Abstract
We prove that the Mann iteration convergence ot a fixed point of an asymptotic hemicontractive map.
Authors
Stefan M. Soltuz
Tiberiu Popoviciu Institute of Numerical analysis
Keywords
Mann iteration; fixed pont; asymptotic hemicontractive map
Paper coordinates
Ș.M. Șoltuz, The convergence of mann iteration for an asymptotic hemicontractive map, Buletinul ştiinţific al Universitatii Baia Mare, Seria B, Fascicola matematică-informatică, 18 (2002) no. 1, pp. 115-118 (Dedicated to Costica Mustata on his 60th anniversary).
About this paper
Journal
Bull. Stiint. Univ. Baia Mare, Matematica-Informatica
Publisher Name
Baia Mare University
DOI
Print ISSN
1222-1201
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Dedeculed to Costica MUSTATA on his 60 th annwersary
THE CONVERGENCE OF MANN ITERATION FOR AN ASYMPTOTIC HEMICONTRACTIVE MAP
Ştefan M. ŞOLTUZ
Abstract. We prove that the Mam iteration converges to a fixed point of an asymptotic hemicontractive map.
MSC: 47H10
Keywords: Mann iteration, fixed point, asymptotic hemicontractive map
Keywords: Mann iteration, fixed point, asymptotic hemicontractive map
1. Introduction.
Let XX be a real Hilbert space, let B sub XB \subset X be a nonempty, convex set. Lot T:B rarr BT: B \rightarrow B be a map. Let N={1,2,dots}\mathbb{N}=\{1,2, \ldots\}. 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)T^(n)x_(n)","AA n in N.:}\begin{equation*}
x_{n+1}=\left(1-\alpha_{n}\right) x_{n}+\alpha_{n} T^{n} x_{n}, \forall n \in N . \tag{1}
\end{equation*}
The sequence (alpha_(n))_(n >= 1)\left(\alpha_{n}\right)_{n \geq 1} satisfies:
(alpha_(n))_(n >= 1) < (0,1),sum_(n=1)^(oo)alpha_(n)=oo,lim_(n rarr oo)alpha_(n)=0\left(\alpha_{n}\right)_{n \geq 1}<(0,1), \sum_{n=1}^{\infty} \alpha_{n}=\infty, \lim _{n \rightarrow \infty} \alpha_{n}=0
A prototype for (kappa_(n))_(n >= 1)\left(\kappa_{n}\right)_{n \geq 1} is (1//n)_(n >= 1)(1 / n)_{n \geqslant 1}. Iteration (1) is known as Mam iteration, sox [8][8]. We consider the following iteration, known as Ishikawa iteration, soe [6]:
{:[(Ish)x_(n+1)=(1-alpha_(n))x_(n)+alpha_(n)T^(n)y_(n+1)],[y_(n)=(1-beta_(n))x_(n)+beta_(n)T^(n)x_(n+1)quad n inN","]:}\begin{gather*}
x_{n+1}=\left(1-\alpha_{n}\right) x_{n}+\alpha_{n} T^{n} y_{n+1} \tag{Ish}\\
y_{n}=\left(1-\beta_{n}\right) x_{n}+\beta_{n} T^{n} x_{n+1} \quad n \in \mathbb{N},
\end{gather*}
where (alpha_(n))_(n),(beta_(n))_(n)sub(0,1)\left(\alpha_{n}\right)_{n},\left(\beta_{n}\right)_{n} \subset(0,1). Choosing beta_(n)=0,AA n inN\beta_{n}=0, \forall n \in \mathbb{N}, from ( TshT s h ) we get (1).
