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Stability and Convergence Results Based on Fixed Point Theory for a Generalized Viscosity Iterative Scheme

Fixed Point Theory and Applications20092009:314581

https://doi.org/10.1155/2009/314581

Received: 18 February 2009

Accepted: 27 April 2009

Published: 4 June 2009

Abstract

A generalization of Halpern's iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern's iteration, a contractive mapping, and a nonexpansive one. The sum of the real coefficient sequences of four of the above terms is not required to be unity at each sample but it is assumed to converge asymptotically to unity. Halpern's iteration solution is proven to converge strongly to a unique fixed point of the asymptotically nonexpansive mapping.

1. Introduction

Fixed point theory is a powerful tool for investigating the convergence of the solutions of iterative discrete processes or that of the solutions of differential equations to fixed points in appropriate convex compact subsets of complete metric spaces or Banach spaces, in general, [112]. A key point is that the equations under study are driven by contractive maps or at least by asymptotically nonexpansive maps. By that reason, the fixed point formalism is useful in stability theory to investigate the asymptotic convergence of the solution to stable attractors which are stable equilibrium points. The uniqueness of the fixed point is not required in the most general context although it can be sometimes suitable provided that only one such a point exists in some given problem. Therefore, the theory is useful for stability problems subject to multiple stable equilibrium points. Compared to Lyapunov's stability theory, it may be a more powerful tool in cases when searching a Lyapunov functional is a difficult task or when there exist multiple equilibrium points, [1, 12]. Furthermore, it is not easy to obtain the value of the equilibrium points from that of the Lyapunov functional in the case that the last one is very involved. A generalization of the contraction principle in metric spaces by using continuous nondecreasing functions subject to an inequality-type constraint has been performed in [2]. The concept of -times reasonable expansive mapping in a complete metric space is defined in [3] and proven to possess a fixed point. In [5], the -stability of Picard's iteration is investigated with being a self-mapping of where ( ) is a complete metric space. The concept of -stability is set as follows: if a solution sequence converges to an existing fixed point of , then the error in terms of distance of any two consecutive values of any solution generated by Picard's iteration converges asymptotically to zero. On the other hand, an important effort has been devoted to the investigation of Halpern's iteration scheme and many associate extensions during the last decades (see, e.g., [4, 6, 9, 10]). Basic Halpern's iteration is driven by an external sequence plus a contractive mapping whose two associate coefficient sequences sum unity for all samples, [9]. Recent extensions of Halpern's iteration to viscosity iterations have been proposed in [4, 6]. In the first reference, a viscosity-type term is added as extraforcing term to the basic external sequence of Halpern's scheme. In the second one, the external driving term is replaced with two ones, namely, a viscosity-type term plus an asymptotically nonexpansive mapping taking values on a left reversible semigroup of asymptotically nonexpansive Lipschitzian mappings on a compact convex subset of the Banach space . The final iteration process investigated in [6] consists of three forcing terms, namely, a contraction on , an asymptotically nonexpansive Lipschitzian mapping taking values in a left reversible semigroup of mappings from a subset of that of bounded functions on its dual. It is proven that the solution converges to a unique common fixed point of all the set asymptotic nonexpansive mappings for any initial conditions on . The objective of this paper is to investigate further generalizations for Halpern's iteration process via fixed point theory by using two more driving terms, namely, an external one taking values on plus a nonlinear term given by a continuous nondecreasing function, subject to an inequality-type constraint as proposed in [2], whose argument is the distance between pairs of points of sequences in certain complete metric space which are not necessarily directly related to the sequence solution taking values in the subset of the Banach space . Another generalization point is that the sample-by-sample sum of the scalar coefficient sequences of all the driving terms is not necessarily unity but it converges asymptotically to unity.

2. Stability and Boundedness Properties of a Viscosity-Type Difference Equation

In this section a real difference equation scheme is investigated from a stability point of view by also discussing the existence of stable limiting finite points. The structure of such an iterative scheme supplies the structural basis for the general viscosity iterative scheme later discussed formally in Section 4 in the light of contractive and asymptotically nonexpansive mappings in compact convex subsets of Banach spaces. The following well-known iterative scheme is investigated for an iterative scheme which generates real sequences.

Theorem 2.1.

