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Fixed point-type results for a class of extended cyclic self-mappings under three
general weak contractive conditions of rational type
Fixed Point Theory and Applications 2011, 2011:102 doi:10.1186/1687-1812-2011-102
Manuel De la Sen ()
Ravi P Agarwal ()
ISSN 1687-1812
Article type Research
Submission date 12 September 2011
Acceptance date 21 December 2011
Publication date 21 December 2011
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1
Fixed point-type results for a class of extended cyclic self-mappings under three
general weak contractive conditions of rational type
Manuel De la Sen
*1
and Ravi P Agarwal
2
1
Instituto de Investigacion y Desarrollo de Procesos, Universidad del Pais Vasco, Campus
of Leioa (Bizkaia) – Aptdo. 644-Bilbao, 48080-Bilbao, Spain

2
Department of Mathematics, Texas A&M University - Kingsville, 700 University Blvd.,
Kingsville, TX 78363-8202, USA
*Corresponding author:
Email address:
RPA:

Abstract
This article discusses three weak φ-contractive conditions of rational type for a class of 2-
cyclic self-mappings defined on the union of two non-empty subsets of a metric space to
itself. If the space is uniformly convex and the subsets are non-empty, closed, and
convex, then the iterates of points obtained through the self-mapping converge to unique
best proximity points in each of the subsets.

1. Introduction
A general contractive condition has been proposed in [1, 2] for mappings on a partially
ordered metric space. Some results about the existence of a fixed point and then its
uniqueness under supplementary conditions are proved in those articles. The rational
2
contractive condition proposed in [3] includes as particular cases several of the
previously proposed ones [1, 4–12], including Banach principle [5] and Kannan fixed
point theorems [4, 8, 9, 11]. The rational contractive conditions of [1, 2] are applicable
only on distinct points of the considered metric spaces. In particular, the fixed point
theory for Kannan mappings is extended in [4] by the use of a non-increasing function
affecting the contractive condition and the best constant to ensure a fixed point is also
obtained. Three fixed point theorems which extended the fixed point theory for Kannan
mappings were stated and proved in [11]. More attention has been paid to the
investigation of standard contractive and Meir-Keeler-type contractive 2-cyclic self-
mappings
BABAT

∪
→
∪
: defined on subsets XBA
⊆
, and, in general, p-cyclic self-
mappings
UU
pi
i
pi
i
AA:T
∈∈
→ defined on any number of subsets XA
i
⊂ ,
{
}
p, ,,pi 21:=∈ , where
(
)
d,X is a metric space (see, for instance [13–22]). More recent
investigation about cyclic self-mappings is being devoted to its characterization in
partially ordered spaces and also to the formal extension of the contractive condition
through the use of more general strictly increasing functions of the distance between
adjacent subsets. In particular, the uniqueness of the best proximity points to which all
the sequences of iterates of composed self-mappings BABAT ∪→∪:
2
converge is

proven in [14] for the extension of the contractive principle for cyclic self-mappings in
uniformly convex Banach spaces (then being strictly convex and reflexive [23]) if the
subsets XBA
⊂
, in the metric space
(
)
d,X , or in the Banach space
(
)
,X
, where the 2-
cyclic self-mappings are defined, are both non-empty, convex and closed. The research in
[14] is centred on the case of the cyclic self-mapping being defined on the union of two
subsets of the metric space. Those results are extended in [15] for Meir-Keeler cyclic
contraction maps and, in general, for the self-mapping
UU
pi
i
pi
i
AA:T
∈∈
→ be a p
(
)
2≥ -
cyclic self-mapping being defined on any number of subsets of the metric space with
3
{

}
p, ,,p 21:= . Also, the concept of best proximity points of (in general) non-self-
mappings BAT,S
→
: relating non-empty subsets of metric spaces in the case that such
maps do not have common fixed points has recently been investigated in [24, 25]. Such an
approach is extended in [26] to a mapping structure being referred to as K-cyclic
mapping with contractive constant
21 /k <
. In [27], the basic properties of cyclic self-
mappings under a rational-type of contractive condition weighted by point-to-point-
dependent continuous functions are investigated. On the other hand, some extensions of
Krasnoselskii-type theorems and general rational contractive conditions to cyclic self-
mappings have recently been given in [28, 29] while the study of stability through fixed
point theory of Caputo linear fractional systems has been provided in [30]. Finally,
promising results are being obtained concerning fixed point theory for multivalued maps
(see, for instance [31–33]).
This manuscript is devoted to the investigation of several modifications of rational type
of the φ-contractive condition of [21, 22] for a class of 2-cyclic self-mappings on non-
empty convex and closed subsets XB,A
⊂
. The contractive modification is of rational
type and includes the nondecreasing function associated with the
ϕ
-contractions. The
existence and uniqueness of two best proximity points, one in each of the subsets
XB,A
⊂
, of 2-cyclic self-mappings
BABAT

∪
→
∪
:
defined on the union of two non-
empty, closed, and convex subsets of a uniformly convex Banach spaces, is proven. The
convergence of the sequences of iterates through
BABAT
∪
→
∪
:
to one of such best
proximity points is also proven. In the case that A and B intersect, both the best proximity
points coincide with the unique fixed point in the intersection of both the sets.

4
2. Basic properties of some modified constraints of 2-cyclic
ϕ
-
contractions
Let
(
)
d,X be a metric space and consider two non-empty subsets
A
and
B
of
X

. Let
BABAT
∪
→
∪
: be a 2-cyclic self-mapping, i.e.,
(
)
BAT ⊆ and
(
)
ABT ⊆ . Suppose, in
addition, that BABAT
∪
→
∪
: is a 2-cyclic modified weak
ϕ
-contraction (see [21, 22])
for some non-decreasing function
++
→
00
: RR
ϕ
subject to the rational modified
ϕ
-
contractive constraint:
( )

( ) ( )
( )
( ) ( )
( )
( ) ( )( )( ) ( )
Dy,xdy,xd
y,xd
Ty,ydTx,xd
y,xd
Ty,ydTx,xd
Ty,Txd
ϕϕβϕα
+−+
















−≤

;
(
)
BAxy,x ∪∈≠∀ (2.1)
where
(
)
(
)
{
}
By,Ax:y,xdinfBAdistD ∈∈== :,: (2.2)
(
)
( )
(
)
( )
( ) ( )
DD
k
kk
Tx,xdklimxT,xTdsuplimD
n
n
n
nn
n
ϕϕ
=









