Received 16 January 2005
Available online 11 July 2005
Submitted by G. Jungck

Keywords: Mann iteration; Ishikawa iteration; $T$$T$TT$T$-stability

Let $X$$X$XX$X$ be a normed space and $T$$T$TT$T$ a selfmap of $X$$X$XX$X$. Let $x_0$$x_0$x_(0)x_{0}$x_0$ be a point of $X$$X$XX$X$, and assume that $x_{n+1}=f(T,x_n)$$x_{n+1}=f\left(T,x_n\right)$x_(n+1)=f(T,x_(n))x_{n+1}=f\left(T, x_{n}\right)$x_{n+1}=f(T,x_n)$ is an iteration procedure, involving $T$$T$TT$T$, which yields a sequence $\left\{x_n\right\}$$\left\{x_n\right\}${x_(n)}\left\{x_{n}\right\}$\left\{x_n\right\}$ of point from $X$$X$XX$X$. Suppose $\left\{x_n\right\}$$\left\{x_n\right\}${x_(n)}\left\{x_{n}\right\}$\left\{x_n\right\}$ converges to a fixed point $x^{\ast }$$x^{\ast }$x^(**)x^{*}$x^{\ast }$ of $T$$T$TT$T$. Let $\left\{{\xi }_n\right\}$$\left\{{\xi }_n\right\}${xi_(n)}\left\{\xi_{n}\right\}$\left\{{\xi }_n\right\}$ be an arbitrary sequence in $X$$X$XX$X$, and set ${ϵ}_n=\Vert {\xi }_{n+1}-f(T,{\xi }_n)\Vert$${ϵ}_n=‖{\xi }_{n+1}-f\left(T,{\xi }_n\right)‖$epsilon_(n)=||xi_(n+1)-f(T,xi_(n))||\epsilon_{n}=\left\|\xi_{n+1}-f\left(T, \xi_{n}\right)\right\|${ϵ}_n=\Vert {\xi }_{n+1}-f(T,{\xi }_n)\Vert$ for all $n\in \mathbb{N}$$n\in \mathbb{N}$n inNn \in \mathbb{N}$n\in \mathbb{N}$.

Definition 1.1. [2] If $((\underset{n\to \mathrm{\infty }}{lim}{ϵ}_n=0)\Rightarrow (\underset{n\to \mathrm{\infty }}{lim}{\xi }_n=p))$$\left(\left(\underset{n\to \mathrm{\infty }}{lim} {ϵ}_n=0\right)\Rightarrow \left(\underset{n\to \mathrm{\infty }}{lim} {\xi }_n=p\right)\right)$((lim_(n rarr oo)epsilon_(n)=0)=>(lim_(n rarr oo)xi_(n)=p))\left(\left(\lim _{n \rightarrow \infty} \epsilon_{n}=0\right) \Rightarrow\left(\lim _{n \rightarrow \infty} \xi_{n}=p\right)\right)$((\underset{n\to \mathrm{\infty }}{lim}{ϵ}_n=0)\Rightarrow (\underset{n\to \mathrm{\infty }}{lim}{\xi }_n=p))$, then the iteration procedure $x_{n+1}=f(T,x_n)$$x_{n+1}=f\left(T,x_n\right)$x_(n+1)=f(T,x_(n))x_{n+1}=f\left(T, x_{n}\right)$x_{n+1}=f(T,x_n)$ is said to be $T$$T$TT$T$-stable with respect to $T$$T$TT$T$.

