Open AccessArticle

Local Convergence of Solvers with Eighth Order Having Weak Conditions

Ramandeep Behl

¹ and

Ioannis K. Argyros

^2,*

Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Department of Mathematical Sciences Lawton, Cameron University, Lawton, OK 73505, USA

Author to whom correspondence should be addressed.

Symmetry 2020, 12(1), 70; https://doi.org/10.3390/sym12010070

Submission received: 25 November 2019 / Revised: 19 December 2019 / Accepted: 25 December 2019 / Published: 2 January 2020

(This article belongs to the Special Issue Iterative Numerical Functional Analysis with Applications)

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Abstract

In particular, the problem of approximating a solution of an equation is of extreme importance in many disciplines, since numerous problems from diverse disciplines reduce to solving such equations. The solutions are found using iterative schemes since in general to find closed form solution is not possible. That is why it is important to study convergence order of solvers. We extended the applicability of an eighth-order convergent solver for solving Banach space valued equations. Earlier considerations adopting suppositions up to the ninth Fŕechet-derivative, although higher than one derivatives are not appearing on these solvers. But, we only practiced supposition on Lipschitz constants and the first-order Fŕechet-derivative. Hence, we extended the applicability of these solvers and provided the computable convergence radii of them not given in the earlier works. We only showed improvements for a certain class of solvers. But, our technique can be used to extend the applicability of other solvers in the literature in a similar fashion. We used a variety of numerical problems to show that our results are applicable to solve nonlinear problems but not earlier ones.

Keywords:

order of convergence; iterative solver; Lipschitz constant; Banach space; local convergence

1. Introduction

A plethora of problems from diverse disciplines such as Applied Mathematics, Mathematical: Biology, chemistry, Economics, Physics, Environmental Sciences and also Engineering are reduced to equations on abstract spaces via mathematical modeling. The closed form solution is obtained only in rare cases. That is why it is important to develop iterates generating a sequence converging to the solution based on some suitable hypotheses on the initial information. Hence, we consider the problem of finding approximate unique solution

α

Γ (μ) = 0,

(1)

is one of the top priorities in the field of numerical analysis. We consider that

Γ : K \subset T_{1} \to T_{2}

is a Fréchet differentiable operator,

T_{1}, T_{2}

are Banach spaces, and

K

is a convex subset of

T_{1}

. The

L (T_{1}, T_{2})

is the space of continuous operators from

T_{1}

T_{2}

We have several examples where researchers demonstrated the applicability of (1). They transformed the real life problems to (1) by adopting mathematical modeling and details can be found in [1,2,3,4,5,6,7]. We have to target on iterative solvers since it is not always feasible to access the solution

α

in an explicit pattern. We have a small number of globally convergent methods that do not require a sufficiently close starting point, e.g., Bisection method or regula falsi method. But, most of the algorithms determine one zero at a time. If the zero has been determined with sufficient accuracy, the polynomial is deflated and the algorithm is applied again on the deflated polynomial. In this way, we can determine all zeros simultaneously and also have theoretical importance [8] for the details of methods can be seen in [9,10,11,12,13,14,15,16,17,18,19]. Therefore, we have extended amount of iterative solvers to solve problems like expression (1). The analysis of solvers involves local convergence that stands on the knowledge around

α

. It also ensures the convergence of iteration procedures. One of the most significant tasks in the analysis of iterative procedures is to yield the convergence region. Hence, we suggest the radius of convergence.

We rewrite for this purpose the iterative solver suggested in [20] in the following way:

\begin{matrix} ν_{τ} & = μ_{τ} - A_{τ}^{- 1} Γ (μ_{τ}), \\ μ_{τ + 1} & = ν_{τ} - 4 B_{τ}^{- 1} Γ (ν_{τ}), \end{matrix},

(2)

where

μ_{0} \in K

is an initial point,

A : K \to L (T_{1}, T_{2})

given as

A (μ_{τ}) = A_{τ} = Γ^{'} (μ_{τ}) + Q (μ_{τ}) Γ (μ_{τ}), B (μ_{τ}, ν_{τ}) = B_{τ} = Γ^{'} (μ_{τ}) + 4 Q (μ_{τ}) Γ (μ_{τ}) + 2 Γ^{'} (\frac{μ_{τ} + ν_{τ}}{2}) + Γ^{'} (ν_{τ})