Let us denote by F(T)F(T) the set of fixed points of the operator TT. We need the following . definition, see for example [10]:
Let us denote by F(T)F(T) the set of fixed points of the operator TT. We need the following . definition, see for example [10]:
Definition 1. The map T:B rarr BT: B \rightarrow B is culled asymptotically bemicumtractme with sequence (k_(n))_(n)\left(k_{n}\right)_{n}, if and only if lim_(n rarr oo),k_(n),-1,F(T)!=O/\lim _{n \rightarrow \infty}, k_{n},-1, F(T) \neq \emptyset such that
{:(2)||T^(n)x-x^(**)||^(2) <= k_(n)||x-x^(**)||^(2)+||x-T^(n)x||^(2)","AA x in B","x^(**)in F(T)","AA n inN.:}\begin{equation*}
\left\|T^{n} x-x^{*}\right\|^{2} \leq k_{n}\left\|x-x^{*}\right\|^{2}+\left\|x-T^{n} x\right\|^{2}, \forall x \in B, x^{*} \in F(T), \forall n \in \mathbb{N} . \tag{2}
\end{equation*}
In this note, we will consider
k_(n) < 1,AA n inN.k_{n}<1, \forall n \in \mathbb{N} .
In context of Hilbert spaces, the convergence of Ishikawa iteration (I ah) to the fixed point of TT, when we deal with s asymptotically hemicontractive map (with k_(n) > 1,Vn ink_{n}>1, \mathrm{~V} n \in N), could be found in [10]. A convergence result in normed spaces for (Ish) (with (alpha_(n))_(n),(beta_(n))_(n)sub(0,1)\left(\alpha_{n}\right)_{n},\left(\beta_{n}\right)_{n} \subset(0,1) being not convergent to zero), could be found in [0]. Anyway, in [5], we deal with an asymptotic hemicontractive like map, which is not the same as in Definition 2 with condition (2). Let us remark that:
(i) In [10][10] and (5)(5), the sequentes (alpha_(n))_(n)(beta_(n))_(n)sub(0,1)\left(\alpha_{n}\right)_{n}\left(\beta_{n}\right)_{n} \subset(0,1) are not convergent to zero, because of the cxistonce of an c > 0c>0 such that c <= alpha_(n) <= beta_(n,)AA n inNc \leq \alpha_{n} \leq \beta_{n,} \forall n \in \mathbb{N}. From the convergence of (Ish) (which is proved in [10] for Hilbert spaces and TT asymptotic hemicontractive), with that sequences (alpha_(n))_(n),(beta_(n))_(n)\left(\alpha_{n}\right)_{n},\left(\beta_{n}\right)_{n}, we can not dednces convergence of Mann iteration. It is impossible to bave beta_(e)=0,AA n inN\beta_{e}=0, \forall n \in \mathbb{N}.
(ii) According to [5], convergence results of (Ish) for asymptotic hemicontractive maps, exist only in [10], (and of course in [5]). So far as we know, no other papers are dealing with asymptotic hemicontractive maps and Mann iteration (1).
(i) In [10][10] and (5)(5), the sequentes (alpha_(n))_(n)(beta_(n))_(n)sub(0,1)\left(\alpha_{n}\right)_{n}\left(\beta_{n}\right)_{n} \subset(0,1) are not convergent to zero, because of the cxistonce of an c > 0c>0 such that c <= alpha_(n) <= beta_(n,)AA n inNc \leq \alpha_{n} \leq \beta_{n,} \forall n \in \mathbb{N}. From the convergence of (Ish) (which is proved in [10] for Hilbert spaces and TT asymptotic hemicontractive), with that sequences (alpha_(n))_(n),(beta_(n))_(n)\left(\alpha_{n}\right)_{n},\left(\beta_{n}\right)_{n}, we can not dednces convergence of Mann iteration. It is impossible to bave beta_(e)=0,AA n inN\beta_{e}=0, \forall n \in \mathbb{N}.
(ii) According to [5], convergence results of (Ish) for asymptotic hemicontractive maps, exist only in [10], (and of course in [5]). So far as we know, no other papers are dealing with asymptotic hemicontractive maps and Mann iteration (1).
This two reasons lead us to remark a lack of a convergenoe result which deal with Mann iteration (1) for an bsymptotic hemicontractive map. In particular for a map asymptotically hemicontractive with sequence (k_(n))_(n),k_(n) < 1\left(k_{n}\right)_{n}, k_{n}<1, ''n in N\in N. Our aim is to prove a result which deal with the convergence of Mamn-iteration (1) for an asymptotic hemicontractive map as in Definition 1.
The following lemma can be found in [11] as Lemma 4. Also, it can be found in [12] as Lemma 1.2, with an other proof. In |1||1| can be found as Lemma 2.
Lemma 1 [1]. , [11]. [12] Let (alpha_(n))_(n >= 1)\left(\alpha_{n}\right)_{n \geq 1} be a nonnegotive sequence which verzfies
a_(n+1) <= (1-lambda_(n))a_(n)+sigma_(n),a_{n+1} \leq\left(1-\lambda_{n}\right) a_{n}+\sigma_{n},
where: (lambda_(n))_(n > 1)sub(0.1),sum_(pi=1)^(oo)lambda_(n)=cos\left(\lambda_{n}\right)_{n>1} \subset(0.1), \sum_{\pi=1}^{\infty} \lambda_{n}=\cos and sigma_(n)=o(lambda_(n))\sigma_{n}=o\left(\lambda_{n}\right). Then lim n rarr oo,a_(n)!=0n \rightarrow \infty, a_{n} \neq 0.
The following result is from 61 :
Lemma 2 [6]. Let XX be a real Hilbert space, the following relation is frue for all x,y in Xx, y \in X, and for all lambda in(0,1)\lambda \in(0,1) :
The following result is from 61 :
Lemma 2 [6]. Let XX be a real Hilbert space, the following relation is frue for all x,y in Xx, y \in X, 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. Main result
We are able now to give the following result:
Theorem 1. Let XX he a real Hilbert syme and let B sub XB \subset X be a nonempty convex bounded set, and T:B rarr BT: B \rightarrow B be an asymptotic homicontractive map with (alpha_(n))_(n >= 1)k_(n) < 1\left(\alpha_{n}\right)_{n \geq 1} k_{n}<1. Let {x^(**)}=F(T)\left\{x^{*}\right\}=F(T). Ir (alpha_(n))_(n)\left(\alpha_{n}\right)_{n} given by (1) verifies alpha_(n)=o((1-k_(n)))\alpha_{n}=o\left(\left(1-k_{n}\right)\right), then the Mann iteration (x_(n))_(n >= 1)\left(x_{n}\right)_{n \geq 1}, given by (1)(1), is comprgent to x^(**)x^{*}, for all x_(1)in Bx_{1} \in B.
Theorem 1. Let XX he a real Hilbert syme and let B sub XB \subset X be a nonempty convex bounded set, and T:B rarr BT: B \rightarrow B be an asymptotic homicontractive map with (alpha_(n))_(n >= 1)k_(n) < 1\left(\alpha_{n}\right)_{n \geq 1} k_{n}<1. Let {x^(**)}=F(T)\left\{x^{*}\right\}=F(T). Ir (alpha_(n))_(n)\left(\alpha_{n}\right)_{n} given by (1) verifies alpha_(n)=o((1-k_(n)))\alpha_{n}=o\left(\left(1-k_{n}\right)\right), then the Mann iteration (x_(n))_(n >= 1)\left(x_{n}\right)_{n \geq 1}, given by (1)(1), is comprgent to x^(**)x^{*}, for all x_(1)in Bx_{1} \in B.
