THE JK ITERATIVE SCHEME FOR GENERALIZED Α-NONEXPANSIVE MAPPINGS: CONVERGENCE THEORY, A NEW NON-SUZUKI EXAMPLE, AND A COST-ADJUSTED NUMERICAL COMPARISON

Abstract

The JK (Junaid–Kifayat) iteration is a three-step scheme reported to converge faster than several classical schemes when approximating fixed points of Suzuki and generalized α-nonexpansive mappings. We give a self-contained treatment of its convergence theory for generalized α-nonexpansive mappings in uniformly convex Banach spaces. The proofs rest on

1−α

 

the inequality  x − T y  ≤ ^3+α x − T x  + x − y  , and they show that only the first averaging

parameter has to be bounded away from 0 and 1. We then construct a new piecewise-affine

self-map of [0, 4] and prove that it is generalized α-nonexpansive with the optimal value α = 1/5, while being neither nonexpansive nor a Suzuki mapping. On this map we compare JK with the Mann, Ishikawa, Noor, Agarwal, Abbas–Nazir and Thakur schemes for six parameter settings, five starting points and two tolerances, using 50-digit arithmetic. For starting points in the affine part the error is multiplied by an exactly computable factor at every iteration; we prove that JK’s factor is strictly the smallest when all parameters coincide, and the predicted iteration counts agree with the observed ones in every tested case. JK needs the fewest iterations in all 60 experiments. However, it uses four evaluations of the map per iteration, and Thakur’s scheme needs fewer evaluations in 47 of the 60 experiments; JK’s advantage in cost appears only for parameters above roughly 0.6