




































©2025 Ada Academica https://adac.eeEur. J. Math. Anal. 5 (2025) 5doi: 10.28924/ada/ma.5.5
Hybrid Iterative Methods for Solving Nonlinear Equations in Banach Spaces

Ioannis K. Argyros1,∗, Santhosh George2, Samundra Regmi3, Michael I. Argyros4
1Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

iargyros@cameron.edu
2Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka,

India-575 025
sgeorge@nitk.edu.in

3Department of Mathematics, University of Houston, Houston, TX 77204, USA
sregmi5@uh.edu

4Department of Computer Science, University of Oklahoma, Norman, OK 73501, USA

argyro01@email.franklin.edu

Abstract. The present article contributes to the solution of equations which carry the symmetryproperty of the problem or not. Iterative methods with in- verses generate sequences convergingfaster to a solution of an equation than methods without inverses. However, the implementationof these methods has drawbacks, since the analytical form of these inverse may be unavailable orcomputationally very expensive. This problem is addressed in this paper by replacing the inversewith a finite sum of linear opera- tors. A convergence analysis is developed for the hybrid methods.The numerical examples demonstrate that the number of iterates is essentially the same between thehybrid and the original method. This technique is also extended to solve generalized equations.

1. Introduction
The letters X, Y denote Banach spaces; Ω ⊂ X is a convex and open subset of X , and F1 : Ω −→

Y stands for a continuous operator. Numerous applications from diverse areas of computationalscience and Engineering can be converted by using mathematical modelling [3, 8, 14, 17,19–21,23,26,28,33,35] to finding a solution s∗ ∈ Ω of the nonlinear in the general equation
F1(x) = 0. (1.1)

The closed form of the solution s∗ is attainable only in special cases. This forces researchers andpractitioners to solve the equation (1.1) iteratively. Single-step methods of high convergence order
Received: 3 Jul 2024.
Key words and phrases. Inverse of an operator, Banach space, Hybrid iterative method, Generalized equations,Continuous operator, Convergence. 1

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 2look like the Newton-type defined for each n = 0, 1, 2, ... by
x0 ∈ Ω, xn+1 = xn − L−1n F1(xn), (1.2)

where Ln ∈ L(X, Y ) which is the space of continuous operators mapping X into Y , and L−1n ∈
L(Y,X) for each n = 0, 1, 2, .... By Ln, we denote L(xn). Some choices for the operator Ln can be

Ln = F ′1(xn) (Newton’s method),

Ln = [xn − F1(xn), xn + F1(xn);F ] (Steffensen’s method),

Ln = I (The Picard method), the identity operator.
Here F ′1, [., .;F1] denote Fréchet-derivative and divided differences of order one for the operator
F1, respectively [22,25].Many other choices are possible. It turns out that the inverse of the operator Ln is costly orimpossible to find in general. This concern with the implementation of these methods constitutesthe motivation for this paper. Our idea is to replace the inverse with a finite sum of linear operatorsconverging to it. The reasoning is explained as follows. Let p ∈ N be fixed.Suppose there exists Γ ∈ L(X, Y ) such that Γ−1 ∈ L(Y,X) and for A = A(x) = Γ−1(Γ− L(x))the operator I−A(x) is also invertible, i.e. (I−A(x))−1 ∈ L(Y,X). In this case, the Newton-typemethod can read as

x0 ∈ D, xn+1 = xn − (I − A)−1Γ−1F1(xn). (1.3)
Note that we have

(I − A)−1Γ−1 = [Γ(I − A)]−1 = L−1n . (1.4)
However, even if the linear operator Γ−1 is known it is still required to find the inverse of (I −A),which is not a fixed operator (in general). But what if we replace this operator with M = Mp(x) =

I + A+ ...+ Ap . Then, method (1.3) can be written as
x0 ∈ Ω, xn+1 = xn −MΓ−1F1(xn). (1.5)

It is clear that (1.5) is a useful alternative for (1.3) because of (1.4). By letting p −→ +∞, weget limp→+∞Mp = L−1n if the limit exists. The condition ‖A‖ < 1 for each x ∈ Ω assures theexistence of such a limit. If the linear operator M is invertible and the sequence {xn} given by(1.5) converges to some s∗, then by (1.5) we get
M−1(xn − xn+1) = Γ−1F1(xn)

leading to
0 = lim

n→+∞
M−1(xn − xn+1) = lim

n→+∞
Γ−1F1(xn),

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 3i.e. F1(s
∗) = 0. Thus, the point s∗ solves the equation (1.1). The same reasoning leads for

A1 = (Γ− L(x))Γ−1 and M1 = I + A1 + ...+ Ap1 to the method
x0 ∈ Ω , xn+1 = xn −M1Γ−1F1(xn). (1.6)

