Abstract
Let \(X\) be a Banach space, \(Y\) a normed space \(P:X\rightarrow Y\) a nonlinear operator and the equation \(P\left( x\right) =0\) with solution \(x^{\ast}\). Consider \(\Sigma:=\left( x_{n}\right) _{n\geq0}\) a sequence from \(X\) and define the convergence order of \(\Sigma\) with respect to the solution of equation \(P\left( x\right) =0\). We give a general result with sufficient conditions such that the sequence \(\Sigma\) converge to the solution \(x^{\ast}\) with a given convergence order.
Authors
Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)
Title
Original title (in French)
Sur l’approximation des solutions des equations à l’aide des suites à éléments dans un espace de Banach
English translation of the title
On the approximation of the solutions of equations in Banach spaces by sequences
Keywords
approximation sequences; nonlinear equations in Banach spaces; convergence order.
Cite this paper as:
I. Păvăloiu, Sur l’approximation des solutions des equations à l’aide des suites à éléments dans un espace de Banach, Anal. Numér. Théor. Approx., 5 (1976) no. 1, pp. 63-67 (in French).
About this paper
Journal
Mathematica – Revue d’Analyse Numérique et de Théorie de l’Approximation
L’Analyse Numérique et la Théorie de l’Approximation
Publisher Name
Editura Academia R. S. Romane
Paper on journal website
Print ISBN
1010-3376
Online ISBN
2457-8118
References
[1] Ghinea, M., Sur la resolution des equations operationnelles dans les espaces de Banach, Revue Francaise de traitement de l’information, 8, 3–22, (1965).
[2] Pavaloiu, I., Sur les procedes iteratifs a un ordre eleve de convergence, Mathematica, 12 (35), 2, 309–324, (1970).
[3] Traub, J. F., Iterative Methods for the Solution of Equations. Prentice-Hall Inc. Englewood Cliffs N. J. (1964).
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ON THE APPROXIMATION OF SOLUTIONS TO EQUATIONS USING SEQUENCES WITH ELEMENTS IN A BANACH SPACE
EitherXXa Banach space andYYa normed linear space.
We consider the equation:
We consider the equation:
{:(1)P(x)=theta:}\begin{equation*} P(x)=\theta \tag{1} \end{equation*}
OrP:X rarr YP: X \rightarrow YAndtheta\thetais the zero element of the spaceYY.
We designate bySigma=(x_(n))_(n=0)^(oo)\Sigma=\left(x_{n}\right)_{n=0}^{\infty}, a sequence with elements in spaceXXand bykkan arbitrary natural number.
We designate bySigma=(x_(n))_(n=0)^(oo)\Sigma=\left(x_{n}\right)_{n=0}^{\infty}, a sequence with elements in spaceXXand bykkan arbitrary natural number.
DEFINITION 1. We say that the sequenceSigma\Sigmato orderkkin relation to the applicationPP, if there exists a non-negative constantrho\rhowhich does not depend onnn, and such that for eachn=0.1 dotsn=0.1 \ldotsthe following inequalities are satisfied:
||P(x_(n+1))|| <= rho||P(x_(n))||^(k)\left\|P\left(x_{n+1}\right)\right\| \leq \rho\left\|P\left(x_{n}\right)\right\|^{k}
Definition 2. We say that the sequenceSigma\Sigmahas the order of convergencekkcompared to the applicationPP, if the following conditions are met:
a) the followingSigma\Sigmato orderkkcompared to the applicationPP;
b) the followingSigma\Sigmathis convergent.
a) the followingSigma\Sigmato orderkkcompared to the applicationPP;
b) the followingSigma\Sigmathis convergent.
IfS in XS \in Xis a set with elements in spaceXX, we will designate byS^(**)=int(S)S^{*}=\int(S), the interior of this set.
Eithers >= 2s \geq 2a given natural number andPPthe approximation that determines equation (1).
In this note we will look for conditions imposed on the applicationPPand followingSigma\Sigma, so that the continuationSigma\Sigmahas the order of convergencesscompared
to the applicationPPand furthermore, if we designate bybar(x)=lim_(n rarr oo)x_(n)\bar{x}=\lim _{n \rightarrow \infty} x_{n}, so that we then haveP( bar(x))=thetaP(\bar{x})=\theta.
to the applicationPPand furthermore, if we designate bybar(x)=lim_(n rarr oo)x_(n)\bar{x}=\lim _{n \rightarrow \infty} x_{n}, so that we then haveP( bar(x))=thetaP(\bar{x})=\theta.
