A simpler and more efficient fixed point iterative scheme
Nida Izhar Mallick, Izhar Uddin
TL;DR
The paper addresses the approximation of fixed points for self-nonexpansive mappings $K: W\to W$ in Banach spaces. It introduces a simpler, two-parameter Ishikawa-like iterative scheme with updates defined by $s_1=(1-\beta_n)s_n+\beta_n Ks_n$ and $s_{n+1}=(1-\alpha_n) r_n+\alpha_n Ks_n$, where $\alpha_n,\beta_n\in(a,b)\subset(0,1)$. The authors prove weak convergence to a fixed point in a uniformly convex Banach space and strong convergence under Condition $(I)$ together with $\liminf_{n\to\infty} d(s_n,F(K))=0$, supplemented by a numerical example showing faster convergence than classical methods (Mann, Ishikawa, Noor). These results indicate a simpler yet faster-converging method for fixed-point problems with potential broad applicability in applied mathematics.
Abstract
Our work presents a new iterative scheme to approximate the fixed points of nonexpansive mapping. The proposed algorithm is constructed to enhance convergence efficiency while preserving theoretical robustness. Under appropriate assumptions on the underlying operator, we establish weak convergence and strong convergence results for the generated sequence. To demonstrate the effectiveness of the proposed scheme, we present a numerical example and perform a detailed comparative study with several well-known iterative methods from the literature. The numerical results clearly indicate that the proposed method exhibits a faster rate of convergence than the existing schemes, thereby confirming its computational advantage. These findings suggest that the new iterative process provides an efficient and reliable alternative for solving fixed point problems arising in applied mathematics and related fields.