On the approximation of the solutions of equations in Banach spaces by sequences

Ion Păvăloiu
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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.

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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

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

byION PĂVĂLOIU(Cluj-Napoca)

Either X X XXXa Banach space and Y Y YYYa normed linear space.
We consider the equation:
(1) P ( x ) = θ (1) P ( x ) = θ {:(1)P(x)=theta:}\begin{equation*} P(x)=\theta \tag{1} \end{equation*}(1)P(x)=θ
Or P : X → Y P : X → Y P:X rarr YP: X \rightarrow YP:X→YAnd θ θ theta\thetaθis the zero element of the space Y Y YYY.
We designate by Σ = ( x n ) n = 0 ∞ Σ = x n n = 0 ∞ Sigma=(x_(n))_(n=0)^(oo)\Sigma=\left(x_{n}\right)_{n=0}^{\infty}Σ=(xn)n=0∞, a sequence with elements in space X X XXXand by k k kkkan arbitrary natural number.
DEFINITION 1. We say that the sequence Σ Σ Sigma\SigmaΣto order k k kkkin relation to the application P P PPP, if there exists a non-negative constant ρ ρ rho\rhoρwhich does not depend on n n nnn, and such that for each n = 0 , 1 … n = 0 , 1 … n=0.1 dotsn=0.1 \ldotsn=0,1…the following inequalities are satisfied:
‖ P ( x n + 1 ) ‖ ≤ ρ ‖ P ( x n ) ‖ k P x n + 1 ≤ ρ P x n k ||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}‖P(xn+1)‖≤ρ‖P(xn)‖k
Definition 2. We say that the sequence Σ Σ Sigma\SigmaΣhas the order of convergence k k kkkcompared to the application P P PPP, if the following conditions are met:
a) the following Σ Σ Sigma\SigmaΣto order k k kkkcompared to the application P P PPP;
b) the following Σ Σ Sigma\SigmaΣthis convergent.
If S ∈ X S ∈ X S in XS \in XS∈Xis a set with elements in space X X XXX, we will designate by S ∗ = ∫ ( S ) S ∗ = ∫ ( S ) S^(**)=int(S)S^{*}=\int(S)S∗=∫(S), the interior of this set.
Either s ≥ 2 s ≥ 2 s >= 2s \geq 2s≥2a given natural number and P P PPPthe approximation that determines equation (1).
In this note we will look for conditions imposed on the application P P PPPand following Σ Σ Sigma\SigmaΣ, so that the continuation Σ Σ Sigma\SigmaΣhas the order of convergence s s ssscompared
to the application P P PPPand furthermore, if we designate by x ¯ = lim n → ∞ x n x ¯ = lim n → ∞   x n bar(x)=lim_(n rarr oo)x_(n)\bar{x}=\lim _{n \rightarrow \infty} x_{n}x¯=limn→∞xn, so that we then have P ( x ¯ ) = θ P ( x ¯ ) = θ P( bar(x))=thetaP(\bar{x})=\thetaP(x¯)=θ.
With respect to the problem posed above, we can state the following result.