With respect to the problem posed above, we can state the following result.
Theorem 1. If the sequenceSigma\Sigma, the applicationPPand the real and positive numberdelta\deltaare such that for each pointx in Int(S)x \in \operatorname{Int}(S), OrS={x in X:||x-x_(0)|| <=delta}S=\left\{x \in X:\left\|x-x_{0}\right\| \leq \delta\right\}, the following conditions are met:
(i) the applicationPPadmits derivatives of the Fréchet type, up to orderss(s >= 2s \geq 2) inclusive, on each point of the set Int(S)(S)And
Theorem 1. If the sequenceSigma\Sigma, the applicationPPand the real and positive numberdelta\deltaare such that for each pointx in Int(S)x \in \operatorname{Int}(S), OrS={x in X:||x-x_(0)|| <=delta}S=\left\{x \in X:\left\|x-x_{0}\right\| \leq \delta\right\}, the following conditions are met:
(i) the applicationPPadmits derivatives of the Fréchet type, up to orderss(s >= 2s \geq 2) inclusive, on each point of the set Int(S)(S)And
su p_(x in Int(S))||P^((s))(x)|| <= M < +oo;\sup _{x \in \operatorname{Int}(S)}\left\|P^{(s)}(x)\right\| \leq M<+\infty ;
ii) There exists a real and non-negative constantalpha\alpha, which does not depend on n, such that the following inequalities are satisfied
||sum_(i=0)^(s-1)(1)/(i!)P^((i))(x_(n))(x_(n+1)-x_(n))^(i)|| <= alpha||P(x_(n))||^(s),\left\|\sum_{i=0}^{s-1} \frac{1}{i!} P^{(i)}\left(x_{n}\right)\left(x_{n+1}-x_{n}\right)^{i}\right\| \leq \alpha\left\|P\left(x_{n}\right)\right\|^{s},
Orx_(n)in Sigma nnS^(**),n=0,1,dotsx_{n} \in \Sigma \cap S^{*}, n=0.1, \ldots;
iii) There exists a real and non-negative constantbeta\betawhich does not depend onnnand such that the following inequalities are satisfied
iii) There exists a real and non-negative constantbeta\betawhich does not depend onnnand such that the following inequalities are satisfied
||x_(n+1)-x_(n)|| <= beta||P(x_(n))||,\left\|x_{n+1}-x_{n}\right\| \leq \beta\left\|P\left(x_{n}\right)\right\|,
Or
x_(n)in Sigma nnS^(**),quad n=0,1,dots;x_{n} \in \Sigma \cap S^{*}, \quad n=0.1, \ldots ;
iv) constantsalpha, beta\alpha, \betaand real numbersMMAnddelta\deltasatisfy the following inequalities:
{:[rho_(0)=v*||P(x_(0))|| < 1" et "(betarho_(0))/((1-rho_(0))*v) <= delta" où "],[v=(alpha+(Mbeta^(s))/(s!))^((1)/(s-1))]:}\begin{aligned}
\rho_{0} & =v \cdot\left\|P\left(x_{0}\right)\right\|<1 \text { et } \frac{\beta \rho_{0}}{\left(1-\rho_{0}\right) \cdot v} \leq \delta \text { où } \\
v & =\left(\alpha+\frac{M \beta^{s}}{s!}\right)^{\frac{1}{s-1}}
\end{aligned}ù
then relatively to equation (1) and the followingSigma\Sigmatake place of the following properties:
j) the sequenceSigma\Sigmahas the order of convergence s and ifbar(x)=lim_(n rarr oo)x_(n)\bar{x}=\lim _{n \rightarrow \infty} x_{n}, SOP( bar(x))=thetaP(\bar{x})=\theta.
