Ijciet 10 01_025

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International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 1, January 2019, pp.
Available online at http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=1
ISSN Print: 0976-6308 and ISSN Online: 0976
©IAEME Publication
GENERALIZED
POINT THEOREMS
COMPATIBLE
Department of Mathematics, College of Education for Pure
University of Baghdad, Ministry of Higher Education and Scientific Research
ABSTRACT
In this paper, a new type of generalized
compatible mappings in Fuzzy
discuss the existence and uniqueness for
mappings having weakly compatible. A
theorems for these mappings
Cite this Article: Zena Hussein Maibed, Generalized Tupl
Theorems For Weakly Compatible Mappings In Fuzzy Metric Space
Journal of Civil Engineering and Technology (IJCIET)
http://www.iaeme.com/IJCIET/issues.asp?JType=IJCIET&VType=10&IType=1
1. INTRODUCTION
Fuzzy set was defined byiZadeh [14
space, George and Veermain [4
continuous t-norms. Many researchers have obtained common
mappings. Pant [9] introduced
established some common fixed point in Fuzzy
11, 15and 7].Were call some definitions and known
Definition (1.1) [12]
A binaryo Generalized Tupled Common Fixed Point Theorems For Weakly Compatible
Mappings In Fuzzy Metric Space
pe rationð: 0,1 → 0,1 is called continuous t
i. ðIsanassociativeand commutative.
ii. ð1 ∀ ∈ 0,1
iii. ð ð Whenever
IJMET/index.asp 255 editor@iaeme.com
International Journal of Civil Engineering and Technology (IJCIET)
Volume 10, Issue 1, January 2019, pp.255–273, Article ID: IJCIET_10_01_025
ISSN Online: 0976-6316
Scopus Indexed
GENERALIZED TUPLED COMMON
OINT THEOREMS FOR WEAKLY
OMPATIBLE MAPPINGS IN FUZZY ME
SPACE
Zena Hussein Maibed
Department of Mathematics, College of Education for Pure Science Ibn Al
Ministry of Higher Education and Scientific Research
type of generalized tupled common fixed point and weakly
in Fuzzy metric space are introduced and studied, we also
and uniqueness for generalized tupled common fixed point
weakly compatible. A generalized tuplet common
mappings are established.
Zena Hussein Maibed, Generalized Tupled Common Fixed Point
Theorems For Weakly Compatible Mappings In Fuzzy Metric Space
Journal of Civil Engineering and Technology (IJCIET), 10 (1), 2018, pp.
Zadeh [14]. Kramosil and Michalek [6] introduced
space, George and Veermain [4] modified the motion of Fuzzy metric space with the help
norms. Many researchers have obtained common fixed point
mappings. Pant [9] introduced the new concept reciprocally continuous
fixed point in Fuzzy metric space can be viewed in [1, 2,
definitions and known results in Fuzzy metric space.
Generalized Tupled Common Fixed Point Theorems For Weakly Compatible
Mappings In Fuzzy Metric Space
is called continuous t – normif the following conditions are satisfy
Isanassociativeand commutative.
.
& , ∀ , , , ∈ 0,1 .
editor@iaeme.com
OMMON FIXED
WEAKLY
MAPPINGS IN FUZZY METRIC
Science Ibn Al-Haitham,
Ministry of Higher Education and Scientific Research, Iraq
fixed point and weakly
studied, we also
common fixed point of
tuplet common fixed point
ed Common Fixed Point
Theorems For Weakly Compatible Mappings In Fuzzy Metric Space, International
, 10 (1), 2018, pp. 255–273.
[6] introducediFuzzy metric
space with the help of
fixed point theorems for
reciprocally continuous mapping and
can be viewed in [1, 2, 5,13, 10,
tric space.
Generalized Tupled Common Fixed Point Theorems For Weakly Compatible
conditions are satisfy

