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Journal of Applied Mathematics
Volume 2013 (2013), Article ID 310106, 12 pages
http://dx.doi.org/10.1155/2013/310106
Research Article

Some Results on Fixed and Best Proximity Points of Precyclic Self-Mappings

Institute of Research and Development of Processes, University of Basque Country, Campus of Leioa (Bizkaia)-Aptdo. Postal 644-Bilbao, 48080 Bilbao, Spain

Received 11 March 2013; Accepted 2 July 2013

Academic Editor: D. R. Sahu

Copyright © 2013 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper is devoted to investigating the limit properties of distances and the existence and uniqueness of fixed points, best proximity points and existence, and uniqueness of limit cycles, to which the iterated sequences converge, of single-valued, and so-called, contractive precyclic self-mappings which are proposed in this paper. Such self-mappings are defined on the union of a finite set of subsets of uniformly convex Banach spaces under generalized contractive conditions. Each point of a subset is mapped either in some point of the same subset or in a point of the adjacent subset. In the general case, the contractive condition of contractive precyclic self-mappings is admitted to be point dependent and it is only formulated on a complete disposal, rather than on each individual subset, while it involves a condition on the number of iterations allowed within each individual subset before switching to its adjacent one. It is also allowed that the distances in-between adjacent subsets can be mutually distinct including the case of potential pairwise intersection for only some of the pairs of adjacent subsets.

1. Introduction

A relevant attention has been recently devoted to the research of existence and uniqueness of fixed points of self-mappings as well as to the investigation of associate relevant properties like, for instance, stability of the iterations [13]. The extension of such topics to the existence of either fixed points of multivalued self-mappings [1, 419], in generalized metric spaces [20, 21], or to the existence of common fixed points of several multivalued mappings or operators is receiving an important attention, for example, [7, 8, 1519, 22] and references therein. Relevant properties on the existence and uniqueness of fixed points and best proximity points for multivalued cyclic self-mappings have been studied under general contractive-type conditions based on the Hausdorff metric between subsets of a metric space. See, for instance, [4, 79], including as a relevant particular case the contractive condition on contractive single-valued self-mappings, [1, 410], as well as concerns related to their extension to cyclic self-mappings. See, for instance, [7, 8, 11] and references there in. The various related performed researches include the cases of strict contractive cyclic self-mappings and Meir-Keeler type cyclic contractions [23, 24]. Also, some of the existing background fixed point results on contractive single and multivalued self-mappings, [1, 4, 5, 9, 10, 2528] and references therein, under some types of contractive conditions, have been revisited and extended in [4]. There is also a wide sample of fixed point type results available on fixed points and asymptotic properties of the iterations for self-mappings satisfying a number of contractive-type conditions while being endowed with partial order conditions. See [18, 19] and references therein.

The main objective of this paper is the investigation of the properties of the distances as well as the existence and uniqueness of fixed points and best proximity points related to contractive so-called single-valued contractive ≥2-precyclic self-mappings . Such a concept extends that of contractive ≥2-cyclic self-mappings.

The concept of precyclic self-mapping generalizes that of cyclic self-mappings in the sense that a finite set of consecutive iterations are optionally allowed within a particular subset of the cyclic disposal of interest before a switching of the image of the self-mapping to the adjacent subset of its pre-image in the iterated sequence. It can also eventually happen that some sequence enters a certain subset and the solution remains permanent within such a subset. The precyclic self-mappings are contractive if they are subject to contractive conditions of similar types to those arising in contractive cyclic self-mappings.

Precyclic contractive self-mappings allow the generation of iterated sequences under constraints of the form for being less than a prescribed positive integer number ; for all , for all which can be set and point dependent, while ; for all . The ordering of the subsets for switching between pairs of adjacent subsets to perform the -precyclic self-mapping is, so-called, in the sequel a -precyclic disposal.

Let   be the set of nonnegative real numbers and   the set of nonnegative integer numbers. Consider a metric space endowed with a metric   and a finite set of nonempty subsets ; of and a so-called (2)-precyclic self-mapping such that ; for all , where for under the congruence relation , that is ; for all , for all . Note that the previous concept of precyclic self-mapping generalizes that of a -cyclic self-mapping which satisfies the stronger constraint ; for all . Let be the distance in-between any two subsets ; for all . Note that, compared to a cyclic self-mapping, an iterated sequence from a precyclic self-mapping might contain iterated subsequences of finite or infinite cardinals in a single subset even if . Also, certain iterated sequences generated from -precyclic self-mappings can converge to a fixed point, rather than oscillate in-between sets of distinct best proximity points, even if , provided that the iterated points all stay in the same subset , for some , after a finite number of iterations.

