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A Picard-Mann hybrid iterative process. (English) Zbl 1317.47065
Summary: We introduce a new iterative process which can be seen as a hybrid of Picard and Mann iterative processes. We show that the new process converges faster than all of Picard, Mann and Ishikawa iterative processes in the sense of V. Berinde [Iterative approximation of fixed points. Baia Mare: Efemeride (2002; Zbl 1036.47037)] for contractions. We support our analytical proof by a numerical example. We prove a strong convergence theorem with the help of our process for the class of nonexpansive mappings in general Banach spaces and apply it to get a result in uniformly convex Banach spaces. Our weak convergence results are proved when the underlying space satisfies Opial’s condition, has FrĂ©chet differentiable norm, or its dual satisfies the Kadec-Klee property.

MSC:
47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
46B20 Geometry and structure of normed linear spaces
65J15 Numerical solutions to equations with nonlinear operators
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