Abstract

This paper relies on the study of fixed points and best proximity points of a class of so-called generalized point-dependent -hybrid -cyclic self-mappings relative to a Bregman distance , associated with a Gâteaux differentiable proper strictly convex function in a smooth Banach space, where the real functions and quantify the point-to-point hybrid and nonexpansive (or contractive) characteristics of the Bregman distance for points associated with the iterations through the cyclic self-mapping. Weak convergence results to weak cluster points are obtained for certain average sequences constructed with the iterates of the cyclic hybrid self-mappings.

1. Introduction and Preliminaries

The following objects are considered through the paper.(1) The Hilbert space on the field (in particular, or ) is endowed with the inner product which maps to , for all which maps to , where is a Banach space when endowed with a norm induced by the inner product and defined by , for all . It is wellknown that all Hilbert spaces are uniformly convex Banach spaces and that Banach spaces are always reflexive.(2) The (≥2)-cyclic self-mapping with is subject to , where are subsets of , for all , that is, a self-mapping satisfying , (3) The function is a proper convex function which is Gâteaux differentiable in the topological interior of the convex set ; , that is, and convex since is proper with since is convex, and for each , there is (the topological dual of ) such that since   is Gâteaux differentiable in where denotes the Gâteaux derivative of at if . On the other hand, is said to be strictly convex if (4) The Bregman distance (or Bregman divergence) associated with the proper convex function   , where , is defined by provided that it is Gâteaux differentiable everywhere in . If is not Gâteaux differentiable at , then (4) is replaced by where and is finite if and only if , the algebraic interior of defined by The topological interior of is , where is the boundary of . It is well known that the Bregman distance does not satisfy either the symmetry property or the triangle inequality which are required for standard distances while they are always nonnegative because of the convexity of the function . The Bregman distance between sets is defined as . If for , then . Through the paper, sequences with are simply denoted by for the sake of notation simplicity.

Fixed points and best proximity points of cyclic self-mappings in uniformly convex Banach spaces have been widely studied along the last decades for the cases when the involved sets intersect or not. See, for instance, [13] and references therein. In parallel, interesting results have been obtained for both nonspreading, nonexpansive, and hybrid maps in Hilbert spaces including also to focus the related problems via iterative methods supported by fixed point theory and the use of more general mappings such as nonspreading and pseudocontractive mappings. See, for instance, recent background [47] and references therein. Let be a nonempty subset of a Hilbert space . On the other hand, it has to be pointed out that the characterization of several classes of iterative computations by invoking results of fixed point theory has received much attention in the background literature. See, for instance, [811] and references therein. In [1218], the existence of fixed points of mappings is discussed when is: nonexpansive; that is, , for all , nonspreading; that is, , for all ,-hybrid [17]; that is, , for all . If , then is referred to as hybrid [14, 15], and if and (1.3) is changed to, , for all ,

where is a Gâteaux differentiable convex function, then is referred to as being point-dependent -hybrid relative to the Bregman distance , [16]. A well-known result is that a nonspreading mapping, and then a nonexpansive one, on a nonempty closed convex subset of a Hilbert space has a fixed point if and only it has a bounded sequence on such a subset [18]. The result has been later on extended to -hybrid mappings, [17] and to point-dependent -hybrid ones [16]. As pointed out in [16], what follows directly from the previous definitions, is nonexpansive if and only if it is 0-hybrid while it is nonspreading if and only if it is 2-hybrid; is hybrid if and only if it is 1-hybrid.

