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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 817193, 29 pages
Fixed and Best Proximity Points of Cyclic Jointly Accretive and Contractive Self-Mappings
Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus of Leioa (Bizkaia) Apertado 644 Bilbao, 48080 Bilbao, Spain
Received 21 November 2011; Accepted 15 December 2011
Academic Editor: Yonghong Yao
Copyright © 2012 M. De la Sen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
-cyclic and contractive self-mappings on a set of subsets of a metric space which are simultaneously accretive on the whole metric space are investigated. The joint fulfilment of the -cyclic contractiveness and accretive properties is formulated as well as potential relationships with cyclic self-mappings in order to be Kannan self-mappings. The existence and uniqueness of best proximity points and fixed points is also investigated as well as some related properties of composed self-mappings from the union of any two adjacent subsets, belonging to the initial set of subsets, to themselves.
In the last years, important attention is being devoted to extend the fixed point theory by weakening the conditions on both the mappings and the sets where those mappings operate [1, 2]. For instance, every nonexpansive self-mappings on weakly compact subsets of a metric space have fixed points if the weak fixed point property holds . Another increasing research interest field relies on the generalization of fixed point theory to more general spaces than the usual metric spaces, for instance, ordered or partially ordered spaces (see, e.g., [3–5]). 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 [6–8]. 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 [9–12]. Another interest of such mappings is their usefulness as formal tool for the study -cyclic contractions even if the involved subsets of the metric space under study of 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. On the other hand, there are close links between contractive self-mappings and Kannan self-mappings [2, 13–16]. In fact, Kannan self-mappings are contractive for values of the contraction constant being less than 1/3, [15, 16] and can be simultaneously -cyclic Meir-Keeler contractive self-mappings. The objective of this paper is the investigation of relevant properties of contractive -cyclic self-mappings of the union of set of subsets of a Banach space which are simultaneously-accretive on the whole , while investigating the existence and uniqueness of potential fixed points on the subsets of if they intersect and best proximity points. For such a purpose, the concept of -accretive self-mapping is established in terms of distances as a, in general, partial requirement of that of an accretive self-mapping. Roughly speaking, the self-mapping from to under study can be locally increasing on but it is still -cyclic contractive on the relevant subsets of . For the obtained results of boundedness of distances between the sequences of 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 points, the sets 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. Consider a metric space associated to the Banach space and a self-mapping such that and , where and are nonempty subsets of . Then, is a 2-cyclic self-mapping. It is said to be a 2-cyclic -contraction self-mapping if it satisfies, furthermore,
for some real . A best proximity point of convex subsets or of is some such that . If and are closed then either (resp., ) or (resp. ) is in (resp., in ). The distance between subsets and of the metric space if either or if either or is open with . In this case, if is a best proximity point either or is not in (in particular, neither nor is in if both of them are open). It turns out that if then ; that is, is a fixed point of since , [9–11]. If then ; , for all and is a 2-cyclic nonexpansive self-mapping, .
superscript denotes vector or matrix transpose, is the set of fixed points of a self-mapping on some nonempty convex subset of a metric space cl and denote, respectively, the closure and the complement in of a subset of , and denote, respectively, the domain and image of the self-mapping and is the family of subsets of , dist denotes the distance between the sets and for a 2-cyclic self-mapping which is simplified as dist; for distances between adjacent subsets of -cyclic self-mappings on .
which 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. Some Definitions and Basic Results about 2-Cyclic Contractive and Accretive Mappings
Let be a normed vector space and be an associate metric space endowed with a metric (or distance function or simply “distance”) . For instance, the distance function may be induced by the norm on. If the metric is homogeneous and translation-invariant, then it is possible conversely to define the norm from the metric. Consider a self-mapping which is a 2-cyclic self-mapping restricted as where and are nonempty subsets of . Such a restricted self-mapping is sometimes simply denoted as . Self-mappings which can be extended by continuity to the boundary of its initial domain as well as compact self-mappings, for instance, satisfy such an extendibility assumption. In the cases that the sets and are not closed, it is assumed that and in order to obtain a direct extension of existence of fixed points and best proximity points. This allows, together with the convexity of and , to discuss the existence and uniqueness of fixed points or best proximity points reached asymptotically through the sequences of iterates of the self-mapping . In some results concerning the accretive property, it is needed to extend the self-mapping in order to define successive iterate points through the self-mapping which do not necessarily belong to . The following definitions are then used to state the main results.
