Abstract
For \(T\) a direct pseudocontractive map, we prove the convergence of Mann iteration to the fixed point of \(T\). In this note we introduce a new class of maps. Let \(X\) be a real normed space, and \(B\subset X\) be a nonemepty set. The map \(T:B\rightarrow B\) is direct pseudocontractive if there exists \(k\in \left( 0,1\right)\) such that%
\[
\left \Vert T_{x}-T_{y}\right \Vert ^{2}\leq k\left \Vert x-y\right \Vert
^{2}+\left \Vert \left( I-T\right) x-\left( I-T\right) y\right \Vert
^{2},\forall x,y\in B.
\]
For \(T\) a direct pseudocontractive map , we prove the convergence of Mann iteration to the fixed point of \(T\).
Authors
S.M. Soltuz
Tiberiu Popoviciu Institute of Numerical Analysis
Keywords
pseudocontractive maps; fix ponts; iterative method; convergence
Paper coordinates
Ş.M. Şoltuz, Mann iteration for direct pseudocontractive maps, Bull. Stiint. Univ. Baia Mare, Ser. B, Fasc. Mat.-Inform., 17 (2001) nos. 1-2, 141-144.
About this paper
Journal
Scientific Bulletin of the University of Baia Mare, Series B, Mathematics-Informatics Fascicola
Publisher Name
University Baia Mare, Romania
DOI
Print ISSN
Online ISSN
3045-1833
google scholar link
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Paper (preprint) in HTML form
MANN ITERATION FOR DIRECT PSEUDOCONTRACTIVE MAPS
Ştefan M. ŞOLTUZ
Abstract
In this note we introduce a new class of maps. Let XX be a real normed space, and B sub XB \subset X be a nonempty set. The map T:B rarr BT: B \rightarrow B is direct pseudocontractive if there exists k in(0,1)k \in(0,1) such that
||Tx-Ty||^(2) <= k||x-y||^(2)+||(I-T)x-(I-T)y||^(2),AA x,y in B.\|T x-T y\|^{2} \leq k\|x-y\|^{2}+\|(I-T) x-(I-T) y\|^{2}, \forall x, y \in B .
For TT a direct pseudocontractive map, we prove the convergence of Mann iteration to the fixed point of TT.
MSC. 47H10
Keywords: preudocontractive maps, fix points, iterative method, convergence
Keywords: preudocontractive maps, fix points, iterative method, convergence
- Introduction. Let HH be a real Hilbert space, let B sub HB \subset H be a nonempty, convex set. Let T:B rarr BT: B \rightarrow B be a map. Let x_(1)in Bx_{1} \in B, be an arbitrary fixed point. We consider the iteration
{:(1)x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)Tx_(n):}\begin{equation*}
x_{n+1}=\left(1-\alpha_{n}\right) x_{n}+\alpha_{n} T x_{n} \tag{1}
\end{equation*}
The sequence (alpha_(n))_(n >= 1)\left(\alpha_{n}\right)_{n \geq 1} satisfics: (alpha_(n))_(n >= 1)sub(0,1),sum_(n=1)^(oo)alpha_(n)=oo\left(\alpha_{n}\right)_{n \geq 1} \subset(0,1), \sum_{n=1}^{\infty} \alpha_{n}=\infty, and sum_(n=1)^(oo)alpha_(n)^(2) < oo\sum_{n=1}^{\infty} \alpha_{n}^{2}< \infty. The last relation implies that lim_(n rarr oo)alpha_(n)=0\lim _{n \rightarrow \infty} \alpha_{n}=0. A prototype for (alpha_(n))_(n >= 1)\left(\alpha_{n}\right)_{n \geq 1} is (1//n)_(n >= 1)(1 / n)_{n \geq 1}.
Definition 1 The map TT is called pseudocontractive if
||Tx-Ty||^(2) <= ||x-y||^(2)+||(I-T)x-(I-T)y||^(2),AA x,y in B.\|T x-T y\|^{2} \leq\|x-y\|^{2}+\|(I-T) x-(I-T) y\|^{2}, \forall x, y \in B .
