Generalization Fixed Points of Multivalued -Admissible Mappings in
2-Metric Spaces
HANY A. ATIA1,* MONA S. BAKRY2, AYA A. ABD-ELRASHED1
1Department of Mathematics, Fauclty of Science, Zagazig University, EGYPT
2Department of Mathematics, Fauclty of Science, Tanta University, EGYPT
Abstract: - In this paper, we create some fixed-point results of multivalue -admissible of 2-metric
spaces. We introduce Hausdorff distance in 2-metric space, use it in our theorem. we investigated the
existence of some fixed point results for new types of contraction. We study the stability of fixed
point set.
Key-Words: - Metric spaces, 2-metric spaces, multivalued admissible mappings, fixed point,
Hausdorff metric, α−ψ-contraction.
Received: March 25, 2021. Revised: December 17, 2021. Accepted: January 10, 2021. Published: February 4, 2022.
1 Introduction
One of the most valuable findings is the popular
Banach contraction mapping principle [1] in
nonlinear analysis. It was used in many different
mathematical branches and in the general physical
sciences. Metric fixed-point theory developed in
various directions by mathematicians over the years.
A comprehensive account is provided of that
development Kirk and Sims in the Handbook [2].
Extended the contraction mapping theory of Banach
by the usage of a legal Contractive Situation by
Dass and Gupta [3]. Abu-Donia establishes some
fixed point theorems in some types of metric spaces
[cf.4-8]. Aubin and Cellina discuss some elements
of this research in their book [9]. Nadler[10]
expanded the Banach principle of contraction to set-
valued mappings by the Hausdorff metric. Driven
by Nadler’s results, much research is done on fixed
points multi-valued functions were conducted using
this Hausdorff metric in different directions by
multiple authors [11-15]. Stability is a approach
associated with the limiting attitude of a system. It
has been studied in various contexts of discrete and
continuous dynamical systems [16,17]. Studies of
the relation between the convergence of a mapping
sequence and its fixed Points, known as stability of
fixed points, were also widely studied in different
settings [18-20]. the set of fixed points of
multivalued mappings becomes bigger and hence
more important for the study of stability. In [21]
Samet et al. presented the definition of -admissible
mapping and a new group contractive mapping type
known as –-contractive mapping type.[21]
Expansion and generalization of current fixed-point
literature results, in fact, the Banach’s contraction
principle. In addition, Karapinar and Samet [22]
widespreaded the - -contractive type mappings
and access assorted fixed point theorems for this
generalized class of contractive mappings. Since
then, fixed point results of -admissible mappings
have been established, such as [23,24]..
The concepts from setvalued analysis that we use in
this paper are as follows. Let be a metric
space. Then
For and , the function , and
for , the function are defined
as follows:
and
is established the Hausdorff metric induced by the
metric on [23]. encourage, if is a
complete then is also complete.
The following lemma Nadler[18] generated by
Nadler[18]
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DOI: 10.37394/23206.2022.21.5
Hany A. Atia, Mona S. Bakry,
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Lemma 1.1 [18] Let be a metric space
and . Let . next for every
, there exists so that
In [18] Nadler certain that Lemma 1.1 is also
accurate for , wherever . Here
we current the lemma
Lemma 1.2 [25] Let be a metric space
and . Let . next for every
, there exists so that
The following is aftereffect of Lemma 1.2
Lemma 1.3 [25] Let two non-empty compact
subsets of a metric space are and and
is a multivalued mapping since
Let . Next for and , there
endure so that .
Definition 1.1 [21]. Let and
be a function. We express that is
an -admissible mapping if ,
.
In the following we characterize multivalued -
admissible mapping. in the interpretation
stand for the collection of all nonempty subsets
of a nonempty set .
Definition 1.2 [25]. Let a
multivalued mapping since is non-empty set
and . For the
mapping called multivalued -admissible if
where
and .
Definition 1.3 [25]. Let be a
multivalued mapping, since are
two metric spaces and is the Hausdorff metric
on . The mapping is called continuous
at if for any sequence in and
when as
Definition 1.4 [25] Let a single-
valued mapping, a multivalued
mapping and is a non-empty set. A point
is a fixed point of (resp. ) iff
(resp. ).
