Abstract
We show that T-stability of Mann and Ishikawa iterations are equivalent.
Authors
B.E. Rhoades
S.M. Soltuz
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)
Keywords
Mann iteration; Ishikawa iteration; T-stability
References
See the expanding block below.
Paper coordinates
B.E. Rhoades, Ş.M . Şoltuz, The equivalence between T-stabilities of Mann and Ishikawa iterations, J. Math. Anal. Appl. 318 (2006), 472-475.
doi: 10.1016/j.jmaa.2005.05.066
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The equivalence between the TT-stabilities of Mann and Ishikawa iterations
Received 16 January 2005
Available online 11 July 2005
Submitted by G. Jungck
Available online 11 July 2005
Submitted by G. Jungck
Abstract
We show that TT-stability of Mann and Ishikawa iterations are equivalent. © 2005 Elsevier Inc. All rights reserved.
Keywords: Mann iteration; Ishikawa iteration; TT-stability
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 1.1. [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. [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\}.
The two most popular iteration procedures for obtaining fixed points of TT, when the Banach principle fails, are Mann iteration [3], defined by
{:(1.1)u_(n+1)=(1-alpha_(n))u_(n)+alpha_(n)Tu_(n)",":}\begin{equation*}
u_{n+1}=\left(1-\alpha_{n}\right) u_{n}+\alpha_{n} T u_{n}, \tag{1.1}
\end{equation*}
and Ishikawa iteration [1], defined by
{:[x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)Tz_(n)],[(1.2)z_(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 z_{n} \\
& z_{n}=\left(1-\beta_{n}\right) x_{n}+\beta_{n} T x_{n} \tag{1.2}
\end{align*}
The sequences {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) satisfy
{:(1.3)lim_(n rarr oo)alpha_(n)=lim_(n rarr oo)beta_(n)=0","quadsum_(n=1)^(oo)alpha_(n)=oo:}\begin{equation*}
\lim _{n \rightarrow \infty} \alpha_{n}=\lim _{n \rightarrow \infty} \beta_{n}=0, \quad \sum_{n=1}^{\infty} \alpha_{n}=\infty \tag{1.3}
\end{equation*}
A reasonable conjecture is that the Ishikawa iteration and the corresponding Mann iteration are equivalent for all maps for which either method provides convergence to a fixed point. In an attempt to verify this conjecture the authors, in a series of papers [4-9] have shown the equivalence for several classes of maps. We shall prove the equivalence between TT-stabilities of (1.1) and (1.2). Throughout this paper, we shall assume that both Mann and Ishikawa iterations converge to a fixed point of TT.
2. The equivalence between T\boldsymbol{T}-stabilities
Let {x_(n)}\left\{x_{n}\right\} be the Ishikawa iteration and {u_(n)}\left\{u_{n}\right\} be the Mann iteration. Let {s_(n)},{p_(n)}sub X\left\{s_{n}\right\},\left\{p_{n}\right\} \subset X be such that s_(0)=p_(0)s_{0}=p_{0}, and let (alpha_(n))_(n)sub(0,1),(beta_(n))_(n)sub[0,1)\left(\alpha_{n}\right)_{n} \subset(0,1),\left(\beta_{n}\right)_{n} \subset[0,1) satisfy (1.3) and
{:(2.1)y_(n)=(1-beta_(n))s_(n)+beta_(n)Ts_(n):}\begin{equation*}
y_{n}=\left(1-\beta_{n}\right) s_{n}+\beta_{n} T s_{n} \tag{2.1}
\end{equation*}
We consider the following nonnegative sequences, for all n inNn \in \mathbb{N} :
{:[(2.2)epsi_(n):=||s_(n+1)-(1-alpha_(n))s_(n)-alpha_(n)Ty_(n)||],[(2.3)delta_(n):=||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Tp_(n)||.]:}\begin{align*}
& \varepsilon_{n}:=\left\|s_{n+1}-\left(1-\alpha_{n}\right) s_{n}-\alpha_{n} T y_{n}\right\| \tag{2.2}\\
& \delta_{n}:=\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T p_{n}\right\| . \tag{2.3}
\end{align*}
Definition 2.1. Definition 1.1 for (2.2) and (2.3) gives:
(i) If lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0 implies that lim_(n rarr oo)s_(n)=x^(**)\lim _{n \rightarrow \infty} s_{n}=x^{*}, then the Ishikawa iteration (1.2), is said to be TT-stable.
