Галерея 2910794
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We introduce new concepts of a-admissible triangular mappings with respect to a function \beta and the concept of ( \alpha , \beta , \psi )-contraction. We also utili... View more
Abstract: We introduce in this paper some concepts of α-admissible triangular mappings with respect to a function β and the concept of (α, β, ψ)-contraction. We also utilize our ne... View more
We introduce in this paper some concepts of α-admissible triangular mappings with respect to a function β and the concept of (α, β, ψ)-contraction. We also utilize our new concept to prove a fixed point theorem based on the contractive condition of type α-admissibility. We apply our main result to derive many fixed point results. Also, we construct some examples to support our work. In the literature, many exciting results are improved and extended using our results.
Published in: IEEE Access ( Volume: 7 )
We introduce new concepts of a-admissible triangular mappings with respect to a function \beta and the concept of ( \alpha , \beta , \psi )-contraction. We also utili... View more
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Since Banach proved his amazing fixed point theorem [1] , many authors established and built many contractive conditions to generalize and extend the Banach contraction theorem in many different ways. The altering distance mapping [2] played a major role to extend the Banach fixed point theorem to new types, for some work in altering distance mapping, we refer the readers to [3] – [9] . Recently, the notion of almost perfect function which considered as a generalization of altering distance function has been introduced by Abodayeh et al. [10] . They gave new generalizations of the Banach fixed point theorem.
The definition of altering distance function was given below:
A self function \psi
on the set \mathbb {R^{+}}
is said to be an altering distance function if the following conditions are satisfied:
The function \psi
is nondecreasing continuous .
Abodayeh et al. [10] introduced the notion of almost perfect function which is given below:
A self function \psi
on the set \mathbb {R^{+}}
is called almost perfect if it is nondecreasing and satisfies the following conditions:
If \{ t_{n} \}
is a sequence in [0,+\infty )
such that \psi (t_{n})\rightarrow 0
, then the sequence \{t_{n} \}
converges to 0 .
The notion of \alpha -
admissibility plays a major role in fixed point theory to generalize and extend the Banach contraction theorem. The definition of \alpha -
admissibility was introduced by Samet et al. [11] . Recently, some authors extended the concept of \alpha -
admissibility to some directions. Karapıner et al. [12] introduced the notion of \alpha -
admissibility triangular for a single valued mapping. While, Abdeljawad [13] extended the definition of \alpha -
admissibility to define it for a pair of functions.
We may advise the readers to give a look at the following [14] – [25] for more work in \alpha -
admissibility.
The \alpha -
admissibility mapping is defined as follows:
Let S:\Psi \rightarrow \Psi
be a mapping and \alpha
be a nonnegative function on \Psi \times \Psi
. Then S
is called \alpha -
admissible if for any a,b\in \Psi
with \alpha (a,b)\geq 1
, then \alpha (Sa,Sb)\geq 1
.
Karapınar et al. [12] introduced the notion of \alpha -
admissibility triangular for a single mapping.
Let T:\Psi \rightarrow \Psi
and \alpha
be nonnegative function on \Psi \times \Psi
. Then T
is called an \alpha -
admissible triangular mapping if it satisfies the following conditions:
if \alpha (a,c)\geq 1
and \alpha (c,b)\geq 1
then we have \alpha (a,b)\geq 1
.
The \alpha -
admissibility for pair of mappings was given by Abdeljawad [13] as follows:
Let S,T
be two self mappings on \Psi
and \alpha
be a nonnegative function on \Psi \times \Psi
. Then the pair (S,T)
is called \alpha -
admissible if for any a,b\in \Psi
with \alpha (a,b)\geq 1
, then \alpha (Sa,Tb)\geq 1
and \alpha (Ta,Sb)\geq 1
.
Now, we recall the definitions of \alpha -\eta -
continuous mappings and \alpha -\eta -
complete metric spaces.
Let (\Psi , \sigma )
be a metric space and let \alpha ,\eta
be two nonnegative functions on \Psi \times \Psi
. Then \Psi
is said to be an \alpha ,\eta -
complete metric space if and only if every Cauchy sequence \{a_{n}\}
in \Psi
with \alpha (a_{n},a_{n+1})\geq \eta (a_{n},a_{n+1})
for all n \in \mathbb {N}
converges in \Psi
.