−
−−
+≤≤
∞→
+
∞→
1
11
1
; BAx
∪
∈
∀
(2.3)
Note that (2.1) is, in particular, a so-called 2-cyclic
ϕ
-contraction if 0
=
α
and
(
)

(
)
tt
αϕ
−= 1 for some real constant
[
)
10 ,∈
α
since
++
→
00
: RR
ϕ
is strictly increasing [1].
We refer to “modified weak
ϕ
-contraction” for (2.1) in the particular case 00
≥
≥
β
α
, ,
1
<
+
β
α
, and

++
→
00
: RR
ϕ
being non-decreasing as counterpart to the term
ϕ
-contraction
(or via an abuse of terminology “modified strong
ϕ
-contraction”) for the case of
++
→
00
: RR
ϕ
in (2.1) being strictly increasing. There are important background results on
the properties of weak contractive mappings (see, for instance, [1, 2, 34] and references
therein). The so-called “
ϕ
-contraction”, [1, 2], involves the particular contractive
5
condition obtained from (2.1) with
0
=
α
, 1
=
β
, and

++
→
00
: RR
ϕ
being strictly
increasing, that is,
(
)
(
)
(
)
(
)
(
)
Dy,xdy,xdTy,Txd
ϕϕ
+−≤ , BAx
∪
∈
∀

In the following, we refer to 2-cyclic self-maps BABAT
∪
→
∪
: simply as cyclic self-
maps. The following result holds:

Lemma 2.1. Assume that BABAT
∪
→
∪
: is a modified weak
ϕ
-contraction, that is, a
cyclic self-map satisfying the contractive condition (2.1) subject to the constraints
(
)
0≥
βα
,min and 1
<
+
β
α
with
++
→
00
: RR
ϕ
being non-decreasing. Then, the following
properties hold:

(i) Assume that
(
)

DD ≥
ϕ

(
)
(
)
(
)
(
)
Dkx,TxkdxTxTdD
nn
ϕ
−+≤≤
+
1,
1
;
{
}
0:
0
∪=∈∀ NNn , BAx
∪
∈
∀
(2.4)
(
)

(
)
(
)
DxT,xTdsuplimxT,xTdinflimD
mnmn
n
mnmn
n
ϕ
≤≤≤
+++
∞→
+++
∞→
11
; BAx
∪
∈
∀
0
N∈∀m (2.5)
and
(
)
(
)
DxT,xTdsuplim
mnmn
n

ϕ
≤
+++
∞→
1
if 0
≠
D . If
(
)
0== DD
ϕ
then
(
)
0
1
=∃
+++
∞→
xT,xTdlim
mnmn
n
; BAx
∪
∈
∀
,
0
N∈∀m .

(ii) Assume that
(
)
(
)
xmTxxd ≤, for any given BAx
∪
∈
. Then
(
)
(
)
( )
D
k
k
k
xmk
x,xTd
n
ϕ
−
+
−
≤
1
1
; BAx
∪

∈
∀
, N
∈
∀
n (2.6)
6
If
(
)
Txxd , is finite and, in particular, if
x
and Tx in
B
A
∪
are finite then the sequences
{
}
0
N∈n
n
xT
and
{
}
0
1
N∈
+

n
n
xT
are bounded sequences where AxT
n
∈ and BxT
n
∈
+1
if Ax
∈

and n is even, BxT
n
∈ and BxT
n
∈
+1
if Bx
∈
and n is even.

Proof: Take Txy
=
so that xTTy
2
= . Since
++
→
00

: RR
ϕ
is non-decreasing
(
)
(
)
Dx
ϕϕ
≥ for
Dx
≥
, one gets for any Ax
∈
and any BTx
∈
or for any Bx
∈
and any ATx
∈
:

(
)
(
)
(
)
(
)

(
)
(
)
(
)
[
]
(
)
DTx,xdTx,xdxT,TxdTx,xTd
ϕϕβαϕα
+−+−≤−
22
1
(
)
(
)
(
)
(
)
(
)
(
)
Tx,xdxT,TxdDTx,xd
βϕαϕϕβ
−−+=

2
; BAx
∪
∈
∀

(
)
( ) ( ) ( ) ( ) ( )
DkTx,xdkDTx,xdkTx,xTd
ϕϕ
α
β
α
−+=
−
−
−
+≤⇔ 1
1
1
2
; BAx
∪
∈
∀
(2.7)
if xTx
≠
where

1
1
: <
−
=
α
β
k
, since BABAT
∪
→
∪
: is cyclic,
(
)
DTx,xd ≥ and
++
→
00
: RR
ϕ

is increasing. Then
(
)
(
)
(
)
(

)
DkTx,xdkxT,xTd
nnnn
ϕ
−+≤
+
1
1
; BAx
∪
∈
∀
; N
∈
∀
n (2.8)
(
)
0≠≥ DD
ϕ
since
(
)
0≥
βα
,min and 1
<
+
β
α

. Proceeding recursively from (2.8), one gets
for any
N
∈
m
:
( )
( ) ( )( ) ( ) ( )
( )
n
n
i
innn
kDx,TxkdkkDx,TxdkxT,xTdD −+≤








−+≤≤
∑
−
=
+
11
1
0

1
ϕϕ
(2.9a)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
Dx,TxdDx,TxkdDkx,Txkd
ϕϕϕ
+<+≤−+≤ 1 ; BAx
∪
∈
∀
(2.9b)
(
)
( ) ( )( )

















−+≤≤
∑
−+
=
+
∞→
+++
∞→
1
0
1
1,
mn
i
imn
n
mnmn

n
kkDx,TxdklimxTxTdsuplimD
ϕ

7
( )( ) ( )
D
k
k
limkD
mn
n
ϕϕ
=








−
−
−≤
+
∞→
1
1
1

; BAx
∪
∈
∀
(2.10)
(
)
0≠≥ DD
ϕ
and if
(
)
0== DD
ϕ
then the
(
)
0
1
=∃
+++
∞→
xT,xTdlim
mnmn
n
; BAx
∪
∈
∀
. Hence,

Property (i) follows from (2.9) and (2.10) since
(
)
DD ≥
ϕ
and
(
)
DTxxd ≥, ; BAx
∪
∈
∀
, since
BABAT
∪
→
∪
: is a 2-cyclic self-mapping and
++
→
00
: RR
ϕ
is non-decreasing. Now, it
follows from triangle inequality for distances and (2.9a) that:

(
)
(
)

( ) ( )
(
)






