\(^\ast \)Department of Applied Mathematics, National University of Mongolia, PB 46/687 Ulaanbaatar 210646, Mongolia, e-mail: tzhanlav@yahoo.com

\(^\S \)Department of Applied Mathematics, National University of Mongolia, PB 46/687 Ulaanbaatar 210646, Mongolia, e-mail: pilpalpil@gmail.com

1 Introduction

Recently, many third order iterative methods free from second derivative have been derived and studied for nonlinear systems [ 1 ] - [ 10 ] . In particular, in [ 1 ] were suggested two Chebyshev-like (CL1, CL2) methods, while in [ 10 ] are considered two families of modifications of Chebyshev method (MOD1, MOD2). In [ 6 ] was also presented a new family of Chebyshev-type methods with a real parameter \(\theta \) \((A2_{\theta }).\) All the above mentioned methods are obtained using different approximations of second derivative in Chebyshev method.

In [ 9 ] it was proposed a family of third-order methods given by

for solving nonlinear scalar equations \(f(x)=0.\) In this study, we consider a generalization of methods (1) in Banach space, which is used to solve the nonlinear operator equation

Suppose that \(F\) is defined on an open convex domain \(\Omega \) of a Banach space \(X\) with values in a Banach space \(Y,\) \(F'(x)\) is a Frechet derivative in \(\Omega ,\) and \(F'(x)^{-1}\) exists. The generalization of methods (1) is

Thus we have a family of methods (3) for solving nonlinear equation (2). We consider some particular cases of (3). Let \(a=-1\). Then (3) leads to

which was proposed by Q.Wu and Y.Zhao in [ 7 ] . They established third-order convergence of this method by using majorizing function and obtained the error estimate. It should be mentioned that the iteration (4) for scalar equation was given also in [ 5 ] . Let \(a=-\tfrac {1}{2}\). Then (3) leads to

This is a generalization of the third order method proposed by the Frontini and Sormani [ 2 ] for scalar case. Thus, the proposed iteration (3) can be considered as a generalization of well known iterations.

We prove Newton-Kantorovich type convergence theorem for the family of methods (3) to show that it has third order convergence by using recurrent relations [ 3 ] - [ 4 ] and get the error bounds. Finally, some examples are provided to show the application of the proposed method.

2 Preliminaries

Let us assume that \(F'(x_0)^{-1} \in L(Y,X)\) exists for some \(x_0 \in \Omega ,\) where \(L(Y,X)\) is a set of bounded linear operators from \(Y\) into \(X.\) Moreover, we suppose that (see [ 4 ] )

Let \(F\) be a nonlinear twice Frechet differentiable operator in an open convex domain \(\Omega .\) We denote \(\Gamma _n= F'(x_n)^{-1},\)

where

and define the sequence

We need following technical lemmas, whose proofs are trivial [ 4 ] .

3 Convergence study

According to (3), we have

where

Using the following formula

in (11), we obtain

where

If we notice that \(M||\Gamma _0||||\Gamma _0F(x_0)||\leq a_0{\lt}\tfrac 23,\) then follows

which shows the existence of \(A_0^{-1}\)

where \(P_0=\tfrac 12 \Gamma _0F''(\xi _0)\Gamma _0F(x_0).\) So, from (10) we get

where \(R=\tfrac {1}{\left(1-\frac{a_0}{2}\right)(1-\gamma \Delta )}\). This means that \(y_0, x_1 \in B(x_0,R\eta )=\{ x\in X: \| x-x_0\| {\lt}R\eta \} .\)

In these conditions, we prove the following statements for \(n\geq 1\):

Assuming \(\tfrac {a_0}{1-a_0/2}{\lt}1\) which is valid for \(a_0{\lt}2/3\) and \(x_1\in \Omega \) we have

Then, by the Banach lemma, \(\Gamma _1\) is defined and satisfies

Taking into account (3) and the Taylor formula of \(x_n, y_n \in \Omega ,\) we have

Also

Substituting the last two expressions into (17), we get

From (3) we also obtain

and

where

Taking into account

we obtain

From (18) for \(n=0,\) we obtain \(||\Gamma _0F(x_n)||\); therefore

So, \((II_1)\) is true. To prove \((III_1)\) and \((IV_1)\), notice that

and

where

Since

there exists \(A_1^{-1}=(I-P_1)^{-1}\Gamma _1,\) thereby we get

Consequently, we obtain

Analogously, we get

i.e. \((IV_1)\) and \((V_1)\) are proved. Now, following an inductive procedure and assuming

the items \((I_n)-(V_n)\) can be proved. Notice that \(\Gamma _n {\gt}0\) for all \(n=0,1,\ldots .\) Indeed if \(\Gamma _k=0\) for some \(k,\) then due to statement \((I_n),\) we have \(\| \Gamma _n\| =0\) for all \(n\geq k.\) As a consequence, the iteration (3) terminated after \(k\)-th step, i.e. the convergence of iteration does not hold. To establish the convergence of \(\{ x_n\} \) we only have to prove that it is a Cauchy sequence and that the above assumptions (19) are true. Note that

As a consequence of Lemma 4, it follows that

So, from \(\Delta {\lt}1\) and \(\gamma {\lt}1,\) we deduce that \(\prod _{k=0}^{n-1}f(a_k)g(a_k)\) converges to zero by letting \(n\to \infty .\) We are now ready to state the main result on convergence for (3).