The convergence of certain iterative methods for solving certain operator equations

9 years ago

Abstract

We consider the solving of the equation \[x=\lambda D\left( x\right)+y,\] where $E$ is a Banach space and $D:E\rightarrow E$, $\lambda\in \mathbb{R}$, $y\in E$. We study the convergence of the iterations \[x_{n+1}=x_{n}-A\left( x_{n}\right)\left[ x_{n}-\lambda D\left( x_{n}\right) -y\right], \ n=0,1,…, \ x_{0}\in E,\] where $A:E\rightarrow E$ is a linear mapping. We assume that the operator $P$ given by $P\left( x\right) =x-\lambda D\left( x\right) -y$ is two times Frechet differentiable, with $P^{\prime}\left( x\right)=I-\lambda D^{\prime}\left( x\right)$, $P^{\prime \prime}\left(x\right) =-\lambda D^{\prime \prime}\left( x\right) $. Under certain assumptions on boundedness of $A$ and $P$ we obtain convergence results for the considered sequences.

Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Authors

Original title (in French)

La convergence de certaines méthodes itératives pour résoudre certaines equations operationnelles

English translation of the title

The convergence of certain iterative methods for solving certain operator equations

Keywords

nonlinear operator equation; Banach space; iterative method;

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Cite this paper as:

I. Păvăloiu, La convergence de certaines méthodes itératives pour résoudre certaines equations operationnelles, Seminar on functional analysis and numerical methods, Preprint no. 1 (1986), pp. 127-132 (in French).

About this paper

Journal

Seminar on functional analysis and numerical methods,
Preprint

Publisher Name

“Babes-Bolyai” University,
Faculty of Mathematics,
Research Seminars

DOI

Not available yet.

References

[1] L.V. Kantorovici, O metodi Niutona Trudi Mat. Inst. V.A. Steklova 28, 104–144 (1949).

[2] A. Diaconu, I. Pavaloiu, Sur quelque methodes iteratives pour la resolution des equations op erationnelles, Rev. Anal. Num´er. Theor. Approx., vol. 1, 45–61 (1972). (journal link )

[3] I. Pavaloiu, Sur les procedes iteratifs a un ordre eleve de convergence, Mathematica (Cluj), 12 (35) 1, 149–158 (1970).

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"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 = l D (x) + and, l \in ℝ,

Or $D : AND \to AND$ 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 (x_{n}) [x_{n} - l D (x_{n}) - and], n = 0, 1, \dots, x_{0} \in AND

Or $A (x) : AND \to AND$ is a linear application for each $x \in AND$ .

We will designate, to fix the ideas, by $P : AND \to AND$ l’application $P (x) = x - l D (x) - and$ .

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

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

Theorem 1 .

If the applications $D$ And $A (x),$ 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 $S (x_{0}, r);$
ii.

$‖ A (x) ‖ \leq b$ for each $x \in S (x_{0}, r),$ Or $b \in ℝ$ And $b > 0;$
iii.

$‖ I - P^{'} (x) A (x) ‖ \leq a$ for each $x \in S (x_{0}, r)$ Or $a \in ℝ$ , $a > 0;$
iv.

$‖ D^{''} (x) ‖ \leq M | l |$ , for each $x \in S (x_{0}, r)$ Or $M \in ℝ$ , $M > 0;$
in.

$(b r_{0}) / (1 - d_{0}) \leq r$ Or $r_{0} = ‖ P (x_{0}) ‖, d_{0} = \frac{M b^{2} r_{0}}{2}$
we.

$d_{0} < 1,$

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

(3)

‖ \bar{x} - x_{n} ‖ \leq \frac{b d_{0}^{n} r_{0}}{1 - d_{0}}, n = 0, 1, \dots .

Demonstration.

