"BABEŞ-BOLYAI" UNIVERSITY

Faculty of Mathematics

Research Seminars

Seminar on Mathematical Analysis

Preprint nr.7, 1991, pp.121-126

On the Convergency of a Steffensen-type Method

Ion Păvăloiu

1. In the paper [2] I.K. Argyros adopts for the divided difference of the mapping $f:X_{1}\rightarrow X_{2},$ where $X_{1}$ and $X_{2}$ are Banach spaces, the following definition:

Definition 1.

One calls divided difference of the application $f$ at the points $x,y,\in X_{1}$ a linear application $\left[x,y;f\right]\in\mathcal{L}\left(X_{1},X_{2}\right)$ which fulfils the following conditions:

(a)

$\left[x,y;f\right]\left(y-x\right)=f\left(y\right)-f\left(x\right)$ for every $x,y\in D\subseteq X_{1};$
(b)

there exist the real constants $l_{1}>0,\ l_{2}>0,$ $l_{3}>0,\ p\in(0,1]$ such that for every $x,y,u\in D$ the following inequality holds:

$\left\|\left[y,u;f\right]-\left[x,y;f\right]\right\|\leq l_{1}\left\|x-u\right% \|^{p}+l_{2}\left\|x-y\right\|^{p}+l_{3}\left\|u-y\right\|^{p}.$

In [5] there are obtained refinements of Argyros results concerning the secant method applied to the solution of the equation:

(1)

f\left(x\right)=0

where $f:X_{1}\rightarrow X_{2}$

2. We shall study further down the convergence of Steffensen’s method for the solution of equation (1), namely the convergence of the sequence $\left(x_{n}\right)_{n\geq 0}$ generated by means of the following procedure:

(2)

x_{n+1}=x_{n}-\left[x_{n},g\left(x_{n}\right);f\right]^{-1}f\left(x_{n}\right)% ,\qquad x_{0}\in x_{1},\ n=0,1,\ldots,

where $g:X_{1}\rightarrow X_{1}$ is an operator having at least one fixed point which coincides with the solution of equation (1).

Obviously, the sequence $\left(x_{n}\right)_{n\geq 0}$ can be generated by means of the procedure (2) if at each iteration step there exists the mapping $\left[x_{n},g\left(x_{n}\right);f\right]^{-1}.$

For our purpose observe firstly that the following identities:

(3)		$\displaystyle x_{n}-\left[x_{n},g\left(x_{n}\right);f\right]^{-1}f\left(x_{n}% \right)=$
(4)		$\displaystyle=g\left(x_{n}\right)-\left[x_{n},g\left(x_{n}\right);f\right]^{-1% }f\left(g\left(x_{n}\right)\right)f\left(x_{n+1}\right)$
	$\displaystyle=f\left(g\left(x_{n}\right)\right)+\left[x_{n},g\left(x_{n}\right% );f\right]\left(x_{n+1}-g\left(x_{n}\right)\right)$
	$\displaystyle\quad+\left(\left[g\left(x_{n}\right),x_{n+1};f\right]-\left[x_{n% },g\left(x_{n}\right);f\right]\right)\ \left(x_{n+1}-g\left(x_{n}\right)\right)$

hold for every $n=0,1,\ldots$

Let $x_{0}\in X_{1}$ be an element, and consider the nonnegative real numbers: $B,\varepsilon_{0},\rho_{0},p\in(0,1],\ \alpha,\beta,q\geq 1,\ l_{1},l_{2}$ and $l_{3}$ , where

\rho_{0}=\beta\alpha\big{(}l_{1}B^{p}+l_{2}B^{p}+l_{3}B^{p}\alpha^{p}\left\|f% \left(x_{0}\right)\right\|^{p\left(q-1\right)}\big{)}

and

\varepsilon_{0}=\rho^{1/\left(p+q-1\right)}\left\|f\left(x_{0}\right)\right\|.

Denote $r=\max\{B,\beta\}$ and suppose that $S\subseteq D,$ where:

S=\Big{\{}x\in X_{1}:\left\|x-x_{0}\right\|\leq\tfrac{r\varepsilon_{0}}{\rho_{% 0}^{1/\left(p+q-1\right)}(1-\varepsilon_{0}^{p+q-1})}\Big{\}}.

The following theorem holds:

Theorem 1.

