Mathematics

In order to find the zeros of nonlinear equations, in this paper, we propose a family of third-order and optimal fourth-order iterative methods. We have also obtained some particular cases of these methods. These methods are constructed through weight function concept. The multivariate case of these methods has also been discussed. The numerical results show that the proposed methods are more efficient than some existing third-and fourth-order methods.

Advancement in network technology provides new ways to utilize telecare medicine information systems (TMIS) for patient care. Although TMIS usually faces various attacks as the services are provided over the public network. Recently, Jiang et al. proposed a chaotic mapbased remote user authentication scheme for TMIS. Their scheme has the merits of low cost and session key agreement using Chaos theory. It enhances the security of the system by resisting various attacks. In this paper, we analyze the security of Jiang et al.'s scheme and demonstrate that their scheme is vulnerable to denial of service attack. Moreover, we demonstrate flaws in password change phase of their scheme. Further, our aim is to propose a new chaos mapbased anonymous user authentication scheme for TMIS to overcome the weaknesses of Jiang et al.'s scheme, while also retaining the original merits of their scheme. We also show that our scheme is secure against various known attacks including the attacks found in Jiang et al.'s scheme. The proposed scheme is comparable in terms of the communication and computational overheads with Jiang et al.'s scheme and other related existing schemes. Moreover, we demonstrate the validity of the proposed scheme through the BAN (Burrows, Abadi, and Needham) logic.

Dynamic ID-based remote user authentication schemes ensure efficient and anonymous mutual authentication between entities. In 2013, Khan et al. proposed an improved dynamic ID-based authentication scheme to overcome the security flaws of Wang et al.'s authentication scheme. Recently, Sun and Cao showed that Khan et al. does not satisfies the claim of the user's privacy and proposed an efficient authentication scheme with user anonymity. The Sun and Cao's scheme achieve improvement over Khan et al.'s scheme in both privacy and performance point of view.

Abstract: The aim of this article is to study the semilocal convergence of a fifth order method in Banach spaces. Using recurrence relations, we have proved convergence, existence and uniqueness theorem, along with a priori error bounds which shows the R-order of convergence. Finally, we have demonstrated the numerical results on nonlinear integral equation.

This article concerned with the issue of solving a nonlinear equation with the help of iterative method where no any derivative evaluation is required per iteration. Therefore, this work contributes to a new class of optimal eighth-order Steffensen-type methods. Theoretical proof has been given to reveal the eighth-order convergence. Numerical comparisons have been carried out to show the effectiveness of contributed scheme.

The prime objective of this paper is to design a new family of eighth-order iterative methods by accelerating the order of convergence and efficiency index of well existing seventh-order iterative method of \cite{Soleymani1} without using more function evaluations for finding simple roots of nonlinear equations. The presented iterative family requires three function and one derivative evaluations and thus agrees with the conjecture of Kung-Traub for the case $n = 4$ (i.e. optimal). We have also discussed the derivative free version of the proposed scheme. Numerical comparisons have been carried out to demonstrate the efficiency and the performances of proposed method.

The object of the present paper is to improve the R-order convergence of with memory method proposed by Eftekhari (Int J Differ Eqn 2014:6, 2014) [1]. To achieve this goal, one more iterative parameter is introduced, which is calculated with the help of Newton's interpolatory polynomial of degree five. It is shown that the R-order convergence of the proposed method is increased from 11.2915 to 13.4031 without any extra evaluation. Smooth as well as nonsmooth examples are presented to confirm theoretical result and superiority of the new scheme.

In the present paper we study the class of harmonic maps on cosymplectic manifolds. First we find the necessary and sufficient condition for the Riemannian map to be harmonic map between two cosymplectic manifolds and then from cosymplectic manifold to Sasakian manifold. Finally, we find the condition for non-existence of harmonic map from cosymplectic manifold to Kenmotsu manifold. Keywords Cosymplectic manifold Á Sasakian manifold Á Kenmotsu manifold Á Holomorphic map Á Harmonic map Á Riemannian map / 2 X ¼ ÀX þ gðXÞn; gðnÞ ¼ 1; gð/XÞ ¼ 0; /n ¼ 0; ð2:1Þ gð/X; /YÞ ¼ gðX; YÞ À gðXÞgðYÞ; ð2:2Þ gðn; nÞ ¼ 1; / n ¼ 0; g / ¼ 0; ð2:3Þ for any X, Y in TM. From Eq. (2.1) and (2.2), it can be seen that

The aim of this article is to investigate the local convergence analysis of the multi-step Homeier-like approach in order to approximate the solution of nonlinear equations in Banach spaces, which fulfilled the Lipschitz as well as H¨older continuity condition. The H¨older condition is more relax than Lipschitz condition. Also, the existence and uniqueness theorem has been derived and found their error bounds. Numerical examples are available to appear the importance of theoretical discussions.

The purpose of this work is to develop two new iterative methods for solving nonlinear equations which does not require any derivative evaluations. These composed formulae have seventh and eighth order convergence and desire only four function evaluations per iteration which support the Kung-Traub conjecture on optimal order for without memory schemes. Finally, numerical comparison is provided to show its e ectiveness and performances over other similar iterative algorithms in high precision computation.