Abstract
We consider a mean value iteration for a family of functions, which corresponds to the Mann iteration with \(\lim_{n\rightarrow \infty}an\neq0\). We prove convergence results for this iteration when applied to strongly pseudocontractive or strongly accretive maps.
Authors
Stefan M. Soltuz
Tiberiu Popoviciu Institute of Numerical Analysis
B.E. Rhoades
Department of Mathematics, Indiana University, Bloomington, IN 47405-7106,USA
Keywords
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Paper coordinates
B.E. Rhoades, Ş.M. Şoltuz , The convergence of mean value iteration for a family of maps, Int. J. Math. Math. Sci. 2005: 21, 3479-3485. (published in 28 dec. 2005)
About this paper
Journal
International Journal of Mathematics and Mathematical Sciences
Publisher Name
Wiley
DOI
Print ISSN
0161-1712
Online ISSN
1687-0425
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[1] C. E. Chidume and S. A. Mutangadura, An example of the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2359–2363.
[2] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), no. 1,147–150.
[3] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506–510.
[4] C. H. Morales and J. S. Jung, Convergence of paths for pseudocontractive mappings in Banachspaces, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3411–3419.
[5] B. E. Rhoades and S. M. Soltuz, The equivalence of Mann iteration and Ishikawa iteration fornon-Lipschitzian operators, Int. J. Math. Math. Sci. 2003 (2003), no. 42, 2645–2651.
[6] , Mean value iteration for a family of functions, to appear in Nonlinear Funct. Anal.Appl.
[7] S. M. Soltuz, Some sequences supplied by inequalities and their applications, Rev. Anal. Numer.Theor. Approx. 29 (2000), no. 2, 207–212.
Paper (preprint) in HTML form
THE CONVERGENCE OF MEAN VALUE ITERATION FOR A FAMILY OF MAPS
Received 3 January 2005 and in revised form 6 September 2005
We consider a mean value iteration for a family of functions, which corresponds to the Mann iteration with lim_(n rarr oo)alpha_(n)!=0\lim _{n \rightarrow \infty} \alpha_{n} \neq 0. We prove convergence results for this iteration when applied to strongly pseudocontractive or strongly accretive maps.
1. Introduction
Let XX be a real Banach space. The map J:X rarr2^(X^(**))J: X \rightarrow 2^{X^{*}} given by
{:(1.1)Jx:={f inX^(**):(:x,f:)=||x||^(2),||f||=||x||}","quad AA x in X:}\begin{equation*}
J x:=\left\{f \in X^{*}:\langle x, f\rangle=\|x\|^{2},\|f\|=\|x\|\right\}, \quad \forall x \in X \tag{1.1}
\end{equation*}
is called the normalized duality mapping. Let y in Xy \in X and j(y)in J(y)j(y) \in J(y); note that (:*,j(y):)\langle\cdot, j(y)\rangle is a Lipschitzian map.
Remark 1.1. The above JJ satisfies
{:(1.2)(:x","j(y):) <= ||x||||y||","quad AA x in X","AA j(y)in J(y).:}\begin{equation*}
\langle x, j(y)\rangle \leq\|x\|\|y\|, \quad \forall x \in X, \forall j(y) \in J(y) . \tag{1.2}
\end{equation*}
Definition 1.2. Let BB be a nonempty subset of XX. The map T:B rarr BT: B \rightarrow B is strongly pseudocontractive if there exist k in(0,1)k \in(0,1) and j(x-y)in J(x-y)j(x-y) \in J(x-y) such that
{:(1.3)(:Tx-Ty","j(x-y):) <= k||x-y||^(2)","quad AA x","y in B.:}\begin{equation*}
\langle T x-T y, j(x-y)\rangle \leq k\|x-y\|^{2}, \quad \forall x, y \in B . \tag{1.3}
\end{equation*}
A map S:B rarr BS: B \rightarrow B is called strongly accretive if there exist k in(0,1)k \in(0,1) and j(x-y)in J(x-y)j(x-y) \in J(x-y) such that
{:(1.4)(:Sx-Sy","j(x-y):) >= k||x-y||^(2)","quad AA x","y in B.:}\begin{equation*}
\langle S x-S y, j(x-y)\rangle \geq k\|x-y\|^{2}, \quad \forall x, y \in B . \tag{1.4}
\end{equation*}
In (1.3), take k=1k=1 to obtain a pseudocontractive map. In (1.4), take k=0k=0 to obtain an accretive map.
Let BB be a nonempty and convex subset of X,T:B rarr BX, T: B \rightarrow B and x_(0),u_(0)in Bx_{0}, u_{0} \in B. The Mann iteration (see [3]) is defined by
{:(1.5)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{1.5}
\end{equation*}
The Ishikawa iteration is defined (see [2]) by
{:[x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)Ty_(n)","],[(1.6)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}, \\
y_{n} & =\left(1-\beta_{n}\right) x_{n}+\beta_{n} T x_{n}, \tag{1.6}
\end{align*}
where {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).