It is clear that the study of the convergence of the method (1.6) is analogous to (1.5). That is whywe study only method (1.5) in Section 2. We deal with two kinds of convergence: the semi-localand the local. The first utilizes knowledge in a neighborhood of x0 and develops estimates relatedto ‖xn+1− xn‖ and ‖s∗− xn‖, and the convergence conditions assure that limn→+∞ xn = s∗. In thesecond kind, knowledge about a neighborhood of s∗ is used to provide the same estimates as inthe semi-local kind and again limn→+∞ xn = s∗. It is worth noting that the iterates generated by(1.3),(1.5) and (1.6) are not the same in general. But we use the same notation for simplicity. Theconvergence for both kinds relies on generalized continuity conditions controlling the operatorsinvolved [5, 6, 9, 18]. In particular, our semi-local convergence analysis depends on the usage ofmajorizing sequences [29,30,34]. Notice that the method (1.5) can also be written for Dn = ΓM−1as
x0 ∈ Ω, F1(xn) +Dn(xn+1 − xn) = 0. (1.7)

In section 3 we also use the developed methodology for solving nonlinear equations to solvegeneralized equations. That is find x ∈ X such that
F1(x) + F2(x) 3 0. (1.8)

Here F2 : X ⇒ Y is a set-valued operator mapping X into Y with closed graph [1–4, 13, 17–19, 21–23, 26, 28, 32, 35] and operator F is as previously defined. A plethora of applications frommathematical programming, variational inequalities, optimal control, or constrained systems arewritten in the form (1.8). There is extensive literature on iterative methods solving the generalizedequation (1.8) [1–4,13,17–19,21–23,26,28,32,35]. Notice that the method used in the literature tosolve (1.8) is defined by
F1(xn) + D̄n(xn+1 − xn) + F2(xn+1) 3 0, (1.9)

where D̄n is a linear operator. It can be chosen as D̄n = Ln, D̄n = F ′1(xn) or D̄n ∈ ∂F1(xn) or otherchoices [9, 11, 23, 24]. These methods have the same problems as the ones for solving nonlinearequations. That is why it is justified to consider the analog of (1.7) defined by
F1(xn) +Dn(xn+1 − xn) + F2(xn+1) 3 0 (1.10)

The semi-local and local convergence of the method (1.10) is developed in section 3 in an analogousway to section 2 for the method (1.5) or (1.7). In numerical section 4, the examples demonstratethat the number of iterations of the hybrid methods to arrive at a predetermined error tolerance

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 4is essentially the same as with the original methods. Moreover, the convergence order is also thesame.In order to achieve all this we redevelop some standard terminology to make the paper as self-contained as possible. More details can be found in [8, 11, 19]. Let S(z, ρ) and its closure S[z, ρ]denote open and closed balls, respectively of center z ∈ X and radius ρ > 0. Let C be a set in X .Define the distance for x ∈ X to C by dist(x, C) = infx∈C ‖x − y‖. The generalized set-valuedoperator G relates with its graph given by gph(G) = {(x, y) ∈ X × Y, y ∈ F2(x)}, and its domain
dom(G) = {x ∈ X|F2(x) 6= 0}. The inverse of G is given as G−1(y) = {x ∈ X, y ∈ F2(x)}. Notethat a set-valued operator H : X ⇒ Y is said to be metrically regular at x0 for y0 if y0 ∈ H(x0)and there exists neighbourhoods V1 of x0 and V2 of y0 and β > 0 such that gph(H ∩ (V1 × V2)) isclosed and for each (x, y) ∈ V1 × V2

dist(x,H−1(y)) ≤ βdist(y ,H(x)). (1.11)
The regularity modulus of H at x0 for y0 is the infimum over all β > 0 and is denoted by
reg(H; x0/y0). Additionally if the operator ∆ : V2 → y → H−1(y) ∩ V1 is not multivalued on
V2, then we say that H is strongly metrically regular. In this case, ∆ is Lipchitz continuous on V2.Finally, Section 5 contains concluding remarks and directions for research.

2. Convergence for the Method (1.5)
We start with the study of the semi-local analysis in this section. Some auxiliary results anddefinitions are useful.

Lemma 2.1. (Banach Lemma on Invertible Operators)( [14, 22, 30, 34]) If P is a bounded linear
operator in X , P−1 exists if and only if there is a bounded linear operator P1 in X such that P−11
exists and

‖I − P1P‖ < 1.

If P−1 exists, then

P−1 =

∞∑
n=0

(I − P1P )nP1

and

‖P−1‖ ≤
‖P1‖

1− ‖I − P1P‖
.

Further, we use majorizing sequences to prove the semi-local convergence. Recall the definitionof majorizing sequence.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 5

Definition 2.1. ( [14,22,30,34]) Let {xn} be a sequence in a normed space X . Then a nonnegative
scalar sequence {un} for which

‖xn+1 − xn‖ ≤ un+1 − un ∀n ≥ 0 (2.1)
holds, is a majorizing sequence for {xn}. Note that any majorizing sequence is necessarily
nondecreasing. Moreover, if the sequence {un} converges, then {xn} converges too, and for
u∗ = limn−→∞ un

‖s∗ − xn‖ ≤ u∗ − un.

Hence, the study of the convergence of the sequence {xn} reduces to that of {un}.

