A SURVEY ON THE FIXED POINT THEOREMS VIA
ADMISSIBLE MAPPING
Erdal Karapınar
Department of Medical Research, China Medical University Hospital, China Medical University, 40402, Taichung,
Taiwan
Department of Mathematics, Çankaya University, 06790, Etimesgut, Ankara, Turkey
E-mail:erdalkarapinar@yahoo.com, karapinar@mail.cmuh.org.tw
ORCID:0000-0002-6798-3254
Reception: 21/08/2022 Acceptance: 05/08/2022 Publication: 29/12/2022
Suggested citation:
Karapınar, E.(2022). A Survey on the Fixed Point Theorems via Admissible Mapping. 3C TIC. Cuadernos de desarrollo
aplicados a las TIC,11 (2), 26-50. https://doi.org/10.17993/3ctic.2022.112.26-50
https://doi.org/10.17993/3ctic.2022.112.26-50
ABSTRACT
In this survey, we discuss the crucial role of the notion of admissible mapping in the metric fixed point
theory. Adding admissibility conditions to the statements leads not only to generalizing the existing
results but also unifying several corresponding results in different settings. In particular, a contraction via
admissible mapping involves and covers contractions defined on partially ordered sets, and contractions
forming cyclic structure.
KEYWORDS
Admissible mapping, fixed point
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
26
A SURVEY ON THE FIXED POINT THEOREMS VIA
ADMISSIBLE MAPPING
Erdal Karapınar
Department of Medical Research, China Medical University Hospital, China Medical University, 40402, Taichung,
Taiwan
Department of Mathematics, Çankaya University, 06790, Etimesgut, Ankara, Turkey
E-mail:erdalkarapinar@yahoo.com, karapinar@mail.cmuh.org.tw
ORCID:0000-0002-6798-3254
Reception: 21/08/2022 Acceptance: 05/08/2022 Publication: 29/12/2022
Suggested citation:
Karapınar, E.(2022). A Survey on the Fixed Point Theorems via Admissible Mapping. 3C TIC. Cuadernos de desarrollo
aplicados a las TIC,11 (2), 26-50. https://doi.org/10.17993/3ctic.2022.112.26-50
https://doi.org/10.17993/3ctic.2022.112.26-50
ABSTRACT
In this survey, we discuss the crucial role of the notion of admissible mapping in the metric fixed point
theory. Adding admissibility conditions to the statements leads not only to generalizing the existing
results but also unifying several corresponding results in different settings. In particular, a contraction via
admissible mapping involves and covers contractions defined on partially ordered sets, and contractions
forming cyclic structure.
KEYWORDS
Admissible mapping, fixed point
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
27
1 INTRODUCTION
The metric fixed point theory is one of the most attractive research topics that lies in the intersection of
three disciplines of the mathematics; topology, applied mathematics and nonlinear functional analysis.
In the literature, it was assumed that the first metric fixed point theorem was proved by Banach [18] in
1922. On the other hand, it was noted that the idea of the fixed point theorem was used before Banach’s
paper. Indeed, the fixed point results were used to prove the existence and uniqueness solution of the
initial value problems in the some research papers of Liouville, Picard, Poincaré and so on. Despite this
fact, the sole purpose of Banach’s theorem is the indicate the existence and uniqueness of the fixed
point.
The Banach’s fixed point theorem stated perfectly and its proof is also awesome: Every contraction
in a complete metric space guarantee not only the existence but also uniqueness of a fixed point. In
the proof, Banach indicates how the desired fixed point should be found. Regarding that it is a core
tool in the solution of the certain differential equation, the existence and uniqueness of the solution of
the differential equations are induced to the existence and uniqueness of a fixed point. Indeed, this
observation can be the main reason why the metric fixed point theory has been investigated heavily.
In the last three decades, thousands of new results have been announced in the framework of the
metric fixed point theory. Most of them claimed that it was the generalization of the existing results.
Mainly, these announced results are slight generalizations or extensions of the existing ones. Indeed,
predominantly, most of the proofs are the mimic of the proof of the pioneer fixed point theorem of
Banach: Construct a sequence (usually, by using the Picard operator); indicate that this sequence is
convergent and finally prove that the limit of this recursive sequence is the required fixed point of
the considered operator. The fact that existing so many publications on this topic has encouraged
researchers to organize and unify this scattered literature. One of the best examples of this trend was
given by Samet et al. [61] in 2012 by involving the notion of admissible mappings.
Now, we shall explain why the new notion of Samet et al. [61], an admissible mapping, is important.
First of all, we need to underline that Banach’s famous fixed point theorem has been generalized and
extended in many different ways in the literature. The most classic approach for generalization and
extension is to change the definitions of contraction. The other approach and method are that a given
contraction function is in cyclic form. One of the other methods for generalization and extension is to
add a partial order on the structure where contraction is defined. Admissible mappings make it possible
to put together these three approaches in a single statement. For the clarification of the fixed point
theory literature, admissible mapping plays one of the key roles.
In this manuscript, we shall revisit the literature by using the auxiliary function: admissible
mapping. We shall provide examples and put several consequences to illustrate how this approach works
successfully.
2 Preliminaries
In this section, we collect some necessary tools (notions, notations) that are essential to express the
results.
First of all, we shall fixed some basic notations: Hereafter,
N
and
N0
denote the set of positive
integers and the set of nonnegative integers. Furthermore, the symbols
R
,
R+
and
R+
0
represent the set
of reals, positive reals and the set of nonnegative reals, respectively. Throughout the manuscript, all
considered sets are non-empty.
We shall first recall the admissible mapping defined by Samet et al. [61]. Let
α
:
X×X→
[0
,∞
)be
a function. Then, the mapping T:X→Xis said to be α-admissible [61] if, for all x, y ∈X,
α(x, y)≥1=⇒α(T x, T y)≥1.
Next, we recollect the notion of triangular
α
-admissible that plays crucial rules in usage of triangle
inequality axiom of the metric.
https://doi.org/10.17993/3ctic.2022.112.26-50
Definition 1. [41] A self-mapping T:X→Xis called triangular α-admissible if
(T1)Tis α−admissible,
(T2)α(x, z)≥1,α(z,y)≥1=⇒α(x, y)≥1, x, y, z ∈X.
Example 1. Suppose that M= (0,+∞). Define T:M→Mand α:M×M→[0,∞)by
(1) T(x)=ln(x+ 1), for all x∈Mand α(x, y)=1,if x≥y;
0,if x<y.
Then Tis α-admissible.
(2) T(x)= 3
√x, for all x∈Mand α(x, y)=ex−y,if x≥y;
0,if x<y.
Then Tis α-admissible.
For more examples of such mappings are presented in [30,40,41,46–48,61] and references therein.
In 2014, Popescu [54] conclude that the notion of triangular
α
-admissible can be refined slightly, as
follows:
Definition 2. [54] Let
T
:
X→X
be a self-mapping and
α
:
X×X→
[0
,∞
)be a function. Then
T
is said to be α-orbital admissible if
(T3) α(x, T x)≥1⇒α(T x, T 2x)≥1.
Definition 3. [54] Let
T
:
X→X
be a self-mapping and
α
:
X×X→
[0
,∞
)be a function. Then
T
is said to be triangular α-orbital admissible if T is α-orbital admissible and
(T4) α(x, y)≥1and α(y, T y)≥1⇒α(x, T y)≥1.
As it is expected, each
α
-admissible mapping is an
α
-orbital admissible mapping and each triangular
α
-admissible mapping is a triangular
α
-orbital admissible mapping. Notice also that the converse is
false, see e.g. ( [54] Example 7).
A metric space (
X, d
)is said
α
-regular [54], if for every sequence
{xn}
in
X
such that
α
(
xn,x
n+1
)
≥
1
for all
n
and
xn→x∈X
as
n→∞
, then there exists a subsequence
{xn(k)}
of
{xn}
such that
α(xn(k),x)≥1for all k.
Lemma 1. [54] Let
T
:
X→X
be a triangular
α
-orbital admissible mapping. Assume that there exists
x0∈X
such that
α
(
x0,Tx
0
)
≥
1. Define a sequence
{xn}
by
xn+1
=
Tx
n
for each
n∈N0
. Then we
have α(xn,x
m)≥1for all m, n ∈Nwith n<m.
In what follows, we recall the following auxiliary family of the functions, namely, comparison and
c
-comparison functions. A mapping
ψ
: [0
,∞
)
→
[0
,∞
)is called a comparison function if it is increasing
and
ψn
(
t
)
→
0,
n→∞
, for any
t∈
[0
,∞
). We denote by Φ, the class of the corporation function
ψ
: [0
,∞
)
→
[0
,∞
). For more details and examples , see e.g. [21, 58]. Among them, we recall the
following essential result.
Lemma 2. (Berinde [21], Rus [58]) If ψ: [0,∞)→[0,∞)is a comparison function, then:
(1) each iterate ψkof ψ,k≥1, is also a comparison function;
(2) ψis continuous at 0;
(3) ψ(t)<t, for any t>0.
Later, Berinde [21] introduced the concept of (c)-comparison function in the following way.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
28
1 INTRODUCTION
The metric fixed point theory is one of the most attractive research topics that lies in the intersection of
three disciplines of the mathematics; topology, applied mathematics and nonlinear functional analysis.
In the literature, it was assumed that the first metric fixed point theorem was proved by Banach [18] in
1922. On the other hand, it was noted that the idea of the fixed point theorem was used before Banach’s
paper. Indeed, the fixed point results were used to prove the existence and uniqueness solution of the
initial value problems in the some research papers of Liouville, Picard, Poincaré and so on. Despite this
fact, the sole purpose of Banach’s theorem is the indicate the existence and uniqueness of the fixed
point.
The Banach’s fixed point theorem stated perfectly and its proof is also awesome: Every contraction
in a complete metric space guarantee not only the existence but also uniqueness of a fixed point. In
the proof, Banach indicates how the desired fixed point should be found. Regarding that it is a core
tool in the solution of the certain differential equation, the existence and uniqueness of the solution of
the differential equations are induced to the existence and uniqueness of a fixed point. Indeed, this
observation can be the main reason why the metric fixed point theory has been investigated heavily.
In the last three decades, thousands of new results have been announced in the framework of the
metric fixed point theory. Most of them claimed that it was the generalization of the existing results.
Mainly, these announced results are slight generalizations or extensions of the existing ones. Indeed,
predominantly, most of the proofs are the mimic of the proof of the pioneer fixed point theorem of
Banach: Construct a sequence (usually, by using the Picard operator); indicate that this sequence is
convergent and finally prove that the limit of this recursive sequence is the required fixed point of
the considered operator. The fact that existing so many publications on this topic has encouraged
researchers to organize and unify this scattered literature. One of the best examples of this trend was
given by Samet et al. [61] in 2012 by involving the notion of admissible mappings.
Now, we shall explain why the new notion of Samet et al. [61], an admissible mapping, is important.
First of all, we need to underline that Banach’s famous fixed point theorem has been generalized and
extended in many different ways in the literature. The most classic approach for generalization and
extension is to change the definitions of contraction. The other approach and method are that a given
contraction function is in cyclic form. One of the other methods for generalization and extension is to
add a partial order on the structure where contraction is defined. Admissible mappings make it possible
to put together these three approaches in a single statement. For the clarification of the fixed point
theory literature, admissible mapping plays one of the key roles.
In this manuscript, we shall revisit the literature by using the auxiliary function: admissible
mapping. We shall provide examples and put several consequences to illustrate how this approach works
successfully.
2 Preliminaries
In this section, we collect some necessary tools (notions, notations) that are essential to express the
results.
First of all, we shall fixed some basic notations: Hereafter,
N
and
N0
denote the set of positive
integers and the set of nonnegative integers. Furthermore, the symbols
R
,
R+
and
R+
0
represent the set
of reals, positive reals and the set of nonnegative reals, respectively. Throughout the manuscript, all
considered sets are non-empty.
We shall first recall the admissible mapping defined by Samet et al. [61]. Let
α
:
X×X→
[0
,∞
)be
a function. Then, the mapping T:X→Xis said to be α-admissible [61] if, for all x, y ∈X,
α(x, y)≥1=⇒α(T x, T y)≥1.
Next, we recollect the notion of triangular
α
-admissible that plays crucial rules in usage of triangle
inequality axiom of the metric.
https://doi.org/10.17993/3ctic.2022.112.26-50
Definition 1. [41] A self-mapping T:X→Xis called triangular α-admissible if
(T1)Tis α−admissible,
(T2)α(x, z)≥1,α(z,y)≥1=⇒α(x, y)≥1, x, y, z ∈X.
Example 1. Suppose that M= (0,+∞). Define T:M→Mand α:M×M→[0,∞)by
(1) T(x)=ln(x+ 1), for all x∈Mand α(x, y)=1,if x≥y;
0,if x<y.
Then Tis α-admissible.
(2) T(x)= 3
√x, for all x∈Mand α(x, y)=ex−y,if x≥y;
0,if x<y.
Then Tis α-admissible.
For more examples of such mappings are presented in [30,40,41,46–48,61] and references therein.
In 2014, Popescu [54] conclude that the notion of triangular
α
-admissible can be refined slightly, as
follows:
Definition 2. [54] Let
T
:
X→X
be a self-mapping and
α
:
X×X→
[0
,∞
)be a function. Then
T
is said to be α-orbital admissible if
(T3) α(x, T x)≥1⇒α(T x, T 2x)≥1.
Definition 3. [54] Let
T
:
X→X
be a self-mapping and
α
:
X×X→
[0
,∞
)be a function. Then
T
is said to be triangular α-orbital admissible if T is α-orbital admissible and
(T4) α(x, y)≥1and α(y, T y)≥1⇒α(x, T y)≥1.
As it is expected, each
α
-admissible mapping is an
α
-orbital admissible mapping and each triangular
α
-admissible mapping is a triangular
α
-orbital admissible mapping. Notice also that the converse is
false, see e.g. ( [54] Example 7).
A metric space (
X, d
)is said
α
-regular [54], if for every sequence
{xn}
in
X
such that
α
(
xn,x
n+1
)
≥
1
for all
n
and
xn→x∈X
as
n→∞
, then there exists a subsequence
{xn(k)}
of
{xn}
such that
α(xn(k),x)≥1for all k.
Lemma 1. [54] Let
T
:
X→X
be a triangular
α
-orbital admissible mapping. Assume that there exists
x0∈X
such that
α
(
x0,Tx
0
)
≥
1. Define a sequence
{xn}
by
xn+1
=
Tx
n
for each
n∈N0
. Then we
have α(xn,x
m)≥1for all m, n ∈Nwith n<m.