Remark 1.2. [2] In practice, such a sequence $\left\{{\xi }_n\right\}$$\left\{{\xi }_n\right\}${xi_(n)}\left\{\xi_{n}\right\}$\left\{{\xi }_n\right\}$ could arise in the following way. Let $x_0$$x_0$x_(0)x_{0}$x_0$ be a point in $X$$X$XX$X$. Set $x_{n+1}=f(T,x_n)$$x_{n+1}=f\left(T,x_n\right)$x_(n+1)=f(T,x_(n))x_{n+1}=f\left(T, x_{n}\right)$x_{n+1}=f(T,x_n)$. Let ${\xi }_0=x_0$${\xi }_0=x_0$xi_(0)=x_(0)\xi_{0}=x_{0}${\xi }_0=x_0$. Now $x_1=f(T,x_0)$$x_1=f\left(T,x_0\right)$x_(1)=f(T,x_(0))x_{1}=f\left(T, x_{0}\right)$x_1=f(T,x_0)$. Because of rounding or discretization in the function $T$$T$TT$T$, a new value ${\xi }_1$${\xi }_1$xi_(1)\xi_{1}${\xi }_1$ approximately equal to $x_1$$x_1$x_(1)x_{1}$x_1$ might be obtained instead of the true value of $f(T,x_0)$$f\left(T,x_0\right)$f(T,x_(0))f\left(T, x_{0}\right)$f(T,x_0)$. Then to approximate ${\xi }_2$${\xi }_2$xi_(2)\xi_{2}${\xi }_2$, the value $f(T,{\xi }_1)$$f\left(T,{\xi }_1\right)$f(T,xi_(1))f\left(T, \xi_{1}\right)$f(T,{\xi }_1)$ is computed to yields ${\xi }_2$${\xi }_2$xi_(2)\xi_{2}${\xi }_2$, an approximation of $f(T,{\xi }_1)$$f\left(T,{\xi }_1\right)$f(T,xi_(1))f\left(T, \xi_{1}\right)$f(T,{\xi }_1)$. This computation is continued to obtain $\left\{{\xi }_n\right\}$$\left\{{\xi }_n\right\}${xi_(n)}\left\{\xi_{n}\right\}$\left\{{\xi }_n\right\}$ an approximate sequence of $\left\{x_n\right\}$$\left\{x_n\right\}${x_(n)}\left\{x_{n}\right\}$\left\{x_n\right\}$.

The two most popular iteration procedures for obtaining fixed points of $T$$T$TT$T$, when the Banach principle fails, are Mann iteration [3], defined by

and Ishikawa iteration [1], defined by

The sequences $\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)${alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1)$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$ satisfy

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 $T$$T$TT$T$-stabilities of (1.1) and (1.2). Throughout this paper, we shall assume that both Mann and Ishikawa iterations converge to a fixed point of $T$$T$TT$T$.

Let $\left\{x_n\right\}$$\left\{x_n\right\}${x_(n)}\left\{x_{n}\right\}$\left\{x_n\right\}$ be the Ishikawa iteration and $\left\{u_n\right\}$$\left\{u_n\right\}${u_(n)}\left\{u_{n}\right\}$\left\{u_n\right\}$ be the Mann iteration. Let $\left\{s_n\right\},\left\{p_n\right\}\subset X$$\left\{s_n\right\},\left\{p_n\right\}\subset X${s_(n)},{p_(n)}sub X\left\{s_{n}\right\},\left\{p_{n}\right\} \subset X$\left\{s_n\right\},\left\{p_n\right\}\subset X$ be such that $s_0=p_0$$s_0=p_0$s_(0)=p_(0)s_{0}=p_{0}$s_0=p_0$, and let ${\left({\alpha }_n\right)}_n\subset (0,1),{\left({\beta }_n\right)}_n\subset [0,1)$${\left({\alpha }_n\right)}_n\subset (0,1),{\left({\beta }_n\right)}_n\subset [0,1)$(alpha_(n))_(n)sub(0,1),(beta_(n))_(n)sub[0,1)\left(\alpha_{n}\right)_{n} \subset(0,1),\left(\beta_{n}\right)_{n} \subset[0,1)${\left({\alpha }_n\right)}_n\subset (0,1),{\left({\beta }_n\right)}_n\subset [0,1)$ satisfy (1.3) and

We consider the following nonnegative sequences, for all $n\in \mathbb{N}$$n\in \mathbb{N}$n inNn \in \mathbb{N}$n\in \mathbb{N}$ :

