Observations Concerning Some Approximation Methods for the Solutions of Operator Equations

Ion Păvăloiu
(Cluj-Napoca)

1. Introduction

The purpose of this paper is to give some completions to some results, recently appeared in the literature, concerning the convergence and the error bounds of some methods for solving operatorial equations, when the Fréchet derivatives or the divided differences of the operators are Hölder continuous.

Let $f:X\rightarrow Y$ be an application, where $X$ and $Y$ are Banach spaces. We shall define the divided difference of a certain order in the following way: let $u_{i}\in X\ i=1,\ n+1,$ where $u_{i}\neq u_{j}$ for $i\neq j$ .

Definition 1.1.

[8]. The divided difference of the first order of the application $f$ at $u_{k},\ u_{s}\in X$ is an application $\left[u_{k},u_{s};f\right]\in\mathcal{L}\left(X,Y\right)$ which verifies:

a)

$\left[u_{k},u_{s};f\right]\left(u_{s}-u_{k}\right)=f\left(u_{s}\right)-f\left(% u_{k}\right)$
b)

if $f$ is Fréchet differentiable, at $u_{s}$ , then

$\left[u_{s},u_{s};f\right]=f^{\prime}\left(u_{s}\right).$

We suppose that there have been defined the applications

\left[u_{k},u_{k+1},\ldots,u_{k+m-1};f\right]\in\mathcal{L}\left(X^{m-1},Y% \right)\ \text{and}

\left[u_{k+1},u_{k+2},\ldots,u_{k+m};f\right]\in\mathcal{L}\left(X^{m-1},Y% \right),\

called the divided differences of the order $m-1$ , where $k+m\leq n$ .

Definition 1.2.

[8]. The divided difference of the order $\ m$ of the application $f$ in $u_{k},u_{k+1},\ldots,u_{k+m}\in X$ , is an application

\left[u_{k},u_{k+1},\ldots u_{k+m};f\right]\in\mathcal{L}\left(X^{m},Y\right)

which verifies:

(a’)		$\displaystyle\left[u_{k},u_{k+1},\ldots,u_{k+m};f\right]\left(u_{k+m}-u_{k}% \right)=$	$\displaystyle\left[u_{k+1},u_{k+2},\ldots,u_{k+m};f\right]$
		$\displaystyle-\left[u_{k},u_{k+1},\ldots,u_{k+m-1};f\right]$

b’)

if $f$ is $m$ time Fréchet differentiable at $u_{k},$ then

$\left[u_{k},u_{k},\ldots,u_{k};f\right]=\tfrac{1}{m!}f^{\left(m\right)}\left(u% _{k}\right).$

2. Considerations on the Secant Method

For the approximation of the solution of the operator equation

(1)

f\left(x\right)=0

consider the iteration

(2)

\displaystyle x_{n+1}

\displaystyle=x_{n}-\left[x_{n-1},x_{n};f\right]^{-1}f\left(x_{n}\right),% \qquad n=1,2,\ldots,x_{0},x_{1}\in X,

It is known that if $f$ satisfies certain conditions, then the sequence $\left(x_{n}\right)_{n\geq 0}$ given by (2) is well defined (there exists $\left[x_{n-1},x_{n};f\right]^{-1}$ for $n=1,2,\ldots$ ) and converges to a solution $x^{\ast}$ of equation (1) (see for example [2], [5], [8], [10].

In the following we shall give some specifications concerning the results obtained in [2]. Then we shall try to obtain conditions that ensure the convergence of the sequence $\left(x_{n}\right)_{n\geq 0}$ to a solution of (1), and, moreover, we shall determine a subset $E\subset X$ that contains this solution.

In paper [2], where the results obtained in [3] are generalized, the convergence of process (2) is studied under the assumptions that $f$ is Fréchet differentiable on a set $D\subset X$ and the Fréchet derivative $f^{\prime}\left(\cdot\right)$ satisfies a Hölder type condition on $D :$ there exist $c\in\mathbb{R}$ , $c>0$ and $p\in(0,1]$ such that:

(3)

\left\|f^{\prime}\left(x\right)-f^{\prime}\left(y\right)\right\|\leq c\left\|x% -y\right\|^{p},\qquad\text{for every }x,y\in D.

Let $H_{D}\left(c,p\right)$ denote the set of all applications $f^{\prime}$ for which (3) holds.

In [2], in addition to the conditions from Definition 1.1, it is assumed that the divided differences of the first order of $f$ satisfy a Hölder type condition, namely there exist $l_{1},l_{2,}l_{3}\geq 0,\ p=\left(0,1\right)$ such that for every $x,y,z\in D$ , the inequality:

(4)

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

holds.

