\(^{\ast }\)Cameron University, Dept. of Mathematics Sciences, Lawton, OK 73505, USA,
e-mail: iargyros@cameron.edu.

\(^{\S }\)Hangzhou Polytechnic, College of Information and Engineering, Hangzhou 311402, Zhejiang, P.R.China, e-mail: rhm65@126.comb.xy.uv.

\(^{\bullet }\)The research of the second author has been supported in part by National Natural Science Foundation of China (Grant No. 10871178), Natural Science Foundation of Zhejiang Province of China (Grant No. Y606154), and Scientific Research Fund of Zhejiang Provincial Education Department of China (Grant No. 20071362).

1 Introduction

In this study we are concerned with the problem of approximating a locally unique solution \(x^{\star }\) of an equation

where \(F\) is a Fréchet–differentiable operator defined on a non–empty, open subset \(\mathcal{D}\) of a Banach space \(\mathcal{X}\) with values in a Banach space \(\mathcal{Y}\).

A large number of problems in applied mathematics and engineering are solved by finding the solutions of certain equations. For example, dynamic systems are mathematically modeled by difference or differential equations, and their solutions usually represent the states of the systems. For the sake of simplicity, we assume that a time–invariant system is driven by the equation \(\dot{x}=Q(x)\), for some suitable operator \(Q\), where \(x\) is the state. Then the equilibrium states are determined by solving equation (1.1). Similar equations are used in the case of discrete systems. The unknowns of engineering equations can be functions (difference, differential, and integral equations), vectors (systems of linear or nonlinear algebraic equations), or real or complex numbers (single algebraic equations with single unknowns). Except in special cases, the most commonly used solution methods are iterative. In fact, starting from one or several initial approximations a sequence is constructed that converges to a solution of the equation. Iteration methods are also applied for solving optimization problems. In such cases, the iteration sequences converge to an optimal solution of the problem at hand. Since all of these methods have the same recursive structure, they can be introduced and discussed in a general framework.

A classic iterative process for solving nonlinear equations is Chebyshev’s method (see [5], [9], [ 14 ] , [ 17 ] ):

This one-point iterative process depends explicitly on the two first derivatives of \(F\) (namely, \(x_{n+1}=\psi (x_{n},F(x_{n}),F^{\prime }(x_{n}),F^{\prime \prime }(x_{n}))\)). Ezquerro and Hernández introduced in  [ 14 ] some modifications of Chebyshev’s method that avoid the computation of the second derivative of \(F\) and reduce the number of evaluations of the first derivative of \(F\). Actually, these authors have obtained a modification of the Chebyshev iterative process which only need to evaluate the first derivative of \(F\), (namely, \(x_{n+1}=\overline{\psi }(x_{n},F^{\prime }(x_{n})\)), but with third-order of convergence [14]. In this paper we recall this method as the Chebyshev–Newton–type method (CNTM) and it is written as follows:

where \(F^{\prime }(x)\) \((x\in \mathcal{D})\) is the Fréchet–derivative of \(F\).

There is an interest in constructing families of iterative processes free of derivatives. To obtain a new family in [9] we considered an approximation of the first derivative of \(F\) from a divided difference of first order, that is, \(F^{\prime }(x_{n})\approx [x_{n-1},x_{n},F]\), where, \([x,y;F]\) is a divided difference of order one for the operator \(F\) at the points \(x\), \(y \in \mathcal{D}\). Then, we introduce the Chebyshev–Secant–type method (CSTM)

where \(a\), \(b\), \(c\) are non–negative parameters to be chosen so that sequence \(\{ x_{k} \} \) converges to \(x^{\star }\). Note that CSTM is reduced to the secant method (SM) if \(a=0\), \(b=c=1/2\), and \(y_{k} = x_{k+1}\). Moreover, if \(x_{k-1} = x_{k}\), and \(F\) is differentiable on \(\mathcal{D}\), then, \(F^{\prime }(x_{k}) = [x_{k}, x_{k} ;F]\), and CSTM reduces to Newton’s method (NM).

We provided a semilocal convergence analysis for CSTM using recurrence sequences, and also illustrated its effectiveness through numerical examples. Bosarge and Falb [ 10 ] , Dennis [ 13 ] , Potra [ 23 ] , Argyros [ 1 ] – [ 5 ] , Hernández et al. [ 15 ] and others [ 16 ] , [ 22 ] , [ 26 ] , have provided sufficient convergence conditions for the SM based on Lipschitz-type conditions on divided difference operator (see, also relevant works in [ 8 ] – [ 13 ] , [ 18 ] , [ 21 ] , [ 24 ] , [ 27 ] ).

