Abstract
We give a correction to the main result from [13].
Authors
Soltuz M. Stefan
Tiberiu Popoviciu Institute of Numerical analysis
Keywords
Ishikawa iteration, strongly pseudocontractive maps
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Ş.M. Şoltuz, A correction for a result on convergence of Ishikawa iteration for strongly pseudocontractive maps, Math. Commun. 7 (2002) no. 1, 61-64.
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Mathematical Communications
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[1] S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung, S. M. Kang, Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl. 224(1998), 149–165.
[2] K. Deimling, Zeroes of Accretive Operators, Manuscripta Math. 13(1974), 365-374.
[3] Gu Feng, Iteration processes for approximating fixed points of operators of monotone type, Proc. Amer. Math. Soc. 129(2001), 2293-2300.
[4] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44(1974), 147-150.
[5] W. R. Mann, Mean value in iteration, Proc. Amer. Math. Soc. 4(1953), 506-510.
[6] C. Morales, J. S. Jung, Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128:11(2000), 3411-3419.
[7] M. O. Osilike, A note on the stability of iteration procedures for strongly pseudocontractions and strongly accretive type equations, J. Math. Anal. Appl.
250(2000), 726-730.
[8] J. A. Park, Mann-iteration for strictly pseudocontractive maps, J. Korean Math. Soc. 31(1994), 333-337.
[9] S. M. S¸oltuz, Some sequences supplied by inequalities and their applications, Revue d’analyse num´erique et de th´eorie de l’approximation, Tome 29(2000),
207-212.
[10] S. M. Soltuz, Three proofs for the convergence of a sequence, OCTOGON Math. Mag. 9(2001), 503-505.
[11] H. Y. Zhou, Y. Jia, Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption, Proc. Amer. Math. Soc. 125(1997), 1705-1709.
[12] H. Y. Zhou, Y. J. Cho, Ishikawa and Mann iterative process with errors for nonlinear φ− strongly quasi-accretive mappings in normed spaces, J. Korean Math. Soc. 36(1999), 1061-1073.
[13] H. Y. Zhou, Stable iteration procedures for strong pseudocontractions and nonlinear equations involving accretive operators without Lipschitz assumption, J. Math. Anal. Appl. 230(1999), 1-30.
Paper (preprint) in HTML form
A correction for a result on convergence of Ishikawa iteration for strongly pseudocontractive maps
Abstract
We give a correction to the main result from [13]. Key words: Ishikawa iteration, strongly pseudocontractive maps AMS subject classifications: Primary 47H10, Secondary 47H06
Received December 11, 2001 Accepted June 27, 2002
1. Introduction
Let XX be a real Banach space. Let BB be a nonempty, convex subset of XX. Let T:B rarr BT: B \rightarrow B be a map. Let x_(1)in Bx_{1} \in B. We consider the following iteration, see [4]:
{:[(I)x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)Ty_(n)","],[y_(n)=(1-beta_(n))x_(n)+beta_(n)Tx_(n)","quad n=1","2","dots.]:}\begin{align*}
x_{n+1} & =\left(1-\alpha_{n}\right) x_{n}+\alpha_{n} T y_{n}, \tag{I}\\
y_{n} & =\left(1-\beta_{n}\right) x_{n}+\beta_{n} T x_{n}, \quad n=1,2, \ldots .
\end{align*}
We suppose that (alpha_(n))_(n),(beta_(n))_(n)sub(0,1)\left(\alpha_{n}\right)_{n},\left(\beta_{n}\right)_{n} \subset(0,1), and the sequence (alpha_(n))_(n)\left(\alpha_{n}\right)_{n} satisfies
{:(1)0 < w <= alpha_(n) <= 1:}\begin{equation*}
0<w \leq \alpha_{n} \leq 1 \tag{1}
\end{equation*}
For beta_(n)=0,AA n in N\beta_{n}=0, \forall n \in N we get Mann iteration, see [5]. Ishikawa iteration with condition (1) is studied in [13]. In [7] it was proven that two assumptions of the main theorem from [13] are contradictory. In this note we will prove that renouncing to one assumption from [13] and supposing true an assumption à la [3], the above theorem from [13] is true.