Definition 2.1. is an accretive mapping if
for any .
Note that, since is also a vector space, is in for all in and all real . This fact facilitates also the motivation of the subsequent definitions as well as the presentation and the various proofs of the mathematical results in this paper. A strong convergence theorem for resolvent accretive operators in Banach spaces has been proved in .Two more restrictive (and also of more general applicability) definitions than Definition 2.1 to be then used are now introduced as follows:
Definition 2.2. is a -accretive mapping, some if for some . A generalization is as followsis -accretive for some if
Definition 2.3. is a weighted -accretive mapping, for some function , if The above concepts of accretive mapping generalize that of a nondecreasing function. Contractive and nonexpansive 2-cyclic self-mappings are defined as follows on unions of subsets of .
Definition 2.4. is a 2-cyclic -contractive (resp., nonexpansive) self-mapping if
for some real (resp. ), [12, 13].
The concepts of Kannan-self mapping and 2-cyclic -Kannan self-mapping which can be also a contractive mapping, and conversely if , , are defined below.
Definition 2.6. is an 2-cyclic -Kannan self-mapping for some real if it satisfies, for some . The relevant concepts concerning 2-cyclic self-mappings are extended to -cyclic self-mappings in Section 3. Some simple explanation examples follow.
Example 2.7. Consider the scalar linear mapping from to as with endowed with the Euclidean distance ; for all . Then, for all for any provided that . In this case, is accretive. It is also -contractive if since ; for all . Also, if , then ; for all if , that is, if . Then, is -accretive and -contractive if .
Example 2.8. Consider the metric space with the distance being homogeneous and translation-invariant and a self-mapping defined by with , and if and . If then is accretive since Furthermore, if , then is the unique fixed point with ; for all . If then, as if and then is again the unique fixed point of . In the general case, implies holds if that is, is weighted -accretive with . The restricted self-mapping is -accretive. Furthermore, if then is -contractive if and the iteration as with being the unique fixed point since Note from the definition of the self-mapping on that it is also a 2-cyclic self-mapping from to itself with the property and .
Assertions 1. (1) If is an accretive mapping, then it is -accretive, for all . (2) If is -accretive, then it is -accretive; for all . (3) Any nonexpansive self-mapping is -accretive and conversely.
Theorem 2.9. Let be a Banach vector space withbeing the associated complete metric space endowed with a norm-induced translation-invariant and homogeneous metric. Consider a self-mapping which restricted tois a 2-cyclic -contractive self-mapping where and are nonempty subsets of . Then, the following properties hold.(i) Assume that the self-mapping satisfies the constraint:
with satisfying the constraint . Then, the restricted self-mapping satisfies
irrespective of and being bounded or not. If, furthermore, and are closed and convex and, then there exists a unique fixed point of such that there exists ; for all for all , implying that . If, in addition, so that , then there exists ; for all , for all for some best proximity points , which depend in general on and . Furthermore, if is a uniformly convex Banach space, then and as ; for all, where and are unique best proximity points in and of .
(ii)Assume that and are nondisjoint. Then, is also contractive and -accretive for any nonnegative and any . It is also nonexpansive and -accretive for any nonnegative .(iii)If then is weighted -accretive for for any and its restriction is 2-cyclic 0-contractive.(iv) is weighted -accretive for satisfying for some . The restricted self-mapping is also -contractive with if with . Also, is nonexpansive and weighted -accretive for satisfying if which implies, furthermore, that is bounded.
Proof. Let us denote Consider that the two following relations are verified simultaneously:
Since the distance is translation-invariant and homogeneous, then the substitution of (2.14) yields if and are disjoint sets, after using the subadditive property of distances, the following chained relationships since :
with . Note from (2.15) that
and, if , then
If then . It is first proven that the existence of the limit of the distance implies that of the limit ; for all . Let be , with. Then,
since being contractive is globally Lipschitz continuous. Then, since, because the fact that the metric is translation-invariant, one gets
As a result, if what implies which ; for all for all , since is globally Lipschitz continuous since it is contractive.
In addition, there exists ; for all , for all . Assume not so that there exists such that and there exists a subsequence on nonnegative integers such that . If so, one gets by taking that which contradicts . Then is a Cauchy sequence for any and then converges to a limit. Furthermore, since for any and as since and are nonempty and closed. It has been proven that ; for all for all .