In [7] we can see an example of a Lipechitz pseudocontractive map with a unique fixed point for which every non trivial Mann sequence fails to converge. The set BB is nonempty, convex and compact.
Definition 2 The map TT is called strongly pseudocontractive if there exists q in(0,1)q \in(0,1) such that
||Tx-Ty||^(2) <= ||x-y||^(2)+q||(I-T)x-(I-T)y||^(2),AA x,y in B.\|T x-T y\|^{2} \leq\|x-y\|^{2}+q\|(I-T) x-(I-T) y\|^{2}, \forall x, y \in B .
In [1], [2], [3], [5], [8], [11] the map TT is considered strongly pseudocontractive. The sequence (x_(n))_(n >= 1)\left(x_{n}\right)_{n \geq 1} given by (1) strongly converges to a fixed point of TT.
We introduce the following class of maps:
Definition 3 The map TT is called direct pseudocontractive if there exists k in(0,1)k \in(0,1) such that
Definition 3 The map TT is called direct pseudocontractive if there exists k in(0,1)k \in(0,1) such that
{:(2)||Tx-Ty||^(2) <= k||x-y||^(2)+||(I-T)x-(I-T)y||^(2)","AA x","y in B.:}\begin{equation*}
\|T x-T y\|^{2} \leq k\|x-y\|^{2}+\|(I-T) x-(I-T) y\|^{2}, \forall x, y \in B . \tag{2}
\end{equation*}
The class of direct pseudocontractive maps is nonempty. If TT is a contraction, then TT is a direct pseudocontractive map.Picard -Banach Theorem can't be used to find the fixed point of a direct psendocontractive map. Instead, Mann iteration (1) can be successfully used. Our aim is to give a convergence result for (1). We denote by F(T):={x in B:Tx=x}F(T):=\{x \in B: T x=x\}.
Remark 1 If TT is a direct pseudocontractive map and has F(T)!=O/F(T) \neq \emptyset, then TT hass a unique fixed point.
Remark 1 If TT is a direct pseudocontractive map and has F(T)!=O/F(T) \neq \emptyset, then TT hass a unique fixed point.
Proof. Let x^(**)x^{*} and y^(**)y^{*} be two distinct fixed points. From (2) we have
{:[||Tx^(**)-Ty^(**)||^(2) <= k||x^(**)-y^(**)||^(2)],[||x^(**)-y^(**)||^(2) <= k||x^(**)-y^(**)||^(2)],[(1-k)||x^(**)-y^(**)||^(2) <= 0","k in(0","1)]:}\begin{aligned}
\left\|T x^{*}-T y^{*}\right\|^{2} & \leq k\left\|x^{*}-y^{*}\right\|^{2} \\
\left\|x^{*}-y^{*}\right\|^{2} & \leq k\left\|x^{*}-y^{*}\right\|^{2} \\
(1-k)\left\|x^{*}-y^{*}\right\|^{2} & \leq 0, k \in(0,1)
\end{aligned}
Hence x^(**)=y^(**)x^{*}=y^{*}. Thus F(T)={x^(**)}F(T)=\left\{x^{*}\right\}.
The following lemma can be found in [10] as Lemma 4. Also, it can be found in [12] as Lemma 1.2, with an other proof. In [1] can be found as Lemma 2, the proof is similar to the proof of Lemma 1 from [8].
Lemma 4 [1], [10], [12] Let (a_(n))_(n >= 1)\left(a_{n}\right)_{n \geq 1} be a nonnegative sequence which verifies where a_(n+1) <= (1-lambda_(n))a_(n)+sigma_(n),(lambda_(n))_(n >= 1)sub(0,1),sum_(n=1)^(oo)lambda_(n)=ooa_{n+1} \leq\left(1-\lambda_{n}\right) a_{n}+\sigma_{n},\left(\lambda_{n}\right)_{n \geq 1} \subset(0,1), \sum_{n=1}^{\infty} \lambda_{n}=\infty and sigma_(n)=o(lambda_(n))\sigma_{n}=o\left(\lambda_{n}\right). Then lim_(n-oo)a_(n)=0\lim _{n-\infty} a_{n}=0.