−+






≤≤
∑∑∑
−
=
−
=
−
=
+
1
1
1
1
1

1
1
1,,
n
i
i
n
i
i
n
i
iin
kDTx,xdkxTxTdxxTd
ϕ

(
)
( ) ( ) ( )
(
)
in
i
n
kDTx,xd
k
kk
∑
−
=
−

−+
−
−
≤
1
1
1
1
1
1
ϕ
(
)
( )
(
)
(
)
(
)
( )
D
k
kk
Tx,xd
k
kk
nn
ϕ
11

111
1
1
−−
−−−
+
−
−
≤

( ) ( )
∞<
−
+
−
≤ D
k
k
Tx,xd
k
k
ϕ
1
1
, BAx
∪
∈
∀
, N
∈

∀
n (2.11)
which leads directly to Property (ii) with
{
}
0
N∈n
n
xT
and
{
}
0
1
N∈
+
n
n
xT
being bounded
sequences for any finite BAx
∪
∈
. □

Concerning the case that A and B intersect, we have the following existence and
uniqueness result of fixed points:

Theorem 2.2. If
(

)
0== DD
ϕ
(i.e., ∅≠∩
00
BA ) then
(
)
0
1
=∃
+++
∞→
xT,xTdlim
mnmn
n
and
(
)
(
)
k
Tx,xdk
x,xTd
n
−
≤
1
; BAx
∪

∈
∀
. Furthermore, if
(
)
d,X is complete and A and B are non-
empty closed and convex then there is a unique fixed point
B
A
z
∩
∈
of BABAT
∪
→
∪
:
8
to which all the sequences
{
}
0
N∈n
n
xT
, which are Cauchy sequences, converge;
BAx
∪
∈
∀

.

Proof: It follows from Lemma 2.1(i)–(ii) for
(
)
0== DD
ϕ
. It also follows that
(
)
(
)
(
)
0,,
121
==
++
∞→
+++
∞→
xTxTdklimxTxTdlim
mmn
n
mnmn
n
; BAx
∪
∈
∀

,
0
N∈∀ m what implies
(
)
0,
1
=
+++
∞→
xTxTdlim
mnmn
m,n
so that
{
}
0
N∈n
n
xT
is a Cauchy sequence, BAx
∪
∈
∀
, then being
bounded and also convergent in BA
∩
as
∞
→

n
since
(
)
d,X is complete and A and B are
non-empty, closed, and convex. Thus,
BAzxTlim
n
n
∩∈=
∞→
and
TzxTlimTxTlimz
n
n
n
n
=






==
+
∞→
+
∞→
11

, since the iterate composed self-mapping
BABAT
n
∪→∪: ,
0
N∈∀n is continuous for any initial point BAx
∪
∈
(since it is
contractive, then Lipschitz continuous in view of (2.9a) with associate Lipschitz constant
10 <≤ k
for
(
)
0== DD
ϕ
). Thus, BAz
∩
∈
is a fixed point of BABAT
∪
→
∪
: . Its
uniqueness is proven by contradiction. Assume that there are two distinct fixed points z
and y of BABAT
∪
→
∪
: in BA

∩
. Then, one gets from (2.1) that either
(
)
(
)
(
)
(
)
(
)
(
)
(
)
y,zdy,zdy,zdy,zdTy,Tzd <≤−≤<
βϕβ
0 or
(
)
(
)
0== y,zdTy,Tzd what contradicts
(
)
0>y,zd since
y
z
≠

. Then,
(
)
(
)
(
)
(
)
y,zdy,xdy,xdTy,Tzd <≤≤
ββ
what leads to the
contradiction
(
)
(
)
00 >==
∞→
y,zdyT,zTdlim
nn
n
. Thus,
y
z
=
. Hence, the theorem. □

Now, the contractive condition (2.1) is modified as follows:

( )
(
)
(
)
( )
(
)
(
)
( )
( ) ( )( )( ) ( )
Dy,xdy,xd
y,xd
Ty,ydTx,xd
y,xd
Ty,ydTx,xd
Ty,Txd
ϕϕβϕα
+−+















−≤
00
(2.12)
9
for
(
)
Xxy,x ∈≠ , where
(
)
0
00
≥
βα
,min ,
(
)
0
00
>
βα
,min , and 1
00
≤+
βα
. Note that in the

former contractive condition (2.1), 1
<
+
β
α
. Thus, for any non-negative real constants
0
αα
≤ and
0
ββ
≤ , (2.12) can be rewritten as

( )
(
)
(
)
( )
(
)
(
)
( )
( ) ( )( )( ) ( )
Dy,xdy,xd
y,xd
Ty,ydTx,xd
y,xd
Ty,ydTx,xd

Ty,Txd
ϕϕβϕα
+−+














−≤
( )
(
)
(
)
( )
(
)
(
)
( )
( ) ( ) ( )( )( )

y,xdy,xd
y,xd
Ty,ydTx,xd
y,xd
Ty,ydTx,xd
ϕββϕαα
−−+














−−+
00
; BAy,x
∪
∈
∀
. (2.13)
The following two results extend Lemma 2.1 and Theorem 2.2 by using constants
0

α
and
0
β
in (2.1) whose sum can equalize unity 1
00
=+
βα
.

Lemma 2.3. Assume that BABAT
∪
→
∪
: is a cyclic self-map satisfying the contractive
condition (2.13) with
(
)
0
00
≥
βα
,min , 1
00
≤+
βα
, and
++
→
00

: RR
ϕ
is non-decreasing.
Assume also that
( )( ) ( )
0
1
1
Mx,Txdx,Txd
βα
α
ϕ
−−
−
−≥
; BAx
∪
∈
∀
(2.14)
For some non-negative real constants
DM
α
β
α
−
−
−
≤
1

1
0
,
0
αα
≤ and
0
ββ
≤ with
1
<
+
β
α
.
Then, the following properties hold:

(i)
(
)
(
)
(
)
DDxT,xTdsuplimD
mnmn
n
βαβαϕ
−−++≤≤
+++

∞→
00
1
; BAx
∪
∈
∀
,
0
N∈∀m (2.15)
for any arbitrarily small
+
∈R
ε
.
10
(ii) If
(
)
(
)
DD
00
1
βαβαϕ
−−++= then
(
)
DxT,xTdlim
mnmn

n
=∃
+++
∞→
1
; BAx
∪
∈
∀
,
0
N∈∀m .
(iii) If
(
)
Txxd , is finite and, in particular, if x and Tx are finite then the sequence
{
}
0
N∈n
n
xT
and
{
}
0
1
N∈
+
n

n
xT
are bounded sequences, where AxT
n
∈ and BxT
n
∈
+1
if Ax
∈

and n is even and BxT
n
∈ and BxT
n
∈
+1
if Bx
∈
and n is even.