For each given , define the following nondecreasing strictly ordered (in general, point dependent) switching sequence of nonnegative integers: containing the numbers of consecutive iterations within each individual subset ; for all , before switching to the successive adjacent subsets , and so forth for of the iterated sequence For all such that

for any given and, either ; for all with (i.e., infinite cardinal of a numerable set) or there is for some existing finite and then is a finite set, that is, with being in the same subset as ; for all ; for all . If for any , for all , then only one iteration stays at each subset before switching to the adjacent one so that the -precyclic self-mapping is a standard -cyclic one. Note, for instance, that if in (2) and , then . If this occurs for each , then on is a usual -precyclic self-mapping.

We will establish the formulation in metric spaces . It might be pointed out that it is usual to state formulations related to differential or dynamic systems and their stability, including those being formulated in a fractional calculus framework, in normed or Banach spaces since their dynamics evolve through time described by their state vectors [14, 2939]. A possibility to focus on the study of their equilibrium points in a formal and structured fashion as well as their limit solutions, provided that they exist, (for instance, the presence of possible limit cycles) is through fixed point theory since the equilibrium points are fixed points of certain mappings and the limit cycles are repeated portions of limit state space trajectories. See, for instance, [33] and references therein. In the subsequently formulated and proved results, where the convexity of sets of is required, it will be assumed that is a normed space with associated metric space for a norm-induced metric , [29]. If is a Banch space, then is a complete metric space. The converses are not true without invoking additional assumptions. For instance, if is a metric space (resp., a complete metric space) endowed with a homogeneous and translation -invariant metric , then the metric-induced norm exists so that is a normed (resp., Banach) space endowed with such a metric-induced norm.

Example 1. For some given , define the real intervals and consider the scalar sequence where with . Note that, if , then and can be a candidate for fixed point depending on certain simple contractive or, at least, nonexpansive conditions on the sequence . If , then the convergence of the sequences and (but not that of which is not convergent) can be possible only to best proximity points . The self-mapping   on defining the solution sequence is a -cyclic one since the solution points are alternately in and . However, the modification where For all with initial value is a -precyclic (but no a -cyclic) self-mapping which generates two consecutive iterations in both and before switching to the other subset. Several extended variants are possible; that is, is constant in this case. For instance, can be dependent on the solution point or on the initial condition. If is infinity, then the trajectory solution remains in either (resp., in ) for if   (resp., ). In this case, depending on conditions on the parameterization sequence , the convergence of the solution in one of the subsets could be possible, even if , when is infinity in at least one of the sets and for some subset of values of the solution so that the solution enters such a set and remains in it for all later iterations. If for any point of the solution sequence at any iteration, then the solution trajectory switches in-between the subsets and so that the -precyclic self-mapping is also a -cyclic one.

2. Convergence of Iterated Sequences to Fixed Points

The following assumptions are made.

(1) There are bounded real functions ; for all fulfilling in under the congruence relation for some and any given such that for where are binary functions; for all such that if   and if ; for all , for all . The notation to be used does not distinguish explicitly the cases when the contractive-like functions are real constants or point-dependent functions, but this can be easily deduced from context.

(2) If , that is, if , then ;  for all .

(3) If , then , where ; for all , for all , provided that   is closed

(4)

If for some , then the constraint (9) is replaced with its following standard counterpart stated for -cyclic self-mappings:

Note that the previous condition holds if and but also if in its particular version , since , and that it contains (9) for iterated sequences generated from as a particular case. Note also the following. (a)A particular pair can satisfy simultaneously several constraints (9). For instance, assume that for some . Then (b)If , then there is some set such that ; for all and some finite for each given . Then, is nonexpansive and is bounded. Note that Assumption 4 is guaranteed directly by Assumption 3 if . If , then Assumption 3 is not invoked; however, Assumption 4 guarantees that is bounded with .(c)Assumption 4 implies that the -precyclic self-mapping is asymptotically nonexpansive.(d)Any -precyclic self-mapping is also a -cyclic self-mapping.

In the following, denotes the set of fixed points of the self-mapping . The following results hold.

Theorem 2. Let be a -precyclic self-mapping. Assume also that the constraint (9) is extended for any and ; for all as follows: where and if    and and , otherwise; for all , for all .
Then, the following properties hold.(i) is a -cyclic self-mapping if and only if  ; for all .(ii)If ; for all and ; for all , then has no fixed point in .(iii)If for some , then has a fixed point in a subset ,   for some , to which the iterated sequence converges if is complete, and is closed. If   for some , then the iterated sequence converges to a fixed point of   for some with such that , provided that is closed, or to a fixed point   for some with and provided that is closed. If, furthermore,   for , respectively,   for are convex then the corresponding fixed point is unique.

