On iterated operators

Ion Păvăloiu
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Abstract

Let \(X,Y\) be normed spaces and \(F:X\rightarrow Y\) a nonlinear operator. Let \(Q:X\rightarrow X.\) We study the convergence orders of iteration operators obtained by composing the given operators. We also study the construction of iterative operators of order \(p+1\), resp. \(2p\), given operators of order \(p\). As particular instances, we consider the Newton, Traub and chord iterative operators.

Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis)

Title

Original title (in Romanian)

Asupra operatorilor iterativi

English translation of the title

On iterated operators

Keywords

Newton method; Traub method; Chord method, convergence order

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Cite this paper as:

I. Păvăloiu, Asupra operatorilor iterativi, Studii şi cercetări matematice, 23 10 (1971), pp. 1567-1574 (in Romanian).

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Studii şi cercetări matematice

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Academia Republicii S.R.

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References

[1] Janko Bela, Despre rezolvarea ecuatiilor operationale. (Lucrare de Doctorat,  Cluj, 1966).

[2] Pavaloiu Ion, Sur la methode de Steffensen  pour  la resolution des equations operationnelles non lineaires. Rev. Roum de math. pures et appl. 13, 6 (1968), 857-861.

[3] Pavaloiu Ion, Interpolation dans des espaces lineaires norme et applications. Mathematica, Cluj, (sub tipar).

[4] Traub, F. j, Sterative Methods for the solution of equation. Pretince-Holl. Inc. Englewood cliffs. N. j (1964).

[5] Ul’m S., Oboscenie methoda Steffensen dlea resenia nelineinth operatornîh uravnenii. Jurnal vîcise  mat. i mat-fiz. 4, 6 (1964), 1093-1097.

