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Journal of Applied Mathematics
VolumeΒ 2011, Article IDΒ 542941, 17 pages
http://dx.doi.org/10.1155/2011/542941
Research Article

On a General Contractive Condition for Cyclic Self-Mappings

Institute of Research and Development of Processes, Faculty of Science and Technology, University of the Basque Country, Campus de Leioa (Bizkaia), Apartado 644 de Bilbao, 48080 Bilbao, Spain

Received 26 April 2011; Accepted 3 October 2011

Academic Editor: J. C.Β Butcher

Copyright Β© 2011 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.

Abstract

This paper is concerned with -cyclic self-mappings in a metric space (, ), with , for , under a general contractive condition which includes as particular cases several of the existing ones in the literature. The existence and uniqueness of fixed points and best proximity points is discussed as well as the convergence to them of the iterates generated by the self-mapping from given initial points.

1. Introduction

There are exhaustive results about fixed point theory concerning the use of general contractive conditions in Banach spaces or in complete metric spaces and in partially ordered metric spaces which include as particular cases previous ones in the background literature. See, for instance, [1, 2] and references therein. On the other hand, important attention is being paid to the study of fixed points and best proximity points of -cyclic contractive mappings and -cyclic Meir-Keeler contractive mappings, [3–7]. Generally speaking, cyclic contract self-mappings on the union of nonempty closed convex subsets of a complete metric space , subject to , have a unique fixed point located in the intersection of such subsets if such subsets intersect, [3, 4]. If the -subsets are disjoint convex closed nonempty subsets of a uniformly convex Banach space, then there is a unique best proximity point at each of the subsets. The above properties also hold for cyclic Meir-Keeler contractions, [5–7]. In this paper, a contractive condition for -cyclic self-mapping on the union of subsets of a metric space which includes as particular cases a number of the existing ones in the background literature is proposed, and their basic associate properties are discussed. It is discussed the existence and uniqueness of fixed points if the metric space is complete and the set of subsets involved in the cyclic contractive condition have nonempty intersections and the existence and uniqueness of best proximity points within each of the subsets if they are convex, closed and disjoint and is a uniformly convex Banach space. The asymptotic convergence of the iterates from given initial point to best proximity points at each subset or to the unique fixed point if the subsets intersect is also discussed.

2. Main Results for a General Contractive Condition

This section contains the main results of the paper for -cyclic self-mapping on the union of a set of nonempty subsets of a metric space under a very general contractive condition which contains as particular cases several previous ones being known in the background of the literature for the noncyclic case .

Theorem 2.1. Let be a metric space with nonempty closed subsets of such that , and let be a continuous -cyclic self-mapping subject to and satisfying the following contractive condition: for all and if fo rall.

Then, the following properties hold:

(i) for any and any given provided that where If, in addition, the distances between any pairs of adjacent subsets are identical; , then is a -cyclic self-mapping. If, furthermore, then all the iterates fulfil the following constraints: where and for any given , with .

If (2.4) is replaced by , that is, then the inequalities (2.6)–(2.8) trivially hold.

(ii) Assume that the contractive condition (2.1) satisfies (2.5). Then, under the necessary condition (2.3).

If, in particular, (2.9) holds, that is, , then for any and some , then If (2.9) holds, and then the limit below exists: which is guaranteed by the condition (2.3).

(iii) If the constants of (2.4) fulfil , then the subsets of , have nonempty intersection (i.e., ), and, if furthermore, the metric space is complete, then and has a unique fixed point in to which all the sequences , which are then bounded, converge, .

If are disjoint, closed, and convex, is uniformly convex and and (2.12) holds with , then all sequences converge to a best proximity point of .

Proof. Let be an arbitrary point in and take and , where , and if and , otherwise. Then, from (2.1), and one gets so that, since Thus, it follows proceeding recursively with (2.14) subject to (2.9), (2.16), and where the first inequality holds irrespective of the identities and it implies directly (2.2) since and and , so that . The second inequality follows in the case that , if (2.9) holds so that if (2.3) leading directly to (2.5)–(2.7). Property (i) has been proven.
Property (ii) is proven by taking for any and proceeding recursively with the first inequality of (2.17) to obtain directly (2.10) and (2.11) since and (2.13) if, in addition, (2.12) holds. To prove Property (iii), note from (2.8), (2.9), (2.12), and (2.13) that ,if and . If, furthermore, is a complete metric space, then each sequence is a Cauchy sequence with a limit in since this set intersection is nonempty and closed since all the intersected sets are nonempty and closed. Since the sequences are convergent to a limit , they are bounded. Also, since is continuous in , then so that . It is proven by contradiction that there exists a unique fixed point. Assume that there exist and subject to in , the set of fixed points of . Then, the subsequent contradiction follows from (2.1) for , by using , and (2.4): so that . Property (iii) has been proven.
Note that it cannot be concluded from Theorem 2.1 that under the contractive condition (2.1) is either a -cyclic nonexpansive self-mapping from Theorem 2.1(i), even if all the contractive constants in (2.7) are identical and all the distances between adjacent subsets of are also identical, or a -cyclic contraction under Theorem 2.1(ii) since (2.6)–(2.8), or its respective versions with strict inequalities, are only guaranteed for the iterates and for any . Assume that the norm of the uniformly convex (Banach) space induces the metric being used. Otherwise, any alternative equivalent metric may be used to conclude the result. If (2.9) and (2.12) hold with distances between adjacent subsets , then all sequences are Cauchy sequences, converge to a best proximity point of for any given from (2.13) and the continuity of ensured by that of .

