On Convergence, Stability, and Data Dependence of a New Kirk-Type Iteration Process

In this study, we introduce a new Kirk-type iteration method that includes some important iteration algorithms in its special cases. Afterward, we investigate the behaviors of strong convergence and fixed point of this iteration method under certain mappings. According to the obtained results, it is theoretically observed that the convergence behavior of the new Kirk-type iteration method is better than some of the existing Kirk-type iteration methods. Furthermore, the rate of convergence studied in this paper is numerically compared with the same Kirk-type iteration methods as well as with some classical iteration methods. Also, new iteration method has been proven to be stable under certain conditions. As another important research topic, the concept of data dependency of Kirk-type iteration method, which has not been studied in the literature, was proved for the first time in this study. It was shown that the solution of a nonlinear equation can be reached more efficiently through this type of iteration method. Finally, we show that the theoretical results obtained throughout the study are valid with interesting numerical examples.