strongly converges to the fixed point of $T$$T$TT$T$.
Key words: Mann iteration, fixed points
AMS subject classifications: $47\mathrm{H}10,47\mathrm{H}06$$47\mathrm{H}10,47\mathrm{H}06$47H10,47H0647 \mathrm{H} 10,47 \mathrm{H} 06$47\mathrm{H}10,47\mathrm{H}06$
Received June 10, 2000
Accepted May 18, 2001

In this note we study the convergence of the Mann iteration process (1) for generalized pseudocontractions. According to [8] the generalized pseudocontractions are more general than the pseudocontractions introduced by Browder.

Definition 1. [8]. Let $X$$X$XX$X$ be a Hilbert space, let $B$$B$BB$B$ be a nonempty subset. $A$$A$AA$A$ map $T:B\to B$$T:B\to B$T:B rarr BT: B \rightarrow B$T:B\to B$ is said to be a generalized pseudocontraction if for $x,y\in B$$x,y\in B$x,y in Bx, y \in B$x,y\in B$ there exists $r>0$$r>0$r > 0r>0$r>0$ such that

Clearly, (2) is equivalent to

The map $T$$T$TT$T$ is a strong pseudocontraction if there exists $k\in (0,1)$$k\in (0,1)$k in(0,1)k \in(0,1)$k\in (0,1)$ such that for all $x,y\in B$$x,y\in B$x,y in Bx, y \in B$x,y\in B$,

see, for example [6]. Remark that both generalized pseudocontractivity and strong pseudocontractivity generalize the pseudocontractivity, but in a different manner. Iteration (1), where $T$$T$TT$T$ is a strong pseudocontraction in Banach spaces, was studied in [1], [2], [3], [4], [6], [9].

The following lemma can be found in [9] as Lemma4. Also, it can be found in [4] as Lemma 1.2, with another proof. A more general case is in Lemma 2 from [5]. The proof from [5] is similar to the proof of Lemma 4 from [9].

Lemma 1. [9], [4]. Let ${\left({\rho }_n\right)}_n$${\left({\rho }_n\right)}_n$(rho_(n))_(n)\left(\rho_{n}\right)_{n}${\left({\rho }_n\right)}_n$ be a nonnegative real sequence satisfying

where ${\lambda }_n\in (0,1),\mathrm{\forall }n\in N,\sum _{n=1}^{\mathrm{\infty }}{\lambda }_n=\mathrm{\infty }$${\lambda }_n\in (0,1),\mathrm{\forall }n\in N,\sum _{n=1}^{\mathrm{\infty }} {\lambda }_n=\mathrm{\infty }$lambda_(n)in(0,1),AA n in N,sum_(n=1)^(oo)lambda_(n)=oo\lambda_{n} \in(0,1), \forall n \in N, \sum_{n=1}^{\infty} \lambda_{n}=\infty${\lambda }_n\in (0,1),\mathrm{\forall }n\in N,\sum _{n=1}^{\mathrm{\infty }}{\lambda }_n=\mathrm{\infty }$ and ${\sigma }_n=o\left({\lambda }_n\right)$${\sigma }_n=o\left({\lambda }_n\right)$sigma_(n)=o(lambda_(n))\sigma_{n}=o\left(\lambda_{n}\right)${\sigma }_n=o\left({\lambda }_n\right)$. Then $\underset{n\to \mathrm{\infty }}{lim}{\rho }_n=0$$\underset{n\to \mathrm{\infty }}{lim} {\rho }_n=0$lim_(n rarr oo)rho_(n)=0\lim _{n \rightarrow \infty} \rho_{n}=0$\underset{n\to \mathrm{\infty }}{lim}{\rho }_n=0$.
The normalized duality mapping $J$$J$JJ$J$ is the identity, when $X$$X$XX$X$ is a Hilbert space, see [4]. Thus Lemma 1.1 from [4] becomes:

Lemma 2. [4]. If $X$$X$XX$X$ is a Hilbert space, then

for all $x,y\in X$$x,y\in X$x,y in Xx, y \in X$x,y\in X$.
The following result is a corollary of Lemma 1 from [7]:
Lemma 3. [7]. If $X$$X$XX$X$ is a real Hilbert space, $B$$B$BB$B$ is a nonempty, bounded, convex and closed subset, and $T:B\to B$$T:B\to B$T:B rarr BT: B \rightarrow B$T:B\to B$ is a generalized pseudocontraction, then the sequence given by (1) satisfies

In [7], the map $T$$T$TT$T$ is nonexpansive. If we consider the proof of Lemma 1 from [7], we see that the result is true, when our assumptions are fulfilled.

