Gen. Math. Notes, Vol. 26, No. 2, February 2015, pp. 104-118
ISSN 2219-7184; Copyright © ICSRS Publication, 2015
www.i-csrs.org
Available free online at http://www.geman.in
On - - -Contraction in Fuzzy Metric Space
and its Application
Noori F. AL-Mayahi1 and Sarim H. Hadi2
1,2
Department of Mathematics, University of AL–Qadissiya
College of Computer Science and Mathematics
1
E-mail: afm2005@yahoo.com
2
E-mail: sarim.h2014@yahoo.com
(Received: 11-8-14 / Accepted: 3-11-14)
Abstract
The aim of this paper is to introduce new concepts of - -complete fuzzy
metric space and - -continuous function and establish fixed point results for - -contraction function in - -complete fuzzy metric space. As an application,
we derive some Suzuki type fixed point theorems, fixed point in orbitally fcomplete. Moreover, we introduce concept - - -contraction function and
application on - - -contraction.
Keywords: - -complete,
- -continuous, - - -contraction, orbitally fcomplete, Suzuki Type Fixed Point Result, - - -Contraction Function,
Application on - - -Contraction.
1
Introduction
The study of fixed points of functions in a fuzzy metric space satisfying certain
contractive conditions has been at the center of vigorous research activity. In
1965, the concept of fuzzy sets was introduced by Zadeh [10]. With the concept of
fuzzy sets, the fuzzy metric space was introduced by I. Kramosil and J. Michalek
On - - -Contraction in Fuzzy Metric Space…
105
[5] in 1975. Helpern [3] in 1981first proved a fixed point theorem for fuzzy
functions. Also M.Grabiec [2] in 1988 proved the contraction principle in the
setting of the fuzzy metric spaces. Moreover, A. George and P. Veeramani [1]
in1994 modified the notion of fuzzy metric spaces with the help of t-norm.
This paper we introduce new concepts of - -complete fuzzy metric space and
- -continuous function and establish fixed point results for - - -contraction
function in - -complete fuzzy metric space. As an application, we derive some
Suzuki type fixed point theorems, fixed point in orbitally -complete. Moreover,
we introduce concept - - -contraction function and application on - - contraction.
2
Preliminaries
Definition 2.1 [4]: Let : → and :
to be
-admissible function if
, )≥1 ⟹
),
Definition 2.2 [6]: Let : → and
said to be , )-admissible function if
, )≥1
, )≥1
),
⟹
),
×
→ [0, ∞) be two function,
) ≥ 1 for all ,
, :
×
is said
∈ .
→ [0, ∞) be two function, is
) ≥1
for all ,
) ≥1
∈ .
Definition 2.3 [4]: Let : → and , : × → [0, ∞) be two function,
said to be -admissible function with respect if
, )≥
, ) ⟹
Not that if we take
Definition 2.4: Let
{
, : ×
% } with
Note:
),
)) ≥
)) , for all ,
∈
, ) = 1 , then this definition reduces to definition (2.1 ).
, ",∗) be a fuzzy metric space and
→ [0, ∞) , is said to be - -complete iff every Cauchy sequence
% , %'( ) ≥
% , %'( ) for all ) ∈ ℕ converge in
is said to be -complete if
Example 2.5: Let
= [0,1+ and "
, ) = 1 for all ,
, )=
+ )8 6 ,
0
6 9. :
-
∈ .
, , ,) = -'|/01| , if , > 0 with 3 ∗ 4 =
56) {3, 4} for every 3, 4 ∈ [0,1+ , define , :
By
),
is
∈
×
→ [0, ∞)
106
Noori F. AL-Mayahi et al.
, )=2
Then
, ",∗) is - -complete fuzzy metric space.
Definition 2.6: Let , ",∗) be a fuzzy metric space and let , : × → [0, ∞)
and : → , is said to be - -continuous function on if for ∈ and { % }
be a sequence in
with % → as ) → ∞ ,
% , %'( ) ≥
% %'( ) for all
)
).
) ∈ ℕ implies
% →
Definition 2.7 [9]:
(1) Let be a function of a fuzzy metric space , ",∗) into itself. , ",∗) is said
to be -orbitally complete if and only if every Cauchy sequence which is
contained in { ,
), 8 ), = ),· · · } for some ∈ converges in .
A
-orbitally complete fuzzy metric space may not be complete.
