




































©2023 Ada Academica https://adac.eeEur. J. Math. Anal. 3 (2023) 26doi: 10.28924/ada/ma.3.26
Two Point Iterative Schemes for Nondifferentiable Equations in Banach Space

Ioannis K. Argyros1,∗, Janak Joshi2, Samundra Regmi3
1Department of Computing and Mathematical Sciences, Cameron University, Lawton, OK 73505, USA

iargyros@cameron.edu
2Department of Mathematics, Dallas Community College, Dallas, TX, USA

janakrajjoshi2036@gmail.com
3Department of Mathematics, University of Houston, Houston, 77024, TX, USA

sregmi5@uh.edu
∗Correspondence: iargyros@cameron.edu

Abstract. The local as well as the semi-local convergence analysis is established for a certain singlestep-two point iterative scheme defined on a Banach space setting. These schemes converge to alocally unique solution of a nonlinear equation. Both types of convergence are based on w-typecontinuity and majorizing functions and sequences. An auxiliary fixed linear operator is utilizedto assure the existence of inverses of the linear operators involved as well as the initial points ofthe iterative scheme. The local analysis provides the radius of convergence, error estimates andinformation on the uniqueness of the solution. Moreover, the semi local analysis provides sufficientconvergence conditions, error estimates and uniqueness of the solution results. Numerical examplesfurther validate the theoretical results.

1. Introduction
Using Mathematical modelling, a plethora of applications from diverse disciplines of science andengineering reduce to determining solutions denoted by x∗ of a nonlinear equation like

F (x) = 0. (1.1)
Here, F : D ⊂ B → B is a continuous operator, B stands for a Banach space and D is an open setin B. The analytic form of the solution for (1.1) can be found only in special cases. That explainswhy researchers and practitioners resort to iterative schemes, when a sequence is generatedapproximating x∗ under certain conditions.
Numerous studies exist in the local as well as the semi-local convergence analysis of iter-ative schemes [1–16]. Recently, there has been a surge in the development of schemes for solving

Received: 4 May 2023.
Key words and phrases. two-point iterative method; Banach space; local-semi local convergences.1

https://adac.ee
https://doi.org/10.28924/ada/ma.3.26


Eur. J. Math. Anal. 10.28924/ada/ma.3.26 2equations or systems of equations involving nondifferentiable operators. Looking towards thisdirection, we develop the Two-point Iterative Scheme (TIPS):
For x0 ∈ D and each n = 0, 1, 2, . . . by

xn+1 = xn − A(xn, xn−1)−1F (xn), (1.2)
where A : D ×D → L(B), the space of bounded linear operators from B into B. TPIS specialtiesto popular schemes.

Case 1: (Secant Scheme [1–4,12–15])
Set A(x, y) = [x, y ;F ], where [·, ·;F ] : D × D → (B) is a divided difference oforder one for the operator F [6–8]. Under this choice the scheme (1.2) specializes to

xn+1 = xn − [xn, xn−1;F ]−1F (xn). (1.3)
Case 2: (Kurchatov’s Scheme [12,13])Set A(x, y) = [2x − y , y ;F ]. Then, the scheme (1.2) becomes:

xn+1 = xn − [2xn − xn−1, xn−1;F ]−1F (xn). (1.4)
Case 3: (Steffensen’s Scheme [2, 8, 15])Set A(x, y) = [x + F (x), y ;F ]. Then, the scheme (1.2) becomes:

xn+1 = xn − [x + F (x), y ;F ]−1F (xn). (1.5)
Case 4: (Picards’s Scheme [1, 3, 8–10,15,16])Set A(x, y) = I , where I stands for the identity operator on B.
Case 5: (Newton’s Scheme [1–3,5, 6, 9, 15,16])Set A(x, y) = F ′(x), where F ′ stands for the Fréchet derivative of the operator F .The local as well as the semi-local convergence results for the aforementioned schemes involveassumptions on derivatives which do not appear on some of these methods (except Case 5).