This paper is focused on the study of fixed points and best proximity points of a class of generalized point-dependent -hybrid (≥2)-cyclic self-mappings , relative to a Bregman distance in a smooth Banach space, where is a point-dependent real function in (1.4) quantifying the “hybrideness” of the (≥2) cyclic self-mapping and is added as a weighting factor in the first right-hand-side term of (1.4). Such a function is defined through a point-dependent product of the particular point -functions while quantifies either the “nonexpansiveness” or the “contractiveness” of the Bregman distance for points associated with the iterates of the cyclic self-mapping in each of the sets for , where are nonempty closed and convex. Thus, the generalization of the hybrid map studied in this paper has two main characteristics, namely, (a) a weighting point-dependent term is introduced in the contractive condition; (b) the hybrid self-mapping is a cyclic self-mappings. Precise definitions and meaning of those functions are given in Definition 2 of Section 2 which are then used to get the main results obtained in the paper. In most of the results obtained in this paper, the Bregman distance is defined associated with a Gâteaux differentiable proper strictly convex function whose domain includes the union of the subsets of the -hybrid (≥2)-cyclic self-mapping which are not assumed, in general, to intersect. Weak convergence results to weak cluster points of certain average sequences built with the iterates of the cyclic hybrid self-mappings are also obtained. In particular, such weak cluster points are proven to be also fixed points of the composite self-mappings on the sets , even if such sets do not intersect, while they are simultaneously best proximity points of the point-dependent -hybrid (≥2)-cyclic self-mapping relative to .

2. Some Fixed Point Theorems for Cyclic Hybrid Self-Mappings on the Union of Intersecting Subsets

The Bregman distance is not properly a distance, since it does not satisfy symmetry and the triangle inequality, but it is always nonnegative and leads to the following interesting result towards its use in applications of fixed point theory.

Lemma 1. If is a proper strictly convex function being Gâteaux differentiable in , then

Proof. By using (4) for and defining , for all by interchanging and in the definition of in (4), which leads to (9) since , for all , [16, 17], if is proper strictly convex, and the fact that , for all .
Equation (7) follows from (9) for leading to . To prove (8), take and proceed by contradiction using (4) by assuming that for such so that which contradicts . Then, , and, hence, (8) follows And, hence, (10) via (7) and (9).

The following definition is then used.

Definition 2. If , for all , and is a proper convex function which is Gâteaux differentiable in , then the -cyclic self-mapping , where and , for all , is said to be a generalized contractive point-dependent -hybrid -cyclic self-mapping relative to if for some given functions and with , for all , where defined by for any , for all .
If, furthermore, , for all , then is said to be a generalized point-dependent -hybrid (≥2)-cyclic self-mapping relative to .

If , it is possible to characterize as a trivial -cyclic self-mapping with which does not need to be specifically referred to as 1-cyclic.

Although depends on , the whole does not depend on so that the cyclic self-mapping is referred to as generalized point-dependent -hybrid in the definition.

The following concepts are useful. is said to be totally convex if the modulus of total convexity ; that is, is positive for . is said to be uniformly convex if the modulus of uniform convexity ; that is, is positive for . It holds that , for all [16]. The following result holds.

Theorem 3. Assume that(1) is a lower-semicontinuous proper strictly totally convex function which is Gâteaux differentiable in ; (2), for all , are bounded, closed, and convex subsets of which intersect and is a generalized point-dependent -hybrid (≥2)-cyclic self-mapping relative to for some given functions and , defined by for any , for all , and some functions , for all , with being bounded;(3)there is a convergent sequence to some for some .
Then, is the unique fixed point of to which all sequences converge for any , for all .

Proof. The recursive use of (14) yields with , with , , for all , where is the identity mapping on . Now, define so that one gets since , for all , since is a generalized contractive point-dependent -hybrid (≥2)-cyclic self-mapping relative to , implies that , for all , since for any , for all (then for any ), where where < 1, since , for all , so that since is bounded, is lower-semicontinuous then with all subgradients in any bounded subsets of being bounded, and and , for all , for all , converge so that they are Cauchy sequences being then bounded, for all , for all , where , since is nonempty and closed, is some fixed point of . As a result, , for all , for all , for all . From a basic property of Bregman distance, , as , for all , for all , for all , if is sequentially consistent. But, since is closed, is sequentially consistent if and only if it is totally convex [19]. Thus, converges also to for any and , for all , so that is a fixed point of . Assume not and proceed by contradiction so as then obtaining ; as from a basic property of Bregman distance. Thus, as since as . As a result, , as from the continuity of , and is a fixed point of . Now, take any so that , then, since is a fixed point of and is a proper strictly totally convex function. As a result, converges to , for all .
It is now proven that is the unique fixed point of . Assume not so that there is . Then, as from (17) for since is a generalized point-dependent -hybrid (≥2)-cyclic self-mapping relative to with and so that , since if since is proper and totally strictly convex, and since , and . Since is closed and convex, it turns out that is the unique fixed point of .
Note that the result also holds for any for all since maps to for some nonnegative integer through the self-mapping so that as since is the unique fixed point of and converges to for any .