Paper (preprint) in HTML form

On-iterated-operators

ON ITERATED OPERATORS

OF

I. PAVĂLOIU(Cluj)

Let the operational equation be given
(1) P ( x ) = θ , (1) P ( x ) = θ , {:(1)P(x)=theta",":}\begin{equation*} P(x)=\theta, \tag{1} \end{equation*}(1)P(x)=I,
where P P PPP is an operator defined on the normed linear space X X XXX and with values in the normed linear space Y Y YYAnd.
For the solution of equation (1) another operator is generally considered Q ( x ) Q ( x ) Q(x)Q(x)Q(x) defined on the X X XXX and with values in itself.
The purpose of this note is to study the properties of the operator Q Q QQQ and the link between it and the operator P P PPP.
Defined 1. We say that the operator Q Q QQQ is an iterative operator attached to equation (1) if any element x ¯ ∈ X x ¯ ∈ X bar(x)in X\bar{x} \in Xx¯∈X for which P ( x ¯ ) = θ P ( x ¯ ) = θ P( bar(x))=thetaP(\bar{x})=\thetaP(x¯)=I, is a fixed point for the operator Q Q QQQ.
Definipta 2. Fie D ⊆ X D ⊆ X D sube XD \subseteq XD⊆X Lots of space elements X X XXX Shi ρ ≧ 0 ρ ≧ 0 rho >= 0\rho \geqq 0R≧0, a real number. We will say that the iterative operator Q Q QQQ attached to the equation (1) has the order k k kkk ( k k kkk - natural number) on the set D D DDD, if for any x ∈ D x ∈ D x in Dx \in Dx∈D Inequality occurs:
(2) ‖ P ( ( Q ( x ) ) ‖ ≤ ρ ‖ P ( x ) ‖ k . (2) ‖ P ( Q ( x ) ) ‖ ≤ ρ ‖ P ( x ) ‖ k . {:(2)||P((Q(x))|| <= rho||P(x)||^(k).:}:}\begin{equation*} \| P\left((Q(x))\|\leq \rho\| P(x) \|^{k} .\right. \tag{2} \end{equation*}(2)‖P((Q(x))‖≤R‖P(x)‖k.
The set of iterative operators, attached to equation (1), that have the order k k kkk on the field D D DDD we will write it down with I k n I k n I_(k)^(n)I_{k}^{n}Ikn.
In relation to the juxtaposition of two iterative operators, we will establish, in a different way than in [4], the following result:
Motto 1. If Q ∈ I k 1 D , R ∈ I k 2 D , Q ( x ) ∈ D Q ∈ I k 1 D , R ∈ I k 2 D , Q ( x ) ∈ D Q inI_(k_(1))^(D),R inI_(k_(2))^(D),Q(x)in DQ \in I_{k_{1}}^{D}, R \in I_{k_{2}}^{D}, Q(x) \in DQ∈Ik1D,R∈Ik2D,Q(x)∈D Yes R ( x ) ∈ D R ( x ) ∈ D R(x)in DR(x) \in DR(x)∈D for anything x ∈ D x ∈ D x in Dx \in Dx∈Dthen R ( Q ) ∈ I k 1 k 2 D R ( Q ) ∈ I k 1 k 2 D R(Q)inI_(k_(1)k_(2))^(D)R(Q) \in I_{k_{1} k_{2}}^{D}R(Q)∈Ik1k2D Yes Q ( R ) ∈ I k 1 k 2 D Q ( R ) ∈ I k 1 k 2 D Q(R)inI_(k_(1)k_(2))^(D)Q(R) \in I_{k_{1} k_{2}}^{D}Q(R)∈Ik1k2D.
Demonstration. Daughter ρ 1 ρ 1 rho_(1)\rho_{1}R1 and ρ 2 ρ 2 rho_(2)\rho_{2}R2 two real and positive numbers for which, according to the hypothesis, inequalities occur
(3) ‖ P ( Q ( x ) ) ‖ ≦ ρ 1 ‖ P ( x ) ‖ k 1 pentru orice x ∈ D , (3) ‖ P ( Q ( x ) ) ‖ ≦ ρ 1 ‖ P ( x ) ‖ k 1  pentru orice  x ∈ D , {:(3)||P(Q(x))|| <= rho_(1)||P(x)||^(k_(1))" pentru orice "x in D",":}\begin{equation*} \|P(Q(x))\| \leqq \rho_{1}\|P(x)\|^{k_{1}} \text { pentru orice } x \in D, \tag{3} \end{equation*}(3)‖P(Q(x))‖≦R1‖P(x)‖k1 for anything x∈D,
‖ P ( R ( x ) ) ‖ ≦ ρ 2 ‖ P ( x ) ‖ k 2 pentru orice x ∈ D . ‖ P ( R ( x ) ) ‖ ≦ ρ 2 ‖ P ( x ) ‖ k 2  pentru orice  x ∈ D . ||P(R(x))|| <= rho_(2)||P(x)||^(k_(2))" pentru orice "x in D.\|P(R(x))\| \leqq \rho_{2}\|P(x)\|^{k_{2}} \text { pentru orice } x \in D .‖P(R(x))‖≦R2‖P(x)‖k2 for anything x∈D.
Because Q ( x ) ∈ D Q ( x ) ∈ D Q(x)in DQ(x) \in DQ(x)∈D Shi R ( x ) ∈ D R ( x ) ∈ D R(x)in DR(x) \in DR(x)∈D for anything x ∈ D x ∈ D x in Dx \in Dx∈D, it follows that R ( Q ( x ) ) ∈ D R ( Q ( x ) ) ∈ D R(Q(x))in DR(Q(x)) \in DR(Q(x))∈D for anything x ∈ D x ∈ D x in Dx \in Dx∈D. Taking into account this fact and inequality (4) results:
‖ P ( R ( Q ( x ) ) ) ‖ ≦ ρ 2 ‖ P ( Q ( x ) ) ‖ k 2 , pentru orice x ∈ D . ‖ P ( R ( Q ( x ) ) ) ‖ ≦ ρ 2 ‖ P ( Q ( x ) ) ‖ k 2 ,  pentru orice  x ∈ D . ||P(R(Q(x)))|| <= rho_(2)||P(Q(x))||^(k_(2))," pentru orice "x in D.\|P(R(Q(x)))\| \leqq \rho_{2}\|P(Q(x))\|^{k_{2}}, \text { pentru orice } x \in D .‖P(R(Q(x)))‖≦R2‖P(Q(x))‖k2, for anything x∈D.
Taking into account this inequality and (3) we deduce
‖ P ( R ( Q ( x ) ) ) ‖ ≦ ρ 2 ⋅ ρ 1 k 2 ‖ P ( x ) ‖ k 1 ⋅ k 2 , pentru orice x ∈ D . ‖ P ( R ( Q ( x ) ) ) ‖ ≦ ρ 2 ⋅ ρ 1 k 2 ‖ P ( x ) ‖ k 1 ⋅ k 2 ,  pentru orice  x ∈ D . ||P(R(Q(x)))|| <= rho_(2)*rho_(1)^(k_(2))||P(x)||^(k_(1)*k_(2))," pentru orice "x in D.\|P(R(Q(x)))\| \leqq \rho_{2} \cdot \rho_{1}^{k_{2}}\|P(x)\|^{k_{1} \cdot k_{2}}, \text { pentru orice } x \in D .‖P(R(Q(x)))‖≦R2⋅R1k2‖P(x)‖k1⋅k2, for anything x∈D.
And with this the motto is demonstrated.
An immediate consequence of this lemma is the following:
Consequence 1. If Q i ∈ I p i D , Q i ( x ) ∈ D Q i ∈ I p i D , Q i ( x ) ∈ D Q_(i)inI_(p_(i))^(D),Q_(i)(x)in DQ_{i} \in I_{p_{i}}^{D}, Q_{i}(x) \in DQi∈IpiD,Qi(x)∈D for anything x ∈ D , i == 1 , 2 , … s x ∈ D , i == 1 , 2 , … s x in D,i==1,2,dots sx \in D, i= =1,2, \ldots sx∈D,i==1,2,…s, then all the operators of the form
H = Q j 1 ( Q j 2 ( … ( Q j g ) … ) ) H = Q j 1 Q j 2 … Q j g … H=Q_(j_(1))(Q_(j_(2))(dots(Q_(j_(g)))dots))H=Q_{j_{1}}\left(Q_{j_{2}}\left(\ldots\left(Q_{j_{g}}\right) \ldots\right)\right)H=Qj1(Qj2(…(Qjg)…))
where ( j 1 , j 2 , … j s ) j 1 , j 2 , … j s (j_(1),j_(2),dotsj_(s))\left(j_{1}, j_{2}, \ldots j_{s}\right)(j1,j2,…js) is some permutation of the numbers ( 1 , 2 , … , s ) ( 1 , 2 , … , s ) (1,2,dots,s)(1,2, \ldots, s)(1,2,…,s), apartin classi I p 1 ⋅ p 2 , … p s D I p 1 ⋅ p 2 , … p s D I_(p_(1)*p_(2),dotsp_(s))^(D)I_{p_{1} \cdot p_{2}, \ldots p_{s}}^{D}Ip1⋅p2,…psD.
The demonstration of this consequence can be done by complete induction.
Theorem 1. Let D be a convex set of space X X XXX. If the operator P P PPP meets the following conditions:
a) The Operator P P PPP Fréchet derivatives up to and including order 2, for any x ∈ D x ∈ D x in Dx \in Dx∈D.
b) Operator [ P ′ ( x ) ] − 1 P ′ ( x ) − 1 [P^(')(x)]^(-1)\left[P^{\prime}(x)\right]^{-1}[P′(x)]−1 exists and is limited, that is, ‖ [ P ′ ( x ) ] − 1 ‖ ≦≤ B < + ∞ P ′ ( x ) − 1 ≦≤ B < + ∞ ||[P^(')(x)]^(-1)||≦≤B < +oo\left\|\left[P^{\prime}(x)\right]^{-1}\right\| \leqq \leq B<+\infty‖[P′(x)]−1‖≦≤B<+∞, for anything x ∈ D x ∈ D x in Dx \in Dx∈D.
c) ‖ P ′ ′ ( x ) ‖ ≦ M < + ∞ P ′ ′ ( x ) ≦ M < + ∞ ||P^('')(x)|| <= M < +oo\left\|P^{\prime \prime}(x)\right\| \leqq M<+\infty‖P′′(x)‖≦M<+∞, for anything x ∈ D x ∈ D x in Dx \in Dx∈D.
d) Operator R ( x ) = x − [ P ′ ( x ) ] − 1 . P ( x ) R ( x ) = x − P ′ ( x ) − 1 . P ( x ) R(x)=x-[P^(')(x)]^(-1).P(x)R(x)=x-\left[P^{\prime}(x)\right]^{-1} . P(x)R(x)=x−[P′(x)]−1.P(x) it transforms the set D into itself.
Then R ∈ I 2 D R ∈ I 2 D R inI_(2)^(D)R \in I_{2}^{D}R∈I2D.
Demonstration. Applying Taylor's generalized formula and taking into account the assumptions of the theorem we have
‖ P ( R ( x ) ) ‖ ≦ ‖ P ( R ( x ) ) − [ P ( x ) − P ′ ( x ) [ P ′ ( x ) ] − 1 P ( x ) ] ‖ + + ‖ P ( x ) − P ′ ( x ) ] [ P ′ ( x ) ] − 1 P ( x ) ) ‖ ≦ ‖ P ′ ′ ( ξ ) ‖ B 2 2 ‖ P ( x ) ‖ 2 ‖ P ( R ( x ) ) ‖ ≦ P ( R ( x ) ) − P ( x ) − P ′ ( x ) P ′ ( x ) − 1 P ( x ) + + ‖ P ( x ) − P ′ ( x ) P ′ ( x ) − 1 P ( x ) ≦ P ′ ′ ( ξ ) B 2 2 P ( x ) ‖ 2 {:[||P(R(x))|| <= ||P(R(x))-[P(x)-P^(')(x)[P^(')(x)]^(-1)P(x)]||+],[{: quad+||P(x)-P^(')(x)][P^(')(x)]^(-1)P(x))|| <= (||P^('')(xi)||B^(2))/(2)||P(x)||^(2)]:}\begin{aligned} & \|P(R(x))\| \leqq\left\|P(R(x))-\left[P(x)-P^{\prime}(x)\left[P^{\prime}(x)\right]^{-1} P(x)\right]\right\|+ \\ & \left.\left.\quad+\| P(x)-P^{\prime}(x)\right]\left[P^{\prime}(x)\right]^{-1} P(x)\right)\left\|\leqq \frac{\left\|P^{\prime \prime}(\xi)\right\| B^{2}}{2}\right\| P(x) \|^{2} \end{aligned}‖P(R(x))‖≦‖P(R(x))−[P(x)−P′(x)[P′(x)]−1P(x)]‖++‖P(x)−P′(x)][P′(x)]−1P(x))‖≦‖P′′(ξ)‖B22‖P(x)‖2
where ξ = x − θ [ P ′ ( x ) ] − 1 . P ( x ) , 0 ≤ θ < 1 ξ = x − θ P ′ ( x ) − 1 . P ( x ) , 0 ≤ θ < 1 xi=x-theta[P^(')(x)]^(-1).P(x),0 <= theta < 1\xi=x-\theta\left[P^{\prime}(x)\right]^{-1} . P(x), 0 \leq \theta<1ξ=x−I[P′(x)]−1.P(x),0≤I<1. Because D D DDD is convex set, then ξ ∈ D ξ ∈ D xi in D\xi \in Dξ∈D and taking into account hypothesis c) we have
‖ P ( R ( x ) ) ‖ ≦ M B 2 2 ‖ P ( x ) ‖ 2 ‖ P ( R ( x ) ) ‖ ≦ M B 2 2 ‖ P ( x ) ‖ 2 ||P(R(x))|| <= (MB^(2))/(2)||P(x)||^(2)\|P(R(x))\| \leqq \frac{M B^{2}}{2}\|P(x)\|^{2}‖P(R(x))‖≦MB22‖P(x)‖2
which proves the stated theorem. This theorem shows us that the Newton–Kantorovich operator [1] has order 2.
An immediate consequence of theorem 1 and lemma 1 is the following:
Consequence 2. Whether the conditions of theorem 1 are met and whether Q ( x ) ∈ I p D , Q ( x ) ∈ D Q ( x ) ∈ I p D , Q ( x ) ∈ D Q(x)inI_(p)^(D),Q(x)in DQ(x) \in I_{p}^{D}, Q(x) \in DQ(x)∈IpD,Q(x)∈D for x ∈ D x ∈ D x in Dx \in Dx∈Dwhere D D DDD is the same set as in theorem 1, then the operator R ( Q ) ∈ I 2 D R ( Q ) ∈ I 2 D R(Q)inI_(2)^(D)R(Q) \in I_{2}{ }^{D}R(Q)∈I2Dwhere R ( x ) = x − [ P ′ ( x ) ] − 1 . P ( x ) R ( x ) = x − P ′ ( x ) − 1 . P ( x ) R(x)=x-[P^(')(x)]^(-1).P(x)R(x)=x-\left[P^{\prime}(x)\right]^{-1} . P(x)R(x)=x−[P′(x)]−1.P(x).
For the results that we will establish below, we will assume that the operator Q Q QQQ has the shape Q ( x ) , = x + φ ( x ) Q ( x ) , = x + φ ( x ) Q(x),=x+varphi(x)Q(x),=x+\varphi(x)Q(x),=x+F(x), where the operator φ ( x ) φ ( x ) varphi(x)\varphi(x)F(x) satisfies the condition
(5) ‖ φ ( x ) ‖ ≦ δ ‖ P ( x ) ‖ , pentru orice x ∈ D (5) ‖ φ ( x ) ‖ ≦ δ ‖ P ( x ) ‖ ,  pentru orice  x ∈ D {:(5)||varphi(x)|| <= delta||P(x)||","" pentru orice "x in D:}\begin{equation*} \|\varphi(x)\| \leqq \delta\|P(x)\|, \text { pentru orice } x \in D \tag{5} \end{equation*}(5)‖F(x)‖≦D‖P(x)‖, for anything x∈D
where δ δ delta\deltaD It is a real and positive number.
