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In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order $ p $ of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order $ p+3 $ (for $ p\geq3 $) and have extended it to a generalized $ (m+2) $-step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to $ p+3m $, $ m\in\mathbb{N} $. We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations.

In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the ω -continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems. [ABSTRACT FROM AUTHOR]

In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order p of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order p + 3 (for p ≥ 3 ) and have extended it to a generalized (m + 2) -step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to p + 3 m , m ∈ N . We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations. In this paper, we have constructed a family of three-step methods with sixth-order convergence and a novel approach to enhance the convergence order of iterative methods for systems of nonlinear equations. Additionally, we propose a three-step scheme with convergence order (for ) and have extended it to a generalized -step scheme by merely incorporating one additional function evaluation, thus achieving convergence orders up to , . We also provide a thorough local convergence analysis in Banach spaces, including the convergence radius and uniqueness results, under the assumption of a Lipschitz-continuous Fréchet derivative. Theoretical findings have been validated through numerical experiments. Lastly, the performance of these methods is showcased through the analysis of their basins of attraction and their application to systems of nonlinear equations. [ABSTRACT FROM AUTHOR]

In this paper, with the aim of approximate the ball convergence of nonlinear equation, we study the local properties of a class of sixth-order Jarratt-type iterative methods in Banach spaces. By utilizing the first-order Fréchet derivative, we establish the local convergence. At last, the nonlinear equations related to the gas-state equation, the continuous stirred tank reactor (CSTR) and the problem of azeotropic point of a binary solutions are solved using this iterative method, and the applicability of the theoretical results is proved. [ABSTRACT FROM AUTHOR]

We carried out a local comparison between two ninth convergence order schemes for solving nonlinear equations, relying on first-order Fréchet derivatives. Earlier investigations require the existence as well as the boundedness of derivatives of a high order to prove the convergence of these schemes. However, these derivatives are not in the schemes. These assumptions restrict the applicability of the schemes, which may converge. Numerical results along with a boundary value problem are given to examine the theoretical results. Both schemes are symmetrical not only in the theoretical results (formation and convergence order), but the numerical and dynamical results are also similar. We calculated the convergence radii of the nonlinear schemes. Moreover, we obtained the extraneous fixed points for the proposed schemes, which are repulsive and are not part of the solution space. Lastly, the theoretical and numerical results are supported by the dynamic results, where we plotted basins of attraction for a selected test function.

We carried out a local comparison between two ninth convergence order schemes for solving nonlinear equations, relying on first-order Fréchet derivatives. Earlier investigations require the existence as well as the boundedness of derivatives of a high order to prove the convergence of these schemes. However, these derivatives are not in the schemes. These assumptions restrict the applicability of the schemes, which may converge. Numerical results along with a boundary value problem are given to examine the theoretical results. Both schemes are symmetrical not only in the theoretical results (formation and convergence order), but the numerical and dynamical results are also similar. We calculated the convergence radii of the nonlinear schemes. Moreover, we obtained the extraneous fixed points for the proposed schemes, which are repulsive and are not part of the solution space. Lastly, the theoretical and numerical results are supported by the dynamic results, where we plotted basins of attraction for a selected test function. [ABSTRACT FROM AUTHOR]

A local convergence analysis is developed for Newton's method in order to approximate a solution of a generalized equations in a Banach space setting. The convergence conditions are based on generalized continuity conditions on the Fréchet derivative of the operator involved and the Aubin property. The specialized cases of our results extend earlier ones using similar information. [ABSTRACT FROM AUTHOR]

The goal in this study is to present a unified local convergence analysis of frozen Steffensen-type methods under generalized Lipschitz-type conditions for Banach space valued operators. We also use our new idea of restricted convergence domains, where we find a more precise location, where the iterates lie leading to at least as tight majorizing functions. Consequently, the new convergence criteria are weaker than in earlier works resulting to the expansion of the applicability of these methods. The conditions do not necessarily imply the differentiability of the operator involved. This way our method is suitable for solving equations and systems of equations.

Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step fifth and multi-step 5+3r order iterative methods obtained using only information from the operators on these methods. Finally, the novelty of our process relates to the fact that the convergence conditions depend only on the functions and operators which are present in the methods. Thus, the applicability is extended to these methods. Numerical applications complement the theory.