Therefore, these results cannot be used to solve nondifferentiable operator equations (seee.g., Example 4.2). Other conditions involve approximations to the divided difference and theselection of an initial point x0 so that the first iteration is computable.
In the present article, the local and semi-local convergence of the scheme (1.2) is investi-gated under w-continuity-type conditions. Moreover, by introducing a certain linear operator P ,then invertibility of the linear operator A is assured in a certain subset of D from which the initial

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Eur. J. Math. Anal. 10.28924/ada/ma.3.26 3point is selected. The rest of the article is structured as follows: The local and the semi-localconvergence of the scheme (1.2) appear in Section 2 and section 3, respectively. The examplescan be found in Section 4. The article is completed the conclusions in Section 5.
2. Convergence I: Local

Some real functions are introduced that play a role in the local convergence analysis of thescheme (1.2). Let T = [0,∞). Assume:
(A1) There exists a continuous and nondecreasing function (CNF) w0 : T × T → T such thatthe equation

w0(t, t)− 1 = 0has a smallest solution r ∈ T0 − {0}.Set T0 = [0, r0) and D1 = D ∪ U(x∗, R). Moreover, there exists CNF w : T0 → T suchthat the equation h(t)− 1 = 0 has a smallest solution r ∈ (0, r0), where
h(t) =

w(t, t)

1− w0(t, t)
.

Let T1 = [0, r).
0 ≤ w0(t, t) < 1 (2.6)and
0 ≤ h(t) < 1 (2.7)From now on we assume that x∗ ∈ D is a solution of the equation F (x) = 0 and the divideddifference [∗, ∗;F ] exists on D×D. The functions w0 and w are connected to the operatorson the scheme (1.2).

(A2) There exists an invertible operator P such that for each x, y ∈ D
‖P−1(A(x, y)− P )‖ ≤ w0(‖x − x∗‖, ‖y − x∗‖).

(A3)

‖P−1(A(x, y)− [x, x∗;F ])‖ ≤ w(‖x − x∗‖, ‖y − x∗‖) for each x, y ∈ D1and
(A4) U[x

∗, r ] ⊂ D.Next, the local convergence of the scheme (1.2) is provided based on the conditions (A1) − (A4)and the developed terminology.
THEOREM 2.1. Assume that the conditions (A1)−(A4) are validated. If the initial points x−1, x0 ∈
U(x∗, r)−{x∗}, then the sequence {xn} generated by the scheme (1.2) is well defined in U(x∗, r)
for each n = 0, 1, 2, 3, . . . and is convergent to the solution x∗ of the equation F (x) = 0, so that

‖xn+1 − x∗‖ ≤ h(‖xn − x∗‖)‖xn − x∗‖ ≤ ‖xn − x∗‖ < r (2.8)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.26 4

where the radius of convergence r is defined in (A1) and the function h is also given in (A1).

Proof. By hypothesis, x−1, x0, (A2) and (A3) we obtain in turn that:
‖P−1(A(x0, x−1)− P )‖ ≤ wo(‖x0 − x∗‖, ‖x−1 − x∗‖) ≤ w0(r, r) < 1. (2.9)

The estimate (2.9) and the Banach Lemma on invertible operators [1–3, 8, 9] assure the existenceof A(x0, x−1)−1L(B) and
‖A(x0, x−1)−1P‖ ≤

1

1− w0(‖x0 − x∗‖, ‖x−1 − x∗‖)
. (2.10)

Moreover, the iterate x1 is well defined by the first subset of the scheme (1.2). Then, we can write:
x1 − x∗ = x0 − x∗ − A(x0 − x−1)−1F (x0)

= A(x0 − x−1)(A(x0, x−1)− [x0, x∗;F ])(x0 − x∗) (2.11)
Using (2.7), (A3), (2.10), (A2) and (2.11) we get in turn that:
‖x1 − x∗‖ =

w(‖x0 − x∗‖, ‖x−1 − x∗‖)‖x0 − x∗‖
1− w0(‖x0 − x∗‖, ‖x−1 − x∗‖)

≤ h(‖x0 − x∗‖)‖x0 − x∗‖ ≤ ‖x0 − x∗‖ < r,(2.12)Thus, the iterate x1 ∈ U(x∗, r) and the item (2.8) holds for n = 0. Simply replace x−1, x0, x1 by
xm−1, xm, xm+1 in the preceding calculations to terminate the induction for items (2.8). Then, fromthe estimation:

