1. Introduction
Consider a nonlinear operator
such that
X and
Y are two Banach spaces,
is a non-empty open convex subset and
t is Fréchet differentiable nonlinear operator. Nonlinear problems has so many applications in the field of chemical engineering, transportation, operational research etc. which can be seen in the form of
To find the solution of Equation (1), Newton’s method defined as
is being preferred though its speed of convergence is low. Newton’s method [
1], is a well known iterative method which converges quadratically, which was initially studied by Kantorovich [
2] and then scrutinized by Rall [
3].
Some Newton-type methods with third-order convergence that do not require the computation of second order derivatives have been developed in the refs [
4,
5,
6,
7]. While the methods of higher
R-order of convergence are generally not executed frequently despite having fast speed of convergence because the operational cost is high. However, the method of higher
R-order of convergence can be used in the problems of stiff system [
2] where fast convergence is required.
From the numerical point of view, the convergence domain plays a crucial role for the stable behaviour of an iterative scheme. Research about the convergence study of Newton methods involves two types: semilocal and local convergence analysis. The semilocal convergence study is based on the information around an initial point to give criteria ensuring the convergence of iterative methods; meanwhile, the local one is, based on the information around a solution, to find estimates for the radii of the convergence balls. Numerous researchers studied the local convergence analysis for Newton-type, Jarratt-type, Weerakoon-type, etc. in Banach space setting in the articles [
8,
9,
10,
11,
12,
13,
14] and reference therein. In most of articles, the local convergence have been discussed using the hypotheses of Lipschitz, Hölder or
w-continuity conditions but sometimes, we will come across that the nonlinear problems do not fulfilled any of these three conditions which limits the applicability of nonlinear equations, but satisfy the generalized Lipschitz condition. Also, the notable feature is that all these three are a particular case of the generalized Lipschitz or
L-average condition.
Here, we discuss the local convergence of the classical third-order modification of two-step Newton’s method [
15] under the
L-average condition which is expressed as:
The important characteristic of the method (3) is that: it is simplest and efficient third-order iterative method, per jth iteration it requires two evaluations of the function , one of the first derivative and no evaluations of the second derivative hence makes it computationally efficient. We find, in the literature, several studies on the weakness and/or extension of the hypotheses made on the underlying operators.
For re-investigating the local convergence of Newton’s method, generalized Lipschitz conditions was constructed by Wang [
16], in which a non-decreasing positive integrable function was used instead of usual Lipschitz constant. Furthermore, Wang and Li [
17] derived some results on convergence of Newton’s method in Banach spaces when derivative of the operators satisfies the radius or center Lipschitz condition but with a weak
L-average. Shakhno [
18] have studied the local convergence of the two step Secant-type method [
2], when the first-order divided differences satisfy the generalized Lipschitz conditions.
As a motivational example let
and
. Define function
t on
D for
by
Then, the Fréchet derivative is
Hence, and (see definitions (5) and (6)). Therefore, replacing L by at the denominator gives the benefits. If L and are not constants then we can take (see definitions (7), (8) and (110)).
Next, the intriguing question strikes out that whether the radius Lipschitz condition with L-average and the non-decreasing of L are necessary for the convergence of the third-order modification of Newton’s method. Motivated and inspired by the above mentioned research works in this direction in the present paper, we derived some theorems for scheme (3). In the first result generalized Lipschitz conditions has been used to study the local convergence which is important to enlarge the convergence region without additional hypotheses along with an error estimate. In the second theorem, the domain of uniqueness of solution has been derived under center Lipschitz condition. In the last two theorems, weak L-average has been used to derive the convergence result of the considered third-order scheme. Also, few corollaries are stated.
The rest part of this paper is structured as follows:
Section 2 contains the definitions related to
L-average conditions. The local convergence and its domain of uniqueness is mentioned in
Section 3 and
Section 4, respectively.
Section 5 deals with the improvement in assumption that the derivative of
t satisfies the radius and center Lipschitz condition with weak
L-average namely
L and
is assumed to belong to some family of positive integrable functions that are not necessarily non-decreasing for convergence theorems. Numerical examples are presented to justify the significance of the results.
2. Generalized Lipschitz Conditions
Here, we denote by
a ball with radius
r and center
. The condition imposed on the function
t
where
is usually called radius Lipschitz condition in the ball
with constant
L. Sometimes, if it is only required to satisfy
We call it the center Lipschitz condition in the ball
with constant
where
. Replacing
L by
in case
leads to wider choice of initial guesses (larger radius of convergence than in traditional studies) and fewer iterates to achieve an error tolerance and the uniqueness of the solution
is also extended in this case [
8,
12]. Furthermore,
L and
in the Lipschitz conditions do not necessarily have to be constant but can be a positive integrable function. In this case, conditions (
5)–(
6) are respectively, replaced by
and
where
and we have
. At the same time, the corresponding ‘Lipschitz conditions’ is referred as to as having the
L-average or generalized Lipschitz conditions. Next, we start with the following lemmas, which will be used later in the main theorems.
Lemma 1. Suppose that t has a continuous derivative in and exists.
(i) If satisfies the radius Lipschitz condition with the L-average:where , and L is non-decreasing, then we have (ii) If satisfies the center Lipschitz condition with the -average:where and is non-decreasing, then we have Proof. The Lipschitz conditions (9) and (11), respectively, imply that
where
and
. □
Lemma 2. [17] Suppose that L is positive integrable. Assume that the function defined by relation (62) is non-decreasing for some a with . Then, , the function defined byis also non-decreasing. 4. The Uniqueness Ball for the Solution of Equations
Here, we derived uniqueness theorem under center Lipschitz condition for Newton-type method (3).
Theorem 2. Suppose that , t has a continuous derivative in , exists and satisfies (8). Let r satisfy the relation Then, the equation has a unique solution in .
Proof. On arbitrarily choosing
,
and considering the iteration, we get
Expanding
along
from Taylor’s expansion, we have
Following the expression (11) and combining the inequalities (37) and (38), we can write
In view of Lemma (1) and expression (39), we obtain
However, this contradicts our assumption. Thus, we see that . This completes the proof of the theorem. □
In particular, assuming that L and are constants, we obtain the following Corollaries 1 and 2 from Theorems 1 and 2, respectively.
Corollary 1. Suppose that satisfies , t has a continuous derivative in , exists and satisfies (5) and (6). Let r satisfy the relation Then, the two-step Newton-type method (3) is convergent for all andwhere the quantitiesare less than 1
. Moreover Corollary 2. Suppose that satisfies , t has a continuous derivative in , exists and satisfies the assumption (6). Let r fulfill the condition Then, the equation has a unique solution in . Moreover, the ball radius r depends only on .
Next, we will apply our main theorems to some special function L and immediately obtain the following corollaries.
Corollary 3. Suppose that satisfies , t has a continuous derivative in , exists and s satisfies (9), (11) where given fixed positive constants γ, and with and i.e.,and, where , . Let r satisfy the relation Then, the two-step Newton-type method (3) is convergent for all andwhere the quantitiesare less than 1
. Moreover Corollary 4. Suppose that satisfies , t has a continuous derivative in , exists and satisfies (11) where given fixed positive constants γ and with i.e.,where . Let r satisfy the relation Then, the equation has a unique solution in . Moreover, the ball radius r depends only on and γ.