We will show by induction that the following properties hold:

a)

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

$‖ P (x_{n}) ‖ \leq d_{0}^{n} \cdot r_{0}, n = 0, 1, \dots .$

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

‖ x_{1} - x_{0} ‖ \leq ‖ A (x_{0}) ‖ \cdot ‖ P (x_{0}) ‖ \leq b r_{0} \leq b r_{0} / (1 - d_{0}) \leq r

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

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

(4)	$‖ P (x_{i}) ‖$	$\leq ‖ P (x_{i}) - P (x_{i - 1}) - P^{'} (x_{i - 1}) (x_{i} - x_{i - 1}) ‖$
	$+ ‖ P (x_{i - 1}) + P^{'} (x_{i - 1}) (x_{i} - x_{i - 1}) ‖$
	$\leq \frac{M b^{2}}{2} {‖ P (x_{i - 1}) ‖}^{2} + a ‖ P (x_{i - 1}) ‖, i = 1, 2, \dots, k$

hence writing $d_{i - 1} = \frac{M b^{2}}{2} ‖ P (x_{i - 1}) ‖ + a, i = 1, 2, \dots$ and taking into account $in i .$ we immediately deduce the following inequalities:

(5)

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

It follows from ( 4 ) and ( 5 )

(6)

‖ P (x_{i}) ‖ \leq d_{0} ‖ P (x_{i - 1}) ‖, i = 1, 2, \dots, k .

It follows

‖ P (x_{i}) ‖ \leq d_{0}^{i} r_{0}, i = 1, 2, \dots, k .

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

	$‖ x_{k + 1} - x_{0} ‖$	$\leq \sum_{i = 0}^{n} ‖ x_{i + 1} - x_{i} ‖ \leq b \sum_{i = 0}^{k} ‖ P (x_{i}) ‖$
		$\leq b r_{0} \sum_{i = 0}^{k} d_{0}^{i} \leq \frac{b r_{0}}{1 - d_{0}} \leq r .$

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

(7)		$‖ x_{n + p} - x_{n} ‖$	$\leq \sum_{i = 1}^{p} ‖ x_{n + i} - x_{n + i - 1} ‖ \leq b \sum_{i = 1}^{p} ‖ P (x_{n + i - 1}) ‖$
		$\leq b r_{0} \sum_{i = 1}^{p} d_{0}^{n + i - 1} \leq \frac{b r_{0} d_{0}^{n}}{1 - d_{0}},$

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

(8)

‖ \bar{x} - x_{n} ‖ \leq \frac{b r_{0} d_{0}^{n}}{1 - d_{0}} .

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

lim_{n \to \infty} P (x_{n}) = P (\bar{x}) = i,

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

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

A (x) = I + l D^{'} (x) .

We then have

I - P^{'} (x) A (x) = I - (I - l D^{'} (x)) (I + l D^{'} (x)) = l^{2} {(D^{'} (x))}^{2} .

If we assume that

‖ D^{'} (x) ‖ \leq b,

it then follows

‖ I - P^{'} (x) A (x) ‖ \leq l^{2} \cdot b^{2}, for each x \in S (x_{0}, r) .

We have

‖ A (x) ‖ \leq 1 + | l | \cdot b,

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

M (1 + | l | b^{2}) \cdot r_{0} + 2 l^{2} \cdot b^{2} - 2 < 0

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

| l | < \frac{b (2 - M r_{0})}{2 + M r_{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 $x \in S (x_{0}, r);$
ii.

$‖ D^{'} (x) ‖ \leq b$ for each $x \in S (x_{0}, r);$
iii.

$‖ D^{''} (x) ‖ \leq M | l |$ for each $x \in S (x_{0}, r);$
iv.

$2 - M r_{0} > 0$ Or $r_{0} = ‖ x_{0} - l D (x_{0}) - and ‖;$
in.

$\frac{r_{0} (1 + | l | b)}{1 - d_{0}} \leq r Or d_{0} = M \frac{1 + | l | b^{2}}{2} r_{0} + l^{2} \cdot b^{2};$
we.

$| l | \leq b \frac{2 - M r_{0}}{2 + M r_{0}},$

then the sequence generated by

x_{n + 1} = x_{n} - [1 - l D^{'} (x_{n})] [x_{n} - l D (x_{n}) - and], n = 0, 1, \dots,

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

‖ \bar{x} - x_{n} ‖ \leq \frac{(1 + | l | b) d_{0}^{n} r_{0}}{1 - d_{0}}, n = 0, 1, \dots

Bibliography

[1] LV Kantorovici, The Newton Trudy Method Mat. Inst. VA Stecklov 28, 104–144 (1949).
[2] ^†^†margin: clickable $\to$ 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 $\to$ I. Păvăloiu, On iterative processes with a high order of convergence , Mathematica (Cluj), 12 (35) 1, 149–158 (1970).

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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 .

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