Abstract

This paper discusses the properties of convergence of sequences to limit cycles defined by best proximity points of adjacent subsets for two kinds of weak contractive cyclic maps defined by composite maps built with decreasing functions with either the so-called -weaker Meir-Keeler or -stronger Meir-Keeler functions in generalized metric spaces. Particular results about existence and uniqueness of fixed points are obtained for the case when the sets of the cyclic disposal have a nonempty intersection. Illustrative examples are discussed.

1. Introduction

The background literature on best proximity points and associated convergence properties in cyclic contractions and proximal contractions in the framework of fixed point theory is abundant. See, for instance, [1–21] and references therein. The literature includes related studies on cyclic contractions and cyclic weak contractions and proximal contractions [1–14, 18–21] and proximal weak contractions [15–17]. See also [22–25] for related results. On the other hand, fixed point theory has a wide amount of applications, for instance, in the study of stability of dynamic systems and differential and difference equations. See, for instance, [21, 22, 26]. In this context, the relevance of cyclic contractions and cyclic nonexpansive mappings is also of interest when strips of the solutions of dynamic systems or difference equations have to lie in different time intervals or due to control actions or external events in distinct defined sets.

The study of contractions in metric and quasi-metric spaces and in generalized metric and quasi-metric spaces has been focused on in a number of papers. See, for instance, [1–4] and references therein. A group of the obtained results are based on the existing background literature on Meir-Keeler contractive-type results. See, for instance, [5, 6]. In particular, the existence of periodic fixed point theorems of weak contractions in the setting of generalized quasi-metric spaces has been studied in [2], while the existence of fixed points for weak contraction mappings in complete generalized metric spaces has been investigated in [3]. The paper has a section of preliminaries where the concepts of Meir-Keeler functions, weaker Meir-Keeler functions, and stronger Meir-Keeler functions are generalized “ad hoc” to be used to define weak generalized contractive mappings involving subsets of a generalized metric space which do not intersect in general. In this context, appropriate nondecreasing functions , generalized weaker Meir-Keeler functions , and stronger Meir-Keeler functions are used to define the generalized - and -weak -cyclic contraction mappings defined and studied in this paper. Section 3 gives and proves a set of main results on -weak -cyclic contraction mappings and on generalized -weak -cyclic contraction mappings. Such results are related to boundedness and to convergence properties of generalized distances of sequences of points built through generalized -weak and through generalized -weak contractive cyclic maps, either in adjacent subsets or in the same subset, and also on the convergences of sequences either to best proximity points or to fixed points in the case when the subsets of the cyclic disposal intersect. Some illustrative examples adapted to the stated and proved results are also discussed.

2. Preliminaries

Let and be the sets of integer numbers and real numbers, respectively, and define their subsets: , , , , and .

Definition 1. For some given , a mapping is said to be a -weaker Meir-Keeler function if, for each real number , there exists a real number such that for some , .

Definition 1 generalizes the two existing definitions below.

Definition 2 (see [1, 2]). A mapping which is a -weaker Meir-Keeler function is said to be a weaker Meir-Keeler function.

Definition 3 (see [1]). A mapping which is a weaker Meir-Keeler function for is said to be a Meir-Keeler function.

Definition 4. For some given , a mapping is said to be a -stronger Meir-Keeler function if, for each real number , there exist real numbers and such that , .

Definition 4 generalizes the existing definition below.

Definition 5 (see [1, 2]). A mapping which is a -stronger Meir-Keeler function is said to be a stronger Meir-Keeler function.

Through the paper, we will use the mappings , , and which belong to the sets of functions defined below.

Definition 6. For some given , the class is the set of -weaker Meir-Keeler functions which satisfy the following: for and . is decreasing for all .For , one has if   for any given real number , andthere exists if .

Definition 7. The class is the set of nondecreasing functions which satisfy the following: for and for . is subadditive; that is, for every , .For all , if and only if .

Definition 8. The class is the set of -stronger Meir-Keeler functions for some real constant which satisfy the following: for and .

Definition 9 (see [2, 3]). Let be a nonempty set. A generalized metric (g.m.) is a mapping which satisfies(1), , and if and only if ;(2), ;(3), , .

Definition 10 (see [2, 3]). Let be a nonempty set and let be a g.m. on . Then, is said to be a generalized metric space (g.m.s.).

