(Received 31 July, 1999)

ABSTRACT In this study we use perturbed-Steffensen- Aitken methods to approximate a locally unique solution of an operator equation in a Banach space. Using projection operators we reduce the problem to solving a linear system of algebraic equations of finite order. Since iterates can rarely be computed exactly we control the residuals to guarantee convergence of the method. Sufficient convergence conditions as well as an error analysis are given for our method.

AMS (MOS) Subject Classification: 65J15, 47H17, 49D15.
Key Words and Phrases Steffensen-Aitken methods, Banach space, projection operator, residuals.
I. INTRODUCTION In this study we are concerned with the problem of
approximating a locally unique fixed point $x^{*}$ of the nonlinear equation.

where $T$ is a continuous operator defined on a convex subset $D$ of a Banach space $E$ with values in $E$. The differentiability of $T$ is not assumed. Let $T_{1}$ be another nonlinear continuous operator from $E$ into $E$, and let $P$ be a projection operator $\left(P^{2}=P\right)$ on $E$.

We introduce the perturbed-Steffensen-Aitken method

where: $[x,y]$ denotes a divided difference of order one of $T_{1}$ at the points $x,y$ satisfying

and

if $T_{1}$ is Frechet-differentiable $D;g_{1},g_{2}:D\rightarrow E$ are continuous operators; the residual points $\left\{x_{n}\right\}(n\geq 0)$ are chosen in such a way that iteration $\left\{x_{n}\right\}(n\geq 0)$ generated by (2) converges to $x^{*}$. The important of studying perturbed SteffensenAitken methods comes from the fact that many commonly used variants can be considered procedures of this type. Indeed the above approximation characterizes any iterative process in which corrections are taken as approximate solutions of the Steffensen-Aitken equations. Moreover we note that if for example an equation on the real line is solved $x_{n}-T\left(x_{n}\right)\geq 0(n\geq 0)$ and $I-PA_{n}$ overestimates the derivative, $x_{n}-\left(I-PA_{n}\right)^{-1}\left(x_{n}-T\left(x_{n}\right)\right)$ is always larger than the corresponding Steffensen-Aitken iterate. In such cases, a positive $z_{n}(n\geq 0)$ correction term is appropriate.

For: $P=I(I$ is the identity operator on $E),T(x)=T_{1}(x)(x\in D),g_{1}(x)=g_{2}(x)(x\in D)$, and $z_{n}=0(n\geq 0)$ we obtain the ordinary Newton method [1], [2]; $P=I,T_{1}(x)=T(x)(x\in D),g_{1}(x)=x(x\in D)$, and $z_{n}=0(n\geq 0)$ we obtain Steffensen method [4], [5]; $P=I,T_{1}(x)=T(x)(x\in D),g_{2}(x)=g_{1}(x-T(x))(x\in D)$, and $z_{n}=0(n\geq 0)$ we obtain Steffensen-Aitken method [4], [5].

It is easy to see that the solution of (2) reduces to solving certain operator equations in the space $E_{p}$. If moreover $E_{p}$ is a finite dimensional space of dimension $N$, we obtain a system of linear algebraic equations of at most order $N$.

We provide sufficient convergence conditions as well as an error analysis for the Steffensen-Aitken method generated by (2).
II. CONVERGENCE ANALYSIS We state the following semilocal convergence theorem.

Theorem Let $T,T_{1},g_{1},g_{2}:D\rightarrow E$ be continuous operators defined on a convex subset $D$ of a Banach space $E$ with values in $E$, and $P$ be a projection operator on E. Moreover, assume:
(a) there exists $x_{0}\in D$ such that $B_{0}=I-PA_{0}$ is invertible;
(b) there exist nonnegative numbers $a_{i},R,\quad i=0,1,2,\cdots,9$ such that

and

for all $x,y,v,w\in U\left(x_{0},R\right)=\left\{x\in E\left\|x-x_{0}\right\|\leq R\right\}\subseteq D;$
(c) the sequence $\left\{z_{n}\right\}(n\geq 0)$ is null;
(d) there exists a minumum nonnegative number $r^{*}$ satisfying