Proof. The sequence (||x_(n)-x^(**)||)_(n)\left(\left\|x_{n}-x^{*}\right\|\right)_{n} is woll-defined, bexause x^(**)x^{*} is unique. Using (3) and (2) we have, w'v inM\in \mathrm{M},
{:[||x_(n+1)-x^(**)||^(2)(1-alpha_(n))(x_(n)-x^(**))+alpha_(n)(T^(n)x_(n)-T^(n)x^(**))|^(2)],[=(1-alpha_(n))||x_(n)-x^(n)||^(2)+alpha_(n)|T^(n)x_(n)-x^(n)||^(2)-alpha_(n)(1-alpha_(n))∣T^(n)x_(n)-x_(n)||^(2):}],[ <= (1-alpha_(n))||x_(n)-x^(**)||^(2)+alpha_(n)(x_(n)||x_(n)-x^(**)||^(2)+∣T^(n)x_(n)-x_(n)||^(2))],[=(1-alpha_(n)(1-alpha_(n))∣T^(n)x_(n)-x_(n)||^(2):}],[-alpha_(n)(1-alpha_(n))||x^(n)||_(n)^(2)+alpha_(n)k_(n)||x_(n)-x_(n)||^(2)+alpha_(n)||T^(n)x_(n)-x_(n)||^(2)],[=(1-alpha_(n)(1-x_(n)))||x_(n)-x^(n)||^(2)+alpha_(n)^(2)||T^(n)x_(n)-x_(n)||^(2)],[ <= (1-alpha_(n)(1-k_(n)))||x_(n)-x^(n)||^(2)+alpha_(n)^(2)M]:}\begin{aligned}
\left\|x_{n+1}-x^{*}\right\|^{2} & \left(1-\alpha_{n}\right)\left(x_{n}-x^{*}\right)+\left.\alpha_{n}\left(T^{n} x_{n}-T^{n} x^{*}\right)\right|^{2} \\
= & \left(1-\alpha_{n}\right)\left\|x_{n}-x^{n}\right\|^{2}+\alpha_{n}\left|T^{n} x_{n}-x^{n}\left\|^{2}-\alpha_{n}\left(1-\alpha_{n}\right) \mid T^{n} x_{n}-x_{n}\right\|^{2}\right. \\
\leq & \left(1-\alpha_{n}\right)\left\|x_{n}-x^{*}\right\|^{2}+\alpha_{n}\left(x_{n}\left\|x_{n}-x^{*}\right\|^{2}+\mid T^{n} x_{n}-x_{n} \|^{2}\right) \\
= & \left(1-\alpha_{n}\left(1-\alpha_{n}\right) \mid T^{n} x_{n}-x_{n} \|^{2}\right. \\
& -\alpha_{n}\left(1-\alpha_{n}\right)\left\|x^{n}\right\|_{n}^{2}+\alpha_{n} k_{n}\left\|x_{n}-x_{n}\right\|^{2}+\alpha_{n}\left\|T^{n} x_{n}-x_{n}\right\|^{2} \\
= & \left(1-\alpha_{n}\left(1-x_{n}\right)\right)\left\|x_{n}-x^{n}\right\|^{2}+\alpha_{n}^{2}\left\|T^{n} x_{n}-x_{n}\right\|^{2} \\
\leq & \left(1-\alpha_{n}\left(1-k_{n}\right)\right)\left\|x_{n}-x^{n}\right\|^{2}+\alpha_{n}^{2} M
\end{aligned}
The last inequality is true, since the sequence (||T^(n)x_(n)-x_(n)||^(2))_(n >= 1)\left(\left\|T^{n} x_{n}-x_{n}\right\|^{2}\right)_{n \geq 1} is bounded, because BB is bounded. There exists M > 0M>0 such that ||T^(n)x_(n)-x_(n)||^(2) < M\left\|T^{n} x_{n}-x_{n}\right\|^{2}<M, for all n inNn \in \mathbb{N}. We denote by a_(n):=||x_(n)-x^(**)||,lambda_(n)=alpha_(n)(1-k_(n)),sigma_(n)=alpha_(n)^(2)M,AA n inNa_{n}:=\left\|x_{n}-x^{*}\right\|, \lambda_{n}=\alpha_{n}\left(1-k_{n}\right), \sigma_{n}=\alpha_{n}^{2} M, \forall n \in \mathbb{N}. Because alpha_(n)=o((1-k_(n)))\alpha_{n}= o\left(\left(1-k_{n}\right)\right), we have lim_(n rarr oo)(kappa_(n))/(1-k_(n))M=0\lim _{n \rightarrow \infty} \frac{\kappa_{n}}{1-k_{n}} M=0. All the conditions from Lemma 2 are fulfilled. Thus lim_(n rarr oo)a_(n)=0\lim _{n \rightarrow \infty} a_{n}=0. The proof is complete.
In Theorem 4 we don't need any Lipschitx condition for TT as in [10]
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