Zena Hussein Maibed
http://www.iaeme.com/IJCIET/index.asp 256 editor@iaeme.com
iv. ðIs continuous.
Definition (1.2). [4]
A triple Ε, , ð is called fuzzy metric space ifΕ ≠ 0, ð is continuousit – norm and : Ε × Ε ×
0, ∞ → 0,1 is a fuzziest and satisfying the following conditions.
1. , , > 0
2. , , 1 ; ! " #
3. , , , ,
4. , ,. : 0, ∞ → 0,1 Is continuous.
5. ,%, &' ≥ , , ð ,%,' ∀ ), * > 0
We will addthe condition lim →. , , 1 ∀", # ∈ Ε
Lemma (1.3). [3]
In any fuzzy metric space Ε, , ð , where ð is a continuous t – norm of H – type. If thereexit∅ ∈ ɸ
such that
, ,∅ 1 , , , ∀ ) > 0Then" #.
Definition(1.4). [8]
For any ∈ 0,1 , thesequence< ð3
>.
345 be defined by:
ð5
678 ð3
ð395
ð . Then a t – norm ð is said to be of H – type if thesequence<
ð3
>.
345 is equip continuousat 1.
Definition (1.5).[4]
Let Ε, , ð be afuzzy metric space.then
(i) A sequence in 3 inΕ is said tobe convergentto a point ∈ Εif
lim →. :, , 1for all) > 0.
(i) A sequence in 3 inΕ is calleda Cauchy sequence if for each 0 < ; <
1 678 ) > 0,thereiexists a positive integer 7< such that :, =, > 1 − ; for
each 7, ? ≥ 7<
2. MAIN RESULTS
Now, we will give the following concepts.

Generalized Tupled Common Fixed Point Theorems For Weakly Compatible Mappings In Fuzzy
Metric Space
Definition (2.1)
Let ℛ5: Ε3
→ 2B
be a multi-valued mapping and ℛ , … … , ℛ3 are self-map on Ε3
. Any element
"5, " , … … , "3 ∈ Ε3
is called a generalized tupled fixed point of these mappings
ifℛ5 Dℛ E… … Eℛ3 F, G,…, :
H… . . HI ∋ "5
ℛ5 Dℛ E… … Eℛ3 G, K,…, F
H… . . HI ∋ "
ℛ5 Dℛ E… … Eℛ3 :, F,…, :LF
H … . . HI ∋ "3
Definition (2.2).
Let ℛ5: Ε3
→ 2B
be a multi-valued mapping, ℛ , … … , ℛ3 are self-map on Ε3
678 M5, M , … … , M3 are
self-map on N.any element "5, " , … … , "3 ∈ Ε3
is called generalized tupled coincidence point of
these mappings if
ℛ5 Dℛ E… … Eℛ3 F, G,…, :
H… . . HI ∋ M5 DM E… … EM3 F
H … . . HI
ℛ5 Dℛ E… … Eℛ3 G, K,…, F
H… . . HI ∋ M5 DM E… … EM3 G
H… . . HI
⋮
ℛ5 Dℛ E… … Eℛ3 :, F,…, :LF
H … . . HI ∋ M5 DM E… … EM3 :
H… . . HI
Definition (2.3).
Let ℛ5: Ε3
→ 2B
678 M5, M , … … , M3are
self-map onN. Any element "5, " , … … , "3 is called generalized tupled common fixed point if
ℛ5 Dℛ E… … Eℛ3 F, G,…, :
H… . . HI ∋ M5 DM E… … EM3 F
H … . . HI "5
ℛ5 Dℛ E… … Eℛ3 G, K,…, F
H… . . HI ∋ M5 DM E… … EM3 G
H… . . HI "
⋮
ℛ5 Dℛ E… … Eℛ3 :, F,…, :LF
H … . HI ∋ M5 DM E… … EM3 :
H… . . HI "3
Definition (2.4):
Let ℛ5: Ε3
→ 2B
, ℛ , … … , ℛ3:Ε3
→ Ε3
678 M5, M , … … , M3: Ε → Ε are mappings. These mappings
are weakly compatible at the point "5, " , … … , "3 if
M5 PM D… … DM3 Qℛ5 Dℛ E… … Eℛ3 F, G,…, :
H … . . HIRI … . . IS ∈
ℛ5 Tℛ U… … U
ℛ3 M5 DM E… … EM3 F
H… . . HI , M5 DM E… … EM3 G
H … . . HI ,
… … , M5 DM E… … EM3 :
H… . . HI
V … . . VW.