Let s >= 2s \geq 2 be fixed. Let T_(i):B rarr B,1 <= i <= sT_{i}: B \rightarrow B, 1 \leq i \leq s, be a family of functions. We consider the following multistep procedure:
Let s >= 2s \geq 2 be fixed. Let T_(i):B rarr B,1 <= i <= sT_{i}: B \rightarrow B, 1 \leq i \leq s, be a family of functions. We consider the following multistep procedure:
{:[x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)T_(1)y_(n)^(1)],[(1.7)y_(n)^(i)=(1-beta_(n)^(i))x_(n)+beta_(n)^(i)T_(i+1)y_(n)^(i+1)","quad i=1","dots","s-2","],[y_(n)^(s-1)=(1-beta_(n)^(s-1))x_(n)+beta_(n)^(s-1)T_(s)x_(n)]:}\begin{gather*}
x_{n+1}=\left(1-\alpha_{n}\right) x_{n}+\alpha_{n} T_{1} y_{n}^{1} \\
y_{n}^{i}=\left(1-\beta_{n}^{i}\right) x_{n}+\beta_{n}^{i} T_{i+1} y_{n}^{i+1}, \quad i=1, \ldots, s-2, \tag{1.7}\\
y_{n}^{s-1}=\left(1-\beta_{n}^{s-1}\right) x_{n}+\beta_{n}^{s-1} T_{s} x_{n}
\end{gather*}
Let A,b in(0,1)A, b \in(0,1) be fixed. The sequence {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfies
{:[(1.8)0 < A <= alpha_(n) <= b < 2(1-k)","quad AA n inN","],[(1.9){beta_(n)^(i)}sub[0","1)","quad i=1","dots","s-1.]:}\begin{gather*}
0<A \leq \alpha_{n} \leq b<2(1-k), \quad \forall n \in \mathbb{N}, \tag{1.8}\\
\left\{\beta_{n}^{i}\right\} \subset[0,1), \quad i=1, \ldots, s-1 . \tag{1.9}
\end{gather*}
Let F(T_(1),dots,T_(s))F\left(T_{1}, \ldots, T_{s}\right) denote the common fixed points set with respect to BB for the family T_(1),dots,T_(s)T_{1}, \ldots, T_{s}. In this paper, we will prove convergence results for iteration (1.7), for strongly pseudocontractive (accretive) maps when {alpha_(n)}\left\{\alpha_{n}\right\} satisfies (1.8). These results improve the recently obtained results from [6], in which {alpha_(n)}\left\{\alpha_{n}\right\} and {beta_(n)}\left\{\beta_{n}\right\} converge to zero. We give two numerical examples in which iteration (1.7), when {alpha_(n)}\left\{\alpha_{n}\right\} satisfies (1.8), converges faster as in [6]. Note that, in both cases, iteration (1.7) converges faster than Ishikawa iteration.
Lemma 1.3 [4]. Let XX be a real Banach space, and let J:X rarr2^(X^(**))J: X \rightarrow 2^{X^{*}} be the duality mapping. Then for any given x,y in Xx, y \in X,
{:(1.10)||x+y||^(2) <= ||x||^(2)+2(:y","j(x+y):)","quad AA x","y in X","AA j(x+y)in J(x+y).:}\begin{equation*}
\|x+y\|^{2} \leq\|x\|^{2}+2\langle y, j(x+y)\rangle, \quad \forall x, y \in X, \forall j(x+y) \in J(x+y) . \tag{1.10}
\end{equation*}
Lemma 1.4 [7]. Let {a_(n)}\left\{a_{n}\right\} be a nonnegative sequence which satisfies the inequality
{:(1.11)a_(n+1) <= (1-t)a_(n)+sigma_(n)",":}\begin{equation*}
a_{n+1} \leq(1-t) a_{n}+\sigma_{n}, \tag{1.11}
\end{equation*}
where t in(0,1)t \in(0,1) is fixed, lim_(n rarr oo)sigma_(n)=0\lim _{n \rightarrow \infty} \sigma_{n}=0. Then lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0.
2. Main result
Theorem 2.1. Let s >= 2s \geq 2 be fixed, XX a real Banach space, and BB a nonempty, closed, convex subset of XX. Let T_(1):B rarr BT_{1}: B \rightarrow B be a strongly pseudocontractive operator and T_(2),dots,T_(s):B rarr BT_{2}, \ldots, T_{s}: B \rightarrow B,
with T_(i)(B)T_{i}(B) bounded for all 1 <= i <= s1 \leq i \leq s, such that F(T_(1),dots,T_(s))!=O/F\left(T_{1}, \ldots, T_{s}\right) \neq \varnothing. If A,b in(0,1),{alpha_(n)}sub(0,1)A, b \in(0,1),\left\{\alpha_{n}\right\} \subset (0,1) satisfies (1.8), x_(0)in Bx_{0} \in B, and the following condition is satisfied:
with T_(i)(B)T_{i}(B) bounded for all 1 <= i <= s1 \leq i \leq s, such that F(T_(1),dots,T_(s))!=O/F\left(T_{1}, \ldots, T_{s}\right) \neq \varnothing. If A,b in(0,1),{alpha_(n)}sub(0,1)A, b \in(0,1),\left\{\alpha_{n}\right\} \subset (0,1) satisfies (1.8), x_(0)in Bx_{0} \in B, and the following condition is satisfied:
{:(2.1)lim_(n rarr oo)||T_(1)x_(n+1)-T_(1)y_(n)^(1)||=0:}\begin{equation*}
\lim _{n \rightarrow \infty}\left\|T_{1} x_{n+1}-T_{1} y_{n}^{1}\right\|=0 \tag{2.1}
\end{equation*}
then iteration (1.7) converges to the unique common fixed point of T_(1),dots,T_(s)T_{1}, \ldots, T_{s}, which is the unique fixed point of T_(1)T_{1}.