Definition 2.1. Definition 1.1 for (2.2) and (2.3) gives:
(i) If $\underset{n\to \mathrm{\infty }}{lim}{\epsilon }_n=0$$\underset{n\to \mathrm{\infty }}{lim} {\epsilon }_n=0$lim_(n rarr oo)epsi_(n)=0\lim _{n \rightarrow \infty} \varepsilon_{n}=0$\underset{n\to \mathrm{\infty }}{lim}{\epsilon }_n=0$ implies that $\underset{n\to \mathrm{\infty }}{lim}{s}_n=x^{\ast }$$\underset{n\to \mathrm{\infty }}{lim} s_n=x^{\ast }$lim_(n rarr oo)s_(n)=x^(**)\lim _{n \rightarrow \infty} s_{n}=x^{*}$\underset{n\to \mathrm{\infty }}{lim}{s}_n=x^{\ast }$, then the Ishikawa iteration (1.2), is said to be $T$$T$TT$T$-stable.
(ii) If $\underset{n\to \mathrm{\infty }}{lim}{\delta }_n=0$$\underset{n\to \mathrm{\infty }}{lim} {\delta }_n=0$lim_(n rarr oo)delta_(n)=0\lim _{n \rightarrow \infty} \delta_{n}=0$\underset{n\to \mathrm{\infty }}{lim}{\delta }_n=0$ implies that $\underset{n\to \mathrm{\infty }}{lim}{p}_n=x^{\ast }$$\underset{n\to \mathrm{\infty }}{lim} p_n=x^{\ast }$lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty} p_{n}=x^{*}$\underset{n\to \mathrm{\infty }}{lim}{p}_n=x^{\ast }$, then the Mann iteration (1.1) is said to be $T$$T$TT$T$-stable.

Remark 2.2. Let $X$$X$XX$X$ be a normed space and $T:X\to X$$T:X\to X$T:X rarr XT: X \rightarrow X$T:X\to X$ a map. The following are equivalent:
(i) for all $\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)${alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1)$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$ satisfying (1.3), the Ishikawa iteration is $T$$T$TT$T$-stable,
(I) for all $\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)${alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1)$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$ satisfying (1.3), $\mathrm{\forall }\left\{s_n\right\}\subset X$$\mathrm{\forall }\left\{s_n\right\}\subset X$AA{s_(n)}sub X\forall\left\{s_{n}\right\} \subset X$\mathrm{\forall }\left\{s_n\right\}\subset X$ :

Remark 2.3. Let $X$$X$XX$X$ be a normed space and $T:X\to X$$T:X\to X$T:X rarr XT: X \rightarrow X$T:X\to X$ a map. The following are equivalent:
(ii) for all $\left\{{\alpha }_n\right\}\subset (0,1)$$\left\{{\alpha }_n\right\}\subset (0,1)${alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1)$\left\{{\alpha }_n\right\}\subset (0,1)$ satisfying (1.3), the Mann iteration is $T$$T$TT$T$-stable,
(II) for all $\left\{{\alpha }_n\right\}\subset (0,1)$$\left\{{\alpha }_n\right\}\subset (0,1)${alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1)$\left\{{\alpha }_n\right\}\subset (0,1)$ satisfying (1.3), $\mathrm{\forall }\left\{p_n\right\}\subset X$$\mathrm{\forall }\left\{p_n\right\}\subset X$AA{p_(n)}sub X\forall\left\{p_{n}\right\} \subset X$\mathrm{\forall }\left\{p_n\right\}\subset X$ :

Theorem 2.4. Let $X$$X$XX$X$ be a normed space and $T:X\to X$$T:X\to X$T:X rarr XT: X \rightarrow X$T:X\to X$ a map. Then the following are equivalent:
(i) for all $\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)${alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1)$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$ satisfying (1.3), the Ishikawa iteration (1.2) is $T$$T$TT$T$-stable,
(ii) for all $\left\{{\alpha }_n\right\}\subset (0,1)$$\left\{{\alpha }_n\right\}\subset (0,1)${alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1)$\left\{{\alpha }_n\right\}\subset (0,1)$, satisfying (1.3), the Mann iteration (1.1) is $T$$T$TT$T$-stable.