In what follows, we recall the following auxiliary family of the functions, namely, comparison and
c
-comparison functions. A mapping
ψ
: [0
,∞
)
→
[0
,∞
)is called a comparison function if it is increasing
and
ψn
(
t
)
→
0,
n→∞
, for any
t∈
[0
,∞
). We denote by Φ, the class of the corporation function
ψ
: [0
,∞
)
→
[0
,∞
). For more details and examples , see e.g. [21, 58]. Among them, we recall the
following essential result.
Lemma 2. (Berinde [21], Rus [58]) If ψ: [0,∞)→[0,∞)is a comparison function, then:
(1) each iterate ψkof ψ,k≥1, is also a comparison function;
(2) ψis continuous at 0;
(3) ψ(t)<t, for any t>0.
Later, Berinde [21] introduced the concept of (c)-comparison function in the following way.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
29
Definition 4. (Berinde [21]) A function
ψ
: [0
,∞
)
→
[0
,∞
)is said to be a (
c
)-comparison function if
(c1)ψis increasing,
(c2)
there exists
k0∈N
,
a∈
(0
,
1) and a convergent series of nonnegative terms
∞
k=1
vk
such that
ψk+1(t)≤aψk(t)+vk, for k≥k0and any t∈[0,∞).
From now on, the letter Ψis reserved to indicate the family of functions (
c
)-comparison function. It
is evident that Lemma 2 is valid for ψ∈Ψ. Further, (c2) yields
+∞
n=1
ψn(t)<∞,
for all t>0, where ψnis the nth iterate of ψ.
Samet et al. [61] prove the first fixed point result via admissible mapping by introducing the following
concepts.
Definition 5. Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping. We say that
T
is an
α−ψcontractive mapping if there exist two functions α:X×X→[0,∞)and ψ∈Ψsuch that
α(x, y)d(T x, T y)≤ψ(d(x, y)),for all x, y ∈X.
It is obvious that, any contractive mapping forms an
α−ψ
contractive mapping with
α
(
x, y
)=1
for all x, y ∈Xand ψ(t)=kt,k∈(0,1).
The following is the interesting fixed point theorem of Samet et al. [61]
Theorem 1. Let (
X, d
)be a complete metric space and
T
:
X→X
be an
α−ψ
contractive mapping.
Suppose that
(i)Tis α−admissible;
(ii)there exists x0∈Xsuch that α(x0,Tx
0)≥1;
(iii)Tis continuous, or
(iii)
if
{xn}
is a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
, then
α(xn,x)≥1for all n.
Then there exists u∈Xsuch that Tu =u.
Note that in this setting, for the uniqueness additional condition is considered.
Theorem 2. Adding to the hypotheses of Theorem 1 (resp. Theorem 1) the condition: For all
x, y ∈X
,
there exists z∈Xsuch that α(x, z)≥1and α(y, z)≥1, we obtain uniqueness of the fixed point.
Another successful attempt to simplify and clarify the literature of the metric fixed point theory
was done by Khojasteh et al. [44] in 2015. In this paper, the authors introduced the notion of the
simulation functions to combine the several existing results. In what follows, we recall the definition of
this auxiliary function.
Definition 6. (See [44]) A function
ζ
: [0
,∞
)
×
[0
,∞
)
→R
is said to be simulation if it satisfies the
following conditions:
https://doi.org/10.17993/3ctic.2022.112.26-50
(ζ1)ζ(0,0) = 0;
(ζ2)ζ(t, s)<s−tfor all t, s > 0;
(ζ3)if {tn},{sn}are sequences in (0,∞)such that lim
n→∞ tn= lim
n→∞ sn>0, then
lim sup
n→∞ ζ(tn,s
n)<0.(1)
The family of all simulation functions
ζ
: [0
,∞
)
×
[0
,∞
)
→R
will be denoted by
Z
. On account of
(ζ2), we observe that
ζ(t, t)<0for all t>0,ζ∈Z.(2)
Notice also that the condition (ζ1)is superfluous due to (ζ2).
Example 2. (See e.g. [12,15,42
–
44,56]) Let
ζi
: [0
,∞
)
×
[0
,∞
)
→R,i∈{
1
,
2
,
3
}
, be mappings defined
by
(i)ζ1
(
t, s
)=
ψ
(
s
)
−ϕ
(
t
)
for all t, s ∈
[0
,∞
)
,
where
ϕ, ψ
: [0
,∞
)
→
[0
,∞
)are two continuous
functions such that ψ(t)=ϕ(t)=0if, and only if, t=0, and ψ(t)<t≤ϕ(t)for all t>0.
(ii)ζ2
(
t, s
)=
s−f(t, s)
g(t, s)tfor all t, s ∈
[0
,∞
)
,
where
f,g
: [0
,∞
)
→
(0
,∞
)are two continuous
functions with respect to each variable such that f(t, s)>g(t, s)for all t, s > 0.
(iii)ζ3
(
t, s
)=
s−φ
(
s
)
−t
for all
t, s ∈
[0
,∞
)
,
where
φ
: [0
,∞
)
→
[0
,∞
)is a continuous function such
that φ(t)=0if, and only if, t=0.
(iv)If φ: [0,∞)→[0,1) is a function such that lim sup
t→r+
φ(t)<1for all r>0, and we define
ζT(t, s)=sφ(s)−tfor all s, t ∈[0,∞),
then ζTis a simulation function.
(v)
If
η
: [0
,∞
)
→
[0
,∞
)is an upper semi-continuous mapping such that
η
(
t
)
<t
for all
t>
0and
η(0) = 0, and we define
ζBW (t, s)=η(s)−tfor all s, t ∈[0,∞),
then ζBW is a simulation function.
(vi)If ϕ: [0,∞)→[0,∞)is a function such that ε
0ϕ(u)du exists and ε
0ϕ(u)du > ε, for each ε>0,
and we define
ζK(t, s)=s−t
0
ϕ(u)du for all s, t ∈[0,∞),
then ζKis a simulation function.
Suppose (
X, d
)is a metric space,
T
is a self-mapping on
X
and
ζ∈Z
. We say that
T
is a
Z-contraction with respect to ζ[44], if
ζ(d(T(x),T (y)),d(x, y)) ≥0,for all x, y ∈X.
It is evident that renowned Banach contraction forms
Z
-contraction with respect to
ζ
where
ζ(t, s)=ks −twith k∈[0,1) and s, t ∈[0,∞).
Theorem 3. On the complete metric space, every Z-contraction possesses a unique fixed point.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
30
Definition 4. (Berinde [21]) A function
ψ
: [0
,∞
)
→
[0
,∞
)is said to be a (
c
)-comparison function if
(c1)ψis increasing,
(c2)
there exists
k0∈N
,
a∈
(0
,
1) and a convergent series of nonnegative terms
∞
k=1
vk
such that
ψk+1(t)≤aψk(t)+vk, for k≥k0and any t∈[0,∞).
From now on, the letter Ψis reserved to indicate the family of functions (
c
)-comparison function. It
is evident that Lemma 2 is valid for ψ∈Ψ. Further, (c2) yields
+∞
n=1
ψn(t)<∞,
for all t>0, where ψnis the nth iterate of ψ.
Samet et al. [61] prove the first fixed point result via admissible mapping by introducing the following
concepts.
Definition 5. Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping. We say that
T
is an
α−ψcontractive mapping if there exist two functions α:X×X→[0,∞)and ψ∈Ψsuch that
α(x, y)d(T x, T y)≤ψ(d(x, y)),for all x, y ∈X.
It is obvious that, any contractive mapping forms an
α−ψ
contractive mapping with
α
(
x, y
)=1
for all x, y ∈Xand ψ(t)=kt,k∈(0,1).
The following is the interesting fixed point theorem of Samet et al. [61]
Theorem 1. Let (
X, d
)be a complete metric space and
T
:
X→X
be an
α−ψ
contractive mapping.
Suppose that
(i)Tis α−admissible;
(ii)there exists x0∈Xsuch that α(x0,Tx
0)≥1;
(iii)Tis continuous, or
(iii)
if
{xn}
is a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
, then
α(xn,x)≥1for all n.
Then there exists u∈Xsuch that Tu =u.
Note that in this setting, for the uniqueness additional condition is considered.
Theorem 2. Adding to the hypotheses of Theorem 1 (resp. Theorem 1) the condition: For all
x, y ∈X
,
there exists z∈Xsuch that α(x, z)≥1and α(y, z)≥1, we obtain uniqueness of the fixed point.
Another successful attempt to simplify and clarify the literature of the metric fixed point theory
was done by Khojasteh et al. [44] in 2015. In this paper, the authors introduced the notion of the
simulation functions to combine the several existing results. In what follows, we recall the definition of
this auxiliary function.
Definition 6. (See [44]) A function
ζ
: [0
,∞
)
×
[0
,∞
)
→R
is said to be simulation if it satisfies the
following conditions:
https://doi.org/10.17993/3ctic.2022.112.26-50
(ζ1)ζ(0,0) = 0;
(ζ2)ζ(t, s)<s−tfor all t, s > 0;
(ζ3)if {tn},{sn}are sequences in (0,∞)such that lim
n→∞ tn= lim
n→∞ sn>0, then
lim sup
n→∞ ζ(tn,s
n)<0.(1)
The family of all simulation functions
ζ
: [0
,∞
)
×
[0
,∞
)
→R
will be denoted by
Z
. On account of
(ζ2), we observe that
ζ(t, t)<0for all t>0,ζ∈Z.(2)
Notice also that the condition (ζ1)is superfluous due to (ζ2).
Example 2. (See e.g. [12,15,42
–
44,56]) Let
ζi
: [0
,∞
)
×
[0
,∞
)
→R,i∈{
1
,
2
,
3
}
, be mappings defined
by
(i)ζ1
(
t, s
)=
ψ
(
s
)
−ϕ
(
t
)
for all t, s ∈
[0
,∞
)
,
where
ϕ, ψ
: [0
,∞
)
→
[0
,∞
)are two continuous
functions such that ψ(t)=ϕ(t)=0if, and only if, t=0, and ψ(t)<t≤ϕ(t)for all t>0.
(ii)ζ2
(
t, s
)=
s−f(t, s)
g(t, s)tfor all t, s ∈
[0
,∞
)
,
where
f,g
: [0
,∞
)
→
(0
,∞
)are two continuous
functions with respect to each variable such that f(t, s)>g(t, s)for all t, s > 0.
(iii)ζ3
(
t, s
)=
s−φ
(
s
)
−t
for all
t, s ∈
[0
,∞
)
,
where
φ
: [0
,∞
)
→
[0
,∞
)is a continuous function such
that φ(t)=0if, and only if, t=0.
(iv)If φ: [0,∞)→[0,1) is a function such that lim sup
t→r+
φ(t)<1for all r>0, and we define
ζT(t, s)=sφ(s)−tfor all s, t ∈[0,∞),
then ζTis a simulation function.
(v)
If
η
: [0
,∞
)
→
[0
,∞
)is an upper semi-continuous mapping such that
η
(
t
)
<t
for all
t>
0and
η(0) = 0, and we define
ζBW (t, s)=η(s)−tfor all s, t ∈[0,∞),
then ζBW is a simulation function.
(vi)If ϕ: [0,∞)→[0,∞)is a function such that ε
0ϕ(u)du exists and ε
0ϕ(u)du > ε, for each ε>0,
and we define
ζK(t, s)=s−t
0
ϕ(u)du for all s, t ∈[0,∞),
then ζKis a simulation function.
Suppose (
X, d
)is a metric space,
T
is a self-mapping on
X
and
ζ∈Z
. We say that
T
is a
Z-contraction with respect to ζ[44], if
ζ(d(T(x),T (y)),d(x, y)) ≥0,for all x, y ∈X.
It is evident that renowned Banach contraction forms
Z
-contraction with respect to
ζ
where
ζ(t, s)=ks −twith k∈[0,1) and s, t ∈[0,∞).
Theorem 3. On the complete metric space, every Z-contraction possesses a unique fixed point.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
31
3 A theorem with many consequences
In this section, we shall consider a theorem that generalizes and hence unifies a number of existing
results. Consequently, we list many corollaries.
Definition 7. Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping. We say that
T
is a
generalized Suzuki type (
α−ψ
)
−Z
-contraction mapping if there exist two functions
α
:
X×X→
[0
,∞
),
ζ∈Zand ψ∈Ψsuch that for all x, y ∈X, we have
1
2d(x, T x)≤d(x, y)implies ζ(ψ(M(x, y)),α(x, y)d(T x, T y)) ≥0,(3)
where M(x, y) = max d(x, y),d(x, T x)+d(y, Ty)
2,d(x, T y)+d(y, T x)
2.
Theorem 4. Let (
X, d
)be a complete metric space. Suppose that
T
:
X→X
is a generalized Suzuki
type (α−ψ)−Z-contraction mapping and satisfies the following conditions:
(i) Tis triangular α-orbital admissible;
(ii) there exists x0∈Xsuch that α(x0,Tx
0)≥1;
(iii) Tis continuous.
Then there exists u∈Xsuch that Tu =u.
Proof. On account of the assumption (ii) of the theorem, there is a point
x0∈X
such that
α(x0,Tx
0)≥1.
Starting with this initial point, we shall built-up a recursive sequence
{xn}
in
X
by
xn+1
=
Tx
n
for
all
n∈N0
. We, first, observe that incase of
xn0
=
xn0+1
for some
n0
, we conclude that
u
=
xn0
is a
fixed point of T. Accordingly, we presume that xn=xn+1 for all n. Hence, we find that
0<1
2d(xn,x
n+1)= 1
2d(xn,Tx
n)≤d(xn,x
n+1),
for all n.
On the other hand, employing that Tis α−admissible, we derive that
α(x0,x
1)=α(x0,Tx
0)≥1⇒α(Tx
0,Tx
1)=α(x1,x
2)≥1.