‖xm+1 − x∗‖ ≤ c‖xm − x∗‖ < r, (2.13)where c = h(‖x0 − x∗‖) ∈ [0, 1). We conclude that limm→∞ xm = x∗ and that the iterate xm+1 ∈
U(x∗, r). �

The uniqueness of the solution region is given in the next result.
PROPOSITION 2.2. Assume: There exists a solution z ∈ U(x∗, R1) of the equation F (x) = 0 for
some R1 ≥ 0; the condition (A2) holds in the ball U(x∗, R1) for A being [·, ·;F ], and there exists
R2 ≥ R1 such that:

w0(R1, R2) < 1. (2.14)
Define the region D2 = D ∪ U[x∗, R2]. Then, x∗ is the only solution of the equation F (x) = 0 in
the region D2.

Proof. If z 6= x∗, then the divided difference E = [x∗, z ;F ] is well defined. Using (A2) and (2.14),we get in turn that
‖P−1(E − P )‖| ≤ w0(‖x∗ − x∗‖, ‖z − x∗‖) ≤ w0(0, R2) < 1.

Thus, E−1 ∈ L(B). Moreover, from the identity
z − x∗ = E−1(F (z)− F (x∗)) = E−1(0) = 0.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.26 5Hence, we conclude that z = x∗. �

3. Convergence II: Semi-local
A certain real sequence is developed that is shown in Theorem 3.1 to be majorizing for thescheme (1.2). Assume:
(H1) : There exists a continuous and nondecreasing function v0 : T → T such that v0(t, t)−1 = 0has a unique positive solution denoted by δ.Let T2 = [0, δ).
(H2) : There exists a CNF v : T2× T2 → T . Define a sequence {γn} for γ−1 = 0, γ0 = α, some

γ1 ≥ α by:
γn+2 = γn+1 +

v(γn+1 − γn, γn − γn+1)(γn+1 − γn)
1− v0(γn − γ0, γn+1 − γ0)

.

(H3) : v0(γn − γ0, γn+1 − γ0) < 1 and γn ≤ γ < δ.Clearly, by the definition of the sequence γn and (H3), this sequence is nondecreasingly convergentto its unique least upper bound denoted by γ∗. The functions v0, v and the limit point γ∗ areconnected to the operators on the scheme as follows:
(H4) : There exists an invertible operator P ∈ L(B) such that P−1 ∈ L(B) and for each x, y ∈ D

‖P−1(A(x, y)− P )‖ ≤ v0(‖x − x0‖, ‖y − x0‖).

It follows by (H1) that: if x−1, x0 ∈ D with
‖x−1 − x0‖ ≤ α, v0(‖x−1 − x0‖, ‖x0 − x0‖) ≤ v0(α, 0) < 1.

Thus A(x0, x−1)−1 ∈ L(B). Let
‖A(x0, x−1)−1F (x0)‖ ≤ γ1 − γ0.

Set D3 = D ∪ U(x0, γ).
‖P−1(A(x, y)− [y , z ;F ])‖ ≤ (‖x − y‖, ‖y − z‖) for each x, y , z ∈ D3

and
(H5) : U[x0, γ∗] ⊂ D.Next, we present the semi-local convergence analysis of the scheme (1.2) under the conditions
(H1)− (H5) and the preceding terminology.
THEOREM 3.1. Assume that the conditions (H1)−(H5) hold. Then the sequence {xn} generated
by the scheme (1.2) is well defined in U(x0, γ0), remains in U(x0, γ0) for each n = 0, 1, 2, . . . and
converges to a solution x∗ ∈ U(x0, γ0) of the equation F (x) = 0. Moreover, the following items
hold:

‖x∗ − xn‖ ≤ γn+1 − γn (3.15)

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Eur. J. Math. Anal. 10.28924/ada/ma.3.26 6

and

‖x∗ − xn‖ ≤ γ∗ − γn, (3.16)
where the sequence {γn} and the point x∗ are defined in (H3).

Proof. The inequality (3.15) holds for n = 0, since:
‖x1 − x0‖ = ‖A(x0, x−1)−1F (x0)‖ ≤ γ1 − γ0 < γ∗ − γ0.