Theorem 1. If the sequence Σ Σ Sigma\SigmaΣ, the application P P PPPand the real and positive number δ δ delta\deltaδare such that for each point x ∈ Int ( S ) x ∈ Int ( S ) x in Int(S)x \in \operatorname{Int}(S)x∈Int(S), Or S = { x ∈ X : ‖ x − x 0 ‖ ≤ δ } S = x ∈ X : x − x 0 ≤ δ S={x in X:||x-x_(0)|| <=delta}S=\left\{x \in X:\left\|x-x_{0}\right\| \leq \delta\right\}S={x∈X:‖x−x0‖≤δ}, the following conditions are met:
(i) the application P P PPPadmits derivatives of the Fréchet type, up to order s s sss( s ≥ 2 s ≥ 2 s >= 2s \geq 2s≥2) inclusive, on each point of the set Int ( S ) ( S ) (S)(S)(S)And
sup x ∈ Int ( S ) ‖ P ( s ) ( x ) ‖ ≤ M < + ∞ ; sup x ∈ Int ( S )   P ( s ) ( x ) ≤ M < + ∞ ; 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 ;supx∈Int(S)‖P(s)(x)‖≤M<+∞;
ii) There exists a real and non-negative constant α α alpha\alphaα, which does not depend on n, such that the following inequalities are satisfied
‖ ∑ i = 0 s − 1 1 i ! P ( i ) ( x n ) ( x n + 1 − x n ) i ‖ ≤ α ‖ P ( x n ) ‖ s , ∑ i = 0 s − 1   1 i ! P ( i ) x n x n + 1 − x n i ≤ α P x n s , ||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},‖∑i=0s−11i!P(i)(xn)(xn+1−xn)i‖≤α‖P(xn)‖s,
Or x n ∈ Σ ∩ S ∗ , n = 0 , 1 , … x n ∈ Σ ∩ S ∗ , n = 0 , 1 , … x_(n)in Sigma nnS^(**),n=0,1,dotsx_{n} \in \Sigma \cap S^{*}, n=0.1, \ldotsxn∈Σ∩S∗,n=0,1,…;
iii) There exists a real and non-negative constant β β beta\betaβwhich does not depend on n n nnnand such that the following inequalities are satisfied
‖ x n + 1 − x n ‖ ≤ β ‖ P ( x n ) ‖ , x n + 1 − x n ≤ β P x n , ||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\|,‖xn+1−xn‖≤β‖P(xn)‖,
Or
x n ∈ Σ ∩ S ∗ , n = 0 , 1 , … ; x n ∈ Σ ∩ S ∗ , n = 0 , 1 , … ; x_(n)in Sigma nnS^(**),quad n=0,1,dots;x_{n} \in \Sigma \cap S^{*}, \quad n=0.1, \ldots ;xn∈Σ∩S∗,n=0,1,…;
iv) constants α , β α , β alpha, beta\alpha, \betaα,βand real numbers M M MMMAnd δ δ delta\deltaδsatisfy the following inequalities:
ρ 0 = v ⋅ ‖ P ( x 0 ) ‖ < 1 And β ρ 0 ( 1 − ρ 0 ) ⋅ v ≤ δ Or v = ( α + M β s s ! ) 1 s − 1 ρ 0 = v ⋅ P x 0 < 1  And  β ρ 0 1 − ρ 0 ⋅ v ≤ δ  Or  v = α + M β s s ! 1 s − 1 {:[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}ρ0=v⋅‖P(x0)‖<1 And βρ0(1−ρ0)⋅v≤δ Or v=(α+Mβss!)1s−1
then relatively to equation (1) and the following Σ Σ Sigma\SigmaΣtake place of the following properties:
j) the sequence Σ Σ Sigma\SigmaΣhas the order of convergence s and if x ¯ = lim n → ∞ x n x ¯ = lim n → ∞   x n bar(x)=lim_(n rarr oo)x_(n)\bar{x}=\lim _{n \rightarrow \infty} x_{n}x¯=limn→∞xn, SO P ( x ¯ ) = θ P ( x ¯ ) = θ P( bar(x))=thetaP(\bar{x})=\thetaP(x¯)=θ.