(2) A function : → is called orbitally continuous at
)
implies ?65%→@ % ) =
The function
∈ .
is orbitally continuous on
if
∈
if ?65%→@
%
)=
is orbitally continuous for all
Remark 2.8:
(1) Let , ",∗) be a fuzzy metric space and : → be a function and let
an orbitally -complete. Define , : × → [0, ∞) by
3 6
, )=A
1
,
∈ C :)
,
9. :
Where C :) is an orbit of a point : ∈
If { % } be a Cauchy sequence with
{ % } ∈ C :)
.
be
, )=1
, ",∗) is an - -complete.
% , %'( )
≥
% , %'( )
for all ) ∈ ℕ then
Now, since is an orbitally -complete fuzzy metric space, then {
. We can say that is - -complete.
%}
converge in
(2) Let , ",∗) and , : × → [0, ∞) is in (1), let : → be an orbitally
continuous function on
, " ∗). Then is - -continuous function. Indeed if
→
3D
)
→
∞
and
%
% , %'( ) ≥
% %'( ) for all ) ∈ ℕ, so % ∈ C :) for
all ) ∈ ℕ , then there exist sequence EF )F∈ℕ of positive integer such that % →
GH
: → as 6 → ∞. Now since is an orbitally continuous on , ",∗), then
GH
:)) →
) 3D 6 → ∞.
%) =
On - - -Contraction in Fuzzy Metric Space…
Denote with
properties:
(1)
(2)
107
: [0,1+ → [0,1+ with the following
the set of all the function
is nondecreasing and continuous
I) = 0 6 I = 1
Definition 2.9: Let , ",∗) be a fuzzy metric space and : → , let J , ) =
), ,), " ,
), ,), " ,
), ,) ∗ " ,
), ,)}
56) {" , , ,), " ,
We say
(1)
is an - - -contraction function if
),
), ,) ≥
, )≥
, )⟹ "
(2)
is - -contraction function if
J , )).
, ) = 1 for all ,
∈
.
Theorem 2.10: Let , ",∗) be a fuzzy metric space and let : → , suppose
that , : × → [0, ∞) are two function. Assume that the following assertions
hold
(i)
(ii)
(iii)
(iv)
(v)
, ",∗) is an - -complete fuzzy metric space .
is an -admissible function with respect to .
is - - -contraction function on .
is an - -continuous function .
There exist K ∈ such that
K,
K )) ≥
K,
Then
has a fixed point in
Proof: Let
K
∈
%
=
%'(
Suppose
Since
%
%}
%
such that
for some ) , then
≠
K)
K,
such that
Define a sequence {
If
.
=
=
%
≥
%0( )
K ))
≥
(, 8)
Then
K)
K,
for all ) ∈ ℕ
is a fixed point of .
%'(
is -admissible function with respect
K,
K )).
K,
=
K ))
K ),
()
≥
and
K ),
()
=
(, 8)
By continuing this process, we get
%,
% ))
=
% , %'( )
≥
% , %'( )
=
%,
% ))
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Noori F. AL-Mayahi et al.
By 1) in definition (2.9)
"
Where
J %0( ,
%)
% , %'( , ,)
= 56) {"
= 56) {"
≥ 56) {"
Since
"
"
%0( , % , ,), "
"
% ), ,)
%0( ,
% ), ,)
%0( ,
% , %'( , ,), "
≥
∗"
∗"
%,
%0( , % )
%,
%0( ), ,)}
% , %'( , ,), "
% , % , ,)}
%0( , % , ,)
% , %'( , ,)}
J
%0( ), ,), "
%0( , % , ,), "
%0( %'( , ,)
%0( , % , ,), "
∗"
% ), ,),
% , %'( , ,),
% , %'( , ,)}
is nondecreasing and continuous, we have
% , %'( , ,)
If 56){"
Then
" %,
%0( , % , ,), "
%0( , % , ,), "
= 56) {"
%0( ),
="
≥
56){"
%0( , % , ,), "
%'( , ,)
%0( , % , ,), "
% , %'( , ,)}
≥ 56){" %0( ,
> " % , %'( , ,)
% , %'( , ,)})
="
% , %'( , ,)
% , ,), "
% , %'( , ,)})
≥
"
% , %'( , ,)
Which is contradiction.