, and

Q (. .) : K \times K \to L (T_{1}, T_{2})

is a bilinear operator. In the special case, when

T_{1} = T_{2} = R

Q (μ, ν) = \frac{G^{'} (μ)}{G (μ)}

, where

G (μ) \neq 0

for each

x \in K - {α}

solver (2) reduces to a fourth-order convergent solver studied in [20]. Shah et al. [20] suggested fourth-order convergence by adopting Taylor series expansions and suppositions up to the ninth-order derivative of the involved function. Such constraints hamper the suitability of solver (2). But, only first-order derivative emerges in the solver (2). Let us assume the succeeding function

Γ

T_{1} = T_{2} = R

K = [- \frac{1}{2}, \frac{3}{2}]

Γ (μ) = \{\begin{matrix} μ^{3} \ln μ^{2} + μ^{5} - μ^{4}, & μ \neq 0 \\ 0, & μ = 0 \end{matrix} .

Then, we have that

Γ^{'} (μ) = 3 μ^{2} \ln μ^{2} + 5 μ^{4} - 4 μ^{3} + 2 μ^{2},

Γ^{″} (μ) = 6 μ \ln μ^{2} + 20 μ^{3} - 12 μ^{2} + 10 μ

and

Γ^{‴} (μ) = 6 \ln μ^{2} + 60 μ^{2} - 24 μ + 22 .

From the above derivatives, it is straightforward to see that the 3rd-order derivative of

Γ

is unbounded in

A

. In the available literature, we have a bulk number of research articles [1,2,3,4,5,6,7,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]. In the majority of these articles, authors mention that starting guess

x_{0}

must be adequately close to

μ

. But this is not offering us an idea of: how to pick

x_{0}

, how much closeness is sufficient for convergence, find radius; bounds on

∥ x_{n} - μ ∥

and results on uniqueness. We deal with all these questions for solver (2) in the next section.

In the present study, we adopt only conditions on the first-order derivative of

Γ

with generalized Lipschitz conditions. In addition, we are avoiding the Taylor series expansions because it proceeds with higher-order derivatives of

Γ

, but we adopt Lipschitz parameters. In this way, we are not committed to adopt higher-order derivatives for convergence order of (2). Further, we adopted the following

(C O C)

and

(A C O C)

for computing the convergence order:

ξ = \frac{l n \frac{∥ μ_{τ + 2} - α ∥}{∥ μ_{τ + 1} - α ∥}}{l n \frac{∥ μ_{τ + 1} - α ∥}{∥ μ_{τ} - α ∥}}, for each = 0, 1, 2, \dots,

(3)

or the approximate computational order of convergence (

A C O C

) [21], defined as

ξ^{*} = \frac{l n \frac{∥ μ_{τ + 2} - μ_{τ + 1} ∥}{∥ μ_{τ + 1} - μ_{τ} ∥}}{l n \frac{∥ μ_{τ + 1} - μ_{τ} ∥}{∥ μ_{τ} - μ_{τ - 1} ∥}}, for each = 1, 2, \dots,

(4)

where the computational order of convergence

C O C

and the approximate computational order of convergence

A C O C

[21], respectively. They do not require higher than one derivatives. It is vital to note that

A C O C

does not need the prior information of exact root

μ

. Finally, we investigate the applicability of our results on several numerical examples, where earlier works did not exhibit this behavior.

The remainder of this paper is coordinated in the succeeding way: We suggest the local convergence study of solver (2) in Section 2. Numerical experimentation is depicted in Section 3. Finally, we make concluding assertions in Section 4.

2. Study of Local Convergence

In this section, we suggest the local convergence study of solver (2). Therefore, we adopt some scalar functions

Δ_{0}, Δ, w_{0}, w, w_{1}

that are non-decreasing continuous functions from

[0, + \infty)

[0, + \infty)

such that

w_{0} (0) = w (0) = 0

. We assume

p (ζ) = 1

(5)

has a minimal positive solution

r_{0}

and

p (ζ) = w_{0} (ζ) + w_{1} (ζ) ζ \int_{0}^{1} Δ_{0} (θ ζ) d θ .