Proof: Since
++
→
00
: RR
ϕ
is non-decreasing then
(
)

(
)
Dx
ϕϕ
≥ for
(
)
Dx ≥∈
+0
R . Note also
that
DM
α
β
α
−
−
−
≤
1
1
0
implies the necessary condition
(
)
(
)
0≥x,Txd
ϕ
and (2.14) implies that

(
)
DD ≤≤
ϕ
0 . Note also for Txy
=
and
2
TxTy = and (2.14), since
(
)
(
)
Dx
ϕϕ
> for Dx
>
, that
for BAx
∪
∈
, one gets from (2.14):
(
)
(
)
(
)
0
22

1
1
,, MTxxTdTxxTd
βα
α
ϕ
−−
−
−≥
; BAx
∪
∈
∀
(2.16)
leading from (2.14) to

( )
(
)
(
)
(
)
[
]
( ) ( ) ( )( )( ) ( )
0000
22
0
1

1
: MMx,Txdx,TxdTx,xTdTx,xTd
βα
α
βαβαϕββϕαα
−−
−
−−+=≤−−+−−
(2.17)
and
(
)
DM
βαβα
−−+≤
00
since
DM
α
β
α
−
−
−
≤
1
1
0
. One gets from (2.13) and (2.17) the
following modifications of (2.9) and (2.10) by taking Txy

=
, xTTy
2
= , and successive
iterates by composition of the self-mapping BABAT
∪
→
∪
: :
( )
( ) ( )( )( ) ( )
( )
( )( )
MDkx,TxdkkkMDx,TxdkxT,xTdD
nn
n
i
innn
+−+≤








−++≤≤
∑
−

=
+
ϕϕ
11
1
0
1

(
)
(
)
MDx,Txkd ++≤
ϕ
; BAx
∪
∈
∀
,
{
}
0:
0
∪=∈∀ NNn (2.18)
11
(
)
(
)
(

)
(
)
DDMDxTxTdsuplimD
nn
n
βαβαϕϕ
−−++≤+≤≤
+
∞→
00
1
,
; BAx
∪
∈
∀
,
0
N∈∀m
(2.19)
(
)
xTxTdsuplimD
mnmn
n
+++
∞→
≤ ,
1

( ) ( ) ( )( )( )
















−−−+++≤
∑
−+
=
+
∞→
1
0
00
1
mn
i

imn
n
kkDDx,Txdklim
βαβαϕ

(
)
(
)
DD
βαβαϕ
−−++≤
00
; BAx
∪
∈
∀
,
0
N∈∀m (2.20)
and Property (i) has been proven. Property (ii) follows from (2.20) directly by replacing
(
)
(
)
DD
00
1
βαβαϕ
−−++= in (2.15). To prove Property (iii), note from (2.18) that

(
)
(
)
( ) ( )( )
(
)






−++






≤≤
∑∑∑
−
=
−
=
−
=
+

1
1
1
1
1
1
1
1,,
n
i
i
n
i
i
n
i
iin
kMDTx,xdkxTxTdxxTd
ϕ

(
)
( ) ( )( ) ( )
(
)
i
n
i
n
kMDTx,xd

k
kk
∑
−
=
−
−++
−
−
≤
1
1
1
1
1
1
ϕ

(
)
( )
(
)
(
)
(
)
( )( )
MD
k

kk
Tx,xd
k
kk
nn
+
−−−
+
−
−
≤
−−
ϕ
11
111
1
1

( ) ( )( )
∞<+
−
+
−
≤ MD
k
k
Tx,xd
k
k
ϕ

1
1
; BAx
∪
∈
∀
, N
∈
∀
n .
Hence,
{
}
0
N∈n
n
xT
and
{
}
0
1
N∈
+
n
n
xT
are bounded for any finite BAx
∪
∈

. Property (iii) has
been proven. Hence, the lemma. □

Theorem 2.4. If
(
)
0== DD
ϕ
then
(
)
0
1
=∃
+++
∞→
xT,xTdlim
mnmn
n
; BAx
∪
∈
∀
. Furthermore, if
(
)
d,X is complete and both A and B are non-empty, closed, and convex then there is a
12
unique fixed point
B

A
z
∩
∈
of BABAT
∪
→
∪
: to which all the sequences
{
}
0
N∈n
n
xT ,
which are Cauchy sequences, converge; BAx
∪
∈
∀
.

Proof guideline: It is identical to that of Theorem 2.2 by using
(
)
0
0
==== MMDD
ϕ
and
the fact that from (2.17)

αα
=
0
and
ββ
=
0
with 10
<
+
≤
β
α
if there is a pair
(
)
ABBATx,x ×∪×∈ such that
(
)
(
)
(
)
x,Txdx,Txd
ϕ
= ;
(
)
(
)

(
)
Tx,xTdTx,xTd
22
ϕ
= ; BAx
∪
∈
∀
.
Hence, the theorem. □

Remark 2.5. Note that Lemma 2.2 (ii) for
(
)
DD ≤
ϕ
(
(
)
DD <
ϕ
if 1
00
≤+≤+
βαβα
) leads to
an identical result as Lemma 2.1 (i) for
(
)

DD =
ϕ
and
1
<
+
β
α
consisting in proving that
(
)
DxT,xTdlim
nn
n
=∃
+
∞→
1
. This result is similar to a parallel obtained for standard 2-cyclic
contractions [2, 5, 8]. □

Remark 2.6. Note from (2.7) that Lemma 2.1 is subject to the necessary condition
(
)
DD
ϕ
≤ since
(
)
DTx,xTd ≥

2
and
(
)
Dx,xTd ≥ ; BAx
∪
∈
∀
. On the other hand, note from
Lemma 2.2, Equation (2.14) that
( )
0
1
1
MDD
βα
α
ϕ
−−
−
−≥
, and one also gets from (2.18) for n
= 1 the dominant lower-bound
( ) ( )
βαβα
βα
α
ϕ
−−+
−−

−
−≥−≥
000
1
1
MDMDD
, that is,
( ) ( )
βαβα
βα
α
ϕ
−−+
−−
−
+≤
000
1
1
MDD
which coincides with the parallel constraint obtained
from Lemma 2.1 if
βαβα
+=+
00
. □

13
Remark 2.7. Note that Lemmas 2.2 and 2.3 apply for non-decreasing functions
++

→
00
: RR
ϕ
. The case of
++
→
00
: RR
ϕ
being monotone increasing, then unbounded, is
also included as it is the case of
++
→
00
: RR
ϕ
being bounded non-decreasing.
□