This condition is useful when divided differences of the second order of $f$ are unbounded on $D$ .

Let $l_{2}^{\prime}=\max\{l_{2},l_{3}\}$ . If $x^{\ast}$ is a simple zero of equation (1), then the application $f^{\prime}\left(x^{\ast}\right)\in\mathcal{L}\left(X,Y\right)$ has a bounded inverse.

From (4) and from the existence and boundness of $\left[f^{\prime}\left(x^{\ast}\right)\right]^{-1}$ there exists $\varepsilon>0$ such that $\left[x,y;f\right]$ has a bounded inverse for every $x,y\in\bar{U}\left(x^{\ast},\varepsilon\right),$ where $\bar{U}\left(x^{\ast},\varepsilon\right)=\{x\in X|\left\|x-x^{\ast}\right\|% \leq\varepsilon\}$ , namely the application $B\left(x,y\right)=\left[x,y;f\right]^{-1}$ is uniformly bounded on $\bar{U}\left(x^{\ast},\varepsilon\right)$ .

In [2] the following theorem was proved:

Theorem 2.1.

Let $D\subset X$ be an open set and $f:X\rightarrow Y$ . If:

i)

$x^{\ast}\in D$ is a simple solution of equation (1).
ii)

there exists $\varepsilon>0$ and $b>0$ such that:

$\big\|\left[x,y;f\right]^{-1}\big\|\leq b,\qquad\text{for every }x,y\in\bar{U}\left(x^{\ast},\varepsilon\right);$
iii)

there exists a convex set $D_{0}$ and a real number $\varepsilon_{1}$ , $0<\varepsilon_{1}<\varepsilon$ , such that:

$f^{\prime}\left(\cdot\right)\in H_{r_{0}}\left(c,p\right),\quad\text{for every\ }x\in D_{0}\ \text{and\ }U\left(x^{\ast},\varepsilon_{1}\right)\subset D% _{0};$
iv)

$x_{0},x_{1}\in\bar{U}\left(x^{\ast},r\right)$ , where $0<r<\min\{\varepsilon_{1},\left[q\left(p\right)\right]^{-\frac{1}{p}}\}$

and

(5) $q\left(p\right)=\tfrac{b}{1-p}\left[2^{p}\left(l_{1}+l_{2}^{\prime}\right)% \left(1+p\right)+c\right],$

then the sequence given by (2) is well defined, and its elements belong to $\bar{U}\left(x^{\ast},r\right)$ . The sequence converges to the unique solution $x^{\ast}$ of (1). Moreover, the following estimation holds

(6)

\left\|x_{n+1}-x^{\ast}\right\|\leq\gamma_{1}\left\|x_{n-1}-x^{\ast}\right\|^{% p}\cdot\left\|x_{n}-x^{\ast}\right\|+\gamma_{2}\left\|x_{n}-x^{\ast}\right\|^{% 1+p}

for $n$ large enough where $\gamma_{1}$ and $\gamma_{2}$ are given by

(7)

\gamma_{1}=b\left(l_{1}+l_{2}^{\prime}\right)2^{p}

(8)

\gamma_{2}=\tfrac{bc}{1+p}

The proof is based on the following two lemmas [2]

Lemma 2.1.

Let $f:X\rightarrow Y$ and $D\subset X$ be an open set. If $f$ is Fréchet differentiable on $D$ and there exists a convex set $D_{0}\subset D$ such that $f^{\prime}\left(\cdot\right)\in H_{D_{0}}\left(c,p\right)$ , then for any $x,y\in D_{0}$ the following inequality holds:

(9)

\left\|f\left(x\right)-f\left(y\right)-f^{\prime}\left(x\right)\left(x-y\right% )\right\|\leq\tfrac{c}{1+p}\left\|x-y\right\|^{p+1}.

Lemma 2.2.

If there exists divided differences $\left[x,y;f\right]$ and inequality (4) is verified for all $x,y,z\in D_{0}$ , then the equality b) from Definition 1.1 holds for every $x\in D_{0}$ and the derivative $f^{\prime}$ of $f$ verifies the relation $f^{\prime}\left(\cdot\right)\in H_{D_{0}}\left[2\left(l_{1}+l_{2}\right),p\right]$ .

