A Semilocal Convergence for a Uniparametric Family of Efficient Secant-Like Methods

We present a semilocal convergence analysis for a uniparametric family of efficient secant-like methods (including the secant and Kurchatovmethod as special cases) in a Banach space setting (Ezquerro et al., 2000–2012). Using our idea of recurrent functions and tighter majorizing sequences, we provide convergence results under the same or less computational cost than the ones of Ezquerro et al., (2013, 2010, and 2012) and Hernández et al., (2000, 2005, and 2002) and with the following advantages: weaker sufficient convergence conditions, tighter error estimates on the distances involved, and at least as precise information on the location of the solution. Numerical examples validating our theoretical results are also provided in this study.


Introduction
Let (, ) and (, ) stand, respectively, for the open and closed ball in X with center  ∈ X and radius  > 0. Denote by L(X, Y) the space of bounded linear operators from X into Y.
In this study, we are concerned with the problem of approximating a locally unique solution  * of nonlinear equation as follows: where  is a Fréchet-differentiable operator defined on a nonempty convex subset D of a Banach space X with values in a Banach space Y.
Many problems from computational sciences, physics and other disciplines can be taken in the form of (1) using mathematical modelling [1][2][3][4][5][6][7].The solution of these equations can rarely be found in closed form.That is why the solution methods for these equations are iterative.In particular, the practice of numerical analysis for finding such solutions is essentially connected to variants of Newton's method [1,2,[4][5][6][7][8][9][10].The study about the convergence of iterative procedures is usually focused on two types: semilocal and local convergence analysis.The semilocal convergence is, based on the information around an initial point, to give criteria ensuring the convergence of iterative procedure; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls.There are a lot of studies on the weakness and/or extension of the hypothesis made on the underlying operators; see, for example,  and the references therein.
Secant-like method (2) can be considered as a combination of the secant and Newton's method.Indeed, if  = 0, we obtain the secant method and if  = 1, we get Newton's method provided that   is Fréchet-differentiable on D, since then   =   and [  ,   ; ] =   (  ).
It was shown in [20,21] that the -order of convergence is at least (1 + √ 5)/2 for  ∈ [0, 1), the same as that of the secant method.Later in [5], another uniparametric family of secant-like methods defined by was studied.It was shown that there exists  0 ≥ 2, and that the -order of convergence is at least , and if  = 2, the -order of convergence is quadratic.Note that if  = 1, we obtain the secant method, whereas if  = 2, we obtain the Kurchatov method [4,5,7,8].
We present a semilocal convergence analysis for secantlike method (2) using our idea of recurrent functions instead of recurrent relations and tighter majorizing sequences.This way, our analysis provided the following advantages () over the work in [5] under the same computational cost.whereas the ones in [5] are given in nonaffine invariant forms.The advantages of affine versus nonaffine results have been explained in [1, 4, 6-8, 14, 15].
Our hypotheses for the semilocal convergence of secantlike method (4) are as follows.
( 4 ) There exists  > 0 such that We will denote by () conditions ( 1 )-( 4 ).In view of ( 4 ), there exist  0 ,  1 ,  > 0 such that Clearly, hold in general and /, / 1 can be arbitrarily large [1,2,4].Note that ( 5 ), ( 6 ), and ( 7 ) are not additional to ( 4 ) hypotheses.In practise, the computation of  requires the computation of  0 ,  1 , and .It also follows from ( 4 ) that  is differentiable [1-3, 8, 9].The paper is organized as follows.In Section 2, we show that under the same hypotheses as in [23] and using recurrent relations, we obtain at least as precise information on the location of the solution.Section 3 contains the semilocal convergence analysis using weaker hypotheses and recurrent functions.We also show the advantages ().The results are also extended to cover the case of equations with nondifferentiable operators.Numerical examples are presented in the concluding Section 4.
Otherwise, that is, if  < , then our Theorem 1 constitutes an improvement over Theorem 3, since where where given in [5] (for  = 1).Hence, (16) justify our claim for this section which was made in the Introduction of this study.

Semilocal Convergence Using Recurrent Functions
We present the semilocal convergence of secant-like methods.First, we need some auxiliary results on majorizing sequences for secant-like method.
Remark 5. (a) Let us consider an interesting choice for .Let  = 1 (secant method).Then, using ( 21) and ( 22), we have that The corresponding condition for the secant method is given by [2,4,9,23] as follows: Condition ( 52) can be weaker than (53) (see also the numerical examples at the end of the study).Moreover, the majorizing sequence {  } for the secant method related to (53) is given by A simple inductive argument shows that if  < , then for each  = 2, 3, . ..: (b) The majorizing sequence {V  } used in [5] is essentially given by Then, again we have Moreover, our sufficient convergence conditions can be weaker than [5].
Proof.The proof until the uniqueness part follows as in Theorem 8 but using the following identity: instead of (74).Finally, for the uniqueness part, let  * ∈ ( 0 , ) be such that ( * ) + ( * ) = 0.Then, we get from (83) the identity This identity leads to

Numerical Examples
Example 1.Let X = Y = C[0, 1], equipped with the max-norm.Consider the following nonlinear boundary value problem: It is well known that this problem can be formulated as the integral equation where Q is the Green function as follows: We observe that max Then, problem (101) is in the form (1), where  : D → Y is defined as The Fréchet derivative of the operator  is given by Then, we have that Hence, if 2 < 5, then       −   ( 0 )      ≤ 2 ( − 2) < 1.
The convergence of the secant-type method is also ensured by Theorem 8.
Notice that the conditions of Theorem 1 and Lemma 3 are satisfied, but since  < , Remark 2 ensures that our uniqueness ball is larger.It is clear as  1 = 1.83333 ⋅ ⋅ ⋅ > 0.193452 ⋅ ⋅ ⋅ =  0 .