Abstract
This paper is devoted to investigating the existence of fixed points and best proximity points of multivalued cyclic selfmappings in metric spaces under a generalized contractive condition involving Hausdorff distances. Some background results for cyclic selfmappings or for multivalued selfmappings in metric fixed point theory are extended to cyclic multivalued selfmappings. An example concerned with the global stability of a timevarying discretetime system is also discussed by applying some of the results obtained in this paper. Such an example includes the analysis with numerical simulations of two particular cases which are focused on switched discretetime control and integrate the associate theory in the context of multivalued mappings.
MSC: 47H10, 55M20, 54H25.
Keywords:
best proximity points; cyclic selfmappings; fixed points; metric space; multicontrol; multivalued selfmappings; uniform convex Banach space1 Introduction
Important attention is being devoted to investigation of fixed point theory for singlevalued and multivalued mappings concerning some relevant properties like, for instance, stability of the iterations, fixed points of contractive and nonexpansive selfmappings and the existence of either common or coupled fixed points of several multivalued mappings or operators. See, for instance, [124] and references therein. Related problems concerning the computational aspects of iterative calculations and best approximations based on fixed point theory have been also investigated. See, for instance, [2123,25,26] and some references therein. On the other hand, a fixed point result for partial metric spaces and partially ordered metric spaces can be found in [2730] and [4,15,31,32], respectively, and references therein.
This paper is devoted to the investigation of some properties of fixed point and best proximity point results for multivalued cyclic selfmappings under a general contractivetype condition based on the Hausdorff metric between subsets of a metric space [13,33,34]. This includes, as a particular case, contractive singlevalued selfmappings [13,25,3336], and similar problems for cyclic (strictly contractive or not) selfmappings [3537] as well. Some previous results on multivalued contractions are retaken by generalizing the contractive condition and extended to cyclic multivalued selfmappings by extending the results of Đorić and Lazović in [1] (then being extended in [6] concerning results on common fixed points of a pair of multivalued maps in a complete metric space) which are based on previous Suzuki et al. and Ćirić’s results for singlevalued selfmappings in some background literature papers. See, for instance, [2,3,33,34] and references therein. Through this paper, we consider a metric space and a multivalued 2cyclic selfmapping (being simply referred to as a multivalued cyclic selfmapping in the sequel), where A and B are nonempty closed subsets of X, so that and and . Let us consider the subset of the set of real numbers , , let the symbols ‘∨’ and ‘∧’ denote the logic disjunction (‘or’) and conjunction (‘and’), and define the functions and as follows:
for some real constants , , where
Note that is nonincreasing since all its partial derivatives with respect to K, α, β exist and are nonpositive; and note also that Δ is the union of the subsets ; .
A general contractive condition is then proposed and discussed based on the Hausdorff metric on subsets of a vector space and the constraints (1.1)(1.4). For this purpose, some preparatory concepts are needed. Let be a family of all nonempty and closed subsets of the vector space X. If , then we can define as the generalized hyperspace of equipped with the Hausdorff metric
with and being nonempty sets. The gap between the nonempty sets A and B is defined by
The proposed general contractive condition to be then discussed is
where is a multivalued cyclic selfmapping on the subset of X, that is, and , where is a complete metric space including the case that is a Banach space with a norminduced metric , so that is a complete metric space, is used, subject to (1.1)(1.4), in the main result Theorem 2.1 below. In this context, Tx is the image set through T of any which is in B, that is, (respectively, ) if (respectively, if ). It is inspired by that proposed in [34] for singlevalued selfmappings while it generalizes that proposed and discussed in [1] for multivalued selfmappings which is based on the Hausdorff generalized metric.
See Figure 1 with the plots of the various involved sets Δ and for and some of their relevant subsets in the contractive condition subject to (1.1)(1.4).
Figure 1. The setsΔand,.