In addition, we describe functions

g_{1}, h_{1}, q

, and

h_{q}

[0, r_{0})

as follows:

\begin{matrix} g_{1} (ζ) & = \frac{\int_{0}^{1} w ((1 - θ) ζ) d θ}{1 - w_{0} (ζ)} + \frac{(\int_{0}^{1} Δ (θ ζ) d θ) (\int_{0}^{1} Δ_{0} (θ ζ) d θ) w_{1} (ζ) ζ}{(1 - w_{0} (ζ)) (1 - p (ζ))}, \\ h_{1} (ζ) & = g_{1} (ζ) - 1, \\ q (ζ) & = \frac{1}{2} [w_{0} (ζ) + 4 \int_{0}^{1} Δ_{0} (θ ζ) d θ w_{1} (ζ) ζ + \int_{0}^{1} Δ (\frac{θ}{2} (1 + g_{1} (ζ)) ζ) d θ (1 + g_{1} (ζ)) ζ + w_{0} (g_{1} (ζ) ζ)], \\ and \\ h_{q} (ζ) & = q (ζ) - 1 . \end{matrix}

We have that

h_{1} (0) = h_{q} (0) = - 1 < 0

and

h_{1} (ζ) \to + \infty

h_{q} (ζ) \to + \infty

ζ \to r_{0}^{-}

. By adopting the intermediate value theorem, we can say that both function

h_{1}

and

h_{q}

have zeros in

(0, r_{0})

. Call as

r_{1}

and

r_{q}

the smallest such zeros in

(0, r_{0})

of the functions

h_{1}

and

h_{q}

, respectively. Further, we represent functions

g_{2}

and

h_{2}

[0, r_{0})

as follows:

\begin{matrix} g_{2} (ζ) & = [1 + \frac{2 \int_{0}^{1} Δ (θ g_{1} (ζ) ζ) d θ}{1 - q (ζ)}] g_{1} (ζ), \\ and \\ h_{2} (ζ) & = g_{2} (ζ) - 1 . \end{matrix}

We have again that

h_{2} (0) = - 1 < 0

and

h_{2} (ζ) \to + \infty

ζ \to r_{q}^{-}

. Let us call

r_{2}

to be minimal zero of

h_{2}

(0, r_{q})

. Finally, we describe the convergence radius r in the following way:

r = min {r_{1}, r_{2}} .

(6)

Then, we have that for each

ζ \in [0, r)

0 \leq p (ζ) < 1,

(7)

0 \leq w_{0} (ζ) < 1,

(8)

0 \leq g_{1} (ζ) < 1,

(9)

0 \leq q (ζ) < 1,

(10)

and

0 \leq g_{2} (ζ) < 1 .

(11)

The

U (λ, ρ)

and

\bar{U} (λ, ρ)

are two open and closed balls, respectively in

T_{1}

centered at

λ \in T_{1}

. Both have the radius

ρ > 0

The local convergence analysis of solver (2) is based on conditions

(A)

(A₁): $Γ : K \subseteq T_{1} \to T_{2}$ is a Fréchet-differentiable operator.
(A₂): $Δ_{0}, Δ, w_{0}, w, w_{1} : [0, \infty) \to [0, \infty)$ with $w_{0} (0) = w (0) = 0$ are non-decreasing continuous functions.
(A₃): There exists a zero $α \in K$ of $Γ$ such that for every $μ \in K$

$Γ (α) = 0, Γ^{'} {(α)}^{- 1} \in L (T_{2}, T_{1})$

(12)

and

$∥ Γ^{'} {(α)}^{- 1} (Γ^{'} (μ) - Γ^{'} (α)) ∥ \leq w_{0} (∥ μ - α ∥) .$

(13)

Set $K_{0} : = K \cap U (α, r_{0})$ .
(A₄): $∥ Γ^{'} {(α)}^{- 1} (Γ^{'} (μ) - Γ^{'} (ν)) ∥ \leq w (∥ μ - ν ∥),$

(14)

$∥ Γ^{'} (μ) ∥ \leq Δ_{0} (∥ μ - α ∥),$

(15)

$∥ Γ^{'} {(α)}^{- 1} Γ^{'} (μ) ∥ \leq Δ (∥ μ - α ∥),$

(16)

$∥ Γ^{'} {(α)}^{- 1} Q (μ, ν) ∥ \leq w_{1} (∥ μ - α ∥), for every μ, ν \in K_{0},$

(17)

and

$\bar{U} (α, r) \subseteq K,$

(18)

where $Q (μ, ν) : K \times K \to L (T_{1}, T_{2})$ .