Zena Hussein Maibed
In this paper, we consider Ψ is the set of all mappings Y: 0, ∞ → 0, ∞ such that:
i. YIs non – decreasing.
ii. YIs upper semi – continuous from the right.
iii. ∑ Y3
< ∞,.
34< ∀) > 0 [ ] Y3&5
Y DY3
I , 7 ∈ ^.
Theorem (2.5):
Let ℛ5: Ε3
→ 2B
678 M5, M , … … , M3 are
self-map on N.if Ε, , ð be a fuzzy metric space such that ð is a t – norm of H – type,
M5EM … … M3 … . . H contained ℛ5Eℛ … … ℛ3 … . . HandY ∈ Ψ satisfying:
ℋ ℛ5 Dℛ E… … Eℛ3 F, G,…, :
H… . . HI , ℛ5 Dℛ E… … Eℛ3 F, G,…, :
H… . . HI , Y ≥
QM5 DM E… … EM3 F
H… . . HI , M5 DM E… … EM3 F
H… . . HI , )R ð
QM5 DM E… … EM3 G
H… . . HI , M5 DM E… … EM3 G
H… . . HI , )R ð … … ð
QM5 DM E… … EM3 :
H … . . HI , M5 DM E… … EM3 :
H … . . HI , )R(1)
Where ) > 0 and "`, #` ∈ Ε ∀ a 1,2, … … , 7
If M5 DM E… … EM3 b H … . . HI is complete subspace of Ε. Then these mappings have a
generalized tupled coincidence point of compose these mappings.
Proof:
Consider"<
5
, "< , … … , "5
3
∈ Ε, since M5EM … … M3 … . . Hcontainedℛ5Eℛ … … ℛ3 … . . H, that
there exists "5
5
, "5 , … … , "5
3
∈ Ε such that
M5 PM D… … DM3E F
FHI … . . IS ∈ ℛ5 Pℛ D… … Dℛ3E c
F, c
G,…, c
:HI … . . IS
M5 PM D… … DM3E F
GHI … . . IS ∈ ℛ5 Pℛ D… … Dℛ3E c
G, c
K,…, c
:, c
FHI … . . IS
⋮
M5 DM E… … EM3 F
: H… . . HI ∈ ℛ5 Pℛ D… … Dℛ3E c
:, c
F,…, c
:LFHI … . . IS
Also,
M5 PM D… … DM3E G
FHI … . . IS ∈ ℛ5 Pℛ D… … Dℛ3E F
F, F
G,…, F
:HI … . . IS
M5 PM D… … DM3E G
GHI … . . IS ∈ ℛ5 Pℛ D… … Dℛ3E F
G, F
K,…, F
:, F
FHI … . . IS
⋮
M5 DM E… … EM3 G
: H … . . HI ∈ ℛ5 Pℛ D… … Dℛ3E F
:, F
F,…, F
:LFHI … . . IS

Metric Space
In general, we can construct the sequences,
< M5 PM D… … DM3E d
FHI … . . IS >, < M5 PM D… … DM3E d
GHI … . . IS >, …,
678 < M5 DM E… … EM3 d
: H… . . HI > as
M5 PM D… DM3E d
FHI … IS ∈ ℛ5 Pℛ D… Dℛ3E dLF
F, dLF
G,…, dLF
:HI . . IS
M5 PM D… DM3E d
GHI … IS ∈ ℛ5 Pℛ D… Dℛ3E dLF
G, dLF
K,…, F
:, dLF
FHI … IS
⋮
M5 DM E… EM3 d
: H … HI ∈ ℛ5 Pℛ D… Dℛ3E dLF
:, dLF
F,…, dLF
:LFHI … IS
We want to show that the above sequences are Cauchy sequences in Ε, , ð , since ðis t –
norm of H – type, this implies
∀ ʎ > 0 ∃ g > 0Such that:
1 − g ð 1 − g ð … ð 1 − g ≥ 1 − ʎ, ∀7 ∈ ^. On other hand. For all ", # ∈ Ε, ", #, . is
continuous and lim →. ", #, ) 1 then there exists )° > 0 such that.
QM5 PM D… … DM3E c
FHI … IS , M5 PM D… … DM3E F
FHI … IS , )<R ≥ 1 − g
GHI … IS , M5 PM D… … DM3E F
GHI … IS , )<R ≥ 1 − g
⋮(2)
QM5 DM E… … EM3 i
: H … HI , M5 DM E… … EM3 F
: H… HI , )<R ≥ 1 − g
By using (1), we get:
QM5 PM D… … DM3E F
FHI … IS , M5 PM D… … DM3E G
FHI … IS , Y c
R ≥
ℋ Qℛ5 Pℛ D… Dℛ3E c
F, c
G,…, c
:HI … IS , ℛ5 Pℛ D… Dℛ3E F
F, F
G,…, F
:HI … IS , Y c
R
≥ QM5 PM D… … DM3E c
FHI … IS , )<R ð
GHI … IS , )<R ð … … ð
QM5 DM E… … EM3 c
: H … HI , M5 DM E… … EM3 F
: H… HI , )<R
Also,
GHI … IS , M5 PM D… … DM3E G
GHI … IS , Y c
R ≥
G, c
K,…, c
:, c
FHI … IS , ℛ5 Pℛ D… Dℛ3E F
G, F
K,…, F
:, c
FHI … IS , Y c
R
GHI … IS , )<R ð