Observation. Whether condition (5) is fulfilled and whether x ¯ ∈ D x ¯ ∈ D bar(x)in D\bar{x} \in Dx¯∈D Shi P ( x ¯ ) = θ P ( x ¯ ) = θ P( bar(x))=thetaP(\bar{x})=\thetaP(x¯)=Ithen φ ( x ¯ ) = θ φ ( x ¯ ) = θ varphi( bar(x))=theta\varphi(\bar{x})=\thetaF(x¯)=I. From the fact that P ( x ¯ ) = θ P ( x ¯ ) = θ P( bar(x))=thetaP(\bar{x})=\thetaP(x¯)=I Results φ ( x ¯ ) == θ φ ( x ¯ ) == θ varphi( bar(x))==theta\varphi(\bar{x})= =\thetaF(x¯)==I and so x ¯ = Q ( x ¯ ) x ¯ = Q ( x ¯ ) bar(x)=Q( bar(x))\bar{x}=Q(\bar{x})x¯=Q(x¯)where Q ( x ) = x + φ ( x ) Q ( x ) = x + φ ( x ) Q(x)=x+varphi(x)Q(x)=x+\varphi(x)Q(x)=x+F(x)in other words x ¯ x ¯ bar(x)\bar{x}x¯ is a fixed point of the operator Q Q QQQ.
Theorem 2. If conditions a), b) and c) of theorem 1 are met and if on the same set D D DDD The following conditions are also met:
a ′ a ′ a^(')\mathrm{a}^{\prime}a′ ) Operator Q ( x ) = x + φ ( x ) Q ( x ) = x + φ ( x ) Q(x)=x+varphi(x)Q(x)=x+\varphi(x)Q(x)=x+F(x) satisface conditia (5).
b ′ ) Q ( x ) ∈ D b ′ Q ( x ) ∈ D {:b^('))Q(x)in D\left.\mathrm{b}^{\prime}\right) Q(x) \in Db′)Q(x)∈D for anything x ∈ D x ∈ D x in Dx \in Dx∈D Yes Q ∈ I p D Q ∈ I p D Q inI_(p)^(D)Q \in I_{p}^{D}Q∈IpD.
с') Operator P P PPP is bordered in D D DDD in other words ‖ P ( x ) ‖ ≦ H < + ∞ ‖ P ( x ) ‖ ≦ H < + ∞ ||P(x)|| <= H < +oo\|P(x)\| \leqq H<+\infty‖P(x)‖≦H<+∞, for anything x ∈ D x ∈ D x in Dx \in Dx∈D.
(d) Operator R ( x ) = Q ( x ) − [ P ′ ( x ) ] − 1 . P ( Q ( x ) ) R ( x ) = Q ( x ) − P ′ ( x ) − 1 . P ( Q ( x ) ) R(x)=Q(x)-[P^(')(x)]^(-1).P(Q(x))R(x)=Q(x)-\left[P^{\prime}(x)\right]^{-1} . P(Q(x))R(x)=Q(x)−[P′(x)]−1.P(Q(x)) Transform the crowd D D DDD in it ı ^ n s a ̆ s i ı ^ n s a ̆ s i hat(ı)nsăsi\hat{\imath} n s a ̆ s iI^nsăsi.
Then the operator R R RRR Apartine Classi I p + 1 D I p + 1 D I_(p+1)^(D)I_{p+1}^{D}Ip+1D.
Demonstration. Taking into account the hypotheses of the theorem and applying the generalized Taylor formula we have
‖ P ( R ( x ) ) ‖ ≦ ‖ P ( R ( x ) ) − [ P ( Q ( x ) ) − P ′ ( Q ( x ) ) [ P ′ ( x ) ] − 1 P ( Q ( x ) ) ] ‖ + + ‖ P ( Q ( x ) ) − P ′ ( x ) [ P ′ ( x ) ] − 1 P ( Q ( x ) ) ‖ + ‖ P ( R ( x ) ) ‖ ≦ P ( R ( x ) ) − P ( Q ( x ) ) − P ′ ( Q ( x ) ) P ′ ( x ) − 1 P ( Q ( x ) ) + + P ( Q ( x ) ) − P ′ ( x ) P ′ ( x ) − 1 P ( Q ( x ) ) + {:[||P(R(x))|| <= ||P(R(x))-[P(Q(x))-P^(')(Q(x))[P^(')(x)]^(-1)P(Q(x))]||+],[+||P(Q(x))-P^(')(x)[P^(')(x)]^(-1)P(Q(x))||+]:}\begin{gathered} \|P(R(x))\| \leqq\left\|P(R(x))-\left[P(Q(x))-P^{\prime}(Q(x))\left[P^{\prime}(x)\right]^{-1} P(Q(x))\right]\right\|+ \\ +\left\|P(Q(x))-P^{\prime}(x)\left[P^{\prime}(x)\right]^{-1} P(Q(x))\right\|+ \end{gathered}‖P(R(x))‖≦‖P(R(x))−[P(Q(x))−P′(Q(x))[P′(x)]−1P(Q(x))]‖++‖P(Q(x))−P′(x)[P′(x)]−1P(Q(x))‖+
where ξ = Q ( x ) − θ . [ P ′ ( x ) ] − 1 . P ( Q ( x ) ) , 0 < θ < 1 ξ = Q ( x ) − θ . P ′ ( x ) − 1 . P ( Q ( x ) ) , 0 < θ < 1 xi=Q(x)-theta.[P^(')(x)]^(-1).P(Q(x)),0 < theta < 1\xi=Q(x)-\theta .\left[P^{\prime}(x)\right]^{-1} . P(Q(x)), 0<\theta<1ξ=Q(x)−I.[P′(x)]−1.P(Q(x)),0<I<1. Because the crowd D D DDD is convex, it follows that ξ ∈ D ξ ∈ D xi in D\xi \in Dξ∈D.
From inequality (5) we deduce
‖ P ′ ( Q ( x ) ) − P ′ ( x ) ‖ ≦ ‖ P ′ ′ ( η ) ‖ ⋅ δ ⋅ ‖ P ( x ) ‖ P ′ ( Q ( x ) ) − P ′ ( x ) ≦ P ′ ′ ( η ) ⋅ δ ⋅ ‖ P ( x ) ‖ ||P^(')(Q(x))-P^(')(x)|| <= ||P^('')(eta)||*delta*||P(x)||\left\|P^{\prime}(Q(x))-P^{\prime}(x)\right\| \leqq\left\|P^{\prime \prime}(\eta)\right\| \cdot \delta \cdot\|P(x)\|‖P′(Q(x))−P′(x)‖≦‖P′′(the)‖⋅D⋅‖P(x)‖
where η = x + θ ⋅ φ ( x ) , 0 < θ < 1 , η ∈ D η = x + θ ⋅ φ ( x ) , 0 < θ < 1 , η ∈ D eta=x+theta*varphi(x),0 < theta < 1,eta in D\eta=x+\theta \cdot \varphi(x), 0<\theta<1, \eta \in Dthe=x+I⋅F(x),0<I<1,the∈D. Hence the fact that ‖ P ( x ) ‖ ‖ P ( x ) ‖ ||P(x)||\|P(x)\|‖P(x)‖. is bordered in D D DDDResults
(6) ‖ P ( R ( x ) ) ‖ ≦ M 2 B 2 ‖ P ( Q ( x ) ) ‖ 2 + M B δ ‖ P ( x ) ‖ ⋅ ‖ P ( Q ( x ) ) ‖ (6) ‖ P ( R ( x ) ) ‖ ≦ M 2 B 2 ‖ P ( Q ( x ) ) ‖ 2 + M B δ ‖ P ( x ) ‖ ⋅ ‖ P ( Q ( x ) ) ‖ {:(6)||P(R(x))|| <= (M)/(2)B^(2)||P(Q(x))||^(2)+MB delta||P(x)||*||P(Q(x))||:}\begin{equation*} \|P(R(x))\| \leqq \frac{M}{2} B^{2}\|P(Q(x))\|^{2}+M B \delta\|P(x)\| \cdot\|P(Q(x))\| \tag{6} \end{equation*}(6)‖P(R(x))‖≦M2B2‖P(Q(x))‖2+MBD‖P(x)‖⋅‖P(Q(x))‖
Because Q ∈ I p D Q ∈ I p D Q inI_(p)^(D)Q \in I_{p}^{D}Q∈IpD, it follows that there is a real number β ≧ 0 β ≧ 0 beta >= 0\beta \geqq 0B≧0 for which we have
‖ P ( Q ( x ) ) ‖ ≦ β ‖ P ( x ) ‖ p ‖ P ( Q ( x ) ) ‖ ≦ β ‖ P ( x ) ‖ p ||P(Q(x))|| <= beta||P(x)||^(p)\|P(Q(x))\| \leqq \beta\|P(x)\|^{p}‖P(Q(x))‖≦B‖P(x)‖p
which replaced in (6) leads us to the following inequality
‖ P ( R ( x ) ) ‖ ≦ α ‖ P ( x ) ‖ p + 1 , ‖ P ( R ( x ) ) ‖ ≦ α ‖ P ( x ) ‖ p + 1 , ||P(R(x))|| <= alpha||P(x)||^(p+1),\|P(R(x))\| \leqq \alpha\|P(x)\|^{p+1},‖P(R(x))‖≦A‖P(x)‖p+1,
where if we do the calculations we find
α 4 = M B 2 β 2 ⋅ H p − 1 2 + M β δ B . α 4 = M B 2 β 2 ⋅ H p − 1 2 + M β δ B . alpha_(4)=(MB^(2)beta^(2)*H^(p-1))/(2)+M beta delta B.\alpha_{4}=\frac{M B^{2} \beta^{2} \cdot H^{p-1}}{2}+M \beta \delta B .A4=MB2B2⋅Hp−12+MBDB.
