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Best Proximity Points of Generalized Semicyclic Impulsive Self-Mappings: Applications to Impulsive Differential and Difference Equations
This paper is devoted to the study of convergence properties of distances between points and the existence and uniqueness of best proximity and fixed points of the so-called semicyclic impulsive self-mappings on the union of a number of nonempty subsets in metric spaces. The convergences of distances between consecutive iterated points are studied in metric spaces, while those associated with convergence to best proximity points are set in uniformly convex Banach spaces which are simultaneously complete metric spaces. The concept of semicyclic self-mappings generalizes the well-known one of cyclic ones in the sense that the iterated sequences built through such mappings are allowed to have images located in the same subset as their pre-image. The self-mappings under study might be in the most general case impulsive in the sense that they are composite mappings consisting of two self-mappings, and one of them is eventually discontinuous. Thus, the developed formalism can be applied to the study of stability of a class of impulsive differential equations and that of their discrete counterparts. Some application examples to impulsive differential equations are also given.
Fixed point theory has an increasing interest in research in the last years especially because of its high richness in bringing together several fields of Mathematics including classical and functional analysis, topology, and geometry [1–8]. There are many fields for the potential application of this rich theory in Physics, Chemistry, and Engineering, for instance, because of its usefulness for the study of existence, uniqueness, and stability of the equilibrium points and for the study of the convergence of state-solution trajectories of differential/difference equations and continuous, discrete, hybrid, and fuzzy dynamic systems as well as the study of the convergence of iterates associated to the solutions. A basic key point in this context is that fixed points are equilibrium points of solutions of most of many of the above problems. Fixed point theory has also been investigated in the context of the so-called cyclic self-mappings [8–20] and multivalued mappings [21–32]. One of the relevant problems under study in fixed point theory is that associated with -cyclic mappings which are defined on the union of a number of nonempty subsets ; of metric or Banach spaces . There is an exhaustive background literature concerning nonexpansive, nonspreading, and contractive -cyclic self-mappings , for example, [8–20], including rational contractive-type conditions and [20, 33], and references therein, and for various kinds of multivalued mappings. See, for instance [21–32] and references therein. A key point in the study of contractive cyclic self-mappings is that if the subsets for are disjoint then the convergence of the sequence of iterates ; , , is only possible to best proximity points. The existence of such fixed points, its uniqueness and associated properties are studied rigorously in [11–13] in the framework of uniformly convex metric spaces, in [14–17], and in [12, 19] for Meir-Keeler type contractive cyclic self-mappings. In this paper, we introduce the notions of nonexpansive and contractive -semicyclic impulsive self-mappings and investigate the best proximity and fixed points of those maps. The properties of boundedness and convergence of distances are studied in metric spaces, while those of the iterated sequences ; , , are studied in uniformly convex Banach spaces. It is also seen through examples that the above combined constraint for distances is relevant for the description of the solutions of impulsive differential equations and discrete impulsive equations and for associate dynamic systems. The boundedness of the sequences of distances between consecutive iterates is guaranteed for nonexpansive -semicyclic self-mappings while its convergence is proved for asymptotically contractive -semicyclic self-mappings. In this case, the existence of a limit set for such sequences is proved. Such a limit set contains best proximity points if the asymptotically contractive -semicyclic self-mapping is asymptotically -cyclic, is a complete metric space which is also a uniformly convex Banach space , and the subsets ; are nonempty, closed, and convex. It has to be pointed out that the standard nonexpansive and contractive cyclic self-mappings may be viewed as a particular case of those proposed in this paper since it suffices to define the map so that any point of a subset is mapped in one of the adjacent subsets in the cyclic disposal and to define the second self-mapping of the composite impulsive one as identity.
2. Nonexpansive and Contractive -Semicyclic and -Cyclic Impulsive Self-Mappings
Consider a metric space and a composite self-mapping of the form , where , are nonempty closed subsets of with ; , (in particular, ) having a distance between any two adjacent subsets and of ; . In order to facilitate the reading of the subsequent formal results obtained in the paper, it is assumed that ; . Some useful types of such composite self-mappings for applications together with some of their properties in metric spaces are studied in this paper according to the following definition and its subsequent extensions.
Definition 1. The composite self-mapping is said to be a -semicyclic impulsive self-mapping if the following conditions hold:(1) is such that ; satisfies the constraint ; , , and for some real constant (2) is such that ; satisfies the constraint for some given bounded function .
Note that -semicyclic impulsive self-mappings satisfy the subsequent combined constraint as follows: then which follows after combining the two ones given in Definition 1.