Zena Hussein Maibed
KHI … IS , M5 PM D… … DM3E F
KHI … IS , )<R ð … … ð
: H … HI , M5 DM E… … EM3 F
: H… HI , )<R
We continue this process in the same way
: H … HI , M5 DM E… … EM3 G
: H… HI , Y c
R ≥
:, c
F,…, c
:LFHI … IS , ℛ5 Pℛ D… Dℛ3E F
:, F
F,…, F
:LFHI … IS , )<R
≥ QM5 DM E… … EM3 c
: H … HI , M5 DM E… … EM3 F
: H… HI , )<R ð
FHI … IS , )<R ð … … ð
:LFHI … IS , M5 PM D… … DM3E F
:LFHI … IS , )<R
As the same way and by using above inequalities,
QM5 PM D… … DM3E G
FHI … IS , M5 PM D… … DM3E K
FHI … IS , Y
c
R ≥
ℋ Qℛ5 Pℛ D… Dℛ3E F
F, F
G,…, F
:HI … IS , ℛ5 Pℛ D… Dℛ3E G
F, G
G,…, G
:HI … IS , Y
c
R
≥ QM5 PM D… … DM3E F
FHI … IS , M5 PM D… … DM3E G
FHI … IS , Y c
R ð
GHI … IS , Y c
R ð … ð
: H… HI , M5 DM E… … EM3 G
: H … HI , Y c
R
FHI … IS , )<R
3
ð
GHI … IS , )<R
3
ð … … ð
: H … HI , M5 DM E… … EM3 F
: H… HI , )<R
3
• QM5 PM D… … DM3E G
GHI … IS , M5 PM D… … DM3E K
GHI … IS , Y
c
R ≥
ℋ Qℛ5 Pℛ D… Dℛ3E F
G, F
K,…, F
:, F
FHI … IS , ℛ5 Pℛ D… Dℛ3E G
G, G
K,…, G
:, F
FHI … IS , Y
c
R
≥ QM5 PM D… … DM3E F
GHI … IS , Y c
R ð
KHI … IS , M5 PM D… … DM3E G
KHI … IS , Y c
R ð … ð
: H … HI , M5 DM E… … EM3 G
: H … HI , Y c
R

Metric Space
: H … HI , M5 DM E… … EM3 F
: H … HI , )<R
3
ð
FHI … IS , )<R
3
ð … … ð
:LFHI … IS , )<R
3
Continue this process, we get
: H … HI , M5 DM E… … EM3 K
: H… HI , Y
c
R ≥
ℋ Qℛ5 Pℛ D… Dℛ3E G
:, G
F,…, G
:LFHI … IS , ℛ5 Pℛ D… Dℛ3E K
:, K
F,…, K
:HI … IS , Y
c
R
≥ QM5 DM E… … EM3 G
: H… HI , M5 DM E… … EM3 K
: H … HI , Y c
R ð
FHI … IS , M5 PM D… … DM3E K
FHI … IS , Y c
R ð … ð
:LFHI … IS , M5 PM D… … DM3E K
:LFHI … IS , Y c
R
FHI … IS , )<R
3
ð
GHI … IS , )<R
3
ð
… … ð QM5 DM E… … EM3 c
: H… HI , M5 DM E… … EM3 F
: H … HI , )<R
3
Similarly
• QM5 PM D… … DM3E d
FHI … IS , M5 PM D… … DM3E djF
FHI … IS , Yk
c
R ≥
ℋ Qℛ5 Pℛ D… Dℛ3E dLF
F, dLF
G,…, dLF
:HI … IS , ℛ5 Pℛ D… Dℛ3E d
F, d
G,…, d
:HI … IS , Yk
c
R
≥ QM5 PM D… … DM3E dLF
FHI … IS , M5 PM D… … DM3E d
FHI … IS , Yk95
c
R ð
QM5 PM D… … DM3E dLF
GHI … IS , M5 PM D… … DM3E d
GHI … IS , Yk95
c
R ð … ð
QM5 DM E… … EM3 dLF
: H … HI , M5 DM E… … EM3 d
: H … HI , Yk95
c
R
⋮
FHI … IS , )<R
3dLF
ð
GHI … IS , )<R
3dLF
ð … … ð