The subsequent result directly extends Theorem 3 to the -composite self-mappings , , defined as ; for all , subject to , for all . The subsets ,   are not required to intersect since the restricted composite mappings as defined earlier are self-mappings on nonempty, closed, and convex sets.

Corollary 4. Assume that (1)   is a proper strictly totally convex function which is lower-semicontinuous and Gâteaux differentiable in , and, furthermore, it is bounded on any bounded subsets of ;  (2)   is bounded and closed, for all , is a -cyclic self-mapping so that   for some is a generalized point-dependent -hybrid (≥2)-cyclic self-mapping relative to for some given functions   and   for some defined by for any , for all , and for some , for all , where , being bounded and being, furthermore, convex for the given ;  (3)  there is a convergent sequence to some for some and .
Then, is a unique fixed point of to which all sequences converge for any for .
Also, if conditions (1)–(3) are satisfied with all the subsets , for all , being nonempty, closed, and convex for some proper strictly convex function which is Gâteaux differentiable in , then , for all , is a unique fixed point of , for all , to which all sequences converge for any , for all . The unique fixed points of each generalized point-dependent -hybrid -cyclic composite self-mappings , for all , fulfil the relations for , for all , for all .

Outline of Proof. Note that . Equation (14) is now extended to for the given leading to since is a trivial 1-cyclic self-mapping on for . The previous relation leads recursively to. with , , for the given with , where is independent of the particular for . One gets by using very close arguments to those used in the proof of Theorem 3 that . Then, converges to some which is proven to be a unique fixed point in the nonempty, closed, and convex set for . The remaining of the proof is similar to that of Theorem 3. The last part of the result follows by applying its first part to each of the generalized point-dependent -hybrid -cyclic composite self-mappings relative to , for all .

Remark 5. If is totally convex if it is a continuous strictly convex function which is Gâteaux differentiable in , and is closed, [20]. In view of this result, Theorem 3 and Corollary 4 are still valid if the condition of its strict total convexity of is replaced by its continuity and its strict convexity if the Banach space is finite dimensional. Since , for all , it turns out that if is uniformly convex, then it is totally convex. Therefore, Theorem 3 and Corollary 4 still hold if the condition of strict total convexity is replaced with the sufficient one of strict uniform convexity. Note that if a convex function is totally convex then it is sequentially consistent in the sense that as if as for any sequences and in .

Some results on weak cluster points of average sequences built with the iterated sequences generated from hybrid cyclic self-mappings relative to a Bregman distance , for and some , are investigated in the following results related to the fixed points of .

Theorem 6. Assume that(1) is a reflexive space and is a lower-semicontinuous strictly convex function, so that it is Gâteaux differentiable in , and it is bounded on any bounded subsets of ;(2)a -cyclic self-mapping is given defining a composite self-mapping with being bounded, convex, and closed, for all , so that its restricted composite mapping to , for some given , is generalized point-dependent -hybrid relative to for some   and the given .

Define the sequence for , where is the identity mapping on so that , for all , and assume that is bounded for . Then, the following properties hold.(i)Every weak cluster point of for is a fixed point of of for the given . Under the conditions of Theorem 3, there is a unique fixed point of which coincides with the unique cluster point of .(ii)Define sequences for any integer and where are bounded, closed, and convex, for all . Thus, converges weakly to for , where is a fixed point of and a weak cluster point of for and () is both a fixed point of and a weak cluster point of for . Furthermore, if is continuous.