We are now able to give the following result:
Theorem 1. If $X$$X$XX$X$ is a real Hilbert space, $B$$B$BB$B$ is a nonempty, bounded, convex and closed subset, and $T:B\to B$$T:B\to B$T:B rarr BT: B \rightarrow B$T:B\to B$ is a generalized pseudocontraction, then the iteration (1) :

strongly converges to the fixed point of $T$$T$TT$T$.
Proof. Theorem 2.1 from [8] gives us the existence and the uniqueness of the fixed point of $T$$T$TT$T$. Let us denote this fixed point by $q$$q$qq$q$. Using Lemma 3 and (2), we have

Let us denote

Thus, we have

We observe that

the last equality is true. From Lemma 4, we have $\underset{n\to \mathrm{\infty }}{lim}\Vert x_{n+1}-x_n\Vert =0$$\underset{n\to \mathrm{\infty }}{lim} ‖x_{n+1}-x_n‖=0$lim_(n rarr oo)||x_(n+1)-x_(n)||=0\lim _{n \rightarrow \infty}\left\|x_{n+1}-x_{n}\right\|=0$\underset{n\to \mathrm{\infty }}{lim}\Vert x_{n+1}-x_n\Vert =0$. The sequence ${\left(\Vert T{x}_n-q\Vert \right)}_n$${\left(‖T{x}_n-q‖\right)}_n$(||Tx_(n)-q||)_(n)\left(\left\|T x_{n}-q\right\|\right)_{n}${\left(\Vert T{x}_n-q\Vert \right)}_n$ is bounded, being in the bounded set $B$$B$BB$B$. Hence we have $\underset{n\to \mathrm{\infty }}{lim}⟨T{x}_n-q,x_{n+1}-x_n⟩=0$$\underset{n\to \mathrm{\infty }}{lim} ⟨T{x}_n-q,x_{n+1}-x_n⟩=0$lim_(n rarr oo)(:Tx_(n)-q,x_(n+1)-x_(n):)=0\lim _{n \rightarrow \infty}\left\langle T x_{n}-q, x_{n+1}-x_{n}\right\rangle=0$\underset{n\to \mathrm{\infty }}{lim}⟨T{x}_n-q,x_{n+1}-x_n⟩=0$. The assumptions from Lemma 2 are fulfilled. Hence ${\rho }_n\to 0$${\rho }_n\to 0$rho_(n)rarr0\rho_{n} \rightarrow 0${\rho }_n\to 0$ as $n\to \mathrm{\infty }$$n\to \mathrm{\infty }$n rarr oon \rightarrow \infty$n\to \mathrm{\infty }$. Thus $x_n\to q$$x_n\to q$x_(n)rarr qx_{n} \rightarrow q$x_n\to q$ as $n\to \mathrm{\infty }$$n\to \mathrm{\infty }$n rarr oon \rightarrow \infty$n\to \mathrm{\infty }$.

A prototype for ${\left({\alpha }_n\right)}_n$${\left({\alpha }_n\right)}_n$(alpha_(n))_(n)\left(\alpha_{n}\right)_{n}${\left({\alpha }_n\right)}_n$ is $(1/\sqrt{n}{)}_{n\ge 1}$$(1/\sqrt{n}{)}_{n\ge 1}$(1//sqrtn)_(n >= 1)(1 / \sqrt{n})_{n \geq 1}$(1/\sqrt{n}{)}_{n\ge 1}$.

[1] C. E. Chidume, Iterative approximation of fixed points of Lipschitzian strictly pseudo-contractive mappings, Proc. Amer. Math. Soc. 99(1987), 283-287.
[2] C. E. Chidume, Global iteration schemes for strongly pseudo-contractive maps, Proc. Amer. Math. Soc. 126(1998), 2641-2649.
[3] C. E. Chidume, C. Moore, Fixed point for pseudocontractive maps, Proc. Amer. Math. Soc. 127(1999), 1163-1170.
[4] Z. Haiyun, J. Yuting, Approximation of fixed points of strongly pseudocontractive maps without Lipschitz assumption, Proc. Amer. Math. Soc. 125(1997), 1705-1709.
[5] L.-S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194(1995), 114-125.
[6] J. A. Park, Mann-iteration for strictly pseudocontractive maps, J. Korean Math. Soc. 31(1994), 333-337.
[7] N. Shioji, W. Takahashi, Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 125(1997), 3641-3645.
[8] R. U. Verma, A fixed point theorem involving Lipschitzian generalised pseudocontractions, Proc. Royal Irish Acad. 97A(1997), 83-86.
[9] X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Amer. Math. Soc. 113(1991), 727-731.