Note that the proposed contractive condition, in fact, considers the worst case, given by the maximum of (1.1), of such a contractive condition of [1], reflected in (1.2a), with one based on a Kannantype contractive condition associated with the choice of possible distinct values for the constants α and β, which is reflected in (1.2b) subject to (1.3)(1.4). In particular, the choice gives a Kannantype contractive condition in (1.2b). Note the importance of Kannantype contractions for singlevalued mappings in the sense that a metric space is complete if and only if each Kannan contraction has a unique fixed point [27,38,39]. The incorporation of (1.2b), (1.3)(1.4) to (1.1) to build the general contractive condition allows an obvious direct generalization of the usual contractive condition, based on the Banach principle combined with a Kannantype constraint, since both of them do not imply each other. In this context, note, for instance, that the simple scalar singlevalued sequence ; , with initial condition , is a strict contraction if . However, it is not a Kannan contraction for all . This is easily seen as follows. Check the Kannan condition for the selfmapping T on R defining the sequence solution and , for instance, for points , , and for any . Then the Kannan contractive test is subject to , which is not fulfilled for given nonzero sufficiently small values of and any real . It is possible also to check in a similar way a failure of the generalized Kannanextended contractive condition with , for given nonzero sufficiently small values of .
In the current approach, a combination of distinct contractive conditions for the pairs of values belonging to some relevant sets constructed from the subsets ; of Δ is itself combined with the two pointtopoint possibilities of combinations of the comparisons for each . The various constraints are used to prove the convergence of the iterated sequences constructed with cyclic selfmappings to best proximity points. On the other hand, the use of ωD in the contractive condition, instead of the distance inbetween subsets, allows via the choice of some real constant to deal with problems where the achievement of limits of sequences at best proximity points is not of particular interest but just their limits superior belonging to certain subsets of the relevant sets ; containing the best proximity points. In this case, the permanence of the relevant sequences after a finite time in subsets of the sets ; after a finite number of steps is guaranteed. The standard analysis can be used for the particular case . The performed study in the manuscript seems to be also promising for its extension to the study of singlevalued and multivalued proximal contraction mappings inbetween subsets of metric spaces because of the close formal relation between cyclic selfmappings and proximal mappings. See, for instance, [40] and references therein.
2 Main results
The first main result follows.
Theorem 2.1Letbe a complete metric space, and letbe, in general, a multivalued cyclic selfmapping, whereare nonempty, closed and subject to the contractive constraint
subject to (1.1)(1.4), for some, and; . Assume also that
Then the following properties hold:
(i) There is a sequenceinsatisfying, such that
IfAandBare bounded sets which intersect, thenandis a Cauchy sequence having its limit in, with; for any given.
IfAandBare not bounded, then the above property still holds if. Furthermore, exists iffor any givenwith the sequencebeing constructed in such a way that.
If, then the sequence of setsconverges to a subsetof best proximity points inA (in the sense that any pointas) and the sequence of setsconverges to a subsetof best proximity points inBwith.
If, i.e., if, then, and any sequencebeing iteratively generated as, for any, is a Cauchy sequence which converges to a fixed pointof.
(ii) Assume that, thatAandBare convex, and that; are fixed points of. Thenand; , that is, the image sets of any fixed points are identical.
(iii) Consider a uniformly convex Banach space, so thatis a complete metric space for the norminduced metric, and letAandBbe nonempty, disjoint, convex and closed subsets ofXwithsatisfying the contractive conditions (2.1)(2.2) with.
Then a sequencebuilt so thatwithis a Cauchy sequence inAifand a Cauchy sequence inBifso that; , and; . Ifand, then the sequences of setsandconverge to unique best proximity pointsandinAandB, respectively.
Proof The proof is organized by firstly splitting it into two parts, namely, the situations: (a) defined in (1.2a), or (b) , defined in (1.2b), gives the maximum for M, defined in (1.1); and then in five distinct cases concerning (1.3), subject to (1.4), as follows.
(a) Assume that . Take, with no loss in generality, and and note that since , which implies that , and since , then it follows that . Since , then one gets from the definition of Hausdorff metric (1.5) and the contractive condition (2.1), which holds for any , that for some ,
since , and ; . Also, since and , then . Thus, there is such that so that
This implies also that and again (2.4) holds for . As a result,
By interchanging the roles of the sets A and B, one also gets by proceeding in a similar way:
Thus,
, where and . Note that since is cyclic, then if and conversely.
Now, construct a sequence in as follows: , , …, , …, which satisfies
so that
and we conclude that is a Cauchy sequence if (i.e., if A and B intersect provided that they are bounded or simply if ) since , which has a limit z in X, since is complete, which is also in which is nonempty and closed since A and B are both nonempty and closed since and . On the other hand, for any distance between A and B,
Note that the sequences and are bounded if and are such that , which is always guaranteed if A and B are bounded. If , then one gets from the above relations that
where , and . Thus, any sequences of sets and contain the best proximity points of A and B, respectively, if and, conversely, of B and A if and converge to them. This follows by contradiction since, if not, for each , there is some , some subsequence of natural numbers with for , and some related subsequences of real numbers and such that so that as is impossible.
