Research Article | Open Access
On the Extensions of Krasnoselskii-Type Theorems to -Cyclic Self-Mappings in Banach Spaces
A set of (≥2)-cyclic and either continuous or contractive self-mappings, with at least one of them being contractive, which are defined on a set of subsets of a Banach space, are considered to build a composed self-mapping of interest. The existence and uniqueness of fixed points and the existence of best proximity points, in the case that the subsets do not intersect, of such composed mappings are investigated by stating and proving ad hoc extensions of several Krasnoselskii-type theorems.
In the last years, important attention is being devoted to extend the fixed point theory by weakening the conditions on both the maps and the sets where those maps operate [1, 2]. For instance, every nonexpansive self-mappings on weakly compact subsets of a metric space has fixed points if the weak fixed point property holds . It has also to be pointed out the relevance of fixed point theory in the stability of complex continuous-time and discrete-time dynamic systems [3–5]. On the other hand, Meir-Keeler self-mappings have received important attention in the context of fixed point theory perhaps due to the associated relaxing in the required conditions for the existence of fixed points compared with the usual contractive mappings [6–10]. Another interest of such maps is their usefulness as formal tool for the study of -cyclic contractions even if the involved subsets of the metric space under study do not intersect . The underlying idea is that the best proximity points are fixed points if such subsets intersect while they play a close role to fixed points otherwise. It has to be pointed out that there are close links between contractive self-mappings and Kannan self-mappings [2, 10–13].
A rich research is being devoted to the existence and uniqueness of best proximity points of cyclic mappings under different assumptions on the vector space to which the subsets involved in the cyclic mapping belong. For instance, in , the concept of 2-cyclic self-mappings is extended to (≥2)-cyclic self-mappings and results about fixed points are derived in the case that the subsets of the considered complete metric space have a nonempty intersection. Some of the ideas in such a manuscript inspired the definition of -cyclic self-mappings from to with being the union of the subsets involved in the cyclic representation. On the other hand, the existence and uniqueness of best proximity points of (≥2)-cyclic -contractive self-mappings are investigated in  borrowing the previous scenario investigated in [16, 17] for 2-cyclic -contractive self-mappings. Generally speaking, the so-called cyclic -contractive self-mappings, which are associated with some strictly increasing unbounded map, are based on a concept of weak contractiveness contrarily to the standard, and commonly used, (strict) contractive concept being inspired in the well-known Banach contraction principle. In those papers, the Banach space under consideration is also assumed to be reflexive and strictly convex. These joint assumptions, which are less restrictive than the assumption that the space is uniformly convex (since uniformly convex Banach spaces are reflexive and strictly convex but the converse is not true in general), are proven to keep intact the essential properties of existence and uniqueness of best proximity points for cyclic -contractive self-mappings if the involved subsets are nonempty, weakly closed, and convex. The above formalism is revisited in  for cyclic -contractive self-mappings in the framework of ordered metric spaces. In , characterization of best proximity points is studied for non-self-mappings , where and are nonempty subsets of a metric space. In general, best proximity points do not fulfil in this context the usual condition . However, they jointly globally optimize the mappings from to the distances and .
In this manuscript, is a complete metric space and is considered associated to a Banach space endowed with translation-invariant and homogeneous metric and is a set of subsets of is a -cyclic self-mapping if it satisfies ; for all . A valid metric is the norm of the Banach space . If the -cyclic self-mapping is nonexpansive (resp., contractive—also referred to as a 2-cyclic contraction) then there exists a real constant (resp., ) such that The self-mapping is said to be a 2-cyclic large contraction if and are two given intersecting subsets of if This concept of 2-cyclic large contraction on intersecting subsets extends that of large contraction , and both concepts extend to -cyclic contractions. A -cyclic large contraction satisfies also (1.2) by replacing , ; for all provided that . In Section 2 of this paper, we consider a mapping defined by such that ; for all (≥2) ; for all , for all and then where ; are 2-cyclic self-mappings. It is not required for most of the obtained results that so that such a map is not required to be 2-cyclic either. For the obtained results related to boundedness of distances between iterates through , it is not required for the set of subsets of to be either closed or convex. For the obtained results concerning fixed points and best proximity point, the sets and are required to be convex but they are not necessarily closed if the self-mapping can be defined on the union of the closures of the sets and . Also, concerning best proximity points in the case that the subsets do not intersect, it is assumed that the space is restricted to be a uniformly convex Banach space [6–9]. It turns out that since uniformly convex Banach spaces are also strictly convex, the results also hold under this more general assumption used in several papers (see, for instance [15–18]). In Section 3, the results of Section 2 are extended to mappings built in a close way via (≥2)-cyclic self-mappings on a set of subsets of .
