A new iteration scheme for approximating fixed points of some generalized nonexpansive mappings

Introduction/purpose: A new three-step iteration process, which converges faster than the Mann iteration and the S-iteration, is introduced, as well as some convergence results for approximation of fixed points of the Suzuki generalized nonexpansive mappings and nearly asymptotically nonexpansive mappings have been established. Methods: The authors provide a specific three-step iterative method {x_n} in a Banach space, defined as a sequence of convex combinations of the current iterate and its images under the mapping T, with control sequences {α_n}, {β_n}, {γ_n} ⊆ (0,1). The results are proved in the setting of uniformly convex Banach spaces, where T is assumed to be either a Suzuki generalized nonexpansive mapping or an nearly asymptotically nonexpansive mapping. The authors obtain both weak and strong convergence theorems by using demiclosedness principles, Suzuki generalized nonexpansive mapping properties, and suitable lemmas on the behavior of the iterates. To compare the rates of convergence, they perform numerical experiments (usually implemented in MATLAB) where the proposed three-step iteration is run in parallel with the Thakur and S-iterations. The iterates are graphed to display the error convergence per iteration. Results: The new iteration scheme converges faster than the S-iteration scheme if the mapping is a contraction. The new iteration scheme converges to a fixed point of a Suzuki generalized nonexpansive mapping under suitable conditions. The new iteration scheme converges to a fixed point of a nearly asymptotically nonexpansive mapping under suitable conditions. Conclusions: The three-step iteration algorithm is proved, both theoretically and numerically, to converge faster than the Mann iteration and the S iteration (and sometimes faster than several other existing methods) for the considered types of mappings. The authors prove weak and strong convergence theorems for fixed points of the Suzuki generalized nonexpansive mappings and nearly asymptotically nonexpansive mappings in uniformly convex Banach spaces, thus generalizing, extending, and unifying several existing fixed-point approximation results in the literature.