In the article, we have proposed a new type of hybrid iterative scheme which is a hybrid of Picard and Thakur et al. repetitive schemes. This new hybrid iterative scheme converges faster than all leading schemes like Picard-S∗ hybrid, Picard-S, Picard-Ishikawa hybrid, Picard-Mann hybrid, Thakur et al. and Abbas and Nazir, S-iterative, Ishikawa and Mann iterative schemes for contraction mapping. By using the Picard-Thakur hybrid iterative scheme, we can find the solution of delay differential equations and also prove some convergence results for nonexpansive mapping in a uniformly convex Banach space.

We have developed a Jungck version of the DK iterative scheme called the Jungck–DK iterative scheme. Our analysis focuses on the convergence and stability of the Jungck–DK scheme for a pair of non-self-mappings using the more general contractive condition. We demonstrate that this iterative scheme converges faster than all other leading Jungck-type iterative schemes. To further illustrate its effectiveness, we provide an example to verify the rate of convergence and prove the data dependence result for the Jungck–DK iterative scheme. Finally, we calculate the escape criteria for generating Mandelbrot and Julia sets for polynomial functions and present visually appealing images of these sets by our modified iteration.

The purpose of this article is to initiate the notion of monotone multivalued generalized ($\alpha$,$\beta$)-nonexpansive mappings and explore the iterative approximation of the fixed points for the mapping in an ordered CAT(0) space. In particular, we employ the S-iteration algorithm in CAT(0) space to prove some convergence results. Moreover, some examples and useful results related to the proposed mapping are provided. Numerical experiments are also provided to illustrate and compare the convergence of the iteration scheme. Finally, an application of the iterative scheme has been presented in finding the solutions of integral differential equation.

In this article, we use the Picard-Thakur hybrid iterative scheme to approximate the fixed points of generalized $ \alpha $-nonexpansive mappings. For generalized $ \alpha $-nonexpansive mappings in hyperbolic spaces, we show several weak and strong convergence results. It is proved numerically and graphically that the Picard-Thakur hybrid iterative scheme converges more faster than other well-known hybrid iterative methods for generalized $ \alpha $-nonexpansive mappings. We also present an application to Fredholm integral equation.

In this paper, we study the four-step iterative process of Garodia and Uddin in the framework of Banach spaces and obtain convergence results for multivalued generalized α-nonexpansive mappings. We provide a numerical example supporting our claim that this process converges faster than other leading iterative schemes. Our results improve, extend, and adjust existing results in the literature.