Also, a self mapping T
on \Psi
is said to be an \alpha ,\eta -
continuous mapping if any sequence \{a_{n}\}
in \Psi
that converges to an element a\in \Psi
and \alpha (a_{n},a_{n+1})\geq \eta (a_{n},a_{n+1})
, for all n \in \mathbb {N}
, implies that the sequence \{ Ta_{n}\}
converges to Ta
.
The concepts of almost perfect function and \alpha -
admissibility will be utilized in this article to introduce and prove a fixed point theorem in the setting of metric spaces. We also derive many fixed point results for a single valued mapping using our main result.
At the start of our work we consider some properties of \alpha -
admissibility triangular with respect to another function \beta
for a self-mapping S
on a set \Psi
.
Let \alpha ,\beta
be two real-valued functions on \Psi \times \Psi
. A self mapping S
on \Psi
is said to be an \alpha -
admissible triangular with respect to \beta
if the following conditions are satisfied:
If a,b\in \Psi
with \alpha (a,b)\geq \beta (a,b)
, then \alpha (Sa,Sb)\geq \beta (Sa,Sb)
.
If a,b,c\in \Psi
with \alpha (a,c)\geq \beta (a,c)
and \alpha (c,b)\geq \beta (c,b)
, then \alpha (a,b)\geq \beta (a,b)
.
Now, we present some examples for a mapping S
which is \alpha -
admissible triangular with respect to \beta
.
Let \Psi =(-\infty ,+\infty )
. Define a self mapping S
on \Psi
by Sa=\frac {a}{1+a^{2}}
. Also, define the functions \alpha ,\beta :\Psi \times \Psi \rightarrow [0,+\infty )
by \alpha (a,b)=e^{a+b}
and \beta (a,b)=e^{a-b}
. Then S
is \alpha -
admissible triangular with respect to \beta
.
To prove condition (2) of Definition 7 , given a,b,c\in \Psi
be such that \alpha (a,b)\geq \beta (a,b)
and \alpha (b,c)\geq \beta (b,c)
. Then, we conclude that b\geq 0
and c\geq 0
and hence a+c\geq a-c
. So \alpha (a,c)\geq \beta (a,c)
.
To prove Condition (1) of Definition 7 , given a,b\in \Psi
be such that \alpha (a,b)\geq \beta (a,b)
. Then b\geq 1
and hence a+b\ge a^{2}+b
. So we conclude that a\in [{0,1}]
. Therefore, a^{2} +b^{2}\geq a^{4} +b^{2}
. So \alpha (Sa,Sb)\geq \beta (Sa,Sb)
.
To verify Condition (2) of Definition 7 , given a,b,c\in \Psi
be such that \alpha (a,b)\geq \beta (a,b)
and \alpha (b,c)\geq \beta (b,c)
. Since \alpha (a,b)\geq \beta (a,b)
, we have b\geq 1
and hence a\in [{0,1}]
. Also, \alpha (b,c)\geq \beta (b,c)
, we have c\geq 1
and hence b\in [{0,1}]
. So, we conclude that a\in [{0,1}], b=1
and c\geq 1
. Therefore, a+c\geq a^{2}+c
and hence \alpha (a,c)\geq \beta (a,c)
.
Now, we introduce the following definition in order to simplify our work:
Now, we will give an example for a mapping S
which is (\alpha ,\beta ,\psi )-
contraction but not \alpha ,\beta -
continuous.
Given a,b\in \Psi
be such that \alpha (a,b)\geq \beta (a,b)
. Then (a,b)\neq (0,0)
.
To show that S
is not \alpha ,\beta -
continuous, we choose a sequence \{ t_{n} \}
in \big(\frac {1}{8},\frac {1}{4}\big]
such that t_{n}\rightarrow \frac {1}{8}
as n\rightarrow +\infty
with \alpha (t_{n},t_{n+1})\geq \beta (t_{n},t_{n+1})
. Note that S(t_{n})=t_{n}^{4}\rightarrow \frac {1}{8^{4}}\neq S\left({\frac {1}{8}}\right)
. So S
is not \alpha ,\beta -
continuous.
On a metric space (\Psi , \sigma )
, let \alpha ,\beta
be two real-valued functions defined on \Psi \times \Psi
, and S:\Psi \rightarrow \Psi
be a mapping. Assume that the following hypotheses hold:
The metric space (\Psi , \sigma )
is \alpha ,\beta -
complete .