Inductively, we have
α(xn,x
n+1)≥1,for all n=0,1,... (4)
From (3) and (4), it follows that for all n≥1, we have
1
2d(xn,Tx
n)≤d(xn,x
n+1),
implies that
ζ(ψ(M(xn,x
n+1)),α(xn,x
n+1))d(Tx
n,Tx
n+1))) ≥0,
which is equivalent to
d(xn+1,x
n)=d(Tx
n,Tx
n−1)≤α(xn,x
n−1)d(Tx
n,Tx
n−1)≤ψ(M(xn,x
n−1)).(5)
Now, we shall simplify the right hand side of the inequality above, as follows
M(xn,x
n−1)=max
d(xn,x
n−1),d(xn,Tx
n)+d(xn−1,Tx
n−1)
2,d(xn,Tx
n−1)+d(xn−1,Tx
n)
2
= max d(xn,x
n−1),d(xn,x
n+1)+d(xn−1,x
n)
2,d(xn−1,x
n+1)
2
≤max d(xn,x
n−1),d(xn,x
n+1)+d(xn−1,x
n)
2
≤max{d(xn,x
n−1),d(xn,x
n+1)}.
https://doi.org/10.17993/3ctic.2022.112.26-50
Consequently, regarding (5) together with the fact that ψis a nondecreasing function, we derive
d(xn+1,x
n)≤ψ(max{d(xn,x
n−1),d(xn,x
n+1)}),(6)
for all n≥1. If for some n≥1, we have d(xn,x
n−1)≤d(xn,x
n+1), from (6), we obtain that
d(xn+1,x
n)≤ψ(d(xn,x
n+1)) <d(xn,x
n+1),
a contradiction. Thus, for all n≥1, we have
max{d(xn,x
n−1),d(xn,x
n+1)}=d(xn,x
n−1).(7)
Using (6) and (7), we get that
d(xn+1,x
n)≤ψ(d(xn,x
n−1)) <d(xn,x
n−1),(8)
for all n≥1. By induction, we get
d(xn+1,x
n)≤ψn(d(x1,x
0)),for all n≥1.(9)
From (9) and using the triangular inequality, for all k≥1, we have
d(xn,x
n+k)≤d(xn,x
n+1)+...+d(xn+k−1,x
n+k)
≤
n+k−1
p=n
ψn(d(x1,x
0))
≤
+∞
p=n
ψn(d(x1,x
0)) →0as n→∞.
This implies that
{xn}
is a Cauchy sequence in (
X, d
). Since (
X, d
)is complete, there exists
u∈X
such that
lim
n→∞ d(xn,u)=0.(10)
Since Tis continuous, we obtain from (10) that
lim
n→∞ d(xn+1,Tu) = lim
n→∞ d(Tx
n,Tu)=0.(11)
From (10), (11) and the uniqueness of the limit, we get immediately that
u
is a fixed point of
T
, that
is, Tu =u.
In what follows, the continuity of the contraction in Theorem 4 is refined.
Theorem 5. Let (
X, d
)be a complete metric space. Suppose that
T
:
X→X
is a generalized Suzuki
type (α−ψ)−Z-contraction mapping and satisfies the following conditions:
(i) Tis triangular α-orbital admissible;
(ii) there exists x0∈Xsuch that α(x0,Tx
0)≥1;
(iii)
if
{xn}
is a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
, then
there exists a subsequence {xn(k)}of {xn}such that α(xn(k),x)≥1for all k.
Then there exists u∈Xsuch that Tu =u.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
32
3 A theorem with many consequences
In this section, we shall consider a theorem that generalizes and hence unifies a number of existing
results. Consequently, we list many corollaries.
Definition 7. Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping. We say that
T
is a
generalized Suzuki type (
α−ψ
)
−Z
-contraction mapping if there exist two functions
α
:
X×X→
[0
,∞
),
ζ∈Zand ψ∈Ψsuch that for all x, y ∈X, we have
1
2d(x, T x)≤d(x, y)implies ζ(ψ(M(x, y)),α(x, y)d(T x, T y)) ≥0,(3)
where M(x, y) = max d(x, y),d(x, T x)+d(y, Ty)
2,d(x, T y)+d(y, T x)
2.
Theorem 4. Let (
X, d
)be a complete metric space. Suppose that
T
:
X→X
is a generalized Suzuki
type (α−ψ)−Z-contraction mapping and satisfies the following conditions:
(i) Tis triangular α-orbital admissible;
(ii) there exists x0∈Xsuch that α(x0,Tx
0)≥1;
(iii) Tis continuous.
Then there exists u∈Xsuch that Tu =u.
Proof. On account of the assumption (ii) of the theorem, there is a point
x0∈X
such that
α(x0,Tx
0)≥1.
Starting with this initial point, we shall built-up a recursive sequence
{xn}
in
X
by
xn+1
=
Tx
n
for
all
n∈N0
. We, first, observe that incase of
xn0
=
xn0+1
for some
n0
, we conclude that
u
=
xn0
is a
fixed point of T. Accordingly, we presume that xn=xn+1 for all n. Hence, we find that
0<1
2d(xn,x
n+1)= 1
2d(xn,Tx
n)≤d(xn,x
n+1),
for all n.
On the other hand, employing that Tis α−admissible, we derive that
α(x0,x
1)=α(x0,Tx
0)≥1⇒α(Tx
0,Tx
1)=α(x1,x
2)≥1.
Inductively, we have
α(xn,x
n+1)≥1,for all n=0,1,... (4)
From (3) and (4), it follows that for all n≥1, we have
1
2d(xn,Tx
n)≤d(xn,x
n+1),
implies that
ζ(ψ(M(xn,x
n+1)),α(xn,x
n+1))d(Tx
n,Tx
n+1))) ≥0,
which is equivalent to
d(xn+1,x
n)=d(Tx
n,Tx
n−1)≤α(xn,x
n−1)d(Tx
n,Tx
n−1)≤ψ(M(xn,x
n−1)).(5)
Now, we shall simplify the right hand side of the inequality above, as follows
M(xn,x
n−1)=max
d(xn,x
n−1),d(xn,Tx
n)+d(xn−1,Tx
n−1)
2,d(xn,Tx
n−1)+d(xn−1,Tx
n)
2
= max d(xn,x
n−1),d(xn,x
n+1)+d(xn−1,x
n)
2,d(xn−1,x
n+1)
2
≤max d(xn,x
n−1),d(xn,x
n+1)+d(xn−1,x
n)
2
≤max{d(xn,x
n−1),d(xn,x
n+1)}.
https://doi.org/10.17993/3ctic.2022.112.26-50
Consequently, regarding (5) together with the fact that ψis a nondecreasing function, we derive
d(xn+1,x
n)≤ψ(max{d(xn,x
n−1),d(xn,x
n+1)}),(6)
for all n≥1. If for some n≥1, we have d(xn,x
n−1)≤d(xn,x
n+1), from (6), we obtain that
d(xn+1,x
n)≤ψ(d(xn,x
n+1)) <d(xn,x
n+1),
a contradiction. Thus, for all n≥1, we have
max{d(xn,x
n−1),d(xn,x
n+1)}=d(xn,x
n−1).(7)
Using (6) and (7), we get that
d(xn+1,x
n)≤ψ(d(xn,x
n−1)) <d(xn,x
n−1),(8)
for all n≥1. By induction, we get
d(xn+1,x
n)≤ψn(d(x1,x
0)),for all n≥1.(9)
From (9) and using the triangular inequality, for all k≥1, we have
d(xn,x
n+k)≤d(xn,x
n+1)+...+d(xn+k−1,x
n+k)
≤
n+k−1
p=n
ψn(d(x1,x
0))
≤
+∞
p=n
ψn(d(x1,x
0)) →0as n→∞.
This implies that
{xn}
is a Cauchy sequence in (
X, d
). Since (
X, d
)is complete, there exists
u∈X
such that
lim
n→∞ d(xn,u)=0.(10)
Since Tis continuous, we obtain from (10) that
lim
n→∞ d(xn+1,Tu) = lim
n→∞ d(Tx
n,Tu)=0.(11)
From (10), (11) and the uniqueness of the limit, we get immediately that
u
is a fixed point of
T
, that
is, Tu =u.
In what follows, the continuity of the contraction in Theorem 4 is refined.
Theorem 5. Let (
X, d
)be a complete metric space. Suppose that
T
:
X→X
is a generalized Suzuki
type (α−ψ)−Z-contraction mapping and satisfies the following conditions:
(i) Tis triangular α-orbital admissible;
(ii) there exists x0∈Xsuch that α(x0,Tx
0)≥1;
(iii)
if
{xn}
is a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
, then
there exists a subsequence {xn(k)}of {xn}such that α(xn(k),x)≥1for all k.
Then there exists u∈Xsuch that Tu =u.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
33
Proof. Following the proof of Theorem 4 line by line, we deduce that the recursive sequence
{xn}
defined by
xn+1
=
Tx
n
for all
n≥
0, converges for some
u∈X
. From (4) and condition (iii), there
exists a subsequence {xn(k)}of {xn}such that α(xn(k),u)≥1for all k.
Now, we use (3) to deduce
u∈X
is the required fixed point. For this purpose, we need to show that
1
2d
(
xn(k),x
n(k)+1
)=
1
2d
(
xn(k),Tx
n(k)
)
≤d
(
xn(k),u
)or
1
2d
(
xn(k)+1,x
n(k)+2
)=
1
2d
(
xn(k)+1,Tx
n(k)+1
)
≤
d
(
xn(k)+1,u
). Suppose, on the contrary, that
1
2d
(
xn(k),x
n(k)+1
)
>d
(
xn(k),u
)and
1
2d
(
xn(k)+1,x
n(k)+2
)
>
d(xn(k)+1,u). On account of the triangle inequality, we have
d(xn(k),x
n(k)+1)≤d(xn(k),u)+d(u, xn(k)+1)
<1
2d(xn(k),x
n(k)+1)+ 1
2d(xn(k)+1,x
n(k)+2)
on account of (8)
<1
2d(xn(k),x
n(k)+1)+ 1
2d(xn(k),x
n(k)+1)=d(xn(k),x
n(k)+1),
(12)
a contradiction. Thus, we have
1
2d(xn(k),Tx
n(k))≤d(xn(k),u)or 1
2d(xn(k)+1,Tx
n(k)+1)≤d(xn(k)+1,u).
After this observation, by applying (3), for all k, we find
1
2d
(
xn(k),Tx
n(k)
)
≤d
(
xn(k),u
)implies
ζ
(
ψ
(
M
(
xn(k),u
))
,α
(
xn(k),u
)
d
(
Tx
n(k),Tu
))
≥
0which yields
that
d(xn(k)+1,Tu)=d(Tx
n(k),Tu)≤α(xn(k),u)d(Tx
n(k),Tu)≤ψ(M(xn(k),u)).(13)
On the other hand, we have
M(xn(k),u) = max d(xn(k),u),d(xn(k),x
n(k)+1)+d(u, T u)
2,d(xn(k),Tu)+d(u, xn(k)+1)
2.
Letting k→∞in the above equality, we obtain that
lim
k→∞ M(xn(k),u)=d(u, T u)
2.(14)
Suppose that
d
(
u, T u
)
>
0. From (14), for
k
large enough, we have
M
(
xn(k),u
)
>
0, which yields that
ψ(M(xn(k),u)) <M(xn(k),u). Hence, from (13), we find
d(xn(k)+1,Tu)<M(xn(k),u).
Letting k→∞in the above inequality, using (14), we obtain that
d(u, T u)≤d(u, T u)
2,
which is a contradiction. Thus we have d(u, T u)=0, that is, u=Tu.
Notice that the proved theorems above are valid when we replace Instead of M(x, y)with N(x, y),
where
N(x, y) = max d(x, y),d(x, T x)+d(y, T y)
2.
The following example indicate that the hypotheses in Theorem 4 and Theorem 5 do not guarantee
uniqueness of the fixed point.
Example 3. Consider the set
X
=
{
(2
,
1)
,
(1
,
2)
}⊂R2
endowed with the standard Euclidean distance
d((x, y),(u, v)) = |x−u|+|y−v|,
https://doi.org/10.17993/3ctic.2022.112.26-50
for all (
x, y
)
,
(
u, v
)
∈X
. It is evident that (
X, d
)is a complete metric space. It is clear that
T
(
x, y
)=
(
x, y
)that is trivially continuous. On account of the inequality 0=
d
(
T
(
x, y
)
,T
(
u, v
))
≤d
((
x, y
)
,
(
u, v
))
we have
ζ(ψ(M((x, y),(u, v))),α((x, y),(u, v))d(T(x, y),T(u, v))) ≥0,
for any ψ∈Ψ. Consequently, we have
α((x, y),(u, v))d(T(x, y),T(u, v)) ≤ψ(M((x, y),(u, v))),
for all (x, y),(u, v)∈X, where
α((x, y),(u, v)) = 1if (x, y)=(u, v),
0if (x, y)=(u, v).
Consequently,
T
is a generalized Suzuki type (
α−ψ
)
−Z
-contraction mapping. On the other hand, for
all (x, y),(u, v)∈X, we have
α((x, y),(u, v)) ≥1→(x, y)=(u, v)→T(x, y)=T(u, v)→α(T(x, y),T(u, v)) ≥1.
Thus, the mapping
T
is
α−
admissible. Furthermore, for all (
x, y
)
∈X
, we have
α
((
x, y
)
,T
(
x, y
))
≥
1.
Hence, the assumptions of Theorem 4 are fulfilled. Note that the assumptions of Theorem 5 are
also satisfied, indeed if
{
(
xn,y
n
)
}
is a sequence in
X
that converges to some point (
x, y
)
∈X
with
α
((
xn,y
n
)
,
(
xn+1,y
n+1
))
≥
1for all
n
, from the definition of
α
, we have (
xn,y
n
)=(
x, y
)for all
n
, that
yields α((xn,y
n),(x, y)) = 1 for all n. Notice that, in this case, Thas two fixed points in X.
For the uniqueness of a fixed point of such mappings, we need to the following additional condition:
(H) For all x, y ∈Fix(T), there exists z∈Xsuch that α(x, z)≥1and α(y, z)≥1.
Here, Fix(T)denotes the set of fixed points of T.
Theorem 6. Adding condition (H) to the hypotheses of Theorem 4 (resp. Theorem 5), we obtain that
uis the unique fixed point of T.
Proof. Suppose, on the contrary, that
v
is another fixed point of
T
. From (H), there exists
z∈X
such that
α(u, z)≥1and α(v, z)≥1.(15)
Due to the fact that Tis α−admissible together with (15), we find
α(u, T nz)≥1and α(v, Tnz)≥1,for all n. (16)
Construct a sequence {zn}in Xby zn+1 =Tz
nfor all n≥0and z0=z.