In the proof of Theorem 2.1 the following inequality is obtained first

(10)

\left\|x_{n+1}-x^{\ast}\right\|\leq\left[M\left(r\right)\right]^{n+1}\left\|x_% {0}-x^{\ast}\right\|,\qquad\text{where }0<M\left(r\right)<1.

from which it follows that the sequence $\left(x_{n}\right)_{n\geq 0}$ is convergent. In the following, by use of inequality (6) obtained in [2], we shall prove that the order of convergence of the sequence given by (2) is $t_{1}=\frac{1+\left(1+4p\right)^{1/2}}{2},$ i.e. it is the positive root of the equation:

(11)

t^{2}-t-p=0.

For this, besides the hypothesis of Theorem 2.1 we shall suppose that $x_{0}$ and $x_{1}$ verify

a’)

$\left\|x^{\ast}-x_{0}\right\|\leq\alpha d_{0};$
b’)

$\left\|x^{\ast}-x_{1}\right\|\leq\min\{\alpha d_{0}^{t_{1}},\left\|x^{\ast}-x_% {0}\right\|\},$

where $0<d_{0}<1$ and $\alpha=\left[q\left(p\right)\right]^{-\frac{1}{p}}$ .

Using Lemmas 2.1 and 2.2 and hypotheses of Theorem 2.1, from (2) we obtain

(12)

\left\|x_{2}-x^{\ast}\right\|\leq\gamma_{1}\left\|x_{0}-x^{\ast}\right\|^{p}% \left\|x_{1}-x^{\ast}\right\|+\gamma_{2}\left\|x_{1}-x^{\ast}\right\|^{p+1}

from which, taking into account a’), b’) it follows

\left\|x_{2}-x^{\ast}\right\|\leq\alpha d_{0}^{t_{1}^{2}}\big(\gamma_{1}+% \gamma_{2}d_{0}^{p\left(t_{1}-1\right)}\big)\alpha^{p},

where

\big(\gamma_{1}+\gamma_{2}d_{0}^{p\left(t_{1}-1\right)}\big)\alpha^{p}=\tfrac{% \gamma_{1}+\gamma_{2}d_{0}^{p\left(t_{1}-1\right)}}{\gamma_{1}+\gamma_{2}}<1

that is,

(13)

\left\|x_{2}-x^{\ast}\right\|\leq d_{0}^{t_{1}^{2}}

Relations (12) and (13) imply $\left\|x_{2}-x^{\ast}\right\|<\left\|x_{1}-x^{\ast}\right\|$ .

Suppose that there exists $n\in N,\ n\geq 2$ , such that

(a”)		$\displaystyle\left\\|x_{n-1}-x^{\ast}\right\\|$	$\displaystyle\leq\alpha d_{0}^{t_{1}^{n-1}}$
(b”)		$\displaystyle\left\\|x_{n}-x^{\ast}\right\\|$	$\displaystyle\leq\min\big\{\alpha d_{0}^{t_{1}^{n}},\ \left\\|x_{n-1}-x^{\ast}% \right\\|\big\}.$

If we repeat the above reasoning and take into account (a”) and (b”) we obtain

\left\|x_{n+1}-x^{\ast}\right\|\leq\alpha^{1+p}\cdot d_{0}^{t_{1}^{n+1}}\big(% \gamma_{1}+\gamma_{2}d_{0}^{p^{t_{1}^{n}\left(t_{1}-1\right)}}\big)\leq\alpha d% _{0}^{t_{1}^{n+1}},

because

\alpha^{p}\big(\gamma_{1}+\gamma_{2}d_{0}^{p^{t_{1}^{n}\left(t_{1}-1\right)}}% \big)<1.

Moreover, it can be easily seen that

\left\|x_{n+1}-x^{\ast}\right\|<\left\|x_{n}-x^{\ast}\right\|.

So far, we have proved the following theorem.

Theorem 2.2.

Under the hypotheses of Theorem 2.1 and if $x_{0}$ and $x_{1}$ verify a’) and b’), where $\alpha=\left(q\left(p\right)\right)^{-\frac{1}{p}}$ and $0<d_{0}<1$ , then for every $n\in\mathbb{N}$ , $x_{n}\in U\left(x^{\ast},\alpha\right)$ and

(2.13’)

\left\|x_{n+1}-x^{\ast}\right\|\leq d_{0}^{t_{1}^{n+1}},\qquad n=0,1,\ldots

One must notice that inequality (2.13’) gives a sharper error bound than (10).

In the following we shall establish a result which ensures not only the convergence of the sequences $\left(x_{n}\right)_{n\geq 0}$ but also the existence of the solution of equation (1) in a determined subset of $X$ .

