### Abstract

This paper discusses a more general contractive condition for a class of extended 2-cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are nonempty, closed and convex, then all the iterations converge to a unique closed limiting finite sequence, which contains the best proximity points of adjacent subsets, and reduce to a unique fixed point if all such subsets intersect.

### 1 Introduction

Strict pseudocontractive mappings and pseudocontractive mappings in the intermediate
sense formulated in the framework of Hilbert spaces have received a certain attention
in the last years concerning their convergence properties and the existence of fixed
points. See, for instance, [1-4] and references therein. Results about the existence of a fixed point are discussed
in those papers. On the other hand, important attention has been paid during the last
decades to the study of the convergence properties of distances in cyclic contractive
self-mappings on *p* subsets
*p* subsets
*M*, as the union of a set of nonempty sets as
*M* to *M* which are cyclic *φ*-contractions, where
*φ*-contraction. In [18], a characterization of best proximity points is studied for individual and pairs
of non-self-mappings
*A* and *B* are nonempty subsets of a metric space. The existence of common fixed points of self-mappings
is investigated in [24] for a class of nonlinear integral equations, while fixed point theory is investigated
in locally convex spaces and non-convex sets in [25-28]. More recently, the existence and uniqueness of best proximity points of more general
cyclic contractions have been investigated in [29,30] and a study of best proximity points for generalized proximal contractions, a concept
referred to non-self-mappings, has been proposed and reported in detail in [31]. Also, the study and characterization of best proximity points for cyclic weaker
Meir-Keeler contractions have been performed in [32] and recent contributions on the study of best proximity and proximal points can be
found in [33-38] and references therein. In general, best proximity points do not fulfill the usual
‘best proximity’ condition
*x* to the distances
*φ*-contractions, which contains the cyclic contraction maps as a subclass, has been
proposed in [18] in order to investigate the convergence and existence results of best proximity points
in reflexive Banach spaces completing previous related results in [6]. Also, the existence and uniqueness of best proximity points of cyclic *φ*-contractive self-mappings in reflexive Banach spaces have been investigated in [19]. This paper is devoted to the convergence properties and the existence of fixed points
of a generalized version of pseudocontractive, strict pseudocontractive and asymptotically
pseudocontractive in the intermediate sense in the more general framework of metric
spaces. The case of 2-cyclic pseudocontractive self-mappings is also considered. The
combination of constants defining the contraction may be different on each of the
subsets and only the product of all the constants is requested to be less than unity.
It is assumed that the considered self-mapping can perform a number of iterations
on each of the subsets before transferring its image to the next adjacent subset of
the 2-cyclic self-mapping. The existence of a unique closed finite limiting sequence
on any sequence of iterations from any initial point in the union of the subsets is
proven if *X* is a uniformly convex Banach space and all the subsets of *X* are nonempty, convex and closed. Such a limiting sequence is of size
*p* of its elements (all of them if

### 2 Asymptotic contractions and pseudocontractions in the intermediate sense in metric spaces

If *H* is a real Hilbert space with an inner product
*A* is a nonempty closed convex subset of *H*, then
*β*-strictly pseudocontractive self-mapping in the intermediate sense for some

for some sequence
*β*-strictly contractive in the intermediate sense, respectively, asymptotically contractive
in the intermediate sense if

then the above concepts translate into
*β*-strictly pseudocontractive self-mapping, an asymptotically pseudocontractive self-mapping
and asymptotically contractive one, respectively. Note that (2.1) is equivalent to

or, equivalently,

where

Note that the high-right-hand-side term

The objective of this paper is to discuss the various pseudocontractive in the intermediate
sense concepts in the framework of metric spaces endowed with a homogeneous and translation-invariant
metric and also to generalize them to the *β*-parameter to eventually be replaced with a sequence
*H* endowed with an inner product

Define

which exists since it follows from (2.7), since the metric is homogeneous and translation-invariant, that

The following result holds related to the discussion (2.7)-(2.9) in metric spaces.

**Theorem 2.1***Let*
*be a metric space and consider a self*-*mapping*
*Assume that the following constraint holds*:

*with*

*for some parameterizing bounded real sequences*
*and*
*of general terms*
*satisfying the following constraints*:

*with*
*and*, *furthermore*, *the following condition is satisfied*:

*if and only if*
*as*

*Then the following properties hold*:

(i)
*for any*
*so that*
*is asymptotically nonexpansive*.