Some basic considerations and properties on a g.m.s. are now quoted from [2] to be then invoked in the body of this paper. Let be a g.m.s. Then, is said to be g.m.s. convergent to if, for each given , such that , , and this is denoted by or as . If, for each given , such that , , , then is called a g.m.s. Cauchy sequence in . If every g.m.s. Cauchy sequence in is g.m.s. convergent in , then is called a complete g.m.s.. It has been pointed out in [2] that a g.m.s. Cauchy sequence is not necessarily a Cauchy sequence and that a g.m.s. convergent sequence is not necessarily either Cauchy or a convergent sequence.

Example 11 (see [2]). Consider the set for some given and define as follows for some given :Then, is a g.m. and then is a g.m.s.. However, is not a metric, and then is not a g.m.s., since .

3. Best Proximity Point and Fixed Point Theorems

We now get some results on -weak contraction mappings and some results on related best proximity points. Given a nonempty abstract set with nonempty subsets , , we say that a self-mapping is a -cyclic self-mapping if , , with the notation convention that , , .

In fact, if we extend the above definition to the case , we find trivially that can be considered as an -cyclic mapping if .

Definition 12. Let be a g.m.s., let be nonempty subsets of , having a common distance in-between adjacent subsets , , and let be a -cyclic self-mapping satisfyingfor some and some . Then, is said to be a generalized -weak -cyclic contraction mapping.

Theorem 13. Let be a g.m.s. and let be a generalized -weak -cyclic contraction mapping for some and some . Then, the following properties hold:
(i) for any sequence constructed from , for some given initial point .
(ii) Any sequence built from any given initial point is bounded.
(iii) Assume, in addition, that is a complete g.m.s. and that has a best proximity point from to (i.e., ) for some given and that is approximatively compact with respect to . Then,, and for any given , and there is such that, for any positive integers , one has , , , and .
Also, all the best proximity points from to are each them unique in if one of them , for some , is unique in and is closed, . Also, each best proximity point is also a fixed point of the respective composite mapping , and then -periodic fixed points of .