where

and

(e) the numbers $r^{*},R$ also satisfy

where

and

where

Then
(i) the scalar sequence $\left\{t_{n}\right\}\quad(n\geq 0)$ generated by

is monotonically increasing, bounded above by $r^{*}$ and $\lim_{n\rightarrow\infty}t_{n}=r^{*}$, with $\beta_{n}=\left[1-a_{0}\left(a_{8}+a_{9}\right)t_{n}\right]^{-1}\quad(n\geq 0)$.
(ii) The perturbed-Steffensen-Aitken method generated by (2) is well defined, remains in $U\left(x_{0},r^{*}\right)$ for all $n\geq 0$, converges to a unique fixed point $x^{*}$ of $T$ in $U\left(x_{0},R\right)$.

Moreover the following error bounds hold:

and

where $\bar{\beta}_{n}=\left[1-a_{0}\left(a_{8}-a_{9}\right)\left\|x_{n}-x_{0}\right\|\right]^{-1}\quad(n\geq 0)$

Proof (i). By (15) and (25) we get $0\leq t_{0}\leq t_{1}\leq r^{*}$. Let us assume $0\leq t_{k-1}\leq t_{k}\leq r^{*}$ for $k=1,2,\cdots,n$. It follows from (18) and (26) that $0\leq t_{k}\leq t_{k+1}$. Hence, the sequence $\left\{t_{n}\right\}(n\geq 0)$ is monotonically increasing. Moreover using (26) we get in turn

That is the sequence $\left\{t_{n}\right\}(n\geq 0)$ is also bounded above by $r^{*}$. Since $r^{*}$ is the minimum nonnegative number satisfying $G\left(r^{*}\right)\leq r^{*}$, it follows that $\lim_{n\rightarrow\infty}t_{n}=r^{*}$.
(ii) By hypothesis (15) and the choice of $a_{1}$ it follows that $x_{1}\in U\left(x_{0},r^{*}\right)$. From (19) and (20) we get $g_{1}\left(x_{0}\right),g_{2}\left(x_{0}\right)\in U\left(x_{0},r^{*}\right)$. Let us assume $x_{k+1},g_{1}\left(x_{k}\right),g_{2}\left(x_{k}\right)\in U\left(x_{0},r^{*}\right)$ for $k=0,1,\cdots,n-1$. Then from (13), (14), (19) and (20) we get

and

Hence $g_{1}\left(x_{n}\right),g_{2}\left(x_{n}\right)\in U\left(x_{0},r^{*}\right)$. Using (5), (13), (14) and (17) we obtain

It follows from the Banach lemma on invertible operators [3] that $B_{k}$ is invertible and

Using (2) we obtain the approximation

From (7), we get

and since by (10), (11), (13), (14)

(32) gives

Moreover from (8), (9) and (12) we obtain respectively

and

Furthermore (31) because of (30), (33)-(36) finally gives (27) for $n=k$.

Estimate (28) is true for $n=0$ by (25). Assume (28) is true for $k=0,1,2,\cdots,n-$ 1. Then from (26), (27) and the induction hypothesis it follows that (28) is true for $k=n$. By (28) and part (i) it follows that iteration $\left\{x_{n}\right\}(n\geq 0)$ is Cauchy in a Banach space $E$ and as such it converges to some $x^{*}\in U\left(x_{0},r^{*}\right)$ (since $U\left(x_{0},r^{*}\right)$ is a closed set). Using hypothesis ( c ) and letting $n\rightarrow\infty$ in (2) we get $x^{*}=T\left(x^{*}\right)$. That is $x^{*}$ is a fixed point of $T$. Estimate (29) follows immediately from (28) using standard majorization techniques [2], [3].

Finally to show uniqueness let us assume $y^{*}\in U\left(x_{0},R\right)$ is a fixed point of equation (1). As in (31) we start from the approximation.

and using (5), (7)-(11), (13), (14), (21), (22) and (24) we get

By letting $n\rightarrow\infty$ as using (21) and (23) we get $\lim_{n\rightarrow\infty}x_{n}=y^{*}$. It follows from the uniqueness of the limit that $x^{*}=y^{*}$.

That completes the proof of the Theorem.