In this respect, we observe that if $\left[x_{n-1},x_{n};f\right]^{-1}$ exists for every $n=1,2,\ldots$ then:

(14)

x_{n}-\left[x_{n-1},x_{n};f\right]^{-1}f\left(x_{n}\right)=x_{n-1}-\left[x_{n-% 1},x_{n};f\right]^{-1}f\left(x_{n-1}\right)

and

(15)		$\displaystyle f\left(x_{n+1}\right)=$	$\displaystyle f\left(x_{n}\right)+\left[x_{n-1},x_{n};f\right]\left(x_{n+1}-x_% {n}\right)+$
		$\displaystyle+\left(\left[x_{n},x_{n+1};f\right]-\left[x_{n-1},x_{n};f\right]% \right)\left(x_{n+1}-x_{n}\right),\qquad n=1,2,\ldots$

hold.

Let $\alpha,B,d_{0}\in\mathbb{R}$ , $\alpha>0,\ B>0,\ d_{0}\in\left(0,1\right)$ and

S=\Big\{x\in X:\left\|x-x_{0}\right\|\leq\tfrac{B\alpha d_{0}}{1-d_{0}^{t_{1}-% 1}}\Big\},

where $t_{1}$ is the positive root of equation (11).

Theorem 2.3.

If the divided differences of the first order of the applicaiton $f$ verify condition (4) for every $x,y,z\in S$ and

i)

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

the initial data $x_{0},x_{1}\in X$ and $f$ verify the inequalities

(16) $\displaystyle\left\|x_{1}-x_{0}\right\|$ $\displaystyle<B\ \alpha d_{0},\quad\left\|f\left(x_{0}\right)\right\|\leq% \alpha d_{0}\quad\text{ and}\ \left\|f\left(x_{1}\right)\right\|\leq\alpha d_{% 0}^{t_{1}}$

where

(17) $\alpha=\tfrac{1}{B^{\frac{1+p}{p}}\left(l_{1}+l_{2}+l_{3}\right)^{\frac{1}{p}}}$

the equation (1) has at least one solution $x^{\ast}\in S$ which is the limit of the sequence $\left(x_{n}\right)_{n\geq 0}$ given by (2) the order of the convergence of this sequence and the error bound are given by

(18) $\left\|x^{\ast}-x_{n}\right\|\leq\tfrac{B\alpha d_{0}^{t_{1}^{n}}}{1-d_{0}^{t_% {1}^{n}\left(t_{1}-1\right)}},\qquad n=1,2,\ldots$

Proof.

From (2), for $n=2$ we have

\left\|x_{2}-x_{1}\right\|\leq B\left\|f\left(x_{1}\right)\right\|\leq B\alpha d% _{0}^{t_{1}}

This inequality, together with the first inequality from (16), implies.

\displaystyle\left\|x_{2}-x_{0}\right\|

\displaystyle\leq\left\|x_{2}-x_{1}\right\|+\left\|x_{1}-x_{0}\right\|\leq B% \alpha d_{0}\big(1+d_{0}^{t_{1}-1}\big)\leq\tfrac{B\alpha d_{0}}{1-d_{0}^{t_{1% }-1}},

and so $x_{2}\in S$ .

Because $x_{2}\in S,$ from (14), (15) and using (4) we obtain:

(19)

\left\|f\left(x_{2}\right)\right\|\leq B^{p+1}\alpha^{p+1}\big(l_{1}+l_{2}+l_{% 3}d_{0}^{t_{1}^{2}-p}\big)d_{0}^{t_{1}^{2}-p}\leq\alpha d_{0}^{t_{1}^{2}},

since

\alpha^{p}B^{p+1}\big(l_{1}+l_{2}+l_{3}d_{0}^{p\left(t_{1}-1\right)}\big)\leq% \alpha^{p}B^{p+1}\left(l_{1}+_{2}+l_{3}\right)\leq 1.

Suppose

(20)		$\displaystyle x_{i}$	$\displaystyle\in S;$
(21)		$\displaystyle\left\\|f\left(x_{i}\right)\right\\|$	$\displaystyle\leq\alpha d_{0}^{t_{1}^{i}},\qquad\text{hold for }i=\overline{1,% k.}$

Then

	$\displaystyle\left\\|x_{n+1}-x_{k}\right\\|$	$\displaystyle\leq B\alpha d_{0}\big(1+d_{0}^{t_{1}-1}+d_{0}^{t_{1}^{2}-1}+% \ldots+d_{0}^{t_{1}^{k}-1}\big)$
		$\displaystyle\leq B\alpha d_{0}\big(1+d_{0}^{\left(t_{1}-1\right)}+d_{0}^{2% \left(t_{1}-1\right)}+\ldots+d_{0}^{k\left(t_{1}-1\right)}\big)\leq\tfrac{B% \alpha d_{0}}{1-d_{0}^{t_{1}-1}},$

that is, $x_{k+1}\in S$ .