Superscript denotes vector or matrix transpose, is the set of fixed points of a self-mapping on some nonempty convex closed subset of a metric space , denotes the closure of a subset of , and denote, respectively, the domain and image of the self-mapping and is the family of subsets of , denotes the distance between the sets and for a 2-cyclic self-mapping what is simplified as ; for all for distances between adjacent subsets of -cyclic self-mappings on where are subsets of .
is the set of best proximity points on a subset of a metric space of a -cyclic self-mapping on , the union of a collection of nonempty subsets of which do not intersect.
2. Results for Mappings Defined by 2-Cyclic Self-Mappings
The following result is concerned with the above-defined map constructed with nonexpansive or contractive 2-cyclic self-mappings for .
Theorem 2.1. The following properties hold.(i)If are both 2-cyclic nonexpansive self-mappings then (ii)If for are both 2-cyclic nonexpansive self-mappings and at least one of them is contractive then Property (i) holds and, furthermore, (iii)If for are both 2-cyclic contractive self-mappings then Properties (i), (ii) hold and, furthermore, (i.e., and are closed and intersect or at least one of them is open while their boundaries intersect).
Proof. Take and . Direct calculation yields by taking into account that ; are 2-cyclic, then both self-mappings satisfy , for , and (1.1) with respective contraction constants and , and that the metric is translation-invariant and homogeneous: If then one gets from (2.8) If ; that is, if both 2-cyclic self-mappings are nonexpansive. Thus, Property (i) follows from (2.9). If only one of them is nonexpansive and the other is contractive then and so that (2.3) follow from (2.9) and Property (ii) is proven. For real constants for some real constants are both contractive and one gets from (2.9) what leads to (2.5a), (2.5b), and to ((2.6)), (2.7) by taking . Property (iii) has been proven.
Corollary 2.2. If are both 2-cyclic asymptotically contractive self-mappings, that is, they are nonexpansive with time-varying real sequences in for which converge asymptotically to two respective real numbers and in as . Assume that the sequences for are such that
is satisfied for some real . Then, (2.1) holds and, furthermore, the following properties hold:
(i)for any bounded positive decreasing real sequence , with arbitrary prescribed upper-bound , which converges asymptotically to zero as for some finite and some bounded nondecreasing positive real sequence of upper-bound .(ii)
Proof. Note that Since the terms in the sum the summation term of the right-hand-side of the above equation are all nonnegative, it follows that: Note also that for , for all , such that ; for all for any given real constant , an infinite subsequence of which is monotone decreasing and for . Thus, if the sequences in for are such that there is some finite for which according to (2.15) then if (2.11) holds, one gets from (2.11) that (2.12) holds for any given and some positive decreasing real sequence and some nondecreasing positive real sequence subject to , so that , and which converges then to zero as since it has an infinite monotone decreasing subsequence for each , satisfying to for any , then . As a result, (2.13) follows, whose upper-bound is zero if .
Remark 2.3. Note that a simple comparison between ((2.6)) in Theorem 2.1 and (2.12) in Corollary 2.2 concludes that in (2.12) can be taken as small as 2 if the sequences in Corollary 2.2 are constant and equal to for . Therefore, a logic procedure of accomplishing with (2.11) is to check for the existence of valid constants possessing a lower-bound 2.