Proof. Let

Since the Mann and Ishikawa iterations converge, $M<\mathrm{\infty }$$M<\mathrm{\infty }$M < ooM<\infty$M<\mathrm{\infty }$. Remarks 2.2 and 2.3 assure that (i) ⇔ (ii) is equivalent to (I) ⇔ (II). We shall prove that (I) ⇒ (II). In (I) and (2.4) set $s_n:=p_n$$s_n:=p_n$s_(n):=p_(n)s_{n}:=p_{n}$s_n:=p_n$ to obtain

Condition (I) assures that $\underset{n\to \mathrm{\infty }}{lim}\Vert p_{n+1}-(1-{\alpha }_n)p_n-{\alpha }_{n}T{y}_n\Vert =0\Rightarrow \underset{n\to \mathrm{\infty }}{lim}{p}_n=x^{\ast }$$\underset{n\to \mathrm{\infty }}{lim} ‖p_{n+1}-\left(1-{\alpha }_n\right)p_n-{\alpha }_{n}T{y}_n‖=0\Rightarrow \underset{n\to \mathrm{\infty }}{lim} p_n=x^{\ast }$lim_(n rarr oo)||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Ty_(n)||=0=>lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty}\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T y_{n}\right\|=0 \Rightarrow \lim _{n \rightarrow \infty} p_{n}=x^{*}$\underset{n\to \mathrm{\infty }}{lim}\Vert p_{n+1}-(1-{\alpha }_n)p_n-{\alpha }_{n}T{y}_n\Vert =0\Rightarrow \underset{n\to \mathrm{\infty }}{lim}{p}_n=x^{\ast }$. Thus, for a $\left\{p_n\right\}$$\left\{p_n\right\}${p_(n)}\left\{p_{n}\right\}$\left\{p_n\right\}$ satisfying $\underset{n\to \mathrm{\infty }}{lim}\Vert p_{n+1}-(1-{\alpha }_n)p_n-{\alpha }_{n}T{p}_n\Vert =0$$\underset{n\to \mathrm{\infty }}{lim} ‖p_{n+1}-\left(1-{\alpha }_n\right)p_n-{\alpha }_{n}T{p}_n‖=0$lim_(n rarr oo)||p_(n+1)-(1-alpha_(n))p_(n)-alpha_(n)Tp_(n)||=0\lim _{n \rightarrow \infty}\left\|p_{n+1}-\left(1-\alpha_{n}\right) p_{n}-\alpha_{n} T p_{n}\right\|=0$\underset{n\to \mathrm{\infty }}{lim}\Vert p_{n+1}-(1-{\alpha }_n)p_n-{\alpha }_{n}T{p}_n\Vert =0$, we have shown that $\underset{n\to \mathrm{\infty }}{lim}{p}_n=x^{\ast }$$\underset{n\to \mathrm{\infty }}{lim} p_n=x^{\ast }$lim_(n rarr oo)p_(n)=x^(**)\lim _{n \rightarrow \infty} p_{n}=x^{*}$\underset{n\to \mathrm{\infty }}{lim}{p}_n=x^{\ast }$.