By the use of the same reasoning as for (19) we obtain

(22)

\left\|f\left(x_{k+1}\right)\right\|\leq B^{p+1}\alpha^{p+1}\Big(l_{1}+l_{2}+l% _{3}^{pt_{1}^{k-1}\left(t_{1}-1\right)}\Big)d_{0}^{t_{1}^{k-1}{\left(t_{1}+p% \right)}}\leq\alpha_{0}d_{0}^{t_{1}^{k+1}}.

The above relations imply that (20) and (21) hold for every $k\in\mathbb{N}$ .

Now notice that $\left(x_{n}\right)_{n\geq 0}$ is a Cauchy sequence, because

(23)

\left\|x_{n+s}-x_{n}\right\|\leq{\sum_{k=n}^{n+s-1}}\left\|x_{k+1}-x_{k}\right% \|\leq B\alpha{\sum_{k=n}^{n+s-1}}d_{0}^{t_{1}^{k}}\leq\tfrac{B\alpha d_{0}^{t% _{1}^{n}}}{1-d_{0}^{t_{1}^{n}\left(t_{1}-1\right)}},

for any $n,s\in\mathbb{N}$ , $t_{1}>1$ and $d_{0}\in\left(0,1\right)$ .

Let $x^{\ast}=\lim\limits_{n\rightarrow\infty}x_{n}$ . Then, taking $s\rightarrow\infty$ in (23) we obtain

(24)

\left\|x^{\ast}-x_{n}\right\|\leq\tfrac{B\alpha d_{0}^{t_{1}^{n}}}{1-d_{0}^{t_% {1}^{n}\left(t_{1}-1\right)}},

$n=0,1,\ldots,$ that is, the inequality (18).

For $n=0$ we obtain $x^{\ast}\in S$ .

If $k\rightarrow\infty$ in (22) then $\left\|f\left(x^{\ast}\right)\right\|=0,$ that is $x^{\ast}$ , is the solution of equation (1). ∎

3. Considerations on Steffensen Method

It is well known that the order of convergence of the secant method can be improved if the elements $x_{n-1}$ and $x_{n}$ form (2) are related by an application $g:X\rightarrow X$ , described in the following.

Consider the sequence $\left(x_{n}\right)n\geq 0$ generated by

(25)

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,

where $g$ is an operator whose fixed points coincide with solutions of equations (1).

Consider $x_{0}\in X$ , the nonnegative real numbers $B,\varepsilon_{0},\rho_{0},\alpha,$ $\beta$ and $q\geq 1$ , where

(26)

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

(27)

\varepsilon_{0}=\rho_{0}\tfrac{1}{\left(p+q-1\right)}\left\|f\left(x_{0}\right% )\right\|,

the numbers $l_{1},\ l_{2},\ l_{3}$ being given by condition (4).

(28)

S=\left\{x\in X:\left\|x-x_{0}\right\|\leq\tfrac{r\varepsilon_{0}}{\rho_{0}^{% \frac{1}{p+q-1}}\left(1-\varepsilon_{0}^{p+q-1}\right)}\right\},

where $r=\max\{B,\beta\}$

Concerning the convergence of method (25), the following theorem holds:

Theorem 3.1.

If the real numbers $B,\ \varepsilon_{0},\rho_{0},\ p_{0},\ p,\ \alpha,\ \beta,\ q,\ l_{1},\ l_{2},% \ l_{3}$ the applications $f$ and $g$ , and the element $x_{0}$ satisfy the conditions:

(i)

for every $x,y\in S$ there exist $[x,y;f]^{-1}$ and $\big\|[x,y;f]^{-1}\big\|\leq B$ ;
(ii)

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

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

the divided differences of the first order of the applications $f$ verify condition (4) for every $x,y,z\in S$ ;
(v)

$\varepsilon_{0}<1$ ,

then the sequence $\left(x_{n}\right),\ n\geq 0$ given by (25) is convergent and if $x^{\ast}=\lim x_{n}$ , then $f\left(x^{\ast}\right)=0$ . Moreover, we have

(29)

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

Proof.

