Convergence, stability, and data dependence of a new iterative algorithm with an application

Abstract

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.