jj)bar(x)in S\bar{x} \in S
jjj)|| bar(x)_(n+1)-x_(n)|| <= (betarho_(0)^(n))/(v)\left\|\bar{x}_{n+1}-x_{n}\right\| \leq \frac{\beta \rho_{0}^{n}}{v}
jv)||( bar(x))-x_(n)|| <= (betarho_(0)^(s^(n)))/(v(1-rho_(0)^(s^(n))))\left\|\bar{x}-x_{n}\right\| \leq \frac{\beta \rho_{0}^{s^{n}}}{v\left(1-\rho_{0}^{s^{n}}\right)}
v)||P(x_(n))|| <= (rho_(0)^(s^(n)))/(v)\left\|P\left(x_{n}\right)\right\| \leq \frac{\rho_{0}^{s^{n}}}{v}
j) the sequenceSigma\Sigmahas the order of convergence s and ifbar(x)=lim_(n rarr oo)x_(n)\bar{x}=\lim _{n \rightarrow \infty} x_{n}, SOP( bar(x))=thetaP(\bar{x})=\theta.
jj)bar(x)in S\bar{x} \in S
jjj)|| bar(x)_(n+1)-x_(n)|| <= (betarho_(0)^(n))/(v)\left\|\bar{x}_{n+1}-x_{n}\right\| \leq \frac{\beta \rho_{0}^{n}}{v}
jv)||( bar(x))-x_(n)|| <= (betarho_(0)^(s^(n)))/(v(1-rho_(0)^(s^(n))))\left\|\bar{x}-x_{n}\right\| \leq \frac{\beta \rho_{0}^{s^{n}}}{v\left(1-\rho_{0}^{s^{n}}\right)}
v)||P(x_(n))|| <= (rho_(0)^(s^(n)))/(v)\left\|P\left(x_{n}\right)\right\| \leq \frac{\rho_{0}^{s^{n}}}{v}
Proof. We will first demonstrate that the elements of the sequenceSigma\Sigmaare contained inS^(**)S^{*}, if the conditions of the stated theorem are met.
Forx_(1)x_{1}we have:
{:(2)||x_(1)-x_(0)|| <= beta||P(x_(0))|| <= (beta v||P(x_(0))||)/(v) <= (betarho_(0))/(v) < (betarho_(0))/(v(1-rho_(0))) <= delta:}\begin{equation*}
\left\|x_{1}-x_{0}\right\| \leq \beta\left\|P\left(x_{0}\right)\right\| \leq \frac{\beta v\left\|P\left(x_{0}\right)\right\|}{v} \leq \frac{\beta \rho_{0}}{v}<\frac{\beta \rho_{0}}{v\left(1-\rho_{0}\right)} \leq \delta \tag{2}
\end{equation*}
from which it follows thatx_(1)inS^(**)x_{1} \in S^{*}. Indeed, using the generalized Taylor formula we deduce
{:[||P(x_(1))|| <= ||P(x_(1))-sum_(i=0)^(s-1)(1)/(i!)P^((i))(x_(0))(x_(1)-x_(0))^(i)||],[+||sum_(i=0)^(s-1)(1)/(i!)P^((i))(x_(0))(x_(1)-x_(0))^(i)||],[ <= (M)/(s!)||x_(1)-x_(0)||^(s)+alpha||P(x_(0))||^(s)],[ <= ((Mbeta^(s))/(s!)+alpha)*||P(x_(0))||^(s)]:}\begin{aligned}
\left\|P\left(x_{1}\right)\right\| \leq & \left\|P\left(x_{1}\right)-\sum_{i=0}^{s-1} \frac{1}{i!} P^{(i)}\left(x_{0}\right)\left(x_{1}-x_{0}\right)^{i}\right\| \\
& +\left\|\sum_{i=0}^{s-1} \frac{1}{i!} P^{(i)}\left(x_{0}\right)\left(x_{1}-x_{0}\right)^{i}\right\| \\
\leq & \frac{M}{s!}\left\|x_{1}-x_{0}\right\|^{s}+\alpha\left\|P\left(x_{0}\right)\right\|^{s} \\
\leq & \left(\frac{M \beta^{s}}{s!}+\alpha\right) \cdot\left\|P\left(x_{0}\right)\right\|^{s}
\end{aligned}
from which we deduce:
{:(3)||P(x_(1))|| <= v^(s-1)*||P(x_(0))||^(s):}\begin{equation*}
\left\|P\left(x_{1}\right)\right\| \leq v^{s-1} \cdot\left\|P\left(x_{0}\right)\right\|^{s} \tag{3}
\end{equation*}
We assume that the following properties are met:
- x_(i)inS^(**),quad i=0,1,dots,nx_{i} \in S^{*}, \quad i=0,1, \ldots, n;