The analysis requires some conditions. Let E = [0,+∞).Suppose(H1) There exists a function φ : E × E × E → [0,+∞) continuous as well nondecreasing in allthree variables and invertible operators M(.) and Γ such that for some x0 ∈ Ω, and each
x, y ∈ Ω the following Mysovskii-like condition holds

‖M(x)Γ−1(F1(y)− F1(x)− ΓM−1(x))‖

≤ φ(‖x − x0‖, ‖y − x0‖, ‖y − x‖)‖y − x‖

Define the real real sequence {αn} for α0 = 0, α1 ≥ η := ‖M(x0)Γ−1F1(x0)‖ and each
n = 0, 1, 2, ... by

αn+1 = αn + φ(αn−1, αn, αn − αn−1)(αn − αn−1), n = 1, 2, ... (2.2)
Notice that the constant η is well defined since the operator Γ is invertible. Moreover, thesequence {αn} defined by the formula (2.2) is proven to be majorizing for the method (1.5)in Theorem 2.3. But let us present convergence conditions for it.(H2) There exists a parameter ρ ≥ η such that for each n = 0, 1, 2, ...

φ(αn−1, αn, αn − αn−1) < 1 and αn ≤ ρ.
It follows by the condition (H2) and the formula (2.2) that

0 ≤ αn−1 ≤ αn ≤ ρ

and there exists α∗ ∈ [η, ρ] such that limn→+∞ αn = α∗. H is well known that this limit isthe unique least upper bound of the sequence {αn} and(H3) S[x0, α
∗] ⊂ Ω.The conditions (H1)-(H3) combined with the terminology are utilized for the convergence of themethod (1.5).

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 6

Theorem 2.1. Suppose that the conditions (H1)-(H3) hold. Then, the sequence {xn} generated by
the method (1.5) exists in the ball S(x0, α

∗) remains in the same ball for each n = 0, 1, 2, ... and
is convergent to a unique solution s∗ ∈ S[x0, α

∗] of the equation F1(x) = 0 such that

‖s∗ − xn‖ ≤ α∗ − αn, n = 0, 1, 2 . . . . (2.3)
Proof. The process of induction is employed to show the assertion

‖xm+1 − xm‖ ≤ αm+1 − αm for m = 0, 1, 2, . . . . (2.4)
The choice of η, (2.2) and the method (1.5) give that

‖x1 − x0‖ = ‖M(x0)Γ−1F1(x0)‖ ≤ η = α1 − α0 < α∗,

so the assertion (2.4) holds if m = 0, and the iterate x1 ∈ S(x0, α
∗). Suppose iterates x0, x1, ..., xmexist and (2.4) holds for all integers smaller or equal to m− 1. Notice that the iterate xm+1 existsby the method (1.5) and the invertability of the operators M(xm) and Γ. Then, we can write by themethod (1.5) the Ostrowski-type representation for F1(xm) as

F1(xm) = F1(xm)− F1(xm−1)− ΓM−1(xm − xm−1). (2.5)
Using the condition (H1), method (1.5), (2.2) and the induction hypothesis on (2.5) we obtain inturn that

‖xm+1 − xm‖ = ‖MΓ−1(F1(xm))‖

= ‖MΓ−1(F1(xm)− F1(xm−1)− ΓM−1(xm − xm−1))‖

≤ φ(‖xm−1 − x0‖, ‖xm − x0‖, ‖xm − xm−1‖)‖xm − xm−1‖

≤ φ(αm−1, αm, αm − αm−1)(αm − αm−1)

= αm+1 − αm, (2.6)
and

‖xm+1 − x0‖ ≤ ‖xm+1 − xm‖+ ‖xm − xm−1‖+ ...+ ‖x1 − x0‖

≤ αm+1 − αm + αm − αm−1 + ...+ α1 − α0

= αm+1 < α∗,

which complete the induction for the assertion (2.4) and show that all iterates {xm+1} ⊂ S(x0, α
∗).But the sequence {αm} is complete by the condition (H2) as convergent. It follows by the estimate(2.4) that {xm} is also complete. But the space X is Banach, so there exists s∗ ∈ S[x0, α

∗] such that
lim

m→+∞xm = s∗. Next, by letting m → +∞ in (2.6), the invertibility of the operators M(.),Γ and

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 7the continuity of the operator F , we deduce that limm→+∞MΓ−1F1(s
∗) = 0. By the invertibilityof M

0 = M−1(0) = M−1( lim
m→+∞

MΓ−1F1(xm)) = lim
m→+∞

MM−1Γ−1F1(xm)

= lim
m→+∞

Γ−1F1(xm) = Γ−1 lim
m→+∞

F1(xm) = Γ−1F1(s
∗).

