On the Local convergence of two-step Newton type Method in Banach Spaces under generalized Lipschitz Conditions

The motive of this paper is to discuss the local convergence of a two-step Newton type method of convergence rate three for solving nonlinear equations in Banach spaces. It is assumed that the first order derivative of nonlinear operator satisfies the generalized Lipschitz i.e. $L$-average condition. Also, some results on convergence of the same method in Banach spaces are established under the assumption that the derivative of the operators satisfies the radius or center Lipschitz condition with a weak $L$-average particularly it is assumed that $L$ is positive integrable function but not necessarily non-decreasing.


Introduction
Consider a nonlinear operator t : Ω ⊆ X → Y such that X and Y are two Banach spaces, Ω 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 t(x) = 0. (1.1) To find the solution of equation (1.1) , Newton's method defined as is being preferred though its speed of convergence is low. Newton's method [9], is a well known iterative method which converges quadratically, has been initially studied by Kantorovich [5] and then scrutinized by Rall [10]. 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], [7], [8] and [14]). While the methods of higher R-order of convergence are generally not executed frequently despite having fast speed of convergence because its operational cost is high. But the method of higher R-order of convergence can be used in the problems of stiff system where fast convergence is required.
In the numerical point of view, the convergence domain plays a crucial role for the stable behavior 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 ( [1], [2], [6] and [13]).
Here, we discuss the local convergence of the classical third-order modification of two-step Newton's method [11] under the L-average condition which is expressed as: (1. 3) The important characteristic of the method (1.3) is that: it is simplest and efficient third-order iterative method, per iteration it requires two evaluations of the function t, one of the first derivative t ′ and no evaluations of the second derivative t ′′ 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 [15], in which a non-decreasing positive integrable function was used instead of usual Lipschitz constant. Further, Wang and Li [16] 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 [12] have studied the local convergence of the two step Secant-type method when the first-order divided differences satisfy the generalized Lipschitz conditions. Now, 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 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 (1.3). 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 the 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 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 is assumed to belong to some family of positive integrable functions that are not necessarily non-decreasing for convergence theorems. An example is also presented to justify the significance of the last result.

Generalized Lipschitz conditions
Here, we denote by V (x * , r) = {x : ||x − x * || < r} a ball with radius r and center x * . The condition imposed on the function t we call it the center Lipschitz condition in the ball V (x * , r) with constant L. Furthermore, L in the Lipschitz conditions does not necessarily have to be constant but can be a positive integrable function. In this case, conditions (2.1) − (2.2) are respectively, replaced by and where ρ(x) = ||x−x * ||. At the same time, the corresponding ' Lipschitz conditions' is referred as to as having the L-average or generalized Lipschitz conditions. Now, we start with the following lemmas, which will be used later in the main theorems.
[16] Suppose that L is positive integrable. Assume that the function L a defined by relation (5.7) is non-decreasing for some a with 0 ≤ a ≤ 1. Then, f or each b ≥ 0, the function ϕ b,a defined by is also non-decreasing.
3. Local convergence of Newton type method (1.3) In this section, we state existence theorem under radius Lipschitz condition for Newton type method (1.3).
satisfies the radius Lipschitz condition with the L-average: where y τ = x * + τ (y − x * ), ρ(x) = ||x − x * || and L is non-decreasing. Let r satisfies the relation Then two step Newton type method (1.3) is convergent for all x 0 ∈ V (x * , r) and where the quantities , (3.5) are less than 1. Furthermore, Proof. On arbitrarily choosing x 0 ∈ V (x * , r), where r satisfies the relation (3.2), q 1 and q 2 , defined according to inequality (3.5) are less than 1. Indeed, since L is monotone, we get Obviously, if x ∈ V (x * , r), then using center Lipschitz condition with the L-average and the relation (3.2), we have Taking into account the Banach Lemma and the below equation we come to following inequality by using the relation (3.7) Expanding t(x n ) along x * from Taylor's Expansion, we attain (3.10) Also, from the expression (3.1) and combining the equations (3.9) and (3.10), it can written as In view of Lemma (2.1) and the above inequality, we can obtain the first inequality of expression (3.3). By similar analogy and using the last sub-step of the scheme (1.3), we can write Using Lemma (2.1) and above expression, we can get the first inequality of expression (3.4). Furthermore, ρ(x n ) and ρ(y n ) are decreasing monotonically, therefore for all n = 0, 1, ..., we have Also, by using second inequality of expression (3.3), we have

13)
Hence, we have the complete inequalities of expressions (3.3) and (3.4). Also, it can be seen that inequality (3.6) may be easily derived from the expression (3.13).