On the basis of this theorem we will construct below an iterative olerator of order 3, which, unlike Chebyshev's operator [1] which also has order 3, will contain only the first-order derivative of the operator P P PPP. Indeed, if we take into account theorem 2 , where we take Q ( x ) = x − Q ( x ) = x − Q(x)=x-Q(x)=x-Q(x)=x− - [ P ′ ( x ) ] − 1 P ( x ) P ′ ( x ) − 1 P ( x ) [P^(')(x)]^(-1)P(x)\left[P^{\prime}(x)\right]^{-1} P(x)[P′(x)]−1P(x), then the operator R ( x ) R ( x ) R(x)R(x)R(x) has the shape
R ( x ) = x − [ P ′ ( x ) ] − 1 ⋅ P ( x ) − [ P ′ ( x ) ] − 1 P [ x − [ P ′ ( x ) ] − 1 ⋅ P ( x ) ] R ( x ) = x − P ′ ( x ) − 1 ⋅ P ( x ) − P ′ ( x ) − 1 P x − P ′ ( x ) − 1 ⋅ P ( x ) R(x)=x-[P^(')(x)]^(-1)*P(x)-[P^(')(x)]^(-1)P[x-[P^(')(x)]^(-1)*P(x)]R(x)=x-\left[P^{\prime}(x)\right]^{-1} \cdot P(x)-\left[P^{\prime}(x)\right]^{-1} P\left[x-\left[P^{\prime}(x)\right]^{-1} \cdot P(x)\right]R(x)=x−[P′(x)]−1⋅P(x)−[P′(x)]−1P[x−[P′(x)]−1⋅P(x)]
This operator has advantages over Chebyshev's operator in that the second derivative of the operator does not intervene in its expression P P PPP which in many cases is difficult to calculate and has a very complicated expression. As we have seen, theorem 2 gives us a method to increase the order of an operator by one unit.
In the following we will study another method of this kind which is at the same time a generalization of Steffensen's iterative operator [2], [5]. This operator has been highlighted by several authors in different ways. Obviously, the simplest way is the one by which Steffensen's operator results from the string method.
In this regard, we will note with the [ x , y ; P ] [ x , y ; P ] [x,y;P][x, y ; P][x,and;P] Operator Split Differences P P PPP [3], on the nodes x x xxx and y y yyand, which we will assume to be symmetrical in relation to x x xxx and y y yyandin other words
[ x , y ; P ] = [ y , x ; P ] [ x , y ; P ] = [ y , x ; P ] [x,y;P]=[y,x;P][x, y ; P]=[y, x ; P][x,and;P]=[and,x;P]
Theorem 3. If the following conditions apply:
a) The Operator Q ( x ) Q ( x ) Q(x)Q(x)Q(x) is an iterative operator attached to equation (1) that belongs to the class I p D I p D I_(p)^(D)I_{p}^{D}IpD and for anything x ∈ D , Q ( x ) ∈ D x ∈ D , Q ( x ) ∈ D x in D,Q(x)in Dx \in D, Q(x) \in Dx∈D,Q(x)∈D.
b) For any x ∈ D x ∈ D x in Dx \in Dx∈D, the linear operator [ x , Q ( x ) ; P ] [ x , Q ( x ) ; P ] [x,Q(x);P][x, Q(x) ; P][x,Q(x);P] admits a bordered inverse, i.e.: ‖ [ x , Q ( x ) ; P ] − 1 ‖ ≦ B < + ∞ [ x , Q ( x ) ; P ] − 1 ≦ B < + ∞ ||[x,Q(x);P]^(-1)|| <= B < quad+oo\left\|[x, Q(x) ; P]^{-1}\right\| \leqq B<\quad+\infty‖[x,Q(x);P]−1‖≦B<+∞.
c) ‖ [ y , z ; P ] − [ x , y ; P ] ‖ = K ‖ x − z ‖ , 0 ≦ K < + ∞ ‖ [ y , z ; P ] − [ x , y ; P ] ‖ = K ‖ x − z ‖ , 0 ≦ K < + ∞ ||[y,z;P]-[x,y;P]||=K||x-z||,0 <= K < +oo\|[y, z ; P]-[x, y ; P]\|=K\|x-z\|, 0 \leqq K<+\infty‖[and,with;P]−[x,and;P]‖=K‖x−with‖,0≦K<+∞, for anything x , y , z ∈ D x , y , z ∈ D x,y,z in Dx, y, z \in Dx,and,with∈D.
d) Operator R ( x ) = x − [ x , Q ( x ) ; P ] − 1 P ( x ) R ( x ) = x − [ x , Q ( x ) ; P ] − 1 P ( x ) R(x)=x-[x,Q(x);P]^(-1)P(x)R(x)=x-[x, Q(x) ; P]^{-1} P(x)R(x)=x−[x,Q(x);P]−1P(x) Transform the crowd D D DDD in itself.
Then the operator R ∈ I p + 1 D R ∈ I p + 1 D R inI_(p+1)^(D)R \in I_{p+1}^{D}R∈Ip+1D. also put in the following form:
R ( x ) = Q ( x ) − [ x , Q ( x ) ; P ] − 1 P ( Q ( x ) ) . R ( x ) = Q ( x ) − [ x , Q ( x ) ; P ] − 1 P ( Q ( x ) ) . R(x)=Q(x)-[x,Q(x);P]^(-1)P(Q(x)).R(x)=Q(x)-[x, Q(x) ; P]^{-1} P(Q(x)) .R(x)=Q(x)−[x,Q(x);P]−1P(Q(x)).
For this we will start from the following obvious equality
(7) x − Q ( x ) = [ x , Q ( x ) ; P ] − 1 [ x , Q ( x ) ; P ] ( x − Q ( x ) ) (7) x − Q ( x ) = [ x , Q ( x ) ; P ] − 1 [ x , Q ( x ) ; P ] ( x − Q ( x ) ) {:(7)x-Q(x)=[x","Q(x);P]^(-1)[x","Q(x);P](x-Q(x)):}\begin{equation*} x-Q(x)=[x, Q(x) ; P]^{-1}[x, Q(x) ; P](x-Q(x)) \tag{7} \end{equation*}(7)x−Q(x)=[x,Q(x);P]−1[x,Q(x);P](x−Q(x))
and taking into account the equality of
P ( x ) − P ( Q ( x ) ) = [ x , Q ( x ) ; P ] ( x − Q ( x ) ) P ( x ) − P ( Q ( x ) ) = [ x , Q ( x ) ; P ] ( x − Q ( x ) ) P(x)-P(Q(x))=[x,Q(x);P](x-Q(x))P(x)-P(Q(x))=[x, Q(x) ; P](x-Q(x))P(x)−P(Q(x))=[x,Q(x);P](x−Q(x))
which replaced in (7) gives us
x − Q ( x ) = [ x , Q ( x ) ; P ] − 1 [ P ( x ) − P ( Q ( x ) ) ] = = − [ x , Q ( x ) ; P ] − 1 ⋅ P ( Q ( x ) ) + [ x , Q ( x ) ; P ] − 1 ⋅ P ( x ) . x − Q ( x ) = [ x , Q ( x ) ; P ] − 1 [ P ( x ) − P ( Q ( x ) ) ] = = − [ x , Q ( x ) ; P ] − 1 ⋅ P ( Q ( x ) ) + [ x , Q ( x ) ; P ] − 1 ⋅ P ( x ) . {:[x-Q(x)=[x","Q(x);P]^(-1)[P(x)-P(Q(x))]=],[=-[x","Q(x);P]^(-1)*P(Q(x))+[x","Q(x);P]^(-1)*P(x).]:}\begin{gathered} x-Q(x)=[x, Q(x) ; P]^{-1}[P(x)-P(Q(x))]= \\ =-[x, Q(x) ; P]^{-1} \cdot P(Q(x))+[x, Q(x) ; P]^{-1} \cdot P(x) . \end{gathered}x−Q(x)=[x,Q(x);P]−1[P(x)−P(Q(x))]==−[x,Q(x);P]−1⋅P(Q(x))+[x,Q(x);P]−1⋅P(x).
hence the direct result of equality
x − [ x , Q ( x ) ; P ] − 1 ⋅ P ( x ) := Q ( x ) − [ x , Q ( x ) ; P ] − 1 ⋅ P ( Q ( x ) ) . x − [ x , Q ( x ) ; P ] − 1 ⋅ P ( x ) := Q ( x ) − [ x , Q ( x ) ; P ] − 1 ⋅ P ( Q ( x ) ) . x-[x,Q(x);P]^(-1)*P(x):=Q(x)-[x,Q(x);P]^(-1)*P(Q(x)).x-[x, Q(x) ; P]^{-1} \cdot P(x):=Q(x)-[x, Q(x) ; P]^{-1} \cdot P(Q(x)) .x−[x,Q(x);P]−1⋅P(x):=Q(x)−[x,Q(x);P]−1⋅P(Q(x)).