Let $x_{0}\in X$ be such that $\varepsilon_{0}$ verifies condition $(v)$ . Using similar relations to (14) and (15) and condition (4), we obtain from (25)

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

which means that $x_{1}\in S$ .

In the above inequality we have admitted the relation $g\left(x_{0}\right)\in S,$ which is implied by $(iii)$ .

From (14), (15), (25), $\left(i\right),$ $\left(ii\right)$ and $\left(iii\right)$ we obtain:

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

From the above inequality there follows

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

and if $\varepsilon_{1}=\rho_{0}^{\frac{1}{p+q-1}}\left\|f\left(x_{1}\right)\right\|$ then,

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

It can be easily seen that $\left\|f\left(x_{1}\right)\right\|\leq\left\|f\left(x_{0}\right)\right\|$ and $\rho_{1}\leq\rho_{0},$ where $\rho_{1}=B\alpha\left[l_{1}B^{p}+l_{2}\beta^{p}+l_{3}B^{p}\alpha^{p}\left\|f% \left(x_{1}\right)\right\|^{p\left(q-1\right)}\right]$ . Suppose now that, for $s=\overline{1,k,}$ the following relations hold $x_{s}\in S,\left\|f\left(x_{s}\right)\right\|\leq\left\|f\left(x_{s-1}\right)% \right\|,$ $\varepsilon_{s}<\varepsilon_{0}^{\left(p+q\right)^{s}},$ where $\varepsilon_{s}=\rho^{\frac{1}{p+q-1}}\left\|f\left(x_{s}\right)\right\|$ .

Using these assumptions and proceeding as above we get

(30)

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

(31)

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

showing that $x_{k+1}\in S$ .

It is also easy to see that

(32)

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

whence

(33)

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

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

We obtain further

(34)

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

whence

(35)

\varepsilon_{k+1}\leq\varepsilon_{0}^{\left(p+q\right)^{k+1}},\ \text{when }\varepsilon_{k+1}=\rho_{0}^{\frac{1}{p+q-1}}\left\|f\left(x_{k+1}\right)% \right\|.

From (30) it follows that, for every $s,n\in\mathbb{N}$ ,

(36)

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

and by $\left(v\right)$ the sequence $\left(x_{n}\right)_{n\geq 0}$ is fundamental, hence convergent. If $x^{\ast}=\lim\limits_{n\rightarrow\infty}x_{n}$ , from (36), for $s\rightarrow\infty$ , we get (29) and from (35) it follows that $x^{\ast}$ is a solution of (1).

From (29), for $n=0$ we have that $x^{\ast}\in S$ . ∎

4. Considerations Concerning Newton’s Method

Consider the sequence given by Newton’s method,

(37)

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

let $S\left(x_{0},r\right)=\{x\in X|\left\|x-x_{0}\right\|\leq r\}$ , where $r\in\mathbb{R}$ , $r>0$ .

Concerning the convergence of this sequence we have the following theorem.

Theorem 4.1.

If the application $f$ is Fréchet differentiable on $S\left(x_{0},r\right),$ the Fréchet derivative $f^{\prime}$ satisfies (3) for every $x,y\in S\left(x_{0},r\right)$ and the following conditions hold:

(i)

$\left[f^{\prime}\left(x_{0}\right)\right]^{-1}$ exists and $\big\|\left[f^{\prime}\left(x_{0}\right)\right]^{-1}\big\|\leq d$ ;
(ii)

$cr^{p}d<1$ ;
(iii)

$\rho_{0}=\alpha^{\frac{1}{p}}\left\|f\left(x_{0}\right)\right\|<1,$ where $\alpha=\frac{c\beta^{1+p}}{1+p}$ and $\beta=\frac{d}{1-cdr^{p}},$
(iv)

$\tfrac{\beta\left\|f\left(x_{0}\right)\right\|}{1-\alpha\left\|f\left(x_{0}% \right)\right\|^{p}}\leq r,$

then,

(j)

$x_{n}\in S\left(x_{0},r\right),\ for$ every $n\in\mathbb{N}$ ,
(jj)

there exists $\Gamma_{n}=\left[f^{\prime}\left(x_{n}\right)\right]^{-1}$ for every n $\in\mathbb{N}$ and $\left\|\Gamma_{n}\right\|<\frac{d}{1-dcr^{p}},$
(jjj)

$\left\|x_{n+1}-x_{n}\right\|\leq\beta\alpha^{-\frac{1}{p}}\rho_{0}^{\left(1+p% \right)^{n}}$ ;
(jv)

the sequence $\left(x_{n}\right)_{n\geq 0}$ is convergent, and if $x^{\ast}=\lim\limits_{n\rightarrow\infty}x_{n}$ then $f\left(x^{\ast}\right)=0$ and

(38) $\left\|x^{\ast}-x_{n}\right\|\leq\tfrac{\beta\alpha^{-\frac{1}{p}}\rho_{0}^{% \left(1+p\right)^{n}}}{1-\rho_{0}^{\left(1+p\right)^{n}}},\qquad n\in\mathbb{N}.$

Proof.

