\(^\ast \)Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA, e-mail: iargyros@cameron.edu.

\(^{\ast \ast }\)Department of Mathematical and Computational Sciences, NIT Karnataka, India-575 025, e-mail: sgeorge@nitk.ac.in.

1 Introduction

Let \(F: {\Omega }\subset \mathcal{B}_1\longrightarrow \mathcal{B}_2\) be a nonlinear operator, \(\mathcal{B}_1,\mathcal{B}_2\) be Banach spaces and \(\Omega \) be a convex set. Numerous iterative methods for solving equation \(F(x)=0\) can be written like

where \(L_n\in \mathcal{L}(\mathcal{B}_2, \mathcal{B}_1)\) the space of bounded linear operators from \(\mathcal{B}_2\) into \(\mathcal{B}_1\) for each \(n\in \mathbb {N}\cup \{ 0\} .\) The most widely used methods like (1) are Newton’s method, where \(L_n= F'(x_n)^{-1},\) and the secant method, where \(L_n=\delta F(x_n, x_{n-1})^{-1}\) and \(\delta F\) stands for a consistent approximation to the Fréchet-derivative of \(F\) [ 6 , 26 ] . A lot of problems in control theory, optimization, inverse problems theory, Mathematical Physics, Chemistry, Economics, Biology and also in engineering can be brought in the form of equation \(F(x)=0\) using Mathematical modeling [ 1 , 2 , 3 , 6 , 11 , 12 , 13 , 14 , 22 , 23 , 26 , 29 ] . Closed form solutions are preferred but this is rarely possible. Consequently, mostly iterative methods are utilized say like method (1) to generate a sequence approximating a locally unique solution \(x^*\) of equation \(F(x)=0\) under some conditions. It is well known from the numerical efficiency that it is not advantageous to change the operator \(L_n\) at each step of the iterative method. If one keeps the operator piecewise constant more efficient iterative methods can be obtained. Optimal recepts can be obtained based on the dimension of the space [ 29 ] . Iterative methods of this type have been studied by Traub [ 29 ] , Potra and Pták [ 26 ] , Bosarg and Falb [ 7 , 8 ] , Dennis [ 9 ] , Potra [ 26 , 27 ] , Amat [ 1 , 2 ] , Ezquerro et al. [ 12 ] , Hernandez et al. [ 14 , 15 , 16 ] , Argyros [ 3 , 4 , 5 , 6 ] (see also the references in the preceding papers).

In this paper we are motivated by the work by Potra [ 25 , 26 ] who improved the work by [ 7 , 8 , 9 , 29 ] . Let us choose

where \(\{ p_n\} \) and \(\{ q_n\} \) are non-decreasing sequences of integers such that

The convergence region of method (1)–(2) is small in general. That is why, we find in this paper a more accurate location containing the iterates \(\{ x_n\} \) than \(\Omega \) leading to tighter Lipschitz constants. This way we obtain: weaker sufficient convergence criteria, tighter error bounds on the distances \(\| x_{n+1}-x_n\| ,\) \(\| x_n-x^*\| \) and at least as precise information on the location of the solution \(x^*.\) It is worth noticing that the preceding improvements are obtained under the same computational effort, since in practice the computation of the old Lipschitz constants requires the computation of the new constants as a special cases.

The rest of the paper is structured as follows. Section 2 and Section 3 contain the semi-local convergence of secant-type and Newton-type methods, respectively.

2 Semi-local convergence \(p_n\neq q_n\) for each \(n\)

We shall study the iterative procedure (1) and (2) in this section for the triplets \((F, x_0, x_{-1})\) belonging to the class \(A(\alpha _0, \alpha , \beta , \gamma )\) defined as follows:

We associate the class \(A(\alpha _0, \alpha , \beta , \gamma )\) with the constant \(\delta \) and sequence \(\{ s_n\} _{n\geq -1}\) given by the formulae:

and \(\delta =\frac{1}{2\alpha }\sqrt{(1-\alpha \gamma )^2-4\alpha \beta }.\)

Using the above notation we present the following semi-local convergence result.