- ||x_(i)-x_(i-1)|| <= (beta )/(v)*rho_(0)^(s^(i-1)),quad i=1,2,dots,n\left\|x_{i}-x_{i-1}\right\| \leq \frac{\beta}{v} \cdot \rho_{0}^{s^{i-1}}, \quad i=1,2, \ldots, n;
- ||P(x_(i))|| <= v^(s-1)||P(x_(i-1))||^(s),quad i=1,2,dots,n\left\|P\left(x_{i}\right)\right\| \leq v^{s-1}\left\|P\left(x_{i-1}\right)\right\|^{s}, \quad i=1,2, \ldots, n,
and in these hypotheses we will demonstrate that:
x_(n+1)inS^(**),quad||x_(n+1)-x_(n)|| <= (beta )/(v)rho_(0)^(s^(n))quad" et "quad||P(x_(n+1))|| <= v^(s-1)*||P(x_(n))||^(s).x_{n+1} \in S^{*}, \quad\left\|x_{n+1}-x_{n}\right\| \leq \frac{\beta}{v} \rho_{0}^{s^{n}} \quad \text { et } \quad\left\|P\left(x_{n+1}\right)\right\| \leq v^{s-1} \cdot\left\|P\left(x_{n}\right)\right\|^{s} .
From what we have demonstrated above, it follows that properties 1)-3) are verified fori=1i=1.
By multiplying byvvinequality 3) and denote byrho_(i)=v||P(x_(i))||,i=1,2,dots,n\rho_{i}=v\left\|P\left(x_{i}\right)\right\|, i= 1,2, \ldots, n, we easily deduce the inequalities:
{:(4)rho_(i) <= rho_(0)^(s^(i))","quad i=1","2","dots","n.:}\begin{equation*}
\rho_{i} \leq \rho_{0}^{s^{i}}, \quad i=1,2, \ldots, n . \tag{4}
\end{equation*}
From inequalities (4) and (iii) we deduce
{:(5)||x_(n+1)-x_(n)|| <= beta||P(x_(n))||=(beta v||p(x_(n))||)/(v) <= (betarho_(0)^(s^(n)))/(v).:}\begin{equation*}
\left\|x_{n+1}-x_{n}\right\| \leq \beta\left\|P\left(x_{n}\right)\right\|=\frac{\beta v\left\|p\left(x_{n}\right)\right\|}{v} \leq \frac{\beta \rho_{0}^{s^{n}}}{v} . \tag{5}
\end{equation*}
From (5) we deduce:
||x_(n+1)-x_(0)|| <= sum_(i=0)^(n)||x_(i+1)-x_(i)|| <= (beta )/(v)sum_(i=0)^(n)rho_(0)^(s^(i)) < (betarho_(0))/(v(1-rho_(0))) <= delta\left\|x_{n+1}-x_{0}\right\| \leq \sum_{i=0}^{n}\left\|x_{i+1}-x_{i}\right\| \leq \frac{\beta}{v} \sum_{i=0}^{n} \rho_{0}^{s^{i}}<\frac{\beta \rho_{0}}{v\left(1-\rho_{0}\right)} \leq \delta
from which it resultsx_(n+1)inS^(**)x_{n+1} \in S^{*}.
Inequality 2) fori=n+1i=n+1results from (5).
For the last inequality we have:
Inequality 2) fori=n+1i=n+1results from (5).
For the last inequality we have:
{:[||P(x_(n+1))|| <= ||P(x_(n+1))-sum_(i=0)^(s-1)(1)/(i!)P^((i))(x_(n))(x_(n+1)-x_(n))^(i)||],[+||sum_(i=0)^(s-1)(1)/(i!)P^((i))(x_(n))(x_(n+1)-x_(n))^(i)||],[ <= (alpha+(Mbeta^(s))/(s!))*||P(x_(n))||=v^(s-1)||P(x_(n))||^(s).]:}\begin{aligned}
\left\|P\left(x_{n+1}\right)\right\| \leq & \left\|P\left(x_{n+1}\right)-\sum_{i=0}^{s-1} \frac{1}{i!} P^{(i)}\left(x_{n}\right)\left(x_{n+1}-x_{n}\right)^{i}\right\| \\
& +\left\|\sum_{i=0}^{s-1} \frac{1}{i!} P^{(i)}\left(x_{n}\right)\left(x_{n+1}-x_{n}\right)^{i}\right\| \\
& \leq\left(\alpha+\frac{M \beta^{s}}{s!}\right) \cdot\left\|P\left(x_{n}\right)\right\|=v^{s-1}\left\|P\left(x_{n}\right)\right\|^{s} .