The uniqueness ball for the solution of equations
Here, we derived uniqueness theorem under center Lipschitz condition for Newton type method (1.3).
Then the equation Proof. On arbitrarily choosing y * ∈ V (x * , r), y * = x * and considering the iteration Expanding t(y * ) along x * from Taylor's expansion, we have (4.4) Following the expression (4.1) and combining the inequalities (4.3) and (4.4), we can write In view of Lemma (2.1) and expression (4.5), we obtain but this contradicts our assumption. Thus, we see that y * = x * . This completes the proof of the theorem.
In particular, assuming that L is a constant, we obtain the following corollaries (4.1) and (4.2) from theorems (3.1) and (4.1), respectively.
where y τ = x * + τ (y − x * ), ρ(x) = ||x − x * || and L is a positive number. Let r satisfies the relation Then two-step Newton type method (1.3) is convergent for all x 0 ∈ V (x * , r) and where the quantities where ρ(x) = ||x − x * || and L is a positive number. Let r fulfills the condition r = 1 L . (4.14) Then the equation t(x) = 0 has a unique solution x * in V (x * , r). Moreover, the ball radius r depends only on L.
Now, we will apply our main theorems to some special function L and immediately obtain the following corollaries.
Then two-step Newton type method (1.3) is convergent for all x 0 ∈ V (x * , r) and

18)
where the quantities

22)
where ρ(x) = ||x − x * ||. Let r satisfies the relation Then the equation t(x) = 0 has a unique solution x * in V (x * , r). Moreover, the ball radius r depends only on L and γ.

Convergence under weak L-average
This section contains the results on re-investigation of the conditions and radius of convergence of considered scheme already presented in the first theorem but L is not taken as non-decreasing function. It has been noticed that the convergence order decreases. The second theorem of this section gives a similar result to theorem (3.1) but under the assumption of center Lipschitz condition.
Then two-step Newton type method (1.3) is convergent for all x 0 ∈ V (x * , r) and

4)
where the quantities

5)
are less than 1. Moreover, Furthermore, suppose that the function L a is defined by is non-decreasing for some a with 0 ≤ a ≤ 1 and r satisfies 1 2r Then the two-step Newton type method (1.3) is convergent for all x 0 ∈ V (x * , r) and where the quantity is less than 1.
Proof. On arbitrarily choosing x 0 ∈ V (x * , r), where r satisfies the relation (5.2), the quantities q 1 and q 2 defined by the equation (5.5) are less than 1. Indeed, since L is positive integrable, we can get Obviously, if x ∈ V (x * , r), then using center Lipschitz condition with the L average and the relation (5.2), we have Taking into account the Banach Lemma and the below equation we come to following inequality using the relation (5.11) Now, if x n ∈ V (x * , r), then we may write from first sub-step of scheme (1.3) Expanding t(x n ) around x * from Taylor's Expansion, it can written as (5.14) Following the hypothesis (5.1) and combining the inequalities (5.13) and (5.14), we may write Using the results of Lemma (2.1) and the inequality (5.12) in the above expression we can obtain the first inequality of (5.3). By similar analogy for the last sub-step of the scheme (1.3), we can write Using Lemma (2.1) in the above expression, we can get the first inequality of (5.4). Furthermore, ρ(x n ) and ρ(y n ) are decreasing monotonically, therefore for all n = 0, 1, ..., we have Using the second inequality of expression (5.3), we arrive at Also, the inequality (5.6) may be easily derived from the expression (5.17). Furthermore, if the function L a defined by the relation (5.7) is non-decreasing for some a with 0 ≤ a ≤ 1 and r is determined by inequality (5.8), it follows from the first inequality of expression (5.3) and Lemma (2.2) that Moreover, from the first inequality of (5.4) and Lemma (2.2), we can write where Q 1 < 1 and Q 2 < 1 are determined by the expression (5.10). Also, the inequality (5.9) may be easily derived and hence x n converges to x * . Thus the proof is completed.
where ρ(x) = ||x − x * || and L is positive integrable function. Let r satisfies Then the two-step Newton type method (1.3) is convergent for all x 0 ∈ V (x * , r) and

20)
where the quantities are less than 1. Moreover,

22)
Furthermore, suppose that the function L a defined by the relation (5.7) is non-decreasing for some a with 0 ≤ a ≤ 1, then and q 1 is given by the first expression of the equation (5.21).