Remark 2.2. If , with some of the being eventually larger than one, then is bounded provided that is finite. If, furthermore, and , then all the composed mappings defined by satisfies ,, so that satisfies and (2.13) holds with some of the being eventually not less than one. The last part of the proof of Theorem 2.1(iii) leads to the conclusion that if is not continuous while the composed self-mapping is continuous, then the convergence of the iterates to a best proximity point in each adjacent subset is still ensured.

The existence of a unique fixed point is guaranteed if the subsets are closed and intersect even if is not continuous and all the constants (2.4) are not less than unity provided that is complete and as follows.

Theorem 2.3. Let be a complete metric space with nonempty closed subsets of , with nonempty intersection satisfying , and let be a -cyclic self-mapping subject to ; and satisfying the contractive condition (2.1) subject to for constants defined in (2.8). Then, all the sequences are bounded and converge to a unique fixed point in of .

Proof. Since (since ), , is nonempty and closed and is complete, it follows that for any , and then for any , from Theorem 2.1(iii) irrespective of being continuous or not. Thus, for any such that from triangle inequality for distances, one has so that each is a Cauchy sequence with a limit in the closed nonempty set which is also bounded since it is convergent. Hence, it follows the existence of fixed points in being each of those limits. The uniqueness of the fixed point follows from its counterpart in the proof of Theorem 2.1(iii) where the continuity of has not been used. Hence, the theorem.

The following result allows to fulfil (2.10) in Theorem 2.1 under some negative scalars and constants exceeding unity provided that a set of necessary conditions involving distances between the adjacent subsets and such scalars are satisfied.

Corollary 2.4. Assume that (2.1) holds and and . Then, (2.3), and thus (2.2), equivalently (2.17), holds if provided that the distances between adjacent subsets and the real scalars satisfy the joint constraints:

Proof. A necessary condition for (2.2)-(2.3) to hold, with a nonnegative second right-hand side term in (2.2) is that the constraints (2.21) hold.

It is wellknown that -cyclic nonexpansive self-maps require that the adjacent subsets have all the same pairwise distances. In the case that the relevant self-mappings are contractive are Meir-Keeler contractions, they have a unique fixed point if all the subsets intersect and the metric space is complete. In the case that the subsets do not intersect, there is a unique convergence best proximity point at each subset, to which the iterates through the self-mapping converge asymptotically, provided that the subsets are nonempty convex and closed and the vector space defining the metric space is uniformly convex, then also being reflexive and strictly convex, [8]. It is still required that all the distances between adjacent subsets be identical so that, otherwise, the self-mappings from the union of the subsets to itself cannot be nonexpansive [9, 10]; hence, they cannot be contractive. In the following, the condition of all the distances between the adjacent subsets being identical is not longer being required. The price to be paid is that the convergence of the iterates through the self-mapping do not necessarily converge to best proximity points located in the boundaries of the sets but to best proximity points located at the boundaries of appropriate nonempty closed convex subsets of the original subsets of . In order to facilitate the formalism for the case of distinct distances between adjacent subsets, the maps of interest are restricting their images as the iterations progress in order to asymptotically reach a new set of adjacent subsets all possessing identical pairwise distances. For such a subsequent analysis, first proceed as follows by first introducing the following hypotheses.

Hypotheses
(H1) Assume that a sequence of nonempty closed sets exists such that and and that satisfying .
(H2) Assume that any sequence of -cyclic self-mappings is subject to while all its elements satisfy the contractive condition (2.1), satisfying (2.8)-(2.9), with with and.
(H3) Define composed self-mapping as follows (with a certain abuse of notation): where is a (nonnecessarily monotone) increasing sequence of natural numbers fulfilling for the given , for some if , with the sequence of natural numberssatisfying . Note that in fact since its image is always restricted to be contained in by construction.

The reason of the abuse of notation when defining the composed mapping is useful since it is explicitly indicated that we are dealing with the composition of times groups of compositions of the sequence of self-mappings . The following result holds.