Zena Hussein Maibed
l QM5 DM E… … EM3 c
: H… HI , M5 DM E… … EM3 F
: H … HI , )<R
3dLF
Also,
QM5 PM D… … DM3E d
GHI … IS , M5 PM D… … DM3E djF
GHI … IS , Yk
c
R ≥
G, dLF
K,…, dLF
:, dLF
FHI … IS ,m
ℛ5 Pℛ D… Dℛ3E d
G, d
K,…, d
:, d
FHI … IS , Yk
c
GHI … IS , Yk95
c
R ð
KHI … IS , M5 PM D… … DM3E d
KHI … IS , Yk95
c
R ð … ð
FHI … IS , ℛ5 Pℛ D… … Dℛ3E d
FHI … IS , Yk95
c
R
⋮
GHI … IS , )<R
3dLF
ð
KHI … IS , M5 PM D… … DM3E F
KHI … IS , )<R
3dLF
ð … … ð
FHI … IS , )<R
3dLF
Continue this process, as the same way we get
QM5 DM E… … EM3 d
: H… HI , M5 DM E… … EM3 djF
: H… HI , Yk
c
R
: H … HI , M5 DM E… … EM3 F
: H… HI , )<R
3dLF
ð
FHI … IS , )<R
3dLF
ð … … ð
:LFHI … IS , )<R
3dLF
Now, by using above inequalities and for each7< 7 < ?, we have
nM5 DM E… … EM3 d
: H… HI , M5 DM E… … EM3 =
: H … HI , o Yk
c
.
k43c
p
≥ nM5 DM E… … EM3 d
: H … HI , M5 DM E… … EM3 =
: H… HI , o Yk&5
c
q95
k43c
p
≥ QM5 DM E… … EM3 d
: H… HI , M5 DM E… … EM3 djF
: H… HI , Yk
c
R ∗
QM5 DM E… … EM3 djF
: H… HI , M5 DM E… … EM3 djG
: H… HI , Yk&5
c
R ð … ð

Metric Space
QM5 DM E… … EM3 =LF
: H … HI , M5 DM E… … EM3 =
: H … HI , Yq95
c
R
⋮
: H … HI , M5 DM E… … EM3 F
: H… HI , )<R
3dLF
ð
FHI … IS , )<R
3dLF
ð … … ð
:LFHI … IS , )<R
3dLF
ð
: H … HI , M5 DM E… … EM3 F
: H… HI , )<R
3d
ð
FHI … IS , )<R
3d
ð … … ð
:LFHI … IS , )<R
3d
ð
… ð QM5 DM E… … EM3 c
: H… HI , M5 DM E… … EM3 F
: H … HI , )<R
3=LG
ð
FHI … IS , )<R
3=LG
ð … … ð
:LFHI … IS , )<R
3=LG
Let s maxv7k95
, 7k
, 7q9 w
: H… HI , M5 DM E… … EM3 F
: H … HI , )<R
x
ð
FHI … IS , )<R
x
ð … … ð
:LFHI … IS , )<R
x
ð
: H… HI , M5 DM E… … EM3 F
: H … HI , )<R
x
ð
FHI … IS , )<R
x
ð … … ð
: H… HI , M5 DM E… … EM3 F
: H … HI , )<R
x
ð
FHI … IS , )<R
x
ð … … ð
:LFHI … IS , )<R
x

Zena Hussein Maibed
> QM5 DM E… … EM3 c
: H … HI , M5 DM E… … EM3 F
: H… HI , )<R
qx
ð
FHI … IS , )<R
qx
ð … … ð
:LFHI … IS , )<R
qx
≥ 1 − g ∗ 1 − g ∗ … … ∗ 1 − g ≥ 1 − ʎ
And hence,
QM5 DM E… … EM3 d
: H… HI , M5EM E… … EM3 =
: H… H, )HR > 1 − ʎ
So, < M5 DM E… … EM3 d
: H… HI > is Cauchy sequence.
As the same way, we get
FHI … IS >, < M5 PM D… … DM3E d
GHI … IS > 678
:LFHI … IS > are Cauchy sequences
Now, to prove that the mappings aregeneralized tuplet common fixed point.
Since M5 DM E… … EM3 b H … HI is complete subspace of Ε then there exists "5, " , … … , "3 ∈
M5 DM E… … EM3 b H … HI and 65, 6 , … … , 63 ∈ Ε such that
lim
k→.
M5 PM D… … DM3E d
FHI … IS
≥ lim
k→.
ℛ5 Pℛ D… Dℛ3E dLF
F,…, dLF
:HI … IS ⟶ M5 DM E… … EM3 zF
H… HI "5
lim
k→.
M5 PM D… … DM3E d
GHI … IS
≥ lim
k→.
G,…, dLF
FHI … IS → M5 DM E… … EM3 zG
H… HI "
⋮
lim
k→.
M5 DM E… … EM3 d
: H… HI
≥ lim
k→.
:,…, dLF
:LFHI … IS → M5 DM E… … EM3 z:
H … HI "3
Now,
F, dLF
G,…, dLF
:HI … IS , ℛ5 Dℛ E… Eℛ3 zF,zG,….,z:
H … HI , Y R
FHI … IS , M5 DM E… … EM3 zF
H … HI , )R ð
GHI … IS , M5 DM E… … EM3 zG
H … HI , )R ð … ð
QM5 DM E… … EM3 dLF
: H … HI , M5 DM E… … EM3 z:
H … HI , )R