Then, we present the main local convergence result.

Theorem 1.

Under the conditions

(A)

sequence

{μ_{τ}}

obtained for

μ_{0} \in U (α, r) - {α}

by solver (2) exists, remains in

U (α, r)

for all

τ = 0, 1, 2, \dots

and converges to α, so that

∥ ν_{τ} - α ∥ \leq g_{1} (∥ μ_{τ} - α ∥) ∥ μ_{τ} - α ∥ \leq ∥ μ_{τ} - α ∥ < r

(19)

and

∥ λ_{τ} - α ∥ \leq g_{2} (∥ μ_{τ} - α ∥) ∥ μ_{τ} - α ∥ \leq ∥ μ_{τ} - α ∥ .

(20)

Furthermore, if

\int_{0}^{1} w_{0} (θ R) d θ < 1, f o r R \geq r,

(21)

then α is the unique root of

Γ (μ) = 0

K_{1} : = K \cap \bar{U} (α, R)

Proof.

We select the mathematical induction to show expressions (19)–(21) are well defined in

U (α, r)

. Further, they converge to required zero

α

. Adopting hypothesis

μ_{0} \in U (α, r) - {α}

, (5)–(7) and (13), we obtain

∥ Γ^{'} {(α)}^{- 1} (Γ^{'} (μ_{0}) - Γ^{'} (α)) ∥ \leq w_{0} (∥ μ_{0} - α ∥) < w_{0} (r) < 1 .

(22)

From the expression (22) and the Banach Lemma on inverse operators [1,2] that

Δ^{'} {(x_{0})}^{- 1} \in L (T_{2}, T_{1})

ν_{0}, λ_{0}

are well defined and

∥ Γ^{'} {(μ_{0})}^{- 1} Γ^{'} (α) ∥ \leq \frac{1}{1 - w_{0} (∥ μ_{0} - α ∥)} .

(23)

To show that

ν_{0}

exists, it suffices to show that

A_{0}^{- 1} \in L (T_{2}, T_{1})

. Using (5), (6), (8), (13) and (15), we get in turn that

\begin{matrix} ∥ Γ^{'} {(μ)}^{- 1} (A_{0} - Γ^{'} (α)) ∥ & \leq ∥ Γ^{'} {(α)}^{- 1} (Γ^{'} (μ_{0}) - Γ^{'} (α)) ∥ + ∥ Γ (μ_{0}) ∥ ∥ Γ^{'} {(α)}^{- 1} Q (μ_{0}) ∥ \\ \leq w (∥ μ_{0} - α ∥) + \int_{0}^{1} Δ_{0} (θ ∥ μ_{0} - α ∥) d θ w_{1} (∥ μ_{0} - α ∥) ∥ μ_{0} - α ∥ \\ = p (∥ μ_{0} - α ∥) < p (r) < 1, \end{matrix}

(24)

A_{0}^{- 1} \in L (T_{2}, T_{1})

is well defined and

∥ A_{0}^{- 1} Γ^{'} (α) ∥ \leq \frac{1}{1 - p (∥ μ_{0} - α ∥)} .

(25)

By the definition of

A_{0}

and the first substep of (2), we can write

\begin{matrix} ν_{0} - α & = μ_{0} - α - Γ_{0}^{'} {(μ_{0})}^{- 1} Γ (μ_{0}) + (Γ_{0}^{'} {(μ_{0})}^{- 1} - A_{0}^{- 1}) Γ (μ_{0}) \\ = Γ_{0}^{'} {(μ_{0})}^{- 1} Γ^{'} (α) \int_{0}^{1} Γ^{'} {(α)}^{- 1} (Γ^{'} (α + θ (μ_{0} - α)) - Γ^{'} (μ_{0})) (μ_{0} - α) d θ \\ + Γ^{'} {(μ_{0})}^{- 1} Γ^{'} (α) Γ^{'} {(α)}^{- 1} (A_{0} - Γ^{'} (μ_{0})) A_{0}^{- 1} Γ^{'} (α) Γ^{'} {(α)}^{- 1} Γ (μ_{0}) . \end{matrix}