In this paper, a Newton-type iterative scheme for solving nonlinear systems is designed. In the process of proving the convergence order, we use the higher derivatives of the function and show that the convergence order of this iterative method is six. In order to avoid the influence of the existence of higher derivatives on the proof of convergence, we mainly discuss the convergence of this iterative method under weak conditions. In Banach space, the local convergence of the iterative scheme is established by using the ω-continuity condition of the first-order Fréchet derivative, and the application range of the iterative method is extended. In addition, we also give the radius of a convergence sphere and the uniqueness of its solution. Finally, the superiority of the new iterative method is illustrated by drawing attractive basins and comparing them with the average iterative times of other same-order iterative methods. Additionally, we utilize this iterative method to solve both nonlinear systems and nonlinear matrix sign functions. The applicability of this study is demonstrated by solving practical chemical problems.

A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ -average (assuming it as a positively integrable function and dropping the essential property of ND) are obtained. In comparison to previous research, in another part, we employ majorizing sequences that are more accurate in their precision along with the certain form of ϰ average Lipschitz criteria. A finer local and semi-local convergence criteria, boosting its utility, by relaxing the assumptions is derived. Applications in engineering to a variety of specific cases, such as object motion governed by a system of differential equations, are illustrated. [ABSTRACT FROM AUTHOR]

We compare the convergence balls and the dynamical behaviors of two efficient weighted-Newton-like equation solvers by Sharma and Arora, and Grau-Sánchez et al. First of all, the results of ball convergence for these algorithms are established by employing generalized Lipschitz constants and assumptions on the first derivative only. Consequently, outcomes for the radii of convergence, measurable error distances and the existence–uniqueness areas for the solution are discussed. Then, the complex dynamical behaviors of these solvers are compared by applying the attraction basin tool. It is observed that the solver suggested by Grau-Sánchez et al. has bigger basins than the method described by Sharma and Arora. Lastly, our ball analysis findings are verified on application problems and the convergence balls are compared. It is found that the method given by Grau-Sánchez et al. has larger convergence balls than the solver of Sharma and Arora. Hence, the solver presented by Grau-Sánchez et al. is more suitable for practical application. The convergence analysis uses the first derivative in contrast to the aforementioned studies, utilizing the seventh derivative not on these methods. The developed process can be used on other methods in order to increase their applicability.

A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates. [ABSTRACT FROM AUTHOR]

Iterative methods which have high convergence order are crucial in computational mathematics since the iterates produce sequences converging to the root of a non-linear equation. A plethora of applications in chemistry and physics require the solution of non-linear equations in abstract spaces iteratively. The derivation of the order of the iterative methods requires expansions using Taylor series formula and higher-order derivatives not present in the method. Thus, these results cannot prove the convergence of the iterative method in these cases when such higher-order derivatives are non-existent. However, these methods may still converge. Our motivation originates from the need to handle these problems. No error estimates are given that are controlled by constants. The process introduced in this paper discusses both the local and the semi-local convergence analysis of two step fifth and multi-step 5 + 3 r order iterative methods obtained using only information from the operators on these methods. Finally, the novelty of our process relates to the fact that the convergence conditions depend only on the functions and operators which are present in the methods. Thus, the applicability is extended to these methods. Numerical applications complement the theory. [ABSTRACT FROM AUTHOR]

The local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires a weak hypothesis where the Fréchet derivative of the nonlinear operator satisfies the ψ -continuity condition, which thereby extends the applicability of the method when both Lipschitz and Hölder conditions fail. The convergence in this study is considered under the hypotheses on the first-order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised here. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some concrete numerical examples showing the performance of the method over some existing schemes are presented in this article. [ABSTRACT FROM AUTHOR]

We compare the convergence balls and the dynamical behaviors of two efficient weighted-Newton-like equation solvers by Sharma and Arora, and Grau-Sánchez et al. First of all, the results of ball convergence for these algorithms are established by employing generalized Lipschitz constants and assumptions on the first derivative only. Consequently, outcomes for the radii of convergence, measurable error distances and the existence–uniqueness areas for the solution are discussed. Then, the complex dynamical behaviors of these solvers are compared by applying the attraction basin tool. It is observed that the solver suggested by Grau-Sánchez et al. has bigger basins than the method described by Sharma and Arora. Lastly, our ball analysis findings are verified on application problems and the convergence balls are compared. It is found that the method given by Grau-Sánchez et al. has larger convergence balls than the solver of Sharma and Arora. Hence, the solver presented by Grau-Sánchez et al. is more suitable for practical application. The convergence analysis uses the first derivative in contrast to the aforementioned studies, utilizing the seventh derivative not on these methods. The developed process can be used on other methods in order to increase their applicability. [ABSTRACT FROM AUTHOR]

The local convergence analysis of multi-step, high-order Jarratt-like schemes is extended for solving Banach space valued systems of equations using the derivative instead of up to the ninth derivative as in previous works. Our idea expands the usage of the scheme in cases not considered earlier and can also be utilized in other schemes, too. Experiments test the theoretical results. [ABSTRACT FROM AUTHOR]

The applicability of an efficient sixth convergence order scheme is extended for solving Banach space valued equations. In previous works, the seventh derivative has been used not appearing on the scheme. But we use only the first derivative that appears on the scheme. Moreover, bounds on the error distances and results on the uniqueness of the solution are provided (not given in earlier works) based on ω–continuity conditions. Numerical examples complete this article.

