Convergence and Stability of a Novel M†Iterative Algorithm with an Application

In this manuscript, we propose a novel three - step iteration scheme called M - iteration to approximate the invariant points for the class of weak contractions in the sense of Berinde and obtain that M - iteration strongly converges to one and only one fixed point for Berinde mappings. Proving ‘almost - stability’ of M - iteration, we compare the M - iterative scheme with other chief iterative algorithms, namely. Picard, Mann, Ishikawa, Noor, S, normal-S, Abbas, Thakur, Varat, Ullah and F∗ and claim that the framed iterative procedure converges to the invariant points of weak contractions at faster rate than other vital algorithms. Some numerical illustrations are adduced to strengthen our claim. We further ascertain data dependency through M - iteration. Finally, we establish the solution of Caputo type fractional differential equation as an application and exhibit that M - fixed point procedure converges to the solution of a fractional differential equation. The obtained results are not only new but, also, extend the scope of previous findings.