In this paper, we continue the study of inverse free iterative processes. We introduce the derivative free method (DFM):

Note that DFM reduces to the Kurchatov-type method (KTM)

if \(a=0,b=c=0.5\), and \(y_{k}=x_{k+1}\) [20], [25].

In this special case the quadratic convergence of KTM was first established in [20], [25] and then in [6], [7] under different sets of sufficient conditions. We provide a semilocal convergence analysis for DFM. Then, we give numerical examples to show that DFM is faster than CSTM. In particular, two numerical examples are also provided. Firstly, we consider a scalar equation where the main study of the paper is applied. Secondly, we discretize a nonlinear integral equation and approximate a numerical solution using DFM.

2 Semilocal convergence analysis of DFM

We shall show the semilocal convergence of DFM under the following conditions

1. \(F \, : \, \mathcal{D} \subseteq \mathcal{X} \longrightarrow \mathcal{Y}\) is a Fréchet–differentiable operator, and there exists divided difference denoted by \([x,y;F]\) satisfying

   \[ [x,y;F] (x-y) = F(x) - F(y) \quad \mathrm{for \, \, all} \quad x,y \in \mathcal{D} ; \]

2. There exist \(x_{-1}\) and \(x_{0}\) in \(\mathcal{D}\) and \(\beta {\gt}0\) such that

   \[ A_{0} ^{-1} = [2x_{0}-x_{-1}, x_{-1};F]^{-1} \in \mathcal{L} (\mathcal{Y} , \mathcal{X}) \]

   exists and

   \[ 0 {\lt} \parallel A_{0} ^{-1} \parallel \leq \beta ; \]

3. There exists \(d {\gt}0\) such that

   \[ \parallel x_{0} - x_{-1} \parallel \leq d ; \]

4. There exists \(\eta {\gt}0\) such that

   \[ 0 {\lt} \parallel A_{0}^{-1} \, F(x_{0} ) \parallel \leq \eta ; \]

5. There exists constant \(M {\gt}0\), such that for all \(x\), \(y\), \(u\), \(v\) in \(\mathcal{D}\)

   \[ \parallel [x,y;F] - [u,v;F] \parallel \leq \displaystyle \tfrac {M}{2 } \, (\parallel x- u \parallel + \parallel y -v \parallel ) ; \]

6. For \(a \in [0, 1]\), \(b \in [0,1]\) and \(c{\gt}0\) given in DFM, we suppose

   \[ (1-a) \, c = 1-b ; \]

1. \[ \alpha =2\, {\bigg(} 1 + d_{0} + a \, c \, \gamma \, (a+ 2\, d_{0}) {\bigg)} \, \gamma {\lt} 1 , \]

   where,

   \[ \gamma = \displaystyle \tfrac {\beta \, M \, \eta }{ 2}, \quad d_{0} = \displaystyle \tfrac {d}{\eta } ; \]
2. \[ \overline{U} (x_{0} , R=r \, \eta ) = \{ x \in \mathcal{X} \, : \, \parallel x- x_{0} \parallel \leq R \} \subseteq \mathcal{D} , \]

   for some \(r{\gt}1\) to be precised later in Theorem 5;

1. \[ x,y\in \mathcal{D}\Rightarrow 2\, y-x\in \mathcal{D}. \]

Delicate condition \((C_{9})\) is certainly satisfied, if \(\mathcal{D}=\mathcal{X}\). It is also satisfied, if \((C_{9})\) is replaced by

1. \[ \overline{U}(x_{0},3\, R)\subseteq \mathcal{D}. \]

Indeed, if \(x,y\in \overline{U}(x_{0},R)\), then

That is, \(2y-x\in \overline{U}(x_{0},3R)\).

We note by (\(\mathcal{C}\)) the set of conditions (\(\mathcal{C}_{1}\))–(\(\mathcal{C}_{9}\)).

We need an Ostrowski–type approximations for DFM. The proof is given in [5], [9].

The following relates DFM with scalar sequences introduced in Definition 1.

We have \(\parallel y_{0} - x_{0} \parallel \leq \eta \), and \(\parallel z_{0} - x_{0} \parallel \leq a \, \eta \), so that \(x_{0}\), \(z_{0} \in \mathcal{D}\).