Therefore 56){"
%0( , % , ,), "
"
"
% , %'( , ,)}
Hence for all ) ∈ ℕ we have
% , %'( , ,)
≥
%0( , % , ,)
Let ), 5 ∈ ℕ with ) > 5 , then
"
% , N , ,)
≥
"
Therefore ?65%→∞ "
Hence {
%}
8
"
≥⋯≥
%
≥⋯≥
%0( , % , ,)
% , N , ,)
≥
%
"
="
%0( , % , ,)
%08 , %0( , ,)
K , ( , ,)
=1
"
K , ( , ,)
is a Cauchy sequence
Since is an - -complete fuzzy metric space there is
)→∞.
Since
is an - -continuous function so
%'(
=
%)
∈
→
such that
%
) as ) → ∞
→
as
On - - -Contraction in Fuzzy Metric Space…
Hence
)=
Suppose
is a fixed point of
),
" , , ,) = "
Hence
=
, )=| − |,
all ,
∈
I) = √I for all I ∈ [0,1+. Define "
and , > 0 , and , :
+ )8 6 ,
0
6 9. :
, )=2 ,
, )=
By
J , ))
= [0,3+ be equipped with the ordinary metric
Example 2.11: Let
O
)=
such that
), ,) ≥
109
×
∈
→ [0, ∞)
, , ,) = R 0
S|TUV|
W
for
Clearly, , ",∗) is an - - complete fuzzy metric space with respect to , −norm
3 ∗ 4 = 34. (by [8])
Let :
→
be defined as
)=
O
),
It follows that "
1
∈ [0,1+
=0/
8
∈ 1,3+
) =|
),
)−
), ,) = R 0
)| =
1
| − |≤| − |=O
2
S|Y T)UY V))|
W
≥ R0
|TUV|
W
=
, )
J , ))
Thus f is α-η-φ-contraction function in fuzzy metric space X, M,∗).
Corollary 2.12: Let , ",∗) be a fuzzy metric space and let : → , suppose
that , : × → [0, ∞) are two function. Assume that the following assertions
hold:
(i)
(ii)
(iii)
(iv)
(v)
, ",∗) is an -complete fuzzy metric space .
is an -admissible function .
is - -contraction function on .
is an -continuous function .
There exist K ∈ such that
K,
K )) ≥ 1.
Then
has a fixed point in
.
110
Noori F. AL-Mayahi et al.
Theorem 2.13: Let , ",∗) be a fuzzy metric space and let : → , suppose
that , : × → [0, ∞) are two function. Assume that the following assertions
hold:
, ",∗) is an -complete fuzzy metric space
is an -admissible function with respect to .
is - - -contraction function on .
There exist K ∈ such that
K,
K )) ≥
K,
)
If { % } is a sequence in
,
≥
,
% %'(
% %'( )
(i)
(ii)
(iii)
(iv)
(v)
%
With
→
as ) → ∞, then
% ),
Either
% ),
8
Or
Then
)≥
)≥
8
K
∈
K,
such that
Define a sequence {
%}
in
8
% ),
.
has a fixed point in
Proof: Let
% ),
%
by
=
=
% ))
% ))
K)
for all ) ∈ ℕ
≥
K)
%
=
∈
Let "
Either
8
%0( ),
)≥
%,
)≥
%0( ),
Then either
Or
Let
"
%'( ,
%,
%'( ,
)≥
)≥
min {"
)≥
as ) → ∞
%0( ),
8
%0( ),
=
% , %'( )
%0( )
for all ) ∈ ℕ.
%'( , % )
≥
%'( , % )
%0( ))
%0( ))
%'( , %'8 )
), ,) = "
%,
8
% , %'( )
Where
J % , ) = 56) {"
=
→
), ,) ≠ 1 , from (v)
,
Or
%
such that
K)
K,
Now is in the proof of theorem (2.10) we have
There exist
K )).
from definition (2.9) condition (1), we get
% ),
), ,) ≥
J
, ,), " % ,
% ), ,), "
), ,) ∗ " ,
" %,
%,
, ,), "
%,
)
), ,),
,
% ), ,)}
), ,),
,
), ,) ∗ " , %'( , ,)})
% , %'( ,), "
"
%,
On - - -Contraction in Fuzzy Metric Space…
> 56){"
%,
, ,), "
% , %'( ,), "
,
By taking ) → ∞ in the above we get
), ,) = 56) {1, " ,
" ,
111
), ,), "
%,
), ,) ∗ " ,
), ,)}) > " ,
%'( , ,)}
), ,)
Which is contradiction.
Hence "
,
), ,) = 1 ⟹
)=
.