Consider the difference equation:
(2.1)
such that the error sequence is generated by
(2.2)

for all , where .

Assume that and are bounded real constants and ; for all . Then, the following properties hold.
  1. (i)

    The real sequences , , and are uniformly bounded if if and if ; for all . If, furthermore, if and , if , with if and only if ; for all , then the sequences , , and converge asymptotically to the zero equilibrium point as and is monotonically decreasing.

     
  2. (ii)

    Let the real sequence be defined by if and if (what implies that from (2.1) and ). Then, is uniformly bounded if ; for all . If, furthermore, ; for all then as .

     
  3. (iii)

    Let and let a positive real sequence (i.e., all its elements are nonnegative real constants). Define if and if . Then, is a positive real sequence and is uniformly bounded if ; for all . If, furthermore, ; for all , then as .

     
  4. (iv)

    If ; for all and , then ; for all . If and ; for all , then ; for all . If and ; for all for some , with , then ; for all and as .

     
  5. (v)

    (Corollary to Venter's theorem, [7]). Assume that for all , as and (what imply as and the sequence has only a finite set of unity values). Assume also that and is a nonnegative real sequence with . Then as .

     
  6. (vi)

    (Suzuki [8]; see also Saeidi [6]). Let be a sequence in with , and let and be bounded sequences. Then, .

     
  7. (vii)

    (Halpern [9]; see Hu [4]). Let be ; for all in (2.1) subject to , ; for all with being a nonexpansive self-mapping on . Thus, converges weakly to a fixed point of in the framework of Hilbert spaces endowed with the inner product , for all , if for any .

     
Proof.
  1. (i)
    Direct calculations with (2.1) lead to
    (2.3)
     
so that if , and equivalently, if and with , and
(2.4)
Thus, , and . If, in addition, and with then and is a monotonically decreasing sequence, and as . Property (i) has been proven.
  1. (ii)
    Direct calculations with (2.2) yield for ,
    (2.5)
     
Since is a convex parabola for all if real constants exist such that ; . The parabola zeros are so that if . If , then with . Thus, if , for all . If , then as . Property (ii) has been proven.
  1. (iii)

    If is positive then is positive from direct calculations through (2.1). The second part follows directly from Property (ii) by restricting for uniform boundedness of and for its asymptotic convergence to zero in the case of nonzero .

     
  2. (iv)
    If ; for all and , then from recursive evaluation of (2.1):
    (2.6)
     
If, and ; for all , then
(2.7)

If and , for all for some , then , for all ; thus, is monotonically strictly decreasing so that it converges asymptotically to zero.

Equation (2.1) under the form

(2.8)

with and being a nonexpansive self-mapping on under the weak or strong convergence conditions of Theorem 2.1(vii) is known as Halpern's iteration [4], which is a particular case of the generalized viscosity iterative scheme studied in the subsequent sections. Theorem 2.1(vi) extends stability Venter's theorem which is useful in recursive stochastic estimation theory when investigating the asymptotic expectation of the norm-squared parametrical estimation error [7]. Note that the stability result of this section has been derived by using discrete Lyapunov's stability theorem with Lyapunov's sequence what guarantees global asymptotic stability to the zero equilibrium point if it is strictly monotonically decreasing on and to global stability (stated essentially in terms of uniform boundedness of the sequence ) if it is monotonically decreasing on . The links between Lyapunov's stability and fixed point theory are clear (see, e.g., [1, 2]). However, fixed point theory is a more powerful tool in the case of uncertain problems since it copes more easily with the existence of multiple stable equilibrium points and with nonlinear mappings. Note that the results of Theorem 2.1 may be further formalized in the context of fixed point theory by defining a complete metric space , respectively, for the particular results being applicable to a positive system under nonnegative initial conditions, with the Euclidean metrics defined by .

3. Some Definitions and Background as Preparatory Tools for Section 4

The four subsequent definitions are then used in the results established and proven in Section 4.

Definition 3.1.

is a left reversible semigroup if ; for all .

It is possible to define a partial preordering relation " " by ; for all for any semigroup . Thus, , for some existing and , such that if is left reversible. The semigroup is said to be left-amenable if it has a left-invariant mean and it is then left reversible, [6, 13].