From the above and from the conditions of the theorem, the following inequalities result:
(8) ‖ R ( x ) − x ‖ ≦ B ‖ P ( x ) ‖ , pentru orice x ∈ D , (8) ‖ R ( x ) − x ‖ ≦ B ‖ P ( x ) ‖ ,  pentru orice  x ∈ D , {:(8)||R(x)-x|| <= B||P(x)||quad","quad" pentru orice "x in D",":}\begin{equation*} \|R(x)-x\| \leqq B\|P(x)\| \quad, \quad \text { pentru orice } x \in D, \tag{8} \end{equation*}(8)‖R(x)−x‖≦B‖P(x)‖, for anything x∈D,
(9) ‖ R ( x ) − Q ( x ) ‖ = B ‖ P ( Q ( x ) ) ‖ ‖ R ( x ) − Q ( x ) ‖ = B ‖ P ( Q ( x ) ) ‖ ||R(x)-Q(x)||=B||P(Q(x))||\|R(x)-Q(x)\|=B\|P(Q(x))\|‖R(x)−Q(x)‖=B‖P(Q(x))‖, for anything x ∈ D x ∈ D x in Dx \in Dx∈D.
Further we consider the following identity to be obvious
P ( u ) = P ( x ) + [ x , Q ( x ) ; P ] ( u − x ) + { [ x , u ; P ] − − [ x , Q ( x ) ; P ] } ( u − x ) P ( u ) = P ( x ) + [ x , Q ( x ) ; P ] ( u − x ) + { [ x , u ; P ] − − [ x , Q ( x ) ; P ] } ( u − x ) {:[P(u)=P(x)+[x","Q(x);P](u-x)+{[x","u;P]-],[-[x","Q(x);P]}(u-x)]:}\begin{aligned} P(u)=P(x) & +[x, Q(x) ; P](u-x)+\{[x, u ; P]- \\ & -[x, Q(x) ; P]\}(u-x) \end{aligned}P(in the)=P(x)+[x,Q(x);P](in the−x)+{[x,in the;P]−−[x,Q(x);P]}(in the−x)
of which replacing the u u uuin the with R ( x ) R ( x ) R(x)R(x)R(x) and taking into account (8), (9) and the hypotheses of the theorem, we
deduce (10) ‖ P ( R ( x ) ) ‖ = K B 2 ‖ P ( x ) ‖ ⋅ ‖ P ( Q ( x ) ) ‖ ‖ P ( R ( x ) ) ‖ = K B 2 ‖ P ( x ) ‖ ⋅ ‖ P ( Q ( x ) ) ‖ quad||P(R(x))||=KB^(2)||P(x)||*||P(Q(x))||\quad\|P(R(x))\|=K B^{2}\|P(x)\| \cdot\|P(Q(x))\|‖P(R(x))‖=KB2‖P(x)‖⋅‖P(Q(x))‖, for anything x ∈ D x ∈ D x in Dx \in Dx∈D.
Condition a) ensures the existence of a real and positive number ρ ρ rho\rhoR for which
‖ P ( Q ( x ) ) ‖ ≦ ρ ‖ P ( x ) ‖ p , pentru orice x ∈ D ‖ P ( Q ( x ) ) ‖ ≦ ρ ‖ P ( x ) ‖ p ,  pentru orice  x ∈ D ||P(Q(x))|| <= rho||P(x)||^(p)," pentru orice "x in D\|P(Q(x))\| \leqq \rho\|P(x)\|^{p}, \text { pentru orice } x \in D‖P(Q(x))‖≦R‖P(x)‖p, for anything x∈D
which replaced in (10) gives us
‖ P ( R ( x ) ) ‖ ≦ K ρ B 2 ‖ P ( x ) ‖ p + 1 , pentru orice x ∈ D ‖ P ( R ( x ) ) ‖ ≦ K ρ B 2 ‖ P ( x ) ‖ p + 1 ,  pentru orice  x ∈ D ||P(R(x))|| <= K rhoB^(2)||P(x)||^(p+1)," pentru orice "x in D\|P(R(x))\| \leqq K \rho B^{2}\|P(x)\|^{p+1}, \text { pentru orice } x \in D‖P(R(x))‖≦KRB2‖P(x)‖p+1, for anything x∈D
which proves the stated theorem.
Using generalized divided differences we have thus constructed an order operator p + 1 p + 1 p+1p+1p+1, using a p p ppp. This operator is more general than those constructed with the help of theorem 2 because it does not appeal to the existence of first- and second-order derivatives. As a matter of fact, Steffensen's operator in the form in which it has been studied, so far, has order 2 , because it has been assumed that Q ( x ) Q ( x ) Q(x)Q(x)Q(x) is a linear iteration operator [2], [5]
I shall continue to consider that the operator Q Q QQQ has the shape Q ( x ) == x + φ ( x ) Q ( x ) == x + φ ( x ) Q(x)==x+varphi(x)Q(x)= =x+\varphi(x)Q(x)==x+F(x) and we will look for conditions on the operator φ φ varphi\varphiF so that Q Q QQQ be an iterative operator of the l l lll attached to equation (1).
Theorem 4. If the following conditions are met:
a) For any x ∈ D , Q ( x ) ∈ D x ∈ D , Q ( x ) ∈ D x in D,Q(x)in Dx \in D, Q(x) \in Dx∈D,Q(x)∈Dwhere D ⊆ X D ⊆ X D sube XD \subseteq XD⊆X It is a convex lot.
b) Operator P P PPP admits derivatives (in the sense of Fréchet) up to the order ℏ ℏ ℏ\hbarħ including, and for any x ∈ D x ∈ D x in Dx \in Dx∈D There is a real and positive number M > 0 M > 0 M > 0M>0M>0 for which sup D ‖ P ( k ) ( x ) ‖ ≦ M sup D   P ( k ) ( x ) ≦ M s u p_(D)||P^((k))(x)|| <= M\sup _{D}\left\|P^{(k)}(x)\right\| \leqq MUD‖P(k)(x)‖≦M.
c) Operator φ φ varphi\varphiF satisfies the condition
‖ P ( x ) + P ′ ( x ) φ ( x ) + P ′ ′ ( x ) 2 ! φ 2 ( x ) + … … + P ( k − 1 ) ( x ) ( k − 1 ) ! φ k − 1 x ‖ ≦ α ‖ P ( x ) ‖ k ‖ P ( x ) + P ′ ( x ) φ ( x ) + P ′ ′ ( x ) 2 ! φ 2 ( x ) + … … + P ( k − 1 ) ( x ) ( k − 1 ) ! φ k − 1 x ‖ ≦ α ‖ P ( x ) ‖ k {:[||P(x)+P^(')(x)varphi(x)+(P^('')(x))/(2!)varphi^(2)(x)+dots],[dots+(P^((k-1))(x))/((k-1)!)varphi^(k-1)x|| <= alpha||P(x)||^(k)]:}\begin{gathered} \| P(x)+P^{\prime}(x) \varphi(x)+\frac{P^{\prime \prime}(x)}{2!} \varphi^{2}(x)+\ldots \\ \ldots+\frac{P^{(k-1)}(x)}{(k-1)!} \varphi^{k-1} x\|\leqq \alpha\| P(x) \|^{k} \end{gathered}‖P(x)+P′(x)F(x)+P′′(x)2!F2(x)+……+P(k−1)(x)(k−1)!Fk−1x‖≦A‖P(x)‖k
for anything x ∈ D x ∈ D x in Dx \in Dx∈D, where a is a real and non-negative number
d) ‖ φ ( x ) ‖ = β ‖ P ( x ) ‖ ‖ φ ( x ) ‖ = β ‖ P ( x ) ‖ ||varphi(x)||=beta||P(x)||\|\varphi(x)\|=\beta\|P(x)\|‖F(x)‖=B‖P(x)‖, for anything x ∈ D x ∈ D x in Dx \in Dx∈Dwhere β > 0 β > 0 beta > 0\beta>0B>0 is a. real number.

