Fibre Contraction Principle with respect to an Iterative Algorithm

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. 1. Introduction We will use the notations and notions from [1]. Let be an operator; then , , , denote the iterate operators of . By we denote the set of all nonempty invariant subsets of . By we denote the fixed point set of the operator . Let be a nonempty set, , a subset of , and an operator. By definition the triple is called an -space if the following conditions are satisfied:(i)if , for all , then and ,(ii)if and , then for all subsequences, , of we have that and . By definition an element of is convergent sequence, is the limit of this sequence and we write as . In what follows we will denote an -space by . Actually, an -space is any set endowed with a structure implying a notion of convergence for sequences. For example, Hausdorff topological spaces, metric spaces, generalized metric spaces in Perov's sense (i.e., ), generalized metric spaces in Luxemburg's sense (i.e., ), -metric spaces (i.e., , where is a cone in an ordered Banach space), gauge spaces, 2-metric spaces, - -spaces ([2, 3]), probabilistic metric spaces, syntopogenous spaces are such -spaces. For more details see Fréchet [4], Blumenthal [5], and Rus [1]. Let be a metric space. We will use the following symbols:？？ , ？？ is nonempty}, is bounded},？？ is closed}, . If is a Banach space, then is convex} Let be an -space. Definition 1. An operator is called a Picard operator (briefly PO) if(i) ;(ii) as , for all . Definition 2. An operator is said to be a weakly Picard operator (briefly WPO) if the sequence converges for all and the limit (which may depend on ) is a fixed point of . If is a WPO, then we may define the operator by If is a PO, then , for all . The following problem has been considered in [1]. Problem 3 (fibre Picard operator problem). Let and be two -spaces. Let be a WPO and let be such that is a WPO for every . Consider the triangular operator defined as follows: In which conditions is a WPO? An answer to this problem is the following result. Theorem 4 (fibre contraction principle (Rus [1, 6])). Let be an -space and let be a complete metric space. Let , be two operators and the triangular operator, . Assume that the following conditions are satisfied:(i) is a WPO;(ii) is an -contraction, for all ;(iii) is continuous. Then is