So, F1(s∗) = 0, since Γ(0) = 0. Let i = 0, 1, 2, .... Then, the triangle inequality and (2.4) give
‖xm+i − xm‖ ≤ αm+i − αm. (2.7)

Hence, by letting i → +∞ in (2.7) we prove (2.3). Finally, to show the uniqueness part, let
w ∈ S[x0, α

∗] with F1(w) = 0 and w 6= s∗. By using the conditions (H1), (H2) we can write inturn that
‖w − s∗‖ = ‖(MΓ−1)(ΓM−1)(w − s∗)‖

= ‖MΓ−1(F1(w)− F1(s∗)− ΓM−1(w − s∗))‖

≤ φ(‖s∗ − x0‖, ‖w − x0‖, ‖w − s∗‖)‖w − s∗‖

≤ φ(α∗, α∗, ‖w − s∗‖)‖w − s∗‖ < ‖w − s∗‖,

which gives a contradiction. Therefore, we conclude that w = s∗. �

Remark 2.1. The condition Mysoskii-type [22] condition in (H1) can be replaced as follows:

(H1)
′ With operator Γ as in condition (H1), suppose that there exists a ∈ (0, 12) such that ‖A‖ < a

and for each x, y ∈ Ω

‖Γ−1(F1(y)− F1(x)− ΓM−1(y − x))‖

≤ φ1(‖x − x0‖, ‖y − x0‖, ‖y − x‖)‖y − x‖,

where the function φ1 is as the function φ. Then, the condition (H1)
′ implies (H1) if we

take φ = a0φ1, where a0 = 1
1−a .

This is the case, since by the definition of the operator M , we have the estimate

‖M‖ = ‖I + A+ ...+ Ap‖

≤ 1 + a + ...+ ap

=
1− ap+1

1− a = a0 <
1

1− a . (2.8)
In this case the invertibility of the operator M is implied by the Banach Lemma 2.1, since

‖I −M‖ ≤ ‖A‖+ ...+ ‖A‖p ≤ a + ...+ ap = a
1− ap

1− a <
a

1− a < 1.

So, M is invertible and

‖M−1‖ ≤ b =
1− a

1− 2a
.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 8

Next, we develop the local convergence. The role of the initial point x0 is exchanged by s∗ and
the function φ by ψ.
Suppose:

(H4) There exists a function ψ : E −→ [0,+∞) continuous and nondecreasing such that the
equation ψ(t)− 1 = 0 has a smallest positive solution denoted by r .

(H5) There exists a solution s∗ ∈ Ω, and invertible linear operators M,Γ such that for each x ∈ Ω

‖MΓ−1(F1(x)− F1(s∗)− ΓM−1(x − s∗))‖

≤ ψ(‖x − s∗‖)‖x − s∗‖

and
(H6) S[s∗, r ] ⊂ Ω.

Next, the constant r is shown to be a radius of convergence for the method (1.5).
Theorem 2.2. Suppose that the conditions (H4) − (H6) hold. Then, the sequence {xn} for x0 ∈
S(s∗, r)− {s∗} exists in S(s∗, r), stays in S(s∗, r) and converges to s∗ so that

‖xn+1 − s∗‖ ≤ ψ(‖xn − s∗‖)‖xn − s∗‖ ≤ ‖xn − s∗‖ < r. (2.9)
Moreover, x∗ is the only solution of the equation F (x) =0 in the ball U(s∗, r).

Proof. The iterates x1, x2, ..., xm+1 are well defined by the method (1.5), and we can write in turnthat
xm+1 − s∗ = xm − s∗ −MΓ−1F1(xm)

= MΓ−1(F1(xm)− F1(s∗)− ΓM−1(xm − s∗)). (2.10)
It follows by the conditions (H4), (H5) and (2.10) that

‖xm+1 − s∗‖ ≤ ψ(‖xm − s∗‖)‖xm − s∗‖

≤ ξ‖xm − s∗‖ ≤ ξm+1‖x0 − s∗‖ < r, (2.11)
where ξ = ψ(‖x0 − s∗‖) ∈ [0, 1) showing the assertion (2.9) for each m = 1, 2, ..., since x0 ∈
S(s∗, r) − {s∗}. By letting m → +∞ in (2.11), we deduce that lim

m→+∞xm. In order to show theuniqueness part, suppose there exists a solution w1 ∈ S(s∗, r) such that w1 6= s∗. Then, as in thesemi-local case, we can write in turn
‖w1 − s∗‖ = ‖(MΓ−1)(ΓM−1(w1 − s∗))‖

= ‖MΓ−1(F1(w1)− F1(s∗)− ΓM−1(w1 − s∗))‖

≤ ψ(‖w1 − s∗‖)‖w1 − s∗‖ < ‖w1 − s∗‖ (2.12)
by the choice of r . Hence, we conclude that w1 = s∗, since (2.12) contradicts the hypothesis
w1 6= s∗. �

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 9

Remark 2.2. Comments similar to the ones in Remark 2.4 can be made provided that x0, ψ are
exchanged by s∗, ψ, respectively. It is also worth noting that the condition (H4) does not neces-
sarily imply the usual condition in local convergence studies that the operator F ′1(s∗) is invertible,
i.e. that s∗ is a simple solution of the equation F1(x) = 0. Consequently the method (1.5) can be
applied to find solutions of multiplicity greater than one.