Now, assume and consider separately the various cases in (1.3)(1.4), by using the contractive condition (2.1), subject to (1.1)(1.4), to prove that there is in to which all sequences converge by using with being a Cauchy sequence since is complete and A and B are nonempty and closed.
Then if . Thus, the contradiction holds if , and . Hence, if with since Tz is closed. If , then so that if . Hence, if and . The proof that if is similar since from the definitions of the sets and , and the fact that distances have the symmetry property.
Then the contractive condition becomes . Then either or and with . But the second possibility is impossible since so that . Hence, since Tz is closed.
Then if , which implies for if that , equivalently, . Since , with is impossible. Hence, since Tz is closed.
Then , which is a contradiction for any . Hence, since Tz is closed.
Then
which is a contradiction if . Hence, since Tz is closed. A combined result of Cases 15 is that for any . Now, assume again that and that there are two distinct fixed points necessary located in to which the sequences and converge to and , respectively, where , for , where . Assume also that . One gets from the contractive condition (2.1), subject to (1.1)(1.4), that
Thus, construct sequences , with and such that and for . Since which is nonempty, closed and convex, for any given , there is such that and are in for . Then () and () as with and . Hence, in contradicting the hypothesis that such sets are distinct. Properties (i)(ii) have been proven.
Property (iii) is proven by using, in addition, [[35], Lemma 3.8], one gets
for any sequence with and since is a uniformly convex Banach space, A and B are nonempty and disjoint closed subsets of X and A is convex. Note that Lemma 3.8 of [35] and its given proof remain fully valid for multivalued cyclic selfmaps since only metric properties were used in its proof. It turns out that is a Cauchy sequence, then bounded, with a limit in A, which is also a best proximity point of in A since
and then converges to some point , which is also a best proximity point in B (then and ), since is a uniformly convex Banach space and A and B are nonempty closed and convex subsets of X. In the same way, . Also, and are bounded sequences since is bounded and . Also, if and B is convex, then the above result holds with , and . Now, for , the reformulated five cases in the proof of Property (i) would lead to contradictions if or if . From Proposition 3.2 of [35], there are and such that since is cyclic satisfying the contractive conditions (2.1)(2.2), where A and B are nonempty and closed subsets of a complete metric space , with convergent subsequences and in both A and B, respectively, for any and in B and A, respectively, for any given . Assume that some given sequence in A is generated from some given with , which converges to the best proximity point in A of . Assume also that there is some sequence , distinct from , in A generated from with which converges to , where is a best proximity point in B of . Consider the complete metric space obtained by using the norminduced metric in the Banach space so that both spaces can be mutually identified to each other. Since for any and , it follows that if , where and are best proximity points of in A and B and is the closure of . Hence, and and then any sequence converges to best proximity points.
It is now proven by contradiction that the best proximity points in A and B are unique. Assume that are two distinct best proximity points of in A. Then there are , , and so that, one gets
which leads to the contradiction , and then . Hence Property (iii) has been proven. □
A special case of Theorem 2.1 is stated and proven in the subsequent result.
Corollary 2.2Assume thatis a uniform Banach space with associate norminduced metric, and letAandBbe nonempty closed and convex subsets ofX. Assume also that, , andin the contractive condition (2.1). If, then there areandsuch that, , i.e., andare, respectively, best proximity points ofinAandB, respectively, and simultaneously, fixed points of, respectively. In addition, if, thenis a fixed point of. The result also holds if (and, in particular, if).
Proof Assume, with no loss in generality, that . Take and by noting that since a multivalued cyclic selfmapping. □
Remark 2.3 Note that the particular case in the contractive condition (2.1) is useful to investigate multivalued cyclic Kannan selfmappings which are contractive with and some of their generalizations [33,34].
The following result follows directly from Theorem 2.1 and Corollary 2.2 without proof.
Corollary 2.4Assume thatis a singlevalued cyclic selfmapping whereAandBare nonempty closed subsets ofXwhereis a complete metric space. Then Theorem 2.1 and Corollary 2.2 still hold mutatismutandis for a fixed pointifAandBare convex and intersect and best proximity points are, with, if, in additionis a uniformly convex Banach space.