Metric Space
As 7 → ∞ and by continuity of l, we get
QM5 DM E… … EM3 zF
H … HI , M5 DM E… EM3 zF,zG,…,z:
H … HI , Y R 1
Also,ℋ Qℛ5 Pℛ D… Dℛ3E dLF
G, dLF
K,…, dLF
FHI … IS , ℛ5 Dℛ E… Eℛ3 zF,zG,…,z:
H … HI , Y R
GHI … IS , M5 DM E… … EM3 zG
H … HI , )R ð
KHI … IS , M5 DM E… … EM3 zK
H … HI , )R ð … ð
H … HI , )R
As 7 → ∞,
QM5 DM E… … EM3 zG
H… HI , ℛ5 Dℛ E… Eℛ3 zG,zK,…,z:
H … HI , Y R 1
Continuity
ℋ Qℛ5 Pℛ D… … Dℛ3E dLF
:, dLF
F,…, dLF
:LFHI … IS , ℛ5 Dℛ E… Eℛ3 z:,zF,…,z:95 H … HI , Y R
≥ QM5 DM E… … EM3 dLF
: H… HI , M5 DM E… … EM3 z:
H… HI , )R ð
H … HI , )R ð … ð
:LFHI … IS , M5 DM E… … EM3 z:LF
H … HI , )R
As7 → ∞, we get
QM5 DM E… … EM3 z:
H… HI , ℛ5 Dℛ E… Eℛ3 z:,zF,…,z:LF
H … HI , Y R 1
⟹ "5 M5 DM E… … EM3 zF
H … HI ∈ ℛ5 Dℛ E… Eℛ3 zF,zG,…,z:
H … HI
" M5 DM E… … EM3 zG
H… HI ∈ ℛ5 Dℛ E… Eℛ3 zG,zK,…,zF
H… HI
⋮
"3 M5 DM E… … EM3 z:
H … HI ∈ ℛ5 Dℛ E… Eℛ3 z:,…,z:LF
H… HI
Therefore, 65, 6 , … … , 63 isgeneralized tupled coincidence point.

Zena Hussein Maibed
Theorem (2.6):
Let Ε, , ð be a fuzzy metric space .Under the same assumptions of theorem(2.5) and all
mappings are weakly compatible at the coincidence point. Then these mappings have a unique
generalized tupled common fixed point of compose these mappings.
Proof:
Since the mappings lies in A and B are weakly compatible, this implies
• M5 PM D… … DM3 Qℛ5 Dℛ E… Eℛ3 zF,zG,…,z:
H … HIRI … IS ∈
ℛ5 Tℛ U… Uℛ3 |
M5 DM E… … EM3 zF
H… HI , M5 DM E… … EM3 zG
H … HI ,
… … , M5 DM E… … EM3 z:
H … HI
}V … VW
• M5 PM D… … DM3 Qℛ5 Dℛ E… Eℛ3 zG,zK,…,zF
M5 DM E… … EM3 zG
H… HI , M5 DM E… … EM3 zK
H … HI ,
… … , M5 DM E… … EM3 zF
H … HI
}V … VW
Continue,
• M5 PM D… … DM3 Qℛ5 Dℛ E… Eℛ3 z:,zF,…,z:LF
M5 DM E… … EM3 z:
H… HI , M5 DM E… … EM3 zF
H … HI ,
… … , M5 DM E… … EM3 z:LF
H … HI
}V … VW
By above inquisitions, we have
• M5 DM E… … EM3 F
H … HI ∈ ℛ5 Dℛ E… Eℛ3 F, G,…, :
H … HI
• M5 DM E… … EM3 G
H … HI ∈ ℛ5 Dℛ E… Eℛ3 G, K,…, F
H … HI
⋮ (3)
• M5 DM E… … EM3 :
H … HI ∈ ℛ5 Dℛ E… Eℛ3 :, F,…, :LF
H … HI
Now, we will prove thatM5 DM E… … EM3 F
H… HI "5
H… HI
⋮
M5 DM E… … EM3 :
H… HI

