\(^\ast \)1Department of Mathematical Sciences, Cameron University, Lawton, OK 73505, USA, e-mail: iargyros@cameron.edu.

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

1 Introduction

Let \(\mathcal{B}_1, \mathcal{B}_2\) be Banach spaces and \(\Omega \) be an open and convex subset of \(\mathcal{B}_1.\) The problem of finding a solution of equation

where \(F:\Omega \longrightarrow \mathcal{B}_2\) is differentiable in the sense of Fréchet is important problem in applied mathematics due its wide applications.

In this paper we study the local convergence of the Euler-Halley-type method (EHTM) defined for each \(n=0,1,2...\) by [ 9 ] – [ 14 ]

where

with \(x_0\) being an initial guess and \(\alpha \in (-\infty , +\infty ).\)

Notice that, method (2) becomes Halley method when \(\alpha =\frac{1}{2}\), becomes Chebyshev-Euler method when \(\alpha =0\) and super-Halley method when \(\alpha =1.\)

The local convergence of the EHTM (2) was studied in [ 14 ] , using the second-order generalized Lipschitz assumption with \(L-\)average (see Section2 in [ 14 ] ). The radius of the optimal convergence ball and the error estimation of method (2) corresponding to the parameter \(\alpha \) are also estimated for each \(\alpha \in (-\infty , +\infty )\) in [ 14 ] . Huang and Guocham in [ 14 ] also shown that the method (2) with \(\alpha \) is better than the one corresponding to \(-\alpha \) for each \(\alpha {\gt}0\) and the Chebyshev-Euler method is best among all methods in the family with \(\alpha \in (-\infty , 0]\) as far as the choice of initial point and error estimates are concerned.

In this study we use second-order generalized Lipschitz condition with \(K_0-\) average (to be precised in Definition 1) to study the local convergence of method (2). Using second-order generalized Lipschitz condition with \(K_0-\) average we improved the results in [ 14 ] . Moreover, our radius of convergence is better than the one in [ 14 ] and the information on the location of the iterates in our study is more precise than that of [ 14 ] .

The paper is structured as follows. In Section 2 we present the local convergence analysis. We also provide a radius of convergence, computable error bounds and uniqueness result not given in the earlier studies [ 1 ] – [ 17 ] . Special cases and numerical examples are presented in the concluding Section 3.

2 Local Convergence

Denote by \(S(\lambda ,\xi ), \bar{S}(\lambda ,\xi ),\) respectively the open and closed balls in \(\mathcal{B}_1\) with center \(\lambda \in \mathcal{B}_1\) and of radius \(\xi {\gt} 0.\)

Let \(R {\gt} 0.\) Set \(R_0=\sup \{ t\in [0,R):S(t, R) \subset \Omega \} .\) Let also \(K_0, \bar{K}, K\) be real valued \(C^1\) functions defined on the interval \([0, R_0],\) increasing on \([0, R_0]\) with \(K_0'(t) \geq 0, \bar{K}'(t) \geq 0, K'(t) \geq 0, K_0(0) {\gt} 0, \bar{K}(0) {\gt} 0\) and \(K(0) {\gt} 0.\) Denote by \(\rho \) the smallest positive solution of equation

Define function \(h_0\) by

Notice that \(h_0'(\rho ) =0\) and \(h_0'(t) {\lt} 0\) for all \(t\in [0,\rho ).\) We need the notion of the second-order generalized center-Lipschitz condition with \(K_0-\)average in \(S(p,\rho ).\)

Next, we introduce the notion of second-order generalized \(K_0-\)restricted Lipschitz condition with \(\bar{K}-\)average in \(S(p,\rho ).\) Define also function \(\bar{h}\) by

Then, we have that

and

Suppose from now on that

Then, the results in [ 14 ] can be written with \(\bar{h}\) replacing \(h\) and estimate (4) replacing (5). If

Then, the results in [ 14 ] can be written with \(h_0\) replacing \(h.\) Hence, we arrived at:

3 Special cases and examples

The numerical examples are presented in this section.

3.1 Special case:

Kantorovich-type hypothesis. Let \(K(t)=\beta t+\gamma \) for some \(\beta \geq 0\) and \(\gamma {\gt} 0.\) The other “\(K\)“ functions can be defined similarly (see also the numerical examples).

3.2 Special case:

(Smale-Wang-type hypothesis). Let \(K(t)=\tfrac {2\delta }{(1-\delta t)^3}\) for some \(\delta {\gt} 0.\) The other ”\(K\)“ functions can be defined similarly.