(26)

We also have by (16)

\begin{matrix} Γ (μ_{0}) = Γ (μ_{0}) - Γ (α) = \int_{0}^{1} Γ^{'} (α + θ (μ_{0} - α)) d θ (μ_{0} - α), \\ so \\ ∥ Γ^{'} {(α)}^{- 1} Γ (μ_{0}) ∥ = ∥ \int_{0}^{1} Γ^{'} {(α)}^{- 1} Γ^{'} (α + θ (μ_{0} - α)) d θ (μ_{0} - α) ∥ \leq \int_{0}^{1} Δ (θ ∥ μ_{0} - α ∥) d θ ∥ μ_{0} - α ∥ . \end{matrix}

(27)

In view of (2), (5), (6), (9), (14), (15), (22) and (24), we obtain

\begin{matrix} ∥ ν_{0} - α ∥ & \leq ∥ Γ^{'} {(μ_{0})}^{- 1} Γ^{'} (α) ∥ ∥\int_{0}^{1} Γ^{'} {(α)}^{- 1} (Γ^{'} (α + θ (μ_{0} - α)) - Γ^{'} (μ_{0})) (μ_{0} - α) d θ∥ \\ + ∥ Γ^{'} {(μ_{0})}^{- 1} Γ^{'} (α) ∥ ∥ Γ^{'} {(μ_{0})}^{- 1} (A_{0} - Γ^{'} (μ_{0})) ∥ ∥ A_{0}^{- 1} Γ^{'} (α) ∥ ∥ Γ^{'} {(α)}^{- 1} Γ (μ_{0}) ∥ \\ \leq \frac{\int_{0}^{1} w ((1 - θ) ∥ μ_{0} - α ∥) d θ ∥ μ_{0} - α ∥}{1 - w_{0} (∥ μ_{0} - α ∥)} \\ + \frac{\int_{0}^{1} Δ_{0} (θ ∥ μ_{0} - α ∥) d θ \int_{0}^{1} Δ (θ ∥ μ_{0} - α ∥) d θ w_{1} (∥ μ_{0} - α ∥) {∥ μ_{0} - α ∥}^{2}}{(1 - w_{0} (∥ μ_{0} - α ∥)) (1 - p (∥ μ_{0} - α ∥))} \\ = g_{1} (∥ μ_{0} - α ∥) ∥ μ_{0} - α ∥ \leq ∥ μ_{0} - α ∥ < r, \end{matrix}

(28)

which illustrates (22) for

τ = 0

and

ν_{0} \in U (α, r)

Next, we have to prove that

B_{0}^{- 1} \in L (T_{2}, T_{1})

. By (5), (6), (10), (13), (15), (16) and (28), we get

\begin{matrix} ∥ {(2 Γ^{'} (α))}^{- 1} (B_{0} - 2 Γ^{'} (α)) ∥ & \leq \frac{1}{2} [∥ Γ^{'} {(α)}^{- 1} (Γ^{'} (μ_{0}) - Γ^{'} (α)) ∥ + 4 ∥ Γ^{'} {(α)}^{- 1} Q (μ_{0}) ∥ ∥ Γ (μ_{0}) ∥ \\ + 2 ∥Γ^{'} {(α)}^{- 1} Γ^{'} (\frac{μ_{0} + ν_{0}}{2})∥ + ∥ Γ^{'} {(α)}^{- 1} (Γ^{'} (ν_{0}) - Γ^{'} (α)) ∥] \\ \leq \frac{1}{2} [w_{0} (∥ μ_{0} - α ∥) + 4 \int_{0}^{1} Δ_{0} (θ ∥ μ_{0} - α ∥) d θ w_{1} (∥ μ_{0} - α ∥) ∥ μ_{0} - α ∥ \\ + \int_{0}^{1} Δ (\frac{θ}{2} (∥ μ_{0} - α ∥ + ∥ ν_{0} - α ∥)) d θ (∥ μ_{0} - α ∥ + ∥ ν_{0} - α ∥) + w_{0} (∥ μ_{0} - α ∥)] \\ \leq q (∥ μ_{0} - α ∥) < q (r) < 1 . \end{matrix}

(29)

Hence,

B_{0}^{- 1} \in L (T_{2}, T_{1})

is valid by solver (2), and

∥ B_{0}^{- 1} Γ^{'} (α) ∥ \leq \frac{1}{2 (1 - q (∥ μ_{0} - α ∥))} .