Proof. Any common fixed point of T_(1),dots,T_(s)T_{1}, \ldots, T_{s}, in particular, is a fixed point of T_(1)T_{1}. However, T_(1)T_{1} can have at most one fixed point since it is strongly pseudocontractive. Let x^(**)=F(T_(1),dots,T_(s))x^{*}=F\left(T_{1}, \ldots, T_{s}\right). Denote
{:(2.2)M=s u p_(n inN){||T_(1)y_(n)^(1)||,||x_(0)||,||x^(**)||}:}\begin{equation*}
M=\sup _{n \in \mathbb{N}}\left\{\left\|T_{1} y_{n}^{1}\right\|,\left\|x_{0}\right\|,\left\|x^{*}\right\|\right\} \tag{2.2}
\end{equation*}
Then if we assume ||x_(n)|| <= M\left\|x_{n}\right\| \leq M, by
{:(2.3)||x_(n+1)|| <= (1-alpha_(n))||x_(n)||+alpha_(n)||T_(1)y_(n)^(1)|| <= M",":}\begin{equation*}
\left\|x_{n+1}\right\| \leq\left(1-\alpha_{n}\right)\left\|x_{n}\right\|+\alpha_{n}\left\|T_{1} y_{n}^{1}\right\| \leq M, \tag{2.3}
\end{equation*}
we get ||x_(n+1)|| <= M\left\|x_{n+1}\right\| \leq M.
From (1.2) and (1.10), with
From (1.2) and (1.10), with
{:[x:=(1-alpha_(n))(x_(n)-x^(**))","],[(2.4)y:=alpha_(n)(T_(1)y_(n)^(1)-T_(1)x^(**))","],[x+y=x_(n+1)-x^(**)","]:}\begin{align*}
x: & =\left(1-\alpha_{n}\right)\left(x_{n}-x^{*}\right), \\
y:= & \alpha_{n}\left(T_{1} y_{n}^{1}-T_{1} x^{*}\right), \tag{2.4}\\
& x+y=x_{n+1}-x^{*},
\end{align*}
we get
{:[||x_(n+1)-x^(**)||^(2)=||(1-alpha_(n))(x_(n)-x^(**))+alpha_(n)(T_(1)y_(n)^(1)-T_(1)x^(**))||^(2)],[ <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)(:T_(1)y_(n)^(1)-T_(1)x^(**),j(x_(n+1)-x^(**)):)],[=(1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)(:T_(1)x_(n+1)-T_(1)x^(**),j(x_(n+1)-x^(**)):)],[+2alpha_(n)(:T_(1)y_(n)^(1)-T_(1)x_(n+1),j(x_(n+1)-x^(**)):)],[(2.5) <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)k||x_(n+1)-x^(**)||^(2)],[+2alpha_(n)(:T_(1)y_(n)^(1)-T_(1)x_(n+1),j(x_(n+1)-x^(**)):)],[ <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)k||x_(n+1)-x^(**)||^(2)],[+2alpha_(n)||T_(1)y_(n)^(1)-T_(1)x_(n+1)||||x_(n+1)-x^(**)||],[ <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)k||x_(n+1)-x^(**)||^(2)],[+4alpha_(n)||T_(1)y_(n)^(1)-T_(1)x_(n+1)||M.]:}\begin{align*}
\left\|x_{n+1}-x^{*}\right\|^{2}= & \left\|\left(1-\alpha_{n}\right)\left(x_{n}-x^{*}\right)+\alpha_{n}\left(T_{1} y_{n}^{1}-T_{1} x^{*}\right)\right\|^{2} \\
\leq & \left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n}\left\langle T_{1} y_{n}^{1}-T_{1} x^{*}, j\left(x_{n+1}-x^{*}\right)\right\rangle \\
= & \left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n}\left\langle T_{1} x_{n+1}-T_{1} x^{*}, j\left(x_{n+1}-x^{*}\right)\right\rangle \\
& +2 \alpha_{n}\left\langle T_{1} y_{n}^{1}-T_{1} x_{n+1}, j\left(x_{n+1}-x^{*}\right)\right\rangle \\
\leq & \left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n} k\left\|x_{n+1}-x^{*}\right\|^{2} \tag{2.5}\\
& +2 \alpha_{n}\left\langle T_{1} y_{n}^{1}-T_{1} x_{n+1}, j\left(x_{n+1}-x^{*}\right)\right\rangle \\
\leq & \left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n} k\left\|x_{n+1}-x^{*}\right\|^{2} \\
& +2 \alpha_{n}\left\|T_{1} y_{n}^{1}-T_{1} x_{n+1}\right\|\left\|x_{n+1}-x^{*}\right\| \\
\leq & \left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n} k\left\|x_{n+1}-x^{*}\right\|^{2} \\
& +4 \alpha_{n}\left\|T_{1} y_{n}^{1}-T_{1} x_{n+1}\right\| M .