jj) x ¯ ∈ S x ¯ ∈ S bar(x)in S\bar{x} \in Sx¯∈S
jjj) ‖ x ¯ n + 1 − x n ‖ ≤ β ρ 0 n v x ¯ n + 1 − x n ≤ β ρ 0 n v || 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}‖x¯n+1−xn‖≤βρ0nv
jv) ‖ x ¯ − x n ‖ ≤ β ρ 0 s n v ( 1 − ρ 0 s n ) x ¯ − x n ≤ β ρ 0 s n v 1 − ρ 0 s n ||( 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)}‖x¯−xn‖≤βρ0snv(1−ρ0sn)
v) ‖ P ( x n ) ‖ ≤ ρ 0 s n v P x n ≤ ρ 0 s n v ||P(x_(n))|| <= (rho_(0)^(s^(n)))/(v)\left\|P\left(x_{n}\right)\right\| \leq \frac{\rho_{0}^{s^{n}}}{v}‖P(xn)‖≤ρ0snv
Proof. We will first demonstrate that the elements of the sequence Σ Σ Sigma\SigmaΣare contained in S ∗ S ∗ S^(**)S^{*}S∗, if the conditions of the stated theorem are met.
For x 1 x 1 x_(1)x_{1}x1we have:
(2) ‖ x 1 − x 0 ‖ ≤ β ‖ P ( x 0 ) ‖ ≤ β v ‖ P ( x 0 ) ‖ v ≤ β ρ 0 v < β ρ 0 v ( 1 − ρ 0 ) ≤ δ (2) x 1 − x 0 ≤ β P x 0 ≤ β v P x 0 v ≤ β ρ 0 v < β ρ 0 v 1 − ρ 0 ≤ δ {:(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*}(2)‖x1−x0‖≤β‖P(x0)‖≤βv‖P(x0)‖v≤βρ0v<βρ0v(1−ρ0)≤δ
from which it follows that x 1 ∈ S ∗ x 1 ∈ S ∗ x_(1)inS^(**)x_{1} \in S^{*}x1∈S∗. Indeed, using the generalized Taylor formula we deduce
‖ P ( x 1 ) ‖ ≤ ‖ P ( x 1 ) − ∑ i = 0 s − 1 1 i ! P ( i ) ( x 0 ) ( x 1 − x 0 ) i ‖ + ‖ ∑ i = 0 s − 1 1 i ! P ( i ) ( x 0 ) ( x 1 − x 0 ) i ‖ ≤ M s ! ‖ x 1 − x 0 ‖ s + α ‖ P ( x 0 ) ‖ s ≤ ( M β s s ! + α ) ⋅ ‖ P ( x 0 ) ‖ s P x 1 ≤ P x 1 − ∑ i = 0 s − 1   1 i ! P ( i ) x 0 x 1 − x 0 i + ∑ i = 0 s − 1   1 i ! P ( i ) x 0 x 1 − x 0 i ≤ M s ! x 1 − x 0 s + α P x 0 s ≤ M β s s ! + α ⋅ P x 0 s {:[||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}‖P(x1)‖≤‖P(x1)−∑i=0s−11i!P(i)(x0)(x1−x0)i‖+‖∑i=0s−11i!P(i)(x0)(x1−x0)i‖≤Ms!‖x1−x0‖s+α‖P(x0)‖s≤(Mβss!+α)⋅‖P(x0)‖s
from which we deduce:
(3) ‖ P ( x 1 ) ‖ ≤ v s − 1 ⋅ ‖ P ( x 0 ) ‖ s (3) P x 1 ≤ v s − 1 ⋅ P x 0 s {:(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*}(3)‖P(x1)‖≤vs−1⋅‖P(x0)‖s
We assume that the following properties are met:
  1. x i ∈ S ∗ , i = 0 , 1 , … , n x i ∈ S ∗ , i = 0 , 1 , … , n x_(i)inS^(**),quad i=0,1,dots,nx_{i} \in S^{*}, \quad i=0,1, \ldots, nxi∈S∗,i=0,1,…,n;
  2. ‖ x i − x i − 1 ‖ ≤ β v ⋅ ρ 0 s i − 1 , i = 1 , 2 , … , n x i − x i − 1 ≤ β v ⋅ ρ 0 s i − 1 , i = 1 , 2 , … , 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‖xi−xi−1‖≤βv⋅ρ0si−1,i=1,2,…,n;
  3. ‖ P ( x i ) ‖ ≤ v s − 1 ‖ P ( x i − 1 ) ‖ s , i = 1 , 2 , … , n P x i ≤ v s − 1 P x i − 1 s , i = 1 , 2 , … , 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‖P(xi)‖≤vs−1‖P(xi−1)‖s,i=1,2,…,n,
    and in these hypotheses we will demonstrate that:
x n + 1 ∈ S ∗ , ‖ x n + 1 − x n ‖ ≤ β v ρ 0 s n et ‖ P ( x n + 1 ) ‖ ≤ v s − 1 ⋅ ‖ P ( x n ) ‖ s . x n + 1 ∈ S ∗ , x n + 1 − x n ≤ β v ρ 0 s n  et  P x n + 1 ≤ v s − 1 ⋅ P x n s . 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} .xn+1∈S∗,‖xn+1−xn‖≤βvρ0sn And ‖P(xn+1)‖≤vs−1⋅‖P(xn)‖s.