Now, modify the modified cyclic φ-contractive constraint (2.1) as follows:

( )
(
)
(
)
( )
( ) ( )( )( )
y,xdy,xd

y,xd
Ty,ydTx,xd
Ty,Txd
ϕβα
−+≤

( )
(
)
(
)
( )
( )
D
y,xd
Ty,ydTx,xd
ϕϕα
+








−+ 1 ; BAx
∪
∈
∀

(2.21)
Thus, the following parallel result to Lemmas 2.1 and 2.2 result holds under a more
restrictive modified weak φ-contraction Assume that BABAT
∪
→
∪
: is modified weak φ-
contraction subject to
++
→
00
: RR
ϕ
subject to the constraint
( )( )
(
)
βα
ϕ
ϕ
−−
>−
+∞→
1
D
xxsuplim
x
and
having a finite limit:

Lemma 2.8. Assume that BABAT
∪
→
∪
: is a cyclic self-map satisfying the contractive
condition (2.21) with
(
)
0≥
βα
,min ,
1
<
+
β
α
, and
++
→
00
: RR
ϕ
is non-decreasing having a
finite limit
(
)
ϕϕ
=
∞→

xlim
x
and subject to
(
)
00 =
ϕ
. Assume also that
++
→
00
: RR
ϕ
satisfies
( )( )
(
)
βα
ϕ
ϕ
−−
>−
+∞→
1
D
xxsuplim
x
. Then, the following properties hold:

(i) The following relations are fulfilled:

14
( )
(
)
(
)
∞<
−−
−
−
≤+
−−
≤≤≤
−−
−
−
+
ϕ
βα
β
α
ϕ
βα
ϕ
ϕ
βα
β
α
1
2

12
1
1
D
xT,xTdDD
nn
; N
∈
∀
n , BAx
∪
∈
∀
(2.22)
( )
(
)
(
)
∞<
−−
−
−
≤+
−−
≤≤≤
−−
−
−
+

→∞
ϕ
βα
β
α
ϕ
βα
ϕ
ϕ
βα
β
α
1
2
12
1
1
D
xT,xTdsuplimDD
nn
n
; BAx
∪
∈
∀
(2.23)

(ii) If, furthermore,
++
→

00
: RR
ϕ
is, in addition, sub-additive and
(
)
Txxd , is finite (in
particular, if
x
and Tx are finite) then the sequences
{
}
0
N∈n
n
xT
and
{
}
0
1
N∈
+
n
n
xT
are both
bounded, where AxT
n
∈ and BxT

n
∈
+1
if Ax
∈
and n is even and BxT
n
∈ and AxT
n
∈
+1
if
Bx
∈
and n is even. If
++
→
00
: RR
ϕ
is identically zero then
(
)
0
1
=∃
+++
∞→
xT,xTdlim
mnmn

n
;
BAx
∪
∈
∀
.
Proof: One gets directly from (2.21):
( )
(
)
(
)
(
)
(
)
(
)
(
)
(
)
( )
DTx,xTdTx,xTdTx,xTdTx,xTd
ϕϕβϕα
+−≤−−
2222
1
; BAx

∪
∈
∀

(2.24)
or, equivalently, one gets for
1
1
<
−
=
α
β
k
that
(
)
(
)
(
)
(
)
(
)
(
)
(
)
α

ϕ
ϕϕ
−
+−≤−
1
2222
D
Tx,xTdTx,xTdkTx,xTdTx,xTd
; BAx
∪
∈
∀
(2.25)
leading to
(
)
(
)
(
)
(
)
(
)
xT,xTdxT,xTdinflimDD
nnnn
n
11
0
++

∞→
−≤−≤
ϕϕ

(
)
(
)
(
)
(
)
(
)
( )( )
(
)
βα
ϕ
α
ϕ
ϕ
−−
=
−−
≤−≤
++
∞→
1
11

11
D
k
D
xT,xTdxT,xTdsuplim
nnnn
n
; BAx
∪
∈
∀
(2.26)
15
what implies the necessary condition
( )
DD
βα
β
α
ϕ
−−
−
−
≥
2
1
leading to
( )
1
1

2
>
−−
−
−
=
βα
β
α
ϕ
D
D
if
0
≠
D and then
(
)
(
)
(
)
(
)
(
)
0
11
≥−≥−
++

∞→
DDxT,xTdxT,xTdinflim
nnnn
n
ϕϕ
; BAx
∪
∈
∀
. Also, since
( )( )
(
)
βα
ϕ
ϕ
−−
>−
+∞→
1
D
xxsuplim
x
;
+
∈∀ Rx , by construction, then
(
)
xT,xTd
nn 1+

is bounded; N
∈
∀
n
since, otherwise, a contradiction to (2.24) holds. Since
++
→
00
: RR
ϕ
is non-decreasing
and has a finite limit
(
)
0≥≥ x
ϕϕ
;
+
∈∀
0
Rx ( 0
=
ϕ
if and only if
++
→
00
: RR
ϕ
is identically

zero), thus
(
)
0≥≥ D
ϕϕ
. Then, (2.22)–(2.23) hold and Property (i) has been proven. On
the other hand, one gets from (2.25), since
++
→
00
: RR
ϕ
is sub-additive and
nondecreasing and has a finite limit, that:

(
)
(
)
(
)
(
)
(
)
(
)







−≤−
+
−
=
+
∑
xT,xTdxTxTdx,xTdxxTd
ii
n
i
iinn 1
1
1
1
,,
ϕϕ

( ) ( )( )( )
(
)
(
)







−
−
+−






≤
∑∑
−
=
−
=
1
1
1
1
1
1
n
i
i
n
i
i
k
D

Tx,xdTx,xdk
α
ϕ
ϕ

(
)
( ) ( )( )( )
(
)
( )
i
n
i
n
k
D
Tx,xdTx,xd
k
kk






−
−
+−
−

−
≤
∑
−
=
−
1
1
1
1
11
1
α
ϕ
ϕ

(
)
( )
(
)
(
)
(
)
(
)
α
ϕ
−

−−−
+
−
−
≤
−−
1
111
1
1
11
D
k
kk
Tx,xd
k
kk
nn

( ) ( )( )( )
(
)
∞<
−
−
+−
−
≤
α
ϕ

ϕ
1
1
1
D
k
k
Tx,xdTx,xd
k
k
; BAx
∪
∈
∀
, N
∈
∀
n (2.27)
(
)
( ) ( )( )( )
(
)
∞<+
−
−
+−
−
≤
∞→