Corollary 2.14: Let , ",∗) be a fuzzy metric space and let : → , suppose
that , : × → [0, ∞) are two function. Assume that the following assertions
hold:
(i)
(ii)
(iii)
(iv)
(v)
With
, ",∗) is an -complete fuzzy metric space.
is an -admissible function.
is - -contraction function on X.
There exist K ∈ such that
K,
K )) ≥ 1.
)
,
If { % } is a sequence in
% %'( ≥ 1
%
→
as ) → ∞, then
Either
Or
Then
8
% ),
% ),
)≥1
) ≥ 1 for all ) ∈ ℕ
has a fixed point in .
Corollary 2.15: Let , ",∗) be a complete fuzzy metric space and let :
be a continuous function such that is contraction function that is
"
Then
2.2
),
), ,) ≥
has a fixed point in .
J , ) for all ,
∈
Fixed Point in Orbitally c- Complete Fuzzy Metric Space
Theorem 2.2.16: Let , ",∗) be a fuzzy metric space and :
following assertion hold:
(i)
(ii)
(iii)
→
→
such that the
, ",∗) is an orbitally -complete fuzzy metric space.
),
), ,) ≥ J , )) for
there exist be a function such that "
all , ∈ C :) for some : ∈ .
If { % } be a sequence .such that { % } ⊆ C :) with % → as ) → ∞,
∈ C :).
Then has fixed point in .
112
Noori F. AL-Mayahi et al.
Proof: Define : × → [0, ∞) from remark (2.8) ,
fuzzy metric space and is an -admissible function.
, ) ≥ 1 for all ,
Let
Then from (ii)
"
),
), ,) ≥
J , ))
is an - -contraction function
Let {
sequence such that
So {
%}
Then
% , %'( )
⊆ C :) from (iii) we have
%,
That is
-complete
∈ C :)
That is
%}
, ",∗) is an
≥ 1 with
∈ C :)
%
→
) ≥ 1 by corollary (2.13)
has unique fixed point in .
Corollary 2.2.17: Let , ",∗) be a fuzzy metric space and :
the following assertion hold:
→
such that
X, M,∗) is an orbitally f-complete fuzzy metric space.
there exist k ∈ 0,1) such that M f x), f y), t) ≥ kJ x, y)
for all x, y ∈ O w) for some w ∈ X.
If {xk } be a sequence
such that {xk } ⊆ O w) with xk → x
as n → ∞. Then x ∈ O w). Then f has fixed point in X.
(i)
(ii)
(iii)
2.3
Suzuki Type Fixed Point Result
From Theorem (2.10) we deduce the following Suzuki type fixed point result
Theorem 2.3.18: Let , ",∗) be a complete fuzzy metric space and let :
continuous function .Assume that there exist E ∈ 0,1) such that
" ,
For all ,
Then
), ,) ≥ " , , ,) ⟹ "
∈ , where
),
J , ) = 56) {" , , ,), " ,
), ,) ∗ "
" ,
has a fixed point in
Proof: Define , :
×
.
), ,) ≥ EJ , ) ____(1)
), ,), " ,
), ,)}
,
→ [0, ∞) and : [0,1+ → [0,1+ by
), ,),
→
On - - -Contraction in Fuzzy Metric Space…
, )="
Such that
,
, )≥
), ,)
, )="
,
, ) for all ,
I) = EI , I ∈ [0,1+
And let
∈
113
, , ,)
By condition (i) –(v) of theorem (2.10 )
Let
Then "
,
,
) ≥
, )
), ,) ≥ " , , ,)
From (1) we have "
),
), ,) ≥ EJ , ) =
J , ))
Then is - - -contraction function by condition of theorem (2.10) and
fixed point, then the unique fixed point follows from (1).
Corollary 2.3.19: Let , ",∗) be a complete fuzzy metric space and let :
continuous function. Assume that there exist E ∈ 0,1) such that
), ,) ≥ " , , ,) ⟹ "
" ,
For all ,
∈ ,
),
.
has a fixed point in
2.4
An - l- m-Contraction Function
Definition 2.4.20: Let , ",∗) be a complete fuzzy metric space. Let :
an -admissible which satisfies the following
),
), ,) ≥
→
), ,) ≥ E" , , ,)
Then
"
has
J , ) −
→
be
→
be
J , ))
Such that
(i)
(ii)
is continuous and decreasing with I) = 0 iff I = 1.
is continuous with I) = 0 iff I = 1
is called - - - contraction function .