\end{aligned}
Therefore properties 1)-3) are fulfilled for eachn=1,2,dotsn=1,2, \ldots
Property 3) shows that the sequenceSigma\Sigmato orderss.
We will demonstrate in the following that the sequenceSigma\Sigmais convergent. To do this we will first demonstrate that the sequenceSigma\Sigmais fundamental.
Property 3) shows that the sequenceSigma\Sigmato orderss.
We will demonstrate in the following that the sequenceSigma\Sigmais convergent. To do this we will first demonstrate that the sequenceSigma\Sigmais fundamental.
LetnnAndpptwo natural numbers.
We have:
We have:
{:(6)||x_(n+p)-x_(n)|| <= sum_(i=n)^(n+p-1)||x_(i+1)-x_(n)|| <= (beta )/(v)sum_(i=n)^(n+p-1)rho_(0)^(s^(i)) <= (betarho_(0)^(s^(n)))/(v(1-rho_(0)^(s^(n)))).:}\begin{equation*}
\left\|x_{n+p}-x_{n}\right\| \leq \sum_{i=n}^{n+p-1}\left\|x_{i+1}-x_{n}\right\| \leq \frac{\beta}{v} \sum_{i=n}^{n+p-1} \rho_{0}^{s^{i}} \leq \frac{\beta \rho_{0}^{s^{n}}}{v\left(1-\rho_{0}^{s^{n}}\right)} . \tag{6}
\end{equation*}
From inequality (6) it follows thatSigma\SigmaThis convergent.
Letbar(x)=lim_(n rarr oo)x_(n)\bar{x}=\lim _{n \rightarrow \infty} x_{n}, then from (6) the inequality results:
Letbar(x)=lim_(n rarr oo)x_(n)\bar{x}=\lim _{n \rightarrow \infty} x_{n}, then from (6) the inequality results:
||( bar(x))-x_(n)|| <= (rho_(0)^(s^(n)))/(v(1-rho_(0)^(s^(n)))),quad n=0,1,dots\left\|\bar{x}-x_{n}\right\| \leq \frac{\rho_{0}^{s^{n}}}{v\left(1-\rho_{0}^{s^{n}}\right)}, \quad n=0,1, \ldots
It follows thatbar(x)in S\bar{x} \in Sand we also obtained the inequality jv). The inequality jjj) results from (6) forp=1p=1.
It remains to be demonstrated thatbar(x)\bar{x}is the solution to equation (1).
We have proved that inequality (4) is true for eachn=0,1,dotsn=0,1, \ldots. So we get:
We have proved that inequality (4) is true for eachn=0,1,dotsn=0,1, \ldots. So we get:
lim_(n rarr oo)rho_(n)=0\lim _{n \rightarrow \infty} \rho_{n}=0
butrho_(n)=v||P(x_(n))||\rho_{n}=v\left\|P\left(x_{n}\right)\right\|and solim_(n rarr oo)P(x_(n))=P( bar(x))=theta\lim _{n \rightarrow \infty} P\left(x_{n}\right)=P(\bar{x})=\theta
This completes the proof of the theorem .◻\square
This completes the proof of the theorem .◻\square
BIBLIOGRAPHY
[1] Ghinea, M., On the resolution of operational equations in Banach spaces, Revue Francaise de traitement de l'information, 8, 3-22, (1965).
[2] Păvăloiu, I, On iterative processes with a high order of conyergency, Mathematica, 12 (35), 2, 309-324, (1970).
[3] Traub, JF, Iterative Methods for the Solution of Equations. Prentice-Hall Inc. Englewood Cliffs NJ (1964).
[2] Păvăloiu, I, On iterative processes with a high order of conyergency, Mathematica, 12 (35), 2, 309-324, (1970).
[3] Traub, JF, Iterative Methods for the Solution of Equations. Prentice-Hall Inc. Englewood Cliffs NJ (1964).
Received on30.VI. 1976