(ii) If lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0 implies that lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty} p_{n}=x^{*}, then the Mann iteration (1.1) is said to be TT-stable.
(i) If lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0 implies that lim_(n rarr oo)s_(n)=x^(**)\lim _{n \rightarrow \infty} s_{n}=x^{*}, then the Ishikawa iteration (1.2), is said to be TT-stable.
(ii) If lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0 implies that lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty} p_{n}=x^{*}, then the Mann iteration (1.1) is said to be TT-stable.
Remark 2.2. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map. The following are equivalent:
(i) for all {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) satisfying (1.3), the Ishikawa iteration is TT-stable,
(I) for all {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) satisfying (1.3), AA{s_(n)}sub X\forall\left\{s_{n}\right\} \subset X :
(i) for all {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) satisfying (1.3), the Ishikawa iteration is TT-stable,
(I) for all {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) satisfying (1.3), AA{s_(n)}sub X\forall\left\{s_{n}\right\} \subset X :
{:(2.4)lim_(n rarr oo)epsi_(n)=lim_(n rarr oo)||s_(n+1)-(1-alpha_(n))s_(n)-alpha_(n)Ty_(n)||=0=>lim_(n rarr oo)s_(n)=x^(**):}\begin{equation*}
\lim _{n \rightarrow \infty} \varepsilon_{n}=\lim _{n \rightarrow \infty}\left\|s_{n+1}-\left(1-\alpha_{n}\right) s_{n}-\alpha_{n} T y_{n}\right\|=0 \Rightarrow \lim _{n \rightarrow \infty} s_{n}=x^{*} \tag{2.4}
\end{equation*}
Remark 2.3. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map. The following are equivalent:
(ii) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfying (1.3), the Mann iteration is TT-stable,
(II) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfying (1.3), AA{p_(n)}sub X\forall\left\{p_{n}\right\} \subset X :
(ii) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfying (1.3), the Mann iteration is TT-stable,
(II) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfying (1.3), AA{p_(n)}sub X\forall\left\{p_{n}\right\} \subset X :
{:(2.5)lim_(n rarr oo)delta_(n)=lim_(n rarr oo)||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Tp_(n)||=0quad=>quadlim_(n rarr oo)p_(n)=x^(**):}\begin{equation*}
\lim _{n \rightarrow \infty} \delta_{n}=\lim _{n \rightarrow \infty}\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T p_{n}\right\|=0 \quad \Rightarrow \quad \lim _{n \rightarrow \infty} p_{n}=x^{*} \tag{2.5}
\end{equation*}
Theorem 2.4. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map. Then the following are equivalent:
(i) for all {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) satisfying (1.3), the Ishikawa iteration (1.2) is TT-stable,
(ii) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1), satisfying (1.3), the Mann iteration (1.1) is TT-stable.
(i) for all {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) satisfying (1.3), the Ishikawa iteration (1.2) is TT-stable,
(ii) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1), satisfying (1.3), the Mann iteration (1.1) is TT-stable.