The following specializations of the -semicyclic impulsive self-mapping of Definition 1 are of interest.(a)It is said to be nonexpansive (resp., contractive) -semicyclic impulsive if, in addition, (resp., if ) and .(b)It is said to be -cyclic impulsive if , . It is said to be a nonexpansive (resp., contractive) -cyclic impulsive if, in addition, (resp., if ) and .(c)It is said to be strictly -semicyclic impulsive self-mapping if it satisfies the more stringent constraint
A motivation for such a concept is direct since is nonexpansive (resp., contractive) if (resp., if ), , , and . This motivates, as a result, the concepts of nonexpansive and contractive strictly -semicyclic impulsive self-mappings and the parallel ones of nonexpansive and contractive strictly -cyclic impulsive self-mappings for the particular case that , .
Remark 2. Note that if , , , and , then , , , and , and this holds if (i.e., ) irrespective of the value of , , , and .
The subsequent result follows directly from Remark 2.
Proposition 3. Assume that any of the two conditions below holds:(1);(2) and , , , and .
Then, the self-mapping is(i)strictly -semicyclic if it is -semicyclic;(ii)strictly nonexpansive (resp., contractive) -semicyclic if it is nonexpansive (resp., contractive) -semicyclic;(iii)strictly -cyclic if it is -cyclic;(iv) strictly nonexpansive (resp., contractive) -cyclic if it is nonexpansive (resp., contractive) -cyclic.
It is of interest the study of weaker properties than the above ones in an asymptotic context to be then able to investigate the asymptotic properties of distances for sequences of iterates built through according to for all and some as well as the existence and uniqueness of fixed and best proximity points.
Lemma 4. Consider the -semicyclic impulsive self-mapping with , and define
for and in adjacent subsets and of for any . Then, the following properties hold.
(i) The sequence is bounded for all , and , if where
If, furthermore, is -cyclic then the lower-bound in (4) is replaced with . If is a nonexpansive -semicyclic impulsive self-mapping (in particular, -cyclic), then is bounded, , and , .
(ii) If, furthermore, , then
If, in addition, is -cyclic, then the lower-bound in (7) is replaced with .
If is contractive -semicyclic, then
If is contractive -cyclic, then there exists , .
Proof. Build a sequence of iterates according to with , , for any given and any that is, so that
Through a recursive calculation with (4), one get:
If , then
Take any , any , and any . Since is finite and (4) holds, it follows that . If, in addition, is -cyclic, then the zero lower-bound of (7) is replaced with . If is -semicyclic (in particular, -cyclic) nonexpansive, then (4) always holds since , so that if and is always bounded; , , and . Property (i) has been proven. If , then
If, in addition, is -cyclic, then the zero lower-bound of (13)-(14) is replaced with .
If is contractive -semicyclic, then (14) becomes from (12). If, in addition, is contractive -cyclic, then , so that there is , . Property (ii) has been proven.
The following result establishes an asymptotic property of the limits superiors of distances of consecutive points of the iterated sequences which implies that is asymptotically contractive, and the limit , , exists. In particular, it is not required that for any , , and as in contractive and, in general, nonexpansive -semicyclic impulsive self-mappings.
Theorem 5. Consider the following generalization of condition 3 of Definition 1:
for any given , , and define . Define
such that . Then, the following properties hold.
(i) where if is -semicyclic and if is -cyclic.
(ii) If, furthermore, there is a real constant such that then
Proof. Since , one has through iterative calculation via (15) with the convention , . Then, one gets (17), and Property (i) has been proven. To prove Property (ii), use the indicator sets (6) and, since , , one also gets from (15)-(16) and (19), and then Property (ii), follows from (18).
Note from (19) in Theorem 5 that if , that is, , and , then , , from (19) since . In this case, is an asymptotically contractive -cyclic (and also -semicyclic since ) self-mapping on the union on intersecting closed subsets of . A close property follows if , and implying from (19) that and leading to such that is a contractive -cyclic self-mapping on the union on disjoint closed subsets of . The above discussion is summarized in the subsequent result.
Corollary 6. Assume that (15) holds with defined in (16) being in , and assume also that
Then, the following properties hold
(i) If , then is an asymptotically contractive -cyclic impulsive self-mapping so that there is the limit
(ii) If , , , , and and the following limit exists: then is an asymptotically contractive -cyclic impulsive self-mapping so that the limit
A particular result got from Theorem 5 follows for contractive -semicyclic and -cyclic impulsive self-mappings .
Corollary 7. Theorem 5 holds with if is contractive -semicyclic and with if the impulsive self-mapping is contractive -cyclic provided that .
Proof. It is a direct consequence of Theorem 5 since implies that since , , , and .