Definition 3.2 (see [6, 13]).

is said to be a representation of a left reversible semigroup as Lipschitzian mappings on if is a Lipschitzian mapping on with Lipschitz constant and, furthermore, ; for all .

The representation may be nonexpansive, asymptotically nonexpansive, contractive and asymptotically contractive according to Definitions 3.3 and 3.4 which follow.

Definition 3.3.

A representation of a left reversible semigroup as Lipschitzian mappings on , a nonempty weakly compact convex subset of , with Lipschitz constants is said to be a nonexpansive (resp., asymptotically nonexpansive, [6]) semigroup on if it holds the uniform Lipschitzian condition (resp., ) on the Lipschitz constants.

Definition 3.4.

A representation of a left reversible semigroup as Lipschitzian mappings on with Lipschitz constants is said to be a contractive (resp., asymptotically contractive) semigroup on if it holds the uniform Lipschitzian condition (resp., ) on the Lipschitz constants.

The iteration process (3.1) is subject to a forcing term generated by a set of Lipschitzian mappings where is a sequence of means on , with the subset (defined in Definition 3.5 below) containing unity, where is the Banach space of all bounded functions on endowed with the supremum norm, such that where is the dual of .

Definition 3.5.

The real sequence is a sequence of means on if .

Some particular characterizations of sequences of means to be invoked later on in the results of Section 4 are now given in the definitions which follow.

Definition 3.6.

The sequence of means on is

(1)left invariant if ; for all , for all , for all in for ;

(2)strongly left regular if , for all , where is the adjoint operator of defined by ; for all , for all .

Parallel definitions follow for right-invariant and strongly right-amenable sequences of means. is said to be left (resp., right)-amenable if it has a left (resp., right)-invariant mean. A general viscosity iteration process considered in [6] is the following:

(3.1)

where

(i)the real sequences , , and have elements in of sum being identity, for all ;

(ii) is a representation of a left reversible semigroup with identity being asymptotically nonexpansive, on a compact convex subset of a smooth Banach space, with respect to a left-regular sequence of means defined on an appropriate invariant subspace of ;

(iii) is a contraction on .

It has been proven that the solution of the sequence converges strongly to a unique common fixed point of the representation which is the solution of a variational inequality [6]. The viscosity iteration process (3.1) generalizes that proposed in [13] for and and also that proposed in [14, 15] with , and ; for all . Halpern's iteration is obtained by replacing and in (3.1) by using the formalism of Hilbert spaces, for all (see, e.g., [4, 9, 10]). There has been proven the weak convergence of the sequence to a fixed point of for any given if for [9], also proven to converge strongly to one such a point if and as , and [10]. On the other hand, note that if , , and with , for all , then the resulting particular iteration process (3.1) becomes the difference equation (2.1) discussed in Theorem 2.1 from a stability point of view provided that the boundedness of the solution is ensured on some convex compact set ; for all .

4. Boundedness and Convergence Properties of a More General Difference Equation

The viscosity iteration process (3.1) is generalized in this section by including two more forcing terms not being directly related to the solution sequence. One of them being dependent on a nondecreasing distance-valued function related to a complete metric space while the other forcing term is governed by an external sequence . Furthermore the sum of the four terms of the scalar sequences , , and and at each sample is not necessarily unity but it is asymptotically convergent to unity.

The following generalized viscosity iterative scheme, which is a more general difference equation than (3.1), is considered in the sequel

(4.1)

for all for a sequence of given finite numbers with (if , then the corresponding sum is dropped off) which can be rewritten as (2.1) if ; for all (except possibly for a finite number of values of the sequence what implies ) by defining the sequence

(4.2)

with , where

(i) is a strongly left-regular sequence of means on , that is, . See Definition 3.5;

(ii) is a left reversible semigroup represented as Lipschitzian mappings on by .

The iterative scheme is subject to the following assumptions.

Assumption 1.

( ) , , and are real sequences in , is a real sequence in , and are sequences in , for all for some given and .

( ) , .

( ) .

( ) .

( ) ; for all with being a bounded real sequence satisfying and .

( ) is a contraction on a nonempty compact convex subset , of diameter of a Banach space , of topological dual , which is smooth, that is, its normalized duality mapping from into the family of nonempty (by the Hahn-Banach theorem [6, 11]), weak-star compact convex subsets of , defined by
(4.3)

is single valued.