ϕ
α
ϕ
ϕ
1
1
1
,
D
k
k
Tx,xdTx,xd
k
k
xxTdsuplim
n
n
; BAx
∪
∈
∀
(2.28)

16
Then the sequences
{
}
0
N∈n
n

xT
and
{
}
0
1
N∈
+
n
n
xT
are both bounded for any BAx
∪
∈
.
Hence, the first part of Property (ii). If
++
→
00
: RR
ϕ
is identically zero then
(
)
0=≡ Tx,x
ϕϕ
; BAx
∪
∈
∀

so that
(
)
0
1
=∃
+++
∞→
xT,xTdlim
mnmn
n
from (2.23). Hence, the
lemma. □

The existence and uniqueness of a fixed point in
B
A
∩
if A and B are non-empty, closed,
and convex and
(
)
d,X is complete follows in the subsequent result as its counterpart in
Theorem 2.2 modified cyclic φ-contractive constraint (2.21):

Theorem 2.9. if
(
)
d,X is complete and A and B intersect and are non-empty, closed, and
convex then there is a unique fixed point

B
A
z
∩
∈
of BABAT
∪
→
∪
: to which all the
sequences
{
}
0
N∈n
n
xT , which are Cauchy sequences, converge; BAx
∪
∈
∀
. □

Remark 2.7. Note that the nondecreasing function
++
→
00
: RR
ϕ
of the contractive
condition (2.21) is not monotone increasing under Lemma 2.5 since it possesses a finite

limit and it is then bounded. □

Remark 2.8. The case of BABAT
∪
→
∪
: being a φ-contraction, namely,
(
)
(
)
(
)
(
)
(
)
Dy,xdy,xdTy,Txd
ϕϕ
+−≤ with strictly increasing
++
→
00
: RR
ϕ
; BAx
∪
∈
∀
, [1, 2]

implies , since
(
)
0=x
ϕ
if and only if 0
=
x , implies the relation
(
)
(
)
(
)
(
)
(
)
Dy,xdDy,xdTy,Txd
ϕϕβ
+<+≤
1
;
(
)
BAxy,x ∪∈≠∀ (2.29)
17
for some real constant
(
)

10
11
<=≤ y,x
ββ
;
(
)
BAxy,x ∪∈≠∀ so that proceeding recursively:
(
)
[
]
(
)
(
)
[
]
(
)
(
)
(
)
DLx,TxdDx,TxdxT,xTd
n
j
n
j
n

i
i
nn
ϕβϕβ
+≤+≤
∑ ∏∏
=
+==
+
1
11
1
l
l
, BAx
∪
∈
∀

(2.30)
(
)
(
)
β
ϕ
γ
−
≤≤
+

∞→
1
1
D
xT,xTdsuplimD
nn
n
; BAx
∪
∈
∀
(2.31)
where
[ ]
( )
1:
1
1
<=
∏
=
∞→
n
n
i
i
n
lim
ββ
and

(
)
0
1
=∃
+
∞→
xT,xTdlim
nn
n
; BAx
∪
∈
∀
if
(
)
0== DD
ϕ
, and
one gets from Lemma 2.1(iii) that
{
}
0
N∈n
n
xT
and
{
}

0
1
N∈
+
n
n
xT are Cauchy sequences
which converge to a unique fixed point in BA
∩
if A and B are non-empty, closed, and
convex and
(
)
d,X is complete [1]. □

Remark 2.9. Note that the constraint (2.1) implies in Lemma 2.1 and Theorem 2.2 that
(
)
(
)
(
)
DD
βαϕβα
−−≤−− 11 what implies
(
)
DD ≤
ϕ
if

(
)
0
>
β
α
,max
since 10
<
+
≤
β
α
.
However, such a constraint in Lemma 2.3 and Theorem 3.4 implies that
(
)
(
)
(
)
DD
0000
11
βαϕβα
−−≤−− . □

3. Properties for the case that A and B do not intersect
This section considers the contractive conditions (2.1) and (2.21) for the case
∅

=
∩
BA .
For such a case, Lemmas 2.1, 2.3, and 2.8 still hold. However, Theorems 2.2, 2.4, and 2.9
do not further hold since fixed points in BA
∩
cannot exist. Thus, the investigation is
centred in the existence of best proximity points. It has been proven in [1] that if
BABAT
∪
→
∪
: is a cyclic φ-contraction with A and B being weakly closed subsets of a
reflexive Banach space
(
)
,X
then,
(
)
BAy,x ×∈∃ such that
(
)
yxy,xdD −==
where
18
++
→
00
RR:d is a norm-induced metric, i.e., x and y are best proximity points. Also, if

BABAT
∪
→
∪
: is a cyclic contraction
(
)
BAy,x ×∈∃ such that
(
)
y,xdD = if A is compact
and B is approximatively compact with respect to A with both A and B being subsets of a
metric space
(
)
dX , (i.e., if
(
)
(
)
(
)
y,zdinfy,BdyxTdlim
Bz
n
n
∈
∞→
== :,
2

for some Ay
∈
and Bx
∈

then the sequence
{
}
0
2
N∈n
n
xT
has a convergent subsequence [14]). Theorem 2.2 extends
via Lemma 2.1 as follows for the case when A and B do not intersect, in general:

Theorem 3.1. Assume that BABAT
∪
→
∪
: is a modified weak φ-contraction, that is, a
cyclic self-map satisfying the contractive condition (2.1) subject to the constraints
(
)
0≥
βα
,min and 1
<
+
β

α
with
++
→
00
: RR
ϕ
being nondecreasing with
(
)
DD =
ϕ
. Assume
also that A and B are non-empty closed and convex subsets of a uniformly convex Banach
space
(
)
,X
. Then, there exist two unique best proximity points
A
z
∈
, By
∈
of
BABAT
∪
→
∪
: such that yTz

=
, zTy
=
to which all the sequences generated by
iterations of BABAT
∪
→
∪
: converge for any BAx
∪
∈
as follows. The sequences
{
}
0
2
N∈n
n
xT and
{
}
0
12
N∈
+
n
n
xT converge to z and y for all Ax
∈
, respectively, to y and z for

all Bx
∈
. If
∅
≠
∩
BA then BAyz
∩
∈
=
is the unique fixed point of BABAT
∪
→
∪
: .