Proof. Since , we can consider equivalently to be an arbitrary point of for some given arbitrary and we can define the sequence inductively by , . Since is a -weak-cyclic contraction mapping, one gets from (2) by induction for each thatwhere , , , and , , . Since , one has from property that is decreasing for all so that is decreasing and converges to some limit , . It is now proved that all the limits , . Since , it is also a -weaker Meir-Keeler function so that, for each real number , there exists a real number such that , for some , for any given , . Thus, for any given arbitrary such that is also arbitrary; one has for each given that if , then since is nondecreasing, and for with equality standing if and only if . Thus, there exist the identical limits , . Then, one gets from (5) and the constraint of the class thatThis also implies from the constraint of the class that , , .
Property (i) has been proved.
To prove Property (ii), we use contradiction arguments by assuming that some sequence , generated by from any , is unbounded. Thus, for any given real constant , there is such that, for any , . There is also some such that, for any , there is such that for all . This is trivial since, by defining , we can choose any real constant such that, for some nonempty interval of positive integers , since the inequality holds by construction for the case . So if is unbounded, then there is some subsequence which diverges so that there are some strictly increasing sequence and some strictly increasing sequence such that and . Then, there is a strictly increasing sequence of natural numbers and a strictly increasing sequence of positive real numbers such thatand then one getsOn the other hand, one has from the rectangular inequality of the g.m.s. which leads to from Property (i). Thus, either or there is some positive real sequence such that . If , then is a bounded subsequence of , a contradiction to its claimed unboundedness. Otherwise, if does not converge to zero while , with , then this contradicts (10). As a result, for any and Property (ii) has been proved.
It remains to prove Property (iii). Since is approximatively compact with respect to if as for some and , then has a convergent subsequence [6]. Since is a best proximity point from to , . If is the unique best proximity point from to , then it is a -periodic point of and a fixed point of . This follows by reformulating (5) with initial points and with being the unique best proximity point from to . One concludes from the properties of the classes and that and . Since has a unique best proximity point to , one concludes that . Next, we prove that . Assume that this property is not true. Thus, there is some for with such that one gets from (5) and the constraints and for the class and the constraint for the class thatwhich leads by taking limits as to the contradiction . Then, , . We now use again (5) with , and an arbitrary to yieldand one concludes that , , and since , , and then as and as since and . Since is approximatively compact with respect to , there is a convergent subsequence for the arbitrary given .
It is now proved that with for any . Assume that this is not the case. Then, there exists a sequence with and some such that the subsequence does not converge to . Since is single-valued, if does not converge to , then it does not converge. It cannot have either a convergent subsequence and since then would have two distinct images in which is impossible.
Thus, one gets from (12) thatfor the given and some subsequences , in with ; since is a -weaker Meir-Keeler function which satisfies and , is defined from to , nondecreasing, and satisfies and . Thus, one gets from (13) the following contradiction:so that , irrespective of the initial point , and is unique if is unique. Since the same contradiction arguments can be used for any claimed nonconvergent subsequence of , one concludes that any such a sequence as well as the whole sequence converges for the given . The sequence is a g.m.s. Cauchy sequence since it is convergent and is a complete g.m.s.. Since is closed, then for the given and it is unique if is unique. Thus, the set of best proximity points is unique if any of them is unique.
It is now proved that , , , and . Proceed by contradiction by assuming that there is and some subsequences of nonnegative integers , , and such that one gets for any two distinct initial points from the subadditivity property of the class and the rectangular inequality of the generalized metricNote that as from Property (i) and alsoIf the convergence of these subsequences fails, then the conclusion got from (12) fails and then Property (i) is not true. For instance, if is false, then as fails. Then,as since is the unique best proximity point in (since is the unique best proximity point in ) and it has been already proved that any sequence in each has to converge to its best proximity point if unique. Also Thus, the subsequent contradiction is got if as is not true for any :Then, one getsand the convergence properties of all subsequences distances and points also hold for the whole sequence so that for such that possesses a unique best proximity point . The two above second limit identities follow since the sequences and may be replaced by any positive integers without altering the got final conclusion. Since is a -weak-cyclic contraction single-valued mapping, then for are unique best proximity points, so equivalently , , are all unique so that the above limit properties can be extended for any sequences with given initial points , . Since , , and are strictly increasing sequences which can be chosen independently of each other, the above conclusion is equivalently enounced as follows: for each given , there is some such that if , then .
It is now proved that . Define , with having a convergent subsequence as it has been proved above and pick up any arbitrary initial point such that for the given arbitrary . Then, one gets by using the rectangular inequality of the g.m.s. that Since , it follows directly thatand is unique since is unique and is single-valued with since is closed. As a result, since is arbitrary, all the best proximity points in-between adjacent subsets are unique if any of them is unique and satisfy for any . It also turns out that ; then it is a fixed point of the composite mapping , , and then a -periodic fixed point of . Property (iii) has been proved.

Note that the classes and (Definitions 6 and 8) can be redefined as sets of functions from to which are identically zero in . This generalization is irrelevant in practice since the domains of the functions of the classes and used to define the weak contractive mappings involve distances of points in-between adjacent subsets of the cyclic disposal, so distances are not smaller than . Theorem 13 and also the relevant subsequent results of the paper can be also got with those extended definitions since , , , implies that In the case that the adjacent subsets are nonempty, closed, and intersecting, we can obtain the subsequent result on the existence of a unique fixed point allocated in their intersection.

Theorem 14. Let be a complete g.m.s. and let be a generalized -weak -cyclic contraction mapping such that that the sets , , are closed and intersect for some and . Then, there is a unique fixed point .

Proof. From Theorem 13(i) for , it follows for any and any thatfor any given and any given . Thus, for any given , there is such that , , and and is g.m.s. Cauchy sequence. Assume on the contrary that there are subsequences and such that . But this contradicts (25) so that is g.m.s. Cauchy sequence for any given initial point . Since is g.m.s. complete, any , , which is g.m.s. Cauchy sequence is convergent to some if for any given , with . It suffices to notice that , , and with satisfying the constraint (equivalently, ). Since is closed, , , , and it follows that , . Also is a fixed point of , . Otherwise, we would get the contradiction . Since , , , and is single-valued, then , , for the given and .
It remains to prove that the fixed point to which the sequences converge is independent on the initial point so that it is unique. Assume that . We consider them to be initial conditions of two distinct sequences in some pair of adjacent subsets and for some so as to use the generalized -weak contractive condition (2), and one getswhich leads to which contradicts .