If the constants $B,$ $\varepsilon_{0},$ $\rho_{0},p,\alpha,\beta,q,l_{1},l_{2},l_{3},$ the mapping $f$ and $g$ , and the initial element $x_{0}\in X_{1}$ , as well, fulfil the conditions:

(I)

for every $x,y\in S$ there exists $\left[x,y;f\right]^{-1}$ , and $\left\|\left[x,y;f\right]^{-1}\right\|\leq B$
(II)

for every $x\in S$ , $\left\|f\left(g\left(x\right)\right)\right\|\leq\alpha\left\|f\left(x\right)% \right\|^{q};$
(III)

for every $x\in S,\ \left\|x-g\left(x\right)\right\|\leq\beta\left\|f\left(x\right)\right\|;$
(IV)

the divided difference of the mapping $f$ fulfils the conditions (a) and (b) specified in the definition given in Section 1;
(V)

$\varepsilon_{0}<1,$

then the sequence $\left(x_{n}\right)_{n\geq 0}$ generated by the procedure (2) is convergent, and, if we denote $\bar{x}=\lim x_{n}$ , then $f\left(\bar{x}\right)=0$ and the following delimitation holds:

\left\|x-x_{n}\right\|\leq\tfrac{r\rho_{0}^{\left(p+q\right)^{n}}}{\rho_{0}^{1% /\left(p+q-1\right)}\left(1-\varepsilon_{0}^{p+q-1}\right)}.

Proof.

Consider $x_{0}\in X_{1}$ for which the condition (V) is fulfilled. Taking into account the condition (b) and the procedure (2), from the identities (3) and (4) it results:

\left\|x_{1}-x_{0}\right\|\leq B\left\|f\left(x_{0}\right)\right\|\leq\tfrac{B% \varepsilon_{0}^{1/\left(p+q-1\right)}}{\rho_{0}^{1/\left(p+q-1\right)}}\left% \|f\left(x_{0}\right)\right\|\leq\tfrac{r\varepsilon_{0}}{\rho_{0}^{1/\left(p+% q-1\right)}\left(1-\varepsilon_{0}^{p+q-1}\right)},

from which follows $x_{1}\in S$ .

Here was used the inequality:

\left\|g\left(x_{0}\right)-x_{0}\right\|\leq\beta\left\|f\left(x_{0}\right)% \right\|\leq\tfrac{r\varepsilon_{0}}{\rho_{0}^{1/\left(p+q-1\right)}\left(1-% \varepsilon_{0}^{p+q-1}\right)},

from which follows that $g\left(x_{0}\right)\in S$ .

Now, considering the above results, we have:

	$\displaystyle\left\\|f\left(x_{1}\right)\right\\|$	$\displaystyle\leq\left\\|\left[g\left(x_{0}\right),x_{1};f\right]-\left[x_{0},g% \left(x_{0}\right);f\right]\right\\|\cdot\left\\|x_{1}-g\left(x_{0}\right)\right\\|$
		$\displaystyle\leq\beta\alpha\left[l_{1}B^{p}+l_{2}B^{p}+l_{3}B^{p}\alpha^{p}% \left\\|f\left(x_{0}\right)\right\\|^{p\left(q-1\right)}\right]\left\\|f\left(x_{% 0}\right)\right\\|^{p+q}$
		$\displaystyle=\rho_{0}\left\\|f\left(x_{0}\right)\right\\|^{p+q}$

This inequality leads to:

\rho_{0}^{1/\left(p+q-1\right)}\left\|f\left(x_{1}\right)\right\|\leq\rho_{0}^% {1/\left(p+q-1\right)}\left\|f\left(x_{0}\right)\right\|^{p+q}

or, using the notation $\varepsilon_{1}=\rho_{0}^{1/\left(p+q-1\right)}\left\|f\left(x_{1}\right)\right\|$ :

\varepsilon_{1}\leq\varepsilon_{0}^{p+q}

From this inequality follows that $\left\|f\left(x_{1}\right)\right\|\leq\left\|f\left(x_{0}\right)\right\|,$ and if

\rho_{1}=\beta\alpha\left(l_{1}B^{p}+l_{2}B^{p}+l_{3}B^{p}\alpha^{p}\left\|f% \left(x_{1}\right)\right\|^{p\left(q-1\right)}\right)

then $\rho_{1}\leq\rho_{0}.$

Suppose now that the following properties hold:

( $\alpha$ )

$x_{p}\in S;$
( $\beta$ )