The aim of this study is to analyze the local convergence of the multi-step Homeier's-like method for solving nonlinear equations in Banach space. Furthermore, we avoid hypotheses on high order derivatives which limit the applicability of the method. Instead, we only use hypotheses on the first derivative. Thus the applicability of the method has been extended by preserving the order of convergence. The convergence of the solution is proved under the weak hypotheses i.e. ω-continuity condition. Some numerical instances where earlier results cannot be applied to solve equations but our results can be applied are provided to validate the theoretical contribution. [ABSTRACT FROM AUTHOR]

A novel local and semi-local convergence theorem for the four-step nonlinear scheme is presented. Earlier studies on local convergence were conducted without particular assumption on Lipschitz constant. In first part, the main local convergence theorems with a weak ϰ-average (assuming it as a positively integrable function and dropping the essential property of ND) are obtained. In comparison to previous research, in another part, we employ majorizing sequences that are more accurate in their precision along with the certain form of ϰ average Lipschitz criteria. A finer local and semi-local convergence criteria, boosting its utility, by relaxing the assumptions is derived. Applications in engineering to a variety of specific cases, such as object motion governed by a system of differential equations, are illustrated.

The main contribution of this study is that the applicability and convergence domain of a fifth-order convergent equation solver is extended. We use ω condition on the first Fréchet derivative to study the local analysis, and this expands the applicability of the formula for such problems where the earlier study based on Lipschitz constants cannot be used. Also, we avoid the use of the extra assumption on boundedness of the first derivative of the nonlinear operator. Our idea can be used on other iterative methods. Numerical tests confirmed that the proposed analysis produces a larger convergence domain, in comparison to the earlier study, without using additional conditions. [ABSTRACT FROM AUTHOR]

A local and semi-local convergence is developed of a class of iterative methods without derivatives for solving nonlinear Banach space valued operator equations under the classical Lipschitz conditions for first-order divided differences. Special cases of this method are well-known iterative algorithms, in particular, the Secant, Kurchatov, and Steffensen methods as well as the Newton method. For the semi-local convergence analysis, we use a technique of recurrent functions and majorizing scalar sequences. First, the convergence of the scalar sequence is proved and its limit is determined. It is then shown that the sequence obtained by the proposed method is bounded by this scalar sequence. In the local convergence analysis, a computable radius of convergence is determined. Finally, the results of the numerical experiments are given that confirm obtained theoretical estimates.

We extend the applicability of a fourth-order convergent nonlinear system solver by providing its local convergence analysis under Lipschitz continuous Fréchet derivative in Banach spaces. Our analysis only uses the first-order Fréchet derivative to ensure the convergence and provides the uniqueness of the solution, the radius of convergence ball and the computable error bounds. This study is applicable in solving such problems for which earlier studies are not effective. Furthermore, the convergence region for the scheme to approximate the zeros of various polynomials is studied using basins of attraction tool. Various computational tests are conducted to validate that our analysis is beneficial when prior studies fail to solve problems. [ABSTRACT FROM AUTHOR]

Different set of criteria based on the seventh derivative are used for convergence of sixth order methods. Then, these methods are compared using numerical examples. But we do not know: if the results of those comparisons are true if the examples change; the largest radii of convergence; error estimates on distance between the iterate and solution, and uniqueness results that are computable. We address these concerns using only the first derivative and a common set of criteria. Numerical experiments are used to test the convergence criteria and further validate the theoretical results. Our technique can be used to make comparisons between other methods of the same order.

We present the local convergence of a Newton-type solver for equations involving Banach space valued operators. The eighth order of convergence was shown earlier in the special case of the k--dimensional Euclidean space, using hypotheses up to the eighth derivative although these derivatives do not appear in the method. We show convergence using only the first derivative. This way we extend the applicability of the methods. Numerical examples are used to show the convergence conditions. Finally, the basins of attraction of the method, on some test problems are presented. [ABSTRACT FROM AUTHOR]

We present the semi-local convergence analysis of a Potra-type method to solve equations involving Banach space valued operators. The analysis is based on our ideas of recurrent functions and restricted convergence region. The study is completed using numerical examples.