Then the operator Q ∈ I k D Q ∈ I k D Q inI_(k)^(D)Q \in I_{k}^{D}Q∈IkD.

Demonstration. Applying Taylor's generalized formula and taking into account the form of the remainder in this formula and hypotheses c) and d) we have
‖ P ( Q ( x ) ) ‖ ≦ ‖ P ( Q ( x ) ) − [ P ( x ) + P ′ ( x ) φ ( x ) + … ⋯ + P ( k − 1 ) ( x ) ( k − 1 ) ! φ k − 1 ( x ) ‖ + ‖ P ( x ) + P ′ ( x ) φ ( x ) + … ⋯ + P ( k − 1 ) ( x ) ( k − 1 ) ! φ k − 1 ( x ) ‖ ‖ P ( Q ( x ) ) ‖ ≦ ‖ P ( Q ( x ) ) − P ( x ) + P ′ ( x ) φ ( x ) + … ⋯ + P ( k − 1 ) ( x ) ( k − 1 ) ! φ k − 1 ( x ) ‖ + ‖ P ( x ) + P ′ ( x ) φ ( x ) + … ⋯ + P ( k − 1 ) ( x ) ( k − 1 ) ! φ k − 1 ( x ) ‖ {:[||P(Q(x))|| <= ||P(Q(x))-[P(x)+P^(')(x)varphi(x)+dots:}],[cdots+(P^((k-1))(x))/((k-1)!)varphi^(k-1)(x)||+||P(x)+P^(')(x)varphi(x)+dots],[cdots+(P^((k-1))(x))/((k-1)!)varphi^(k-1)(x)||]:}\begin{gathered} \|P(Q(x))\| \leqq \| P(Q(x))-\left[P(x)+P^{\prime}(x) \varphi(x)+\ldots\right. \\ \cdots+\frac{P^{(k-1)}(x)}{(k-1)!} \varphi^{k-1}(x)\|+\| P(x)+P^{\prime}(x) \varphi(x)+\ldots \\ \cdots+\frac{P^{(k-1)}(x)}{(k-1)!} \varphi^{k-1}(x) \| \end{gathered}‖P(Q(x))‖≦‖P(Q(x))−[P(x)+P′(x)F(x)+…⋯+P(k−1)(x)(k−1)!Fk−1(x)‖+‖P(x)+P′(x)F(x)+…⋯+P(k−1)(x)(k−1)!Fk−1(x)‖
Where to Deduce
P ( Q ( x ) ) = ‖ P ( k ) ( ξ ) ‖ k ! ‖ φ ( x ) ‖ k + α ‖ P ( x ) ‖ k P ( Q ( x ) ) = P ( k ) ( ξ ) k ! ‖ φ ( x ) ‖ k + α ‖ P ( x ) ‖ k P(Q(x))=(||P^((k))(xi)||)/(k!)||varphi(x)||^(k)+alpha||P(x)||^(k)P(Q(x))=\frac{\left\|P^{(k)}(\xi)\right\|}{k!}\|\varphi(x)\|^{k}+\alpha\|P(x)\|^{k}P(Q(x))=‖P(k)(ξ)‖k!‖F(x)‖k+A‖P(x)‖k
where ξ = x + θ φ ( x ) , 0 < θ < 1 , ξ ∈ D ξ = x + θ φ ( x ) , 0 < θ < 1 , ξ ∈ D xi=x+theta varphi(x),0 < theta < 1,xi in D\xi=x+\theta \varphi(x), 0<\theta<1, \xi \in Dξ=x+IF(x),0<I<1,ξ∈Din other words
‖ P ( Q ( x ) ) ‖ ≦ ( M β k k ! + α ) ‖ P ( x ) ‖ k ‖ P ( Q ( x ) ) ‖ ≦ M β k k ! + α ‖ P ( x ) ‖ k ||P(Q(x))|| <= ((Mbeta^(k))/(k!)+alpha)||P(x)||^(k)\|P(Q(x))\| \leqq\left(\frac{M \beta^{k}}{k!}+\alpha\right)\|P(x)\|^{k}‖P(Q(x))‖≦(MBkk!+A)‖P(x)‖k for anything x ∈ D x ∈ D x in Dx \in Dx∈D. Inequality d) shows us that Q ( x ) Q ( x ) Q(x)Q(x)Q(x) is an iterative operator attached to equation (1) and if we take into account the last inequality it follows that Q ( x ) ∈ I k D Q ( x ) ∈ I k D Q(x)inI_(k)^(D)Q(x) \in I_{k}^{D}Q(x)∈IkD.
Theorem 5. Daughter x ¯ ∈ D ⊆ X x ¯ ∈ D ⊆ X bar(x)in D sube X\bar{x} \in D \subseteq Xx¯∈D⊆X a solution of equation (1), where D D DDD this o o ooor convex crowd. If the following conditions are met:
a) The Operator Q Q QQQ admits derivatives (in the sense of Fréchet) for any x ∈ D x ∈ D x in Dx \in Dx∈D, until the order of the k k kkk including these derivatives satisfies the following conditions:
Q ′ ( x ¯ ) = θ 1 , Q ′ ′ ( x ¯ ) = θ 2 , … , Q ( k − 1 ) ( x ¯ ) = θ k − 1 , Q ( k ) ( x ¯ ) ≠ θ k Q ′ ( x ¯ ) = θ 1 , Q ′ ′ ( x ¯ ) = θ 2 , … , Q ( k − 1 ) ( x ¯ ) = θ k − 1 , Q ( k ) ( x ¯ ) ≠ θ k Q^(')( bar(x))=theta_(1),quadQ^('')( bar(x))=theta_(2),dots,quadQ^((k-1))( bar(x))=theta_(k-1),quadQ^((k))( bar(x))!=theta_(k)Q^{\prime}(\bar{x})=\theta_{1}, \quad Q^{\prime \prime}(\bar{x})=\theta_{2}, \ldots, \quad Q^{(k-1)}(\bar{x})=\theta_{k-1}, \quad Q^{(k)}(\bar{x}) \neq \theta_{k}Q′(x¯)=I1,Q′′(x¯)=I2,…,Q(k−1)(x¯)=Ik−1,Q(k)(x¯)≠Ik
and
sup x ∈ D ¯ ‖ Q ( k ) ( x ) ‖ ≤ M < + ∞ sup x ∈ D ¯   Q ( k ) ( x ) ≤ M < + ∞ s u p_(x in bar(D))||Q^((k))(x)|| <= M < +oo\sup _{x \in \bar{D}}\left\|Q^{(k)}(x)\right\| \leq M<+\inftyUx∈D¯‖Q(k)(x)‖≤M<+∞
where θ i ( i = 1 , 2 , … k ) θ i ( i = 1 , 2 , … k ) theta_(i)(i=1,2,dots k)\theta_{i}(i=1,2, \ldots k)Ii(i=1,2,…k) are the operators i i iii - null linear.
b) For any x ∈ D , Q ( x ) ∈ D x ∈ D , Q ( x ) ∈ D x in D,Q(x)in Dx \in D, Q(x) \in Dx∈D,Q(x)∈D Yes Q ( x ¯ ) = x ¯ Q ( x ¯ ) = x ¯ Q( bar(x))= bar(x)Q(\bar{x})=\bar{x}Q(x¯)=x¯.
c) Operator [ x ¯ , x ; P ] [ x ¯ , x ; P ] [ bar(x),x;P][\bar{x}, x ; P][x¯,x;P] exists and admits the reverse for everything x ∈ D x ∈ D x in Dx \in Dx∈D.
d) Operators [ x ¯ , x ; P ] s i [ x ¯ , x ; P ] − 1 [ x ¯ , x ; P ] s i [ x ¯ , x ; P ] − 1 [ bar(x),x;P]si[ bar(x),x;P]^(-1)[\bar{x}, x ; P] s i[\bar{x}, x ; P]^{-1}[x¯,x;P]si[x¯,x;P]−1 they are limited, that is, there are two real and non-negative constants N N NNN Shi E E EEAnd for which ‖ [ x ¯ , x ; P ] ‖ ≤ N ‖ [ x ¯ , x ; P ] ‖ ≤ N ||[ bar(x),x;P]|| <= N\|[\bar{x}, x ; P]\| \leq N‖[x¯,x;P]‖≤N, ‖ [ x ¯ , x ; P ] − 1 ‖ ≦ E [ x ¯ , x ; P ] − 1 ≦ E ||[( bar(x)),x;P]^(-1)|| <= E\left\|[\bar{x}, x ; P]^{-1}\right\| \leqq E‖[x¯,x;P]−1‖≦And, for anything x ∈ D x ∈ D x in Dx \in Dx∈D.
Then Q ∈ I k D Q ∈ I k D Q inI_(k)^(D)Q \in I_{k}^{D}Q∈IkD.
Demonstration. From the definition of the divided difference it follows
P ( Q ( x ) ) = P ( Q ( x ) ) − P ( x ¯ ) = P ( Q ( x ) ) − P ( Q ( x ¯ ) ) = P ( Q ( x ) ) = P ( Q ( x ) ) − P ( x ¯ ) = P ( Q ( x ) ) − P ( Q ( x ¯ ) ) = P(Q(x))=P(Q(x))-P( bar(x))=P(Q(x))-P(Q( bar(x)))=P(Q(x))=P(Q(x))-P(\bar{x})=P(Q(x))-P(Q(\bar{x}))=P(Q(x))=P(Q(x))−P(x¯)=P(Q(x))−P(Q(x¯))=
= [ Q ( x ¯ ) , Q ( x ) ; P ] × [ Q ( x ) − Q ( x ¯ ) ] = [ Q ( x ¯ ) , Q ( x ) ; P ] × [ Q ( x ) − Q ( x ¯ ) ] =[Q( bar(x)),Q(x);P]xx[Q(x)-Q( bar(x))]=[Q(\bar{x}), Q(x) ; P] \times[Q(x)-Q(\bar{x})]=[Q(x¯),Q(x);P]×[Q(x)−Q(x¯)] where, taking into account d), we have
(11) ‖ P ( Q ( x ) ) ‖ = N . ‖ Q ( x ) − Q ( x ) ‖ , pentru orice x ∈ D . (11) ‖ P ( Q ( x ) ) ‖ = N . ‖ Q ( x ) − Q ( x ) ‖ ,  pentru orice  x ∈ D . {:(11)||P(Q(x))||=N.quad||Q(x)-Q(x)||","" pentru orice "x in D.:}\begin{equation*} \|P(Q(x))\|=N . \quad\|Q(x)-Q(x)\|, \text { pentru orice } x \in D . \tag{11} \end{equation*}(11)‖P(Q(x))‖=N.‖Q(x)−Q(x)‖, for anything x∈D.
But hypothesis a) leads us to the following inequality
(12) ‖ Q ( x ) − Q ( x ¯ ) ‖ ≦ M k ! ‖ x − x ¯ ‖ k , pentru orice x ∈ D (12) ‖ Q ( x ) − Q ( x ¯ ) ‖ ≦ M k ! ‖ x − x ¯ ‖ k ,  pentru orice  x ∈ D {:(12)||Q(x)-Q( bar(x))|| <= (M)/(k!)||x- bar(x)||^(k)","" pentru orice "x in D:}\begin{equation*} \|Q(x)-Q(\bar{x})\| \leqq \frac{M}{k!}\|x-\bar{x}\|^{k}, \text { pentru orice } x \in D \tag{12} \end{equation*}(12)‖Q(x)−Q(x¯)‖≦Mk!‖x−x¯‖k, for anything x∈D
From the fact that P ( x ¯ ) = θ P ( x ¯ ) = θ P( bar(x))=thetaP(\bar{x})=\thetaP(x¯)=I Get
P ( x ) = [ x ¯ , x ; P ] ( x − x ¯ ) P ( x ) = [ x ¯ , x ; P ] ( x − x ¯ ) P(x)=[ bar(x),x;P](x- bar(x))P(x)=[\bar{x}, x ; P](x-\bar{x})P(x)=[x¯,x;P](x−x¯)
that gives us
‖ x − x ¯ ‖ ≦ E ‖ P ( x ) ‖ , pentru orice x ∈ D ‖ x − x ¯ ‖ ≦ E ‖ P ( x ) ‖ ,  pentru orice  x ∈ D ||x- bar(x)|| <= E||P(x)||," pentru orice "x in D\|x-\bar{x}\| \leqq E\|P(x)\|, \text { pentru orice } x \in D‖x−x¯‖≦And‖P(x)‖, for anything x∈D
From the last inequality and from (11) and (12) we deduce
‖ P ( Q ( x ) ) ‖ ≦ M E k ⋅ N k ! ‖ P ( x ) ‖ k ‖ P ( Q ( x ) ) ‖ ≦ M E k ⋅ N k ! ‖ P ( x ) ‖ k ||P(Q(x))|| <= (ME^(k)*N)/(k!)||P(x)||^(k)\|P(Q(x))\| \leqq \frac{M E^{k} \cdot N}{k!}\|P(x)\|^{k}‖P(Q(x))‖≦MAndk⋅Nk!‖P(x)‖k
what had to be shown.
From the above theorem it follows that the conditions imposed by us in the definition of the order of convergence for an iterative operator are at least as general as the conditions in hypotheses a) and b) which are essential to the proof of theorem 5. On the other hand, the condition imposed by us in definition 2 does not require the existence of the Fréchet derivative and it can also be applied when it is replaced by the divided difference as follows from theorem 3. In this way, definition 2 allows us to classify 0 as a broader class of iterative operators.
Received at the editorial office on June 23, 1970
Academy of the Socialist Republic of Romania
Cluj
Branch Institute of Computing.