$\left\|f\left(x_{p}\right)\right\|\leq\left\|f\left(x_{p-1}\right)\right\|;$
( $\gamma$ )

$\varepsilon_{p}\leq\varepsilon_{0}^{\left(p+q\right)^{p}},\ \varepsilon_{p}=% \rho_{0}^{1/\left(p+q-1\right)}\left\|f\left(x_{p}\right)\right\|,\ p=1,2,% \ldots,k$

From (2) for $n=k$ we obtain:

\left\|x_{k+1}-x_{k}\right\|\leq B\left\|f\left(x_{k}\right)\right\|\leq\tfrac% {r\varepsilon_{k}}{\rho_{0}^{1/\left(p+q-1\right)}}\leq\tfrac{r\varepsilon_{0}% ^{\left(p+q\right)^{k}}}{\rho_{0}^{1/\left(p+q-1\right)}},

which leads to:

	$\displaystyle\left\\|x_{k+1}-x_{0}\right\\|$	$\displaystyle\leq\tfrac{r}{\rho_{0}^{1/\left(p+q-1\right)}}\left(\varepsilon_{% 0}+\varepsilon_{0}^{p+q}+\varepsilon_{0}^{\left(p+q\right)^{2}}+\ldots+% \varepsilon_{0}^{\left(p+q\right)^{k}}\right)$
		$\displaystyle\leq\tfrac{r\varepsilon_{0}}{\rho_{0}^{1/\left(p+q-1\right)}\left% (1-\varepsilon_{0}^{p+q-1}\right)},$

namely $x_{k+1}\in S$ .

Here was used the inequality:

\left\|g\left(x_{k}\right)-x_{k}\right\|\leq B\left\|f\left(x_{k}\right)\right% \|\leq\tfrac{r\rho_{0}^{1/\left(p+q-1\right)}}{\rho_{0}^{1/\left(p+q-1\right)}% }\left\|f\left(x_{k}\right)\right\|\leq\tfrac{r\varepsilon_{0}^{\left(p+q% \right)^{k}}}{\rho_{0}^{1/\left(p+q-1\right)}},

from which follows immediately:

\left\|g\left(x_{k}\right)-x_{0}\right\|\leq\tfrac{r\varepsilon_{0}}{\rho_{0}^% {1/\left(p+q-1\right)}\left(1-\varepsilon_{0}^{p+q-1}\right)},

that is, $g\left(x_{k}\right)\in S$ .

As to $\left\|f\left(x_{k+1}\right)\right\|$ we have:

\left\|f\left(x_{k+1}\right)\right\|\leq\beta\alpha\left(l_{1}B^{p}+l_{2}B^{p}% +l_{3}\alpha^{p}B^{p}\left\|f\left(x_{k}\right)\right\|^{p\left(q-1\right)}% \right)\left\|f\left(x_{k}\right)\right\|^{p+q}

namely

\left\|f\left(x_{k+1}\right)\right\|\leq\rho_{0}\left\|f\left(x_{k}\right)% \right\|^{p+q},

which yields:

\varepsilon_{k+1}\leq\varepsilon_{k}^{p+q}\leq\varepsilon_{0}^{\left(p+q\right% )^{k+1}}.

By virtue of the above proved results follows that the properties ( $\alpha$ )–( $\gamma$ ) hold for every $p\in\mathbb{N}$ .

We prove further down that the sequence $\left(x_{n}\right)_{n\geq 0}$ is a fundamental sequence. Indeed, we have:

	$\displaystyle\left\\|x_{n+s}-x_{n}\right\\|$	$\displaystyle\leq\left\\|x_{n+s}-x_{n+s-1}\right\\|+\left\\|x_{n+s-1}-x_{n+s-2}% \right\\|+\ldots+\left\\|x_{n+1}-x_{n}\right\\|$
		$\displaystyle\leq B\left(\left\\|f\left(x_{n}\right)\right\\|+\left\\|f\left(x_{n% +1}\right)\right\\|+\ldots+\left\\|f\left(x_{n+s-1}\right)\right\\|\right)$
		$\displaystyle\leq\tfrac{B}{\rho_{0}^{1/\left(p+q-1\right)}}\left(\varepsilon_{% 0}^{\left(p+q\right)^{n}}+\varepsilon_{0}^{\left(p+q\right)^{n+1}}+\ldots+% \varepsilon_{0}^{\left(p+q\right)^{n+s-1}}\right)$
		$\displaystyle\leq\tfrac{B\varepsilon_{0}^{\left(p+q\right)^{n}}}{\rho_{0}^{1/% \left(p+q-1\right)}}\left(1+\varepsilon^{p+q-1}+\varepsilon^{\left(p+q\right)^% {2}-1}+\varepsilon^{\left(p+q\right)^{s-1}-1}\right)$
		$\displaystyle\leq\tfrac{B\varepsilon_{0}^{\left(p+q\right)^{n}}}{\rho_{0}^{1/% \left(p+q-1\right)}\left(1-\varepsilon_{0}^{p+q-1}\right)},$