Items \(\mathbf{(I_{0})}\) and \(\mathbf{(II_{0})}\) hold by (\(\mathcal{C} _{2}\)) and (\(\mathcal{C} _{4}\)), respectively. To prove \(\mathbf{(III_{0})}\), we use Lemma 2 for \(n=0\) to obtain by (\(\mathcal{C} _{2}\))–(\(\mathcal{C} _{5}\))

Moreover,

which implies \(\mathbf{(IV_{0})}\). Note also that \(z_{1} \in \mathcal{D}\). Following an inductive argument, assume \(x_{k} \in \mathcal{D}\), and \(2\, \gamma \, \mu _{k} \, (q _{k-1} + q_{k}) {\lt} 1\). Then, we have

It follows from the Banach lemma on invertible operators [1], [5], [19] that \(A_{k+1} ^{-1} \) exists, and

which shows \(\mathbf{(I_{k+1})}\). Using Lemma 2, (\(\mathcal{C} _{5}\)), and the induction hypotheses, we get

Then, we get

Moreover, by Lemma 2, we have

and consequently,

This completes the proof of Lemma 3.

We shall establish the convergence of sequence \(\{ x_{n} \} \) generated by DFM. This can be achieved by showing that \(\{ q_{n} \} \) is a Cauchy sequence, if the following conditions hold for \(n \geq 0\):

1. \(x_{n} \in \mathcal{D}\),

and

1. \(2\, \gamma \, \mu _{n} \, (q_{n-1} + q_{n} ) {\lt} 1 .\)

In the next result, we show the Cauchy property for sequence \(\{ q_{n} \} \).

1. We show using induction that all scalar sequences involved are positive. By Definition 1, and (\(\mathcal{C} _{8}\)), we have for \(j=0\): \(\mu _{j} \), \(p_{j}\), \(q_{j}\), \(w_{j}\), \(c_{j}\), and \(1- 2\, \gamma \, \mu _{j} \, (q_{j-1} + q_{j} )\) are positive. Assume \(\mu _{k} \), \(p_{k}\), \(q_{k}\), \(w_{k}\), \(c_{k}\), and \(1- 2\, \gamma \, \mu _{k} \, (q_{k-1} + q_{k} )\) are positive for all \(k \leq n\). Since \(c_{k} {\gt}0\), it follows from the definition of the scalar sequences that \(w_{k+1}\), \(\mu _{k+1}\), \(p_{k+1}\), \(c_{k+1}\) have the same sign. Assume the common sign to be negative. Then

   \[ \begin{array}[c]{l}q_{k-1}+ q_{k} +q_{k+1} {\lt} q_{k-1} + q_{k}\\ \Longrightarrow 1- 2\, \gamma \, \mu _{k} \, (q_{k-1}+ q_{k} +q_{k+1}) {\gt} 1 -2\, \gamma \, \mu _{k} \, (q_{k-1}+ q_{k} )\\ \Longrightarrow \displaystyle \tfrac {1- 2\, \gamma \, \mu _{k} \, (q_{k-1}+ q_{k} +q_{k+1})}{ 1 - 2\, \gamma \, \mu _{k} \, (q_{k-1}+ q_{k} )} {\gt} 1 . \end{array} \]

   But it follows from the definition of sequence \(\{ \mu _{k} \} \) that

   \[ \begin{array}[c]{l}1- 2\, \gamma \, \mu _{k+1} \, (q_{k} + q_{k+1} ) = \displaystyle \tfrac {1-2\, \gamma \, \mu _{k} \, (q_{k-1}+ 2\, q_{k} +q_{k+1})}{ 1 - 2\, \gamma \, \mu _{k} \, (q_{k-1}+ q_{k} )}\\ \begin{array}[c]{lll}\Longrightarrow 1-2\, \gamma \, \mu _{k+1} \, q_{k+1} & = & \displaystyle \tfrac {1-2\, \gamma \, \mu _{k} \, (q_{k-1}+ 2\, q_{k} +q_{k+1})}{ 1 -2\, \gamma \, \mu _{k} \, (q_{k-1}+ q_{k} )} + 2\, \gamma \, \mu _{k+1} \, q_{k}\\ & = & \displaystyle \tfrac {1- 2\, \gamma \, \mu _{k} \, (q_{k-1}+ q_{k} +q_{k+1})}{ 1 -2\, \gamma \, \mu _{k} \, (q_{k-1}+ q_{k} )} {\gt} 1, \end{array}\end{array} \]

   which is a contradiction, since we get \(2\, \gamma \, \mu _{k+1} \, q_{k+1} {\lt} 0\), but \(\mu _{k+1}, \, q_{k+1}\) have the same sign, and \(\gamma {\gt}0\). The induction is then completed.