Proof. for some ; then such that . Then, is not a cyclic self-mapping. Hence, Property (i) follows.
Note that ; for all ; for all . If , then is an infinite sequence of switches in-between adjacent subsets of which are never intersecting. Thus, has no fixed point in . Hence, Property (ii) follows.
To prove Property (iii), first note that such that
Note also that, if for some , then the iterated sequence built from such an initial point through remains in , for such a , for all and some finite integer . Then, the asymptotically nonexpansive self-mapping is asymptotically contractive from Assumption 4 and also contractive after a finite number of iterations since . Thus, is bounded and has a Cauchy convergent subsequence since is complete. Since the subset is nonempty and closed for such , there is a fixed point of from Banach contraction principle and such a fixed point is unique if the subset is, furthermore, convex.

Now, the following result is proven for a class of contractive -precyclic self-mapping .

Theorem 3. Assume that is a normed space with associated metric space for a norm-induced metric . Let be a -precyclic self-mapping and ; for all . If are nonempty, bounded, closed, and convex; for all and , where . Then has a unique fixed point . If is not closed for some while is a Banach space, and then is a complete metric space, then has a unique fixed point in .

Proof. It follows that since for any finite , since is bounded, where that is, the distance between any two consecutive elements of any such a sequence converges asymptotically to zero. Furthermore, since the subsets are nonempty, closed, all of them intersect and the composite self-mapping is uniformly Lipschitz -continuous in , since it is contractive with constant , the sequence converges to for some , which is a unique fixed point of the composite in ; for all . To prove uniqueness, proceed by contradiction. Assume that there are two fixed points , in of .
Then, which leads to the contradiction that . Thus, there is a unique fixed point of in . Also, one gets from (16) that for any . As a result,
and with some unique for some ;  for all . Now, assume that there are two distinct fixed points and of for some . Since is nonempty, closed, and convex for any , then is nonempty, closed, and convex; for all . From the convexity of , there is such that with some real constant and . Then, so that as and, since is uniformly Lipschitz continuous in , the sequence converges to . But so that we can repeat the previous reasoning with , , and some real constant to conclude that as and converges to which is a contradiction to its convergence to . Then, there is a unique fixed point in the nonempty, closed, and convex set . By extending the same reasoning to any pair of subsets and of , one concludes that the composite self-mapping has a unique fixed point .
It remains to be proved that the unique fixed point of the composite mapping is also the unique fixed point of . Since the subsets intersect, one gets from (16) that since , where . Also, so that and it is the unique fixed point of . If some is not closed, then all Cauchy sequences have a limit in if is complete so that there is still a unique fixed point in of and .

The following result is now proven for a class of nonexpansive -precyclic self-mapping .

Corollary 4. Assume the hypothesis of Theorem 3 with the modified weaker condition of nonexpansive self-mapping . Then, is bounded; for all there is a subsequence , for some strictly ordered subset   of  , such that   as ; for all for some real . If, in addition, ; for all , then as ; for all .

Proof. is always finite since is bounded and then is a bounded sequence; for all . Also, ; for all , for all since . Thus, ;  for all since is bounded and ;  for all from the properties of distances since is finite. Thus, there is a subsequence for which converges as . If, in addition, ; for all , then ; for all , so that as ; for all .

The subsequent result is related to convergence to a unique fixed point in one of the subsets of the precyclic disposal even if the subsets do not intersect, provided that the self-mapping is asymptotically contractive in one of the subsets.

Corollary 5. Assume the hypothesis of Theorem 3 with the subsequent further hypothesis on the -precyclic self-mapping : (1); (2) (nonnecessarily unique; that is, can have a cardinal greater than one) such that for some given initial point .Then, the iterated sequence , (2), converges to a fixed point in   for a unique , where is a strictly ordered finite set of , containing all the elements of the set , with the strict order relation defined by its enumeration order defined by for any if a precedes in the previous enumeration definition of the set .

Proof. Since ; such that for the given as if , or as if for any nonnegative integer such that . This is obvious since, if such a does not exist, then the iterated sequence , (2), with starting point for the given has infinitely many switches in-between consecutive adjacent subsets ; then the switching sequence associated with such an iterated sequence is finite so that .
From the extra previous hypothesis 2, there is a nonempty set for which for the given . Note that the -precyclic self-mapping is asymptotically contractive, for the given on , where is unique (even if is of cardinal greater than one) from the fact that in (23) is of finite cardinal, strictly ordered, then with unique minimal and maximal elements which are then a unique minimum and maximum , respectively. From the previous part of this proof, it also turns out that   is, furthermore, (strictly) contractive on such a subset for any infinite subsequence . Therefore, such a subsequence is itself a Cauchy sequence fulfilling and then being bounded and convergent in (since satisfies a Lipschitz-continuous property) to a fixed point , since is nonempty, bounded (then is bounded), and closed, which is unique since is convex. Since the sequence contains, by construction, all the elements of the subsequence alter a finite number of iterations, it also converges to the same unique fixed point .