Abstract We extend the applicability of a cubically convergent nonlinear system solver using Lipschitz continuous first-order Fréchet derivative in Banach spaces. This analysis avoids the usual application of Taylor expansion in convergence analysis and extends the applicability of the scheme by applying the technique based on the first-order derivative only. Also, our study provides the radius of convergence ball and computable error bounds along with the uniqueness of the solution. Furthermore, the generalization of this analysis using Hölder condition is provided. Various numerical tests confirm that our analysis produces better results and it is useful in solving such problems where previous methods can not be implemented.

The local convergence analysis of the multi-step seventh order method to solve nonlinear equations is presented in this paper. The point of this paper is that our proposed study requires a weak hypothesis where the Fréchet derivative of the nonlinear operator satisfies the ψ-continuity condition, which thereby extends the applicability of the method when both Lipschitz and Hölder conditions fail. The convergence in this study is considered under the hypotheses on the first-order derivative without involving derivatives of the higher-order. To find a subset of the original convergence domain, a strategy is devised here. As a result, the new Lipschitz constants are at least as tight as the old ones, allowing for a more precise convergence analysis in the local convergence case. Some concrete numerical examples showing the performance of the method over some existing schemes are presented in this article.

The local convergence analysis of multi-step, high-order Jarratt-like schemes is extended for solving Banach space valued systems of equations using the derivative instead of up to the ninth derivative as in previous works. Our idea expands the usage of the scheme in cases not considered earlier and can also be utilized in other schemes, too. Experiments test the theoretical results.

In earlier studies different methods of same convergence order are campared using numerical examples. The drawback of this approach is that, we do not know: if the results of those comparisons are true if the examples change; the largest radii of convergence; error estimates on distance between the iterate and solution, and uniqueness results that are computable. In this paper we campare the ball convergence of two-step iterative methods for solving the equation G (x) = 0 using only the first derivative and a common set of criteria. Numerical experiments are used to test the convergence criteria and further validate the theoretical results. Our technique can be used to make comparisons between other methods of the same order. [ABSTRACT FROM AUTHOR]

We study the local convergence analysis of two higher-order methods using Hölder continuity condition on the first Frechet derivative to solve nonlinear equations in Banach spaces. Hölder continuous first derivative is used to extend the applicability of the method on such problems for which Lipschitz condition fails. Also, this convergence analysis generalizes the local convergence analysis based on Lipschitz continuity condition. Our analysis provides the radius of convergence ball and error bounds along with the uniqueness of the solution. Numerical examples like Hammerstein integral equation and a system of nonlinear equations are solved to verify our theoretical results. [ABSTRACT FROM AUTHOR]

We are motivated by a Kung-Traub-type method for solving equations on the real line. In particular, we extend this method for Banach space valued operators. The radius of convergence is also obtained as well as error bounds on the distances involved and a uniqueness result. Our convergence analysis avoids Taylor expansions and the computation of higher order than one derivatives. [ABSTRACT FROM AUTHOR]

In this manuscript, the study of local convergence analysis for the cubically convergent Weerakoon's method using Hölder continuity condition is presented to solve nonlinear equations in Banach spaces. Hölder continuity condition on the first derivative is assumed to extend the applicability of the method on such problems for which Lipschitz condition fails. This convergence analysis generalises the local convergence with Lipschitz continuity condition. A theorem showing existence and uniqueness of the solution with the error bounds is established. To verify our theoretical findings some numerical examples like Hammerstein integral equation and a system of nonlinear equations are solved. [ABSTRACT FROM AUTHOR]

There is a plethora of k‐step solvers for equations involving operators on Banach spaces. Their convergence is estimated by adopting hypotheses on high‐order derivatives which are not even in these iterative solvers. In addition, no computable error bounds or information on the uniqueness of the solution based on Lipschitz‐type functions are given. Moreover, the choice of the initial guess is like shooting in the dark. Finally, the criteria of convergence differ from solver to solver, so no comparison can be made between their convergence domains. The novelty of our work is that we address these problems by introducing a generalized k‐step solver containing all previous k‐step solvers. Moreover, we address all previously stated problems in the earlier papers and utilizing weaker conditions that require only the continuity of the involved operator. Applications are also presented where we test the convergence criteria. [ABSTRACT FROM AUTHOR]