Proof. Using (14) with being a generalized point-dependent -hybrid (≥2)-cyclic self-mapping relative to for , with , for all , yields Summing up from to and taking yields since is bounded for , its subsequence is then bounded for , and is also bounded on the bounded subset of . Then, converges to zero since is bounded for . Since is lower-semicontinuous, the set of subgradients is bounded in all bounded subsets , for all . As a result, is bounded for and has a subsequence being weakly convergent to some for since is reflexive. One gets by taking that Hence, it follows from Lemma 1 that is a fixed point of   for the given . Now, consider ,  , for any integers so that for . Hence, Property (i) follows with the uniqueness and the coincidence of the weak cluster point of and fixed point of under Theorem 3. The previous reasoning remains valid so that is weakly convergent to some since is bounded for (since is bounded for and is finite), its subsequence for is also bounded so that converges to zero since in is bounded for , . Now, take so that , and then , , is both a fixed point of and a weak cluster point of . Now, take so that . Then, as for and, in addition, if is continuous. Hence, Property (ii) follows.

3. Extensions for Generalized Point-Dependent Cyclic Hybrid Self-Mappings on Nonintersecting Subsets: Weak Convergence to Weak Cluster Points of a Class of Sequences

Some of the results of Section 2 are now generalized to the case when the subsets of the cyclic mapping do not intersect , in general, by taking advantage of the fact that best proximity points of such a self-mapping are fixed points of the restricted composite mapping for . Weak convergence of averaging sequences to weak cluster points and their links with the best proximity points in the various subsets of the -cyclic self-mappings is discussed. Firstly, the following result follows from a close proof to that of Theorem 6 which is omitted.

Theorem 7. Let be a reflexive space, and let be a lower-semicontinuous strictly convex function so that it is Gâteaux differentiable in and it is bounded on any bounded subsets of . Consider the generalized point-dependent -cyclic hybrid self-mapping being relative to for some such that are all bounded, convex, closed, and with nonempty intersection. Define the sequence for , where is the identity mapping on , and assume that is bounded for . Then, the following properties hold. (i) Every weak cluster point of for is a fixed point of .  (ii) Define the sequence for which is bounded, closed, and convex, for all and any integer . Thus, converges weakly to the fixed point of for which is also a weak cluster point of .

Remark 8. The results of Theorems 6 and 7 are extendable without difficulty to the weak cluster points of other related sequences to the considered ones.
Define sequences , , for any given finite non-negative integer under all the hypotheses of Theorem 7. With this notation, the sequence considered in such a corollary is . Direct calculation yields for as since , and then , is bounded. Then, weakly which is the same fixed point of in which is a weak cluster point of for for any finite non-negative integer .
Consider all the hypotheses of Theorem 7 and now define sequences , , for any given finite non-negative integer . With this notation, the sequence considered in the corollary is . Direct calculation yields weakly for as since , and then , is bounded. Then, weakly which is the same fixed point of in which is a weak cluster point of for and for any finite non-negative integer .
Now consider the hypotheses of Theorem 6. It turns out that the sequence for satisfies for any integer , weakly as , where since , is bounded, and for and . Thus, is a fixed point of which is also a weak cluster point of the sequences for . However, it is not guaranteed that without additional hypotheses on such as its continuity, or at least that of the composite mapping allowing to equalize the function of the limit with the limit of the function at such a fixed point.
Now, define for . Note that for , , weakly as since is finite, which is a fixed point in of the composite mapping and a weak cluster point of for finite .

Note that Theorem 6 are supported by boundedness constraints for the sequences of iterates obtained through the cyclic self-mapping which is generalized point-dependent with respect to some convex function. The results of identification of weak cluster points of some average sequences with fixed points of the cyclic self-mapping or its composite mappings do not guarantee uniqueness of fixed points and weak cluster points because the cyclic self-mapping is not restricted to be contractive. By incorporating some background contractive-type conditions for the cyclic self-mapping, the previous results can be extended to include uniqueness of fixed points as follows.

Theorem 9. Assume that.(1)Assumption 1 of Theorem 6 holds with the restriction of to be a uniformly convex Banach space;(2)Assumption 2 of Theorem 6 holds, and, furthermore, all the -cyclic composite mappings with restricted domain ; for all are either contractive or Meir-Keeler contractions.