The map J:X rarr2^(X^(**))J: X \rightarrow 2^{X^{*}} given by
Jx:={f inX^(**):(:x,f:)=||x||^(2),||f||=||x||},AA x in X,J x:=\left\{f \in X^{*}:\langle x, f\rangle=\|x\|^{2},\|f\|=\|x\|\right\}, \forall x \in X,
is called the normalized duality mapping. The Hahn-Banach theorem assures that Jx!=O/,AA x in XJ x \neq \emptyset, \forall x \in X. It is easy to see that we have
{:(2)(:j(x)","y:) <= ||x||||y||","AA x","y in X","AA j(x)in J(x):}\begin{equation*}
\langle j(x), y\rangle \leq\|x\|\|y\|, \forall x, y \in X, \forall j(x) \in J(x) \tag{2}
\end{equation*}
Definition 1. Let XX be a real Banach space, let BB be a nonempty subset. AA map T:B rarr BT: B \rightarrow B is called strongly pseudocontractive if for all x,y in Bx, y \in B, there exists j(x-y)in J(x-y)j(x-y) \in J(x-y) such that
{:(3)EE gamma in(0","1):(:Tx-Ty","j(x-y):) <= gamma||x-y||^(2).:}\begin{equation*}
\exists \gamma \in(0,1):\langle T x-T y, j(x-y)\rangle \leq \gamma\|x-y\|^{2} . \tag{3}
\end{equation*}
The following Lemma could be found in [6], [12], with different proofs. A particular form of this lemma is in [11].
Lemma 1. [6], [11], [12] If XX is a real Banach space, then the following relation is true
{:(4)||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{4}
\end{equation*}
The following result is from [9]. Three other proofs could be found in [10].
Proposition 1. [9], [10]. Let (a_(n))_(n)\left(a_{n}\right)_{n} be a nonnegative sequence which satisfies
Proposition 1. [9], [10]. Let (a_(n))_(n)\left(a_{n}\right)_{n} be a nonnegative sequence which satisfies
{:(5)a_(n+1) <= (1-w)a_(n)+sigma_(n)S", ":}\begin{equation*}
a_{n+1} \leq(1-w) a_{n}+\sigma_{n} S \text {, } \tag{5}
\end{equation*}
where w in(0,1),S > 0w \in(0,1), S>0 are fixed numbers, sigma_(n) >= 0,AA n in N,lim_(n rarr oo)sigma_(n)=0\sigma_{n} \geq 0, \forall n \in N, \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
We are able now to give the following result.
Theorem 1. Let XX be a real Banach space, and let T:X rarr XT: X \rightarrow X be a continuous, strongly pseudocontractive with bounded range map. If lim_(n rarr oo)||Ty_(n)-Tx_(n+1)||=0\lim _{n \rightarrow \infty}\left\|T y_{n}-T x_{n+1}\right\|=0, and gamma in(0,1//2)\gamma \in(0,1 / 2), then the iteration (I)(I) strongly converges to the unique fixed point of TT.
Theorem 1. Let XX be a real Banach space, and let T:X rarr XT: X \rightarrow X be a continuous, strongly pseudocontractive with bounded range map. If lim_(n rarr oo)||Ty_(n)-Tx_(n+1)||=0\lim _{n \rightarrow \infty}\left\|T y_{n}-T x_{n+1}\right\|=0, and gamma in(0,1//2)\gamma \in(0,1 / 2), then the iteration (I)(I) strongly converges to the unique fixed point of TT.
Proof. . The existence follows from [2] and the uniqueness from strongly pseudocontractivity. Let x^(**)=Tx^(**)x^{*}=T x^{*}. Using (4) for the first inequality, (2) and (3) for the third one, we can see:
{:[||x_(n+1)-x^(**)||^(2)=||(1-alpha_(n))(x_(n)-x^(**))+alpha_(n)(Ty_(n)-x^(**))||^(2)],[ <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)(:Ty_(n)-x^(**),j(x_(n+1)-x^(**)):)],[ <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)(:Ty_(n)-Tx_(n+1),j(x_(n+1)-x^(**)):)],[+2alpha_(n)(:Tx_(n+1)-x^(**),J(x_(n+1)-x^(**)):)],[ <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)||Ty_(n)-Tx_(n+1)||||x_(n+1)-x^(**)||],[+2alpha_(n)gamma||x_(n+1)-x^(**)||^(2)","quad AA j(x_(n+1)-x^(***))in J(x_(n+1)-x^(***)).]:}\begin{aligned}
& \left\|x_{n+1}-x^{*}\right\|^{2}=\left\|\left(1-\alpha_{n}\right)\left(x_{n}-x^{*}\right)+\alpha_{n}\left(T y_{n}-x^{*}\right)\right\|^{2} \\
& \leq\left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n}\left\langle T y_{n}-x^{*}, 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}\left\langle T y_{n}-T x_{n+1}, j\left(x_{n+1}-x^{*}\right)\right\rangle \\
& +2 \alpha_{n}\left\langle T x_{n+1}-x^{*}, 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}\left\|T y_{n}-T x_{n+1}\right\|\left\|x_{n+1}-x^{*}\right\| \\
& +2 \alpha_{n} \gamma\left\|x_{n+1}-x^{*}\right\|^{2}, \quad \forall j\left(x_{n+1}-x^{\star}\right) \in J\left(x_{n+1}-x^{\star}\right) .