( ) The representation of the left reversible semigroup with identity is asymptotically nonexpansive on (see Definition 3.3) with respect to , with which is strongly left regular so that it fulfils .

( ) .

( ) is a complete metric space and is a self-mapping satisfying the inequality
(4.4)

where , for all are continuous monotone nondecreasing functions satisfying if and only if ; for all .

( ) is a sequence in generated as , with and is a finite given number.

Note that Assumption 1( ) is stronger than the conditions imposed on the sequence in Theorem 2.1 for (2.1). However, the whole viscosity iteration is much more general than the iterative equation (2.1). Three generalizations compared to existing schemes of this class are that an extracoefficient sequence is added to the set of usual coefficient sequences and that the exact constraint for the sum of coefficients being unity for all is replaced by a limit-type constraint as while during the transient such a constraint can exceed unity or be below unity at each sample (see Assumption 1( ). Another generalization is the inclusion of a nonnegative term with generalized contractive mapping involving another iterative scheme evolving on another, and in general distinct, complete metric space (see Assumptions 1( ) and 1( ). Some boundedness and convergence properties of the iterative process (4.1) are formulated and proven in the subsequent result.

Theorem 4.1.

The difference iterative scheme (4.1) and equivalently the difference equation (2.1) subject to (4.2) possess the following properties under Assumption 1.

(i) . Also, and for any norm defined on the smooth Banach space and there exists a nonempty bounded compact convex set such that the solution of (4.2) is permanent in , for all and some sufficiently large finite with .

(ii) and as .
  1. (iii)
    (4.5)
     

(iv)Assume that such that each sequence element (the first closed orthant of ); for all , for some so that (4.1) is a positive viscosity iteration scheme. Then,

(iv.1) is a nonnegative sequence (i.e., all its components are nonnegative for all , for all ), denoted as ; for all .

(iv.2)Property (i) holds for and Property (ii) also holds for a limiting point .

(iv.3)Property (iii) becomes
(4.6)
what implies that either
(4.7)
or
(4.8)

Proof.

From (4.2) and substituting the real sequence from the constraint Assumption 1( ), we have the following:
(4.9)
Thus,
(4.10)
where is an arbitrary finite sufficiently large integer, and
(4.11)
since the functions are continuous on with and as , [2] with being prefixed and arbitrarily small. The constants and are finite for sufficiently large since (Assumption 1( ), is a contraction on (Assumption 1( ), and is a self-mapping on satisfying Assumption 1( ). Since , and as from Assumptions 1( ) and 1( ) and is finite, as and ; for all being arbitrarily small since is arbitrarily large. Since from Assumption 1( ), is an asymptotically nonexpansive semigroup on , and , , and as :
(4.12)
with as . One gets from (4.12) into (4.10),
(4.13)
what implies that
(4.14)
(see [8]) since as since is in and is a strongly left-regular sequence of means on such that ; furthermore, , , , , as and as . Thus, from (4.14) and using the above technical result in [8] for difference equations of the class (2.1) (see also [2]), it follows that
(4.15)
since from Assumption 1( ) since , , and as . From (4.1),
(4.16)
so that
(4.17)
Using Assumption 1 and using (4.15) into (4.17) yield
(4.18)
since , , , as . Also, it follows that as from (4.15) and (4.18). Note that it has not been proven yet that the sequences and converge to a finite limit as since it has not been proven that they are bounded. Thus, the four sequences and converge asymptotically to the same finite or infinite real limit. Proceed recursively with the solution of (4.1). Thus, for a given sufficiently large finite and for all , one gets
(4.19)
(4.20)
for all , for some positive real sequences , , and satisfying and , and as with and being arbitrarily small for sufficiently large , and
(4.21)
for sufficiently large and a sufficiently small which exists from Assumptions 1( ) and 1( ). Note that the sequences and may be chosen to satisfy and ; for all . Now, proceed by complete induction by assuming that for given sufficiently large and finite . Then, one gets from (4.20) that for any prescribed if
(4.22)

with and which always holds for sufficiently large finite since as . It has been proven by complete induction that the first part of Property (i) holds with the set being built such that for the given initial condition . For a set of initial conditions with any set convex and bounded, a common set might be defined for any initial condition of (4.1) in with a redefinition of the constant as . The second part of Property (i) follows for any norm on from the property of equivalence of norms. Furthermore, the real sequences and converge strongly to a finite limit in since they are uniformly bounded so that Property (ii) has also been proven. Property (iii) follows directly from (4.1) and Property (ii). Property (iv. ) follows since is a nonnegative -vector sequence provided that if what follows from simple inspection of (4.1). Properties (iv. )-(iv. ) follow directly from separating nonnegative positive and nonpositive terms in the right-hand side of the expression in Property (iii).