\end{align*}
Using (1.8), we obtain
{:(2.6)(1-alpha_(n))^(2) <= 1-2alpha_(n)+alpha_(n)b < 1-2alpha_(n)+alpha_(n)2(1-k)=1-2alpha_(n)k:}\begin{equation*}
\left(1-\alpha_{n}\right)^{2} \leq 1-2 \alpha_{n}+\alpha_{n} b<1-2 \alpha_{n}+\alpha_{n} 2(1-k)=1-2 \alpha_{n} k \tag{2.6}
\end{equation*}
thus,
{:(2.7)||x_(n+1)-x^(**)||^(2) <= ((1-alpha_(n))^(2))/(1-2alpha_(n)k)||x_(n)-x^(**)||^(2)+(4alpha_(n)M)/(1-2alpha_(n)k)||T_(1)y_(n)^(1)-T_(1)x_(n+1)||:}\begin{equation*}
\left\|x_{n+1}-x^{*}\right\|^{2} \leq \frac{\left(1-\alpha_{n}\right)^{2}}{1-2 \alpha_{n} k}\left\|x_{n}-x^{*}\right\|^{2}+\frac{4 \alpha_{n} M}{1-2 \alpha_{n} k}\left\|T_{1} y_{n}^{1}-T_{1} x_{n+1}\right\| \tag{2.7}
\end{equation*}
The following inequality is satisfied:
{:[((1-alpha_(n))^(2))/(1-2alpha_(n)k)=((1-alpha_(n))^(2)(1-2alpha_(n)k+2alpha_(n)k))/(1-2alpha_(n)k)=(1-alpha_(n))^(2)(1+(2alpha_(n)k)/(1-2alpha_(n)k))],[=(1-alpha_(n))^(2)+(2alpha_(n)k(1-alpha_(n))^(2))/(1-2alpha_(n)k) <= (1-alpha_(n))^(2)+2alpha_(n)k <= 1-2alpha_(n)+alpha_(n)b+2alpha_(n)k],[(2.8)=1-(2(1-k)-b)alpha_(n) <= 1-(2(1-k)-b)A]:}\begin{align*}
\frac{\left(1-\alpha_{n}\right)^{2}}{1-2 \alpha_{n} k} & =\frac{\left(1-\alpha_{n}\right)^{2}\left(1-2 \alpha_{n} k+2 \alpha_{n} k\right)}{1-2 \alpha_{n} k}=\left(1-\alpha_{n}\right)^{2}\left(1+\frac{2 \alpha_{n} k}{1-2 \alpha_{n} k}\right) \\
& =\left(1-\alpha_{n}\right)^{2}+\frac{2 \alpha_{n} k\left(1-\alpha_{n}\right)^{2}}{1-2 \alpha_{n} k} \leq\left(1-\alpha_{n}\right)^{2}+2 \alpha_{n} k \leq 1-2 \alpha_{n}+\alpha_{n} b+2 \alpha_{n} k \\
& =1-(2(1-k)-b) \alpha_{n} \leq 1-(2(1-k)-b) A \tag{2.8}
\end{align*}
Substituting (2.6) and (2.8) into (2.7), we obtain
{:(2.9)||x_(n+1)-x^(**)||^(2) <= (1-(2(1-k)-b)A)||x_(n)-x^(**)||^(2)+(4bM)/(1-2bk)||T_(1)y_(n)^(1)-T_(1)x_(n+1)||:}\begin{equation*}
\left\|x_{n+1}-x^{*}\right\|^{2} \leq(1-(2(1-k)-b) A)\left\|x_{n}-x^{*}\right\|^{2}+\frac{4 b M}{1-2 b k}\left\|T_{1} y_{n}^{1}-T_{1} x_{n+1}\right\| \tag{2.9}
\end{equation*}
Set
{:[a_(n):=||x_(n)-x^(**)||^(2)],[(2.10)t:=(2(1-k)-b)A in(0","1)],[sigma_(n):=(4bM)/(1-2bk)||T_(1)y_(n)^(1)-T_(1)x_(n+1)||]:}\begin{gather*}
a_{n}:=\left\|x_{n}-x^{*}\right\|^{2} \\
t:=(2(1-k)-b) A \in(0,1) \tag{2.10}\\
\sigma_{n}:=\frac{4 b M}{1-2 b k}\left\|T_{1} y_{n}^{1}-T_{1} x_{n+1}\right\|
\end{gather*}
From (2.1), we know that lim_(n rarr oo)sigma_(n)=0\lim _{n \rightarrow \infty} \sigma_{n}=0; all the assumptions of Lemma 1.4 are fulfilled and consequently we have lim_(n rarr oo)||x_(n)-x^(**)||=0\lim _{n \rightarrow \infty}\left\|x_{n}-x^{*}\right\|=0.
In Theorem 2.1, {alpha_(n)}\left\{\alpha_{n}\right\} does not converge to zero, while in [6], {alpha_(n)}\left\{\alpha_{n}\right\} converges to zero.
Theorem 2.2 [6]. Let s >= 2s \geq 2 be fixed, XX a real Banach space with a uniformly convex dual, and BB a nonempty, closed, convex subset of XX. Let T_(1):B rarr BT_{1}: B \rightarrow B be a strongly pseudocontractive operator and T_(2),dots,T_(s):B rarr BT_{2}, \ldots, T_{s}: B \rightarrow B, with T_(i)(B)T_{i}(B) bounded for all 1 <= i <= s1 \leq i \leq s, such that F(T_(1),dots,T_(s))!=O/F\left(T_{1}, \ldots, T_{s}\right) \neq \varnothing. If {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfies lim_(n rarr oo)alpha_(n)=0,sum_(n=1)^(oo)alpha_(n)=+oo\lim _{n \rightarrow \infty} \alpha_{n}=0, \sum_{n=1}^{\infty} \alpha_{n}=+\infty, and {beta_(n)^(i)}sub[0,1),i=1,dots,s-1\left\{\beta_{n}^{i}\right\} \subset [0,1), i=1, \ldots, s-1, satisfy lim_(n rarr oo)beta_(n)^(1)=0\lim _{n \rightarrow \infty} \beta_{n}^{1}=0, then iteration (1.7) converges to a fixed point of T_(1),dots,T_(s)T_{1}, \ldots, T_{s}.