(30)

Then, by the last sub step of solver (2), (5), (6), (11), (15), (28) and (30), we have in turn that

\begin{matrix} ∥ μ_{1} - α ∥ & \leq ∥ ν_{0} - α ∥ + 4 ∥ B_{0}^{- 1} Γ^{'} (α) ∥ ∥ Γ^{'} {(α)}^{- 1} Γ (ν_{0}) ∥ \\ \leq g_{1} (∥ μ_{0} - α ∥) ∥ μ_{0} - α ∥ + \frac{2 \int_{0}^{1} Δ (θ ∥ ν_{0} - α ∥) d θ g_{1} (∥ μ_{0} - α ∥) ∥ μ_{0} - α ∥}{1 - q (∥ μ_{0} - α ∥)} \\ = g_{2} (∥ μ_{0} - α ∥) ∥ μ_{0} - α ∥ \leq ∥ μ_{0} - α ∥ < r, \end{matrix}

(31)

which illustrates (20) for

τ = 0

and

λ_{0} \in U (α, r)

. By restoring

μ_{0}

ν_{0}, μ_{1}

μ_{σ}

ν_{σ}

μ_{σ + 1}

in the succeeding estimates, we attain (19) and (20). Then, in view of the estimates

∥ μ_{σ + 1} - α ∥ \leq c ∥ μ_{σ} - α ∥ < r, c = g_{2} (∥ μ_{0} - α ∥) \in [0, 1),

(32)

that attain

lim_{σ \to \infty} μ_{σ} = α

and

μ_{σ + 1} \in U (α, r)

. Finally, the uniqueness of solution is required. Therefore, we assume that

ν^{*} \in D_{1}

with

Γ (ν^{*}) = 0

and

Q = \int_{0}^{1} Γ^{'} (α + θ (α - ν^{*})) d θ

By adopting (9) and (16), we yield

\begin{matrix} ∥ Γ^{'} {(α)}^{- 1} (Q - Γ^{'} (α)) ∥ \leq ∥ \int_{0}^{1} w_{0} (θ ∥ ν^{*} - α ∥) d θ \\ \leq \int_{0}^{1} w_{0} (θ R) d θ < 1 . \end{matrix}

(33)

So, Q is invertible in view of

0 = Γ (α) - Γ (ν^{*}) = Q (α - ν^{*}),

(34)

that yields

α = ν^{*}

. ☐

Remark 1.

(a): It is straightforward from the expression of (14) that we can drop the hypothesis (16) and restore as

$Δ (ζ) = 1 + w_{0} (ζ) or Δ (ζ) = 1 + w_{0} (r_{0}),$

(35)

since,

$\begin{matrix} ∥ Γ^{'} {(α)}^{- 1} [(Γ^{'} (μ) - Γ^{'} (α)) + Γ^{'} (α)] ∥ & \leq 1 + ∥ Γ^{'} {(α)}^{- 1} (Γ^{'} (μ) - Γ^{'} (α)) ∥ \\ \leq 1 + w_{0} (∥ μ - α ∥) \\ = 1 + w_{0} (ζ) for ∥ μ - α ∥ \leq r_{0} . \end{matrix}$

(36)
(b): We can choose

$r_{0} = w_{0}^{- 1} (1),$

(37)

instead of (5) provided the function $w_{0}$ is strictly increasing.
(c): If $w_{0}, w, Δ$ are constants functions, then we have

$r_{1} = \frac{2}{2 w_{0} + w}$

(38)

and

$r \leq r_{1} .$

(39)

The $r_{1}$ stands for the radius of the following Newton’s solver

$μ_{τ + 1} = μ_{τ} - Γ^{'} {(μ_{τ})}^{- 1} Γ (μ_{τ}) .$

(40)

Rheindoldt [22] and Traub [5] also suggested convergence radius instead of $r_{1}$

$r_{T R} = \frac{2}{3 w_{1}},$

(41)

and by Argyros [1,2]