ON ITERATIVE OPERATORS

(SUMMARY)
We define the notion of iterative ọperator and the order of an iterative operator. With the help of these concepts, the following problems are studied: a) The characterization of the order of an operator which arises from the juxtaposition of two or more operators; b) Methods of constructing iterative order operators p + 1 p + 1 p+1p+1p+1 and 2 p 2 p 2p2 p2p with the help of order operators p ; c ) p ; c ) p;c)p ; \mathrm{c})p;c) Sufficient conditions that must be fulfilled by a given operator for it to be an iterative operator attached to an operational equation that must admit a given order.

BIBLIOGRAPHY

  1. Jankó, Béla, On the Solution of Operational Equations. (Ph.D. thesis, Cluj, 1966).
  2. Păvăloru, Ion, On Steffensen's Method for Solving Nonlinear Operational Equations. Rev. Roum. de math. pures et appl., 13, 6 (1968), 857-861.
  3. Păvăloiu, Ion, Interpolation in norm linear spaces and applications. Mathematica, Cluj, (sub tipar).
  4. Traub, F. j, Sterative Methods for the solution of Equations. Pretince - HolI, Inc., Englewood cliffs. N. j (1964).
  5. Ul'm, S., Obobscenie metoda Steffensena dlea reşenia nelineinth operotornih uravnenii. Jurnal vicise. mat. i mat-fiz., 4, 6 (1964), 1093-1097.
1971

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