n)||=0,\lim _{n \rightarrow \infty}\left\|p_{n}-T p_{n}\right\|=0, \lim _{n \rightarrow \infty}\left\|x_{n}-T x_{n}\right\|=0 \text { and } \lim _{n \rightarrow \infty}\left\|u_{n}-T u_{n}\right\|=0,
then the following are equivalent:
(i) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) and {beta_(n)}sub[0,1)\left\{\beta_{n}\right\} \subset[0,1), satisfying (4), the Ishikawa iteration is TT-stable,
(ii) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1), satisfying (4), the Mann iteration is TT-stable,
(iii) the Picard iteration is TT-stable.
(i) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) and {beta_(n)}sub[0,1)\left\{\beta_{n}\right\} \subset[0,1), satisfying (4), the Ishikawa iteration is TT-stable,
(ii) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1), satisfying (4), the Mann iteration is TT-stable,
(iii) the Picard iteration is TT-stable.
3 Applications
The following example is from [2] and [4]. For sake of completeness we give here the whole proof.
EXAMPLE 1. Let T:[0,1]rarr[0,1],Tx=xT:[0,1] \rightarrow[0,1], T x=x.
- [2] Picard iteration converges but is not TT-stable. Then every point in ( 0,1 ] is a fixed point of TT. Let b_(0)b_{0} be a point in ( 0,1 ], then b_(n+1)=Tb_(n)=T^(n)b_(0)=b_(0)b_{n+1}=T b_{n}=T^{n} b_{0}=b_{0}. Thus lim_(n rarr oo)b_(n)=b_(0)\lim _{n \rightarrow \infty} b_{n}=b_{0}. Take p_(0)=0p_{0}=0 and p_(n)=(1)/(n)p_{n}=\frac{1}{n}. Thus
delta_(n)=|p_(n+1)-Tp_(n)|=(1)/(n(n+1))rarr0\delta_{n}=\left|p_{n+1}-T p_{n}\right|=\frac{1}{n(n+1)} \rightarrow 0
but lim_(n rarr oo)p_(n)=0!=b_(0)\lim _{n \rightarrow \infty} p_{n}=0 \neq b_{0}.
- [4] Mann iteration converges but is not TT-stable. Let e_(0)e_{0} be a point in (0,1](0,1], then e_(n+1)=(1-alpha_(n))e_(n)+alpha_(n)e_(n)=e_(n)=dots=e_(0)e_{n+1}=\left(1-\alpha_{n}\right) e_{n}+\alpha_{n} e_{n}=e_{n}=\ldots=e_{0}. Take u_(0)=e_(0),u_(n)=(1)/(n+1)u_{0}=e_{0}, u_{n}=\frac{1}{n+1} to obtain
{:[epsi_(n)=|u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)|=|(1)/(n+2)-(1-alpha_(n))(1)/(n+1)-alpha_(n)(1)/(n+1)|],[=|(1)/(n+2)-(1)/(n+1)|=(1)/((n+1)(n+2))rarr0]:}\begin{aligned}
\varepsilon_{n} & =\left|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right|=\left|\frac{1}{n+2}-\left(1-\alpha_{n}\right) \frac{1}{n+1}-\alpha_{n} \frac{1}{n+1}\right| \\
& =\left|\frac{1}{n+2}-\frac{1}{n+1}\right|=\frac{1}{(n+1)(n+2)} \rightarrow 0
\end{aligned}
but lim_(n rarr oo)u_(n)=0!=e_(0)\lim _{n \rightarrow \infty} u_{n}=0 \neq e_{0}.
EXAMPLE 2. Let T:[0,oo)rarr[0,oo)T:[0, \infty) \rightarrow[0, \infty) be given by Tx=(x)/(3)T x=\frac{x}{3}. Then the Mann iteration converges to the fixed point of x^(**)=0x^{*}=0 but is not TT-stable, and applying Theorem 1, the Picard iteration is not TT-stable while it converges.
(i) Mann iteration converges because the sequence e_(n)rarr0e_{n} \rightarrow 0 as we can see:
(i) Mann iteration converges because the sequence e_(n)rarr0e_{n} \rightarrow 0 as we can see:
{:[e_(n+1)=(1-alpha_(n))e_(n)+alpha_(n)(e_(n))/(3)=(1-(2alpha_(n))/(3))e_(n)],[=prod_(k=1)^(n)(1-(2alpha_(k))/(3))e_(0) <= exp(-(2)/(3)sum_(k=1)^(n)alpha_(k))rarr0]:}\begin{aligned}
e_{n+1} & =\left(1-\alpha_{n}\right) e_{n}+\alpha_{n} \frac{e_{n}}{3}=\left(1-\frac{2 \alpha_{n}}{3}\right) e_{n} \\
& =\prod_{k=1}^{n}\left(1-\frac{2 \alpha_{k}}{3}\right) e_{0} \leq \exp \left(-\frac{2}{3} \sum_{k=1}^{n} \alpha_{k}\right) \rightarrow 0
\end{aligned}
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 supplied by (4).