Note that Theorem 14 does not require the sets , , to be convex sets so as to prove the uniqueness of the fixed point. Simply, the weak contractive condition (2) is used by considering that two claimed distinct fixed points which are in the nonempty intersection of , , belong in any case to two adjacent subsets , while the weak contractive condition guarantees that they are identical. We can extend easily Definition 12 and Theorem 13 to the case when the distances in-between adjacent subsets are distinct as follows. The existence and uniqueness of fixed points for generalized -weak (noncyclic) contraction mappings of the type (2) on a g.m.s. have been proved in [3].

Sufficient conditions for the uniqueness of uniqueness of the best proximity points in each of the subsets , , are discussed in the subsequent result which is related to the consideration of a complex structure on the metric space.

Theorem 15. Let be a complete g.m.s. admitting midpoints which has nonempty closed subsets , with , , where is an abstract set, with one of the subsets , for some , being strictly convex. Assume that there is some strictly increasing function such that has the following uniform convexity property for all , , , and :Let be a generalized -weak -cyclic contraction mapping for some and some . Then, has a unique best proximity point , , which is also a -periodic point of and a fixed point of the composite mapping , .
If , then , , is the unique fixed point of .

Proof. Assume that for some is a strictly convex set. Since is a generalized -weak -cyclic contraction mapping, it follows from Theorem 13 that as , , for any given . Note that there is some integer such that for any given (i.e., belongs to the strictly convex subset ) for the given initial point and with , , , and some for any given . We now take initial conditions for sequences built with for any given such that and for some . Then, it follows thatwith , , , and . Then, for every given , there exists an integer such that , .
One now firstly proves that as , , ; equivalently, as , so that, for every , there is such that, for all integer numbers , , . Assume that the property as , , does not hold. Thus, there exists some such that, for each , there exist integers for which . Choose such that and choose such that . For such a chosen arbitrary constant , there exists such that for all integer numbers , for all , and also there exists such that for and all , , , from Theorem 13(i). Similar distance constraints also hold from Theorem 13(i) for appropriate subsequences of nonnegative integers and (with ) exceeding certain finite thresholds, dependent on , for each initial conditions ( being a strictly convex subset of ) and ; that is, for all for and some , , and for all , some , , and all . Now, note that since the set is strictly convex, and , it follows thatNow, choose so that one gets from (28) for all integers that the following contradiction holds:Since is arbitrary, it can be chosen so that since is strictly increasing in . Then, for each given , there is some such that, for all integer numbers , , , such that the set is strictly convex for some , and then , , , and for any given implying that , are sequences in . From Theorem 13(i) (see (5)), it follows that as , . From Theorem 13(ii), it follows that , are bounded sequences in the strictly convex set . From the above discussion in this proof, it follows that as . Since is a generalized metric and is a complete g.m.s., it follows that and are bounded and g.m.s. convergent sequences in to and , respectively, and then g.m.s. Cauchy sequences and . Finally, since is strictly convex, there cannot exit two distinct with the minimum distance to , and thus . As a result, there exists a unique best proximity point such that is strictly convex and closed. Since is single-valued, then is the unique best proximity point in , , which satisfies , . Then, has a unique best proximity point , (since is closed, ), which is also, by construction, a -periodic point of and a fixed point of the composite mapping , . It turns out that if , then , is the unique fixed point of .