Theorem 2.2 [6]. Let s >= 2s \geq 2 be fixed, XX a real Banach space with a uniformly convex dual, and BB a nonempty, closed, convex subset of XX. Let T_(1):B rarr BT_{1}: B \rightarrow B be a strongly pseudocontractive operator and T_(2),dots,T_(s):B rarr BT_{2}, \ldots, T_{s}: B \rightarrow B, with T_(i)(B)T_{i}(B) bounded for all 1 <= i <= s1 \leq i \leq s, such that F(T_(1),dots,T_(s))!=O/F\left(T_{1}, \ldots, T_{s}\right) \neq \varnothing. If {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1) satisfies lim_(n rarr oo)alpha_(n)=0,sum_(n=1)^(oo)alpha_(n)=+oo\lim _{n \rightarrow \infty} \alpha_{n}=0, \sum_{n=1}^{\infty} \alpha_{n}=+\infty, and {beta_(n)^(i)}sub[0,1),i=1,dots,s-1\left\{\beta_{n}^{i}\right\} \subset [0,1), i=1, \ldots, s-1, satisfy lim_(n rarr oo)beta_(n)^(1)=0\lim _{n \rightarrow \infty} \beta_{n}^{1}=0, then iteration (1.7) converges to a fixed point of T_(1),dots,T_(s)T_{1}, \ldots, T_{s}.
The Banach space in Theorem 2.1 contains no restrictions.
3. Further results
Denote by II the identity map.
Remark 3.1. Let T,S:X rarr XT, S: X \rightarrow X and let f in Xf \in X be given. Then,
(i) a fixed point for the map Tx=f+(I-S)xT x=f+(I-S) x, for all x in Xx \in X, is a solution for Sx=fS x=f;
(ii) a fixed point for Tx=f-SxT x=f-S x is a solution for x+Sx=fx+S x=f.
Remark 3.1. Let T,S:X rarr XT, S: X \rightarrow X and let f in Xf \in X be given. Then,
(i) a fixed point for the map Tx=f+(I-S)xT x=f+(I-S) x, for all x in Xx \in X, is a solution for Sx=fS x=f;
(ii) a fixed point for Tx=f-SxT x=f-S x is a solution for x+Sx=fx+S x=f.
Remark 3.2 [5]. The following are true.
(i) The operator T:X rarr XT: X \rightarrow X is a (strongly) pseudocontractive map if and only if ( I-T):X rarr XI- T): X \rightarrow X is (strongly) accretive.
(ii) If S:X rarr XS: X \rightarrow X is an accretive map, then T=f-S:X rarr XT=f-S: X \rightarrow X is a strongly pseudocontractive map.
(i) The operator T:X rarr XT: X \rightarrow X is a (strongly) pseudocontractive map if and only if ( I-T):X rarr XI- T): X \rightarrow X is (strongly) accretive.
(ii) If S:X rarr XS: X \rightarrow X is an accretive map, then T=f-S:X rarr XT=f-S: X \rightarrow X is a strongly pseudocontractive map.
We consider iteration (1.7), with T_(i)x=f_(i)+(I-S_(i))x,1 <= i <= sT_{i} x=f_{i}+\left(I-S_{i}\right) x, 1 \leq i \leq s and s >= 2,{alpha_(n)}sub(0,1)s \geq 2,\left\{\alpha_{n}\right\} \subset(0,1), {beta_(n)^(i)}sub[0,1),i=1,dots,s-1\left\{\beta_{n}^{i}\right\} \subset[0,1), i=1, \ldots, s-1 satisfying (1.8):
{:[x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)(f_(1)+(I-S_(1))y_(n)^(1))],[(3.1)y_(n)^(i)=(1-beta_(n)^(i))x_(n)+beta_(n)^(i)(f_(i+1)+(I-S_(i+1))y_(n)^(i+1))","quad i=1","dots","s-2],[y_(n)^(s-1)=(1-beta_(n)^(s-1))x_(n)+beta_(n)^(s-1)(f_(s-1)+(I-S_(s))x_(n))]:}\begin{gather*}
x_{n+1}=\left(1-\alpha_{n}\right) x_{n}+\alpha_{n}\left(f_{1}+\left(I-S_{1}\right) y_{n}^{1}\right) \\
y_{n}^{i}=\left(1-\beta_{n}^{i}\right) x_{n}+\beta_{n}^{i}\left(f_{i+1}+\left(I-S_{i+1}\right) y_{n}^{i+1}\right), \quad i=1, \ldots, s-2 \tag{3.1}\\
y_{n}^{s-1}=\left(1-\beta_{n}^{s-1}\right) x_{n}+\beta_{n}^{s-1}\left(f_{s-1}+\left(I-S_{s}\right) x_{n}\right)
\end{gather*}
Theorem 2.1, Remark 3.1(i), and Remark 3.2(i) lead to the following result.