Remark 8. Note that if is a nonexpansive -cyclic impulsive self-mapping, the following constraints hold:
implying that(a), , , and if ; that is, if the sets intersect .(b) if ; that is, for best proximity points associated with any two adjacent disjoint subsets , for .
On the other hand, note that Corollary 6 (ii) implies the asymptotic convergence of distances in-between consecutive points of the iterated sequences generated via to the distance between adjacent sets. This property does not imply , , and , as required for nonexpansive (and, in particular, for contractive) -cyclic impulsive self-mappings. However, it implies as from (25), since the sequence defining its left-hand-side sequence has to converge asymptotically to zero.
Define recursively global functions to evaluate the nonexpansive and contractive properties of the impulsive self-mapping which take into account the most general case that the constant in Definition 1 (1) can be generalized to be set dependent and point-dependent leading to a combined extended constraint as follows: so that with and initial, in general, point-dependent value for each iterated sequence constructed through the impulsive self-mapping . The following related result follows.
Theorem 9. Consider the -semicyclic impulsive self-mapping under the constraint (29) subject to (30)-(31). If , , then the following properties hold.
(i) If then so that is asymptotically contractive -semicyclic cyclic in the sense that, given , there is a sufficiently large such that, together with (32), , for .
(ii) If and the limit below exists: then and is asymptotically contractive -cyclic in the sense that, given , together with (34), there is a sufficiently large such that, together with (34), , for .(iii) The limit (33) exists and then (34) holds if : satisfies the identity
Proof. One gets from (20), (29)–(31) that
where . If , and (33) holds, then as , , , and . This leads directly to Property (i) for if (without the constraint (33) being needed) and to Property (ii) for if .
Consider that converges to zero as if for some real sequence which converges to zero, the function satisfies (35). This proves Property (iii).
Theorem 9 has a counterpart in terms of asymptotically strict -semicyclic and cyclic versions established as follows.
Corollary 10. Assume that the following strict-type contractive condition holds:
subject to the constraints (30) and (31). If , , then (34) holds, and is a strictly asymptotically contractive -cyclic impulsive self-mapping in the sense that, given any , there is a sufficiently large such that, together with (34), , for all if .
If , then is (at least) strictly asymptotically contractive -semicyclic in the sense that there is a sufficiently large such that, together with (32), , for for any given .
Proof (outline of proof). It follows directly by replacing (37) with so that there is the limit , , and .
3. Convergence of the Iterations to Best Proximity Points and Fixed Points
Important results about convergence of iterated sequences of 2-cyclic self-mappings to unique best proximity points were firstly stated and proven in  and then widely used in the literature. Some of them are quoted here to be then used in the context of this paper. Consider a metric space with nonempty subsets , such that . The following basic results have been proven in the existing background literature.
Result 1 (see ). Let be a metric space, and let and be subsets of . Then, if is compact and is approximatively compact with respect to (i.e., as for each sequence for some ), then and are nonempty.
It is known that if and are both compact, then is approximatively compact which respect to .
Result 2 (see ). Let be a reflexive Banach space, let be a nonempty, closed, bounded, and convex subset of and let be a nonempty, closed and convex subset of . Then, the sets of best proximity points and are nonempty.
Result 3 (see ). Let be a metric space, let and be nonempty closed subsets of , and let be a -cyclic contraction. If either is boundedly compact (i.e., if any bounded sequence has a subsequence converging to a point of ) or is boundedly compact, then there is such that .
Remark 11. It is known that if is boundedly compact, then it is approximatively compact. Also, a closed set of a normed space is boundedly compact if it is locally compact (the inverse is not true in separable Hilbert spaces ); equivalently, if and only if the closure of each bounded subset is compact and contained in . If is a linear metric space, a closed subset is boundedly compact if each bounded is relatively compact. It turns out that if is closed and bounded then it is relatively compact . It also turns out that if is a complete metric space and the metric is homogeneous and translation-invariant, then is a linear metric space and is also a Banach space with being the norm induced by the metric . Note that, since the metric is homogeneous and translation-invariant and since is a linear metric space, such a metric induces a norm. In such a Banach space, if is bounded and closed, then is boundedly compact and thus approximatively compact.
Result 4 (see ). Let be a uniformly convex Banach space, let be a nonempty closed and convex subset of , and let be a nonempty closed subset of . Let sequences , and satisfy and as . Then as .
It is known that a uniformly convex Banach space is reflexive and that a Banach space is a complete metric space with respect to the norm-induced distance.
Result 5 (see ). If is a complete metric space, is a -cyclic contraction, where and are nonempty closed subsets of , and the sequence