The purpose of this article is to introduce a new two-step iterative algorithm, called \(F^{*}\) algorithm, to approximate the fixed points of weak contractions in Banach spaces. It is also showed that the proposed algorithm converges strongly to the fixed point of weak contractions. Furthermore, it is proved that \(F^{*}\) iterative algorithm is almost-stable for weak contractions, and converges to a fixed point faster than Picard, Mann, Ishikawa, S, normal-S, and Varat iterative algorithms. Moreover, a data dependence result is obtained via \(F^{*}\) algorithm. Some numerical examples are presented to support the main results. Finally, the solution of the nonlinear quadratic Volterra integral equation is approximated by utilizing our main result. The results of the paper are new and extend several relevant results in the literature.

中文翻译：

---

一种新的迭代算法及其应用的收敛性，稳定性和数据依赖性

本文的目的是介绍一种新的两步迭代算法，称为\（F ^ {*} \）算法，以逼近Banach空间中弱收缩的不动点。还表明，所提出的算法强烈收敛到弱收缩的固定点。此外，证明了（F ^ {*} \）迭代算法对于弱收缩几乎稳定，并且收敛速度比Picard，Mann，Ishikawa，S，normal-S和Varat迭代算法快。此外，通过\（F ^ {*} \）获得数据依赖结果算法。提出了一些数值示例来支持主要结果。最后，利用我们的主要结果，对非线性二次Volterra积分方程的解进行了近似。本文的结果是新的，并扩展了文献中的一些相关结果。