\end{aligned}
There results
{:[(1-2alpha_(n)gamma)||x_(n+1)-x^(**)||^(2) <= (1-alpha_(n))^(2)||x_(n)-x^(**)||^(2)+2alpha_(n)||Ty_(n)-Tx_(n+1)||||x_(n+1)-x^(**)||","],[||x_(n+1)-x^(**)||^(2) <= ((1-alpha_(n))^(2))/((1-2alpha_(n)gamma))||x_(n)-x^(**)||^(2)+(2alpha_(n))/((1-2alpha_(n)gamma))||Ty_(n)-Tx_(n+1)||||x_(n+1)-x^(**)||.]:}\begin{aligned}
& \left(1-2 \alpha_{n} \gamma\right)\left\|x_{n+1}-x^{*}\right\|^{2} \leq\left(1-\alpha_{n}\right)^{2}\left\|x_{n}-x^{*}\right\|^{2}+2 \alpha_{n}\left\|T y_{n}-T x_{n+1}\right\|\left\|x_{n+1}-x^{*}\right\|, \\
& \left\|x_{n+1}-x^{*}\right\|^{2} \leq \frac{\left(1-\alpha_{n}\right)^{2}}{\left(1-2 \alpha_{n} \gamma\right)}\left\|x_{n}-x^{*}\right\|^{2}+\frac{2 \alpha_{n}}{\left(1-2 \alpha_{n} \gamma\right)}\left\|T y_{n}-T x_{n+1}\right\|\left\|x_{n+1}-x^{*}\right\| .
\end{aligned}
Because gamma in(0,1//2),alpha_(n)in(0,1)=>(2(1-gamma)-alpha_(n))/(1-2alpha_(n)gamma) >= 1\gamma \in(0,1 / 2), \alpha_{n} \in(0,1) \Rightarrow \frac{2(1-\gamma)-\alpha_{n}}{1-2 \alpha_{n} \gamma} \geq 1 i.e. -((2(1-gamma)-alpha_(n))/(1-2alpha_(n)gamma)) <= -1-\left(\frac{2(1-\gamma)-\alpha_{n}}{1-2 \alpha_{n} \gamma}\right) \leq-1, we have
{:[((1-alpha_(n))^(2))/((1-2alpha_(n)gamma))=(1-2alpha_(n)+alpha_(n)^(2))/((1-2alpha_(n)gamma))=(((1-2alpha_(n)gamma)+2alpha_(n)gamma-2alpha_(n)+alpha_(n)^(2)))/((1-2alpha_(n)gamma))=],[(6)=1-((2(1-gamma)-alpha_(n))/(1-2alpha_(n)gamma))alpha_(n) <= 1-alpha_(n)]:}\begin{align*}
\frac{\left(1-\alpha_{n}\right)^{2}}{\left(1-2 \alpha_{n} \gamma\right)} & =\frac{1-2 \alpha_{n}+\alpha_{n}^{2}}{\left(1-2 \alpha_{n} \gamma\right)}=\frac{\left(\left(1-2 \alpha_{n} \gamma\right)+2 \alpha_{n} \gamma-2 \alpha_{n}+\alpha_{n}^{2}\right)}{\left(1-2 \alpha_{n} \gamma\right)}= \\
& =1-\left(\frac{2(1-\gamma)-\alpha_{n}}{1-2 \alpha_{n} \gamma}\right) \alpha_{n} \leq 1-\alpha_{n} \tag{6}
\end{align*}
Also, the sequence (x_(n))\left(x_{n}\right) is bounded. We will prove that by induction. Let us denote by d:=s u p{||Tx||:x in B}+||x^(**)||d:=\sup \{\|T x\|: x \in B\}+\left\|x^{*}\right\|. Because the range of TT is bounded we have d < ood<\infty. We denote by M:=d+||x_(0)-x^(**)||+1M:=d+\left\|x_{0}-x^{*}\right\|+1. Observe that
{:[||x_(1)-x^(**)|| <= (1-alpha_(0))||x_(0)-x^(**)||+alpha_(0)||Ty_(0)-x^(**)||],[ <= (1-alpha_(0))M+alpha_(0)(||Ty_(0)||+||x^(**)||) <= (1-alpha_(0))M+alpha_(0)M=M]:}\begin{aligned}