By (37), for $n=1$ we get $x_{1}=x_{0}-\left[f^{\prime}\left(x_{0}\right)\right]^{-1}f\left(x_{0}\right)$ , and from $\left(i\right)$ $\left\|x_{1}-x_{0}\right\|\leq d\left\|f\left(x_{0}\right)\right\|\leq\beta% \left\|f\left(x_{0}\right)\right\|\leq r$ , that is, $x_{1}\in S\left(x_{0},r\right)$ .

By (3) and $ii)$ it follows

\left\|[f^{\prime}\left(x_{0}\right)]^{-1}\left[f^{\prime}\left(x_{0}\right)-f% ^{\prime}\left(x_{1}\right)\right]\right\|\leq dc\left\|x_{1}-x_{0}\right\|^{p% }\leq dcr^{p}<1,

whence $\left[f^{\prime}\left(x_{1}\right)\right]^{-1}$ exists and

\big\|\left[f^{\prime}\left(x_{1}\right)\right]^{-1}\big\|\leq\frac{d}{1-dcr^{% p}}=\beta.

From (3) it follows

	$\displaystyle\left\\|f\left(x_{1}\right)\right\\|$	$\displaystyle=\left\\|f\left(x_{1}\right)-f\left(x_{0}\right)-f^{\prime}\left(x% _{0}\right)\left(x_{1}-x_{0}\right)\right\\|$
		$\displaystyle\leq\tfrac{c}{p+1}\left\\|x_{1}-x_{0}\right\\|^{p+1}\leq\tfrac{c% \beta^{p+1}}{p+1}\left\\|f\left(x_{0}\right)\right\\|^{p+1}$

and if

\rho_{1}=\alpha^{\frac{1}{p}}\left\|f\left(x_{1}\right)\right\|

then

\rho_{1}\leq\rho_{0}^{1+p},\quad\left\|x_{2}-x_{1}\right\|\leq\beta\alpha^{-% \frac{1}{p}}\rho_{0}^{1+p}.

If $\rho_{i}=\alpha^{\frac{1}{p}}\left\|f\left(x_{i}\right)\right\|$ and

(a)	$\displaystyle x_{i}$	$\displaystyle\in S\left(x_{0},r\right),\qquad i=\overline{1,k},$
(b)	$\displaystyle\left\\|x_{i+1}-x_{i}\right\\|$	$\displaystyle\leq\beta\alpha^{-\frac{1}{p}}\rho_{0}^{\left(1+p\right)^{i}},% \qquad i=\overline{1,k-1},$
(c)	$\displaystyle\rho_{i}$	$\displaystyle\leq\rho_{0}^{\left(1+p\right)^{i}},\qquad i=\overline{1,k,}$

then we get by $\left(a\right),$ (3) and $\left(i\right)$

\big\|\left[f^{\prime}\left(x_{0}\right)\right]^{-1}\left[f^{\prime}\left(x_{k% }\right)-f^{\prime}\left(x_{0}\right)\right]\big\|\leq dc\left\|x_{k}-x_{0}% \right\|^{p}\leq cdr^{p}<1.

It follows that

\big\|\left[f^{\prime}\left(x_{k}\right)\right]^{-1}\big\|\leq\tfrac{d}{1-dcr^% {p}}=\beta.

From (37) and from the above inequality we get that:

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

which by $\left(b\right)$ implies

\left\|x_{k+1}-x_{0}\right\|\leq\tfrac{\beta\left\|f\left(x_{0}\right)\right\|% }{1-\alpha\left\|f\left(x_{0}\right)\right\|^{p}}\leq r,

that is, $x_{k+1}\in S\left(x_{0},r\right)$ .