3. Convergence for the Method (1.10)
Let λ > 0, µ > 0, δ ≥ 0 and β > 0 be given parameters.Suppose:(C1) There exists a function

ϕ1 : [0, λ]× [0, λ]× [0, λ] −→ [0, δ]

which is continuous and nondecreasing. Define the scalar sequence {hn} for h0 = 0, some
h1 ≥ 0, β1 > β and each n = 1, 2, ... by

hn+1 = hn + β1ϕ1(hn−1, hn, hn − hn−1)(hn − hn−1). (3.1)
The scalar sequence {hn} is shown to be majorizing for {xn} is generated by the formulain the Theorem 3.2. However, let us present a convergence criterion for it.(C2) There exists λ0 ∈ [0, j ] such that for each n = 0, 1, 2, ... hn ≤ λ0.It follows by this condition and (3.1) that 0 ≤ hn−1 ≤ hn ≤ λ0 and there exists h∗ ∈ [0, λ0]such that l imn→∞ = h∗.The limit point h∗ is the unique least upper bound of the sequence {hn}.(C3) There exists x0 ∈ X and y0 ∈ F1(x0) + F2(x0) such that
βδ < 1 and ‖y0‖ ≤ (1− βδ) min{λβ , µ}.Choose h1 ≤ β1‖y0‖.(C4) The operator

x −→ QDn(x) := F1(x0) +Dn(x − x0) + F2(x) (3.2)
is metrically regular at x0 for y0 with constant β and neighborhoods S(x0, λ) and S(y0, µ),respectively.The mapping ϕ1 relates to the operators on the method (1.10).(C5)
‖F1(x)− F1(xn)−Dn(x − xn)‖ ≤ ϕ1(‖x − x0‖, ‖xn − x0‖, ‖x − xn‖)‖x − xn‖,

for each x ∈ S(x0, λ).Next, the semi-local analysis of convergence is developed using the conditions (C1)− (C5).

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 10

Theorem 3.1. Suppose that the conditions (C1)−(C5) hold. Then, for each γ ∈ (βδ, 1) there
exists a sequence {xn} generated by the method (1.10) which is well defined in S(x0, λ),

remains in S(x0, λ) for each n = 0, 1, 2, ... and convergence to a solution s∗ ∈ S[x0, λ] of
the generated equation (1.8). Moreover, the following assertion hold

‖s∗ − xn‖ ≤ γnλ (3.3)
and

dist(0, F1(xn) + F2(xn)) ≤ γn‖y0‖ (3.4)
for each n = 0, 1, 2, ...

Thus, the convergence rate is R-linear. Furthermore, If the operator QDn is stronglymetrically regular with constant β and neighbourhoods S(x0, λ) and S(y0, µ), respectively,then the sequence {xn} is the only one satisfying (1.10), and staying in S(x0, λ).
Remark 3.1. The proof is similar to the one in [11, Theorem 2.2]. But it uses (1.10), (C4)

and (C5), instead of stronger (1.9),(C4)’ x −→ QAn(x) := F1(x0) + An(x − x0) + F2(x),(C5)’ ‖F1(x)− F1(xn)− Ln(x − xn)‖ ≤ w(‖x − xn‖)‖x − xn‖
for each x ∈ S(x0, λ), where w : [0.λ] −→ [0, δ] satisfies lim

n→+∞w(t) = 0

Proof. Pick γ ∈ (βδ, 1) and β1 so that β < β1 ≤ γ
δ and

‖y0‖ ≤ (1− γ)min

{
λ

β1
, µ

}
. (3.5)

The choice of β1 is certainly possible since there are infinitely many numbers between βand γ
δ . We shall show the existence of the sequence {xn} using mathematical induction for

n = 1, 2, ... satisfying(In) ‖xn − x0‖ ≤ 1−γn
1−γ β1‖y0‖ ≤ (1− γn)λ < λ.(IIn) ‖xn − xn−1‖ ≤ hn − hn−1.(IIIn) 0 ∈ F1(xn−1) +Dn−1(xn − xn−1) + F2(xn),where Dn−1 = Dn−1(x0, ..., xn−1).By hypothesis 0 ∈ S(y0, µ) and y0 ∈ QD0(x0). Using the condition (C4) for QD0 we get
dist(x0, Q

−1
D0

(0)) ≤ β dist(0, QD0(x0)) ≤ β‖y0‖.

If y0 = 0, pick x1 = x0. Otherwise, It follows that
dist(x0, Q

−1
D0

(0)) ≤ β1‖y0‖.

Thus, there exists x1 ∈ Q−1D0 (0) satisfying
‖x1 − x0‖ < β1‖y0‖ < (1− γ)λ.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 11So, (I1), (II1) and (III1) hold.Suppose that for some natural number m the element xm is defined so that the inductionhypothesis (Im), (IIm) and (IIIm) hold. We shall show that the iterate xm+1 is defined anddthe assertions (Im+1), (IIm+1) and (IIIm+1) hold. It follows by (Im) that xm ∈ S(x0, λ).Set em = F1(x0)− F1(xm)−Dm(x0 − xm).By (C5) for x = x0, (Im), (IIm) and the properties of the function ϕ1 we can write
‖em − y0‖ ≤ ‖y0‖+ ‖F1(x0)− F1(xm)−Dm(x0 − xm)‖

≤ ‖y0‖+ ϕ1(‖x0 − x0‖, ‖xm − x0‖, ‖xm − x0‖)‖x0 − xm‖

≤ ‖y0‖+ ϕ1(0, hm, hm)‖x0 − xm‖

≤ ‖y0‖+ δ‖x0 − xm‖ ≤ ‖y0‖+
1− γm

1− γ β‖y0‖

≤ (1 +
1− γm

1− γ γ)‖y0‖ =
1− γm+1

1− γ ‖y0‖ < µ.