The following lemma can be found in [10] as Lemma 4. Also, it can be found in [12] as Lemma 1.2, with an other proof. In [1] can be found as Lemma 2, the proof is similar to the proof of Lemma 1 from [8].
Lemma 4 [1], [10], [12] Let (a_(n))_(n >= 1)\left(a_{n}\right)_{n \geq 1} be a nonnegative sequence which verifies where a_(n+1) <= (1-lambda_(n))a_(n)+sigma_(n),(lambda_(n))_(n >= 1)sub(0,1),sum_(n=1)^(oo)lambda_(n)=ooa_{n+1} \leq\left(1-\lambda_{n}\right) a_{n}+\sigma_{n},\left(\lambda_{n}\right)_{n \geq 1} \subset(0,1), \sum_{n=1}^{\infty} \lambda_{n}=\infty and sigma_(n)=o(lambda_(n))\sigma_{n}=o\left(\lambda_{n}\right). Then lim_(n-oo)a_(n)=0\lim _{n-\infty} a_{n}=0.
The following result is proved in [4].
Lemma 5 (4) Let HH be a Hilbert space, the following relation is true for all x,y in Hx, y \in H, and for all lambda in(0,1)\lambda \in(0,1) :
Lemma 5 (4) Let HH be a Hilbert space, the following relation is true for all x,y in Hx, y \in H, and for all lambda in(0,1)\lambda \in(0,1) :
{:(3)||(1-lambda)x+lambda y||^(2)=(1-lambda)||x||^(2)+lambda||y||^(2)-lambda(1-lambda)||x=y||^(2).:}\begin{equation*}
\|(1-\lambda) x+\lambda y\|^{2}=(1-\lambda)\|x\|^{2}+\lambda\|y\|^{2}-\lambda(1-\lambda)\|x=y\|^{2} . \tag{3}
\end{equation*}
2.The main result.
We are able now to give the main result:
Theorem 6 Let HH be a real Hilbert space, let B sub HB \subset H be a nonempty, convex, bounded and closed set and let T:B rarr BT: B \rightarrow B be a continuous, direct pseudocontructive map, with F(T)!=O/F(T) \neq \emptyset. Then for each x_(1)x_{1} a fixed point in BB, the sequence (x_(n))_(n >= 1)\left(x_{n}\right)_{n \geq 1} given by (1) converges strongly to the unique fixed point of TT.
Proof. Let x^(**)in F(T)x^{*} \in F(T). From remark 2 we know that F(T)={x^(**)}F(T)=\left\{x^{*}\right\}. Using (2) and (3) we get
{:[||x_(n+1)-x^(**)||^(2)],[=||(1-alpha_(n))x_(n)+alpha_(n)Tx_(n)-x^(**)||^(2)],[=||(1-alpha_(n))(x_(n)-x^(**))+alpha_(n)(Tx_(n)-x^(**))||^(2)],[=(1-alpha_(n))||x_(n)-x^(**)||^(2)+alpha_(n)||Tx_(n)-x^(**)||^(2)-alpha_(n)(1-alpha_(n))||Tx_(n)-x_(n)||^(2)],[ <= (1-alpha_(n))||x_(n)-x^(**)||^(2)+alpha_(n)k||x_(n)-x^(**)||^(2)+],[+alpha_(n)||Tx_(n)-x_(n)||^(2)-alpha_(n)(1-alpha_(n))||Tx_(n)-x_(n)||^(2)],[ <= [1-(1-k)alpha_(n)]||x_(n)-x^(**)||^(2)+alpha_(n)^(2)||Tx_(n)-x_(n)||^(2).]:}\begin{aligned}
& \left\|x_{n+1}-x^{*}\right\|^{2} \\
= & \left\|\left(1-\alpha_{n}\right) x_{n}+\alpha_{n} T x_{n}-x^{*}\right\|^{2} \\
= & \left\|\left(1-\alpha_{n}\right)\left(x_{n}-x^{*}\right)+\alpha_{n}\left(T x_{n}-x^{*}\right)\right\|^{2} \\