that is, for every $s,n\in\mathbb{N}$ the following inequality holds:

\left\|x_{n+s}-x_{n}\right\|\leq\frac{B\varepsilon_{0}^{\left(p+q\right)^{n}}}% {\rho_{0}^{1/\left(p+q-1\right)}\left(1-\varepsilon_{0}^{p+q-1}\right)},

from which, since $\varepsilon_{0}<1$ , it results that the sequence $\left(x_{n}\right)_{n\geq 0}$ is fundamental. Since $X_{1}$ is a Banach space, there exists $\lim\limits_{n\rightarrow\infty}x_{n}=\bar{x}$ ,

and

\left\|\bar{x}-x_{n}\right\|\leq\tfrac{B\varepsilon_{0}^{\left(p+q\right)^{n}}% }{\rho_{0}^{1/\left(p+q-1\right)}\left(1-\varepsilon_{0}^{p+q-1}\right)},

which leads, for $n=0,$ to $\bar{x}\in S$ .

From the inequality $\varepsilon_{n}\leq\varepsilon_{0}^{\left(p+q\right)^{n}},$ for $n\rightarrow\infty,$ we obtain:

f\left(\bar{x}\right)=\lim_{n\rightarrow\infty}f\left(x_{n}\right)=0,

and one sees that $\bar{x}$ is the solution of the equation (1). ∎

References

[1]
[2] Argyros, I.K., The secant method and fixed points of nonlinear operators, Mh. Math. 106, 85–94 (1988).
[3] ^†^†margin: clickable $\rightarrow$ Păvăloiu, I., Sur la méthode de Steffensen pour la résolution des équations operationnelles non linéaires, Revue Roumaine des Mathematiques pures et appliquees, 1, XIII, 149–158 (1968).
[4] Păvăloiu, I., ^†^†margin: clickable $\rightarrow$ Introduction in the Theory of Approximation of Equations Solutions, Dacia Ed., Cluj-Napoca 1976 (in Romanian).
[5] ^†^†margin: clickable $\rightarrow$ Păvăloiu, I., Remarks on the secant method for the solution of nonlinear operatorial equations, Research Seminars, Seminar on Mathematical Analysis, Preprint no. 7, (1991), pp. 127–132.
[6] Ul’m, S., Ob obobscenie metod Steffensen dlea resenia nelineinîh operatornîh urnavnenii, Journal Vîcisl., mat. i mat.-fiz. 4, 6 (1964).

Institutul de Calcul

Oficiul Poştal 1

C.P. 68

3400 Cluj-Napoca

Romania

This paper is in final form and no version of it is or will be sumitted for publication elsewhere.

	$\displaystyle\left\\|f\left(x_{1}\right)\right\\|$	$\displaystyle\leq\left\\|\left[g\left(x_{0}\right),x_{1};f\right]-\left[x_{0},g% \left(x_{0}\right);f\right]\right\\|\cdot\left\\|x_{1}-g\left(x_{0}\right)\right\\|$
		$\displaystyle\leq\beta\alpha\left[l_{1}B^{p}+l_{2}B^{p}+l_{3}B^{p}\alpha^{p}% \left\\|f\left(x_{0}\right)\right\\|^{p\left(q-1\right)}\right]\left\\|f\left(x_{% 0}\right)\right\\|^{p+q}$
		$\displaystyle=\rho_{0}\left\\|f\left(x_{0}\right)\right\\|^{p+q}$

On the convergency of a Steffensen-type method

Abstract

Authors

Keywords

PDF

Cite this paper as:

About this paper

Journal

Publisher Name

DOI

References

Paper (preprint) in HTML form

On the Convergency of a Steffensen-type Method

Definition 1.

Theorem 1.

Proof.

References

Related Posts