�

Thus, the iterate x1 ∈ U(x0, γ0). Let xn, xn−1 ∈ U(x0, γ0). Using (H3) and (H4), we have inturn,
‖P−1(A(xn, xn−1)− P )‖ ≤ v0(‖xn − x0‖, ‖xn−1 − x0‖) ≤ v0(γ0, γ0) < 1.So, A(xn, xn−1)−1L(B) and

‖A(xn, xn−1)−1P‖ ≤
1

1− v0(‖xn − x0‖, ‖xn−1 − x0‖)
. (3.17)

Moreover, the iterate xn+1 is well defined by the scheme (1.2) and (3.17). Further we can writerby the scheme (1.2)
F (xn+1) = F (xn+1)− F (xn)− A(xn, xn−1)(xn+1 − xn)

= ([xn+1, xn;F ]− A(xn, xn−1))(xn+1 − xn). (3.18)
In view of (H4) and (3.18) we get:

‖P−1F (xn+1)‖ ≤ ‖P−1([xn+1, xn;F ]− A(xn, xn−1))‖‖xn+1 − xn‖

≤ v(‖xn+1 − xn‖, ‖xn − xn−1‖)‖xn+1 − xn‖

≤ v(‖γn+1 − γn‖, ‖γn − γn−1‖)‖γn+1 − γn‖. (3.19)
Then, by the scheme (1.2) for n replaced by n + 1, we obtain:
‖xn+2 − xn+1‖ ≤ ‖A(xn+1, xn)−1P‖‖PF (xn+1)‖

≤
v(γn+1 − γn, γn − γn−1)

1− v0(‖xn+1 − x0‖, ‖xn − x0‖)
≤
v(γn+1 − γn, γn − γn−1)(γn+1 − γn)

1− v0(γn+1, γn)and
‖xn+2 − x0‖ ≤ ‖xn+2 − xn+1‖+ ‖xn+1 − x0‖ ≤ γn+2 − γn+1 + γn+1 − γ0 < γ∗ − γ0.

Therefore, the induction for the item (3.15) is terminated and {xn} ⊂ U(x0, γ
∗ − γ0). But thesequence {γn} is complete. Thus, the sequence {xn} is also complete in a Banach space B. Hencethere exists x∗ ∈ U[x0, γ∗ − γ0] such that limn→∞ xn = x∗.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.26 7By letting n → +∞ in (3.19) and using the continuity of the operator F , we deduce
F (x∗) = 0. Then from the estimation:

‖xn+i − xn‖ ≤ γn+i − γn, (3.20)
the item (3.16) is obtained by letting i → +∞ in (3.20).
A Uniqueness of the solution region is determined in the next result.
PROPOSITION 3.2. Assume: There exists a solution z ∈ U(x0, δ1) of the equation F (x) = 0 for
some δ1 > 0; the condition (H4) holds for [∗, ∗, ;F ] replacing A on the ball U(x0, δ1) and there
exists δ2 ≥ δ1 such that

v0(δ1, δ2) < 1. (3.21)
Define the region D4 = D ∩ U[x0, δ2]. Then, the only solution of the equation F (x) = 0 in the
region D4 is z .

Proof. As in Proposition 2.2, consider z1 ∈ D4 with F (z1) = 0 and define E1 = [z, z1;F ] for z 6= z1.Then, the application of (H4) and (3.21) give
‖P−1(E1 − P )‖ ≤ v0(‖z − z0‖, ‖z1 − x0‖) ≤ v0(δ1, δ2) < 1.

Consequently, it follows again that z1 = z0. �

REMARK 3.3. (i) Possible choices but not the only ones for the linear operator P are:
– Differential case: P = F ′(x∗)

and
– Non-Differential case: P = [x−1, x0;F ].

P should be chosen in general, so the majorant functions are as tight as possible in both
the local and semi-local analysis (see the Numerical Section 4 that follows).(ii) The limit point γ∗ in (H5) is replaced by δ given in (H1).(iii) Notice that only (H4) out of conditions (H1)− (H5) is used in Proposition 3.2. However,
if all conditions are used, let δ1 = γ∗ and z = x∗.