From what we have demonstrated above, it follows that properties 1)-3) are verified for i = 1 i = 1 i=1i=1i=1.
By multiplying by v v vvvinequality 3) and denote by ρ i = v ‖ P ( x i ) ‖ , i = 1 , 2 , … , n ρ i = v P x i , i = 1 , 2 , … , n rho_(i)=v||P(x_(i))||,i=1,2,dots,n\rho_{i}=v\left\|P\left(x_{i}\right)\right\|, i= 1,2, \ldots, nρi=v‖P(xi)‖,i=1,2,…,n, we easily deduce the inequalities:
(4) ρ i ≤ ρ 0 s i , i = 1 , 2 , … , n . (4) ρ i ≤ ρ 0 s i , i = 1 , 2 , … , n . {:(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*}(4)ρi≤ρ0si,i=1,2,…,n.
From inequalities (4) and (iii) we deduce
(5) ‖ x n + 1 − x n ‖ ≤ β ‖ P ( x n ) ‖ = β v ‖ p ( x n ) ‖ v ≤ β ρ 0 s n v . (5) x n + 1 − x n ≤ β P x n = β v p x n v ≤ β ρ 0 s n v . {:(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*}(5)‖xn+1−xn‖≤β‖P(xn)‖=βv‖p(xn)‖v≤βρ0snv.
From (5) we deduce:
‖ x n + 1 − x 0 ‖ ≤ ∑ i = 0 n ‖ x i + 1 − x i ‖ ≤ β v ∑ i = 0 n ρ 0 s i < β ρ 0 v ( 1 − ρ 0 ) ≤ δ x n + 1 − x 0 ≤ ∑ i = 0 n   x i + 1 − x i ≤ β v ∑ i = 0 n   ρ 0 s i < β ρ 0 v 1 − ρ 0 ≤ δ ||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‖xn+1−x0‖≤∑i=0n‖xi+1−xi‖≤βv∑i=0nρ0si<βρ0v(1−ρ0)≤δ
from which it results x n + 1 ∈ S ∗ x n + 1 ∈ S ∗ x_(n+1)inS^(**)x_{n+1} \in S^{*}xn+1∈S∗.
Inequality 2) for i = n + 1 i = n + 1 i=n+1i=n+1i=n+1results from (5).
For the last inequality we have:
‖ P ( x n + 1 ) ‖ ≤ ‖ P ( x n + 1 ) − ∑ i = 0 s − 1 1 i ! P ( i ) ( x n ) ( x n + 1 − x n ) i ‖ + ‖ ∑ i = 0 s − 1 1 i ! P ( i ) ( x n ) ( x n + 1 − x n ) i ‖ ≤ ( α + M β s s ! ) ⋅ ‖ P ( x n ) ‖ = v s − 1 ‖ P ( x n ) ‖ s . P x n + 1 ≤ P x n + 1 − ∑ i = 0 s − 1   1 i ! P ( i ) x n x n + 1 − x n i + ∑ i = 0 s − 1   1 i ! P ( i ) x n x n + 1 − x n i ≤ α + M β s s ! ⋅ P x n = v s − 1 P x n s . {:[||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}‖P(xn+1)‖≤‖P(xn+1)−∑i=0s−11i!P(i)(xn)(xn+1−xn)i‖+‖∑i=0s−11i!P(i)(xn)(xn+1−xn)i‖≤(α+Mβss!)⋅‖P(xn)‖=vs−1‖P(xn)‖s.
