The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu (J. Math. Anal. Appl. 224:91-101, 1998) for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces.

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In this note, by taking an counter example, we prove that the iteration process due to Agarwal et al. (J. Nonlinear Convex. Anal. 8 (1), [61][62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78][79] 2007) is faster than the Mann and Ishikawa iteration processes for Zamfirescu operators.

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We establish a new second-order iteration method for solving nonlinear equations. The efficiency index of the method is 1.4142 which is the same as the Newton-Raphson method. By using some examples, the efficiency of the method is also discussed. It is worth to note that (i) our method is performing very well in comparison to the fixed point method and the method discussed in Babolian and Biazar (2002) and (ii) our method is so simple to apply in comparison to the method discussed in and involves only first-order derivative but showing second-order convergence and this is not the case in , where the method requires the computations of higher-order derivatives of the nonlinear operator involved in the functional equation.

Complex graphics of dynamical system have striking features of fractals and become a wide area of research due to their beauty and complexity of their nature. The aim of this paper is to study dynamics of relative superior tricorns and multicorns usingS-iteration schemes. Several examples are presented to explore the geometry of relative superior tricorns and multicorns for antipolynomialz→z¯n+cof complex polynomialzn+cforn≥2.

The purpose of this paper is to introduce Kirk-type new iterative schemes called Kirk-SP and Kirk-CR schemes and to study the convergence of these iterative schemes by employing certain quasi-contractive operators. By taking an example, we will compare Kirk-SP, Kirk-CR, Kirk-Mann, Kirk-Ishikawa, and Kirk-Noor iterative schemes for aforementioned class of operators. Also, using computer programs in C++, we compare the above-mentioned iterative schemes through examples of increasing, decreasing, sublinear, superlinear, and oscillatory functions.

We establish a strong convergence for the hybridS-iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces.