- Research Article
- Open Access
Stable Iteration Procedures in Metric Spaces which Generalize a Picard-Type Iteration
© M. De la Sen. 2010
Received: 25 March 2010
Accepted: 11 July 2010
Published: 27 July 2010
This paper investigates the stability of iteration procedures defined by continuous functions acting on self-maps in continuous metric spaces. Some of the obtained results extend the contraction principle to the use of altering-distance functions and extended altering-distance functions, the last ones being piecewise continuous. The conditions for the maps to be contractive for the achievement of stability of the iteration process can be relaxed to the fulfilment of being large contractions or to be subject to altering-distance functions or extended altering functions.
Banach contraction principle is a very basic and useful result of Mathematical Analysis [1–7]. Basic applications of this principle are related to stability of both continuous-time and discrete-time dynamic systems [4, 8], including the case of high-complexity models for dynamic systems consisting of functional differential equations by the presence of delays [4, 9]. Several generalizations of the contraction principle are investigated in  by proving that the result still holds if altering-distance functions  are replaced with a difference of two continuous monotone nondecreasing real functions which take zero values only at the origin. The so-called -times reasonable expansive mappings and the associated existence of unique fixed points are investigated in . The so-called Halpern's iteration  and several of its extensions in the context of fixed-point theory have been investigated in [11–13]. Further extended viscosity iteration schemes with nonexpansive mappings based on the above one have been investigated in [9, 10, 12–18], while proving the common existence of unique fixed points for the related schemes and the strong convergence of the iterations to those points for any arbitrary initial conditions. The stability of Picard iteration has been investigated exhaustively (see, e.g., [5, 19–22]). The Picard and approximate Picard methods have been also used in classical papers for proving the existence and uniqueness of solutions in many differential equations including those of Sobolev type (see, e.g., ).
This paper presents some generalizations of results concerning the stability of iterations in the sense that the iteration scheme subject to error sequences converges asymptotically to its nominal fixed point provided that the iteration error converges asymptotically to zero. Several generalizations are discussed in the framework of stability of iteration schemes in complete metric spaces including:
(a)the use of altering-distance functions (Definition 1.1) [1, 2], and the so-called then defined extended altering functions (Definition 2.1 in Section 2) where the continuous altering functions are allowed to be piecewise continuous;
(c)the removal of the common hypothesis in the context of -stability that the set of fixed points of the iteration scheme is nonempty by guaranteeing that this is in fact true under contractive mappings, large contractions, or altering- and extended altering-distance functions, [1–4, 6].
Definition 1.1 (see  (altering-distance function)).
for some monotone nondecreasing function satisfying . Those results are directly extended to monotone nondecreasing piecewise continuous functions being continuous at "0" after a preliminary "ad hoc" definition in the subsequent section.
2. Fixed Point Properties Related to Altering- and Extended Altering-Distance Functions
Since but continuous at it can possess bounded isolated discontinuities on and it is necessary to reflect this fact in the notation as follows. The left (resp., right) limit of at is simply denoted by , instead of using the more cumbersome classical notation (resp., by instead of using the more cumbersome ). Since is an extended altering-distance function, then continuous at , . If is continuous at a given , then . If is has a discontinuity point (of second class), then , with .
Definition 2.1 (extended altering-distance function).
(i)Assume that is an altering-distance function such that , for all for some real constant . Then has a unique fixed point .
Thus, there is a unique which is in , the set of fixed points of , that is, , for all . Since is a complete metric space and the sequence with is a Cauchy sequence, for all then . It holds trivially that all the above proof also holds for special case and some monotone nondecreasing function satisfying (i.e., is a weak contraction) as may be proven  (see also ). Property (ii) has been fully proven.
Theorem 2.2 might be linked to the concept of large contraction which is less restrictive than that of contraction. The related discussion follows.
Definition 2.3 (see  (large contraction)).
It turns out that a contraction is also a large contraction with being independent of in Definition 2.3. The following result proves that the self-mapping on satisfying Theorem 2.2(ii) is a large contraction.
which is a contradiction unless , as , as , since is continuous at and , if and only if But, if the subsequence has a zero limit as , then it is a bounded sequence. Thus, as is false and then , being unbounded fails so that the contradiction follows. The right-limit convergence leads to the same conclusion. As a result, there is no such that , is unbounded and the result is fully proven.
An alternative proof to that of Theorem 2.2(ii) related to the existence of a unique fixed point in , follows directly by using Theorem in  since is a large contraction and the sequence is uniformly bounded (Propositions 2.4 and 2.5).
is a large contraction from Proposition 2.4, since it fulfils Theorem 2.2(ii). Also, , for all and from Proposition 2.5. Thus, from [4, Theorem ], has a unique fixed point in .
The following result is a direct consequence of Theorem 2.2, Propositions 2.4 and 2.5.
3. -Stability Related to a Class of Nonlinear Iterations Related to Distance and Altering-Distance Functions
according to Hypothesis ( ) since and from Hypothesis ( ). Since is everywhere continuous and satisfies , then as . Also, as as . The remaining of the proof follows with the same arguments as in that of Theorem 3.1.
Since and are both continuous and monotone nondecreasing then Hypothesis ( ) implies that , for all and ; for all with and which is Hypothesis ( ) of Theorem 3.1. Furthermore, from the continuity of everywhere within its definition domain and its property . Thus, the proof follows as in Theorem 3.1 since Hypothesis ( ) to ( ) of this theorem hold.
The following direct particular result of Theorems 3.1 to 3.3 follows.
Corollary 3.4 referred to Theorem 3.1 was first proven in . It is now of interest the removal of the condition of the set of fixed points to be nonempty by guaranteeing that is in fact nonempty consisting of a unique element under extra contractive properties of the pair . The following result holds.
The following two properties hold.
Theorem 3.5(ii) allows directly extending Theorem 3.1 as follows by removing the requirement of the set of fixed points to be nonempty with a unique element since this is guaranteed by the Banach contractive mapping principle.
with being any fixed point; that is, and some real constants , , and , and, furthermore, , for all . Then, the Picard iteration procedure , for all , is -stable with being its unique fixed point, that is, .
Theorems 3.5 and 3.6 and Corollary 3.7 are directly extendable to the case that the pair is a large contraction. Also, Theorem 3.6 and Corollary 3.7 can be extended directly for the use of distance functions or extended altering-distance functions as follows.
The author is very grateful to the Spanish Ministry of Education for its partial support of this work through Grant DPI2009-07197. He is also grateful to the Basque Government for its support through Grants IT378-10, SAIOTEK S-PE08UN15, and SAIOTEK SPE07UN04.
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