Abstract
In this paper, we consider the iteration method called ‘Picard-Mann hybrid iterative process’ for finding a fixed point of continuous functions on an arbitrary interval. We give a necessary and sufficient condition for convergence of this iteration for continuous functions on an arbitrary interval. Also, we compare the rate of convergence of the Picard-Mann hybrid iteration with the other iterations and prove that it is better than the others under the same computational cost. Moreover, we present numerical examples.
Highlights
1 Introduction and preliminaries Let E be a closed interval on the real line, and let f : E → E be a continuous mapping
Βn)zn γn)xn βnf γnf, for all n ≥, where x is an arbitrary initial value, {αn}, {βn} and {γn} are sequences in [, ]. They proved that ( . ) iteration method converges to a fixed point of continuous function f on an arbitrary interval E ⊂ R
The purpose of this paper is to prove that the PMH-iteration process converges to a fixed point of continuous function f on an arbitrary interval E, and compare the convergence speed of ( . ) with the other iteration processes under the suitable conditions and the same computational cost
Summary
Introduction and preliminaries LetE be a closed interval on the real line, and let f : E → E be a continuous mapping. ) iteration method converges to a fixed point of continuous function f on an arbitrary interval E ⊂ R.
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