Taking (16) into account, for all n, we have
0=1
2d(u, T u)≤d(u, zn)implies ζ(ψ(M(u, zn)),α(u, zn)d(Tu,Tz
n)) ≥0,(17)
which is equivalent to
d(u, zn+1)=d(Tu,Tz
n)≤α(u, zn)d(Tu,Tz
n)≤ψ(M(u, zn)).(18)
On the other hand, we find
M(u, zn)=max
d(u, zn),d(zn,z
n+1)
2,d(u, zn+1)+d(zn,u)
2
≤max d(u, zn),d(zn,u)+d(u, zn+1)
2
≤max{d(u, zn),d(u, zn+1)}.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
34
Proof. Following the proof of Theorem 4 line by line, we deduce that the recursive sequence
{xn}
defined by
xn+1
=
Tx
n
for all
n≥
0, converges for some
u∈X
. From (4) and condition (iii), there
exists a subsequence {xn(k)}of {xn}such that α(xn(k),u)≥1for all k.
Now, we use (3) to deduce
u∈X
is the required fixed point. For this purpose, we need to show that
1
2d
(
xn(k),x
n(k)+1
)=
1
2d
(
xn(k),Tx
n(k)
)
≤d
(
xn(k),u
)or
1
2d
(
xn(k)+1,x
n(k)+2
)=
1
2d
(
xn(k)+1,Tx
n(k)+1
)
≤
d
(
xn(k)+1,u
). Suppose, on the contrary, that
1
2d
(
xn(k),x
n(k)+1
)
>d
(
xn(k),u
)and
1
2d
(
xn(k)+1,x
n(k)+2
)
>
d(xn(k)+1,u). On account of the triangle inequality, we have
d(xn(k),x
n(k)+1)≤d(xn(k),u)+d(u, xn(k)+1)
<1
2d(xn(k),x
n(k)+1)+ 1
2d(xn(k)+1,x
n(k)+2)
on account of (8)
<1
2d(xn(k),x
n(k)+1)+ 1
2d(xn(k),x
n(k)+1)=d(xn(k),x
n(k)+1),
(12)
a contradiction. Thus, we have
1
2d(xn(k),Tx
n(k))≤d(xn(k),u)or 1
2d(xn(k)+1,Tx
n(k)+1)≤d(xn(k)+1,u).
After this observation, by applying (3), for all k, we find
1
2d
(
xn(k),Tx
n(k)
)
≤d
(
xn(k),u
)implies
ζ
(
ψ
(
M
(
xn(k),u
))
,α
(
xn(k),u
)
d
(
Tx
n(k),Tu
))
≥
0which yields
that
d(xn(k)+1,Tu)=d(Tx
n(k),Tu)≤α(xn(k),u)d(Tx
n(k),Tu)≤ψ(M(xn(k),u)).(13)
On the other hand, we have
M(xn(k),u) = max d(xn(k),u),d(xn(k),x
n(k)+1)+d(u, T u)
2,d(xn(k),Tu)+d(u, xn(k)+1)
2.
Letting k→∞in the above equality, we obtain that
lim
k→∞ M(xn(k),u)=d(u, T u)
2.(14)
Suppose that
d
(
u, T u
)
>
0. From (14), for
k
large enough, we have
M
(
xn(k),u
)
>
0, which yields that
ψ(M(xn(k),u)) <M(xn(k),u). Hence, from (13), we find
d(xn(k)+1,Tu)<M(xn(k),u).
Letting k→∞in the above inequality, using (14), we obtain that
d(u, T u)≤d(u, T u)
2,
which is a contradiction. Thus we have d(u, T u)=0, that is, u=Tu.
Notice that the proved theorems above are valid when we replace Instead of M(x, y)with N(x, y),
where
N(x, y) = max d(x, y),d(x, T x)+d(y, T y)
2.
The following example indicate that the hypotheses in Theorem 4 and Theorem 5 do not guarantee
uniqueness of the fixed point.
Example 3. Consider the set
X
=
{
(2
,
1)
,
(1
,
2)
}⊂R2
endowed with the standard Euclidean distance
d((x, y),(u, v)) = |x−u|+|y−v|,
https://doi.org/10.17993/3ctic.2022.112.26-50
for all (
x, y
)
,
(
u, v
)
∈X
. It is evident that (
X, d
)is a complete metric space. It is clear that
T
(
x, y
)=
(
x, y
)that is trivially continuous. On account of the inequality 0=
d
(
T
(
x, y
)
,T
(
u, v
))
≤d
((
x, y
)
,
(
u, v
))
we have
ζ(ψ(M((x, y),(u, v))),α((x, y),(u, v))d(T(x, y),T(u, v))) ≥0,
for any ψ∈Ψ. Consequently, we have
α((x, y),(u, v))d(T(x, y),T(u, v)) ≤ψ(M((x, y),(u, v))),
for all (x, y),(u, v)∈X, where
α((x, y),(u, v)) = 1if (x, y)=(u, v),
0if (x, y)=(u, v).
Consequently,
T
is a generalized Suzuki type (
α−ψ
)
−Z
-contraction mapping. On the other hand, for
all (x, y),(u, v)∈X, we have
α((x, y),(u, v)) ≥1→(x, y)=(u, v)→T(x, y)=T(u, v)→α(T(x, y),T(u, v)) ≥1.
Thus, the mapping
T
is
α−
admissible. Furthermore, for all (
x, y
)
∈X
, we have
α
((
x, y
)
,T
(
x, y
))
≥
1.
Hence, the assumptions of Theorem 4 are fulfilled. Note that the assumptions of Theorem 5 are
also satisfied, indeed if
{
(
xn,y
n
)
}
is a sequence in
X
that converges to some point (
x, y
)
∈X
with
α
((
xn,y
n
)
,
(
xn+1,y
n+1
))
≥
1for all
n
, from the definition of
α
, we have (
xn,y
n
)=(
x, y
)for all
n
, that
yields α((xn,y
n),(x, y)) = 1 for all n. Notice that, in this case, Thas two fixed points in X.
For the uniqueness of a fixed point of such mappings, we need to the following additional condition:
(H) For all x, y ∈Fix(T), there exists z∈Xsuch that α(x, z)≥1and α(y, z)≥1.
Here, Fix(T)denotes the set of fixed points of T.
Theorem 6. Adding condition (H) to the hypotheses of Theorem 4 (resp. Theorem 5), we obtain that
uis the unique fixed point of T.
Proof. Suppose, on the contrary, that
v
is another fixed point of
T
. From (H), there exists
z∈X
such that
α(u, z)≥1and α(v, z)≥1.(15)
Due to the fact that Tis α−admissible together with (15), we find
α(u, T nz)≥1and α(v, Tnz)≥1,for all n. (16)
Construct a sequence {zn}in Xby zn+1 =Tz
nfor all n≥0and z0=z.
Taking (16) into account, for all n, we have
0=1
2d(u, T u)≤d(u, zn)implies ζ(ψ(M(u, zn)),α(u, zn)d(Tu,Tz
n)) ≥0,(17)
which is equivalent to
d(u, zn+1)=d(Tu,Tz
n)≤α(u, zn)d(Tu,Tz
n)≤ψ(M(u, zn)).(18)
On the other hand, we find
M(u, zn)=max
d(u, zn),d(zn,z
n+1)
2,d(u, zn+1)+d(zn,u)
2
≤max d(u, zn),d(zn,u)+d(u, zn+1)
2
≤max{d(u, zn),d(u, zn+1)}.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
35
On account of the inequality above, the expression (18) and the monotone property of
ψ
, we derive that
d(u, zn+1)≤ψ(max{d(u, zn),d(u, zn+1)}),(19)
for all
n
. Without restriction to the generality, we can suppose that
d
(
u, zn
)
>
0for all
n
. If
max{d(u, zn),d(u, zn+1)}=d(u, zn+1), we get from (19) that
d(u, zn+1)≤ψ(d(u, zn+1)) <d(u, zn+1),
which is a contradiction. Thus we have max{d(u, zn),d(u, zn+1)}=d(u, zn), and
d(u, zn+1)≤ψ(d(u, zn)),
for all n. This implies that
d(u, zn)≤ψn(d(u, z0)),for all n≥1.
Letting n→∞in the above inequality, we obtain
lim
n→∞ d(zn,u)=0.(20)
Similarly, one can show that
lim
n→∞ d(zn,v)=0.(21)
From (20) and (21), it follows that u=v. Thus we proved that uis the unique fixed point of T.
3.1 Immediate consequences
. The first immediate consequence is obtained by removing the Suzuki condition, as in the following
definition.
Definition 8. Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping. We say that
T
is
a generalized type (
α−ψ
)
−Z
-contraction mapping if there exist two functions
α
:
X×X→
[0
,∞
),
ζ∈Zand ψ∈Ψsuch that for all x, y ∈X, we have
ζ(ψ(M(x, y)),α(x, y)d(T x, T y)) ≥0,(22)
where M(x, y) = max d(x, y),d(x, T x)+d(y, Ty)
2,d(x, T y)+d(y, T x)
2.
Theorem 7. Let (
X, d
)be a complete metric space. Suppose that
T
:
X→X
is a generalized Suzuki
type (α−ψ)−Z-contraction mapping and satisfies the following conditions:
(i) Tis triangular α-orbital admissible;
(ii) there exists x0∈Xsuch that α(x0,Tx
0)≥1;
(iii)* Tis continuous
or
(iii)**
if
{xn}
is a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
, then
there exists a subsequence {xn(k)}of {xn}such that α(xn(k),x)≥1for all k,
(iv) the condition (H) holds.
Then there exists u∈Xsuch that Tu =u.
We skipped the proof. Indeed, it is verbatim of the combinations of the proofs of Theorem 4, Theorem
5 and Theorem 6, by removing the related lines about the Suzuki condition.
Taking Example 2 into account, both Theorem 6 and Theorem 7 yields several consequences. In this
direction, one of the first example, by considering the case (i) Example 2 is the following theorem
https://doi.org/10.17993/3ctic.2022.112.26-50
Definition 9. Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping. We say that
T
is a
generalized
α−ψ
contractive mapping if there exist two functions
α
:
X×X→
[0
,∞
)and
ψ∈
Ψsuch
that for all x, y ∈X, we have
α(x, y)d(T x, T y)≤ψ(M(x, y)),(23)
where M(x, y) = max d(x, y),d(x, T x)+d(y, Ty)
2,d(x, T y)+d(y, T x)
2.
Theorem 8. Let (
X, d
)be a complete metric space. Suppose that
T
:
X→X
is a generalized
α−ψ
contractive mapping and satisfies the following conditions:
(i) Tis α−admissible ;
(ii) there exists x0∈Xsuch that α(x0,Tx
0)≥1;
(iii)* Tis continuous
or
(iii)**
if
{xn}
is a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
, then
there exists a subsequence {xn(k)}of {xn}such that α(xn(k),x)≥1for all k,
(iv) the condition (H) holds.
Then there exists u∈Xsuch that Tu =u.
It is evident that by taking, the other cases of Example 2, into account, one can further consequences.
We prefer to skip these consequences by the sake of the length of the manuscript.
4 Further Consequences
In this section, we shall indicate that several existing results in the literature can be deduced easily
from our Theorem 6.
4.1 Standard fixed point theorems
Taking Theorem 6 into account, employing
α
(
x, y
)=1for all
x, y ∈X
, we obtain immediately the
following fixed point theorem.
Corollary 1. Let (
X, d
)be a complete metric space and
T
:
X→X
be a given mapping. Suppose that
there exists a function ψ∈Ψsuch that
1
2d(x, T x)≤d(x, y)implies ζ(ψ(M(x, y)),d(Tx,Ty)) ≥0,
for all
x, y ∈X
, where
M
(
x, y
)=
max d(x, y),d(x, T x)+d(y, T y)
2,d(x, T y)+d(y, T x)
2
. Then
T
has a unique fixed point.
In the same way, by letting
α
(
x, y
)=1, for all
x, y ∈X
, in Theorem 7, we find the following result:
Corollary 2. Let (
X, d
)be a complete metric space and
T
:
X→X
be a given mapping. Suppose that
there exists a function ψ∈Ψsuch that
ζ(ψ(M(x, y)),d(Tx,Ty)) ≥0,
for all
x, y ∈X
, where
M
(
x, y
)=
max d(x, y),d(x, T x)+d(y, T y)
2,d(x, T y)+d(y, T x)
2
. Then
T
has a unique fixed point.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
36
On account of the inequality above, the expression (18) and the monotone property of
ψ
, we derive that
d(u, zn+1)≤ψ(max{d(u, zn),d(u, zn+1)}),(19)
for all
n
. Without restriction to the generality, we can suppose that
d
(
u, zn
)
>
0for all
n
. If
max{d(u, zn),d(u, zn+1)}=d(u, zn+1), we get from (19) that
d(u, zn+1)≤ψ(d(u, zn+1)) <d(u, zn+1),
which is a contradiction. Thus we have max{d(u, zn),d(u, zn+1)}=d(u, zn), and
d(u, zn+1)≤ψ(d(u, zn)),
for all n. This implies that
d(u, zn)≤ψn(d(u, z0)),for all n≥1.
Letting n→∞in the above inequality, we obtain
lim
n→∞ d(zn,u)=0.(20)
Similarly, one can show that
lim
n→∞ d(zn,v)=0.(21)
From (20) and (21), it follows that u=v. Thus we proved that uis the unique fixed point of T.
3.1 Immediate consequences
. The first immediate consequence is obtained by removing the Suzuki condition, as in the following
definition.
Definition 8. Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping. We say that
T
is
a generalized type (
α−ψ
)
−Z
-contraction mapping if there exist two functions
α
:
X×X→
[0
,∞
),
ζ∈Zand ψ∈Ψsuch that for all x, y ∈X, we have
ζ(ψ(M(x, y)),α(x, y)d(T x, T y)) ≥0,(22)
where M(x, y) = max d(x, y),d(x, T x)+d(y, Ty)
2,d(x, T y)+d(y, T x)
2.
Theorem 7. Let (
X, d
)be a complete metric space. Suppose that
T
:
X→X
is a generalized Suzuki
type (α−ψ)−Z-contraction mapping and satisfies the following conditions:
(i) Tis triangular α-orbital admissible;
(ii) there exists x0∈Xsuch that α(x0,Tx
0)≥1;
(iii)* Tis continuous
or
(iii)**
if
{xn}
is a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
, then
there exists a subsequence {xn(k)}of {xn}such that α(xn(k),x)≥1for all k,
(iv) the condition (H) holds.
Then there exists u∈Xsuch that Tu =u.
We skipped the proof. Indeed, it is verbatim of the combinations of the proofs of Theorem 4, Theorem
5 and Theorem 6, by removing the related lines about the Suzuki condition.