Corollary 2.4. The following properties hold.(i) for are both 2-cyclic contractive self-mappings with time-varying real sequences () such that any element of the sum sequence is in for with and where is some infinite subset of . Then, where if , if , and if and (ii)Let be both 2-cyclic nonexpansive and asymptotically contractive self-mappings for with the sequences in for and having limits for satisfying . Then, where .
Proof. (i) First, note that the integer subset exists since the time-varying contraction sequences converge to real constants being less than unity. One gets from (2.5a) for where and (2.17) holds. Property (i) has been proven.
(ii) The proof of Property (ii) is close to that of Property (i) by using a close method to that of the proof of Corollary 2.2(ii): for all , for all , where as , for all, and ; are sequences of positive finite real constants whose integer arguments and are related to the iterate of the left and right hand sides of (2.20), respectively, defined by Since ; for all then ; for all and since the limit as , is uniformly bounded and ; for all . Thus, combining (2.20) and (2.21), one gets and then
Some Krasnoselskii-type fixed point results follow for the map defined trough 2-cyclic binary self-mappings for the case when and intersect.
Theorem 2.5. Assume that is a Banach space which has an associate complete metric space with the metric being translation-invariant and homogeneous. Assume that and are nonempty, convex, and closed subsets of which intersect. Assume also that () are both 2-cyclic contractive self-mappings and fulfilling . Then, there is a unique fixed point of in which satisfies , where are also the respective unique fixed points of ().
Proof. One gets from Theorem 2.1(iii), (2.9) since implies . Since is a Banach space then is a complete metric space, since are 2-cyclic contractive self-mappings then satisfying (1.1) with contraction constants and (since ) and subject for , , , and since , the unique limits below for Cauchy sequences exist where are unique in since (and then ) are nonempty, convex, and closed. It follows that by making in , for all . Since and then is unique so that .
Corollary 2.6. Theorem 2.5 also holds if () are both 2-cyclic asymptotically contractive self-mappings with respective time-varying contraction sequences being in with limits in .
Theorem 2.7. Assume that is a Banach space which has an associate complete metric space with the metric being translation-invariant and homogeneous. Assume that and are nonempty, convex, and closed subsets of , which intersect and have a convex union. Assume that () are both 2-cyclic self-mappings, the first one being continuous, the second one being contractive, and, furthermore, the mapping satisfies Then, there is a (in general, nonunique) fixed point of which has the properties below: where , for all , is the unique fixed point of .
Proof. Note that the following properties hold(1)By hypothesis, is a nonempty convex set which is, furthermore, closed since and are both nonempty and closed.(2) is nonempty, convex, and closed since and are nonempty, closed, and convex with nonempty intersection.(3) is invariant through the mapping by hypothesis.(4) is a nonempty compact set which is trivially invariant under the 2-cyclic continuous self-mapping .(5) is a 2-cyclic contraction which has then a unique fixed point from Theorem 2.1 since is nonempty, convex, and closed.
Thus, from the standard Krasnoselskii fixed point theorem [20, 21], under the above Properties (1)–(4), it exists which satisfies ; for all . Also, one gets from Property (5) that are limits of Cauchy sequences so that as leading to .
The following results follow from Krasnoselskii fixed point theorem extended to 2-cyclic self-mappings since is a compact set .
Theorem 2.8. Theorem 2.7 holds if is a 2-cyclic large contraction instead of being a contraction.
Theorem 2.9. Theorem 2.7 holds if the hypothesis is replaced by .
Corollary 2.10. Theorem 2.7 also follows with the replacement of the 2-cyclic contractive self-mapping by a 2-cyclic asymptotic contractive self-mapping for with time-varying contraction sequences in with limits in .
Corollary 2.10 follows by using Corollary 2.2 instead of Theorem 2.1. Also, Theorems 2.8 and 2.9 can be extended for being asymptotically contractive closely to the extension Corollary 2.10 to Theorem 2.7.
