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Journal of Operators
Volume 2013 (2013), Article ID 408791, 6 pages
http://dx.doi.org/10.1155/2013/408791
Research Article

Fibre Contraction Principle with respect to an Iterative Algorithm

Department of Mathematics, Babeş-Bolyai University, 1 M. Kogălniceanu, 400048 Cluj-Napoca, Romania

Received 6 March 2013; Accepted 2 September 2013

Academic Editor: Lingju Kong

Copyright © 2013 Marcel-Adrian Şerban. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 a WPO and Moreover, if is a PO, then is a PO and , where and .

Theorem 4 generalizes the result of Hirsch and Pugh [7]. The fibre contraction principle is used in order to prove the differentiability of the solutions for some operatorial equations with respect to parameters. For more considerations on fiber WPOs and applications see Sotomayor [8], Tămăşan [9], Rus [6, 10], Şerban [1113], Andrász [14], Bacoţiu [15], Petruşel et al. [16, 17], Chiş-Novac et al. [18], Ilea and Otrocol [19], Dobriţoiu [2022], Olaru [23], and Barreira and Valls [24].

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.

2. Admissible Perturbations of an Operator

Let be a nonempty set, and , be two operators. We consider the operator defined by

Definition 5 (Rus [25]). We call an admissible perturbation of corresponding to if satisfies()  , ;()  , implies .

We remark that but, in general,

Example 6 (Krasnoselskii). Let be a vectorial space, a convex subset, , and , defined by
Then is an admissible perturbation of . We will denote by and call it the Krasnoselskii perturbation of . The corresponding iterative algorithm generated by the Krasnoselskii perturbation of is

It is known that the Krasnoselskii iteration is convergent to a fixed point of the operators in the case when is a bounded closed convex subset of a Hilbert space and is a nonexpansive and demicompact operator (see [26]).

Let be an -space, , and .

Example 7 (GK-algorithm (Rus [25])). We consider the iterative algorithm
By definition, this iterative process is convergent if and only if We remark that . So, this algorithm is convergent if and only if is WPO. If is WPO and an admissible perturbation of , then is a set retraction.

We call this algorithm, Krasnoselskii algorithm corresponding to or -algorithm.

For other examples of iterative algorithms see Rus [25].

3. Fibre Contraction Principle for Admissible Perturbations

Let and be two nonempty sets, , , and the triangular operator Let and satisfying and . Then , is an admissible perturbation of and, for , , is an admissible perturbation of . In these settings, we have the corresponding iterative algorithm for starting points , , and the operator is an admissible perturbation of and From the fibre contraction principle, we have the following.

Theorem 8. Let be an -space and let be a complete metric space. Let , be two operators, the triangular operator , , and satisfying and . One supposes that(i) is a WPO;(ii) is an -contraction, for all ;(iii) is continuous.
Then(a); (b) is a WPO; that is, the iterative algorithm, defined by (15), is convergent to a fixed point of ;(c)if is a PO, then is a PO.

Proof. (a) From (i) and the fact that satisfies and we have that . From (ii) we have that is PO for a fixed , so and for all . But , also, satisfies and ; therefore . Thus, It is easy to see that the triangular operator satisfies the conditions from the fibre contraction principle and, thus, we get the conclusion.

Let us consider, in particular, the following iterative algorithm for starting points , , , , , , , , are vectorial spaces. The operator is an admissible perturbation of and

Corollary 9. Let be a Hilbert space, a Banach space, and , . Let , be two operators and the triangular operator . One supposes that(i)there exists such that is WPO;(ii)there exists such that is an -contraction, for all ;(iii) and are continuous.
Then(a);(b) is a WPO; that is, the iterative algorithm defined by (19) is convergent to a fixed point of ;(c)if is a PO, then is a PO.

Proof. In this case we take , , , . Since and are continuous, then is continuous and from Theorem 8 we get the conclusion.

Corollary 10. Let be a real Hilbert space, a Banach space, and , . Let , be two operators and the triangular operator . One supposes that:(i) is a nonexpansive and demicompact operator;(ii)there exists such that is an -contraction, for all ;(iii) is continuous.
Then(a), for any ;(b) is a WPO; that is, the iterative algorithm, defined by (19), is convergent to a fixed point of ;(c)if then is a PO.

Proof. In this case we take , , , . Condition (i) implies that converges (strongly) to a fixed point of for any , (see Theorem 3.2 in [26]), so is WPO. Since is nonexpansive and is continuous, then, also, is continuous and from Corollary 9 we obtain the conclusion.

Corollary 11. Let be a real Hilbert space with the inner product and the norm , a Banach space and , . Let , be two operators and the triangular operator . One supposes that:(i)   is a generalized pseudocontraction; that is, there exists, for all , with ;(ii)   is -Lipschitz, with ;(iii)  there exists such that is an -contraction, for all ;(iv)   is continuous.
Then is a PO, for all , satisfying that is, the iterative algorithm, defined by (19), is convergent to a unique fixed point of , .

Proof. Conditions (i) and (ii) imply that is -contraction for any satisfying (23), with (see Theorem 3.6, in [26]), so is PO and . From the continuity of and we have that, also, is continuous and from Corollary 9 we get the conclusion.

The following example is a simple illustration of the above abstract results and also shows the importance of the admissible perturbations for a faster convergence of Picard iterations.

Example 12. Let us consider the following system on . We have the operators and It is clear that a solution of the system (24) is a fixed point of the triangular operator The system (24) has a unique solution , , and the iterative algorithm (19) is convergent to for any and .

Proof. We have that is -Lipschitz, with , so is not a contraction. Notice that, for any and , the Picard iteration is an oscillatory sequence so it is not convergent. For we have therefore
We have the following cases:(i)if , then and ;(ii)if , then and ;(iii)if , then and ;(iv)if , then , so is not a contraction.
For any we get that is a contraction, so is PO. The smallest contraction constant is obtained for , so for this choice we get the best rate convergence for .
For we have for all ; thus which implies that is not a contraction. For we obtain so We have the following cases:(i)if , then and ;(ii)if , then , and ;(iii)if , then , and ;(iv)if , then , so is not a contraction.
For any we get that is a contraction. The smallest contraction constant is obtain for , so the fastest iterative algorithm (19) is obtained for and .

4. Application

Let us consider the following functional equation with parameter where and is a compact interval. Such type of equations arise from several classes of integral equations with parameter, initial value and boundary value problems for ordinary differential equations with parameter.

From Theorem 8 we have the following.

Theorem 13. Corresponding to (34) one supposes that(i)  ;(ii)   for all ;(iii)  there exists such that for all , , .
Then (34) has in a unique solution and for all .

Proof. Let and be defined by We consider on the supremum norm . From (i), (ii), and (iii) we have that is -contraction, so is PO; that is, (34) has in a unique solution and as for all .
Let us suppose that . We have from that This leads us to consider the following operator defined by So, we have the operator and the operator In our conditions, from Theorem 8 we have that is PO and .
Let us take such that and consider the sequence with . Since , we can prove by induction that . So, From this it follows that and .

Remark 14. For (34) in the case where the admissible perturbation is not used, see [18].

Acknowledgments

The author would like to thank Professor Ioan A. Rus for giving useful suggestions and comments for the improvement of this paper. This work was supported by a Grant of the Romanian National Authority for Scientific Research, CNCS—UEFISCDI, Project no. PN-II-ID-PCE-2011-3-0094.

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