^{1}

^{2}

^{1}

^{2}

This paper investigates the boundedness and convergence properties of two general iterative processes which involve sequences of self-mappings on either complete metric or Banach spaces. The sequences of self-mappings considered in the first iterative scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. The sequences of self-mappings of the second iterative scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are also given to the important problem of global stability of a class of extended nonlinear polytopic-type parameterizations of certain dynamic systems.

The problems of boundedness and convergence of sequences of iterative schemes are very important in numerical analysis and the numerical implementation of discrete schemes; see [

This paper is firstly devoted to giving a framework for the contractive properties of two general iterative schemes which are constructed via combinations of elementary self-maps in appropriate metric or Banach spaces. The sequences of self-mappings of the first scheme are constructed by linear combinations of a set of self-mappings, each of them being a weighted version of a certain primary self-mapping on the same space. Such weights are nonnegative real sequences in general. The single parameterizations of the first iterative scheme include polytopic-type ones, where a set of real scalar sequences define both the sequence of self-mappings of interest and the individual parameterizations as a particular case. The second iterative scheme is a generalization of De Figueiredo scheme [

Consider the following iterative scheme under a sequence of self-mappings

Consider the iterative scheme (

Either

There exists the limit

If, in addition,

If either

The “a priori” and “a posteriori” error estimates and the convergence rate are, respectively, given by the subsequent relations:

Define the

On the other hand,

To prove property (iii), note that the assumption of uniform convergence

Property (iv) is well known for Picard iterations.

Note that the parameterization sequences

Note also that if

The following result relaxes condition (3) of strict contraction mappings in the sequence

Consider the iterative scheme (

There exists

If, in addition,

If

Note that a metric space is compact if and only if it is complete and totally bounded. Note also that

Note that a metric space is compact if and only if it is complete and totally bounded. Equivalently, a metric space is compact if and only if every family of closed subsets of

An extension of Theorem

Consider the iterative scheme (

There exists the limit

If, in addition,

If either

The “a priori” and “a posteriori” error estimates and the convergence rate are, respectively, given by the subsequent relations:

Note from (

The variation in the proof development of the concerns derived from the assumption

Note that assumption 4 of Theorem

Note that Theorems

Now, consider the iterative scheme

Let the iterative scheme (

Either

There exist the following limits:

If

As in the proof of Theorem

If

If

On the other hand, if

Theorem

Let the iterative scheme (

If

Property (i) follows from Theorem

The iterative scheme (

Let the iterative scheme (

If

As in Theorem

In a similar way as Corollary

Consider the iterative scheme (

If

Note that in Theorem

Note also that Theorem

This section contains two numerical examples. The first one is related to the Iterative Scheme 1 introduced in Section

Consider the iterative scheme defined by (

Evolution of the iterates for different initial conditions.

Furthermore, Theorem

Upper-bounding of the convergence rate of iteration.

Consider the time-varying parameterization under the time-varying weights given by

As it can be appreciated in Figure

Evolution of the time-varying weights

Iterates for different initial conditions and weights defined by (

One advantage of the results in Theorem

Evolution of the weights under (

Figure

Numerical verification of the stability condition (

Convergence of the iterates to zero for time-varying weights given by (

Also,

This second example is concerned with the iterative scheme defined by (

Evolution of

From Theorem

Evolution of the iterates through time.

This paper has investigated the boundedness and convergence properties of two general iterative processes built with sequences of self-mappings in either complete metric or Banach spaces. The self-mappings of the first iterative scheme are built with linear combinations of a set of self-mappings each of them being a weighted version of a self-mapping on the same space. Those of the second scheme are powers of an iteration-dependent scaled version of the primary self-mapping. Some applications are given for global stability of a class of nonlinear polytopic-type parameterizations of dynamic systems.

The authors declare that they have no conflict of interests regarding the publication of this paper.

The authors are very grateful to the Spanish and Basque Governments and to UPV/EHU for Grants DPI2012-30651, IT378-10, SAIOTEK S-PE13UN039, and UFI 2011/07. They thank the reviewers for their comments.