The convergence properties of Theorem 4.1(ii) are now related to the limits being fixed points of the asymptotically nonexpansive semigroup which is the representation as Lipschitzian mappings on of a left reversible semigroup with identity.

Theorem 4.2.

The following properties hold.
  1. (i)

    Let be the set of fixed points of the asymptotically nonexpansive semigroup on . Then, the common strong limit of the sequences and in Theorem 4.1(ii) is a fixed point of located in and, thus, a stable equilibrium point of the iterative scheme (4.1) provided that , and then , is sufficiently large.

     

(ii) .

Proof.
  1. (i)
    Proceed by contradiction by assuming that so that there exists such that
    (4.23)
     

since , where the above two limits exist and are zero from Theorem 4.1(ii). Then, , with being nonempty since, at least one such finite fixed point exists in .

Property (ii) follows directly from Theorem 4.1(iii)-(iv).

Remark 4.3.

Note that the boundedness property of Theorem 4.1(i) does not require explicitly the condition of Assumption 1( ) that is asymptotically nonexpansive. On the other hand, neither Theorem 4.1 nor Theorem 4.2 requires Assumption 1( ).

Definition 4.4 (see [8]).

Let the sequence of means be in , and let be a representation of a left reversible semigroup . Then is -stable if the functions and on are also in ; for all , for all .

Definition 4.5 (see [8, 11]).

Let and be convex subsets of the Banach space , with under proper inclusion, and let be a retraction of onto . Then is said to be sunny if ; for all , for all provided that .

Definition 4.6.

is said t be a sunny nonexpansive retract of if there exists a sunny nonexpansive retraction of onto .

It is known that if is weakly compact, is a mean on (see Definition 3.5), and is in for each , then there is a unique such that for each . Also, if is smooth, that is, the duality mapping of is single valued then a retraction of onto is sunny and nonexpansive if and only if , for all [6, 11].

Remark 4.7.

Note that Theorem 4.2 proves the convergence to a fixed point in , with being constructively proven to be nonempty by first building a sufficiently large convex compact so that the solution of the iterative scheme (4.1) is always bounded on . Note also that Theorems 4.1 and 4.2 need not the assumption of being a left-invariant -stable subspace of containing " " and to be a left-invariant mean on , although it is assumed to be strongly left regular so that it fulfils ; for all (Assumption 1( ), see Definition 3.6. However, the convergence to a unique fixed point in the set is not proven under those less stringent assumptions. Note also that Assumption 1( ) required by Theorem 4.1 and also by Theorem 4.2 as a result is one of the two properties associated with the -stability of .

The results of Theorems 4.1 and 4.2 with further considerations by using Definitions 4.4 and 4.5 allow to obtain the convergence to a unique fixed point under more stringent conditions for the semigroup of self-mappings , as follows.

Theorem 4.8.

If Assumption 1 hold and, furthermore, is a left-invariant -stable subspace of then the sequence , generated by (4.1), converges strongly to a unique ; for all , for all , for all which is the unique solution of the variational inequality . Equivalently, where is the unique sunny nonexpansive retraction of onto .

The proof follows under similar tools as those used in [6] since is a nonempy sunny nonexpansive retract of which is unique since is nonexpansive for all .

Proof.

Let be the sequence solution generated by the particular iterative scheme resulting from (4.1) for any initial conditions when all the functions and are zeroed. It is obvious by the calculation of the recursive solution of (4.1) from (4.19) that the error from both solutions satisfies
(4.24)
Since the convergence of the solution to fixed points of Theorems 4.1, 4.2, and 4.8 follows also for the sequence it follows that a unique fixed point exists satisfying
(4.25)
where is unique since is also unique from Theorem 4.8. Assume that with . Then,
(4.26)
If and ; for all and the -functions are zero then both fixed points are related by the constraint . Thus, consider a representation of a left reversible semigroup as Lipschitzian mappings on (see Definitions 3.2 and 3.3), a nonempty compact subset of the smooth Banach space with Lipschitz constants which is asymptotically nonexpansive. Consider the iteration scheme:
(4.27)
(4.28)

with , where

(i) is a strongly left-regular sequence of means on , that is, (the dual of ). See Definitions 3.5 and 3.6;

(ii) is a left reversible semigroup represented as Lipschitzian mappings on by .