If em ∈ QDm(xm), set xm+1 = xm. Otherwise by the condition (C4) we have
dist(xm, Q

−1
Dm

(em)) ≤ β dist(em, QDm(xn))

≤ β1 dist(em, QDm(xm)).

Next, there exists an element xm+1 ∈ Q−1Dm(em) satisfying
‖xm+1 − xm‖ ≤ β1 dist(em, QDm(xm)).

By (IIIm) it follows
QDm(xm) = F1(x0) +Dm(xm − x0) + F2(xm)

3 F1(x0) +Dm(xm − x0)− F1(xm−1 −Dm−1(xm − xm−1).

In view of the condition (C4) with x = xm, (3.5) and (IIm), we obtain in turn that
‖xm+1 − xm‖ ≤ β1

∥∥em − [F1(x0)− F1(xm−1) +Dm(xm − x0)

−Dm−1(xm − xm−1)
]∥∥

= β1‖F1(xm)− F1(xm−1)−Dm−1(xm − xm−1)‖

≤ β1ϕ1(‖xm−1 − x0‖, ‖xm − x0‖, ‖xm − xm−1‖

≤ β1ϕ1(hm−1, hm, hm − hm−1) = hm+1 − hm, (3.6)
and

‖xm+1 − xm‖ ≤ β1δ‖xm − xm−1‖ = γ‖xm − xm−1‖ ≤ γm‖x1 − x0‖

≤ γmβ1‖y0‖.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 12Thus, the condition (IIIm) holds if m + 1 replaces m. Moreover, by the selection of xm+1,we get
em ∈ QDm(xm+1) = F1(x0) +Dm(xm+1− − x0) + F2(xm+1).

So, the assertion (IIIn) holds if m + 1 replaces m. Moreover, by (Im) we get in turn that
‖xm+1 − x0‖ ≤ ‖xm+1 − xm‖+ ‖xm − x0‖

≤ γmβ1‖y0‖+
1− γm

1− γ β1‖y0‖

=
1− γm+1

1− γ β1‖y0‖,

which terminates the induction for (Im). Then, by (C2) the sequence is complete as con-vergent. It follows by (IIm) that the sequence {xm} is also complete in a Banach space Xand as such it converges to some s∗. By (Im) we have s∗ ∈ S(x0, λ). We must show thatthe limit point s∗ solves (1.8).Set ym := F1(xm) − F1(xm−1) − Dm−1(xm − xm−1). It follows by (IIIm) that ym ∈
F1(xm) + F2(xm). By using (IIm) and the condition (C4) for x = xm, we get in turnthat

‖ym‖ = ‖F1(xm)− F1(xm−1)−Dm−1(xm − xm−1)‖

≤ ψ1(‖xm−1 − x0‖, ‖xm − x0‖, ‖xm − xm−1‖)‖xm − xm−1‖

≤ ψ1(hm−1, hm, hm − hm−1)(αm − αm−1) = αm+1 − αm −→ 0

as m → +∞.Consequently, we deduce that (xm, ym)→ (s∗, 0) as m → +∞.But F is continuous whereas G has a closed graph. Thus, we conclude 0 ∈ F1(s∗) +F2(s
∗).Finally, If QDm is a strongly metrically regular operator. It follows that the iterate xm+1 isunique and is obtained from xm. �

Next, we develop the local convergence analysis of the method (1.10).Suppose :(C6) There exists a parameter β > 1 and a function ϕ2 : R+ → R+ which is continuous andnondecreasing such that the equation β1ϕ2(t) − 1 = 0 has a smallest positive solution.Denote such a solution by r0.(C7) There exists a solution s∗ ∈ X of the generalized equation (1.8) and parameters β > 0 and
δ ≥ 0 such that βδ < 1.(C8) The operator x −→ TDn(x) := F1(s

∗) + Dn(x − s∗) + F2(x) is a metrically regular at s∗for 0 with constant β and neighbourhoods S(s∗, λ) and (0, µ) for some µ > 0, respectively.The function ϕ2 relates to the operators on the method (1.10).

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 13(C9)
‖F1(s∗)− F1(xn)−Dn(s∗ − xn)‖ ≤ ϕ2(‖s∗ − xn‖)‖s∗ − xn‖ (3.7)

for x0 ∈ S(s∗, λ). Next, the local convergence analysis of the method (1.10) is presentedusing the conditions (C6)− (C4).

Theorem 3.2. Suppose that the conditions (C6)− (C9) hold. Then, for each γ ∈ (βδ, 1) the
re exists a sequence {xn} generated by the method (1.10) which is well defined in S(s∗, λ),

remains in S(s∗, λ) for each n = 0, 1, 2, .. and converges to s∗ so that

‖s∗ − xn‖ ≤ d‖s∗ − xn−1‖ ≤ dn‖s∗ − x0‖ < λ,

where d = β1ϕ2(‖s∗−x0)‖ ∈ [0, 1). Additionally, If the operator TDn is strongly metrically
regular with constant β and neighbourhoods S(s∗, λ) and S(0, µ), respectively, then the
sequence {xn} is the only one satisfying (1.8), and satisfying in (s∗, λ).