Remark 16. Note that in Theorem 15 is a uniformly convex metric space (not specifically a uniformly convex Banach space) and might be an abstract set (nonnecessarily being a linear space). However, it is obvious that any uniformly convex Banach space is also a uniformly convex metric space. In this case, it is assumed that one of the subsets of the cyclic disposal, for which the generalized -weak -cyclic contraction mapping is defined, is strictly convex and subject to a uniform convexity-type condition (28) for the elements of such a set with respect to the points of . That condition is related to the uniformly convex structure of the metric space and, equivalently, to the existence of midpoints in the sense that if (28) fails to hold, then the metric space is not uniformly convex. Such a condition, supported by the existence of midpoints, is close to the one satisfied on uniformly convex Banach spaces [27] and used in Lemmas 7-8 and Theorem of [6] to prove the existence and uniqueness of fixed points in a -cyclic contractive condition on the union of two nonempty closed and convex subsets of . See also [23–25]. In particular, could be endowed with a simpler structure than a linear space as, for instance, a group endowed with a composition law, provided that the subsets of the cyclic disposal are closed while just one of them is strictly convex.
Note also that could even be just an abstract set with nonempty closed subsets with one of them being strictly convex.
Note that Theorem 15 is applicable to the case that is a uniformly convex Banach space with a norm-induced metric so that with nonempty closed convex subsets , . This is a direct consequence of the fact that if is a uniformly convex Banach space, then it is also complete and the norm triangle inequality can be trivially upper-bounded by quadrangular-type sum of norms. This allows adding distances related to distinct subsequences converging to zero to the usual triangle inequality leading to be able to characterize the complete also as being a complete g.m.s. [28].
Finally, note that the assumption of the existence of midpoints [27] can also be focused on under the framework of convex structures perhaps at the expenses of a more involved presentation. It is said that a mapping is a convex structure on if, for each , , . See, for instance, [29–33]. Note that for any and for any . If and , then satisfying is the midpoint of the segment . The convex structure allows characterizing uniformly convex metric spaces as triples associated with metric spaces [32]. See also [28, 34–36]. For a generalized metric and its associate quadrangular constraint of distances, we can define in the same way a generalized convex structure such that, for , one hasand the choice , , and leads to Thus, if is the midpoint of , then is located at 1/3 from and 2/3 from of the length .

Example 17. Consider the set of real numbers for some given and consider nonempty closed subsets and of . For some given real constants , and , a g.m. is defined as follows:Note that and that is not a metric since , and then is not a metric space, while it is a g.m.s. since where is such that . For instance, take . , for the above , is such that .
For instance, take . Define a -cyclic mapping as follows:which is a generalized -weak -cyclic contraction for some , for instance, it can be defined so as to satisfy the constraints and for , and some chosen to satisfy and for for some so as to be nondecreasing and subadditive. Then, is a generalized -weak -cyclic contraction which converges to a limit cycle with the two best proximity points and . Theorem 13 leads to the above conclusion.
Note that a -weak -cyclic contraction on is defined by the sequence , with and defined under the same above constraints, leading to the distances in-between adjacent subsets given by the sequence .

Example 18. Reconsider Example 17 with the same set and the same metric with and . Then, and . The map is defined by for , , and . Then, is a generalized -cyclic contraction which converges to its fixed point . Theorem 14 leads to that conclusion.

Example 19. How to combine Examples 17 and 18 with the constraint (28) is now checked, assumed as hypothesis in Theorem 15. Such a constraint is close to the property of uniformly convexity of Banach spaces 8 (see also Remark 16). Now, for all real constant such that , there is some real constant such that . Several cases can occur:(a) for , . This is solved with strictly increasing for if ; , that is, if , and , .(b), for and for with for which leads to .(c), for (and, equivalently, for ) with , with the strictly increasing , for and some real constant .The various above parametrical constraints are compatible with those in Example 17, and the constraint (28) of Theorem 15 holds, if However, due to the discrete nature of the set , the assumption of Theorem 15 on strict convexity of sets towards the proof of existence of unique best proximity points is not fulfilled by this example.

Definition 20. Let be a g.m.s., let be a set of nonempty subsets of distances in-between adjacent subsets , , and let be a -cyclic self-mapping satisfyingfor some and some and each , where the classes and are defined with functions and , .
Then, is said to be a generalized -weak -cyclic contraction mapping.

Theorem 21. Let be a g.m.s. and let be a generalized -weak -cyclic contraction mapping. Then, the following properties hold:
(i) , , , and for any given and any given arbitrary , where is a composite mapping of the functions for .
(ii) Assume that is a complete g.m.s. and that has a best proximity point from to (i.e., ) for some given and that is approximatively compact with respect to . Then, there is a best proximity point from to which is unique and in if is unique and is closed.
Also, if all the subsets , , are closed, then there are best proximity points , , which are unique if anyone of them is unique. Each is also a fixed point of the composite mapping , , and then a -periodic fixed point of .