4. Numerical Examples
EXAMPLE 4.1. Let B = R× R and D = U[x∗, 1] with x∗ = (0, 0, 0)T . Define the mapping F on
D for y = (y1, y2, y3)T as:

F (y) =

(
ey1 − 1, y2,

e − 1
2

y23 + y3

)T
.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.26 8

Then, by the definition of the Fréchet derivative F ′ of F is given by:

F ′(y) =

e
y1 0 0

0 1 0

0 0 (e − 1)y3 + 1


It follows that F ′(x∗) = I . Then, the convergence conditions (A3)−(A5) are validated, respectively
for:

Case 1:

w0(s1, s2) =
1

2
(e − 1)(s1 + s2), w(s1, s2) =

1

2
(e − 1)s1, and U[x∗, r ] ⊂ D.

Case 2:

w0(s1, s2) =
1

2
(e − 1)(s1 + s2), w(s1, s2) =

1

2
(e − 1)(2s1 + s2), and U[x∗, 3r ] ⊂ D.

Case 3:

w0(s1, s2) =
1

2
(e − 1)(s1 + s2), w(s1, s2) =

1

2
(e − 1)(s1 + 2s2), and U[x∗, r ] ⊂ D,

where

r = max{r, f (r)} and f (t) =
(
‖I + P‖+

e − 1
2

t

)
t or f (t) =

(
2 +

e − 1
t

)
t.

Case 4:
w0(s1, s2) = 0, w(s1, s2) =

1

2
(e − 1)s1, and U[x∗, r ] ⊂ D.

Case 5:

w0(s1, s2) =
1

2
(e − 1)(s1 + s2), w(s1, s2) =

1

2
(e − 1)s1, and U[x∗, 3r ] ⊂ D.

The exact radius r can be then computed immediately using the condition (A2). As an example,
for the case 5, we must solve:

1
2(e − 1)t
1− (e − 1)t = 0,

leading to

t = r =
2

3(e − 1) .Concerning the semi-local case and the application of the method (1.2), we present anotherexample:
EXAMPLE 4.2. Let B = R × R × R. The two by two nonlinear and nondifferentiable system to
be solved is:

3s21 s2 + s
2
2 − 1 + |s1 − 1| = 0,

s41 + s1s
3
2 − 1 + |s2| = 0.

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Eur. J. Math. Anal. 10.28924/ada/ma.3.26 9

Then, the system can be described as Q = (Q1, Q2), where

Q1(s1, s2) = 3s
2
1 s2 + s

2
1 − 1 + |s1 − 1|

and

Q2(s1, s2) = s
4
1 + s1s

3
2 − 1 + |s2|.

The system becomes Q(s1, s2) = 0. Then as A = [∗, ∗;Q], which is an 2 × 2 real matrix defined
for s = [s1, s2]T and s̃ = [s3, s4]T by:

[s̃ , s̃;Q]i ,1 =
Qi [(s1, s4)−Qi(s1, s2)]

s4 − s2
for s2 6= s4, i = 1, 2.

Otherwise, set [∗, ∗;Q] = 0. Let us choose s0 = (5, 5) and s−1 = (1, 0) to be the starters for the
scheme (1.2). Then,

n x
(1)
n x

(2)
n ‖xn − xn−1‖

0 5 5
1 1 0 5
2 0.98909090909090909 0.363636363636364 3.636E-01
3 0.894886945874111 0.329098638203090 3.453E-02
4 0.894655531991499 0.327827544745569 1.271E-03
5 0.894655373334793 0.327826521746906 1.022E-06
6 0.894655373334687 0.327826421746298 6.089E-13
7 0.894655373334687 0.327826421746298 2.701E-20

Therefore, the solution s∗ = (s∗1 , s∗2)T of the system is

s∗1 = 0.894655373334687 and s∗2 = 0.327826421746298.

REMARK 4.3. A more targeted choice for L than the two mentioned already can give a larger
radius of convergence. Indeed, assume the conditions instead of (A3) and (A4)

(A3)
′

‖P−1(F ′(x)− P )‖ < 1

and
(A4)

′

‖P−1(F ′(x∗ + θ(x − x∗))− F ′(x))‖ ≤ g4(θ)‖x − x∗‖ for each x ∈ D.