Corollary 3.3. Let s >= 2s \geq 2 be fixed, XX a real Banach space, and S_(1):X rarr XS_{1}: X \rightarrow X a strongly accretive operator, S_(2),dots,S_(s):X rarr XS_{2}, \ldots, S_{s}: X \rightarrow X, such that the equations S_(i)x=f_(i),1 <= i <= sS_{i} x=f_{i}, 1 \leq i \leq s, have a common solution and T_(i)(X),1 <= i <= sT_{i}(X), 1 \leq i \leq s, are bounded. If A,b in(0,1),{alpha_(n)}sub(0,1)A, b \in(0,1),\left\{\alpha_{n}\right\} \subset(0,1) satisfies (1.8), and condition (2.1) is satisfied, then iteration (3.1) converges to a common solution of S_(i)x=f_(i)S_{i} x=f_{i}, 1 <= i <= s1 \leq i \leq s.
Corollary 3.3. Let s >= 2s \geq 2 be fixed, XX a real Banach space, and S_(1):X rarr XS_{1}: X \rightarrow X a strongly accretive operator, S_(2),dots,S_(s):X rarr XS_{2}, \ldots, S_{s}: X \rightarrow X, such that the equations S_(i)x=f_(i),1 <= i <= sS_{i} x=f_{i}, 1 \leq i \leq s, have a common solution and T_(i)(X),1 <= i <= sT_{i}(X), 1 \leq i \leq s, are bounded. If A,b in(0,1),{alpha_(n)}sub(0,1)A, b \in(0,1),\left\{\alpha_{n}\right\} \subset(0,1) satisfies (1.8), and condition (2.1) is satisfied, then iteration (3.1) converges to a common solution of S_(i)x=f_(i)S_{i} x=f_{i}, 1 <= i <= s1 \leq i \leq s.
We consider iteration (1.7), with T_(i)x=f_(i)-S_(i)x,1 <= i <= sT_{i} x=f_{i}-S_{i} x, 1 \leq i \leq s, and s >= 2,{alpha_(n)}sub(0,1)s \geq 2,\left\{\alpha_{n}\right\} \subset(0,1), {beta_(n)^(i)}sub[0,1),i=1,dots,s-1\left\{\beta_{n}^{i}\right\} \subset[0,1), i=1, \ldots, s-1, satisfying (1.8):
{:[x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)(f_(1)-S_(1)y_(n)^(1))],[(3.2)y_(n)^(i)=(1-beta_(n)^(i))x_(n)+beta_(n)^(i)(f_(i+1)-S_(i+1)y_(n)^(i+1))","quad i=1","dots","s-2],[y_(n)^(s-1)=(1-beta_(n)^(s-1))x_(n)+beta_(n)^(s-1)(f_(s-1)-S_(s)x_(n))]:}\begin{gather*}
x_{n+1}=\left(1-\alpha_{n}\right) x_{n}+\alpha_{n}\left(f_{1}-S_{1} y_{n}^{1}\right) \\
y_{n}^{i}=\left(1-\beta_{n}^{i}\right) x_{n}+\beta_{n}^{i}\left(f_{i+1}-S_{i+1} y_{n}^{i+1}\right), \quad i=1, \ldots, s-2 \tag{3.2}\\
y_{n}^{s-1}=\left(1-\beta_{n}^{s-1}\right) x_{n}+\beta_{n}^{s-1}\left(f_{s-1}-S_{s} x_{n}\right)
\end{gather*}
Theorem 2.1, Remark 3.1(ii), and Remark 3.2(ii) lead to the following result.
Corollary 3.4. Let s >= 2s \geq 2 be fixed, XX a real Banach space, and S_(1):X rarr XS_{1}: X \rightarrow X an accretive operator, S_(2),dots,S_(s):X rarr XS_{2}, \ldots, S_{s}: X \rightarrow X, such that the equations x+S_(i)x=f_(i),1 <= i <= sx+S_{i} x=f_{i}, 1 \leq i \leq s, have a common solution and S_(i)(X),1 <= i <= sS_{i}(X), 1 \leq i \leq s, are bounded. If A,b in(0,1),{alpha_(n)}sub(0,1)A, b \in(0,1),\left\{\alpha_{n}\right\} \subset(0,1) satisfies (1.8), and condition (2.1) is satisfied, then iteration (3.2) converges to a common solution of x+S_(i)x=f_(i),1 <= i <= sx+S_{i} x= f_{i}, 1 \leq i \leq s.
Corollary 3.4. Let s >= 2s \geq 2 be fixed, XX a real Banach space, and S_(1):X rarr XS_{1}: X \rightarrow X an accretive operator, S_(2),dots,S_(s):X rarr XS_{2}, \ldots, S_{s}: X \rightarrow X, such that the equations x+S_(i)x=f_(i),1 <= i <= sx+S_{i} x=f_{i}, 1 \leq i \leq s, have a common solution and S_(i)(X),1 <= i <= sS_{i}(X), 1 \leq i \leq s, are bounded. If A,b in(0,1),{alpha_(n)}sub(0,1)A, b \in(0,1),\left\{\alpha_{n}\right\} \subset(0,1) satisfies (1.8), and condition (2.1) is satisfied, then iteration (3.2) converges to a common solution of x+S_(i)x=f_(i),1 <= i <= sx+S_{i} x= f_{i}, 1 \leq i \leq s.