Taking Example 2 into account, both Theorem 6 and Theorem 7 yields several consequences. In this
direction, one of the first example, by considering the case (i) Example 2 is the following theorem
https://doi.org/10.17993/3ctic.2022.112.26-50
Definition 9. Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping. We say that
T
is a
generalized
α−ψ
contractive mapping if there exist two functions
α
:
X×X→
[0
,∞
)and
ψ∈
Ψsuch
that for all x, y ∈X, we have
α(x, y)d(T x, T y)≤ψ(M(x, y)),(23)
where M(x, y) = max d(x, y),d(x, T x)+d(y, Ty)
2,d(x, T y)+d(y, T x)
2.
Theorem 8. Let (
X, d
)be a complete metric space. Suppose that
T
:
X→X
is a generalized
α−ψ
contractive mapping and satisfies the following conditions:
(i) Tis α−admissible ;
(ii) there exists x0∈Xsuch that α(x0,Tx
0)≥1;
(iii)* Tis continuous
or
(iii)**
if
{xn}
is a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
, then
there exists a subsequence {xn(k)}of {xn}such that α(xn(k),x)≥1for all k,
(iv) the condition (H) holds.
Then there exists u∈Xsuch that Tu =u.
It is evident that by taking, the other cases of Example 2, into account, one can further consequences.
We prefer to skip these consequences by the sake of the length of the manuscript.
4 Further Consequences
In this section, we shall indicate that several existing results in the literature can be deduced easily
from our Theorem 6.
4.1 Standard fixed point theorems
Taking Theorem 6 into account, employing
α
(
x, y
)=1for all
x, y ∈X
, we obtain immediately the
following fixed point theorem.
Corollary 1. Let (
X, d
)be a complete metric space and
T
:
X→X
be a given mapping. Suppose that
there exists a function ψ∈Ψsuch that
1
2d(x, T x)≤d(x, y)implies ζ(ψ(M(x, y)),d(Tx,Ty)) ≥0,
for all
x, y ∈X
, where
M
(
x, y
)=
max d(x, y),d(x, T x)+d(y, T y)
2,d(x, T y)+d(y, T x)
2
. Then
T
has a unique fixed point.
In the same way, by letting
α
(
x, y
)=1, for all
x, y ∈X
, in Theorem 7, we find the following result:
Corollary 2. Let (
X, d
)be a complete metric space and
T
:
X→X
be a given mapping. Suppose that
there exists a function ψ∈Ψsuch that
ζ(ψ(M(x, y)),d(Tx,Ty)) ≥0,
for all
x, y ∈X
, where
M
(
x, y
)=
max d(x, y),d(x, T x)+d(y, T y)
2,d(x, T y)+d(y, T x)
2
. Then
T
has a unique fixed point.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
37
Analogously, by letting α(x, y)=1, for all x, y ∈X, in Theorem 8, we find the following result:
Corollary 3. Let (
X, d
)be a complete metric space and
T
:
X→X
be a given mapping. Suppose that
there exists a function ψ∈Ψsuch that
d(Tx,Ty)≤ψ(M(x, y)),
for all x, y ∈X. Then Thas a unique fixed point.
The following fixed point theorems follow immediately from Corollary 3.
Corollary 4 (see Berinde [22]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a given
mapping. Suppose that there exists a function ψ∈Ψsuch that
d(T x, T y)≤ψ(d(x, y)),
for all x, y ∈X. Then Thas a unique fixed point.
Corollary 5 (see Ćirić [26]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a given mapping.
Suppose that there exists a constant λ∈(0,1) such that
d(T x, T y)≤λ max d(x, y),d(x, T x)+d(y, T y)
2,d(x, T y)+d(y, T x)
2,
for all x, y ∈X. Then Thas a unique fixed point.
Corollary 6 (see Hardy and Rogers [33]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a
given mapping. Suppose that there exist constants A, B, C ≥0with (A+2B+2C)∈(0,1) such that
d(T x, T y)≤Ad(x, y)+B[d(x, T x)+d(y, T y)] + C[d(x, T y)+d(y, Tx)],
for all x, y ∈X. Then Thas a unique fixed point.
Corollary 7 (Banach Contraction Principle [18]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a given mapping. Suppose that there exists a constant λ∈(0,1) such that
d(T x, T y)≤λd(x, y),
for all x, y ∈X. Then Thas a unique fixed point.
Corollary 8 (see Kannan [36]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a given
mapping. Suppose that there exists a constant λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T x)+d(y, T y)],
for all x, y ∈X. Then Thas a unique fixed point.
Corollary 9 (see Chatterjea [24]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a given
mapping. Suppose that there exists a constant λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T y)+d(y, T x)],
for all x, y ∈X. Then Thas a unique fixed point.
https://doi.org/10.17993/3ctic.2022.112.26-50
Corollary 10 (Dass-Gupta Theorem [29]).Let (
X, d
)be a metric space and
T
:
X→X
be a given
mapping. Suppose that there exist constants λ, µ ≥0with λ+µ<1such that
d(T x, T y)≤µd(y, T y)1+d(x, T x)
1+d(x, y)+λd(x, y),for all x, y ∈X. (24)
Then Thas a unique fixed point.
Sketch of the Proof. Consider the functions ψ: [0,∞)→[0,∞)and α:X×X→Rdefined by
ψ(t)=λ t, t ≥0(25)
and
α(x, y)=
1−µd(y, Ty)(1 + d(x, T x))
(1 + d(x, y))d(T x, T y),if Tx =T y,
0,otherwise.
(26)
The rest is simple evaluation. For the detailed proof, we refer to Samet [60].
Corollary 11 (Jaggi theorem [35]).Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping.
Suppose there exist constants λ, µ ≥0with λ+µ<1such that
d(T x, T y)≤µd(x, T x)d(y, T y)
d(x, y)+λd(x, y),for all x, y ∈X, x =y. (27)
Then there exist
ψ∈
Ψand
α
:
X×X→R
such that
T
is an
α
-
ψ
contraction. Then
T
has a unique
fixed point.
Sketch of the Proof.
Consider the functions ψ: [0,∞)→[0,∞)and α:X×X→Rdefined by
ψ(t)=λ t, t ≥0(28)
and
α(x, y)=
1−µd(x, T x)d(y, Ty)
d(x, y)d(T x, T y),if Tx =Ty,
0,otherwise.
(29)
The rest is a simple evaluation. For the detailed proof, we refer to Samet [60].
Theorem 9 ( Berinde Theorem [23]).Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping.
Suppose that there exists λ∈(0,1) and L≥0such that
d(T x, T y)≤λd(x, y)+Ld(y, T x),for all x, y ∈X. (30)
Then Thas a fixed point.
Sketch of the Proof.
Consider the functions ψ: [0,∞)→[0,∞)and α:X×X→Rdefined by
ψ(t)=λ t, t ≥0
and
α(x, y)=
1−Ld(y, Tx)
d(T x, T y),if Tx =Ty,
0,otherwise.
(31)
The rest is straightforward. For more details for the proof, we refer to Samet [60].
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
38
Analogously, by letting α(x, y)=1, for all x, y ∈X, in Theorem 8, we find the following result:
Corollary 3. Let (
X, d
)be a complete metric space and
T
:
X→X
be a given mapping. Suppose that
there exists a function ψ∈Ψsuch that
d(Tx,Ty)≤ψ(M(x, y)),
for all x, y ∈X. Then Thas a unique fixed point.
The following fixed point theorems follow immediately from Corollary 3.
Corollary 4 (see Berinde [22]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a given
mapping. Suppose that there exists a function ψ∈Ψsuch that
d(T x, T y)≤ψ(d(x, y)),
for all x, y ∈X. Then Thas a unique fixed point.
Corollary 5 (see Ćirić [26]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a given mapping.
Suppose that there exists a constant λ∈(0,1) such that
d(T x, T y)≤λ max d(x, y),d(x, T x)+d(y, T y)
2,d(x, T y)+d(y, T x)
2,
for all x, y ∈X. Then Thas a unique fixed point.
Corollary 6 (see Hardy and Rogers [33]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a
given mapping. Suppose that there exist constants A, B, C ≥0with (A+2B+2C)∈(0,1) such that
d(T x, T y)≤Ad(x, y)+B[d(x, T x)+d(y, T y)] + C[d(x, T y)+d(y, Tx)],
for all x, y ∈X. Then Thas a unique fixed point.
Corollary 7 (Banach Contraction Principle [18]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a given mapping. Suppose that there exists a constant λ∈(0,1) such that
d(T x, T y)≤λd(x, y),
for all x, y ∈X. Then Thas a unique fixed point.
Corollary 8 (see Kannan [36]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a given
mapping. Suppose that there exists a constant λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T x)+d(y, T y)],
for all x, y ∈X. Then Thas a unique fixed point.
Corollary 9 (see Chatterjea [24]).Let (
X, d
)be a complete metric space and
T
:
X→X
be a given
mapping. Suppose that there exists a constant λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T y)+d(y, T x)],
for all x, y ∈X. Then Thas a unique fixed point.
https://doi.org/10.17993/3ctic.2022.112.26-50
Corollary 10 (Dass-Gupta Theorem [29]).Let (
X, d
)be a metric space and
T
:
X→X
be a given
mapping. Suppose that there exist constants λ, µ ≥0with λ+µ<1such that
d(T x, T y)≤µd(y, T y)1+d(x, T x)
1+d(x, y)+λd(x, y),for all x, y ∈X. (24)
Then Thas a unique fixed point.
Sketch of the Proof. Consider the functions ψ: [0,∞)→[0,∞)and α:X×X→Rdefined by
ψ(t)=λ t, t ≥0(25)
and
α(x, y)=
1−µd(y, Ty)(1 + d(x, T x))
(1 + d(x, y))d(T x, T y),if Tx =T y,
0,otherwise.
(26)
The rest is simple evaluation. For the detailed proof, we refer to Samet [60].
Corollary 11 (Jaggi theorem [35]).Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping.
Suppose there exist constants λ, µ ≥0with λ+µ<1such that
d(T x, T y)≤µd(x, T x)d(y, T y)
d(x, y)+λd(x, y),for all x, y ∈X, x =y. (27)
Then there exist
ψ∈
Ψand
α
:
X×X→R
such that
T
is an
α
-
ψ
contraction. Then
T
has a unique
fixed point.
Sketch of the Proof.
Consider the functions ψ: [0,∞)→[0,∞)and α:X×X→Rdefined by
ψ(t)=λ t, t ≥0(28)
and
α(x, y)=
1−µd(x, T x)d(y, Ty)
d(x, y)d(T x, T y),if Tx =Ty,
0,otherwise.
(29)
The rest is a simple evaluation. For the detailed proof, we refer to Samet [60].
Theorem 9 ( Berinde Theorem [23]).Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping.
Suppose that there exists λ∈(0,1) and L≥0such that
d(T x, T y)≤λd(x, y)+Ld(y, T x),for all x, y ∈X. (30)
Then Thas a fixed point.
Sketch of the Proof.
Consider the functions ψ: [0,∞)→[0,∞)and α:X×X→Rdefined by
ψ(t)=λ t, t ≥0
and
α(x, y)=
1−Ld(y, Tx)
d(T x, T y),if Tx =Ty,
0,otherwise.
(31)
The rest is straightforward. For more details for the proof, we refer to Samet [60].
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
39
Theorem 10 ( Ćirić’s non-unique fixed point theorem [27].).Let (
X, d
)be a metric space and
T:X→Xbe a given mapping. there exists λ∈(0,1) such that for all x, y ∈X, we have
min{d(T x, T y),d(x, T x),d(y, T y)}−min{d(x, T y),d(y, Tx)}≤λd(x, y).(32)
Then Thas a fixed point.
Sketch of the Proof. Consider the functions ψ: [0,∞)→[0,∞)and α:X×X→Rdefined by
ψ(t)=λ t, t ≥0(33)
and
α(x, y)=
min 1,d(x,T x)
d(T x,T y),d(y,T y)
d(T x,T y)−min d(x,T y)
d(T x,T y),d(y,T x)
d(T x,T y),if Tx =Ty,
0,otherwise.
(34)
The rest is simple evaluation. For the detailed proof, we refer to Samet [60].
Theorem 11 (Suzuki Theorem [63]).Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping.
Suppose that there exists r∈(0,1) such that
(1 + r)−1d(x, T x)≤d(x, y)=⇒d(T x, T y)≤rd(x, y),for all x, y ∈X. (35)
Then Thas a unique fixed point.
Sketch of the Proof. Consider the functions ψ: [0,∞)→[0,∞)and α:X×X→Rdefined by
ψ(t)=r t, t ≥0
and
α(x, y)=
1,if (1 + r)−1d(x, T x)≤d(x, y),
0,otherwise.
(36)
From (35), we have
α(x, y)d(Tx,Ty)≤ψ(d(x, y)),for all x, y ∈X.
Then Tis an α-ψcontraction.
4.2 Fixed point theorems on metric spaces endowed with a partial order
In the last two decades, one of the trend in fixed point theory is the revisit the well-known fixed
point theorem on metric spaces endowed with partial orders [65]. Among all, Ran and Reurings in [55]
revisited the Banach contraction principle in partially ordered sets with some applications to matrix
equations. Another version of the generalization of Banach contraction principle in partially ordered
sets was proposed by Nieto and Rodríguez-López in [49]. Later, this trend was supported by several
authors, see e.g. [2,13,28,34,53,59] and the references cited therein . In this section, we shall show that
Theorem 6 implies easily various fixed point results on a metric space endowed with a partial order. At
first, we need to recall some concepts.
Definition 10. Let (
X, ⪯
)be a partially ordered set and
T
:
X→X
be a given mapping. We say that
Tis nondecreasing with respect to ⪯if
x, y ∈X, x ⪯y=⇒Tx ⪯Ty.
Definition 11. Let (
X, ⪯
)be a partially ordered set. A sequence
{xn}⊂X
is said to be nondecreasing
with respect to ⪯if xn⪯xn+1 for all n.
https://doi.org/10.17993/3ctic.2022.112.26-50
Definition 12. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
. We say that (
X, ⪯,d
)is
regular if for every nondecreasing sequence
{xn}⊂X
such that
xn→x∈X
as
n→∞
, there exists a
subsequence {xn(k)}of {xn}such that xn(k)⪯xfor all k.
We have the following result.
Corollary 12. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a function
ψ∈Ψsuch that 1
2d(x, T x)≤d(x, y)implies ζ(ψ(M(x, y)),d(Tx,Ty)) ≥0,
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Proof. Define the mapping α:X×X→[0,∞)by
α(x, y)=1if x⪯yor x⪰y,
0otherwise.