= & \left(1-\alpha_{n}\right)\left\|x_{n}-x^{*}\right\|^{2}+\alpha_{n}\left\|T x_{n}-x^{*}\right\|^{2}-\alpha_{n}\left(1-\alpha_{n}\right)\left\|T x_{n}-x_{n}\right\|^{2} \\
\leq & \left(1-\alpha_{n}\right)\left\|x_{n}-x^{*}\right\|^{2}+\alpha_{n} k\left\|x_{n}-x^{*}\right\|^{2}+ \\
& +\alpha_{n}\left\|T x_{n}-x_{n}\right\|^{2}-\alpha_{n}\left(1-\alpha_{n}\right)\left\|T x_{n}-x_{n}\right\|^{2} \\
\leq & {\left[1-(1-k) \alpha_{n}\right]\left\|x_{n}-x^{*}\right\|^{2}+\alpha_{n}^{2}\left\|T x_{n}-x_{n}\right\|^{2} . }
\end{aligned}
The sequence (||Tx_(n)-x_(n)||^(2))_(n >= 1)\left(\left\|T x_{n}-x_{n}\right\|^{2}\right)_{n \geq 1} is bounded, because BB is bounded. There exists M > 0M>0 such that ||Tx_(n)-x_(n)||^(2) < M\left\|T x_{n}-x_{n}\right\|^{2}<M, for all n >= 1n \geq 1. We denote a_(n):=||x_(n)-x^(**)||^(2)a_{n}:= \left\|x_{n}-x^{*}\right\|^{2}, and we get:
a_(n+1) <= [1-(1-k)alpha_(n)]a_(n)+alpha_(n)^(2)M.a_{n+1} \leq\left[1-(1-k) \alpha_{n}\right] a_{n}+\alpha_{n}^{2} M .
Let us denote by
{:[lambda_(n):=(1-k)alpha_(n)],[sigma_(n):=alpha_(n)^(2)M.]:}\begin{aligned}
& \lambda_{n}:=(1-k) \alpha_{n} \\
& \sigma_{n}:=\alpha_{n}^{2} M .
\end{aligned}
Observe that lambda_(n)=(1-k)alpha_(n)sub(0,1)\lambda_{n}=(1-k) \alpha_{n} \subset(0,1), for all n >= 1n \geq 1. We have sum_(n-1)^(oo)lambda_(n)=(1-k)sum_(n=1)^(oo)alpha_(n)=oo\sum_{n-1}^{\infty} \lambda_{n}= (1-k) \sum_{n=1}^{\infty} \alpha_{n}=\infty. The following relation is true
lim_(n rarr oo)(sigma_(n))/(lambda_(n))=lim_(n rarr oo)(alpha_(s)^(2)M)/((1-k)alpha_(n))=(M)/(1-k)lim_(n rarr oo)alpha_(n)=0\lim _{n \rightarrow \infty} \frac{\sigma_{n}}{\lambda_{n}}=\lim _{n \rightarrow \infty} \frac{\alpha_{s}^{2} M}{(1-k) \alpha_{n}}=\frac{M}{1-k} \lim _{n \rightarrow \infty} \alpha_{n}=0
Thus, we have sigma_(n)=o(lambda_(n))\sigma_{n}=o\left(\lambda_{n}\right). From Lemma 1 we get lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0. Hence lim_(n rarr oo)||x_(n)-x^(**)||=0\lim _{n \rightarrow \infty}\left\|x_{n}-x^{*}\right\|=0. The proof is complete.
Using the Schauder fixed point theorem we give the following corollary:
Corollary 7 Let HH be a real Hilbert space, let B sub HB \subset H be a nonempty, convex, compact set and let T:B rarr BT: B \rightarrow B be a continuous, direct pseudocontractive map. Then for exach x_(1)x_{1} a fixed point in BB, the sequence (x_(n))_(n >= 1)\left(x_{n}\right)_{n \geq 1} given by (1) converges strongly to the unique fixed poinl of TT.