In this paper, an efficient sixth order multi step method is proposed in Banach spaces and its local convergence is studied in order to approximate a locally unique solution of a nonlinear or a system of nonlinear equations. Based on Lipschitz's continuity condition, new convergence analysis provides radius of convergence, bounds on error and estimates on the uniqueness of solution under assumptions on the first derivative only. Hence, the method can be applied to a wider class of functions that expands its applicability. Numerical examples are presented to illustrate the theoretical results. In the end, the basins of attraction are given to show the convergence behavior in complex plane. [ABSTRACT FROM AUTHOR]

In a Banach space setting, we provide the local convergence analysis of a cubically convergent nonlinear system solver assuming that the first-order Fréchet derivative belongs to the Lipschitz class. The significance of this study is that it avoids the standard practice of Taylor expansion in the analysis of convergence and extends the applicability of the algorithm by using the theory based on the first-order derivative only. Also, our analysis provides the radii of convergence balls and computable error bounds along with the uniqueness of the solution. Furthermore, the generalization of this analysis using Hölder condition is studied. Different numerical tests confirm that the new technique produces better results, and it is useful in solving such problems where previous studies fail to solve. [ABSTRACT FROM AUTHOR]

The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

The aim of this article is to provide the local convergence analysis of two novel competing sixth convergence order methods for solving equations involving Banach space valued operators. Earlier studies have used hypotheses reaching up to the sixth derivative but only the first derivative appears in these methods. These hypotheses limit the applicability of the methods. That is why we are motivated to present convergence analysis based only on the first derivative. Numerical examples where the convergence criteria are tested are provided. It turns out that in these examples the criteria in the earlier works are not satisfied, so these results cannot be used to solve equations but our results can be used.

The aim of this study is to extend the applicability of an eighth convergence order method from the k−dimensional Euclidean space to a Banach space setting. We use hypotheses only on the first derivative to show the local convergence of the method. Earlier studies use hypotheses up to the eighth derivative although only the first derivative and a divided difference of order one appear in the method. Moreover, we provide computable error bounds based on Lipschitz-type functions.

We present a local as well as a semi-local convergence analysis of a two-step secant-type method for solving nonlinear equations involving Banach space valued operators. By using weakened Lipschitz and center Lipschitz conditions in combination with a more precise domain containing the iterates, we obtain tighter Lipschitz constants than in earlier studies. This technique lead to an extended convergence domain, more precise information on the location of the solution and tighter error bounds on the distances involved. These advantages are obtained under the same computational effort, since the new constants are special cases of the old ones used in earlier studies. The new technique can be used on other iterative methods. The numerical examples further illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We introduce a new faster two-step King–Werner-type iterative method for solving nonlinear equations. The methodology is based on rational Hermite interpolation. The local as well as semi-local convergence analyses are presented under weak center Lipschitz and Lipschitz conditions. The convergence order is increased from 1 + 2 to 3 without any additional function calculations. Another advantage is the convenient fact that this method does not use derivatives. Numerical examples further validate the theoretical results. [ABSTRACT FROM AUTHOR]

We study the local convergence analysis of a fifth order method and its multi-step version in Banach spaces. The hypotheses used are based on the first Fréchet-derivative only. The new approach provides a computable radius of convergence, error bounds on the distances involved, and estimates on the uniqueness of the solution. Such estimates are not provided in the approaches using Taylor expansions of higher order derivatives, which may not exist or may be very expensive or impossible to compute. Numerical examples are provided to validate the theoretical results. Convergence domains of the methods are also checked through complex geometry shown by drawing basins of attraction. The boundaries of the basins show fractal-like shapes through which the basins are symmetric.

We present a local convergence analysis for inexact Newton methods in order to approximate a solution of a nonlinear equation in a Banach space. Our sufficient convergence conditions involve only hypotheses on the first Fréchet-derivative of the operator involved. Numerical examples are also provided in this study. [ABSTRACT FROM AUTHOR]

In this paper, we present a local convergence analysis of some iterative methods to approximate a locally unique solution of nonlinear equations in a Banach space setting. In the earlier study [Babajee et al. (2015) "On some improved harmonic mean Newton-like methods for solving systems of nonlinear equations," Algorithms8(4), 895–909], demonstrate convergence of their methods under hypotheses on the fourth-order derivative or even higher. However, only first-order derivative of the function appears in their proposed scheme. In this study, we have shown that the local convergence of these methods depends on hypotheses only on the first-order derivative and the Lipschitz condition. In this way, we not only expand the applicability of these methods but also proposed the theoretical radius of convergence of these methods. Finally, a variety of concrete numerical examples demonstrate that our results even apply to solve those nonlinear equations where earlier studies cannot apply. [ABSTRACT FROM AUTHOR]