$r_{A} = \frac{2}{2 w_{0} + w_{1}},$

(42)

where $w_{1}$ is a Lipschitz parameter for (10) on $K$ . Hence, we have

$w \leq w_{1}, w_{0} \leq w_{1},$

(43)

so

$r_{T R} \leq r_{A} \leq r_{1}$

(44)

and

$\frac{r_{T R}}{r_{A}} \to \frac{1}{3} a s \frac{w_{0}}{w} \to 0 .$

(45)

The convergence radius q suggested by Dennis and Schabel [1] is smaller than the radius $r_{D S}$

$q < r_{S D} = \frac{1}{2 w_{1}} < r_{T R} .$

(46)

However, q can not be calculated by the Lipschitz conditions.
(d): By adopting conditions on the ninth-order derivative of operator Γ, the order of convergence of solver (2) was provided by Shah et al. [20]. But, we assume hypotheses only on first-order derivative of operator Γ. For obtaining the computational order of convergence $(C O C)$ , we adopted expressions (3) and (4).
(e): Assume [1,2] satisfying the autonomous differential equation

$Γ^{'} (μ) = P (Γ (μ))$

(47)

where P is a given and continuous operator. Then, $Γ^{'} (α) = P (Γ (α)) = P (0),$ our results apply. But, without knowledge of α and choose $Γ (μ) = e^{μ} - 1 .$ Hence, we select $P (μ) = μ + 1$ .

3. Numerical Experimentation

Here, we illustrate the theoretical consequences suggested in Section 2. Next, we choose

Q = I

in the first four examples.

Example 1.

Let

T_{1} = T_{2} = H

and

H = C [0, 1]

. We study the mixed Hammerstein-like equation [6,23], defined by

μ (s) = 1 + \int_{0}^{1} G (s, ζ) (μ {(ζ)}^{\frac{3}{2}} + \frac{μ {(ζ)}^{2}}{2}) d ζ

(48)

where

Γ (s, ζ) = \{\begin{matrix} (1 - s) ζ, ζ \leq s, \\ s (1 - ζ), s \leq ζ, \end{matrix}

(49)

defined in

[0, 1] \times [0, 1]

. The solution

α (s) = 0

is the same as zero of (1), where

Γ : H \to H

, given as:

Γ (μ) (s) = μ (s) - \int_{0}^{ζ} G (s, ζ) (μ {(ζ)}^{\frac{3}{2}} + \frac{μ {(ζ)}^{2}}{2}) d ζ .

(50)

But

∥\int_{0}^{ζ} G (s, ζ) d ζ∥ \leq \frac{1}{8} .

(51)

Then, we have that

Γ^{'} (μ) ν (s) = ν (s) - \int_{0}^{ζ} G (s, ζ) (\frac{3}{2} μ {(ζ)}^{\frac{1}{2}} + μ (ζ)) d ζ,

and since

Γ^{'} (α (s)) = I

∥ Γ^{'} {(α)}^{- 1} (Γ^{'} (μ) - Γ^{'} (ν)) ∥ \leq \frac{1}{8} [\frac{3}{2} {∥ μ - ν ∥}^{\frac{1}{2}} + ∥ μ - ν ∥] .

(52)

Therefore, we can choose

w_{0} (ζ) = w (ζ) = \frac{1}{8} [\frac{3}{2} ζ^{\frac{1}{2}} + ζ] .

Hence, by Remark 2.2(a), we can set

Δ_{0} (ζ) = Δ (ζ) = 1 + w_{0} (ζ) a n d w_{1} (ζ) = 1 .

But, theorems in [20] can not be utilized to solve this problem because

Γ^{'}

is not Lipschitz. Notice though that our theorems can be utilized. We have the following radii for Example 1:

\begin{matrix} r_{1} = 0.321768, r_{q} = 0.284919, r_{2} = 0.119079, \end{matrix}

r = 0.119079 .

Example 2.

Consider, setting

T_{1} = T_{2} = R^{3}

and

Ω = Ω (0, 1)

. Then, for

w = {(μ, ν, λ)}^{T}

define a function

Γ : Ω \to R^{3}

as follows:

Γ (w) = {(e^{μ} - 1, \frac{e - 1}{2} ν^{2} + ν, λ)}^{T} .