Corollary 7 Let HH be a real Hilbert space, let B sub HB \subset H be a nonempty, convex, compact set and let T:B rarr BT: B \rightarrow B be a continuous, direct pseudocontractive map. Then for exach x_(1)x_{1} a fixed point in BB, the sequence (x_(n))_(n >= 1)\left(x_{n}\right)_{n \geq 1} given by (1) converges strongly to the unique fixed poinl of TT.
References
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[2] C. E. Chidume, Approximation of Fixed Points of Strongly Pseudocontractive Mappings, Proc. Amer. Math. Soc. 120 (1994), 546-551.
[3] C. E. Chidume, C. Moore, Fixed Point Iteration for Strongly Pseudocontractive Maps, Proc. Amer. Math. Soc. 127 (1999), 1163-1170.
[4] S. Ishikawa, Fized Points by a New Iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147-150.
[5] G. G. Johnson, Fixed Points by Mean value iterations, Proc. Amer. Math. Soc. 34 (1972), 193-195.
[6] R. W. Mann, Mean Value Methods in Iteration, Proc. Amer. Math. Soc. 4 (1953), 504-510.
[7] S.A. Mutangadura, C.E. Chidume, An Example of the Mann Iteration Method for Lipschitz Pseudocontractions, internal report ICTP Trieste (2000), http://www.ictp.trieste.it
[8] J. A. Park, Mann-Ileration for Strictly Pseudocontructive Maps, J. Korean Math. Soc. 31 (1994), 333-337.
[9] R. U. Verma, A Ficed Point Theorem Involving Lipschitzian Generalized Pseudo-contractions, Proc. Royal Irish Acad. 97A (1997), 83-86.
[10] X. Weng, Pired Point Iteration for Local Strictly Pseudocontractive Mapping, Proc. Amer. Math. Soc. 113 (1991), 727-731.
[11] II. 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.
[12] H. Zhou, J. Yuting, Approximation of Fixed Points of Strongly Pseudocontractive Maps without Lipschitz Assumption, Proc. Amer. Math. Soc.
[2] C. E. Chidume, Approximation of Fixed Points of Strongly Pseudocontractive Mappings, Proc. Amer. Math. Soc. 120 (1994), 546-551.
[3] C. E. Chidume, C. Moore, Fixed Point Iteration for Strongly Pseudocontractive Maps, Proc. Amer. Math. Soc. 127 (1999), 1163-1170.
[4] S. Ishikawa, Fized Points by a New Iteration Method, Proc. Amer. Math. Soc. 44 (1974), 147-150.
[5] G. G. Johnson, Fixed Points by Mean value iterations, Proc. Amer. Math. Soc. 34 (1972), 193-195.
[6] R. W. Mann, Mean Value Methods in Iteration, Proc. Amer. Math. Soc. 4 (1953), 504-510.
[7] S.A. Mutangadura, C.E. Chidume, An Example of the Mann Iteration Method for Lipschitz Pseudocontractions, internal report ICTP Trieste (2000), http://www.ictp.trieste.it
[8] J. A. Park, Mann-Ileration for Strictly Pseudocontructive Maps, J. Korean Math. Soc. 31 (1994), 333-337.
[9] R. U. Verma, A Ficed Point Theorem Involving Lipschitzian Generalized Pseudo-contractions, Proc. Royal Irish Acad. 97A (1997), 83-86.
[10] X. Weng, Pired Point Iteration for Local Strictly Pseudocontractive Mapping, Proc. Amer. Math. Soc. 113 (1991), 727-731.
[11] II. 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.
[12] H. Zhou, J. Yuting, Approximation of Fixed Points of Strongly Pseudocontractive Maps without Lipschitz Assumption, Proc. Amer. Math. Soc.
125 (1997), 1705-1709.
Received: 12.03.2001
"T. Popoviciu" Institute of
Numerical Analysis
Gh. Bilascu 37, P.O. Box 68-1,
3400 Cluj-Napoca, Romania.
Received: 12.03.2001
"T. Popoviciu" Institute of
Numerical Analysis
Gh. Bilascu 37, P.O. Box 68-1,
3400 Cluj-Napoca, Romania.