Then, the following properties hold.(i)Theorem 6 holds. Furthermore, each of the mappings has a unique fixed point which are also best proximity points of in so that ; for all , for all .(ii)If, in addition, , then, there is a unique fixed point of and , for all .

Proof. Note that uniformly convex Banach spaces are also reflexive spaces required by Theorem 6. Each mapping has a unique fixed point , for all , irrespective of being empty or not if is either a cyclic contraction or a Meir-Keeler contraction [13], since are non-empty, closed, and convex, and is a uniformly convex Banach space so that each ; for all is a best proximity point in of . It follows from the hypothesis that there is a unique weak cluster point of for which is the unique fixed point of , for all , and also the unique best proximity point of in for .
It is now proven that if then , for all . Take some for some . Thus, and as since is the unique fixed point of and is the unique fixed point of . Then, , for all ; for all , weakly as , for all , and is the unique weak cluster point of , for all .

Theorem 9 can be also extended “mutatis-mutandis” to the convergence of weak cluster points of the alternative sequences discussed in Remark 8. It is now proven that the sets of fixed points of the restricted composite mapping , some , are convex if such mappings are quasi-nonexpansive with respect to in the sense that it has (at least) a fixed point in and , for all , and is a proper strictly convex function, [16]. The concept of quasi-nonexpansive mapping is addressed in the subsequent result to discuss the topology of fixed points and best proximity points of composite mappings of cyclic self-mappings.

Theorem 10. Let be a proper strictly convex function on the Banach space so that it is Gâteaux differentiable in , and consider the restricted composite mapping for some given built from the -cyclic self-mapping so that is nonempty, convex, and closed. Assume that , for all , and that the composite mapping is quasi-nonexpansive with respect to for the given .

Then, the following properties hold.(i)The set of fixed points of is a closed and convex subset of for the given .(ii)Assume, in addition, that , for all , are nonempty convex closed subsets of subject to , and assume also that are quasi-nonexpansive with respect to , for all . Then, the set of best proximity points in of the -cyclic self-mapping coincides with , and it is then a closed and convex subset of , for all . Furthermore, if , then which is then nonempty, closed, and convex.

Proof. Take and so that as . Note that and are nonempty sets since is quasi-nonexpansive with respect to then possessing at least a fixed point. By the continuity of and that of , the strict convexity of , and the assumption , one has and , from the strict convexity of and Lemma 1, which is in which is then a closed subset of as a result. Now, it is proven that is convex. Following the steps of a parallel result proven in [16] for noncyclic self-mappings, take and consider for some arbitrary real constant a point which is in since such a set is convex. Since is quasi-nonexpansive with respect to leading to and, since and for any , that which implies from Lemma 1 that for any since is strictly convex and . Thus, is a convex subset of . Hence, Property (ii) follows. The first part of property (ii) is a direct consequence of property (i) if the composite self-mappings on all the sets are quasi-nonexpansive with respect to since the respective sets of fixed points are the best proximity points of the -cyclic self-mapping in each of the sets , for all . If, furthermore, the sets , for all , have a nonempty intersection, then its set of fixed points coincides with the intersection of the sets of best proximity points of the composite mappings which are all identical for . The proof is trivial. Take any . Then, the sequence of iterates obtained through the composite converges to some . This implies that , closed and convex from property (i). Thus, . Then, the set of fixed points of is . Property (ii) has been proven.

Concerning that Theorem 10(ii), note that the set inclusion does not guarantee, in general, that the identity is not guaranteed for the case when except for cases under extra conditions such as the contractiveness of the composite mappings built from the -cyclic one leading, for instance, to the uniqueness of the fixed point of the cyclic self-mapping. See, for instance, Theorem 3 and Corollary 4.

It is direct to give sufficient conditions for the restricted composite mappings of the -cyclic self-mapping to be quasi-nonexpansive under the relevant conditions of Theorem 3, Corollary 4, Theorem 6, and Theorem 7 (see Proposition 3.5 of [16]) for noncyclic self-mappings, as follows.

