"Babeş-Bolyai" University, Faculty of Mathematics

Research Seminars

Seminar on Functional Analysis and Numerical Methods

Preprint Nr.1, 1986, pp.127-132

The Convergence of Certain Iterative Methods for Solving Certain Operator Equations

Ion Pavaloiu

Let us designate for $AND$ a Banach space and consider an operational equation

(1)

x=\lambda D\left(x\right)+y,\qquad\lambda\in{\mathbb{R}},

Or $D:E\rightarrow E$ is a nonlinear application.

In the following and to solve equation ( 1 ), we consider the following iterative process:

(2)

x_{n+1}=x_{n}-A\left(x_{n}\right)\left[x_{n}-\lambda D\left(x_{n}\right)-y% \right],\qquad n=0,1,\ldots,x_{0}\in E

Or $A\left(x\right):E\rightarrow E$ is a linear application for each $x\in E$ .

We will designate, to fix the ideas, by $P:E\rightarrow E$ l’application $P\left(x\right)=x-\lambda D\left(x\right)-y$ .

If we assume that the application $D$ admits derivatives up to and including second order on the space $AND$ , SO $P^{\prime}\left(x\right)=I-\lambda D^{\prime}\left(x\right)$ And $P^{\prime\prime}\left(x\right)=-\lambda D^{\prime\prime}\left(x\right)$ And $r\in\mathbb{R}$ And $r>0,$ we write $S\left(x_{0},r\right)=\{x\in E;\left\|x-x_{0}\right\|\leq r\}$ .

Concerning the convergence of the process ( 2 ) we have the following theorem:

Theorem 1 .

If the applications $D$ And $A\left(x\right),$ the initial element $x_{0}$ and the real number $r>0$ meet the conditions:

i.

L’application $D$ admits Frechet derivatives up to second order incusively on $S\left(x_{0},r\right);$
ii.

$\left\|A\left(x\right)\right\|\leq\beta$ for each $x\in S\left(x_{0},r\right),$ Or $\beta\in\mathbb{R}$ And $\beta>0;$
iii.

$\left\|I-P^{\prime}\left(x\right)A\left(x\right)\right\|\leq\alpha$ for each $x\in S\left(x_{0},r\right)$ Or $\alpha\in\mathbb{R}$ , $\alpha>0;$
iv.

$\left\|D^{\prime\prime}\left(x\right)\right\|\leq M\left|\lambda\right|$ , for each $x\in S\left(x_{0},r\right)$ Or $M\in\mathbb{R}$ , $M>0;$
in.

$\left(\beta\rho_{0}\right)/\left(1-d_{0}\right)\leq r$ Or $\rho_{0}=\left\|P\left(x_{0}\right)\right\|,\ d_{0}=\frac{M\beta^{2}\rho_{0}}{2}$
we.

$d_{0}<1,$

so the rest $\left(x_{n}\right)_{n=0}^{\infty}$ generated by ( 2 ) is convergent and if we write $\bar{x}=\lim\limits_{n\rightarrow\infty}x_{n},$ SO $P\left(\bar{x}\right)=\theta$ . We have the following delimitation:

(3)

\left\|\bar{x}-x_{n}\right\|\leq\frac{\beta d_{0}^{n}\rho_{0}}{1-d_{0}},\ n=0,% 1,\ldots\ .

Demonstration.

We will show by induction that the following properties hold:

a)

$x_{n}\in S\left(x_{0},r\right)$ for each $n=0,1,\ldots\$ ;
b)

$\left\|P\left(x_{n}\right)\right\|\leq d_{0}^{n}\cdot\rho_{0},\ n=0,1,\ldots\ .$

Indeed, we deduce from ( 2 ) that $n=0$

\left\|x_{1}-x_{0}\right\|\leq\left\|A\left(x_{0}\right)\right\|\cdot\left\|P% \left(x_{0}\right)\right\|\leq\beta\rho_{0}\leq\beta\rho_{0}/\left(1-d_{0}% \right)\leq r

from which it follows that $x_{0}\in B\left(x_{1},r\right)$ .