Remark 2.5 The results of this section can be extended mutatismutandis to multivalued cyclic selfmaps , where , , and with being a complete metric space. See [2,3,36,37] and references therein for some background results for singlevalued cyclic sselfmappings.
3 Example of application to timevarying discretetime dynamic systems
3.1 Multicontrol discretetime linear dynamic system
The problems of stability in differential equations, difference equations and related dynamic systems are closely related to fixed point theory of singlevalued functions since stable equilibrium points are fixed points [4144]. Also, fixed point theory of a class of cyclic selfmappings has been recently applied to differential and difference impulsive equations in a stability context study [44]. On the other hand, some typical applications of multivalued maps can be located in the framework of dynamic programming techniques for optimal control of dynamic systems [26,45]. Several tentative controls are tested to obtain the one which minimizes a suitable cost function on a certain ahead timeinterval. One of them is selected as the optimal one. Đorić and Lazović discussed in [1] a dynamic programming application in the continuoustime domain of contractive multivaluedself maps under the theoretical results of their paper. Switches among distinct parameterizations of a dynamic system and the associate stabilization problem have been discussed in the literature. Also, switching processes among different estimators of unknown systems according to the optimization or suboptimization of some appropriate loss function have been described so as to improve the estimation error. See, for instance, [4143] and included references. On the other hand, fixed point theory has been shown to be useful to discuss the stability of iterative sequences and, in general, for the analysis of the stability of discrete dynamic systems. See, for instance, [46] and references therein. We now discuss a linear timevarying discrete control problem under several tentative controls at each stage with the purpose of selecting the control sequence which guarantees a prescribed stability degree of the feedback system. The problem is stated in such a way that the tentative statetrajectory solution is formally stated as a multivalued function generating several pointtopoint iterated sequences and one of them is being selected. In particular, each current state generates a set of tentative ones at the next sampling time which belongs to the image set of the current sampled state. The convergence to fixed points or to best proximity points, if the trajectory solution sequence has a cyclic nature, describes the convergence either to equilibrium points or to a limit cycle of the solution. This second case occurs when the mapping defining the statetrajectory solution is cyclic and the subsets on whose union such a mapping is defined do not intersect. Consider the discrete timevarying control system:
where is the state vector sequence for any under some nonzero initial state and for some and all is the linear timevarying control where ; is a sequence of control gain matrices in which is chosen from an admissible set of cardinal values for each . System (3.1) is said to be an uncontrolled (or openloop) system if the control sequence is identically zero [26]. The controlled (or closedloop) system for any timevarying control being generated by a statefeedback control law under a gain matrix sequence results to be
where is a sequence of matrices in of closedloop dynamics. The stabilization via linear statefeedback of (3.2) and its links to fixed point theory via Theorem 2.1 are now discussed. For any sequence of natural numbers with , the following relation is obtained from (3.2):
where
where the superscript ‘T’ denotes matrix transposition. Note that if for , then is the controllability matrix of (3.1) on the sequence of samples . Any prefixed state is reachable in any given prefixed number of samples from a null initial condition by some linear timeinvariant statefeedback control in at most p samples if and only if (3.1) is reachable, that is, if
for any sequence of integers with , with such that is uniformly bounded. It is controllable to the origin if and only if it is reachable, that is, (3.6) holds and, furthermore, are all nonsingular for . It is well known that if the dynamic system (3.1) is controllable to the origin, then it is also stabilizable in the sense that some linear timevarying statefeedback control sequence is such that as for any . The controllability assumption can be weakened while keeping the stabilizability property as follows.
Proposition 3.1Assume that (3.1) is stabilizable (which is guaranteed if it is controllable to the origin).
Then the following properties hold:
(i) There is a sequence of control gain matricessuch that all the matrices in the subsequenceare convergent matrices withwithbeing some existing sequence withbeing a uniformly bounded sequence for any.
(ii) The subsequenceof states of the closedloop system (3.2) converges to zero as. As a result, the sequenceof states of the closedloop system also converges to zero as.
Proof One gets from (3.2) that if such a sequence of finite natural numbers exists for with , then
as for some existing sequence of stabilizing controller gains since , and then are all convergent matrices, i.e., with all their eigenvalues being of modulus less than one. Note that, since system (3.1) is stabilizable, then such a sequence of nonnegative integers always exists since it exists with for all . Now, it follows from (3.7) for any vectorinduced matrix norm that
as for any integer since as , and
since the sequence is uniformly bounded. Thus, converges to zero for any given . □
Proposition 3.1 is linked to Theorem 2.1 of Section 2 in the subsequent result.