4. Numerical examples
Let X=R^(2)X=\mathbb{R}^{2} be the euclidean plane, consider x=(a,b)inR^(2)x=(a, b) \in \mathbb{R}^{2}, with x^(_|_)=(b,-a)inR^(2)x^{\perp}=(b,-a) \in \mathbb{R}^{2}. We know that (:x,x^(_|_):)=0,||x||=||x^(_|_)||,(:x^(_|_),y^(_|_):)=(:x,y:),||x^(_|_)-y^(_|_)||=||x-y||\left\langle x, x^{\perp}\right\rangle=0,\|x\|=\left\|x^{\perp}\right\|,\left\langle x^{\perp}, y^{\perp}\right\rangle=\langle x, y\rangle,\left\|x^{\perp}-y^{\perp}\right\|=\|x-y\|, and (:x^(_|_),y:)+(:x,y^(_|_):)=0\left\langle x^{\perp}, y\right\rangle+\left\langle x, y^{\perp}\right\rangle=0, for all x,y inR^(2)x, y \in \mathbb{R}^{2}. Denote by BB the closed unit ball. In [1], we can get the following example in which Ishikawa iteration (1.6) converges and (1.5) is not convergent.
Table 4.1
| \Iteration (1.7) | Case 1 | Case 2 |
| Step 10 | (0.2217,0.1480) | (0.0151, -0.0023) |
| Step 15 | (0.1837, 0.1184) | (0.0017,-0.0006) |
| Step 20 | (0.1603, 0.1015) | (0.0002, -0.0001) |
| Step 21 | (0.1566, 0.0989) | 10^(-3)*(0.1156,-0.0686)10^{-3} \cdot(0.1156,-0.0686) |
| Step 22 | (0.1531, 0.0965) | 10^(-4)*(0.7406,-0.4641)10^{-4} \cdot(0.7406,-0.4641) |
| Step 23 | (0.1499, 0.0942) | 10^(-4)*(0.4743,-0.3129)10^{-4} \cdot(0.4743,-0.3129) |
| Step 24 | (0.1468, 0.0921) | 10^(-4)*(0.3037,-0.2103)10^{-4} \cdot(0.3037,-0.2103) |
| Step 25 | (0.1440, 0.0902) | 10^(-4)*(0.1945,-0.1409)10^{-4} \cdot(0.1945,-0.1409) |
Example 4.1 [1]. Let H=R^(2)H=\mathbb{R}^{2} and let
{:(4.1)B_(1)={x inR^(2):||x|| <= (1)/(2)}","quadB_(2)={x inR^(2):(1)/(2) <= ||x|| <= 1}.:}\begin{equation*}
B_{1}=\left\{x \in \mathbb{R}^{2}:\|x\| \leq \frac{1}{2}\right\}, \quad B_{2}=\left\{x \in \mathbb{R}^{2}: \frac{1}{2} \leq\|x\| \leq 1\right\} . \tag{4.1}
\end{equation*}
The map T:B rarr BT: B \rightarrow B is given by
{:(4.2)Tx={[x+x^(_|_)",",x inB_(1)],[(x)/(||x||)-x+x^(_|_)",",x inB_(2).]:}:}T x= \begin{cases}x+x^{\perp}, & x \in B_{1} \tag{4.2}\\ \frac{x}{\|x\|}-x+x^{\perp}, & x \in B_{2} .\end{cases}
Then the following are true:
(i) TT is Lipschitz and pseudocontractive;
(ii) for all (alpha_(n))_(n)sub(0,1)\left(\alpha_{n}\right)_{n} \subset(0,1), the Mann iteration does not converge to the fixed point of TT (which is the point (0,0)(0,0) and it is unique).
(i) TT is Lipschitz and pseudocontractive;
(ii) for all (alpha_(n))_(n)sub(0,1)\left(\alpha_{n}\right)_{n} \subset(0,1), the Mann iteration does not converge to the fixed point of TT (which is the point (0,0)(0,0) and it is unique).
The main result from [2] assures the convergence of the Ishikawa iteration (1.6) applied to the map TT given by (4.2). The convergence is very slow. In [6], for the same TT, it was shown that iteration (1.7) converges faster. Here, we give an example for which (1.7) with {alpha_(n)}\left\{\alpha_{n}\right\} satisfying (1.8) converges even faster as in [6].
Case 1 [6]. Consider now T_(1)(x,y)=0.5*(x,y)T_{1}(x, y)=0.5 \cdot(x, y), for all (x,y)in B,T_(2)=T(x, y) \in B, T_{2}=T, and s=2s=2, where TT is given by (4.2), the initial point x_(0)=(0.5,0.7)x_{0}=(0.5,0.7), and alpha_(n)=beta_(n)=1//(n+1)\alpha_{n}=\beta_{n}=1 /(n+1) in (1.7). The main result from [6] assures the convergence of (1.7).
