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We study the local convergence of a family of fifth and sixth convergence order derivative free methods for solving Banach space valued nonlinear models. Earlier results used hypotheses up to the seventh derivative to show convergence. However, we only use the first divided difference of order one as well as the first derivative in our analysis. We also provide computable radius of convergence, error estimates, and uniqueness of the solution results not given in earlier studies. Hence, we expand the applicability of these methods. The dynamical analysis of the discussed family is also presented. Numerical experiments complete this article.

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In this article, we suggest the local analysis of a uni-parametric third and fourth order class of iterative algorithms for addressing nonlinear equations in Banach spaces. The proposed local convergence is established using an ω-continuity condition on the first Fréchet derivative. In this way, the utility of the discussed schemes is extended and the application of Taylor expansion in convergence analysis is removed. Furthermore, this study provides radii of convergence balls and the uniqueness of the solution along with the calculable error distances. The dynamical analysis of the discussed family is also presented. Finally, we provide numerical explanations that show the suggested analysis performs well in the situation where the earlier approach cannot be implemented.

Abstract. In this paper we have studied some new inequalities similar to Hardy-Hilbert's inequality. As applications, we have considered the associated integral inequalities.Mathematics subject classification (2010): 26D15.

In this paper, we deal with the construction, analysis and applications of a modified uniparametric family of methods to solve nonlinear equations in R. We study the convergence of new methods which shows the order of convergence is at least five and for a particular value 3/2 of the parameter γ, the method is sixth order convergent. We discuss several applications such as Max Planck’s conservative law, chemical equilibrium and multi-factor effect to demonstrate the productiveness and capability of the suggested method (for γ = 3/2 ). At every iteration our method is compared with Maroju et al. method[1] and Parhi and Gupta method[2] in terms of the values |f (xn)| and |xn − xn−1|. From the numerical experiments, advantages of our method is observed. Furthermore, we study the complex dynamics to determine the stability and dynamical properties of the methods.

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