Proof. Simply follow the proof of Theorem 3.1 for γ ∈ (βδ, 1)x0 = s∗ and β < β1 ≤ γ
δ toobtain as in (3.6) but using (C8) and (3.7) instead of (C4) and (3.6), respectively to obtain

‖s∗ − xm‖ ≤ β1ϕ2(‖s∗ − xm−1‖)‖s∗ − xm−1‖

≤ d‖s∗ − xm−1‖ ≤ ... ≤ dm‖s∗ − x0‖ < λ.

Therefore, we conclude that lim
m−→+∞xm = s∗ and the iterate xm ∈ S(s∗, λ).Finally, if the operator TDm is strongly metrically regular, it follows that the iterate xmis unique in S(s∗, λ) by the way the iterate xm is derived from xm−1. �

4. Numerical Examples
The examples use Ln = F ′1(xn), Γ = I which is independent of x0 and s∗.

Example 4.1. The solution sought for the nonlinear system

f1 = x − 0.1 sin x − 0.3 cos y + 0.4

f2 = y − 0.2 cos x + 0.1 sin y + 0.3

Let F1 = (f1, f2). Then, the system becomes

F1(s) = 0 f or s = (x, y)T .

Then

F ′1((x, y)) =

[
1− 0.1 cos(x) 0.3 sin(y)

0.2 sin(x) 0.1 cos(y) + 1

]
.

Method (1.2)
xp+1 = xp − F ′1(xp)−1F1(xp).

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 14

Method (1.5), p = 1, Γ = I,

M1(x) = I + (I − F ′1(x)),

Q1(x) = x − (I + (I − F ′1(x)))F1(x), (4.1)
xn+1 = Q1(xn).

Method (1.5), p = 2, Γ = I,

M2(x) = I + (I − F ′1(x)) + (I − F ′1(x))2,

Q2(x) = x −M2(x)F1(x), (4.2)
xn+1 = Q2(xn).

Method (1.5), p = 3, Γ = I,

M3(x) = I + (I − F ′1(x)) + (I − F ′1(x))2 + (I − F ′1(x))3,

Q3(x) = x −M3(x)F1(x), (4.3)
xn+1 = Q3(xn).

Method (1.5), p = 4, Γ = I,

M4(x) = I + (I − F ′1(x)) + (I − F ′1(x))2 + (I − F ′1(x))3 + (I − F ′1(x))4,

Q4(x) = x −M4(x)F1(x), (4.4)
xn+1 = Q4(xn).

Method (1.5), p = 5, Γ = I,

M5(x) = I + (I − F ′1(x)) + (I − F ′1(x))2 + (I − F ′(x))3 + (I − F ′1(x))4 + (I − F ′1(x))5,

Q5(x) = x −M5(x)F1(x), (4.5)
xn+1 = Q5(xn).

Method (1.5), p = 1, 5 , Γ = F ′1(x0),

xn+1 = xn −MΓ−1F1(xn),

A = B−1(B − F ′1(x)), (4.6)
M = I +

p∑
i=1

Ai .

Thus, the comparison shows that the behavior of the method (1.5) is essentially the same as
Newton’s method (1.2). However, the iterates of the method (1.5) are cheaper to obtain than
Newton’s. As observed in Table 1 - Table 4, the number of iterations required for the proposed
methods with k ranging from 3 to 5 closely aligns with those of Newton’s method.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 15

Method Iterations Method Iterations
(1.2) Newton 4 (1.2) Newton 4
(4.1), p = 1 6 (4.6), p = 1 8
(4.2), p = 2 5 (4.6), p = 2 6
(4.3), p = 3 4 (4.6), p = 3 5
(4.4), p = 4 4 (4.6), p = 4 5
(4.5), p = 5 4 (4.6), p = 5 4Table 1. The number of iterations to reach error tolerance ε = 10−9 with initialguess x0 = (1, 1) and ‖I − F ′1(x0)‖ = 0.3129 < 1.

Method Iterations Method Iterations
(1.2) Newton 3 (1.2) Newton 3
(4.1), p = 1 5 (4.6), p = 1 3
(4.2), p = 2 4 (4.6), p = 2 3
(4.3), p = 3 3 (4.6), p = 3 3
(4.4), p = 4 3 (4.6), p = 4 3
(4.5), p = 5 3 (4.6), p = 5 3Table 2. The number of iterations to reach error tolerance ε = 10−9 with initialguess x0 = (0, 0) and ‖I − F ′1(x0)‖ = 0.1414 < 1.