Theorem 11. Assume that.(1) is a reflexive space and is a proper strictly convex function, so that it is Gâteaux differentiable in , and it is bounded on any bounded subsets of . (2)A -cyclic self-mapping is given defining a composite self-mapping with the subsets being bounded, closed, and convex, for all .(3)The restricted composite mapping to for some given , that is, , is generalized point-dependent -hybrid relative to for some function and the given which possesses a bounded sequence for some point .
Then, the restricted composite mapping is quasi-nonexpansive with respect to so that is a nonempty closed convex subset of .

Remark 12. The well-known concepts of nonexpansive, nonspreading, hybrid, and contractive cyclic self-mappings [1216, 19] are useful in the context of particular cases of interest of (14) within the given framework for generalized nonexpansive -cyclic self-mappings relative to . (1) If (14) holds and , for all , for each , for all , then is said to be a generalized nonexpansive -cyclic self-mapping relative to . (2) If (14) holds and , for all , for each , for all , then is said to be a generalized nonspreading -cyclic self-mapping relative to . (3) If (14) holds, for all ,  for  all  ,  for  all   with and , for all , then is said to be a generalized nonexpansive -hybrid -cyclic self-mapping relative to . (4) If (14) holds, for all , for all , for all with and for some , for all , then is said to be a generalized nonexpansive (point-dependent if some for some is not constant) -hybrid -cyclic self-mapping relative to . (5) If (14) holds, for all with and for some ; then is said to be a generalized nonexpansive (point-dependent if some for some is not constant) -cyclic self-mapping relative to ; (6) If (14) holds, for all , for all , for all with and for some , for all , then is said to be a generalized contractive (point-dependent if some for some is not constant) -hybrid -cyclic self-mapping relative to .(7)  If (14) holds and , for all for each , for all , then is said to be a generalized -contractive -cyclic self-mapping relative to .
The various given results can be easily focused on these particular cases.

4. Examples

Dynamic systems are a very important tool to describe and design control systems in applications. Fixed point theory has been found useful to study their controllability and stability properties. See, for instance, [2126] and references there in. Two examples are now given related to discrete dynamic systems in order to illustrate the theoretical aspects of this paper.