Consider an example
F (x) = ex − 1

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Eur. J. Math. Anal. 10.28924/ada/ma.3.26 10with D = U[x∗, 1]. Then x∗ = 0. Choose P = 1
be
x for b ∈ (0, 2). Then the Newton’s scheme gives:

xn+1 − x∗ = xn − x∗ − F ′(xn)−1F (xn)

= s0F
′(xn)

−1 (F ′(x∗ + θ(xn − x∗))− F ′(xn)) (xn − x∗)
Leading by (A3)′ and (A4)′ to

‖xn+1 − x∗‖ ≤
|b|(e − 2)‖xn − x∗‖2

1− |1− b| ,

where we also used
‖P−1(F ′(x)− P )‖ =

∥∥∥∥be−x (ex − 1bex
)∥∥∥∥ = |b − 1| < 1.

Thus,
‖P−1(F ′(x)− P )‖ ≤

1

1− |1− b| ,and
‖P−1(F ′(x∗ + θ(x − x∗))− F ′(x))‖ ≤ |b|(e − 2).

The last estimate is obtained, since for y = x∗ + θ(x − x∗)
e−x(e−y − ex) = ey − 1 = 1 + y +

y2

2!
+ · · ·+

y k

k!
+ · · · − 1 = y

(
1 +

y

2!
+ · · ·+

y k−1

k!
+ . . .

)
.

Hence,
‖ex(ey − ex)‖ = (1− θ)Gn‖x − x∗‖,where,

Gn = 1 +
1− θ
2!
+ · · ·+

(1− θ)n−1

n!
+ . . . ,

and ∫ 1
0

(1− θ)ndθ =
1

n + 1
.

But, ∫ 1
0

(1− θ)nGndθ =
∫ 1
0

(1− θ)dθt +
∫ 1
0

(1− θ)2

2!
dθt2 + · · ·+

∫ 1
0

(1− θ)n

n!
dθtn−1 + . . .

=
1

2
t +

1

3 · 2!t
2 + · · ·+

1

(n + 1)n!
tn−1 + · · · ≤ e − 2.

Consequently, it follows from the error estimate that:
rA =

1− |1− b|
|b|(e − 2) .Let us compare the new radius with the ones already in the literature developed independently byRheinboldt [11] and Traub [15]. The condition used is:

‖F ′(x∗)−1(F ′(x)− F ′(y))‖ ≤ l‖x − y‖

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Eur. J. Math. Anal. 10.28924/ada/ma.3.26 11for each x, y ∈ D to obtain the error estimate:
‖xn+1 − x∗‖ ≤

l‖xn − x∗‖2

2(1− ‖xn − x∗‖)
,

and the radius is
rTR =

2

3l
.But l = e for the example, so:

rTR =
3

2e
< rA, say for b = 1.

Therefore, the new radius of convergence is larger allowing for a wider choice of initial points.Other choices of P can lead to even larger radius of convergence. We leave the detail to themotivated reader.
5. Conclusion

A finer and more flexible local and semi-local convergence analysis for the scheme (1.2) is devel-oped involving an invertible operator P , which if chosen appropriately leads to weaker convergenceconditions, better uniqueness of the solution and a larger radius of convergence than if P is chosento be as in earlier studies F ′(x∗) or F ′(x0) or [x0, x−1;F ]. This idea can be extended to multistepand multipoint schemes [1–16]. This is the direction of our future research.
References

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[13] S.M. Shakhno, On an iterative algorithm with superquadratic convergence for solving nonlinear operator equations.J. Comp. Appl. Math. 231 (2009) 222–235.[14] J.R. Sharma, H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems ofnonlinear equations, Numer. Algor. 67 (2014) 917–933.[15] J.F. Traub, Prentice-Hall, Iterative methods for the solution of equations, Prentice-Hall, Englewood Cliffs, NewJersey, (1964).[16] T. Yamamoto, A convergence theorem for Newton-like methods in Banach spaces, Numer. Math. 51 (1987) 545–557.

https://doi.org/10.28924/ada/ma.3.26

	1. Introduction
	2. Convergence I: Local
	3. Convergence II: Semi-local
	4. Numerical Examples
	5. Conclusion
	References