The linear operator \(M=\delta F(u,v)\) is invertible for each \(u, v\in \Omega ^\circ \) with

It follows from (7) that

Hence, by the Banach Lemma [ 6 , 28 , 30 , 31 , 32 ] \(M\) is invertible and

Note that the condition (7) implies the following Lipschitz condition for \(F'\)

Using the integral representation

we obtain that

for all \(x, y \in \Omega _0^\circ .\)

Finally form (9) and (17) we have

for all \(x, y, u,v\in \Omega _0^\circ .\) Estimates (16), (17),(18), by a continuity argument, remain valid if \(x\) and/or \(y\) belong to \(\Omega _\gamma .\) Using the above inequalities we shall prove that

for \(n=-1,0,1,\ldots .\)

Clearly, the sequence \(\{ s_n\} _{n\geq 1}\) given by (11) is decreasing and converges to \(\delta .\) Hence, if (1)-(2) is well defined for \(n=0,1,2,\ldots , k, \) and if (20) holds for \(n\leq k\) than

for \(n\leq k.\) That is (14) is satisfied for \(u=x_i\) and \(v=x_j\) with \(i, j \leq k.\) Therefore, (1)-(2) will be well defined for \(n=k+1\) as well.

For \(n=-1\) and \(n=0\) (20) reduces to \(\| x_{-1}-x_0\| \leq \gamma \) and \(\| x_0-x_{-1}\| \leq \beta \) (see (5) and (7)). Suppose (20) holds for \(n=-1,0,1,\ldots , k,\) where \( k\geq 0.\) Denote \(M_n=L_n^{-1},\) where \(L_n\) is given by (2). Notice that

Then, using (14) and (18) we can write

So, (19) holds for each \(n.\) \(\mathcal{B}_1\) is a complete space. Hence, the sequence \(\{ x_n\} _{n\geq 0}\) converges to \(x^*\) and

Next, by (18) and (20), we obtain

so it follows that \(F(x^*)=0.\)

Let \(x=x_n\) and \(y=x^*\) in (17) and denote \(A=\int _0^1F'(x^*+\theta (x_n-x^*))d\theta .\) Then, by (19) and (21) we get in turn that

In view of (8) and the Banach’s lemma we show as in (15) that, \(A\) is invertible and

Using (22) and (24), we have

It is easy to see that the above inequality together with (10) and \(\| x_n-x^*\| {\lt} s_0\) imply the estimate (12).

By the identity

(20) and (19), we obtain

so (12) is shown.

The existence has been shown in the Theorem2. Suppose \(\delta {\gt} 0\) and let \(x^*\in U\) and \(y^*\in Q_1\) be solutions of the equation \(F(x)=0.\) Let us denote \(S_*=\int _0^1 F'(y^*+\theta (x^*-y^*))d\theta .\) Using (8), we have

That is \(S_*\) is invertible. Then, by (17) we deduce that \(x^*=y^*.\)

Case \(\delta =0.\) Let \(M_n=0\) and \(q_n=-1\) for \(n=0,1,2,\ldots \) then the iterative procedure (1)-(2) becomes

By Theorem 2 it follows that the sequence \(\{ x_n\} _{n\geq 0}\) given by (25) converges to a solution \(x^*\) of the equation \(F(x)=0.\) It also follows that

where

Using induction we get that

Let \(y^*\in Q_2\) be a solution of the equation \(F(x)=0\) and denote \(A_n=\int _0^1F'(y^*+\theta (x_n-y^*))d \theta .\) According to (8), (14), (24) and (25) we have

By (28), we deduce that \(\lim _{n\longrightarrow \infty }\prod _{j=1}^n(1-\alpha _0 s_j) =0.\) Hence, we conclude that \(y^*=\lim _{n\longrightarrow \infty }x_n=x^*.\)

Next, we show that the results obtained in this section are sharp within the class \(A(\alpha _0, \alpha , \beta , \gamma ).\)

Concerning the domain of uniqueness of the solution \(x^*\) established in the Corollary 3 we have

Next, we shall consider some particular cases of the iterative procedure (1)-(2) and shall compare the results obtained in the preceding section with some known results.

3 Semi-local convergence analysis \(\lowercase {p}_{\lowercase {n}}=\lowercase {q}_{\lowercase {n}}\) for each \(\lowercase {n}\)

If \(x_{-1}=x_0\) and \(p_n=q_n\) for each \(n=0,1,2,\ldots ,\) then the iterative procedure (1)-(2) becomes

In [ 9 , 10 ] , Dennis proved that this iterative procedure converges under the hypotheses of the Kantorovich theorem. This fact follows by taking \(\gamma =0\) in Theorem 2. To be more precise let us consider the class \(A'(\alpha _0,\alpha ,\beta )\) defined below.

It is easy to see that \((F, x_0)\in A'(\alpha _0,\alpha ,\beta )\) if and only if \((F, x_0,x_0)\in A(\alpha _0,\alpha ,\beta ).\) In this case from Theorem 2 it follows that the iterative procedure (36) converges and the following estimates hold:

and