Example 1. Consider the scalar discrete dynamic system for given initial condition with , being a state disturbance which can include combined effects of parametrical disturbances and unmodeled dynamics (roughly speaking, the neglected dynamic effects of describing a higher-order difference equation by the previous first-order one). The solution sequence is defined by the self-mapping ( being the extended real line including the infinity points) given by , for all . It is assumed that a fixed point exists for some ; that is, a sequence for some initial point is bounded; That is, with . In particular, this holds for the unperturbed system with and which possess a unique globally asymptotically stable equilibrium which is also a unique fixed point of the solution. If , then there is a stable constant solution for each initial condition which is also a (nonunique) fixed point. In both cases, the mapping is trivially nonexpansive and, in the first case, it is also contractive. Note that the previous mapping is also a trivial cyclic self-mapping for . Under a cyclic repetition of the sequence with for some   being constant with and . In this case, we can describe the given difference equation equivalently as for the same initial condition. Then, the composite self-mapping generating the subsequence of the solution has a unique fixed point for any given . If, furthermore, there are finite limits , as , for all, then as which is a fixed point of . If the second set of limits exists being all finite, but arbitrary, that is, the identities do not all hold for , then the solution sequence converges to a cycle . We now retake the example under the point if view of a point-dependent -hybrid map where . The considered Banach space is which is a Hilbert space for the inner product being the Euclidean scalar product and the norm is the Euclidean norm. Then, Condition (14) for   to be point-dependent 1-cyclic -hybrid becomes in particular for some real functions and for and, equivalently, Note that if or if , equivalently, if or if , then the previous equivalent constraints (35)-(36) cannot be satisfied by a choice of some finite value of unless so that any arbitrary value of would satisfy the inequalities. Note that both inequalities hold directly for any fixed points.
If the previous constraint (36) holds, subject to (37), for some real sequence , then any weak cluster point of is a fixed point of according to Theorem 6. If there is a unique fixed point according to Theorem 3, then the unique fixed point of and weak cluster point of coincide. The same property holds for the weak cluster point of the average sequences referred to in Remark 8. Note in particular the following.
If and , then the previous constraint leads to which guarantees that Theorem 6 holds for any real sequence satisfying if and taking any arbitrary real value, otherwise for each . Since , a choice of independent of is as follows:
Thus, any cluster point of is a fixed point of  . There is a unique such fixed point if   and , which is also a globally asymptotically stable equilibrium point of the solution, and there are finite limits , as for some given and such a fixed point is also a fixed point of the composite self-mapping . If , then the constraint of point-dependent -hybrid self-mapping is satisfied for if , for all . If (i.e., both and have the same sign or one of them is zero) then is -hybrid for , for all . However, Theorem 6 is not applicable for cluster fixed points of the averaging sequence since there is no fixed point of the difference equation, in general.
If is not identically zero, then the constraint is satisfied if and if and taking any arbitrary real value, otherwise.
In the general case, the constraint is satisfied if with provided that the denominator of (41) is nonzero or if so (37) hold so that may take an arbitrary value.
Assume that , for all , as with and satisfying , and and that . Thus, the difference equation can be described equivalently by , and summing up both sides from to yields since the solution sequence is non-negative for nonnegative initial conditions and since : for some real constant . Thus, which implies that ,  , and . But if , then which is a contradiction. Then, is bounded and . Thus, is a fixed point of   and the previous particular cases (1)–(3) can be applied for weak cluster points of the average sequences.
Now, consider the (≥2)-dimensional dynamic system , for all ,  where , , for all , and the convex function with (i.e., a positive definite square -matrix with the superscript standing for transposes). The Bregman distance becomes resulting in the point-dependent -hybrid constraint: where is the th identity matrix. A finite, in general nonunique, real sequence exists satisfying the previous constraint if for any , which is a generalization of (37) to the -the dimensional case. Thus,
The previous discussion for the particular scalar difference equation may be generalized for this case with the replacement , for all , where stands for the maximum (real) eigenvalue of the symmetric matrix leading to the results for point-hybrid mappings to hold if there is a sequence satisfying the previous constraint (45a) and (45b).

Example 2. It is direct to extend Example 1 to a -cyclic self-mapping as follows. For instance, consider a scalar difference equation of the form for a given initial condition where is a control sequence. Recursive computation for two consecutive samples yields where , ; . Define the sets as so that . If the control sequence is chosen as , for all , for some constant , then , , for all , as , and the sequences , both converge to the unique fixed point of both   and . Now, suppose that the control sequence is changed to , for all for some positive real constant , then , and , as with being redefined as . In this case, and are the best proximity points of in and , respectively, while and are also fixed points of and , respectively.
Now, note that and if and and if . Thus, one gets with and if and and if . The Bregman constraint for the composite self-mapping to be -hybrid relative to holds in a similar way as (36), subject to (37), by replacing the subscripts and the sequences , , , , and “mutatis-mutandis” performed replacements for subscripts for the composite self-mapping for . If , then the modifications in the Bregman constraint (36), subject to (37), are referred to , .
Then, we have the following. (a) If and the control sequence is chosen as ; for all , then has a unique weak cluster point for any real   which is the unique fixed point and best proximity point of , and a fixed point of   provided that the previous modified Bregman constraint (36), subject to (37), holds. (b) Take a control , for all , and are redefined as for some positive real constant , then , , and , as if . Then, has a unique weak cluster point which is the unique fixed point of and the unique best proximity point in of   for any . Also, has a unique weak cluster point which is the unique fixed point of and the unique best proximity point in of   for any provided that the mentioned modified Bregman constraint (36), subject to (37), holds.

Acknowledgments

The author thanks the Spanish Ministry of Economy and Competitiveness for its support for this work through Grant DPI2012-30651. He is also grateful to the Basque Government for its support through Grants IT378-10 and SAIOTEK SPE12UN15 and to UPV by its Grant UFI 2011/07.