Case 2. Consider T_(1)(x,y)=0.5*(x,y)T_{1}(x, y)=0.5 \cdot(x, y), for all (x,y)in B,T_(2)=T(x, y) \in B, T_{2}=T, and s=2s=2, where TT is given by (4.2), the initial point x_(0)=(0.5,0.7),alpha_(n)=0.7x_{0}=(0.5,0.7), \alpha_{n}=0.7, for all n inNn \in \mathbb{N}, and beta_(n)=1//(n+1)\beta_{n}=1 /(n+1) in (1.7). The fixed point for both functions is (0,0)(0,0). Observe that k=0.5k=0.5, and {alpha_(n)}\left\{\alpha_{n}\right\} satisfies (1.8):
{:(4.3)A=0.7=alpha_(n)=b <= 2(1-k)=1","quad AA n inN.:}\begin{equation*}
A=0.7=\alpha_{n}=b \leq 2(1-k)=1, \quad \forall n \in \mathbb{N} . \tag{4.3}
\end{equation*}
Note that Mann iteration does not converge for any {alpha_(n)}sub(0,1)\left\{\alpha_{n}\right\} \subset(0,1). Using a Matlab program, we obtain Table 4.1.
Case 3. Consider in (1.7) the same T_(1),T_(2),s=2T_{1}, T_{2}, s=2, and x_(0)x_{0} as in Case 1 and alpha_(n)=beta_(n)=1//sqrt(n+1)\alpha_{n}=\beta_{n}= 1 / \sqrt{n+1}.
Table 4.2
| \Iteration | Case 3 (1.7) | Case 4 (1.7) | Ishikawa iteration |
| Step 10 | (0.0631, -0.0333) | (0.0044, -0.0164) | (0.4545, 0.2689) |
| Step 15 | (0.0256, -0.0221) | (-0.0010, -0.0018) | (0.1289, -0.4827) |
| Step 20 | (0.0117,-0.0139) | 10^(-5)*(-22.6516,-11.0267)10^{-5} \cdot(-22.6516,-11.0267) | (-0.4456, -0.1532) |
| Step 11 | (0.0101, -0.0126) | 10^(-5)*(-15.5657,-5.4373)10^{-5} \cdot(-15.5657,-5.4373) | (-0.4651,-0.0274) |
| Step 22 | (0.0087,-0.0115) | 10^(-5)*(-10.5234,-2.3727)10^{-5} \cdot(-10.5234,-2.3727) | (-0.4511, 0.0941) |
| Step 23 | (0.0075,-0.0105) | 10^(-5)*(-7.0134,-0.7743)10^{-5} \cdot(-7.0134,-0.7743) | (-0.4077, 0.2037) |
| Step 24 | (0.0066, -0.0096) | 10^(-5)*(-4.6140,-0.0022)10^{-5} \cdot(-4.6140,-0.0022) | (-0.3407, 0.2954) |
| Step 25 | (0.0057,-0.0088) | 10^(-5)*(-2.9993,0.3215)10^{-5} \cdot(-2.9993,0.3215) | (-0.2562, 0.3654) |
| Step 1500 | - | - | (0.0790, -0.0311) |
Case 4. Consider in (1.7) T_(1),T_(2),s=2T_{1}, T_{2}, s=2, and x_(0)x_{0} as above and alpha_(n)=0.7\alpha_{n}=0.7, for all n inN,beta_(n)=1//sqrt(n+1)n \in \mathbb{N}, \beta_{n}= 1 / \sqrt{n+1}.
Also, consider the Ishikawa iteration with the same TT as in (4.2), x_(0)=(0.5,0.7),alpha_(n)=beta_(n)=1//sqrt(n+1)x_{0}=(0.5,0.7), \alpha_{n}= \beta_{n}=1 / \sqrt{n+1}, for all n inNn \in \mathbb{N}. The main result from [2] assures the convergence of Ishikawa iteration. Note that in this case the convergence is very slow. Eventually, Example 4.1 assures that for the same map, Mann iteration does not converge. A Matlab program leads to the evaluations illustrated in Table 4.2.
Acknowledgment
The authors are indebted to the referee for carefully reading the paper and for making useful suggestions.
References
[1] C. E. Chidume and S. A. Mutangadura, An example of the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc. 129 (2001), no. 8, 2359-2363.
[2] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), no. 1, 147-150.
[3] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
[4] C. H. Morales and J. S. Jung, Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3411-3419.
[5] B. E. Rhoades and Ş. M. Şoltuz, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Int. J. Math. Math. Sci. 2003 (2003), no. 42, 2645-2651.
[6] -, Mean value iteration for a family of functions, to appear in Nonlinear Funct. Anal. Appl.
[7] Ş. M. Şoltuz, Some sequences supplied by inequalities and their applications, Rev. Anal. Numér. Théor. Approx. 29 (2000), no. 2, 207-212.
[2] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), no. 1, 147-150.
[3] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
[4] C. H. Morales and J. S. Jung, Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3411-3419.
[5] B. E. Rhoades and Ş. M. Şoltuz, The equivalence of Mann iteration and Ishikawa iteration for non-Lipschitzian operators, Int. J. Math. Math. Sci. 2003 (2003), no. 42, 2645-2651.
[6] -, Mean value iteration for a family of functions, to appear in Nonlinear Funct. Anal. Appl.
[7] Ş. M. Şoltuz, Some sequences supplied by inequalities and their applications, Rev. Anal. Numér. Théor. Approx. 29 (2000), no. 2, 207-212.
- B. E. Rhoades: Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USAE-mail address: rhoades@indiana.edu
Ştefan M. Şoltuz: Institute of Numerical Analysis, P.O. Box 68-1, 400110 Cluj-Napoca, Romania
E-mail address: smsoltuz@gmail.com