It is now proven that. Assume not, then, from triangle inequality, which contradicts so that . It is now proven that . Assume not, such that, for instance, and . If so, since , then the existing limit fulfils which is impossible so that there would be no existing limit in , contradicting the former result of its existence. Then, implying that Fix.
It is now proven by contradiction that ; for all is the unique fixed point of . Assume that , then for some with no loss in generality and all . Thus, which contradicts so that Fix.
Now, assume that and do not intersect so that dist. Then, one gets from the first inequality in (2.15) that for all , , one gets
Note that since , and , then and if is even and and if is oddand if is even and and if is odd.
Then, and are not both in either or if and are not both in either or for any . As a result, is impossible so that for some best proximity points and or conversely. Then, where Thus, . It turns out that and dist as . Otherwise, it would exist an infinite subsequence of with being an infinite subset of such that for . On the other hand, since is a normed space, then by taking the norm-translation invariant and homogeneous induced metric and since there exists , it follows that there exist and such that for any given ; for all for all with for any even and , for any odd . As a result, by choosing the positive real constant arbitrarily small, one gets that (a best proximity point of ) and (a best proximity point of ), or vice-versa, as for any given and . A best proximity point fulfils . Best proximity points are unique in and as it is now proven by contradiction. Assume not, for instance, and with no loss in generality, assume that there exist two distinct best proximity points and in . Then and contradict so that necessarily . Since is a uniformly convex Banach space, we take the norm-induced metric to consider such a space as the complete metric space to obtain the following contradiction: since is also a strictly convex Banach space and and are nonempty closed and convex sets. Then, is the unique best proximity point of in and is its unique best proximity point in . Then, Property (i) has been fully proven. Since and are not disjoint, then , and is -contractive and -accretive if with . By taking , note that is nonexpansive and -accretive. Property (ii) has been proven.
To prove Property (iii), we now discuss if is possible with and . Note that for some . Define , if for some , where and . Three cases can occur in (2.26), namely,(a)If then which is untrue if and and it holds for either or ,(b), then (2.26) is equivalent to Take to be a best proximity point with so that which is untrue if and true for ,(c), then (2.16) is equivalent to ; for all for all , but . Thus, the above constraint is guaranteed to hold in the worst case if which is a contradiction.Property (iii) follows from the above three cases (a)–(c).
To prove Property (iv), consider again (2.26) by replacing the real constants and with the real functions and . Note that (2.26) holds through direct calculation if ; for all for all for some . Thus, the self-mapping is weighted -accretive for satisfying for some ; and it is also -contractive with if with and nonexpansive if . On the other hand, note that . If and are bounded and , then Property (iv) has been proven.
Remark 2.10. Note that Theorem 2.9 (iii) allows to overcome the weakness of Theorem 2.9 (ii) when and are disjoint by introducing the concept of weighted accretive mapping since for best proximity points , .
Remark 2.11. Note that the assumption that is a uniformly convex Banach space could be replaced by a condition of strictly convex Banach space since uniformly convex Banach spaces are reflexive and strictly convex, . In both cases, the existence and uniqueness of best proximity points of the 2-cyclic in and are obtained provided that both sets are nonempty, convex, and closed.
Remark 2.12. Note that if either or is not closed, then its best proximity point of is in its closure since , leads to and for finitely many and for infinitely many iterations through the self-mapping and Theorem 2.9 is still valid under this extension.
Note that the relevance of iterative processes either in contractive, nonexpansive and pseudocontractive mappings is crucial towards proving convergence of distances and also in the iterative calculations of fixed points of a mapping or common fixed points of several mappings. See, for instance, [19–25] and references therein. Some results on recursive multiestimation schemes have been obtained in . On the other hand, some recent results on Krasnoselskii-type theorems and related to the statement of general rational cyclic contractive conditions for cyclic self-maps in metric spaces have been obtained in  and , respectively. Finally, the relevance of certain convergence properties of iterative schemes for accretive mappings in Banach spaces has been discussed in  and references therein. The following result is concerned with norm constraints related to 2-cyclic accretive self-mappings which can eventually be also contractive or nonexpansive.