Proof. The proof is close to that of Theorem 13. Let be an arbitrary point of for some given arbitrary and define the sequence inductively by , . Since is a -weak -cyclic contraction mapping, one gets from (39) by induction for each thatsince and , , where , . Since , one has from property and the composite mapping structure that is decreasing for all so that is decreasing and converges to some limit . It is now proved that all the limits , . Since , it is also a -weaker Meir-Keeler function so that, for each real number , there exist a real number such that , for some , for any given , . Thus, for any given arbitrary such that is also arbitrary; one has for each given that if , then , and , and then there exist the identical limits , . Then, one gets from (42) that

The notion of -weak -cyclic contraction mapping is given below.

Definition 22. Let be a g.m.s., let be nonempty subsets of , having a common distance in-between adjacent subsets , , and let be a -cyclic self-mapping satisfyingfor some and some . Then, is said to be a generalized -weak -cyclic contraction mapping.

Theorem 23. Let be a g.m.s. and let be a generalized -weak -cyclic contraction mapping for some function , some , and some . Then, the following properties hold:
(i) for any sequence constructed from , for any given initial point .
(ii) Any sequence built from any given initial point is bounded.
(iii) Assume, in addition, that is a complete g.m.s. and that has a best proximity point from to (i.e., ) for some given and that is approximatively compact with respect to . Then, there is a best proximity point from to which is unique in if is unique in which is closed for the given .
Also, if all the subsets of ; are closed, then there are best proximity points , , which are unique if one of them is unique, with each of them being also a fixed point of the respective composite mapping , , and then -periodic fixed points of .

Proof. Since , we can consider equivalently to be an arbitrary point of for some given arbitrary and we can define the sequence inductively by , . Since is a -weak-cyclic contraction mapping, one gets from (44) that, , provided that , and then the sequences , , bounded from below are strictly decreasing, so convergent to some , since , for and since and the property of the class . Note that , , , and , . It is be proved from (46), since , that so thatAssume that this is not the case, so that , and proceed by contradiction to conclude that . Suppose that for some . Thus, for each , there is and such that for all , since and since is a stronger Meir-Keeler function. Then, one gets from (46) since and , , , thatsince , a contradiction. Then, for any ,Property (i) has been proved. Property (ii) is proved in the same ways as its counterpart in Theorem 13 since it is based on Theorem 13(i) which still holds for this theorem and from the rectangular property of distances of the g.m.s. . Finally, Property (iii) follows from the given assumptions, similar to those of Theorem 13(ii) since (50) has been got from (46) which is a similar property to that used in the proof of Theorem 13(iii).

Based on (50) and (46) obtained for the proof of Theorem 23, it is direct to prove in a similar way as it has been done for Theorem 14, Theorem 15 (see also Remark 16), and Theorem 21, the following results.

Theorem 24. Let be a complete g.m.s. and let be a generalized -weak -cyclic contraction mapping such that that the sets ; are closed and intersect for some and . Then, there is a unique fixed point .

Theorem 25. Let be a complete g.m.s. admitting midpoints which has nonempty closed subsets , with , , where is an abstract set, with one of the subsets , for some , being strictly convex. Assume that there is some strictly increasing function such that has the property (28) for all , , and . Let be a generalized -weak -cyclic contraction mapping for some , some , and some . Then, has a unique best proximity point , , which is also a -periodic point of and a fixed point of the composite mapping , .
If , then , , is the unique fixed point of .

Theorem 26. Let be a g.m.s. and let be a generalized -weak -cyclic contraction mapping. Then, the following properties hold:
(i) , ,  , and for any given and any given arbitrary for some set of functions , , some real constants , , where is a composite mapping of such functions.
(ii) Assume that is a complete g.m.s. and that has a best proximity point from to (i.e., ) for some given and that is approximatively compact with respect to . Then, there is a best proximity point from to which is unique and in if is unique and is closed.
Also, if all the subsets , , are closed, then there are best proximity points , , which are unique if anyone of them is unique. Each is also a fixed point of the composite mapping , , and then a -periodic fixed point of .

Example 27. Examples 17–19 can be reformulated for generalized -weak -cyclic contraction mappings with the replacement with some since and .

Competing Interests

The author declares that he has no competing interests.

Acknowledgments

The author thanks the University of the Basque Country for its partial support of the work through Grant UFI 11/07 and the Spanish Government by Grant DPI2015-64766-R.