(53)

Then, we obtain

Γ^{'} (w) = [\begin{matrix} e^{μ} & 0 & 0 \\ 0 & (e - 1) ν + 1 & 0 \\ 0 & 0 & 1 \end{matrix}] .

Hence, for

μ = {(0, 0, 0)}^{T}

we can choose

w_{0} (ζ) = (e - 1) ζ, w (ζ) = e^{\frac{1}{e - 1}} ζ

Δ_{0} (ζ) = Δ (ζ) = e^{\frac{1}{e - 1}}

, and

w_{1} (ζ) = 1

. By adopting these functions and parameters, we obtain the following radii for Example 2:

\begin{matrix} r_{1} = 0.121854, r_{q} = 0.134127, r_{2} = 0.0370321, \end{matrix}

r = 0.0370321 .

Example 3.

Let us choose that

T_{1} = T_{2} = H

, facilitated by the max norm. In addition, we consider

B (x) = Γ^{″} (μ)

and

K = \bar{U} (0, 1)

for every

μ \in K

. Choose a function Γ on

K

Γ (φ) (μ) = ϕ (μ) - 5 \int_{0}^{1} μ θ φ {(θ)}^{3} d θ,

(54)

which yields

Γ^{'} (φ (ξ)) (μ) = ξ (μ) - 15 \int_{0}^{1} μ θ φ {(θ)}^{2} ξ (θ) d θ, for each ξ \in Ω .

(55)

Then, we have that

w_{0} (ζ) = 7.5 ζ, w (ζ) = 15 ζ

and

Δ_{0} (ζ) = Δ (ζ) = 2

, and

w_{1} (ζ) = 1

. We have the following radii for Example 3:

\begin{matrix} r_{1} = 0.0453881, r_{q} = 0.0587713, r_{2} = 0.0133343, \end{matrix}

r = 0.0133343 .

Example 4.

By the academic problem that we considered in the introduction. We can choose

w_{0} (ζ) = w (ζ) = 96.662907 ζ

and

Δ_{0} (ζ) = 6, Δ (ζ) = 2

, and

w_{1} (ζ) = \frac{1}{3}

. By adopting these functions and parameters, we yield the following radii, for Example 4:

\begin{matrix} r_{1} = 0.00641476, r_{q} = 0.00741294, r_{2} = 0.00247724, \end{matrix}

r = 0.00247724 .

4. Concluding Assertions

We first generalized solver (2) from functions on the real line to Banach space valued operators. Then, we presented a local convergence analysis in this setting and by using generalized-continuity conditions. Our analysis uses only the first derivative appearing in the solver. In the special case of the real line, derivatives up to the order seven were used. Notice that these high order derivatives do not appear in the solver (2) and also limit the applicability of the solver, as we saw in the introduction. Hence, the applicability of solver (2) has been significantly extended. Numerical examples and applications complete the paper.

Author Contributions

Both authors have equal contribution for this paper. All authors have read and agreed to the published version of the manuscript.

Funding

This work was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under grant No. (D-274-130-1440). The authors, therefore, acknowledge with thanks DSR technical and financial support.

Conflicts of Interest

The authors declare no conflict of interest.

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Behl, R.; Argyros, I.K. Local Convergence of Solvers with Eighth Order Having Weak Conditions. Symmetry 2020, 12, 70. https://doi.org/10.3390/sym12010070

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Behl R, Argyros IK. Local Convergence of Solvers with Eighth Order Having Weak Conditions. Symmetry. 2020; 12(1):70. https://doi.org/10.3390/sym12010070

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Behl, Ramandeep, and Ioannis K. Argyros. 2020. "Local Convergence of Solvers with Eighth Order Having Weak Conditions" Symmetry 12, no. 1: 70. https://doi.org/10.3390/sym12010070

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Behl, R., & Argyros, I. K. (2020). Local Convergence of Solvers with Eighth Order Having Weak Conditions. Symmetry, 12(1), 70. https://doi.org/10.3390/sym12010070

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Local Convergence of Solvers with Eighth Order Having Weak Conditions

Abstract

1. Introduction

2. Study of Local Convergence

3. Numerical Experimentation

4. Concluding Assertions

Author Contributions

Funding

Conflicts of Interest

References

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