Clearly, Tis a generalized α−ψcontractive mapping, that is,
α(x, y)d(T x, T y)≤ψ(M(x, y)),
for all
x, y ∈X
. From condition (i), we have
α
(
x0,Tx
0
)
≥
1. Moreover, for all
x, y ∈X
, from the
monotone property of T, we have
α(x, y)≥1=⇒x⪰yor x⪯y=⇒Tx ⪰Ty or Tx ⪯Ty =⇒α(T x, T y)≥1.
Thus
T
is
α−
admissible. Now, if
T
is continuous, the existence of a fixed point follows from Theorem 4.
Suppose now that (
X, ⪯,d
)is regular. Let
{xn}
be a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
. From the regularity hypothesis, there exists a subsequence
{xn(k)}
of
{xn}
such that
xn(k)⪯x
for all
k
. This implies from the definition of
α
that
α
(
xn(k),x
)
≥
1for all
k
. In this
case, the existence of a fixed point follows from Theorem 5. To show the uniqueness, let
x, y ∈X
. By
hypothesis, there exists
z∈X
such that
x⪯z
and
y⪯z
, which implies from the definition of
α
that
α(x, z)≥1and α(y, z)≥1. Thus we deduce the uniqueness of the fixed point by Theorem 6.
Corollary 13. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a function
ψ∈Ψsuch that
ζ(ψ(M(x, y)),d(Tx,Ty)) ≥0,
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
40
Theorem 10 ( Ćirić’s non-unique fixed point theorem [27].).Let (
X, d
)be a metric space and
T:X→Xbe a given mapping. there exists λ∈(0,1) such that for all x, y ∈X, we have
min{d(T x, T y),d(x, T x),d(y, T y)}−min{d(x, T y),d(y, Tx)}≤λd(x, y).(32)
Then Thas a fixed point.
Sketch of the Proof. Consider the functions ψ: [0,∞)→[0,∞)and α:X×X→Rdefined by
ψ(t)=λ t, t ≥0(33)
and
α(x, y)=
min 1,d(x,T x)
d(T x,T y),d(y,T y)
d(T x,T y)−min d(x,T y)
d(T x,T y),d(y,T x)
d(T x,T y),if Tx =Ty,
0,otherwise.
(34)
The rest is simple evaluation. For the detailed proof, we refer to Samet [60].
Theorem 11 (Suzuki Theorem [63]).Let (
X, d
)be a metric space and
T
:
X→X
be a given mapping.
Suppose that there exists r∈(0,1) such that
(1 + r)−1d(x, T x)≤d(x, y)=⇒d(T x, T y)≤rd(x, y),for all x, y ∈X. (35)
Then Thas a unique fixed point.
Sketch of the Proof. Consider the functions ψ: [0,∞)→[0,∞)and α:X×X→Rdefined by
ψ(t)=r t, t ≥0
and
α(x, y)=
1,if (1 + r)−1d(x, T x)≤d(x, y),
0,otherwise.
(36)
From (35), we have
α(x, y)d(Tx,Ty)≤ψ(d(x, y)),for all x, y ∈X.
Then Tis an α-ψcontraction.
4.2 Fixed point theorems on metric spaces endowed with a partial order
In the last two decades, one of the trend in fixed point theory is the revisit the well-known fixed
point theorem on metric spaces endowed with partial orders [65]. Among all, Ran and Reurings in [55]
revisited the Banach contraction principle in partially ordered sets with some applications to matrix
equations. Another version of the generalization of Banach contraction principle in partially ordered
sets was proposed by Nieto and Rodríguez-López in [49]. Later, this trend was supported by several
authors, see e.g. [2,13,28,34,53,59] and the references cited therein . In this section, we shall show that
Theorem 6 implies easily various fixed point results on a metric space endowed with a partial order. At
first, we need to recall some concepts.
Definition 10. Let (
X, ⪯
)be a partially ordered set and
T
:
X→X
be a given mapping. We say that
Tis nondecreasing with respect to ⪯if
x, y ∈X, x ⪯y=⇒Tx ⪯Ty.
Definition 11. Let (
X, ⪯
)be a partially ordered set. A sequence
{xn}⊂X
is said to be nondecreasing
with respect to ⪯if xn⪯xn+1 for all n.
https://doi.org/10.17993/3ctic.2022.112.26-50
Definition 12. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
. We say that (
X, ⪯,d
)is
regular if for every nondecreasing sequence
{xn}⊂X
such that
xn→x∈X
as
n→∞
, there exists a
subsequence {xn(k)}of {xn}such that xn(k)⪯xfor all k.
We have the following result.
Corollary 12. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a function
ψ∈Ψsuch that 1
2d(x, T x)≤d(x, y)implies ζ(ψ(M(x, y)),d(Tx,Ty)) ≥0,
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Proof. Define the mapping α:X×X→[0,∞)by
α(x, y)=1if x⪯yor x⪰y,
0otherwise.
Clearly, Tis a generalized α−ψcontractive mapping, that is,
α(x, y)d(T x, T y)≤ψ(M(x, y)),
for all
x, y ∈X
. From condition (i), we have
α
(
x0,Tx
0
)
≥
1. Moreover, for all
x, y ∈X
, from the
monotone property of T, we have
α(x, y)≥1=⇒x⪰yor x⪯y=⇒Tx ⪰Ty or Tx ⪯Ty =⇒α(T x, T y)≥1.
Thus
T
is
α−
admissible. Now, if
T
is continuous, the existence of a fixed point follows from Theorem 4.
Suppose now that (
X, ⪯,d
)is regular. Let
{xn}
be a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
. From the regularity hypothesis, there exists a subsequence
{xn(k)}
of
{xn}
such that
xn(k)⪯x
for all
k
. This implies from the definition of
α
that
α
(
xn(k),x
)
≥
1for all
k
. In this
case, the existence of a fixed point follows from Theorem 5. To show the uniqueness, let
x, y ∈X
. By
hypothesis, there exists
z∈X
such that
x⪯z
and
y⪯z
, which implies from the definition of
α
that
α(x, z)≥1and α(y, z)≥1. Thus we deduce the uniqueness of the fixed point by Theorem 6.
Corollary 13. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a function
ψ∈Ψsuch that
ζ(ψ(M(x, y)),d(Tx,Ty)) ≥0,
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
41
Corollary 14. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a function
ψ∈Ψsuch that
d(Tx,Ty)≤ψ(M(x, y)),
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
The following results are immediate consequences of Corollary14.
Corollary 15. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a function
ψ∈Ψsuch that
d(T x, T y)≤ψ(d(x, y)),
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Corollary 16. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a constant
λ∈(0,1) such that
d(T x, T y)≤λ max d(x, y),d(x, T x)+d(y, T y)
2,d(x, T y)+d(y, T x)
2,
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Corollary 17. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exist constants
A, B, C ≥0with (A+2B+2C)∈(0,1) such that
d(T x, T y)≤Ad(x, y)+B[d(x, T x)+d(y, T y)] + C[d(x, T y)+d(y, Tx)],
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
https://doi.org/10.17993/3ctic.2022.112.26-50
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Corollary 18 (see Ran and Reurings [55], Nieto and López [49]).Let (
X, ⪯
)be a partially ordered set
and
d
be a metric on
X
such that (
X, d
)is complete. Let
T
:
X→X
be a nondecreasing mapping with
respect to ⪯. Suppose that there exists a constant λ∈(0,1) such that
d(T x, T y)≤λd(x, y),
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Corollary 19. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a constant
λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T x)+d(y, T y)],
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Corollary 20. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a constant
λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T y)+d(y, T x)],
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
4.3 Fixed point theorems for cyclic contractive mappings
The notion of "cyclic contraction mapping"to find a fixed point was proposed by Kirk, Srinivasan and
Veeramani [45]. In this paper, they revisited the famous Banach Contraction Mapping Principle was
proved by Kirk, Srinivasan and Veeramani [45] via cyclic contraction. Following this pioneer work [45],
several fixed point theorems in the framework of cyclic contractive mappings have appeared (see, for
instant, [1,37, 39, 51,52,57]). In this section, we shall proved that Theorem 6 implies several fixed point
theorems in the context of cyclic contractive mappings.
We have the following result.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
42
Corollary 14. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a function
ψ∈Ψsuch that
d(Tx,Ty)≤ψ(M(x, y)),
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
The following results are immediate consequences of Corollary14.
Corollary 15. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a function
ψ∈Ψsuch that
d(T x, T y)≤ψ(d(x, y)),
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Corollary 16. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a constant
λ∈(0,1) such that
d(T x, T y)≤λ max d(x, y),d(x, T x)+d(y, T y)
2,d(x, T y)+d(y, T x)
2,
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Corollary 17. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exist constants
A, B, C ≥0with (A+2B+2C)∈(0,1) such that
d(T x, T y)≤Ad(x, y)+B[d(x, T x)+d(y, T y)] + C[d(x, T y)+d(y, Tx)],
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
https://doi.org/10.17993/3ctic.2022.112.26-50
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Corollary 18 (see Ran and Reurings [55], Nieto and López [49]).Let (
X, ⪯
)be a partially ordered set
and
d
be a metric on
X
such that (
X, d
)is complete. Let
T
:
X→X
be a nondecreasing mapping with
respect to ⪯. Suppose that there exists a constant λ∈(0,1) such that
d(T x, T y)≤λd(x, y),
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Corollary 19. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a constant
λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T x)+d(y, T y)],
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
Corollary 20. Let (
X, ⪯
)be a partially ordered set and
d
be a metric on
X
such that (
X, d
)is complete.
Let
T
:
X→X
be a nondecreasing mapping with respect to
⪯
. Suppose that there exists a constant
λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T y)+d(y, T x)],
for all x, y ∈Xwith x⪰y. Suppose also that the following conditions hold:
(i) there exists x0∈Xsuch that x0⪯Tx
0;
(ii) Tis continuous or (X, ⪯,d)is regular.
Then
T
has a fixed point. Moreover, if for all
x, y ∈X
there exists
z∈X
such that
x⪯z
and
y⪯z
,
we have uniqueness of the fixed point.
4.3 Fixed point theorems for cyclic contractive mappings
The notion of "cyclic contraction mapping"to find a fixed point was proposed by Kirk, Srinivasan and
Veeramani [45]. In this paper, they revisited the famous Banach Contraction Mapping Principle was
proved by Kirk, Srinivasan and Veeramani [45] via cyclic contraction. Following this pioneer work [45],
several fixed point theorems in the framework of cyclic contractive mappings have appeared (see, for
instant, [1,37, 39, 51,52,57]). In this section, we shall proved that Theorem 6 implies several fixed point
theorems in the context of cyclic contractive mappings.
We have the following result.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
43
Corollary 21. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2and ζ∈Z. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a function ψ∈Ψsuch that
1
2d(x, T x)≤d(x, y)implies ζ(ψ(M(x, y)),d(T x, T y)) ≥0;
Then Thas a unique fixed point that belongs to A1∩A2.
Proof. Since
A1
and
A2
are closed subsets of the complete metric space (
X, d
), then (
Y,d
)is complete.
Define the mapping α:Y×Y→[0,∞)by
α(x, y)=1if (x, y)∈(A1×A2)∪(A2×A1),
0otherwise.
From (II) and the definition of α, we can write
α(x, y)d(T x, T y)≤ψ(M(x, y)),
for all x, y ∈Y. Thus Tis a generalized α−ψcontractive mapping.
Let (x, y)∈Y×Ysuch that α(x, y)≥1. If (x, y)∈A1×A2, from (I), (T x, T y)∈A2×A1, which
implies that
α
(
T x, T y
)
≥
1. If (
x, y
)
∈A2×A1
, from (I), (
T x, T y
)
∈A1×A2
, which implies that
α(T x, T y)≥1. Thus in all cases, we have α(T x, T y)≥1. This implies that Tis α−admissible.
Also, from (I), for any a∈A1, we have (a, T a)∈A1×A2, which implies that α(a, T a)≥1.
Now, let
{xn}
be a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
.
This implies from the definition of αthat
(xn,x
n+1)∈(A1×A2)∪(A2×A1),for all n.
Since (A1×A2)∪(A2×A1)is a closed set with respect to the Euclidean metric, we get that
(x, x)∈(A1×A2)∪(A2×A1),
which implies that
x∈A1∩A2
. Thus we get immediately from the definition of
α
that
α
(
xn,x
)
≥
1
for all n.
Finally, let
x, y ∈Fix
(
T
). From (I), this implies that
x, y ∈A1∩A2
. So, for any
z∈Y
, we have
α(x, z)≥1and α(y, z)≥1. Thus condition (H) is satisfied.
Now, all the hypotheses of Theorem 6 are satisfied, we deduce that
T
has a unique fixed point that
belongs to A1∩A2(from (I)).
Corollary 22. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2and ζ∈Z. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a function ψ∈Ψsuch that
ζ(ψ(M(x, y)),d(T x, T y)) ≥0;
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 23. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2. Suppose that the following conditions hold:
https://doi.org/10.17993/3ctic.2022.112.26-50
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a function ψ∈Ψsuch that
d(T x, T y)≤ψ(M(x, y)),for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
The following results are immediate consequences of Corollary 23.
Corollary 24 (see Pacurar and Rus [51]).Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric
space (
X, d
)and
T
:
Y→Y
be a given mapping, where
Y
=
A1∪A2
. Suppose that the following
conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a function ψ∈Ψsuch that
d(T x, T y)≤ψ(d(x, y)),for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 25. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a constant λ∈(0,1) such that
d(T x, T y)≤λ max d(x, y),d(x, T x)+d(y, T y)
2,d(x, T y)+d(y, T x)
2,for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 26. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exist constants A, B, C ≥0with (A+2B+2C)∈(0,1) such that
d(T x, T y)≤Ad(x, y)+B[d(x, T x)+d(y, T y)] + C[d(x, T y)+d(y, Tx)],for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 27 (see Kirk et al. [45]).Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space
(
X, d
)and
T
:
Y→Y
be a given mapping, where
Y
=
A1∪A2
. Suppose that the following conditions
hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a constant λ∈(0,1) such that
d(T x, T y)≤λd(x, y),for all (x, y)∈A1×A2.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
44
Corollary 21. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2and ζ∈Z. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a function ψ∈Ψsuch that
1
2d(x, T x)≤d(x, y)implies ζ(ψ(M(x, y)),d(T x, T y)) ≥0;
Then Thas a unique fixed point that belongs to A1∩A2.
Proof. Since
A1
and
A2
are closed subsets of the complete metric space (
X, d
), then (
Y,d
)is complete.