Using the assumptions of the theorem we get that

	$\displaystyle\left\\|f\left(x_{k+1}\right)\right\\|$	$\displaystyle=\left\\|f\left(x_{k+1}\right)-f\left(x_{k}\right)-f^{\prime}\left% (x_{k}\right)\left(x_{k+1}-x_{k}\right)\right\\|$
		$\displaystyle\leq\tfrac{c}{p+1}\left\\|x_{k+1}-x_{k}\right\\|^{p+1}\leq\alpha^{-% \frac{1}{p}}\rho_{k}^{p+1},$

whence

\rho_{k+1}\leq\rho_{0}^{\left(1+p\right)^{k+1}}.

For every $m,n\in\mathbb{N}$

\left\|x_{m+n}-x_{n}\right\|\leq\tfrac{\beta\alpha^{-\frac{1}{p}}\rho_{0}^{% \left(1+p\right)^{n}}}{1-\rho_{0}^{p\left(1+p\right)^{n}}}

which, together with $\rho_{0}<1$ , show that $\left(x_{n}\right)_{n\geq 0}$ is a Cauchy sequence. If $x^{\ast}=\lim\limits_{n\rightarrow\infty}x_{n}$ then for $m\rightarrow\infty$ in the above inequality, we get (38), and from

\left\|f\left(x_{n}\right)\right\|\leq\alpha^{-\frac{1}{p}}\rho_{0}^{\left(1+p% \right)^{n}},

for $n\rightarrow\infty,$ we get $f\left(x^{\ast}\right)=0$ . ∎

References

[1] Argyros, K.I., Concerning the convergence of Newton’s method, The Renjab University Journal of Mathematics, Vol. XXI (1988), pp.1–11.
[2] Argyros, K.I, The secant method and fixed points of nonlinear operrators, Mh. Math., 106 (1988) pp. 85–94.
[3] Dennis, J. E., Toward a unified convergence theory for Newton like methods, Nonlinear Functional Analysis and Applications. (Ed. by L.B. Rall). John Wiley, New York (1986).
[4] Lazăr, I., ^†^†margin: available soon,
refresh and click here $\rightarrow$ On Newton’s method for solving operator equations with Hölder continuous derivative. Rev. Anal. Numér. Théor. Approx., v. 23. no. 2 (1993), pp. 177–187.
[5] Ortega, J. M. and Rheinboldt, W., Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York and London, 1970.
[6] ^†^†margin: clickable $\rightarrow$ Păvăloiu, I., Remarks on the secant method for the solution of nonlinear operational equations. Research Seminars, Seminar on Mathematical analysis. Preprint nr.7 (1991) pp.127–132.
[7] ^†^†margin: clickable $\rightarrow$ Păvăloiu, I., On the convergence of a Steffensen-type method, Research Seminars. Seminar of Mathematical Analysis. Preprint nr.7 (1991), pp.121–126.
[8] Păvăloiu, I., ^†^†margin: available soon,
refresh and click here $\rightarrow$ Introduction in the theory of approximation of equations solutions. Dacia Ed., Cluj-Napoca, (1976) (in Romanian).
[9] ^†^†margin: clickable $\rightarrow$ Păvăloiu, I., Sur une généralisation de la méthode de Steffensen, Rev. Anal. Numér. Théor. Approx. v. 21, no. 1, (1992), pp.59–65.
[10] Schmidt, J. W. Konvergenzgesch windigkert der Regula falsi und der Steffensen Verfahrens in Banachraum. Z.A.M.M. 46, 2, (1996) pp. 146–148.
[11] U’lm, S., Ob. obobschennyh razdelennih raznostiakh I., Izv. Acad. Nauk Estonskoi SSR, 16 (1967), 13-36.
[12] U’lm, S., Ob. obobschennyh razdelennih raznostiakh II., Izv. Acad. Nauk Estonskoi SSR, 16 (1967), 146-155.
[13] U’lm, S., Ob. obobschenie metoda Steffensena dlea reshenia nelineingh-operatornih urovnenii. Jurnal vicisl. mat. i mat.-fiz. 4, 6, (1969).

Received 15 X 1993

Institutul de Calcul

Str. Republicii 37

P.O. Box 68

3400 Cluj-Napoca

România

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

Observations concerning some approximation methods for the solutions of operator equations

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Observations Concerning Some Approximation Methods for the Solutions of Operator Equations

1. Introduction

Definition 1.1.

Definition 1.2.

2. Considerations on the Secant Method

Theorem 2.1.

Lemma 2.1.

Lemma 2.2.

Theorem 2.2.

Theorem 2.3.

Proof.

3. Considerations on Steffensen Method

Theorem 3.1.

Proof.

4. Considerations Concerning Newton’s Method

Theorem 4.1.

Proof.

References

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