We will assume that $x_{i}\in S\left(x_{0},r\right)$ for each $i=0,1,\ldots,k$ . We then have

(4)	$\displaystyle\left\\|P\left(x_{i}\right)\right\\|$	$\displaystyle\leq\left\\|P\left(x_{i}\right)-P\left(x_{i-1}\right)-P^{\prime}% \left(x_{i-1}\right)\left(x_{i}-x_{i-1}\right)\right\\|$
	$\displaystyle\quad+\left\\|P\left(x_{i-1}\right)+P^{\prime}\left(x_{i-1}\right)% \left(x_{i}-x_{i-1}\right)\right\\|$
	$\displaystyle\leq\tfrac{M\beta^{2}}{2}\left\\|P\left(x_{i-1}\right)\right\\|^{2}% +\alpha\left\\|P\left(x_{i-1}\right)\right\\|,\qquad i=1,2,\ldots,k$

hence writing $d_{i-1}=\frac{M\beta^{2}}{2}\left\|P\left(x_{i-1}\right)\right\|+\alpha,\ i=1,% 2,\ldots$ and taking into account $in i .$ we immediately deduce the following inequalities:

(5)

d_{0}\geq d_{1}\geq d_{2}\geq\cdots\geq d_{k}.

It follows from ( 4 ) and ( 5 )

(6)

\left\|P\left(x_{i}\right)\right\|\leq d_{0}\left\|P\left(x_{i-1}\right)\right% \|,\qquad i=1,2,\ldots,k.

It follows

\left\|P\left(x_{i}\right)\right\|\leq d_{0}^{i}\rho_{0},\qquad i=1,2,\ldots,k.

We will now show that $x_{k+1}\in S\left(x_{0},r\right)$ . Indeed, we have

	$\displaystyle\left\\|x_{k+1}-x_{0}\right\\|$	$\displaystyle\leq\sum_{i=0}^{n}\left\\|x_{i+1}-x_{i}\right\\|\leq\beta\sum_{i=0}% ^{k}\left\\|P\left(x_{i}\right)\right\\|$
		$\displaystyle\leq\beta\rho_{0}\sum_{i=0}^{k}d_{0}^{i}\leq\tfrac{\beta\rho_{0}}% {1-d_{0}}\leq r.$

Let's suppose that $n,p\in{\mathbb{N}}$ , we then have

(7)		$\displaystyle\left\\|x_{n+p}-x_{n}\right\\|$	$\displaystyle\leq\sum_{i=1}^{p}\left\\|x_{n+i}-x_{n+i-1}\right\\|\leq\beta\sum_{% i=1}^{p}\left\\|P\left(x_{n+i-1}\right)\right\\|$
		$\displaystyle\leq\beta\rho_{0}\sum_{i=1}^{p}d_{0}^{n+i-1}\leq\tfrac{\beta\rho_% {0}d_{0}^{n}}{1-d_{0}},$

from which it follows that the following $\left(x_{n}\right)_{n=0}^{\infty}$ generated by ( 2 ) is convergent. If we write $\bar{x}=\lim\limits_{n\rightarrow\infty}x_{n}$ and let us pass to the limit in the inequality ( 7 ) when $p\rightarrow\infty$ , then we have

(8)

\left\|\bar{x}-x_{n}\right\|\leq\frac{\beta\rho_{0}d_{0}^{n}}{1-d_{0}}.

We immediately deduce from ( 8 ) that $n=0$ what $\bar{x}\in S\left(x_{0},r\right)$ . If we take into account that $P$ is a continuous application, it results from b)

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

that is to say that $\bar{x}$ is a solution to equation ( 1 ).

We now deal with the case where the application $A\left(x\right)$ is given by the equality

A\left(x\right)=I+\lambda D^{\prime}\left(x\right).

We then have

I-P^{\prime}\left(x\right)A\left(x\right)=I-\left(I-\lambda D^{\prime}\left(x% \right)\right)\left(I+\lambda D^{\prime}\left(x\right)\right)=\lambda^{2}\left% (D^{\prime}\left(x\right)\right)^{2}.

If we assume that

\left\|D^{\prime}\left(x\right)\right\|\leq b,

it then follows

\left\|I-P^{\prime}\left(x\right)A\left(x\right)\right\|\leq\lambda^{2}\cdot b% ^{2},\qquad\text{pour chaque\ }x\in S\left(x_{0},r\right).

