Abstract
We apply the fibre contraction principle in the case of a general iterative algorithm to approximate the fixed point of triangular operator using the admissible perturbation. A simple example and an application to a functional equation with parameter are given in order to illustrate the abstract results and to show the role of admissible perturbations.
Highlights
We will use the notations and notions from [1]
The aim of this paper is to establish some new fixed point theorems for triangular operators using the admissible perturbations
This notion was introduced by Rus in [25] and gives the advantage to obtain new iterative approximations of the fixed point for such operators
Summary
We will use the notations and notions from [1]. Let f : X → X be an operator; f0 = 1X, f1 = f, . . . , fn+1 = f ∘ fn, n ∈ N, denote the iterate operators of f. Let X be a nonempty set, s(X) := {(xn)n∈N | xn ∈ X, n ∈ N}, c(X) ⊂ s(X) a subset of s(X), and Lim : c(X) → X an operator. An operator f : X → X is said to be a weakly Picard operator (briefly WPO) if the sequence (fn(x))n∈N converges for all x ∈ X and the limit (which may depend on x) is a fixed point of f. The aim of this paper is to establish some new fixed point theorems for triangular operators using the admissible perturbations. This notion was introduced by Rus in [25] and gives the advantage to obtain new iterative approximations of the fixed point for such operators
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