(ii) *Let*
*be complete*,
*be*, *in addition*, *a translation*-*invariant homogeneous norm and let*
*with*
*being the metric*-*induced norm from*
*be a uniformly convex Banach space*. *Assume also that*
*is continuous*. *Then any sequence*
*is bounded and convergent to some point*
*being in general dependent on**x*, *in some nonempty bounded*, *closed and convex subset**C**of**A*, *where**A**is any nonempty bounded subset of**X*. *Also*,
*is bounded*;
*and*
*is a fixed point of the restricted self*-*mapping*
*Furthermore*,

*Proof* Consider two possibilities for the constraint (2.10), subject to (2.11), to hold
for each given

(A)

which holds from (2.12)-(2.13) if

However,

(B)

where

which holds from (2.12) and

Thus, (2.15)-(2.16), with the second option in the logic disjunction being true if
and only if
*X*. From Property (i),

Then any sequence
*C*;

First of all, note that Property (ii) of Theorem 2.1 applies to a uniformly convex
space which is also a complete metric space. Since the metric is homogeneous and translation-invariant,
a norm can be induced by such a metric. Alternatively, the property could be established
on any uniformly convex Banach space by taking a norm-induced metric which always
exists. Conceptually similar arguments are used in later parallel results throughout
the paper. Note that the proof of Theorem 2.1(i) has two parts: Case (A) refers to
an asymptotically nonexpansive self-mapping which is contractive for any number of
finite iteration steps and Case (B) refers to an asymptotically nonexpansive self-mapping
which is allowed to be expansive for a finite number of iteration steps. It has to
be pointed out concerning such a Theorem 2.1(ii) that the given conditions guarantee
the existence of at least a fixed point but not its uniqueness. Therefore, the proof
is outlined with the existence of a
*A* of *X*. Note that the set *C*, being in general dependent on the initial set *A*, is bounded, convex and closed by construction while any taken nonempty set of initial
conditions
*A* in *X*) so that some set
*X*. The following result extends Theorem 2.1 for a modification of the asymptotically
nonexpansive condition (2.10).

**Theorem 2.2***Let*
*be a metric space and consider the self*-*mapping*
*Assume that the constraint below holds*:

*with*

*for some parameterizing real sequences*
*and*
*satisfying*, *for any*

*Then the following properties hold*:

(i)
*so that*
*is asymptotically nonexpansive*, *and then*
*if*

*and the following limit exists*:

(ii) *Property* (ii) *of Theorem *2.1 *if*
*is complete and*
*is a uniformly convex Banach space under the metric*-*induced norm*

*Sketch of the proof* Property (i) follows in the same way as the proof of Property (i) of Theorem 2.1
for Case (B). Using proving arguments similar to those used to prove Theorem 2.1,
one proves Property (ii). □

The relevant part in Theorem 2.1 being of usefulness concerning the asymptotic pseudocontractions
in the intermediate sense and the asymptotic strict contractions in the intermediate
sense relies on Case (B) in the proof of Property (i) with the sequence of constants

**Definition 2.3** Assume that
*β*-strictly pseudocontractive in the intermediate sense if

for
*i.e.*,

**Definition 2.4**

**Definition 2.5**
*β*-strictly contractive in the intermediate sense if

**Definition 2.6**

**Remark 2.7** Note that Definitions 2.3-2.5 lead to direct interpretations of their role in the
convergence properties under the constraint (2.22), subject to (2.23), by noting the
following:

(1) If
*β*-strictly pseudocontractive in the intermediate sense (Definition 2.3), then the real
sequence

(2) If

(3) If
*β*-strictly contractive in the intermediate sense (Definition 2.5), then the sequence
of asymptotically contractive constants is defined by
*α* and *β* are arbitrarily close to zero, since

(4) If

with
*β*-strictly contractive in the intermediate sense than if it is just asymptotically
contractive in the intermediate sense if

(5) The above considerations could also be applied to Theorem 2.1 for the case

The subsequent result, being supported by Theorem 2.2, relies on the concepts of asymptotically
contractive and pseudocontractive self-mappings in the intermediate sense. Therefore,
it is assumed that

**Theorem 2.8***Let*
*be a complete metric space endowed with a homogeneous translation*-*invariant metric*
*and consider the self*-*mapping*
*Assume that*
*is a uniformly convex Banach space endowed with a metric*-*induced norm*
*from the metric*
*Assume that the asymptotically nonexpansive condition* (2.22), *subject to* (2.23), *holds for some parameterizing real sequences*
*and*
*satisfying*, *for any*

*Then*
*for any*
*satisfying the conditions*

*Furthermore*, *the following properties hold*:

(i)
*is asymptotically**β*-*strictly pseudocontractive in the intermediate sense for some nonempty*, *bounded*, *closed and convex set*
*and any given nonempty*, *bounded and closed subset*
*of initial conditions if* (2.29) *hold with*
*and*
*as*
*Also*,
*has a fixed point for any such set**C**if*
*is continuous*.

(ii)
*is asymptotically pseudocontractive in the intermediate sense for some nonempty*, *bounded*, *closed and convex set*
*and any given nonempty*, *bounded and closed subset*
*of initial conditions if* (2.29) *hold with*
*and*
*as*
*Also*,
*has a fixed point for any such set**C**if*
*is continuous*.

(iii) *If* (2.29) *hold with*
*and*
*as*
*then*
*is asymptotically**β*-*strictly contractive in the intermediate sense*. *Also*,
*has a unique fixed point*.