Assumption 2.

The iterative scheme (4.27) keeps the applicable parts of Assumptions 1( )–1( ), 1( ) for the nonidentically zero parameterizing sequences , and . Assumptions 1( ) and 1( ) are modified with the replacements , , and .

Theorems 4.1 and 4.8 result in the following result for the iterative scheme (4.27) for , :

Theorem 4.9.

The following properties hold under Assumption 2.

(i) ; for all . Also, and for any norm defined on the smooth Banach space and there exists a nonempty bounded compact convex set such that the solution of (4.2) is permanent in , for all and some sufficiently large finite with .

(ii) and as .

(iii)

(iv)Assume that the nonempty convex subset of the smooth Banach space , which contains the sequence of means on , is such that each element ; for all , for some so that (4.1) is a positive viscosity iteration scheme (4.27). Then,

(iv.1) is a nonnegative sequence (i.e., all its components are nonnegative for all , for all ), denoted as ; for all .

(iv.2)Property (i) holds for and Property (ii) also holds for a limiting point .

(iv.3)Property (iii) becomes
(4.29)
what implies that either
(4.30)
or
(4.31)
(v)If, furthermore, is a left-invariant -stable subspace of , then the sequence , generated by (4.27), converges strongly to a unique ; for all , for all which is the unique solution of the variational inequality . Equivalently, where is the unique sunny nonexpansive retraction of onto . Furthermore, the unique fixed points of the iterative schemes (4.1) and (4.27) are related by
(4.32)

If, in addition, and and the -functions are identically zero in the iterative scheme (4.1), then .

Remark 4.10.

Note that the results of Section 4 generalize those of Section 2 since the iterative process (4.1) possesses simultaneously a nonlinear contraction and a nonexpansive mapping plus terms associated to driving terms combining both external driving forces plus the contribution of a nonlinear function evaluating distances over, in general, distinct metric spaces than that generating the solution of the iteration process. Therefore, the results about fixed points in Theorem 2.1(vi)-(vii) are directly included in Theorem 4.1.

Venter's theorem can be used for the convergence to the equilibrium points of the solutions of the generalized iterative schemes (4.1) and (4.27), provided they are positive, as follows.

Corollary 4.11.

Assume that

( ) are both contractive mappings with being compact and convex, , such that is a left-invariant -stable subspace of with being a left reversible semigroup;

( ) , with being compact and convex, , , and ; for all for some real constants , and if ;

( ) and .

Then, the sets of fixed points of the positive iteration schemes (4.1) and (4.27) contain a common stable equilibrium point which is a unique solution to the variational equations of Theorems 4.8 and 4.9; that is, and that .

Outline of Proof

The fact that the mappings are both contractive, and imply that the generated sequences , are both nonnegative and bounded for any and they have unique zero limits from Theorem 2.1(v).

The following result is obvious since if the representation is nonexpansive, contractive or asymptotically contractive (Definitions 3.3 and 3.4), then it is also asymptotically nonexpansive as a result.

Corollary 4.12.

If the representation is nonexpansive, contractive or asymptotically contractive, then Theorems 4.1, 4.2, and 4.8 still hold under Assumption 1, and Theorem 4.9 still holds under Assumption 2.

Declarations

Acknowledgments

The author is very grateful to the Spanish Ministry of Education by its partial support of this work through Grant DPI2006-00714. He is also grateful to the Basque Government by its support through Grants GIC07143-IT-269-07 and SAIOTEK S-PE08UN15. The author is also grateful to the reviewers for their interesting comments which helped him to improve the final version of the manuscript.

Authors’ Affiliations

(1)
IIDP. Faculty of Science and Technology, University of the Basque Country, Campus of Leioa (Bizkaia)

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Copyright

© M. De la Sen. 2009

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