Metric Space
Since ð is a t – norm of H – type, we have, ∀ ʎ > 0 ∃ g > 0 such that
1 − g ð … … ð 1 − g ≥ 1 − ʎ .But ", #, ð is continuous and lim →. , , 1 , ∀ ", # ∈ Ε
then there exits )< > 0 such that
DM5 DM E… … EM3 F
H … HI , " , )<I ≥ 1 − g
DM5 DM E… … EM3 G
H … HI , "~, )<I ≥ 1 − g
⋮
DM5 DM E… … EM3 :
H … HI , "5, )<I ≥ 1 − g
SinceY ∈ Ψ, by properties of Ψ we get,∑ Y3
c
< ∞.
345 which is implies, ∀ ) > 0 ∃ 7< ∈ ^ such
that, ) > ∑ Yk
c
.
k43c
On other hand,
• DM5 DM E… … EM3 F
H … HI , M5 PM D… … DM3E d
GHI … IS , Y c
I
Dℛ5 Dℛ E… Eℛ3 F, G,…, :
H … HI , ℛ5 Pℛ D… Dℛ3E dLF
G, dLF
K,…, dLF
FHI … IS , Y c
I
≥ DM5 DM E… … EMM3 F
H… HI , M5 PM D… … DM3E dLF
GHI … IS , )<I ð
H … HI , M5 PM D… DM3E dLF
KHI … IS , )<I ð … … ð
⋮
H … HI , M5 PM D… … DM3E dLF
FHI … IS , )<I
• DM5 DM E… … EM3 G
KHI … IS , Y c
I
Dℛ5 Dℛ E… Eℛ3 G, K,…, F
H … HI , ℛ5 Pℛ D… Dℛ3E dLF
K, dLF
•,…, dLF
GHI … IS , Y c
I
≥ DM5 DM E… … EM3 G
KHI … IS , )<I ð
DM5 DM E… … EM3 K
•HI … IS , )<I ð … … ð
⋮
GHI … IS , )<I
Continues,
FHI … IS , Y c
I

Zena Hussein Maibed
Dℛ5 Dℛ E… Eℛ3 :, F,…, :LF
H… HI , ℛ5 Pℛ D… Dℛ3E dLF
F, dLF
G,…, dLF
:HI … IS , Y c
I
≥ DM5 DM E… … EM3 :
FHI … IS , )<I ð
GHI … IS , )<I ð … … ð
⋮
DM5 DM E… … EM3 :LF
H … HI , M5 DM E… … EM3 dLF
: H … HI , )<I
As 7 → ∞ in the above equalities, we obtain
i. DM5 DM E… … EM3 F
H … HI , " , Y c
I ≥ DM5 DM E… … EM3 :
H … HI , "5, )<I ð
ii. DM5 DM E… … EM3 F
H … HI , " , )<I ð … … ð DM5 DM E… … EM3 :LF
H … HI , "3, )<I
iii. DM5 DM E… … EM3 G
H … HI , "~, Y c
I ≥
H … HI , "~, )<I ð DM5 DM E… … EM3 K
H … HI , "€, )<I ð
iv. … … ð DM5 DM E… … EM3 F
H … HI , " , )<I
v. DM5 DM E… … EM3 :
H … HI , "5, Y c
I ≥
H … HI , "5, )<I ð DM5 DM E… … EM3 F
H … HI , " , )<I ð … …
vi. ð DM5 DM E… … EM3 :LF
H … HI , "3, )<I
By i, ii and iii we have
H… HI , " , Y c
I ð
H… HI , "~, Y c
I ð … … ð DM5 DM E… … EM3 3 H… HI , "5, Y c
I ≥
H … HI , "5, Y c
R
3
ð QM5 DM E… … EM3 F
H… HI , " , Y c
R
3
ð … … ð QM5 DM E… … EM3 :LF
H … HI , "3, Y c
R
3
By induction,
H … HI , " , Yk
c
I ð DM5 DM E… … EM3 G
H … HI , "~, Yk
c
I ð … … ð
H… HI , "5, Yk
c
I
≥ QM5 DM E… … EM3 :
H… HI , "5, Yk95
c
R
3
H… HI , " , Yk95
c
R
3
ð … … ð
QM5 DM E… … EM3 :LF
H … HI , "3, Yk95
c
R
3