Proof: If 0
=
D , i.e., A and B intersect then this result reduces to Theorem 2.2 with the
best proximity points being coincident and equal to the unique fixed point. Consider the
case that A and B do not intersect, that is, 0
>
D and take BAx
∪
∈
. Assume with no loss
in generality that Ax
∈
. It follows, since A and B are non-empty and closed, A is convex
and Lemma 3.1 (i) that:

19

(
)
(
)
[
]
( )
(
)
0;
222212212
→⇒→→
++++
xT,xTdDxT,xTdDxT,xTd
npnnnnn
as
∞
→
n
(3.1)
(proven in Lemma 3.8 [14]). The same conclusion arises if Bx
∈
since B is convex. Thus,
{
}
0
2
N∈n

n
xT
is bounded [Lemma 2.1 (ii)] and converges to some point
(
)
xzz = , being
potentially dependently on the initial point x, which is in A if Ax
∈
, since A is closed, and
in B if Bx
∈
since B is closed. Take with no loss in generality the norm-induced metric
and consider the associate metric space
(
)
d,X which can be identified with
(
)
,X
in this
context. It is now proven by contradiction that for every
+
∈R
ε
, there exists
00
N∈n such
that
(
)

ε
+≤
+
DxTxTd
nm 122
, for all
0
nnm
≥
>
. Assume the contrary, that is, given some
+
∈R
ε
, there exists
00
N
∈
n
such that
(
)
ε
+>
+
DxT,xTd
kk
nm 122
for all
0

nnm
kk
≥>
0
N∈∀k .
Then, by using the triangle inequality for distances:

(
)
(
)
(
)
xTxTdxTxTdxTxTdD
kkkkkk
nmmmnm 1222222122
,,,
++++
+≤<+
ε
as
∞
→
n
(3.2)

One gets from (3.1) and (3.2) that
(
)
(

)
(
)
(
)
ε
+>=+
++
∞→
+++
∞→
DxTxTdinflimxTxTdxTxTdinflim
kkkkkk
nm
k
nmmm
k
12221222222
,,,
(3.3)

Now, one gets from (3.1), (3.3),
(
)
DD ≥
ϕ
, and Lemma 2.1 (i) the following contradiction:
(
)
(

)
(
)
xTxTdsuplimxTxTdsuplimxTxTdsuplimD
kkkk
k
kk
nm
k
nn
n
nm
k
222212221222
,,,
++
∞→
++
∞→
++
∞→
+≤<+
ε

(
)
DxTxTdsuplim
kk
k
nn

n
==
++
∞→
1222
,
(3.4)
20
As a result,
(
)
ε
+≤
+
DxTxTd
nm 122
, for every given
+
∈R
ε
and all
0
nnm
≥
>
for some
existing
00
N∈n . This leads by a choice of arbitrarily small
ε

to
(
)
(
)
DxTxTdlimDxTxTdsuplimD
nm
n
nm
n
=∃⇒≤≤
+
∞→
+
∞→
122122
,,
(3.5)
But
{
}
0
2
N∈n
n
xT
is a Cauchy sequence with a limit
z
T
z

2
=
in A (respectively, with a limit
yTy
2
= in B) if Ax
∈
(respectively, if Bx
∈
) such that
(
)
Tz,zdzTzD =−=
(Proposition
3.2 [14]). Assume on the contrary that Ax
∈
and
{
}
zTzxT
n
n 22
0
≠→
∈N
as
∞
→
n
so that

yzTzzTzzT −≠−=−
2
so that since A is convex and
(
)
,X
is uniformly convex Banach
space, then strictly convex, one has
( )
222
22
TzzTzzT
Tz
zzT
dTz,zdD
−
+
−
=








−
+
== D

DDTzzTzzT
=+<
−
+
−
≤
2222
2
(3.6)
which is a contradiction so that
z
T
z
2
=
is a best approximation point in A of
BABAT
∪
→
∪
: . In the same way,
{
}
0
2
N∈n
n
xT
is a Cauchy sequence with a limit
ByyT ∈=

2
which is a best approximation point in B of BABAT
∪
→
∪
: if Bx
∈
since B is
convex and
(
)
,X
is strictly convex. We prove now that Tzy
=
. Assume, on the contrary
that Tzy
≠
with BzTTz,yTy ∈==
32
, AzTz ∈=
2
,
(
)
Dy,zd > ,
(
)
DTy,Tzd ≥ ,
(
)

(
)
Dy,Tydz,Tzd == , and
(
)
DD =
ϕ
. One gets from (2.1) since
++
→
00
: RR
ϕ
is non-
decreasing the following contradiction:
( )
( )
(
)
(
)
( )
(
)
(
)
( )
( ) ( )( )( )
DTy,TzdTy,Tzd
Ty,Tzd

Ty,yTdTz,zTd
Ty,Tzd
Ty,yTdTz,zTd
yT,zTdy,zdD +−+
















−≤=<
ϕβϕα
2222
22

(
)
(
)
DDD

=
−
−
+
+
β
α
β
α
1
(3.7)
21
Thus,
yTzTTyz
32
=== and zTyTTzy
32
=== are the best proximity points of
BABAT
∪
→
∪
: in A and B. Finally, we prove that the best proximity points
A
z
∈
and
By
∈
are unique. Assume that

(
)
Azz ∈≠
21
are two distinct best proximity points of
BABAT
∪
→
∪
: in A. Thus,
(
)
BTzTz ∈≠
21
are two distinct best proximity points in B.
Otherwise,
212
2
1
2
21
zzzTzTTzTz =⇒=⇒=
, since
1
z and
1
z are best proximity points,
contradicts
21
zz ≠ . One gets from Lemma 2.1(i) and

(
)
(
)
(
)
(
)
DTz,zdTz,zdzT,TzdzT,Tzd ====
12211
2
22
2
1
. Through a similar argument to that
concluding with (3.6) with the convexity of A and the strict convexity of
(
)
,X
,
guaranteed by its uniform convexity, one gets the contradiction:
(
)
21
2
Tz,zTdD =
D
DDTzzTzzT
=+<
−

+
−
≤
2222
2211
2
(3.8)
since
1111
2
zTzTzzT −≠− . Thus, z
1
is the unique best proximity point in A while
1
Tz is the
unique best proximity point in
B
. □

In a similar way, Theorem 2.4 extends via Lemma 2.3 as follows from the modification
(2.12) of the contractive condition (2.1):

Theorem 3.2. Assume the following hypotheses:
(1) BABAT
∪
→
∪
: is a modified weak ϕ-contraction, that is, a cyclic self-map
satisfying the contractive condition (2.12) subject to the constraints
(

)
0
00
≥
βα
,min ,
(
)
0
00
>
βα
,min , and 1
00
≤+
βα
.
22
(2)
++
→
00
: RR
ϕ
is non-decreasing subject to
( )( ) ( )
0
1
1
Mx,Txdx,Txd

βα
α
ϕ
−−
−
−≥
;
BAx
∪
∈
∀
and
(
)
(
)
DD
00
1
βαβαϕ
−−++= for some non-negative real constants
DM
α
β
α
−
−
−
≤
1

1
0
,
0
0
αα
≤≤ and
0
0
ββ
≤≤ with
1
<
+
β
α
.
(3) A and B are non-empty closed and convex subsets of a uniformly convex Banach
space
(
)
,X
.