We have

\left\|A\left(x\right)\right\|\leq 1+\left|\lambda\right|\cdot b,

for each $x\in B\left(x_{0},r\right)$ . In this case the condition vi of theorem 1 becomes for $\alpha=\lambda^{2}\cdot b^{2}$ And $\beta=1+\left|\lambda\right|\cdot b$

M\left(1+\left|\lambda\right|b^{2}\right)\cdot\rho_{0}+2\lambda^{2}\cdot b^{2}% -2<0

which, assuming $2-M\rho_{0}>0$ , leads to inequality

\left|\lambda\right|<\tfrac{b\left(2-M\rho_{0}\right)}{2+M\rho_{0}}.

∎

Taking the above into account, the following follows from Theorem 1 :

Theorem 2 .

If the application $D,$ the initial element $x_{0}$ and the real number $r>0$ meet the following conditions:

i.

L’application $D$ admits Fréchet derivatives up to and including second order for each $x\in S\left(x_{0},r\right);$
ii.

$\left\|D^{\prime}\left(x\right)\right\|\leq b$ for each $x\in S\left(x_{0},r\right);$
iii.

$\left\|D^{\prime\prime}\left(x\right)\right\|\leq M\left|\lambda\right|$ for each $x\in S\left(x_{0},r\right);$
iv.

$2-M\rho_{0}>0$ Or $\rho_{0}=\left\|x_{0}-\lambda D\left(x_{0}\right)-y\right\|;$
in.

$\tfrac{\rho_{0}\left(1+\left|\lambda\right|b\right)}{1-d_{0}}\leq r\ \text{o\`% {u} \ }d_{0}=M\tfrac{1+\left|\lambda\right|b^{2}}{2}\rho_{0}+\lambda^{2}\cdot b% ^{2};$
we.

$\left|\lambda\right|\leq b\tfrac{2-M\rho_{0}}{2+M\rho_{0}},$

then the sequence generated by

x_{n+1}=x_{n}-[1-\lambda D^{\prime}\left(x_{n}\right)]\left[x_{n}-\lambda D% \left(x_{n}\right)-y\right],\qquad n=0,1,\ldots,

converges to the solution $\bar{x}$ of equation ( 1 ) and we have the delimitation:

\left\|\bar{x}-x_{n}\right\|\leq\frac{\left(1+\left|\lambda\right|b\right)d_{0% }^{n}\rho_{0}}{1-d_{0}},\qquad n=0,1,\ldots

Bibliography

[1] LV Kantorovici, The Newton Trudy Method Mat. Inst. VA Stecklov 28, 104–144 (1949).
[2] ^†^†margin: clickable $\rightarrow$ A. Diaconu, I. Păvăloiu, On some iterative methods for the resolution of operational equations , Rev. Anal. Numér. Theor. Approx., vol. 1, 45–61 (1972). ( journal link )
[3] ^†^†margin: clickable $\rightarrow$ I. Păvăloiu, On iterative processes with a high order of convergence , Mathematica (Cluj), 12 (35) 1, 149–158 (1970).

(4)	$\displaystyle\left\\|P\left(x_{i}\right)\right\\|$	$\displaystyle\leq\left\\|P\left(x_{i}\right)-P\left(x_{i-1}\right)-P^{\prime}% \left(x_{i-1}\right)\left(x_{i}-x_{i-1}\right)\right\\|$
	$\displaystyle\quad+\left\\|P\left(x_{i-1}\right)+P^{\prime}\left(x_{i-1}\right)% \left(x_{i}-x_{i-1}\right)\right\\|$
	$\displaystyle\leq\tfrac{M\beta^{2}}{2}\left\\|P\left(x_{i-1}\right)\right\\|^{2}% +\alpha\left\\|P\left(x_{i-1}\right)\right\\|,\qquad i=1,2,\ldots,k$

The convergence of certain iterative methods for solving certain operator equations

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The Convergence of Certain Iterative Methods for Solving Certain Operator Equations

Theorem 1 .

Demonstration.

Theorem 2 .

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