\left\|x_{1}-x^{*}\right\| & \leq\left(1-\alpha_{0}\right)\left\|x_{0}-x^{*}\right\|+\alpha_{0}\left\|T y_{0}-x^{*}\right\| \\
& \leq\left(1-\alpha_{0}\right) M+\alpha_{0}\left(\left\|T y_{0}\right\|+\left\|x^{*}\right\|\right) \leq\left(1-\alpha_{0}\right) M+\alpha_{0} M=M
\end{aligned}
Supposing ||x_(n)-x^(**)|| <= M\left\|x_{n}-x^{*}\right\| \leq M, we will prove that ||x_(n+1)-x^(**)|| <= M\left\|x_{n+1}-x^{*}\right\| \leq M. Indeed we have
{:[||x_(n+1)-x^(**)|| <= (1-alpha_(n))||x_(n)-x^(**)||+alpha_(n)||Ty_(n)-x^(**)||],[ <= (1-alpha_(n))M+alpha_(n)(||Ty_(n)||+||x^(**)||) <= (1-alpha_(n))M+alpha_(n)M=M]:}\begin{aligned}
\left\|x_{n+1}-x^{*}\right\| & \leq\left(1-\alpha_{n}\right)\left\|x_{n}-x^{*}\right\|+\alpha_{n}\left\|T y_{n}-x^{*}\right\| \\
& \leq\left(1-\alpha_{n}\right) M+\alpha_{n}\left(\left\|T y_{n}\right\|+\left\|x^{*}\right\|\right) \leq\left(1-\alpha_{n}\right) M+\alpha_{n} M=M
\end{aligned}
Thus we have
{:(7)EE M > 0:||x_(n+1)-x^(**)|| <= M","AA n >= 0:}\begin{equation*}
\exists M>0:\left\|x_{n+1}-x^{*}\right\| \leq M, \forall n \geq 0 \tag{7}
\end{equation*}
Conditions (6), (7) and (8) lead us to
||x_(n+1)-x^(**)||^(2) <= (1-alpha_(n))||x_(n)-x^(**)||^(2)+||Ty_(n)-Tx_(n+1)||(2alpha_(n))/((1-2alpha_(n)gamma))M\left\|x_{n+1}-x^{*}\right\|^{2} \leq\left(1-\alpha_{n}\right)\left\|x_{n}-x^{*}\right\|^{2}+\left\|T y_{n}-T x_{n+1}\right\| \frac{2 \alpha_{n}}{\left(1-2 \alpha_{n} \gamma\right)} M
But (1-alpha_(n)) <= (1-w)\left(1-\alpha_{n}\right) \leq(1-w), and (2alpha_(n))/((1-2alpha_(n)gamma)) <= (2)/(1-2gamma)\frac{2 \alpha_{n}}{\left(1-2 \alpha_{n} \gamma\right)} \leq \frac{2}{1-2 \gamma}. So, we have
||x_(n+1)-x^(**)||^(2) <= (1-w)||x_(n)-x^(**)||^(2)+||Ty_(n)-Tx_(n+1)||(2)/(1-2gamma)M\left\|x_{n+1}-x^{*}\right\|^{2} \leq(1-w)\left\|x_{n}-x^{*}\right\|^{2}+\left\|T y_{n}-T x_{n+1}\right\| \frac{2}{1-2 \gamma} M
Let us denote be a_(n):=||x_(n)-x^(**)||^(2),sigma_(n):=||Ty_(n)-Tx_(n+1)||a_{n}:=\left\|x_{n}-x^{*}\right\|^{2}, \sigma_{n}:=\left\|T y_{n}-T x_{n+1}\right\|, and S:=(2)/(1-2gamma)MS:=\frac{2}{1-2 \gamma} M. Then we have lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0. Thus lim_(n rarr oo)x_(n)=x^(**)\lim _{n \rightarrow \infty} x_{n}=x^{*}.