Conversely, we prove (II) $\Rightarrow$$\Rightarrow$=>\Rightarrow$\Rightarrow$ (I). In (II) and (2.5) set $p_n:=s_n$$p_n:=s_n$p_(n):=s_(n)p_{n}:=s_{n}$p_n:=s_n$ to obtain

Condition (II) assures that $\underset{n\to \mathrm{\infty }}{lim}\Vert s_{n+1}-(1-{\alpha }_n)s_n-{\alpha }_{n}T{s}_n\Vert =0\Rightarrow \underset{n\to \mathrm{\infty }}{lim}{s}_n=x^{\ast }$$\underset{n\to \mathrm{\infty }}{lim} ‖s_{n+1}-\left(1-{\alpha }_n\right)s_n-{\alpha }_{n}T{s}_n‖=0\Rightarrow \underset{n\to \mathrm{\infty }}{lim} s_n=x^{\ast }$lim_(n rarr oo)||s_(n+1)-(1-alpha_(n))s_(n)-alpha_(n)Ts_(n)||=0=>lim_(n rarr oo)s_(n)=x^(**)\lim _{n \rightarrow \infty}\left\|s_{n+1}-\left(1-\alpha_{n}\right) s_{n}-\alpha_{n} T s_{n}\right\|=0 \Rightarrow \lim _{n \rightarrow \infty} s_{n}=x^{*}$\underset{n\to \mathrm{\infty }}{lim}\Vert s_{n+1}-(1-{\alpha }_n)s_n-{\alpha }_{n}T{s}_n\Vert =0\Rightarrow \underset{n\to \mathrm{\infty }}{lim}{s}_n=x^{\ast }$. Thus, for a $\left\{s_n\right\}$$\left\{s_n\right\}${s_(n)}\left\{s_{n}\right\}$\left\{s_n\right\}$ satisfying $\underset{n\to \mathrm{\infty }}{lim}\Vert s_{n+1}-(1-{\alpha }_n)s_n-{\alpha }_{n}T{y}_n\Vert =0$$\underset{n\to \mathrm{\infty }}{lim} ‖s_{n+1}-\left(1-{\alpha }_n\right)s_n-{\alpha }_{n}T{y}_n‖=0$lim_(n rarr oo)||s_(n+1)-(1-alpha_(n))s_(n)-alpha_(n)Ty_(n)||=0\lim _{n \rightarrow \infty}\left\|s_{n+1}-\left(1-\alpha_{n}\right) s_{n}-\alpha_{n} T y_{n}\right\|=0$\underset{n\to \mathrm{\infty }}{lim}\Vert s_{n+1}-(1-{\alpha }_n)s_n-{\alpha }_{n}T{y}_n\Vert =0$, we have shown that $\underset{n\to \mathrm{\infty }}{lim}{s}_n=x^{\ast }$$\underset{n\to \mathrm{\infty }}{lim} s_n=x^{\ast }$lim_(n rarr oo)s_(n)=x^(**)\lim _{n \rightarrow \infty} s_{n}=x^{*}$\underset{n\to \mathrm{\infty }}{lim}{s}_n=x^{\ast }$.

Set in (1.1) and (1.2), $T:=T^n$$T:=T^n$T:=T^(n)T:=T^{n}$T:=T^n$, to obtain the modified Mann and modified Ishikawa iterations. We suppose that both modified Mann and modified Ishikawa iterations converge to a fixed point of $T$$T$TT$T$. Note that Definition 2.1, Remarks 2.2 and 2.3, and Theorem 2.4 hold in this case too.

Corollary 2.5. Let $X$$X$XX$X$ be a normed space and $T:X\to X$$T:X\to X$T:X rarr XT: X \rightarrow X$T:X\to X$ a map. Then the following are equivalent:
(i) for all $\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)${alpha_(n)}sub(0,1),{beta_(n)}sub[0,1)\left\{\alpha_{n}\right\} \subset(0,1),\left\{\beta_{n}\right\} \subset[0,1)$\left\{{\alpha }_n\right\}\subset (0,1),\left\{{\beta }_n\right\}\subset [0,1)$ satisfying (1.3), the modified Ishikawa iteration is $T$$T$TT$T$-stable,
(ii) for all $\left\{{\alpha }_n\right\}\subset (0,1)$$\left\{{\alpha }_n\right\}\subset (0,1)${alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1)$\left\{{\alpha }_n\right\}\subset (0,1)$, satisfying (1.3), the modified Mann iteration is $T$$T$TT$T$-stable.

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