This paper is devoted to investigate the fixed points and best proximity points of multivalued cyclic self-mappings on a set of subsets of complete metric spaces endowed with a partial order under a generalized contractive condition involving a Hausdorff distance. The existence and uniqueness of fixed points of both the cyclic self-mapping and its associate composite self-mappings on each of the subsets are investigated, if the subsets in the cyclic disposal are nonempty, bounded and of nonempty convex intersection. The obtained results are extended to the existence of unique best proximity points in uniformly convex Banach spaces.

Important attention is being devoted recently to the investigation of fixed points of self-mappings as well as to the investigation of associate relevant properties like, for instance, stability of the iterations [

Assume that

If

Let

There exist

If

There are some

There is a partially ordered subsequence

If

If assumption (3) is removed and (

Let

so that

The remaining proof of property (ii) follows by contradiction. Suppose that the limit (

If assumption (3) is removed, while

Note that (

Note that Theorem

In addition to assumptions (1)–(4) of Theorem

the limit

The sequence

If, in addition,

If assumption (4) of Theorem

If

The property (i) follows from Theorem

To prove property (ii), assume that there are two distinct limits

It remains to be proven that

Thus,

On the other hand, since any fixed point of the self-mapping

The property (iii) is proven as follows. Assume that there are

The proof of property (iv) follows directly from the above properties (i)–(iii) and property (iii) of Theorem

(1) Note that the restricted composite multivalued self-mapping

(2) The convexity of the subsets

(3) Finally, note that a convex set in a Euclidean space is convex metric space under the Euclidean induced norm and that closed subsets of Euclidean spaces are convex metric spaces if and only if they are convex. This property is used in the proof of property (v) of Theorem

An “ad hoc” version of Theorem

It is well known that a norm defines a metric. In this sense, a Banach space

The next result is an “ad hoc” version for this paper of previous technical results. See Lemma

Let

For every

Then, the following properties hold:

For every

If

The proof of Lemma

Let

Let

The complete metric space

Then, the following properties hold.

There are unique best proximity points

Take any

If assumption (4) of Theorem

Note from the various hypothesis the uniformly convex Banach space

(1) Theorem

(2) The value of the individual contractive constants being less than, equal to, or larger than one for each pair of adjacent subsets is irrelevant in Theorem

(3) Note also that, for Euclidean metric, the convexity of

(4) It can be observed that the metric convexity of the space

(5) Note that the results of Sections

Consider two bounded and closed real subsets

Consider also a scalar discrete dynamic system of state

for some

It follows from Theorem

The following extension of the example is direct. Assume that (

for some

The author is very grateful to the Spanish Government for its support of this research through Grant DPI2012-30651 and to the Basque Government by its support of this research through Grants IT378-10 and SAIOTEK S-PE12UN015. He is also grateful to UPV/EHU for its financial support through UFI 2011/07 and to the referees for their useful comments.