Theorem 3.2The following properties hold:
(i) Assume that system (3.1) is stabilizable and a linear timevarying feedback controlis used wherefor any. Assume also that for eachfor some sequence of nonnegative integers, such thatis uniformly bounded, for anywith, there is a controller gainfor some integerwithfor allsuch that any matrix in the subsequence of matricesis convergent for eachfor some uniformly bounded sequence of samplesand some set of upperbounded positive integer numbersfor all.
(ii) If, in addition, the elements of the subsequence of pairsare all controllable for some sequence of nonnegative integers, with the sequence of natural numbersbeing uniformly bounded, then the closedloop system can be exponentially stabilized via timevarying linear control with prescribed stability degree.
Outline of proof Property (i) follows directly from Proposition 3.1. Property (ii) follows since all the pairs being controllable implies that the matrices of the closedloop dynamics satisfy at the subsequence of samples the following matrix relation:
where the superscript ‘’ stands for the ith row vector of matrix, ‘≈’ stands for matrix similarity, denotes the identity matrix and denotes some prefixed preal row vector by the appropriate choice of the real controller matrix , since is controllable, towards the achievement of a suitable closedloop stability degree. Note that the closedloop matrix of dynamics at the sample is similar by a similarity transformation to its companion block partitioned form in (3.10). Thus, both matrices have the same characteristic monic polynomial, thus the same characteristic roots which are also the prefixed eigenvalues of the closedloop dynamics given by (3.10), which can be arbitrarily fixed via such that its nonleading real coefficients are the components of the real row vector . Thus, the sequence of closedloop matrices can be chosen with the sequence having a stability degree such that the stability degree of . This follows since since is uniformly bounded. Thus, the timevarying closedloop system is exponentially stable with prescribed stability degree ρ and for any integer and . □
The stability degree is defined by the modulus of the dominant eigenvalue of the matrix of dynamics if the dominant eigenvalue is simple and such a number is a strict upperbound of the stability degree, otherwise. At samples which are not in the subsequence , the controller gains may be chosen arbitrarily. The exponential stabilization of the closedloop system is now related to Theorem 2.1 as follows. Assume that the sequence of sets of matrices contains at least a stabilizing matrix such that Theorem 3.2(ii) holds via stabilization with such stabilizing matrices.
Then Theorem 2.1 is applicable to some subset being a nonempty bounded set about such that the initial condition of (3.1) satisfies of with since . Take the distance function equal to the Euclidean norm so that we can consider the complete metric space to be identical to the Banach space . Redenote the sequence of points in A as the states (the replacement is made following the notation of Theorem 2.1), . If Theorem 3.2(ii) holds, then there is some bounded such that defined by for is a contractive mapping which defines the state trajectory solution at the points of the sampling subsequence . Take and in Theorem 2.1. Note that if the stabilizing matrix is chosen within the sequence of matrices, then is singlevalued. If all the matrices in the sequence are tested, then the multivalued map is defined as such that satisfies the Hausdorff particular contractive condition of Theorem 2.1. Also, one of the points of the sequence of sets satisfies the pointtopoint contractive particular condition of Theorem 2.1, by virtue of such a theorem, according to the constraints
obtained from the stabilizing control matrix sequence, and, furthermore, for all samples given by the integers . Note that the matrix norm of any real matrix M of any order satisfies , where and stand, respectively, for the maximum and minimum eigenvalues of the matrix with all its eigenvalues being real. A weak result is obtained with the particular case and for in the contractive condition of Theorem 2.1. In this case, we have a multivalued contractive Kannan selfmapping. In both cases, is a fixed point of for any which is also a stable equilibrium point of the closedloop dynamic system. Now assume , that is, the uncontrolled system (3.1) is scalar subject to a scalar control with and for some given . Take , then , and note that . The tentative controller gains used are if and if for the bounded sets of integers for , where the nonnegative real sequences of sets are uniformly bounded and contain a strictly decreasing positive real sequence with and some existing difference sequence of integers being uniformly upperbounded for .
The formalism of Section 2 is applicable to bounded sets and with . If , then a particular case of the above result follows for . If , then the closedloop statetrajectory solutions and converge to the best proximity points ε and −ε, respectively, if the initial condition is in A and, conversely, if it is in B under the sequence of stabilizing matrices .