Define the mapping α:Y×Y→[0,∞)by
α(x, y)=1if (x, y)∈(A1×A2)∪(A2×A1),
0otherwise.
From (II) and the definition of α, we can write
α(x, y)d(T x, T y)≤ψ(M(x, y)),
for all x, y ∈Y. Thus Tis a generalized α−ψcontractive mapping.
Let (x, y)∈Y×Ysuch that α(x, y)≥1. If (x, y)∈A1×A2, from (I), (T x, T y)∈A2×A1, which
implies that
α
(
T x, T y
)
≥
1. If (
x, y
)
∈A2×A1
, from (I), (
T x, T y
)
∈A1×A2
, which implies that
α(T x, T y)≥1. Thus in all cases, we have α(T x, T y)≥1. This implies that Tis α−admissible.
Also, from (I), for any a∈A1, we have (a, T a)∈A1×A2, which implies that α(a, T a)≥1.
Now, let
{xn}
be a sequence in
X
such that
α
(
xn,x
n+1
)
≥
1for all
n
and
xn→x∈X
as
n→∞
.
This implies from the definition of αthat
(xn,x
n+1)∈(A1×A2)∪(A2×A1),for all n.
Since (A1×A2)∪(A2×A1)is a closed set with respect to the Euclidean metric, we get that
(x, x)∈(A1×A2)∪(A2×A1),
which implies that
x∈A1∩A2
. Thus we get immediately from the definition of
α
that
α
(
xn,x
)
≥
1
for all n.
Finally, let
x, y ∈Fix
(
T
). From (I), this implies that
x, y ∈A1∩A2
. So, for any
z∈Y
, we have
α(x, z)≥1and α(y, z)≥1. Thus condition (H) is satisfied.
Now, all the hypotheses of Theorem 6 are satisfied, we deduce that
T
has a unique fixed point that
belongs to A1∩A2(from (I)).
Corollary 22. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2and ζ∈Z. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a function ψ∈Ψsuch that
ζ(ψ(M(x, y)),d(T x, T y)) ≥0;
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 23. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2. Suppose that the following conditions hold:
https://doi.org/10.17993/3ctic.2022.112.26-50
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a function ψ∈Ψsuch that
d(T x, T y)≤ψ(M(x, y)),for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
The following results are immediate consequences of Corollary 23.
Corollary 24 (see Pacurar and Rus [51]).Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric
space (
X, d
)and
T
:
Y→Y
be a given mapping, where
Y
=
A1∪A2
. Suppose that the following
conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a function ψ∈Ψsuch that
d(T x, T y)≤ψ(d(x, y)),for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 25. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a constant λ∈(0,1) such that
d(T x, T y)≤λ max d(x, y),d(x, T x)+d(y, T y)
2,d(x, T y)+d(y, T x)
2,for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 26. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exist constants A, B, C ≥0with (A+2B+2C)∈(0,1) such that
d(T x, T y)≤Ad(x, y)+B[d(x, T x)+d(y, T y)] + C[d(x, T y)+d(y, Tx)],for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 27 (see Kirk et al. [45]).Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space
(
X, d
)and
T
:
Y→Y
be a given mapping, where
Y
=
A1∪A2
. Suppose that the following conditions
hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a constant λ∈(0,1) such that
d(T x, T y)≤λd(x, y),for all (x, y)∈A1×A2.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
45
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 28. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a constant λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T x)+d(y, T y)],for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 29. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a constant λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T y)+d(y, T x)],for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
5 Conclusion
The fixed point theorem is one of the most actively studied research fields of recent times. Naturally,
there are many publications on this subject and several new results are announced. This causes the
literature to become rather disorganized and dysfunctional. The more troublesome situation is that the
existing theorems are rediscovered again and again due to this messiness. More accurately, the results
have been repeated. It is therefore essential to organize the fixed-point theory literature, weeding out
false and/or repetitive results, and, if possible, combining and unifying existing results into a more
general framework. In this work, we show that using admissible mapping, many existing fixed point
theorems can be written as a consequence of the main theorem we have given.
Note that the consequences of the main result of the paper, Theorem 6 is not complete. It is possible
to add several corollaries. On the other hand, we prefer to skip these possible consequences, since it is
clear how the possible result can be concluded from our main theorem and how can be proved. Further,
we underline that the main theorem can be derived in the distinct abstract spaces, such as, partial
metric space, b-metric space, quasi-metric space, and so on.
REFERENCES
[1]
Agarwal R.P.,Alghamdi M.A. and Shahzad N. (2012). Fixed point theory for cyclic generalized
contractions in partial metric spaces. Fixed Point Theory Appl., 2012:40.
[2]
Agarwal R.P.,El-Gebeily M.A. and O’Regan D. (2008). Generalized contractions in partially
ordered metric spaces. Appl. Anal. 87, 109–116.
[3]
Aksoy U., Karapinar E. and Erhan I. M. (2016). Fixed points of generalized alpha-admissible
contractions on b-metric spaces with an application to boundary value problems. Journal of Nonlinear
and Convex Analysis, Volume 17, Number 6, 1095-1108
https://doi.org/10.17993/3ctic.2022.112.26-50
[4]
Alharbi A.S. S.,Alsulami H. H., and Karapinar E. (2017). On the Power of Simulation and
Admissible Functions in Metric Fixed Point Theory. Journal of Function Spaces, Volume 2017 ,
Article ID 2068163, 7 pages
[5]
Ali M.U., Kamram T. and Karapinar E. (2014). An approach to existence of fixed points of
generalized contractive multivalued mappings of integral type via admissible mapping. Abstract and
Applied Analysis, Article Id: 141489
[6]
Ali M.U., Kamram T. and Karapinar E. (2014). Fixed point of
α−ψ
-contractive type mappings
in uniform spaces. Fixed Point Theory and Applications.
[7]
Ali M.U.,Kamram T., Karapinar E. (2014). A new approach to (
α, ψ
)-contractive nonself
multivalued mappings. Journal of Inequalities and Applications, 2014:71.
[8]
Ali M.U., Kamram T. and Karapinar E. (2014). (
α, ψ, ξ
)-contractive multi-valued mappings.
Fixed Point Theory and Applications, 2014, 2014:7
[9]
Ali M.U., Kamram T., Karapinar E. (2016). Discussion on
ϕ
-Strictly Contractive Nonself
Multivalued Maps. Vietnam Journal of Mathematics June 2016, Volume 44, Issue 2, pp 441-447
[10]
Al-Mezel S.,Chen C. M.,Karapinar E. and Rakocevic V. (2014). Fixed point results
for various
α
-admissible contractive mappings on metric-like spaces. Abstract and Applied Analysis
Volume 2014 , Article ID 379358
[11]
Alsulami H.,Gulyaz S.,Karapinar E. and Erhan I.M. (2014). Fixed point theorems for a
class of alpha-admissible contractions and applications to boundary value problem. Abstract and
Applied Analysis, Article Id: 187031
[12]
Alsulami H., Karapınar E., Khojasteh F., Roldán-López-de-Hierro A. F. (2014). A
proposal to the study of contractions in quasi-metric spaces. Discrete Dynamics in Nature and Society
2014, Article ID 269286, 10 pages.
[13]
Altun I.,Simsek H. (2010). Some fixed point theorems on ordered metric spaces and application.
Fixed Point Theory Appl. 2010, Article ID 621492, 17 pages.
[14]
Arshad M., Ameer E. and Karapinar E. (2016). Generalized contractions with triangular
alpha-orbital admissible mapping on Branciari metric spaces. Journal of Inequalities and Applications
2016, 2016:63 (16 February 2016).
[15]
Aydi H., Felhi A., Karapinar E. and Alojail F.A. (2018). Fixed points on quasi-metric spaces
via simulation functions and consequences. J. Math. Anal., 9, 10-24
[16]
Aydi H.,Karapinar E. and Yazidi H. (2017). Modified F-Contractions via alpha-Admissible
Mappings and Application to Integral Equations. FILOMAT Volume: 31 Issue: 5 , Pages: 1141- 148
Published: 2017.
[17]
Aydi H., Karapinar E. and Zhang D. (2017). A note on generalized admissible-Meir-Keeler-
contractions in the context of generalized metric spaces. Results in Mathematics, February 2017,
Volume 71, Issue 1, pp 73-92.
[18]
Banach S. (1922). Sur les opérations dans les ensembles abstraits et leur application aux equations
itegrales. Fund. Math. 3 133-181.
[19]
Berinde V. (1993). Generalized contractions in quasimetric spaces. Seminar on Fixed Point
Theory, Preprint no. 3(1993), 3-9.
[20]
Berinde V. (1996). Sequences of operators and fixed points in quasimetric spaces. Stud. Univ.
"Babeş-Bolyai", Math., 16(4), 23-27.
[21]
Berinde V. (1997). Contracţii generalizate şi aplicaţii. Editura Club Press 22, Baia Mare, 1997.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
46
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 28. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a constant λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T x)+d(y, T y)],for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
Corollary 29. Let
{Ai}2
i=1
be nonempty closed subsets of a complete metric space (
X, d
)and
T
:
Y→Y
be a given mapping, where Y=A1∪A2. Suppose that the following conditions hold:
(I) T(A1)⊆A2and T(A2)⊆A1;
(II) there exists a constant λ∈(0,1/2) such that
d(T x, T y)≤λ[d(x, T y)+d(y, T x)],for all (x, y)∈A1×A2.
Then Thas a unique fixed point that belongs to A1∩A2.
5 Conclusion
The fixed point theorem is one of the most actively studied research fields of recent times. Naturally,
there are many publications on this subject and several new results are announced. This causes the
literature to become rather disorganized and dysfunctional. The more troublesome situation is that the
existing theorems are rediscovered again and again due to this messiness. More accurately, the results
have been repeated. It is therefore essential to organize the fixed-point theory literature, weeding out
false and/or repetitive results, and, if possible, combining and unifying existing results into a more
general framework. In this work, we show that using admissible mapping, many existing fixed point
theorems can be written as a consequence of the main theorem we have given.
Note that the consequences of the main result of the paper, Theorem 6 is not complete. It is possible
to add several corollaries. On the other hand, we prefer to skip these possible consequences, since it is
clear how the possible result can be concluded from our main theorem and how can be proved. Further,
we underline that the main theorem can be derived in the distinct abstract spaces, such as, partial
metric space, b-metric space, quasi-metric space, and so on.
REFERENCES
[1]
Agarwal R.P.,Alghamdi M.A. and Shahzad N. (2012). Fixed point theory for cyclic generalized
contractions in partial metric spaces. Fixed Point Theory Appl., 2012:40.
[2]
Agarwal R.P.,El-Gebeily M.A. and O’Regan D. (2008). Generalized contractions in partially
ordered metric spaces. Appl. Anal. 87, 109–116.
[3]
Aksoy U., Karapinar E. and Erhan I. M. (2016). Fixed points of generalized alpha-admissible
contractions on b-metric spaces with an application to boundary value problems. Journal of Nonlinear
and Convex Analysis, Volume 17, Number 6, 1095-1108
https://doi.org/10.17993/3ctic.2022.112.26-50
[4]
Alharbi A.S. S.,Alsulami H. H., and Karapinar E. (2017). On the Power of Simulation and
Admissible Functions in Metric Fixed Point Theory. Journal of Function Spaces, Volume 2017 ,
Article ID 2068163, 7 pages
[5]
Ali M.U., Kamram T. and Karapinar E. (2014). An approach to existence of fixed points of
generalized contractive multivalued mappings of integral type via admissible mapping. Abstract and
Applied Analysis, Article Id: 141489
[6]
Ali M.U., Kamram T. and Karapinar E. (2014). Fixed point of
α−ψ
-contractive type mappings
in uniform spaces. Fixed Point Theory and Applications.
[7]
Ali M.U.,Kamram T., Karapinar E. (2014). A new approach to (
α, ψ
)-contractive nonself
multivalued mappings. Journal of Inequalities and Applications, 2014:71.
[8]
Ali M.U., Kamram T. and Karapinar E. (2014). (
α, ψ, ξ
)-contractive multi-valued mappings.
Fixed Point Theory and Applications, 2014, 2014:7
[9]
Ali M.U., Kamram T., Karapinar E. (2016). Discussion on
ϕ
-Strictly Contractive Nonself
Multivalued Maps. Vietnam Journal of Mathematics June 2016, Volume 44, Issue 2, pp 441-447
[10]
Al-Mezel S.,Chen C. M.,Karapinar E. and Rakocevic V. (2014). Fixed point results
for various
α
-admissible contractive mappings on metric-like spaces. Abstract and Applied Analysis
Volume 2014 , Article ID 379358
[11]
Alsulami H.,Gulyaz S.,Karapinar E. and Erhan I.M. (2014). Fixed point theorems for a
class of alpha-admissible contractions and applications to boundary value problem. Abstract and
Applied Analysis, Article Id: 187031
[12]
Alsulami H., Karapınar E., Khojasteh F., Roldán-López-de-Hierro A. F. (2014). A
proposal to the study of contractions in quasi-metric spaces. Discrete Dynamics in Nature and Society
2014, Article ID 269286, 10 pages.
[13]
Altun I.,Simsek H. (2010). Some fixed point theorems on ordered metric spaces and application.
Fixed Point Theory Appl. 2010, Article ID 621492, 17 pages.
[14]
Arshad M., Ameer E. and Karapinar E. (2016). Generalized contractions with triangular
alpha-orbital admissible mapping on Branciari metric spaces. Journal of Inequalities and Applications
2016, 2016:63 (16 February 2016).
[15]
Aydi H., Felhi A., Karapinar E. and Alojail F.A. (2018). Fixed points on quasi-metric spaces
via simulation functions and consequences. J. Math. Anal., 9, 10-24
[16]
Aydi H.,Karapinar E. and Yazidi H. (2017). Modified F-Contractions via alpha-Admissible
Mappings and Application to Integral Equations. FILOMAT Volume: 31 Issue: 5 , Pages: 1141- 148
Published: 2017.
[17]
Aydi H., Karapinar E. and Zhang D. (2017). A note on generalized admissible-Meir-Keeler-
contractions in the context of generalized metric spaces. Results in Mathematics, February 2017,
Volume 71, Issue 1, pp 73-92.
[18]
Banach S. (1922). Sur les opérations dans les ensembles abstraits et leur application aux equations
itegrales. Fund. Math. 3 133-181.
[19]
Berinde V. (1993). Generalized contractions in quasimetric spaces. Seminar on Fixed Point
Theory, Preprint no. 3(1993), 3-9.
[20]
Berinde V. (1996). Sequences of operators and fixed points in quasimetric spaces. Stud. Univ.