Method Iterations Method Iterations
(1.2) Newton 5 (1.2) Newton 5
(4.1), p = 1 7 (4.6), p = 1 9
(4.2), p = 2 5 (4.6), p = 2 7
(4.3), p = 3 5 (4.6), p = 3 6
(4.4), p = 4 5 (4.6), p = 4 6
(4.5), p = 5 5 (4.6), p = 5 5Table 3. The number of iterations to reach error tolerance ε = 10−9, where x0 =

(−15,−15) and ‖I − F ′1(x0)‖ = 0.257 < 1.
Table 5 shows the results of calculations to determine the Computational Order of Convergence

(COC) and the Approximated Computational Order of Convergence (ACOC) aiming to compare the
convergence order of method (1.5) with the convergence order of Newton’s method (1.2).

Definition 4.1. Computational order of convergence of a sequence {xj}j≥0 is defined by

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 16

Method Iterations Method Iterations
(1.2) Newton 7 (1.2) Newton 7
(4.1), p = 1 8 (4.6), p = 1 12
(4.2), p = 2 7 (4.6), p = 2 8
(4.3), p = 3 7 (4.6), p = 3 8
(4.4), p = 4 7 (4.6), p = 4 7
(4.5), p = 5 7 (4.6), p = 5 7Table 4. The number of iterations to reach error tolerance ε = 10−12, where x0 =

(−15,−15) and ‖I − F ′1(x0)‖ = 0.257 < 1.

Υj =
ln |ej+1/ej |
ln |ej/ej−1|

,

where xj−1, xj , xj+1 are three consecutive iterations near the root α and ej = xj − α [6].

Definition 4.2. The approximated computational order of convergence of a sequence {xj}j≥0 is
defined by

Υ̂n =
ln |êj+1/êj |
ln |êj/êj−1|

,

where êj = xj − xj−1. xj , xj−1, xj−2 are three consecutive iterates [6].

Method COC ACOC
(1.2) Newton 1.8624 1.9697
(4.1), p = 1 0.863 1
(4.2), p = 2 0.2695 1.0438
(4.3), p = 3 1.9714 2.3569
(4.4), p = 4 1.8354 1.9453
(4.5), p = 5 1.8642 1.9661
(4.6), p = 1 0.9065 1.0118
(4.6), p = 2 0.5912 0.999
(4.6), p = 3 0.7321 0.9926
(4.6), p = 4 1.933 2.0151
(4.6), p = 5 1.8679 1.9578Table 5. The computational Order of Convergence and the Approximated Compu-tational Order of Convergence, where x0 = (−15,−15), ε = 10−12.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 17

Table 5 shows that the convergence of the proposed methods closely corresponds with the con-
vergence of Newton’s method, particularly for values of k ranging from 4 to 5 with the convergence
order closely approximating 2.

Example 4.2. Let X = Y = R3 and Ω = S[s∗, 1]. The mapping F is defined on Ω for a =

(a1, a2, a3)
tr ∈ R3 as

F1(a) = (a1, e
a2 − 1,

e − 1

2
a23 + a3)

tr .

Then, the definition of the derivative according to Fréchet [22,25] is given for the mapping F1

F ′1(a) =

 1 0 0

0 ea2 0

0 0 (e − 1)a3 + 1

 .
The point s∗ = (0, 0, 0)tr solves the equation F1(a) = 0. Moreover, F ′1(s∗) = I. The conditions of
the Theorem 2.2 hold for p = 1, if ϕ(t) = a(t) = (e−1)t. Then, we can have r ∈ (0, 0.2909883534).

Example 4.3. Let H[0, 1] stand for the space of continuous functions mapping the interval [0, 1]

into the real numbers. Let X = Y = H[0, 1] and Ω = S[s∗, 1] with s∗(v) = 0. The operator F1 is
defined on H[0, 1] as

F1(z)(v) = z(v)− 4

∫ 1
0

vz(τ)3dτ.

Then, of the derivative according to Fréchet [1, 10,15,22,30] is given below for the operator F1

F ′1(z(w))(v) = w(v)− 12

∫ 1
0

vτz(τ)2w(τ)dτ

for each w ∈ H[0, 1]. Therefore, the conditions of the Theorem 2.2 hold if, since for s∗ = 0,

F ′1(x
∗(v)) = I hold for that p = 1, if ϕ(t) = a(t) = 6t. Then, again by the definition of r, we can

choose r ∈ (0, 0.83̄).

5. Concluding Remarks
The paper addresses the issue with the inverses appearing in the study of the convergenceof simple-step iterative methods. It is shown that the inverse can be replaced by a finite sumof linear operators related to the operator involved. The resulting hybrid methods demonstratethe effectiveness of these methods since the number of iterations is essentially the same as wellas the convergence order of the methods. However, the hybrid method is cheaper to implement.This idea can be used for multiple steps and multiple point methods with the same advantages[3, 5–7,9, 14,23,24,31]. This is the direction of future research.

Acknowledgement:
We would like to thank graduate student Mr. Mykhailo Havdiak from The Department of OptimalProcesses, Ivan Franko National University of Lviv, Lviv Ukraine for providing Example 4.1.

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Eur. J. Math. Anal. 10.28924/ada/ma.5.5 18References
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https://doi.org/10.28924/ada/ma.5.5

	1. Introduction
	2. Convergence for the Method (1.5)
	3. Convergence for the Method (1.10)
	4. Numerical Examples
	5. Concluding Remarks
	Acknowledgement:
	References

