Best proximity and fixed point results for cyclic multivalued mappings under a generalized contractive condition
© Sen et al.; licensee Springer. 2013
Received: 26 July 2013
Accepted: 5 November 2013
Published: 27 November 2013
This paper is devoted to investigating the existence of fixed points and best proximity points of multivalued cyclic self-mappings in metric spaces under a generalized contractive condition involving Hausdorff distances. Some background results for cyclic self-mappings or for multivalued self-mappings in metric fixed point theory are extended to cyclic multivalued self-mappings. An example concerned with the global stability of a time-varying discrete-time 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 discrete-time control and integrate the associate theory in the context of multivalued mappings.
MSC:47H10, 55M20, 54H25.
Important attention is being devoted to investigation of fixed point theory for single-valued and multivalued mappings concerning some relevant properties like, for instance, stability of the iterations, fixed points of contractive and nonexpansive self-mappings and the existence of either common or coupled fixed points of several multivalued mappings or operators. See, for instance, [1–24] 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, [21–23, 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 [27–30] and [4, 15, 31, 32], respectively, and references therein.
Note that is non-increasing since all its partial derivatives with respect to K, α, β exist and are non-positive; and note also that Δ is the union of the subsets ; .
where is a multivalued cyclic self-mapping on the subset of X, that is, and , where is a complete metric space including the case that is a Banach space with a norm-induced 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  for single-valued self-mappings while it generalizes that proposed and discussed in  for multivalued self-mappings which is based on the Hausdorff generalized metric.
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 , reflected in (1.2a), with one based on a Kannan-type 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 Kannan-type contractive condition in (1.2b). Note the importance of Kannan-type contractions for single-valued 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 Kannan-type constraint, since both of them do not imply each other. In this context, note, for instance, that the simple scalar single-valued 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 self-mapping 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 Kannan-extended 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 point-to-point possibilities of combinations of the comparisons for each . The various constraints are used to prove the convergence of the iterated sequences constructed with cyclic self-mappings to best proximity points. On the other hand, the use of ωD in the contractive condition, instead of the distance in-between 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 single-valued and multivalued proximal contraction mappings in-between subsets of metric spaces because of the close formal relation between cyclic self-mappings and proximal mappings. See, for instance,  and references therein.
2 Main results
The first main result follows.
Then the following properties hold:
If A and B are bounded sets which intersect, then and is a Cauchy sequence having its limit in , with ; for any given .
If A and B are not bounded, then the above property still holds if . Furthermore, exists if for any given with the sequence being constructed in such a way that .
If , then the sequence of sets converges to a subset of best proximity points in A (in the sense that any point as ) and the sequence of sets converges to a subset of best proximity points in B with .
If , i.e., if , then , and any sequence being iteratively generated as , for any , is a Cauchy sequence which converges to a fixed point of .
(ii) Assume that , that A and B are convex, and that ; are fixed points of . Then and ; , that is, the image sets of any fixed points are identical.
(iii) Consider a uniformly convex Banach space , so that is a complete metric space for the norm-induced metric , and let A and B be nonempty, disjoint, convex and closed subsets of X with satisfying the contractive conditions (2.1)-(2.2) with .
Then a sequence built so that with is a Cauchy sequence in A if and a Cauchy sequence in B if so that ; , and ; . If and , then the sequences of sets and converge to unique best proximity points and in A and B, 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.
since ; .
, where and . Note that since is cyclic, then if and conversely.
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.
Case 1: , .
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.
Case 2: , .
Then the contractive condition becomes . Then either or and with . But the second possibility is impossible since so that . Hence, since Tz is closed.
Case 3: , .
Then if , which implies for if that , equivalently, . Since , with is impossible. Hence, since Tz is closed.
Case 4: , .
Then , which is a contradiction for any . Hence, since Tz is closed.
Case 5: , .
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.
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 , 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 norm-induced 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.
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 2.2 Assume that is a uniform Banach space with associate norm-induced metric , and let A and B be nonempty closed and convex subsets of X. Assume also that , , and in the contractive condition (2.1). If , then there are and such that , , i.e., and are, respectively, best proximity points of in A and B, respectively, and simultaneously, fixed points of , respectively. In addition, if , then is 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 self-mapping. □
Remark 2.3 Note that the particular case in the contractive condition (2.1) is useful to investigate multivalued cyclic Kannan self-mappings 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 2.4 Assume that is a single-valued cyclic self-mapping where A and B are nonempty closed subsets of X where is a complete metric space. Then Theorem 2.1 and Corollary 2.2 still hold mutatis-mutandis for a fixed point if A and B are convex and intersect and best proximity points are , with , if, in addition is a uniformly convex Banach space.
Remark 2.5 The results of this section can be extended mutatis-mutandis to multivalued -cyclic self-maps , where , , and with being a complete metric space. See [2, 3, 36, 37] and references therein for some background results for single-valued cyclic s-self-mappings.
3 Example of application to time-varying discrete-time dynamic systems
3.1 Multi-control discrete-time linear dynamic system
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 non-singular 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 time-varying state-feedback control sequence is such that as for any . The controllability assumption can be weakened while keeping the stabilizability property as follows.
Proposition 3.1 Assume 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 matrices such that all the matrices in the subsequence are convergent matrices with with being some existing sequence with being a uniformly bounded sequence for any .
(ii) The subsequence of states of the closed-loop system (3.2) converges to zero as . As a result, the sequence of states of the closed-loop system also converges to zero as .
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 3.2 The following properties hold:
(i) Assume that system (3.1) is stabilizable and a linear time-varying feedback control is used where for any . Assume also that for each for some sequence of nonnegative integers , such that is uniformly bounded, for any with , there is a controller gain for some integer with for all such that any matrix in the subsequence of matrices is convergent for each for some uniformly bounded sequence of samples and some set of upper-bounded positive integer numbers for all .
(ii) If, in addition, the elements of the subsequence of pairs are all controllable for some sequence of nonnegative integers , with the sequence of natural numbers being uniformly bounded, then the closed-loop system can be exponentially stabilized via time-varying linear control with prescribed stability degree.
where the superscript ‘’ stands for the i th row vector of matrix, ‘≈’ stands for matrix similarity, denotes the identity matrix and denotes some prefixed p-real row vector by the appropriate choice of the real controller matrix , since is controllable, towards the achievement of a suitable closed-loop stability degree. Note that the closed-loop 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 closed-loop dynamics given by (3.10), which can be arbitrarily fixed via such that its non-leading real coefficients are the components of the real row vector . Thus, the sequence of closed-loop 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 time-varying closed-loop 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 upper-bound 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 closed-loop 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.
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 self-mapping. In both cases, is a fixed point of for any which is also a stable equilibrium point of the closed-loop 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 upper-bounded 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 closed-loop state-trajectory 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 vector-valued discrete-time dynamic system with multiple parameterizations
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 discrete-time dynamic system with multiple parameterization
contains at least once,
contains an infinite (countable) number of times.
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 DPI2012-30651, and to the Basque Government for its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. 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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