3.2 Numerical example: a vectorvalued discretetime dynamic system with multiple parameterizations
A numerical simulation of the abovepresented example (3.1) is given now. Consider the discretetime dynamic switched system described by
with
and being the socalled switching function, which selects one of the dynamic systems subscripted by 1, 2 or 3 which parameterize the timevarying system (3.12) at each discretetime instant (or sample), n. This dynamic system is a simplified version of an automobile roll dynamics enhancement control system given in [15]. The switching function is assumed, for simulation purposes, to be the 1sample periodic (cyclic) sequence . The following statefeedback gains are considered:
The control design problem can be formulated as how to select the appropriate feedback gain at each sample, , from the set in order to guarantee the asymptotic stability of the closedloop. For this purpose, a dynamic optimization procedure can be used. Therefore, n, one considers the multivalued map for for each sample from to . The Banach space can be identified with the metric space by taking the distance to be the Euclidean norm. Thus, and so that it is direct to apply the formalism and results of Section 2. The multivalued composite map represents the set of reachable states starting from for all potential feedback gains at each sample. Figure 2 displays graphically this concept. The starting point is depicted with a circle. The application of the multivalued map T to this point produces the three points (each one corresponding to one of the feedback matrices ), labeled as first iteration in Figure 2. A second application of T generates three more points from each previous one, providing nine new points, which are depicted in Figure 2 as the second iteration. This procedure can be continued to generate the complete set of reachable states from . The ‘plus’ symbols are used to represent the image for , dots are used for , while squares are used to represent the image for .
Figure 2. Graphical representation of two iterations of the multivalued mapT.
The control algorithm generates all the images of the multivalued map T and then chooses the gain in such a way that the null vector, , is a fixed point of the multivalued map T. In this example, the choice for all allows stabilizing asymptotically the system. Then, according to Proposition 3.1, all the states are bounded and the norm of the state converges to zero asymptotically as Figures 3 and 4 show.
Figure 3. Evolution of the states with initial conditionand.
Figure 4. Evolution of the norm of the state,.
In addition, it can be verified that the following matrices are convergent as Proposition 3.1(i) states:
while the eigenvalues of the matrix product , which describes the evolution of the discrete dynamics, converge asymptotically to zero as .
3.3 Numerical example: a scalar discretetime dynamic system with multiple parameterization
Now consider the controlled singleinput singleoutput (SISO) dynamic system of the control sequence given by
with the statefeedback control law , where . This system defines the multivalued map
Note that (3.14) is a 2cyclic selfmapping with nondisjoint semiclosed sets () of of whose intersection is . Consider the complete metric space which is also a Banach space if the defined distance is the Euclidean norm. Thus, from each value of , (3.12) generates an image set of dimension 3, the points being labeled as , and . If the iteration process goes on, then each one of these values generates three more ones as depicted in Figure 5.
Figure 5. Several iterations of the multivalued map (3.14).
The multivalued map generates three images at from the starting value at , . Then at three more values are obtained from each previous one. However, note that only four different values are obtained at and five at . Thus, the image of possesses repeated values. Moreover, note that as the number of iterations increases, there are a larger number of points approaching zero since the use of the stabilizing gain forces some of the previously obtained points to approach zero. The particular numerical values for the first iterations showed in Figure 5 are as follows:
where the subscript denotes the sample while the superscript denotes the sequence of gains used to reach the point from . System (3.11) is asymptotically stabilizable provided that at least one of the following conditions for the sequence of feedback gains is met:
(ii) contains an infinite (countable) number of times.
For instance, the sequence for all satisfies the above condition (ii) and, according to Proposition 3.1, the norm of the state will converge to zero as shown in Figure 6.
Figure 6. Evolution of the norm of.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All the authors participated actively in the ellaboration of the whole paper. All authors read and approved the final manuscript.
Acknowledgements
The first and fourth authors (M De la Sen and A Ibeas) are grateful to the Spanish Government for its support of this research through Grant DPI201230651, and to the Basque Government for its support of this research through Grants IT37810 and SAIOTEK SPE12UN015. They are also grateful to the University of Basque Country for its financial support through Grant UFI 2011/07. The second author (SL Singh) acknowledges the support by the UGC New Delhi under Emeritus Fellowship. The authors are also grateful to the referees for their valuable comments which helped to improve the manuscript.
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