Comparison Rate of the Convergence of Single Step and Triple Steps Iteration Schemes

A fixed point of a function f:X → X is defined as an element k ∈ X such that f(k) = k. In this study, we analyze fixed point iterative procedures, which are essential for solving equations in various physical formulations. We rigorously establish and compare the convergence and convergence rates of single-step and triple-step iterative schemes with errors in Banach spaces, employing the Zamfirescu operator. Specifically, we demonstrate that for a contraction mapping T:X → X, the sequences generated by these iterative schemes converge to a unique fixed point p ∈ X. Additionally, we explore the existence and stability of Mann iterations defined by the iterative scheme xn+1 = (1 − αn )xn + αnT(xn) and Noor iterations given by xn+1 = (1 − βn )xn + βnT(T((xn )), where αn, βn are appropriate step sizes. Our results not only elucidate the effectiveness of these iterative methods but also contribute to the broader understanding of fixed point theory in Banach spaces.