If

*H* is a real Hilbert space with an inner product

and a norm

and

*A* is a nonempty closed convex subset of

*H*, then

is said to be an asymptotically

*β*-strictly pseudocontractive self-mapping in the intermediate sense for some

if

for some sequence

,

as

[

1–

4,

23]. Such a concept was firstly introduced in [

1]. If (2.1) holds for

, then

is said to be an asymptotically pseudocontractive self-mapping in the intermediate sense. Finally, if

as

, then

is asymptotically

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

. If (2.1) is changed to the stronger condition

then the above concepts translate into

being an asymptotically

*β*-strictly pseudocontractive self-mapping, an asymptotically pseudocontractive self-mapping and asymptotically contractive one, respectively. Note that (2.1) is equivalent to

Note that the high-right-hand-side term

of (2.3) is expanded as follows for any

:

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

in

. Now, if instead of a real Hilbert space

*H* endowed with an inner product

and a norm

, we deal with any generic Banach space

, then its norm induces a homogeneous and translation invariant metric

defined by

;

so that (2.6) takes the form

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*:

*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)

- (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
and
as follows:

(A)

for any

,

. Then one gets from (2.10)

,

, where

which holds from (2.12)-(2.13) if

since

as

in (2.13) is equivalent to (2.16). Note that

is ensured either with

or with

if

However,
with
has to be excluded because of the unboundedness or nonnegativity of the second right-hand-side term of (2.15).

(B)

for some

,

. Then one gets from (2.10)

which holds from (2.12) and

if

, and

Thus, (2.15)-(2.16), with the second option in the logic disjunction being true if and only if

together with (2.18)-(2.20), are equivalent to (2.12)-(2.13) by taking

to be either

or

for each

. It then follows that

;

from (2.15)-(2.19) since

and

;

as

. Thus,

is asymptotically nonexpansive. Thus, Property (i) has been proven. Property (ii) is proven as follows. Consider the metric-induced norm

equivalent to the translation-invariant homogeneous metric

. Such a norm exists since the metric is homogeneous and translation-invariant so that norm and metric are formally equivalent. Rename

and define a sequence of subsets

of

*X*. From Property (i),

is bounded;

if

is finite, since it is bounded for any finite

and, furthermore, it has a finite limit as

. Thus, all the collections of subsets

;

are bounded since

is bounded. Define the set

which is nonempty bounded, closed and convex by construction. Since

is complete,

is a uniformly convex Banach space and

is asymptotically nonexpansive from Property (i), then it has a fixed point

[

1,

23]. Since the restricted self-mapping

is also continuous, one gets from Property (i)

Then any sequence
is convergent (otherwise, the above limit would not exist contradicting Property (i)), and then bounded in *C*;
. This also implies
is bounded;
,
and
;
,
. This implies also
as
;
such that
;
which is then a fixed point of
(otherwise, the above property
;
,
would be contradicted). Hence, Property (ii) is proven. □

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
for any nonempty, bounded and closed subset *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
is not required to be convex. However, the property that all the sequences converge to fixed points opens two potential possibilities depending on particular extra restrictions on the self-mapping
, namely: (1) the fixed point is not unique so that
for any
(and any *A* in *X*) so that some set
for some
contains more than one point. In other words,
as
;
has not been proven although it is true that
;
; (2) there is only a fixed point in *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*:

*for some parameterizing real sequences*
,

*and*
*satisfying*,

*for any*
,

*Then the following properties hold*:

- (i)
*and the following limit exists*:

- (ii)

*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
;
,
and
; as
,
. The concepts of an asymptotic pseudocontraction and an asymptotic strict pseudocontraction in the intermediate sense motivated in Theorem 2.1 by (2.7)-(2.9), under the asymptotically nonexpansive constraints (2.10) subject to (2.11) and in Theorem 2.2 by (2.22) subject to (2.23) are revisited as follows in the context of metric spaces.

**Definition 2.3** Assume that

is a complete metric space with

being a homogeneous translation-invariant metric. Thus,

is asymptotically

*β*-strictly pseudocontractive in the intermediate sense if

for

;

and some real sequences

,

being, in general, dependent on the initial points,

*i.e.*,

,

and

**Definition 2.4**
is asymptotically pseudocontractive in the intermediate sense if (2.30) holds with
,
,
,
,
,
as
and the remaining conditions as in Definition 2.3 with
,
and
.

**Definition 2.5**
is asymptotically *β*-strictly contractive in the intermediate sense if
,
,
;
,
,
as
, in Definition 2.3 with
,
.

**Definition 2.6**
is asymptotically contractive in the intermediate sense if
,
,
;
,
,
, and
as
in Definition 2.3 with
,
and
.