(iv) *If* (2.29) *hold with*
*and*
*as*
*then*
*is asymptotically strictly contractive in the intermediate sense*. *Also*,
*has a unique fixed point*.

*Proof* (i) It follows from Definition 2.3 and the fact that Theorem 2.2 holds under the
particular nonexpansive condition (2.22), subject to (2.23), so that

**Example 2.9** Consider the time-varying *p*th order nonlinear discrete dynamic system

*ad hoc* re-definition of the mapping which generates the trajectory solution from given initial
conditions) do not require such a consistency constraint. The dynamic system (2.31)
is asymptotically linear if
*i.e.*, it equalizes the absolute value of its dominant eigenvalue. We have to check the
condition

provided, for instance, that the distance is the Euclidean distance, induced by the
Euclidean norm, then both being coincident, and provided also that we take the metric
space

(a)
*β*-pseudocontraction in the intermediate sense. This also implies that (2.31) is globally
stable as it is proven as follows. Assume the contrary so that there is an infinite
subsequence
*M*, if such a strictly increasing sequence
*A* is convex since the union of convex sets is not necessarily convex (so that
*A* is convex). Consider now the self-mapping
*A* from Theorem 2.8(i).

(b) If the self-mapping is asymptotically pseudocontractive in the intermediate sense,
then the above conclusions still hold with the modification
*β*-strict pseudocontraction in the intermediate sense than for an asymptotic pseudocontraction
in the intermediate sense with a sequence
*A* from Theorem 2.8(ii).

(c) If
*β*-strictly contractive in the intermediate sense, then

(d) If

**Remark 2.10** Note that conditions like (2.32) can be tested on dynamic systems being different
from (2.31) by redefining, in an appropriate way, the self-mapping which generates
the solution sequence from given initial conditions. This allows to investigate the
asymptotic properties of the self-mapping, the convergence of the solution to fixed
points, then the system stability, *etc.* in a unified way for different dynamic systems. Close considerations can be discussed
for different dynamic systems and convergence of the solutions generated by the different
cyclic self-mappings defined on the union of several subsets to the best proximity
points of each of the involved subsets.

### 3 Asymptotic contractions and pseudocontractions of cyclic self-mappings in the intermediate sense

Let
*X*.

with

with
*A* and *B* are closed and convex, then there is a unique fixed point of

**Theorem 3.1***Let*
*be a metric space and let*
*be a cyclic self*-*mapping*, *i*.*e*.,
*and*
*where**A**and**B**are nonempty subsets of**X*. *Define the sequence*
*of asymptotically nonexpansive iteration*-*dependent constants as follows*:

*provided that*
*satisfies the constraint* (3.1), *subject to* (3.2), *and*

*and*

*for*
*and for*
*provided that*
*satisfies the constraint* (3.3) *subject to* (3.4) *provided that the parameterizing bounded real sequences*
*and*
*of general terms*
*and*
*fulfill the following constraints*:

*and assuming that the following limits exist*:

*Then*, *the following properties hold*:

(i)
*satisfies* (3.3) *subject to* (3.4)-(3.9);
*Then*

*so that*
*is a cyclic asymptotically nonexpansive self*-*mapping*. *If*
*is a best proximity point of**A**and*
*is a best proximity point of**B*, *then*
*and*
*and*
*which are best proximity points of**A**and**B* (*not being necessarily identical to**x**and**y*), *respectively if*
*is continuous*.

(ii) *Property* (i) *also holds if*
*satisfies* (3.1) *subject to* (3.2), (3.7), (3.8)-(3.9) *and* (3.5b) *provided that*

*Proof* The second condition of (2.18) now becomes under either (3.1)-(3.2) and (3.8)-(3.9)

and it now becomes under (3.3)-(3.4) and (3.8)-(3.9)

since

Also, if

**Remark 3.2** Note that Theorem 3.1 does not guarantee the convergence of

The following result specifies Theorem 3.1 for asymptotically nonexpansive mappings
with

**Theorem 3.3***Let*
*be a metric space and let*
*be a cyclic self*-*mapping which satisfies the asymptotically nonexpansive constraint* (3.1), *subject to* (3.2), *where**A**and**B**are nonempty subsets of**X*. *Let the sequence*
*of asymptotically nonexpansive iteration*-*dependent constants be defined by a general term*
*under the constraints*
*and*
*Then the subsequent properties hold*:

(i) *The following limits exist*:

(ii) *Assume*, *furthermore*, *that*
*is complete*, *A**and**B**are closed and convex and*
*is translation*-*invariant and homogeneous and*
*is uniformly convex where*
*is the metric*-*induced norm*. *Then*

*and*
*where**z**and**Tz**are unique best proximity points in**A**and**B*, *respectively*. *If*
*then*
*is the unique fixed point of*

*Proof* Note from (3.9), under (3.6) and (3.7), that there is no division by zero on the
right-hand side of (3.10) and

There are several possible cases as follows.

Case A:

Case B:

Case C:

but the above expression is equivalent, for
*A* or *B*, to

which contradicts