"Babeş-Bolyai", Math., 16(4), 23-27.
[21]
Berinde V. (1997). Contracţii generalizate şi aplicaţii. Editura Club Press 22, Baia Mare, 1997.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
47
[22] Berinde V. (2002). Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare.
[23]
Berinde V. (2004). Approximating fixed points of weak contractions using the Picard iteration.
Nonlinear Anal. Forum. 9 43-53.
[24] Chatterjea S.K. (1972). Fixed point theorems. C.R. Acad. Bulgare Sci. 25, 727-730.
[25]
Chen C.-M., Abkar A., Ghods S. and Karapinar E. (2017). Fixed Point Theory for the
ϕ
-Admissible Meir-Keeler Type Set Contractions Having KKM* Property on Almost Convex Sets.
Appl. Math. Inf. Sci. 11, No. 1, 171-176.
[26]
Ćirić Lj.B. (1972). Fixed points for generalized multi-valued mappings. Mat. Vesnik. 9 (24),
265-272.
[27] Ćirić Lj. (1974). On some maps with a nonunique fixed point. Pub. Inst. Math. 17, 52-58.
[28]
Ćirić Lj.B.,Cakić N.,Rajović M. and Ume J.S. (2008). Monotone generalized nonlinear
contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008 Article ID 131294, 11
pages.
[29]
Dass B. K. and Gupta S. (1975). An extension of Banach contraction principle through rational
expression. Indian. J. Pure. Appl. Math., 6, 1455-1458.
[30]
Eshraghisamani M., Vaezpour S. M., Asadi M. (2018). New fixed point results with
αqsp
-
admissible contractions on
b
–Branciari metric spaces. Journal of Inequalities and Special Functions,
9(3):38-46, 2018.
[31]
Gulyaz S., Karapinar E. and Erhanand I. M. (2017) Generalized
ψ
–Meir-Keeler contraction
mappings on Branciari b-metric spaces. Filomat, vol. 31, no. 17, pp. 5445-5456, 2017.
[32]
Hammache K.,Karapinar E.,Ould-Hammouda A. (2017). On Admissible Weak Contractions
in b-Metric-Like Space. Journal Of Mathematical Analysis Volume: 8 Issue: 3, Pages: 167-180
Published: 2017
[33]
Hardy G.E. and Rogers T.D. (1973). A generalization of a fixed point theorem of Reich. Canad.
Math. Bull. 16, 201-206.
[34]
Harjani J. and Sadarangani K. (2008). Fixed point theorems for weakly contractive mappings
in partially ordered sets. Nonlinear Anal. 71, 3403-3410.
[35]
Jaggi D. S. (1977). Some unique fixed point theorems. Indian. J. Pure. Appl. Math., Vol. 8, No
2, 223-230.
[36] Kannan R. (1968). Some results on fixed points. Bull. Clacutta. Math. Soc. 10, 71-76.
[37]
Karapınar E. (2011). Fixed point theory for cyclic weak
ϕ
-contraction. Appl. Math. Lett. 24 (6),
822-825.
[38]
Karapinar E. (2013). On best proximity point of
ψ
-Geraghty contractions. Fixed Point Theory
and Applications, 2013:200.
[39]
Karapınar E. and Sadaranagni K. (2011). Fixed point theory for cyclic (
ϕ
-
ψ
)-contractions.
Fixed Point Theory Appl. , 2011:69.
[40]
Karapinar E. and B. Samet, Generalized (
α, ψ
)contractive type mappings and related fixed
point theorems with applications, Abstr. Appl. Anal , 2012 (2012) Article id: 793486
[41]
Karapinar E., Kumam P., Salimi P. (2013). On
α
-
ψ
-Meir-Keeler contractive mappings. Fixed
Point Theory and Applications 2013, 94.
[42]
Karapinar E. (2016). Fixed points results via simulation functions. Filomat 2016, 30, 2343–2350
https://doi.org/10.17993/3ctic.2022.112.26-50
[43]
Karapinar E., Khojasteh F. (2017). An approach to best proximity points results via simulation
functions. J. Fixed Point Theory Appl. 2017, 19, 1983-1995.
[44]
Khojasteh F., Shukla S. and Radenović S. (2015). A new approach to the study of fixed point
theorems via simulation functions. Filomat 29:6 , 1189-1194.
[45]
Kirk W.A.,Srinivasan P.S. and Veeramani P. (2003). Fixed points for mappings satisfying
cyclical contractive conditions. Fixed Point Theory. 4(1), 79–89.
[46]
Monfared H., Asadi M., Azhini M.
F
(
ψ, φ
)-contractions for
α
–admissible mappings on metric
spaces and related fixed point results. Communications in Nonlinear Analysis (CNA), 2(1):86-94, 2.
[47]
Monfared H., Asadi M., Azhini M., and O’regan D. (2018)
F
(
ψ, φ
)-contractions for
α
–admissible mappings on
m
-metric spaces. Fixed Point Theory and Applications, 2018(1):22, 2018.
[48]
Monfared H., Asadi M., Farajzadeh A. (2020). New generalization of Darbo’s fixed point theo-
rem via
α
-admissible simulation functions with application. Sahand Communications in Mathematical
Analysis, 17(2):161-171, 2020.
[49]
Nieto J.J. and Rodríguez-López R. (2005). Contractive Mapping Theorems in Partially Ordered
Sets and Applications to Ordinary Differential Equations. Order. 22, 223–239.
[50]
Ozyurt S. G. (2017). On some alpha-admissible contraction mappings on Branciari b-metric
spaces. Advances in the Theory of Nonlinear Analysis and its Applications, vol. 1, no. 1, pp. 1–13,
2017.
[51]
Pacurar M. and Rus I.A. (2010). Fixed point theory for cyclic
φ
-contractions. Nonlinear Anal.,
72, 1181–1187.
[52]
Petric M.A. (2010). Some results concerning cyclical contractive mappings. General Mathematics.
18 (4), 213-226.
[53]
Petruşel A. and Rus I.A. (2006). Fixed point theorems in ordered L-spaces. Proc. Amer. Math.
Soc. 134, 411–418.
[54]
Popescu O. (2014). Some new fixed point theorems for
α
-Geraghty contraction type maps in
metric spaces. Fixed Point Theory Appl. 2014, 2014:190
[55]
Ran A.C.M. and Reurings M.C.B. (2003). A fixed point theorem in partially ordered sets and
some applications to matrix equations. Proc. Amer. Math. Soc., 132, 1435-1443.
[56]
Roldán-López-de-Hierro A. F., Karapınar E., Roldán-López-de-Hierro C., Martínez-
Moreno J. (2015). Coincidence point theorems on metric spaces via simulation functions. J. Comput.
Appl. Math. 275, 345–355.
[57]
Rus I.A. (2005). Cyclic representations and fixed points. Ann. T. Popoviciu, Seminar Funct. Eq.
Approx. Convexity 3, 171-178.
[58]
Rus, I. A. (2001). Generalized contractions and applications. Cluj University Press, Cluj-Napoca,
2001.
[59]
Samet B. (2010). Coupled fixed point theorems for a generalized Meir-Keeler contraction in
partially ordered metric spaces. Nonlinear Anal. 72. 4508-4517.
[60]
Samet B. (2014). Fixed Points for
α−ψ
Contractive Mappings With An Application To Quadratic
Integral Equations. Electronic Journal of Differential Equations, Vol. 2014 , No. 152, pp. 1–18.
[61]
Samet B.,Vetro C. and Vetro P. (2012). Fixed point theorem for
α−ψ
contractive type
mappings. Nonlinear Anal. 75, 2154-2165.
https://doi.org/10.17993/3ctic.2022.112.26-50
3C TIC. Cuadernos de desarrollo aplicados a las TIC. ISSN: 2254-6529
Ed. 41 Vol. 11 N.º 2 August - December 2022
48
[22] Berinde V. (2002). Iterative Approximation of Fixed Points. Editura Efemeride, Baia Mare.
[23]
Berinde V. (2004). Approximating fixed points of weak contractions using the Picard iteration.
Nonlinear Anal. Forum. 9 43-53.
[24] Chatterjea S.K. (1972). Fixed point theorems. C.R. Acad. Bulgare Sci. 25, 727-730.
[25]
Chen C.-M., Abkar A., Ghods S. and Karapinar E. (2017). Fixed Point Theory for the
ϕ
-Admissible Meir-Keeler Type Set Contractions Having KKM* Property on Almost Convex Sets.
Appl. Math. Inf. Sci. 11, No. 1, 171-176.
[26]
Ćirić Lj.B. (1972). Fixed points for generalized multi-valued mappings. Mat. Vesnik. 9 (24),
265-272.
[27] Ćirić Lj. (1974). On some maps with a nonunique fixed point. Pub. Inst. Math. 17, 52-58.
[28]
Ćirić Lj.B.,Cakić N.,Rajović M. and Ume J.S. (2008). Monotone generalized nonlinear
contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2008 Article ID 131294, 11
pages.
[29]
Dass B. K. and Gupta S. (1975). An extension of Banach contraction principle through rational
expression. Indian. J. Pure. Appl. Math., 6, 1455-1458.
[30]
Eshraghisamani M., Vaezpour S. M., Asadi M. (2018). New fixed point results with
αqsp
-
admissible contractions on
b
–Branciari metric spaces. Journal of Inequalities and Special Functions,
9(3):38-46, 2018.
[31]
Gulyaz S., Karapinar E. and Erhanand I. M. (2017) Generalized
ψ
–Meir-Keeler contraction
mappings on Branciari b-metric spaces. Filomat, vol. 31, no. 17, pp. 5445-5456, 2017.
[32]
Hammache K.,Karapinar E.,Ould-Hammouda A. (2017). On Admissible Weak Contractions
in b-Metric-Like Space. Journal Of Mathematical Analysis Volume: 8 Issue: 3, Pages: 167-180
Published: 2017
[33]
Hardy G.E. and Rogers T.D. (1973). A generalization of a fixed point theorem of Reich. Canad.
Math. Bull. 16, 201-206.
[34]
Harjani J. and Sadarangani K. (2008). Fixed point theorems for weakly contractive mappings
in partially ordered sets. Nonlinear Anal. 71, 3403-3410.
[35]
Jaggi D. S. (1977). Some unique fixed point theorems. Indian. J. Pure. Appl. Math., Vol. 8, No
2, 223-230.
[36] Kannan R. (1968). Some results on fixed points. Bull. Clacutta. Math. Soc. 10, 71-76.
[37]
Karapınar E. (2011). Fixed point theory for cyclic weak
ϕ
-contraction. Appl. Math. Lett. 24 (6),
822-825.
[38]
Karapinar E. (2013). On best proximity point of
ψ
-Geraghty contractions. Fixed Point Theory
and Applications, 2013:200.
[39]
Karapınar E. and Sadaranagni K. (2011). Fixed point theory for cyclic (
ϕ
-
ψ
)-contractions.
Fixed Point Theory Appl. , 2011:69.
[40]
Karapinar E. and B. Samet, Generalized (
α, ψ
)contractive type mappings and related fixed
point theorems with applications, Abstr. Appl. Anal , 2012 (2012) Article id: 793486
[41]
Karapinar E., Kumam P., Salimi P. (2013). On
α
-
ψ
-Meir-Keeler contractive mappings. Fixed
Point Theory and Applications 2013, 94.
[42]
Karapinar E. (2016). Fixed points results via simulation functions. Filomat 2016, 30, 2343–2350
https://doi.org/10.17993/3ctic.2022.112.26-50
[43]
Karapinar E., Khojasteh F. (2017). An approach to best proximity points results via simulation
functions. J. Fixed Point Theory Appl. 2017, 19, 1983-1995.
[44]
Khojasteh F., Shukla S. and Radenović S. (2015). A new approach to the study of fixed point
theorems via simulation functions. Filomat 29:6 , 1189-1194.
[45]
Kirk W.A.,Srinivasan P.S. and Veeramani P. (2003). Fixed points for mappings satisfying
cyclical contractive conditions. Fixed Point Theory. 4(1), 79–89.
[46]
Monfared H., Asadi M., Azhini M.
F
(
ψ, φ
)-contractions for
α
–admissible mappings on metric
spaces and related fixed point results. Communications in Nonlinear Analysis (CNA), 2(1):86-94, 2.
[47]
Monfared H., Asadi M., Azhini M., and O’regan D. (2018)
F
(
ψ, φ
)-contractions for
α
–admissible mappings on
m
-metric spaces. Fixed Point Theory and Applications, 2018(1):22, 2018.
[48]
Monfared H., Asadi M., Farajzadeh A. (2020). New generalization of Darbo’s fixed point theo-
rem via
α
-admissible simulation functions with application. Sahand Communications in Mathematical
Analysis, 17(2):161-171, 2020.
[49]
Nieto J.J. and Rodríguez-López R. (2005). Contractive Mapping Theorems in Partially Ordered
Sets and Applications to Ordinary Differential Equations. Order. 22, 223–239.
[50]
Ozyurt S. G. (2017). On some alpha-admissible contraction mappings on Branciari b-metric
spaces. Advances in the Theory of Nonlinear Analysis and its Applications, vol. 1, no. 1, pp. 1–13,
2017.
[51]
Pacurar M. and Rus I.A. (2010). Fixed point theory for cyclic
φ
-contractions. Nonlinear Anal.,
72, 1181–1187.
[52]
Petric M.A. (2010). Some results concerning cyclical contractive mappings. General Mathematics.
18 (4), 213-226.
[53]
Petruşel A. and Rus I.A. (2006). Fixed point theorems in ordered L-spaces. Proc. Amer. Math.
Soc. 134, 411–418.
[54]
Popescu O. (2014). Some new fixed point theorems for
α
-Geraghty contraction type maps in
metric spaces. Fixed Point Theory Appl. 2014, 2014:190
[55]
Ran A.C.M. and Reurings M.C.B. (2003). A fixed point theorem in partially ordered sets and
some applications to matrix equations. Proc. Amer. Math. Soc., 132, 1435-1443.
[56]
Roldán-López-de-Hierro A. F., Karapınar E., Roldán-López-de-Hierro C., Martínez-
Moreno J. (2015). Coincidence point theorems on metric spaces via simulation functions. J. Comput.
Appl. Math. 275, 345–355.
[57]
Rus I.A. (2005). Cyclic representations and fixed points. Ann. T. Popoviciu, Seminar Funct. Eq.
Approx. Convexity 3, 171-178.
[58]
Rus, I. A. (2001). Generalized contractions and applications. Cluj University Press, Cluj-Napoca,
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