On the other hand, note that if then the convergence through the mapping to a fixed point under the conditions of Theorem 2.5, or those of Corollary 2.6, cannot be achieved since fixed points, if any, are in since . See Theorem 2.1, ((2.6)), Corollary 2.2, (2.13), Corollary 2.4, (2.18), or (2.19). It is not either guaranteed the convergence to best proximity points because the guaranteed upper-bounds for the limit superiors of distances of the iterates exceed the distance between the adjacent subsets and since the lower upper-bounds of the above respective referred to limit superiors are of the form for some real constant defined directly by inspection of ((2.6)), (2.13), or (2.18), or (2.19). Then, the following result follows instead of Theorem 2.5.
Theorem 2.11. Assume that is a Banach space which has an associate complete metric space with the metric being translation-invariant and homogeneous. Assume also that and are nonempty subsets of , which do not intersect. Assume that () are both 2-cyclic either contractive under Theorem 2.1 or under Corollary 2.4 (i) (or asymptotically contractive under Corollary 2.2 or under Corollary 2.4(ii)) self-mappings, and, furthermore, Then, is asymptotically permanent as ; for all , for all entering a compact subset of , where: and is defined directly from (2.6), under Theorem 2.1; from (2.13), under Corollary 2.2, or from either (2.18) or (2.19), under Corollary 2.4.
Note that (2.29) which restricts the image of , used in Theorem 2.5 and in Corollary 2.6 is essential for the existence of the residual set where the iterates enter asymptotically as . It is now proven how Corollary 2.4 may be improved to obtain in Theorem 2.11 so that the convergence of the iterates converge to a best proximity point if and are nonempty, disjoint, convex, and closed. The part of Theorem 2.11 referred to the fulfilment of Corollary 2.4 (i.e., either the sum of both contraction constants or the sum of their limits if they are time-varying sequences is less than unity) is improved as follows.
Corollary 2.12. Assume that is a Banach space which has an associate complete metric space with the metric being translation-invariant and homogeneous. Assume that and are nonempty, disjoint, convex, and closed subsets of . Assume also that () are both 2-cyclic either contractive under Corollary 2.4(i) with (or asymptotically contractive under Corollary 2.4(ii)) self-mappings and, furthermore, (see (2.29)). Then, any iterates , respectively, converge to a best proximity point of either , respectively, , or conversely, as for any given , .
Proof. Take some real constant and note from (1.1) that (2.8) is modified as follows: which holds since Thus, so that the limit exists modifying (2.18) in Corollary 2.4(i). Then, Theorem 2.11 holds with and is a subset of the boundary of containing the best proximity points of to the set () with . Thus, since is a Cauchy sequence since is complete, then converges to a best proximity point of (resp., ) if and is odd (resp., even). A close proof leading to a similar modified limit follows by modifying (2.19) in Corollary 2.4(ii) by using the limits of the time-varying asymptotically contractive sequences for , instead of the constants , satisfying implying for .
Some illustrative examples follow.
Example 2.13. Consider the scalar differential equations with for . The solution trajectories are defined by contractive self-mappings for as follows: Both solutions converge to respective unique fixed points as which are also stable equilibrium points, for since by using the Euclidean metric, it follows trivially for any two initial conditions and with as for . Note that the same property holds by redefining the self-mappings for so as to pick up by successive iterates starting from initial conditions the sequence of points of the solutions for some as which are both contractive since as for and with as for . Define the self-mapping by for any ; for all . Note that and for any with . Now, assume that the initial conditions are restricted to fulfil the constraint for . Consider a convex closed real interval . Then, and are in such an interval and converge to the fixed point for any initial conditions satisfying the constraint. Note that and are trivially 2-cyclic contractive self-mappings from to (Theorem 2.5).
Example 2.14. Now, assume the replacement in Example 2.13 where it exists the integral . Then, the above results still hold with the individual and combined resulting mappings still being contractive leading to respective unique fixed points , and with , subject to , and .
Example 2.15. Consider self-mappings () defined for any given , , and () by