Proof. Let
M:=max{s u p_(n inN){||T(y_(n))||},s u p_(n inN){||T(s_(n))||},s u p_(n inN){||T(p_(n))||}}M:=\max \left\{\sup _{n \in \mathbb{N}}\left\{\left\|T\left(y_{n}\right)\right\|\right\}, \sup _{n \in \mathbb{N}}\left\{\left\|T\left(s_{n}\right)\right\|\right\}, \sup _{n \in \mathbb{N}}\left\{\left\|T\left(p_{n}\right)\right\|\right\}\right\}
Since the Mann and Ishikawa iterations converge, M < ooM<\infty. Remarks 2.2 and 2.3 assure that (i) ⇔ (ii) is equivalent to (I) ⇔ (II). We shall prove that (I) ⇒ (II). In (I) and (2.4) set s_(n):=p_(n)s_{n}:=p_{n} to obtain
{:[||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Tp_(n)||],[quad <= ||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Ty_(n)||+||alpha_(n)Ty_(n)-alpha_(n)Tp_(n)||],[quad <= ||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Ty_(n)||+alpha_(n)(||Ty_(n)||+||Tp_(n)||)],[(2.6)quad <= ||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Ty_(n)||+2alpha_(n)M rarr0quad" as "n rarr oo.]:}\begin{align*}
& \left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T p_{n}\right\| \\
& \quad \leqslant\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T y_{n}\right\|+\left\|\alpha_{n} T y_{n}-\alpha_{n} T p_{n}\right\| \\
& \quad \leqslant\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T y_{n}\right\|+\alpha_{n}\left(\left\|T y_{n}\right\|+\left\|T p_{n}\right\|\right) \\
& \quad \leqslant\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T y_{n}\right\|+2 \alpha_{n} M \rightarrow 0 \quad \text { as } n \rightarrow \infty . \tag{2.6}
\end{align*}
Condition (I) assures that lim_(n rarr oo)||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Ty_(n)||=0=>lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty}\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T y_{n}\right\|=0 \Rightarrow \lim _{n \rightarrow \infty} p_{n}=x^{*}. Thus, for a {p_(n)}\left\{p_{n}\right\} satisfying lim_(n rarr oo)||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Tp_(n)||=0\lim _{n \rightarrow \infty}\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T p_{n}\right\|=0, we have shown that lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty} p_{n}=x^{*}.
Conversely, we prove (II) =>\Rightarrow (I). In (II) and (2.5) set p_(n):=s_(n)p_{n}:=s_{n} to obtain
{:[||s_(n+1)-(1-alpha_(n))s_(n)-alpha_(n)Ty_(n)||],[quad <= ||s_(n+1)-(1-alpha_(n))s_(n)-alpha_(n)Ts_(n)||+||alpha_(n)Ty_(n)-alpha_(n)Ts_(n)||],[(2.7)quad <= ||s_(n+1)-(1-alpha_(n))s_(n)-alpha_(n)Ts_(n)||+2alpha_(n)M rarr0quad" as "n rarr oo]:}\begin{align*}
& \left\|s_{n+1}-\left(1-\alpha_{n}\right) s_{n}-\alpha_{n} T y_{n}\right\| \\
& \quad \leqslant\left\|s_{n+1}-\left(1-\alpha_{n}\right) s_{n}-\alpha_{n} T s_{n}\right\|+\left\|\alpha_{n} T y_{n}-\alpha_{n} T s_{n}\right\| \\
& \quad \leqslant\left\|s_{n+1}-\left(1-\alpha_{n}\right) s_{n}-\alpha_{n} T s_{n}\right\|+2 \alpha_{n} M \rightarrow 0 \quad \text { as } n \rightarrow \infty \tag{2.7}
\end{align*}
Condition (II) assures that lim_(n rarr oo)||s_(n+1)-(1-alpha_(n))s_(n)-alpha_(n)Ts_(n)||=0=>lim_(n rarr oo)s_(n)=x^(**)\lim _{n \rightarrow \infty}\left\|s_{n+1}-\left(1-\alpha_{n}\right) s_{n}-\alpha_{n} T s_{n}\right\|=0 \Rightarrow \lim _{n \rightarrow \infty} s_{n}=x^{*}. Thus, for a {s_(n)}\left\{s_{n}\right\} satisfying lim_(n rarr oo)||s_(n+1)-(1-alpha_(n))s_(n)-alpha_(n)Ty_(n)||=0\lim _{n \rightarrow \infty}\left\|s_{n+1}-\left(1-\alpha_{n}\right) s_{n}-\alpha_{n} T y_{n}\right\|=0, we have shown that lim_(n rarr oo)s_(n)=x^(**)\lim _{n \rightarrow \infty} s_{n}=x^{*}.