Therefore properties 1)-3) are fulfilled for each n = 1 , 2 , … n = 1 , 2 , … n=1,2,dotsn=1,2, \ldotsn=1,2,…
Property 3) shows that the sequence Σ Σ Sigma\SigmaΣto order s s sss.
We will demonstrate in the following that the sequence Σ Σ Sigma\SigmaΣis convergent. To do this we will first demonstrate that the sequence Σ Σ Sigma\SigmaΣis fundamental.
Let n n nnnAnd p p ppptwo natural numbers.
We have:
(6) ‖ x n + p − x n ‖ ≤ ∑ i = n n + p − 1 ‖ x i + 1 − x n ‖ ≤ β v ∑ i = n n + p − 1 ρ 0 s i ≤ β ρ 0 s n v ( 1 − ρ 0 s n ) . (6) x n + p − x n ≤ ∑ i = n n + p − 1   x i + 1 − x n ≤ β v ∑ i = n n + p − 1   ρ 0 s i ≤ β ρ 0 s n v 1 − ρ 0 s n . {:(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*}(6)‖xn+p−xn‖≤∑i=nn+p−1‖xi+1−xn‖≤βv∑i=nn+p−1ρ0si≤βρ0snv(1−ρ0sn).
From inequality (6) it follows that Σ Σ Sigma\SigmaΣThis convergent.
Let x ¯ = lim n → ∞ x n x ¯ = lim n → ∞   x n bar(x)=lim_(n rarr oo)x_(n)\bar{x}=\lim _{n \rightarrow \infty} x_{n}x¯=limn→∞xn, then from (6) the inequality results:
‖ x ¯ − x n ‖ ≤ ρ 0 s n v ( 1 − ρ 0 s n ) , n = 0 , 1 , … x ¯ − x n ≤ ρ 0 s n v 1 − ρ 0 s n , n = 0 , 1 , … ||( 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‖x¯−xn‖≤ρ0snv(1−ρ0sn),n=0,1,…
It follows that x ¯ ∈ S x ¯ ∈ S bar(x)in S\bar{x} \in Sx¯∈Sand we also obtained the inequality jv). The inequality jjj) results from (6) for p = 1 p = 1 p=1p=1p=1.
It remains to be demonstrated that x ¯ x ¯ bar(x)\bar{x}x¯is the solution to equation (1).
We have proved that inequality (4) is true for each n = 0 , 1 , … n = 0 , 1 , … n=0,1,dotsn=0,1, \ldotsn=0,1,…. So we get:
lim n → ∞ ρ n = 0 lim n → ∞   ρ n = 0 lim_(n rarr oo)rho_(n)=0\lim _{n \rightarrow \infty} \rho_{n}=0limn→∞ρn=0
but ρ n = v ‖ P ( x n ) ‖ ρ n = v P x n rho_(n)=v||P(x_(n))||\rho_{n}=v\left\|P\left(x_{n}\right)\right\|ρn=v‖P(xn)‖and so lim n → ∞ P ( x n ) = P ( x ¯ ) = θ lim n → ∞   P x n = P ( x ¯ ) = θ lim_(n rarr oo)P(x_(n))=P( bar(x))=theta\lim _{n \rightarrow \infty} P\left(x_{n}\right)=P(\bar{x})=\thetalimn→∞P(xn)=P(x¯)=θ
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).
Received on30.VI. 1976
1976

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