The mapping S
is (\alpha ,\beta ,\psi )-
contraction .
S
is \alpha -
admissible triangular with respect to \beta
.
There exists a_{0}\in \Psi
such that \alpha (a_{0},Sa_{0})\geq \beta (a_{0},Sa_{0})
. Then S
has a fixed point .
By induction one may prove the validity of the following inequality \alpha (a_{n},a_{n+1} )\geq \beta (a_{n},a_{n+1})
for all n\in \mathbb {N}
.
If there exists m\in \mathbb {N}
such that a_{m} =a_{m+1}
, then a_{m} =Sa_{m}
and hence a_{m}
is a fixed point of S
. Thus, we may assume that a_{n}\neq a_{n+1}
for all n\in \mathbb {N}
.
Now, we shall show that \{ a_{n} \}
is a Cauchy sequence in \Psi
. Take n,m\in \mathbb {N}
with m>n
.
The following corollary is a consequence result of Theorem 12 .
Let (\Psi , \sigma )
be a metric space and let \alpha ,\beta
be two real-valued functions on \Psi \times \Psi
. Let S:\Psi \rightarrow \Psi
be an \alpha ,\beta
-continuous mapping. Assume the following assertions:
(\Psi , \sigma )
is \alpha ,\beta -
complete .
S
is \alpha -
admissible triangular with respect to \beta
.
There exists a_{0}\in \Psi
such that \alpha (a_{0},Sa_{0})\geq \beta (a_{0},Sa_{0})
.
On the metric space (\Psi , \sigma )
, let \alpha ,\beta
be two real-valued functions defined on \Psi \times \Psi
be and S:\Psi \rightarrow \Psi
a mapping. Assume that the following hypotheses hold:
The metric space (\Psi , \sigma )
is an \alpha ,\beta -
complete metric space .
S
is \alpha -
admissible triangular with respect to \beta
.
There exists a_{0}\in \Psi
such that \alpha (a_{0},Sa_{0})\geq \beta (a_{0},Sa_{0})
.
The proof follows from Theorem 12 by noting that each altering distance function is an almost perfect function.
The following corollary is a consequence result of Corollary 13 .
On the metric space (\Psi ,\sigma )
, let \alpha ,\beta
be two real valued functions defined on \Psi \times \Psi
and the mapping S:\Psi \rightarrow \Psi
is \alpha ,\beta
continuous. Assume the following assertions:
(\Psi , \sigma )
be \alpha ,\beta -
complete .
S
is \alpha -
admissible triangular with respect to \beta
.
There exists a_{0}\in \Psi
such that \alpha (a_{0},Sa_{0})\geq \beta (a_{0},Sa_{0})
.
On the metric space (\Psi , \sigma )
, let \alpha ,\beta
be two real-valued functions defined on \Psi \times \Psi
and S:\Psi \rightarrow \Psi
a mapping. Assume that the following hypotheses hold:
The metric space (\Psi , \sigma )
is an \alpha ,\beta -
complete metric space .
S
is \alpha -
admissible triangular with respect to \beta
.
There exists a_{0}\in \Psi
such that \alpha (a_{0},Sa_{0})\geq \beta (a_{0},Sa_{0})
.
On the metric space (\Psi , \sigma )
, let \alpha ,\beta
be two real-valued functions defined on \Psi \times \Psi
and S:\Psi \rightarrow \Psi
a mapping. Assume that the following hypotheses hold:
The metric space (\Psi , \sigma )
is an \alpha ,\beta -
complete metric space .
S
is \alpha -
admissible triangular with respect to \beta
.
There exists a_{0}\in \Psi
such that \alpha (a_{0},Sa_{0})\geq \beta (a_{0},Sa_{0})
.
Follows from Corollary 16 by noting that each altering distance function is an almost perfect function.
By defining \beta :\Psi \times \Psi \rightarrow \mathbb {R}
via \beta (s,t)=1
, we derive many results.
Before that we will rewrite and formulate definitions of \alpha
- \beta -
complete metric spaces and \alpha -\beta -
continuous mappings to \alpha
-complete metric spaces and \alpha
-continuous mapping.
Let (\Psi , \sigma )
be a metric space and \alpha :\Psi \times \Psi \rightarrow [0,\infty )
be a function. Then \Psi
is said to be an \alpha -
complete metric space if and only if every Cauchy sequence \{a_{n}\}
in \Psi
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