2 Main Results
for and , for and
the functions
,
where
.
is known as the Hausdorff metric induced by
the 2-metric on .If is complete
then is also complete.
We established the following lemma.
Lemma 2.1 Let be a 2-metric space and
. Let . next for every
, there exists and so that
proof. Let and . and . Since
and , the resultbis true if .
So, we shall prove the result for . Now,
we know that
.
From the definition,
. Then there exists a sequence
in such that as .
Since is compact, has a convergent
subsequence Hunce there exists
such that as . As is
compact, it is closed and . Now,
implies that
that is,
.
Hence the proof is completed.
the following is consequence of Lemma 2.1
Lemma 2.2 Let and are non-empty
compact subsets of a 2-metric space ,
where and is a multivalued
mapping since Let . Next for
there endure
and so that .
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Definition 2.1 Let and
be a function. We say that is an
-admissible mapping if ,
.
Definition 2.2 Let be a 2-metric space
and a multivalued mapping since
. For and
the mapping called multivalued -admissible
if
where and
Definition 2.3 Let are two 2-
metric spaces, and is the
Hausdorff metric on . The mapping is
said to be continuous at if for any
sequence in ,
whenever as .
Theorem 2.1 Let be a complete 2-metric
space, and
a multivalued mapping. Let be multivalued -
admissible and continuous. Let
be a nondecreasing function and
continuos with
and
for each . Suppose that for all ,
if there exist and
such that , then has a fixed
point in .
Proof From the condition, there exist
and such that
. By lemma 2.2, for
there exists such that
.
Employ (1) and applying the monotone
property of , we have
It follows that
Now,if
max
Then from (2) and property of
,
which is a contradiction. Hence
Then from (2), we
have
Since and and
, the -admissibility of
implies that By Lemma (2.2),
for there exists such that
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Hany A. Atia, Mona S. Bakry,
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Volume 21, 2022
employ (1) and using the monotone property of
, we have
Suppose that
Then and it folows by (4) and
property of that
,
which is a contradiction. Then from (4) we have
Since and and
, the -admissibility of
implies that . Continuing this
process, we build up a sequence such that
for all
and
By copied operation (8) and monotone property
of , we have
Then by a property of , we have
This appearance that is a Cauchy
sequence. From the completness of , there
exists such that
as
Since , we have
Taking limit as in the raised inequality,
and accepting (9) and the continuity of , we
have
that is,
Since is compact and hence is
closed, that is, , where denotes the
closure of Now, implies
that , that is, is a fixed point ot
Theorem 2.2 Let be a 2-metric space,
, be two multivalued
mapping and . Let each
be continuous and multivalued -
admissible. Let be a
continuous and nondecreasing function with
, as
and for each . Suppose that
(i) each satisfing (1), that is, for all
and ,
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(ii) for any ,and , we have
whenever ; and for any
and , we have
whenever .
Then where
.
Proof From Theorem 2.1, the set of fixed point
of are non-empty, that is,
, for . Let , that is,
. Then by Lemma 2.1, there exists
and such that
Since and , by
condition (ii) ,we have . By
lemma 2.2, for there
exists such that
.
Then contend similarly as in the proof of
Theorem 2.1, we construct a sequence such
that for all
and
Contend similarly as in the proof of Theorem
2.1, we prove that is a Cauchy sequence
and there exists such that
as
further v is a fixed point of , that is, .
Now, from (10) and the definition of , we
have
Repeatedly, by the triangle inequality and using
(14), we have
.
Taking limit in the above inequality,
using (15),(16) and the propertyies of , we
have
.
Thus, given arbitrary , we can find
for which
Similarly, we can prove that for arbitrary
, there exists a such that
. Hence, we conclude that
.
3 Conclusion
In this paper we established the existence of fixed
points of multivalued -admissible mappings in 2-
metric spaces. and we investigated the stability of
fixed point, also we introduced and studied the
notion of multivalued -admissible in 2-metric
spaces
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