Theorem 2.4.21: Let , ",∗) be a complete fuzzy metric space . Let :
an -admissible which satisfies the following
"
),
), ,) ≥
J , ) −
J , ))
114
Noori F. AL-Mayahi et al.
Such that
, ",∗) is an -complete.
is -continuous function
There exist K ∈ such that
(i)
(ii)
(iv)
Proof: Let
K
∈
Define a sequence {
If
=
%
%
%}
%
such that
for some ) , then
%'(
Suppose
K,
such that
≠
=
K,
K ))
=
%
≥1
K ))
%0( )
K,
K ))
%'(
=
(, 8)
Then
for all ) ∈ ℕ
is a fixed point of .
is -admissible function with respect
Since
≥ 1.
K, ()
≥1
K ),
=
()
and
K ),
≥
()
(, 8)
=
By continuing this process, we get
% ))
%,
Where
J %0( ,
%)
= 56) {"
"
% , %'( )
≥ 1 for all ) ∈ ℕ
% , %'( , ,)
≥
=
J
%0( , % , ,), "
= 56) {"
= 56) {"
≥ 56) {"
=
"
%0( ,
%0( , % , ,), "
"
%0( , % , ,), "
%0( , % , ,), "
"
%0( , % )
%0( ,
),
% ,) ∗ "
%0( , % , ,), "
%0( %'( , ,)
% , %'( , ,), "
∗"
% , %'( , ,)}
Now taking ) → ∞ we obtain for all , > 0
? ,) −
−
% ), ,))
J
%0( ), ,), "
%0( , % )
%,
),
,)}
%0(
%,
% , %'( , ,), "
% , % , ,)}
%0( , % , ,)
Since decreasing, then " %0( , % , ,) < "
is an increasing sequence thus there
?65%→∞ " % , %'( , ,) = ? ,) < 1
? ,) ≥
%0( ),
∗"
% , %'( , ,),
% , %'( , ,)}
% , %'( , ,)
exist
% ), ,),
that is {" % , %'( , ,)}
? ,) ∈ 0,1+ such that
? ,)) which is contradiction ? ,) = 1
On - - -Contraction in Fuzzy Metric Space…
Thus we conclude for all , > 0
{
%}
?65%→∞ "
% , %'( , ,)
is Cauchy sequence .since
"
), ,)) =
%'( ,
115
=1
%
→
%,
)
is -complete ,then
% ),
), ,)
J %, ) −
"
≥
J
Letting ) → ∞ in the above inequality using properties ,
Thus ?65%→∞ "
Hence
→
%
%,
?65%→∞ "
)⟹
Now we assume
), ,) = 1
)=
1) −
.
is a fixed point of
" , , ,) = "
⟹ " , , ,) = 1 ⟹ =
2.5
), ,)) ≥
%,
.
),
)=
such that
), ,) ≥
1) = 0
J , ) −
J , ) =0
Application On - - -Contraction
Definition 2.5.22: Let
, ",∗) be a fuzzy metric space and
, )-admissible function, is said to be
:
→
be an
, )-contraction function of type ( I ) , if
),
), ,) ≥ J , ) .
, )
, )"
, )-contraction function of type (II) , if there exist 0 < ℓ ≤ 1 such that
, )
, ) + ℓ)p q /),q 1),-) ≥ 1 + ℓ)r J /,1))
(a)
(b)
Theorem 2.5.23: Let , ",∗) be a complete fuzzy metric space and let : →
be an -continuous and , )-contraction function of type (I), (II), if there exist
has a unique fixed point in .
K,
K )) ≥ 1 and
K,
K )) ≥ 1 , then
Proof: Let
K
∈
such that
Define a sequence {
Since
Then
is
%}
K ))
K,
such that
%
=
, )-admissible function and
(, 8)
=
K ),
()
By continuing this process, we get
≥1
≥ 1 and
%0( )
K,
K,
K ))
for all ) ∈ ℕ
K ))
≥1
≥1
116
Noori F. AL-Mayahi et al.
% , %'( )
=
% , %'( )
Similarly we have
If
=
%
(a)
for some ) , then
%'(
Suppose
%
≠
% , %'( )
Where
J %0( ,
" %0( ,
=
≥
If 56){"
=
=
% ))
%,
%
≥1
is a fixed point of .