Theorem 2.5. Any composed mapping from to , defined by (2.22) under the Hypothesis (H1)–(H3), has the following properties.
(i) If the sequence , defined for some , is unbounded then,
(ii) Assume that the sequence , with , converges uniformly on some nonempty subset (under proper set inclusion) as . Then, any sequence converges uniformly to a limit self-mapping, dependent on the sequence as provided that . Also,
(iii) The limit of any composed sequence of mappings generated by some unbounded sequence has a best proximity point between the adjacent sets and at each which is also in to which all the sequences , which are then bounded, converge; .
If , then all sequences converges to a best proximity point of .
If, in addition, the metric space is complete and then all sequences converge to a unique fixed point of .
(iv) Assume that is a uniformly convex space and that the subsets are convex and closed. Then, the best proximity points of the adjacent subsets and are unique for each . Furthermore, and with .

Proof. Note that the necessary condition (2.3) for Property (i) holds by construction and take, with no loss of generality, for and for for any given finite positive natural numbers , dependent on and , so that , if in (2.22), and also , if , as in (2.22) for being some unbounded sequence of natural numbers as . Since with and and subject to , it follows from Cantor’s intersection theorem that any arbitrary intersection is nonempty and closed; and since and . Then, the composed mapping from to , whose nonempty image is restricted by construction to the subset of , is well-posed for any arbitrary sequence and for any . Thus, one gets from Theorem 2.1, (2.10), that for any given such that: where the sequence of natural numbers being subject to for and for . Thus, it follows that (2.23) holds from (2.25) since that implies for , and for so that diverges to infinity as taking values in a subset of of infinite cardinal. Hence, Property (i).
To prove Property (ii), note first that the composed self-mapping map has an image restricted to by construction. If , satisfying , converges uniformly as on some subset of satisfying , then can converge uniformly in as . Proceed by contradiction. If do not converge uniformly, while converges does, then, for any given , there exist some , some and some and some sequences of nonnegative ordered integers and of minimal element such that: after using the triangle property for distances, for some and satisfying The chained inequalities (2.25) contradict a choice of satisfying . Since is arbitrary, converges uniformly in as , or trivially, if is finite, for all . On the other hand, the triangle inequality yields  , since a uniform convergence of the sequence as , has been proven, so that (2.24) follows. Hence, Property (ii).
To prove Property (iii), remember that form Property (ii), and the composed fulfil: From (2.24), such that for any integers , Proceed by contradiction to prove the convergence of the iterates to best proximity points of . Assume that there is no such that ,and so that there exist some integers and such that Since is arbitrary, one can choose which yields a contradiction in (2.31). Then, such that . Furthermore, since . If all the distances are nonzero and identical then the best proximity points are at the boundaries of the closed subsets . If, in addition, the metric space is complete and the subsets intersect (i.e., all the distances between adjacent subsets are zero), then and the best proximity points are also coincident in a unique fixed point (Theorem 2.1(iii)) in the nonempty intersection of all such subsets. Property (iii) has been proven.
Property (iv) is proven by contradiction. Assume that there are two distinct best proximity points for any given . Define the following real sequences , , and by its general terms as follows: generated by a uniformly convergent mapping for some , some as the integer sequences. The limits above exist from Property (iii). Also, one gets from Property (i), (2.23), and Lemma 3.8 of [4], since is uniformly convex and are nonempty and closed , that since the uniformly convex space with a norm is a Banach space so that we could rewrite the above constraints by replacing the metric by a metric , induced by , which is always equivalent to even in the case that both metric functions are not coincident. Equation (2.34) contradicts , since there is no for any given such that implying that for some . Thus, (2.33) is false unless . It follows also as a result that the best proximity points satisfy and , with , , .

Note that the existence of nonempty subsets ,, in Theorem 2.5 are guaranteed if are bounded (although nonnecessarily closed) and ; . Note also that is guaranteed since .

3. Links with General Kannan Mappings and -Cyclic Kannan Mappings

The next result is concerned with -cyclic Kannan self-mappings [7, 11, 12] which can eventually satisfy contractive conditions within the class (2.1).

Theorem 3.1. Assume that a -cyclic self-mapping satisfies the contractive condition (2.1) with . Then, the following properties hold.
(i) If and, furthermore, with the constants defined in (2.4), then, is a -cyclic Kannan self-mapping.
(ii) If, in addition, (2.9) is replaced by the constraints and, furthermore, then which is guaranteed by the condition (2.3). If the subsets of , have nonempty intersection and if the metric space is complete, then and have a unique fixed point in to which all the sequences , which are then bounded, converge; . If the above subsets of are disjoint, closed, and convex, , is uniformly convex and then all sequences converge to a best proximity point of and .

Proof. If , then the contractive condition (2.1) may be upper-bounded as follows for all ; any by using the triangle inequality where necessary and upper-bounding the necessary fractions by unity when the denominator is not less than the numerator in the right-hand side of (2.1): and, since , one gets with . Thus, is a -cyclic Kannan self-mapping. Hence, Property (i).
Furthermore, since (2.9) is replaced by (3.1) for so that if in (2.4) and in (2.1), it follows from (3.5) that (3.1)–(3.3) guarantee that under the necessary condition , [11, 12]. Then, the self-mapping satisfies the following contractive condition for and any :