Theorem 2.13. The following properties hold.(i)Let be a metric space endowed with a norm-induced translation-invariant and homogeneous metric . Consider the -accretive mapping for some which restricted as is 2-cyclic, where and are nonempty subsets of subject to . Then, If, furthermore, is -contractive, then is guaranteed to be nonexpansive (resp., asymptotically nonexpansive) if respectively, (ii)Let be a normed vector space. Consider a -accretive mapping for some which restricted to is 2-cyclic, where and are nonempty subsets of subject to then If, furthermore, is -contractive, then is nonexpansive (resp., asymptotically nonexpansive, ) if respectively,
Proof. To prove Property (i), define an induced by the metric norm as follows since the metric is homogeneous and translation-invariant. Define the norm of , that is, the norm of on restricted to as follows: with the above set being closed, nonempty, and bounded from below. Since is 2-cyclic and is -accretive (Definition 2.2), one gets by proceeding recursively since the metric is homogeneous and , and is the identity operator on , where with the above set being closed, nonempty, and bounded from below. If for some , then we get the contradiction ; for all , for all in (2.38). Thus, ; for all , , for all . If now and are replaced with and for any in (2.30), one gets if is a 2-cyclic -contractive for some real and -accretive mapping: for all , for all , for all , for all , for all . Then, ; for all , for all , for all . If ; for all , for all , for all , it turns out that is -accretive and is a 2-cyclic nonexpansive self-mapping. It is asymptotically nonexpansive if ; for all , . Property (i) has been proven. The proof of Property (ii) for being a normed vector space is identical to that of Property (i) without associating the norms to a metric.
Example 2.14. Assume that , , and the 2-cyclic self-mapping defined by the iteration rule with , ; for all , and . Let be the Euclidean metric.(a)If, then so that for any , ; for all as with , but it is not in which is empty. If and (i.e., there are infinitely many values being less than unity), then the conclusion is identical. If and are redefined as , , then .(b)If ; for all the self-mapping is not expansive and there is no fixed point. (c)If for some , then for,
so that is also -accretive and -contractive with . (d)Now, define closed sets and for any given so that . The 2-cyclic self-mapping is re-defined by the iterationif and , for , otherwise, wherefor with the real sequence being subject to , ; , for all and . Then, for any and any , there are two best proximity points and fulfilling and .(e)Redefine so that with , ; ,. In the case that then and are open disjoint subsets (resp., , are closed nondisjoint subsets with ).The 2-cyclic self-mapping is re-defined by the iteration rule:
with the real sequence being subject to, ; for ; for all and .
The same parallel conclusions to the above ones (a)–(c) follow related to the existence of the unique fixed point in the closure of and but not in its empty intersection if either or is open, respectively, in the intersection of and (the vertical real line of zero abscissa) if they are closed. The same conclusion of (d) is valid for the best proximity points if .
The following result which leads to elementary tests is immediate from Theorem 2.13.
Corollary 2.15. The following properties hold.(i) Let be a normed vector space with being the associate metric space endowed with a norm-induced translation-invariant and homogeneous metric and consider the self-mapping so that the restricted is -accretive for some , where and are nonempty subsets of subject to , and the restricted is 2-cyclic.Then, If, furthermore, is -contractive, then is guaranteed to be nonexpansive (resp., asymptotically nonexpansive) if(ii)Let be a normed vector space. Then if is a -accretive mapping and is 2-cyclic for some where and are nonempty subsets of subject to , then If, furthermore, is 2-cyclic k -contractive, then
Outline of Proof
It follows since the basic constraint of being -accretive holds if while it fails if
Remark 2.16. Theorem 2.13 and Corollary 2.15 are easily linked to Theorem 2.9 as follows. Assume that is 2-cyclic -contractive and is a-accretive mapping. Assume that there exists such that . Then, ; for all from (2.47). This is guaranteed under sufficiency-type conditions with with for some real constants, . It is direct to see that if .
Example 2.17. Constraint (2.50) linking Theorem 2.13 and Corollary 2.15 to Theorem 2.9 is tested in a simple case as follows. Let . is a vector space endowed with the Euclidean norm induced by the homogeneous and translation-invariant Euclidean metric. is a linear self-mapping from to represented by a nonsingular constant matrix in . Then, is the spectral (or -) norm of the -contractive self-mapping which is the matrix norm induced by the corresponding vector norm (the vector Euclidean norm being identical to the vector norm as it is wellknown) fulfilling with the symmetric matrix being a matrix having all its eigenvalues positive and less than one, since is nonsingular, upper-bounded by a real constant which is less than one. Thus, is also -accretive for any real constant and -contractive for any real . Assume now that for some integer with ; for all . If , then . Also, for any integer (then is singular) but the last -components of any are zeroed at the first iteration via so that if is the th unit vector in with its th component being one, then Now, assume that the matrix is of rank one with its first column being of the form