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.

We provide an iterative algorithm for fixed point issues in vector spaces in the paragraphs that follow. We demonstrate that, compared to the Ishikawa technique in Banach spaces, the iterative algorithm presented in this study performs better under weaker conditions. In order to achieve this, we compare the convergence behavior of iterations, and taking into account a few offered cases, we support the major findings.