Then, there exist two unique best proximity points
A
z
∈
, By
∈

of BABAT
∪
→
∪
: such
that yTz
=
, zTy
=
to which all the sequences generated by iterations of BABAT
∪
→
∪
:
converge for any BAx
∪
∈
as follows. The sequences
{
}
0
2
N∈n
n
xT and
{
}
0
12
N∈

+
n
n
xT
converge to z and y for all Ax
∈
, respectively, to y and z for all Bx
∈
. If
∅
≠
∩
BA then
BAyz
∩
∈
=
is the unique fixed point of BABAT
∪
→
∪
: .

Outline of proof: It is similar to that of Theorem 3.1 since (3.1) to (3.3) still hold, (3.4)
and (3.5) still hold as well from Lemma 2.3(ii) as well as the results from the
contradictions (3.6)–(3.8). □

The following result may be proven using identical arguments to those used in the proof
of Theorem 3.1 by using Lemma 2.8 starting with its proven convergence property (2.23)
for distances:

23
Theorem 3.3. Assume that
BABAT
∪
→
∪
: is a cyclic self-map satisfying the contractive
condition (2.21) with
(
)
0≥
βα
,min ,
1
<
+
β
α
, and
++
→
00
: RR
ϕ
is non-decreasing having a
finite limit
(
)
ϕϕ

=
∞→
xlim
x
and subject to
(
)
00 =
ϕ
. Assume also that
++
→
00
: RR
ϕ
satisfies
( )( )
(
)
βα
ϕ
ϕ
−−
>−
+∞→
1
D
xxsuplim
x
. Finally, assume that A and B are non-empty closed and convex

subsets of a uniformly convex Banach space
(
)
,X
. Then, there exist two unique best
proximity points
A
z
∈
, By
∈
of BABAT
∪
→
∪
: such that yTz
=
, zTy
=
to which all the
sequences generated by iterations of BABAT
∪
→
∪
: converge for any BAx
∪
∈
as
follows. The sequences
{

}
0
2
N∈n
n
xT and
{
}
0
12
N∈
+
n
n
xT converge to z and y for all Ax
∈
,
respectively, to y and z for all Bx
∈
. If
∅
≠
∩
BA then BAyz
∩
∈
=
is the unique fixed
point of BABAT
∪

→
∪
: . □

Example 3.4. The first contractive condition (2.1) is equivalent to
(
)
( ) ( )
(
)
(
)
( )( )
(
)
Tx,xdxT,TxdDTx,xdxT,Txd
ϕβϕαϕβ
α
−−+
−
≤
22
1
1
. (3.9)
To fix ideas, we first consider the trivial particular case
(
)
0≡x
ϕ

(
)
(
)
0=⇒ D
ϕ
;
+
∈∀
0
Rx .
This figures out that BABAT
∪
→
∪
: is a strict contraction if BA
∩
is non-empty and
closed,
(
)
0≥
βα
,min , and 1
<
+
β
α
. Then, it is known from the contraction principle that
there is a unique fixed point in BA

∩
. Note that in this case 0:
0
→
+
R
ϕ
. If 1
=
+
β
α
then
BABAT
∪
→
∪
: is non-expansive fulfilling
(
)
(
)
TxxdxTxTd
pp
,,
1
=
+
; BAx
∪

∈
∀
,
+
∈∀
0
Zp .
The convergence to fixed points cannot be proven. It is of interest to see if
BABAT
∪
→
∪
: being a weak contraction with
++
→
00
: RR
ϕ
being non-decreasing
guarantees the convergence to a fixed point if 1
=
+
β
α
and
(
)
0D0 ==
ϕ
according to the

24
modified contractive condition (2.12). In this case, if
(
)
0>x
ϕ
;
+
∈∀ Rx then convergence
to a fixed point is still potentially achievable since
(
)
( )
(
)
(
)
( )( )
(
)
( )
Tx,xdTx,xdxT,TxdTx,xdxT,Txd <+
−
−≤
ϕβϕα
α
22
1
1
if Txx

≠
. (3.10)
Now, consider the discrete scalar dynamic difference equation of respective state and
control real sequences
{
}
+
∈
0
Zk
k
x and
{
}
+
∈
0
Zk
k
u and dynamics and control parametrical
real sequences
{
}
+
∈
0
Zk
k
a and
{

}
+
∈
≠
0
0
Zk
k
b , respectively:

kkkkkk
ubxax
η
+
+
=
+1
;
+
∈∀
0
Zk , R∈
0
x (3.11)
where
{
}
+
∈
0

Zk
k
x , of general term defined by
(
)
kk
x, ,x,xx
10
:
=
, is a sequence of real kth
tuples built with state values up till the kth sampled value such that the real sequence
{
}
+
∈
0
Zk
k
η
with
(
)
kkk
x
ηη
= is related to non-perfectly modeled effects which can include,
for instance, contributions of unmodeled dynamics (if the real order of the difference
equation is larger than one), parametrical errors (for instance, the sequences of
parameters are not exactly known), and external disturbances. It is assumed that upper-

and lower-bounding real sequences
{
}
+
∈
0
Zk
k
η
and
{
}
+
∈
0
0
Zk
k
η
are known which satisfy
(
)
(
)
kkkkkkk
xx
00
ηηηηη
=≥≥=
;

+
∈∀
0
Zk . Define a 2-cyclic self-mapping BB:
∪
→
∪
AAT
with
(
)
BAT ⊆ and
(
)
ABT ⊆ for some sets
{
}
0::
0
≥∈=⊂
+
zzA RR and
{
}
0::
0
≤∈=⊂
−
zzB RR
being non-empty bounded connected sets containing

{
}
0 , so that 0
=
D , such that
1+
=
kk
xxT ;
+
∈∀
0
Zk for the control sequence
{
}
+
∈
0
Zk
k
u lying in some appropriate class to
be specified later on. Note from (3.11) that
11112 +++++
+
+
=
kkkkkk
ubxax
η