References
[1] S. S. Chang, Y. J. Cho, B. S. Lee, J. S. Jung, S. M. Kang, Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl. 224(1998), 149-165.
[2] K. Deimling, Zeroes of Accretive Operators, Manuscripta Math. 13(1974), 365-374.
[3] Gu Feng, Iteration processes for approximating fixed points of operators of monotone type, Proc. Amer. Math. Soc. 129(2001), 2293-2300.
[4] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44(1974), 147-150.
[5] W. R. Mann, Mean value in iteration, Proc. Amer. Math. Soc. 4(1953), 506510.
[6] C. Morales, J. S. Jung, Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128:11(2000), 3411-3419.
[7] M. O. Osilike, A note on the stability of iteration procedures for strongly pseudocontractions and strongly accretive type equations, J. Math. Anal. Appl. 250\mathbf{2 5 0} (2000), 726-730.
[8] J. A. Park, Mann-iteration for strictly pseudocontractive maps, J. Korean Math. Soc. 31(1994), 333-337.
[9] Ş. M. Şoltuz, Some sequences supplied by inequalities and their applications, Revue d'analyse numérique et de théorie de l'approximation, Tome 29(2000)\mathbf{2 9}(2000), 207-212.
[10] Ş. M. Şoltuz, Three proofs for the convergence of a sequence, OCTOGON Math. Mag. 9(2001), 503-505.
[11] H. Y. Zhou, Y. Jia, Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption, Proc. Amer. Math. Soc. 125(1997), 1705-1709.
[12] H. Y. Zhou, Y. J. Cho, Ishikawa and Mann iterative process with errors for nonlinear phi\phi - strongly quasi-accretive mappings in normed spaces, J. Korean Math. Soc. 36(1999), 1061-1073.
[13] H. Y. Zhou, Stable iteration procedures for strong pseudocontractions and nonlinear equations involving accretive operators without Lipschitz assumption, J. Math. Anal. Appl. 230(1999), 1-30.
[2] K. Deimling, Zeroes of Accretive Operators, Manuscripta Math. 13(1974), 365-374.
[3] Gu Feng, Iteration processes for approximating fixed points of operators of monotone type, Proc. Amer. Math. Soc. 129(2001), 2293-2300.
[4] S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44(1974), 147-150.
[5] W. R. Mann, Mean value in iteration, Proc. Amer. Math. Soc. 4(1953), 506510.
[6] C. Morales, J. S. Jung, Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128:11(2000), 3411-3419.
[7] M. O. Osilike, A note on the stability of iteration procedures for strongly pseudocontractions and strongly accretive type equations, J. Math. Anal. Appl. 250\mathbf{2 5 0} (2000), 726-730.
[8] J. A. Park, Mann-iteration for strictly pseudocontractive maps, J. Korean Math. Soc. 31(1994), 333-337.
[9] Ş. M. Şoltuz, Some sequences supplied by inequalities and their applications, Revue d'analyse numérique et de théorie de l'approximation, Tome 29(2000)\mathbf{2 9}(2000), 207-212.
[10] Ş. M. Şoltuz, Three proofs for the convergence of a sequence, OCTOGON Math. Mag. 9(2001), 503-505.
[11] H. Y. Zhou, Y. Jia, Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption, Proc. Amer. Math. Soc. 125(1997), 1705-1709.
[12] H. Y. Zhou, Y. J. Cho, Ishikawa and Mann iterative process with errors for nonlinear phi\phi - strongly quasi-accretive mappings in normed spaces, J. Korean Math. Soc. 36(1999), 1061-1073.
[13] H. Y. Zhou, Stable iteration procedures for strong pseudocontractions and nonlinear equations involving accretive operators without Lipschitz assumption, J. Math. Anal. Appl. 230(1999), 1-30.
- *Ştefan M. Şoltuz, Kurt Schumacher Str. 48, Ap. 38, 67663 Kaiserslautern, Germany, e-mail: ssoltuz@yahoo.com, soltuz@itwm.fhg.de