**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)

- (2)

- (3)

- (4)

- (5)
The above considerations could also be applied to Theorem 2.1 for the case
(Case (B) in the proof of Property (i)) being asymptotically nonexpansive for the asymptotically nonexpansive condition (2.10) subject to (2.11).

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*
,

*Furthermore*,

*the following properties hold*:

- (i)

- (ii)

- (iii)

- (iv)

*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
is asymptotically nonexpansive (see Remark 2.7(1)). Property (ii) follows in a similar way from Definition 2.4 (see Remark 2.7(2)). Properties (iii)-(iv) follow from Theorem 2.2 and Definitions 2.5-2.6 implying also that the asymptotically nonexpansive self-mapping
is also a strict contraction, then continuous with a unique fixed point, since
(see Remark 2.7(3)) and
with
(see Remark 2.7(4)), respectively. (The above properties could also be got from Theorem 2.1 for Case (B) of the proof of Theorem 2.1(ii) - see Remark 2.7(5).) □

**Example 2.9** Consider the time-varying

*p*th order nonlinear discrete dynamic system

for some given nonempty bounded set

, where

is a

matrix sequence of elements

with

and

with

;

, and

defines the state-sequence trajectory solution

. Equation (

2.13) requires the consistency constraint

to calculate

. However, other discrete systems being dealt with in the same way as, for instance, that obtained by replacing

in (2.31) with the initial condition

(and appropriate

*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

as

;

. Note that for the Euclidean distance (and norm),

;

. Assume that the squared spectral norm of

is upper-bounded by

for some parameterizing scalar sequences

,

and

which can be dependent, in a more general case, on the state

. This holds, for instance, if

, where

is a real positive sequence satisfying

and

both being potentially dependent on the state as the rest of the parameterizing sequences. Since the spectral norm equalizes the spectral radius if the matrix is symmetric, then

can be taken exactly as the spectral radius of

in such a case,

*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
which holds, in particular, if

(a)
,
,
,
;
,
and
,
, as
;
. This implies that
;
and
as
;
. Thus,
is asymptotically nonexpansive being also an asymptotic strict *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
of
which is unbounded, and then there is also an infinite subsequence
which is strictly increasing. Since
and
as
;
, one has that for
, any given
and some sufficiently large
,
,
,
such that
and
;
,
. Now, take
and
. Then
;
and any given
. If
, then stability holds trivially. Assume not, and there are unbounded solutions. Thus, take
such that
for any given
,
and some
. Note that since
is a strictly increasing real sequence
implying
as
, which leads to a contradiction to the inequality
for
for some sufficiently large
, then for some sufficiently large *M*, if such a strictly increasing sequence
exists. Hence, there is no such sequence, and then no unbounded sequence
for any initial condition in
. As a result, for any initial condition in any given subset
of
(even if it is unbounded), any solution sequence of (2.31) is bounded, and then (2.31) is globally stable. The above reasoning implies that there is an infinite collection of numerable nonempty bounded closed sets
, which are not necessarily connected, such that
;
and any given
. Assume that the set
of initial conditions is bounded, convex and closed and consider the collection of convex envelopes
, define constructively the closure convex set
which is trivially bounded, convex and closed. Note that it is not guaranteed that
is either open or closed since there is a union of infinitely many closed sets involved. Note also that the convex hull of all the convex envelopes of the collection of sets is involved to ensure that *A* is convex since the union of convex sets is not necessarily convex (so that
is not guaranteed to be convex while *A* is convex). Consider now the self-mapping
which defines exactly the same solution as
for initial conditions in
so that
is identified with the restricted self-mapping
from a nonempty bounded, convex and closed set to itself. Note that
for the Euclidean distance is a convex metric space which is also complete since it is finite dimensional. Then
and
are both continuous, then
is also continuous and has a fixed point in *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
and
as
;
. From Remark 2.7(2),
and
for any
. Thus the convergence is guaranteed to be faster for an asymptotic *β*-strict pseudocontraction in the intermediate sense than for an asymptotic pseudocontraction in the intermediate sense with a sequence
such that
;
with the remaining parameters and parametrical sequences being identical in both cases. If
and
;
are both continuous, then
is continuous and has a fixed point in *A* from Theorem 2.8(ii).

(c) If
is asymptotically *β*-strictly contractive in the intermediate sense, then
;
so that it is asymptotically strictly contractive and has a unique fixed point from Theorem 2.8(iii).

(d) If
is asymptotically contractive in the intermediate sense,
;
. Thus,
is an asymptotic strict contraction and has a unique fixed point from Theorem 2.8(iv).

**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.