(ii) Mann iteration is not TT-stable. Take u_(n)=(n)/(n+1)u_{n}=\frac{n}{n+1}, note that u_(n)rarr1!=x^(**)=0u_{n} \rightarrow 1 \neq x^{*}=0, and epsi_(n)=||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||rarr0\varepsilon_{n}=\left\|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right\| \rightarrow 0 because
(ii) Mann iteration is not TT-stable. Take u_(n)=(n)/(n+1)u_{n}=\frac{n}{n+1}, note that u_(n)rarr1!=x^(**)=0u_{n} \rightarrow 1 \neq x^{*}=0, and epsi_(n)=||u_(n+1)-(1-alpha_(n))u_(n)-alpha_(n)Tu_(n)||rarr0\varepsilon_{n}=\left\|u_{n+1}-\left(1-\alpha_{n}\right) u_{n}-\alpha_{n} T u_{n}\right\| \rightarrow 0 because
{:[epsi_(n)=|(n+1)/(n+2)-(1-alpha_(n))(n)/(n+1)-alpha_(n)(n)/(3(n+1))|],[=(3+2alpha_(n)n^(2)+4alpha_(n)n)/(3(n+1)(n+2))]:}\begin{aligned}
\varepsilon_{n} & =\left|\frac{n+1}{n+2}-\left(1-\alpha_{n}\right) \frac{n}{n+1}-\alpha_{n} \frac{n}{3(n+1)}\right| \\
& =\frac{3+2 \alpha_{n} n^{2}+4 \alpha_{n} n}{3(n+1)(n+2)}
\end{aligned}
(iii) Picard iteration converges to fixed point x^(**)=0x^{*}=0, because b_(n+1)=Tb_(n)=T^(n)b_(0)=(b_(0))/(3^(n))rarr0b_{n+1}=T b_{n}=T^{n} b_{0}= \frac{b_{0}}{3^{n}} \rightarrow 0.
REMARK. Take again T:[0,oo)rarr[0,oo),Tx=(x)/(3)T:[0, \infty) \rightarrow[0, \infty), T x=\frac{x}{3}, and x_(n)=(n)/(n+1)x_{n}=\frac{n}{n+1} to note that lim_(n rarr oo)xi_(n)=0\lim _{n \rightarrow \infty} \xi_{n}=0 and lim_(n rarr oo)x_(n)=1!=x^(**)=0\lim _{n \rightarrow \infty} x_{n}=1 \neq x^{*}=0, and to conclude that Ishikawa iteration is not TT-stable. Remark (analogously to Mann iteration, see also [5]) that it converges while TT is a contraction.
References
[1] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44(1974), 147-150.
[2] A. M. Harder and T. Hicks, Stability results for fixed point iteration procedures, Math. Japonica, 33 (1988), 693-706.
[3] W. R. Mann, Mean value in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
[4] M. O. Osilike, Stability of the Mann and Ishikawa iteration procedures for phi\phi-strong pseudocontractions and nonlinear equations of the phi\phi-strongly accretive type, J. Math. Anal. Appl., 227 (1998), 319-334.
[5] B. E. Rhoades and Ş. M. Şoltuz, On the equivalence of Mann and Ishikawa iteration methods, Int. J. Math. Math. Sci., 2003(2003), 451-459.
[6] B. E. Rhoades and Ş. M. Şoltuz, The equivalence between the TT-stabilities of Mann and Ishikawa iterations, J. Math. Anal. Appl., 318(2006), 472-475.
[2] A. M. Harder and T. Hicks, Stability results for fixed point iteration procedures, Math. Japonica, 33 (1988), 693-706.
[3] W. R. Mann, Mean value in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
[4] M. O. Osilike, Stability of the Mann and Ishikawa iteration procedures for phi\phi-strong pseudocontractions and nonlinear equations of the phi\phi-strongly accretive type, J. Math. Anal. Appl., 227 (1998), 319-334.
[5] B. E. Rhoades and Ş. M. Şoltuz, On the equivalence of Mann and Ishikawa iteration methods, Int. J. Math. Math. Sci., 2003(2003), 451-459.
[6] B. E. Rhoades and Ş. M. Şoltuz, The equivalence between the TT-stabilities of Mann and Ishikawa iterations, J. Math. Anal. Appl., 318(2006), 472-475.
- *Mathematics Subject Classifications: 47H10
^(†){ }^{\dagger} Institute of Numerical Analysis of Romanian Academy, P.O. Box 68-1, Cluj-Napoca, Romania, and Departamento de Matematicas, Universidad de los Andes, Carrera 1 No. 18A-10, Bogota, Colombia.