Remark 6. (1) Note that Corollary 5 is stated for a certain iterated sequence being built from a starting point since the contractive conditions a point-to-point condition. Point-to-point contractivity-type conditions have been also used in the literature for the characterization of fixed point properties of contractive self-mappings. See, for instance, [40, 41] and references therein. It can be generalized directly under generalization for any starting point in any of the subsets or in some subset of its union. It can be reformulated, in particular, if ; for all . In such a case, it follows the convergence of the sequence of iterates to the same unique fixed point in built from any initial point .
(2) Note that the uniqueness of the final set from the initial set , such that , arises from the fact that the first subset where the iterations remain after a finite number of them is the relevant one for the final reached limit if the precyclic self-mapping stops to transfer the iterated sequence to the next adjacent subset.
(3) Note that, if the convexity assumption is only made on the subset , then Corollary 5 still holds.
(4) Note also that, if the convexity assumption on the subsets is removed in Corollary 5, then the existence of the fixed point is still proven although that one is not necessarily unique.

The subsequent result is a consequence of direct proof of Corollaries 4 and 5.

Corollary 7. Assume the hypothesis of Corollary 5 under the weaker condition of asymptotic nonexpansivness of the self-mapping in some of the subsets for some given initial point . Then, and   for some finite and a unique as defined in Corollary 5.

Some extensions of Corollary 7 can be directly obtained from Remark 6 (3)-(4).

3. Convergence to Best Proximity Points

The following preliminary technical result concerning the convergence for distances in-between consecutive iterates through the self-mapping is now given in the case that the subsets do not necessarily intersect.

Theorem 8. Assume that the following constraints hold.(1); for all .(2) is constant within each subset and the switching sequences are not point dependent on the given for all so that, in addition, (3)(4)
Then, the following properties hold for any : for all , , for all .
If, in addition, Assumption 5(5)Holds; then,

Proof. Take any . Thus, one gets from (9) where
Then, since , with . Now, since
it follows that
Thus, if , then one gets from the previous relationships
In the same way, if the constraint is extended to ; ; for all , then one gets from (34) and (38) that for all . Thus, with . Then, (31)–(33) follow and the result is proven for . Such a choice can be made with no loss in generality since, if, instead, for any given , then the previous result still holds with with for a unique integer . On the other hand (28)-(29) are a consequence of (36) which is independent of the constraints ; for all .

Remark 9. (1) Note that Theorem 8 requires a set of necessary constraints on the and ; for all which are induced by Assumptions 45 as follows: with .
(2) Note that if , and ; for all , then is a contractive -cyclic and
If (29) is evaluated in this case, then the limit superior is also the limit inferior of the obtained expression leading to the existence of the limit being equal to , the distance in-between al, the adjacent subsets ; for all . This is a well- known result for contractive -cyclic self-mappings , [23, 24, 42].
(3) Note that the first property of (32) is a convergence of the iterated sequences of distances to best proximity points of adjacent subsets provided that Assumptions 1–5 of Theorem 8 hold implying, in particular, a strict contractive constant on a composite self-mapping (see Assumption 3) and a set of further constraints on the distances in-between adjacent subsets even if such distances are not identical (see Assumptions 45). The following result is supported by parallel results in [42] (see also [43]) for cyclic self-mappings. It establishes the asymptotic convergence of the iterated sequences to cycles containing best proximity points in-between adjacent subsets in uniformly convex Banach spaces.

Theorem 10. Assume that is a uniformly convex Banach space with the subsets being all disjoint, bounded, closed, and convex for all and that is a -precyclic self-mapping subject to Assumptions 1–5 of Theorem 8.
Then, the subsequences and converge to respective unique best proximity points in and in for each starting point ; for all . The sequence converges asymptotically to a unique cycle of points: which contains the p best proximity points.

Proof (Outline of Proof). Note that is a complete metric space for a (norm-) induced metric since is a Banach space. Thus, Theorem 8 remains true for such a metric. Since the subsets are nonempty, bounded, and closed (then compact and also boundedly compact), there exist such that for each so that and are the best proximity points in to and from to , respectively [42]. If and , then both of them are confluent in the fixed point ; . From the relation (32) of Theorem 8, it follows that
The composite self-mapping