These authors contributed equally to this work.

The goal is to extend the applicability of Newton-Traub-like methods in cases not covered in earlier articles requiring the usage of derivatives up to order seven that do not appear in the methods. The price we pay by using conditions on the first derivative that actually appear in the method is that we show only linear convergence. To find the convergence order is not our intention, however, since this is already known in the case where the spaces coincide with the multidimensional Euclidean space. Note that the order is rediscovered by using ACOC or COC, which require only the first derivative. Moreover, in earlier studies using Taylor series, no computable error distances were available based on generalized Lipschitz conditions. Therefore, we do not know, for example, in advance, how many iterates are needed to achieve a predetermined error tolerance. Furthermore, no uniqueness of the solution results is available in the aforementioned studies, but we also provide such results. Our technique can be used to extend the applicability of other methods in an analogous way, since it is so general. Finally note that local results of this type are important, since they demonstrate the difficulty in choosing initial points. Our approach also extends the applicability of this family of methods from the multi-dimensional Euclidean to the more general Banach space case. Numerical examples complement the theoretical results.

Let

Moreover, they used method (

However, in this article we do not necessarily assume that

For example, consider

Then, we calculate

Notice that

Another problem is that there are no error bounds on

It is worth noticing that the

The layout for the rest of the article includes the convergence analysis of method (

Let

Assume

By these definitions

Assume

By these definitions

Define a radius of convergence

By the preceding we have for all

Define

Define

Define

If

This estimation together with the lemma by Banach for operators that are invertible [

It also follows that iterate

We get the estimates by (

We use

Then, we can choose functions

In this article we extended the applicability of Newton-Traub-like methods in cases not covered before requiring the usage of derivatives up to order seven that do not appear in the methods. The price we pay by using conditions on the first derivative that actually appears on the method is that we show only linear convergence. To find the convergence order is not, however, our intention, since this is already known in the case of spaces that coincide with the multidimensional Euclidean space. Notice that the order is rediscovered by using ACOC or COC, which require only the first derivative. Moreover, in earlier studies using Taylor series, no computable error distances were available based on generalized Lipschitz conditions. Therefore, we do not know, for example, in advance, how many iterates are needed to achieve a predetermined error tolerance. Furthermore, no uniqueness of the solution results is available in the aforementioned studies, but we also provide such results. Our technique can be used to extend the applicability of other methods in an analogous way, since it is so general. Finally notice that local results of this type are important, since they demonstrate the difficulty in choosing initial points.

Conceptualization, S.R., C.I.A., I.K.A. and S.G.; methodology, S.R., C.I.A., I.K.A. and S.G.; software, S.R., C.I.A., I.K.A. and S.G.; validation, S.R., C.I.A., I.K.A. and S.G.; formal analysis, S.R., C.I.A., I.K.A. and S.G.; investigation, S.R., C.I.A., I.K.A. and S.G.; resources, S.R., C.I.A., I.K.A. and S.G.; data curation, S.R., C.I.A., I.K.A. and S.G.; writing—original draft preparation, S.R., C.I.A., I.K.A. and S.G.; writing—review and editing, S.R., C.I.A., I.K.A. and S.G.; visualization, S.R., C.I.A., I.K.A. and S.G.; supervision, S.R., C.I.A., I.K.A. and S.G.; project administration, S.R., C.I.A., I.K.A. and S.G.; funding acquisition, S.R., C.I.A., I.K.A. and S.G. All authors have read and agreed to the published version of the manuscript.

This research received no external funding.

No new data were created or analyzed in this study. Data sharing is not applicable to this article.

The authors declare no conflict of interest.

Plot of motivational function.