   By the definition of sequence \(\{ \mu _{n}\} \) and \(\mu _{0} =1\), we have

   \[ \begin{array}[c]{l}1- 2\, \gamma \, \mu _{k} \, ( q_{k-1} + q_{k}) = \displaystyle \tfrac {\mu _{k}}{\mu _{k+1}}\\ \Longrightarrow q_{k-1} + q_{k} = \displaystyle \tfrac {1}{2\, \gamma } \, (\displaystyle \tfrac {1}{ \mu _{k}} - \displaystyle \tfrac {1}{\mu _{k+1} })\\ \Longrightarrow \displaystyle \sum _{i=0} ^{k-1} ( q_{i-1} + q_{i} ) = \displaystyle \tfrac {1}{2\, \gamma } \, (\displaystyle \tfrac {1}{ \mu _{0}} - \displaystyle \tfrac {1}{\mu _{k} }) = \displaystyle \tfrac {1}{2\, \gamma } \, (1 - \displaystyle \tfrac {1}{\mu _{k} })\\ \Longrightarrow \mu _{k} = \displaystyle \tfrac {1}{1 -2\, \gamma \, \displaystyle \sum _{i=0} ^{k-1} ( q_{i-1} + q_{i} ) } . \end{array} \]

   But \(1- 2\, \gamma \, \displaystyle \sum _{i=0} ^{k-1} ( q_{i-1} + q_{i} )\) decreases. Therefore, sequence \(\{ \mu _{k} \} \) increases, and consequently \(\mu _{k} \geq \mu _{0} =1\).

2. We have that sequence \(\mu _{k} {\gt}1\) is increasing, so that \(0 \leq \displaystyle \tfrac {1}{\mu _{k}} \leq 1\). Since \(\{ \displaystyle \tfrac {1}{\mu _{k}} \} \) is monotonic on a compact set, it converges to \(\displaystyle \tfrac {1}{\mu _{\infty }}\). Then, we have

   \[ \displaystyle \lim _{k \longrightarrow \infty } (q_{k-1} + q_{k}) = \displaystyle \tfrac {1}{2\, \gamma } \, \displaystyle \lim _{k \longrightarrow \infty } (\displaystyle \tfrac {1}{\mu _{k}} - \displaystyle \tfrac {1}{\mu _{k+1}}) = \displaystyle \tfrac {1}{2\, \gamma } \, (\displaystyle \tfrac {1}{\mu _{\infty }} - \displaystyle \tfrac {1}{\mu _{\infty }})=0 . \]

This completes the proof of Lemma 4.

We can show the main semilocal convergence theorem for DFM.

and \(\displaystyle \lim _{n \longrightarrow \infty } q_{n} =0\), which imply \(\displaystyle \lim _{n \longrightarrow \infty } c_{n} =0\). By letting \(n \longrightarrow \infty \) in (2.4), we get \(F(x^{\star }) =0\).

We also have

which imply \(x_{n} \in \overline{U} (x_{0} , r \, \eta ) \). Consequently, we obtain \(x^{\star }\in \overline{U} (x_{0} , r \, \eta )\).

Finally, we shall show the uniqueness of the solution \(x^{\star }\) in \(U(x_{0} ,r_{0})\). Let \(y^\star \) be a solution of equation \(F(x)=0\) in \(U(x_{0} , r_{0})\). Define linear operator

We shall show \({\mathcal{L} } ^{-1}\) exists. Using (\({\mathcal{C} _{2}}\)) and (\({\mathcal{C} _{7}}\)), we get

It follows from (2.8), and the Banach lemma on invertible operators, that \(\mathcal{L} \) is invertible.

Finally, in view of the equality

we obtain

This completes the proof of Theorem 5.

3 Numerical examples

To illustrate the theoretical results introduced previously, we present some numerical examples. In these examples we show some situations where the results provided in the paper can be applied.

Conclusion

We provided a semilocal convergence analysis of DFM for approximating a locally unique solution of an equation in a Banach space, which shows that DFM is faster than CSTM given in [9]. These advantages are obtained under the same computational cost as in [9]. DFM is also a useful derivative free alternative to the usage of Newton’s method (NW). The latter method requires Fréchet-derivative operator at each step but its computation may be too expensive or impossible, especially if the analytic representation of Fréchet-derivative operator involved is unavailable [5]–[7], [14]–[17], [19], [20], [22], [25].