Set in (1.1) and (1.2), T:=T^(n)T:=T^{n}, to obtain the modified Mann and modified Ishikawa iterations. We suppose that both modified Mann and modified Ishikawa iterations converge to a fixed point of TT. Note that Definition 2.1, Remarks 2.2 and 2.3, and Theorem 2.4 hold in this case too.
Corollary 2.5. Let XX be a normed space and T:X rarr XT: X \rightarrow X a map. Then the following are equivalent:
(i) for all {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) satisfying (1.3), the modified Ishikawa iteration is TT-stable,
(ii) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1), satisfying (1.3), the modified Mann iteration is TT-stable.
(i) for all {alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1) satisfying (1.3), the modified Ishikawa iteration is TT-stable,
(ii) for all {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1), satisfying (1.3), the modified Mann iteration is TT-stable.
References
[1] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974) 147-150.
[2] A.M. Harder, T. Hicks, Stability results for fixed point iteration procedures, Math. Japonica 33 (1988) 693706.
[3] W.R. Mann, Mean value in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-510.
[4] B.E. Rhoades, Ş.M. Şoltuz, On the equivalence of Mann and Ishikawa iteration methods, Internat. J. Math. Math. Sci. 2003 (2003) 451-459.
[5] B.E. Rhoades, Ş.M. Şoltuz, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Internat. J. Math. Math. Sci. 2003 (2003) 2645-2652.
[6] B.E. Rhoades, Ş.M. Şoltuz, The equivalence of Mann and Ishikawa iteration for psi\psi-uniformly pseudocontractive or psi\psi-uniformly accretive maps, Internat. J. Math. Math. Sci. 46 (2004) 2443-2452.
[7] B.E. Rhoades, Ş.M. Şoltuz, The equivalence of Mann and Ishikawa iteration for a Lipschitzian psi-uniformly pseudocontractive and psi-uniformly accretive maps, Tamkang J. Math. 35 (2004) 235-245.
[8] B.E. Rhoades, Ş.M. Şoltuz, The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289 (2004) 266-278.
[9] B.E. Rhoades, Ş.M. Şoltuz, The equivalence between Mann-Ishikawa iteration and the multistep iteration, Nonlinear Anal. 58 (2004) 219-228.
[2] A.M. Harder, T. Hicks, Stability results for fixed point iteration procedures, Math. Japonica 33 (1988) 693706.
[3] W.R. Mann, Mean value in iteration, Proc. Amer. Math. Soc. 4 (1953) 506-510.
[4] B.E. Rhoades, Ş.M. Şoltuz, On the equivalence of Mann and Ishikawa iteration methods, Internat. J. Math. Math. Sci. 2003 (2003) 451-459.
[5] B.E. Rhoades, Ş.M. Şoltuz, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Internat. J. Math. Math. Sci. 2003 (2003) 2645-2652.
[6] B.E. Rhoades, Ş.M. Şoltuz, The equivalence of Mann and Ishikawa iteration for psi\psi-uniformly pseudocontractive or psi\psi-uniformly accretive maps, Internat. J. Math. Math. Sci. 46 (2004) 2443-2452.
[7] B.E. Rhoades, Ş.M. Şoltuz, The equivalence of Mann and Ishikawa iteration for a Lipschitzian psi-uniformly pseudocontractive and psi-uniformly accretive maps, Tamkang J. Math. 35 (2004) 235-245.
[8] B.E. Rhoades, Ş.M. Şoltuz, The equivalence between the convergences of Ishikawa and Mann iterations for asymptotically nonexpansive in the intermediate sense and strongly successively pseudocontractive maps, J. Math. Anal. Appl. 289 (2004) 266-278.
[9] B.E. Rhoades, Ş.M. Şoltuz, The equivalence between Mann-Ishikawa iteration and the multistep iteration, Nonlinear Anal. 58 (2004) 219-228.
- Corresponding author.
E-mail addresses: rhoades@indiana.edu (B.E. Rhoades), soltuzul@yahoo.com, stefanmsoltuz@yahoo.com (Ş.M. Şoltuz).