% , %'( , ,)
% , %'( )
J
%0( ,
% , %'( )"
% ))
= 56) {" %0( , % , ,), "
% ), ,) ∗ " % ,
%0( ), ,)}
= 56) {"
= 56) {"
≥1
%'(
% , %'( )"
%)
≥ 56) {"
% ))
%,
%0( , % , ,), "
"
%0(
%0( , % , ,), "
%0( , % , ,), "
%0( , % , ,), "
%0( ),
%0( ), ,), "
%0( ,
%0( , % , ,), "
%'( , ,) ∗ "
% , %'( , ,), "
% , %'( , ,)}
% , %'( , ,)}
%,
% , ,)}
∗"
% ), ,),
%,
% , %'( , ,), "
%0( , % , ,)
="
% ), ,)
% , %'( , ,),
% , %'( , ,)}
% , %'( , ,)
Then
"
% , %'( , ,)
≥ 56){" %0( ,
> " % , %'( , ,)
% , ,), "
% , %'( , ,)})
≥
"
% , %'( , ,)
Which is contradiction
Therefore 56){"
%0( , % , ,), "
"
"
Hence for all ) ∈ ℕ we have
% , %'( , ,)
≥
%0( , % , ,)
≥⋯≥
Let ), 5 ∈ ℕ with ) > 5 , then
"
% , N , ,)
=
%,
≥⋯≥
Therefore ?65%→∞ "
%'( )
%
"
% , N , ,)
% , %'( , ,)}
≥
%
8
"
% , %'( )"
K , ( , ,)
=1
"
="
%0( , % , ,)
%08 , %0( , ,)
K , ( , ,)
%0( ),
% ), ,)
≥
J
%0( , % )
On - - -Contraction in Fuzzy Metric Space…
Hence {
%}
117
is a Cauchy sequence
Since is an - -complete fuzzy metric space there is ∈ such that % →
)→∞.
), ,) =
), ,) ≥ J % , )
" %,
%, )
% , )"
% ),
Hence
)=
Suppose
is a fixed point of
, )
" , , ,) =
= .
Hence
(b)
, )"
% , %'( )
Where
J %0( ,
%)
= 56) {"
If 56){"
Then
" %,
),
), ,) ≥
+ ℓ)p /s ,/stu ,-)
p q
=
% , %'( )
% , %'( ) + ℓ)
≥ 1 + ℓ)r J /sUu ,/s ))
%0( , % , ,), "
"
%0( , % , ,), "
%0( , % , ,), "
≥
% , %'( , ,), "
% , %'( , ,)}
56){"
Therefore 56){"
%0( , % , ,), "
"
"
Hence for all ) ∈ ℕ we have
≥
="
>"
%0( , % , ,)
≥⋯≥
Let ), 5 ∈ ℕ with ) > 5 , then
%,
% , % , ,)}
% , %'( , ,)})
% , %'( , ,)}
%
8
"
% , %'( , ,),
% , %'( , ,)}
% , %'( , ,)
% , %'( , ,)
≥
∗"
% ), ,),
%,
%0( ), ,)}
% , %'( , ,), "
%0( , % , ,)
%0( , % , ,), "
Which is contradiction
% , %'( , ,)
∗"
% , %'( , ,)}
/sUu ),q /s ),-)
%0( ), ,), "
%0( , % , ,), "
%0( %'( , ,)
%0( , % , ,), "
%'( , ,)
J , )).
% , %'( )
= 56) {" %0( , % , ,), " %0( ,
" %0( ,
% ), ,) ∗ "
= 56) {"
≥ 56) {"
)=
such that
"
="
≥
%0( , % , ,)
%08 , %0( , ,)
K , ( , ,)
"
% , %'( , ,)
as
118
Noori F. AL-Mayahi et al.
"
% , N , ,)
Therefore ?65%→∞ "
Hence {
%}
p q /sUu ),q /s ),-)
=
% , %'( )
% , %'( ) + 1)
s
≥ 1 + ℓ)r J /sUu ,/s ) ≥ ⋯ ≥ 1 + ℓ)r p /v ,/u,-)
% , N , ,)
=1
is a Cauchy sequence
Since is an - -complete fuzzy metric space there is
)→∞.
%,
)
) + ℓ)p /s ,q /),-) =
≥ 1 + ℓ)r J /s ,/))
%,
Hence
)=
Suppose
is a fixed point of
, )
Hence
, ) + ℓ)p
= .
such that
q /),q 1),-)
%,
)
%,
∈
such that
) + ℓ)p
%
→
as
q /s ),q /),-)
)=
≥ 1 + ℓ)r
J /,1))
.
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