Metric Space
⋮
≥ QM5 DM E… … EM3 :
H … HI , "5, )<R
3d
H … HI , " , )<R
3d
ð
… … ð QM5 DM E… … EM3 :LF
H … HI , "3, )<R
3d
But∑ ∅3
c
< ).
345 , ∀ 7 ∈ ^, we get
H … HI , " , )R
3d
ð QM5 DM E… … EM3 H … HI , "~, )R
3d
ð … …
ð QM1 DM2E… …EM7 "7
H … HI , "1,)R
7•
≥ UM1 DM2E… … EM7 "1
H … HI , "2, o Y•
)0
∞
• 70
V ð UM1 DM2E… … EM7 "2
H … HI , "3, o Y•
)0
∞
• 70
V ð…
… ð UM5 DM E… … EM3 :
H … HI , "5, o Yk
c
.
k43c
V
≥ DM5 DM E… … EM3 F
H … HI , " , Y3c
c
I ð DM5 DM E… … EM3 G
H … HI , "~, Y3c
c
I
ð … … ð DM5 DM E… … EM3 :
H … HI , "5, Y3c
c
I
≥ QM5 DM E… … EM3 :
H … HI , "5, )<R
3:c
H … HI , " , )<R
3:c
ð … … ð QM5 DM E… … EM3 :LF
H … HI , "3, )<R
3:c
≥ 1 − g ð 1 − g ð … … ð 1 − g ≥ 1 − ʎ
Therefore, DM5 DM E… … EM3 F
H … HI , " , )I ð DM5 DM E… … EM3 G
H … HI , "~, )I ð … … ð
H … HI , "5, )I ≥ 1 − ʎ
That is,M5 DM E… … EM3 F
H … HI "
H … HI "~
⋮ (4)
M5 DM E… … EM3 :
H … HI "5
Now, we will show that "5 " . . . … "3.As that same way, since ", #, . is
continuous and lim →. , , 1 ∀ ", # ∈ Ε ⟹ ∃ )< > 0 such that
F, G, c
≥ 1 − g, G, K, c
≥ 1 − g
⋮

Zena Hussein Maibed
:, F, c
≥ 1 − g
But ∑ Y3
c
.
345 < ∞ ⟹ ∑ Yk
c
.
k43c
< ∞,for some 7< ∈ ^.
Now,
• DM5 PM D… … DM3E d
FHI … IS , M5 PM D… … DM3E d
GHI … IS , Y c
I
Dℛ5 Pℛ D… Dℛ3E dLF
F,…, dLF
:HI … IS , ℛ5 Pℛ D… Dℛ3E dLF
G,…, dLF
FHI … IS , Y c
I
≥ ƒM5 PM D… … DM3E dLF
FHI … IS , M5 PM D… … DM3E dLF
GHI … IS„ ð … … ð
ƒM5 DM E… … EM3 dLF
: H… HI , M5 PM D… … DM3E dLF
FHI … IS„
Also,
• DM5 PM D… … DM3E d
KHI … IS , Y c
I
G,…, dLF
FHI … IS , ℛ5 Pℛ D… Dℛ3E dLF
K,…, dLF
GHI … IS , Y c
I
≥ DM5 PM D… … DM3E dLF
GHI … IS , M5 PM D… … DM3E dLF
KHI … IS , )<I ð … … ð
DM5 PM D… … DM3E dLF
FHI … IS , M5 PM D… … DM3E dLF
GHI … IS , )<I
Continue,
• DM5 DM E… … EM3 d
: H … HI , M5 PM D… … DM3E d
FHI … IS , Y c
I
:,…, dLF
:LFHI … IS , ℛ5 Pℛ D… Dℛ3E dLF
F,…, dLF
:HI … IS , Y c
I
≥ DM5 DM E… … EM3 dLF
: H … HI , M5 PM D… … DM3E dLF
FHI … IS , )<I ð … … ð
DM5 PM D… … DM3E dLF
:LFHI … IS , M5 DM E… … EM3 dLF
: H … HI , )<I
As7 → ∞, we get,
E"5, " , Y c
H ≥ "5, " , )< ð " , "~, )< ð … … ð "3, "5, )<
E" , "~, Y c
H ≥ " , "~, )< ð "~, "€, )< ð … … ð "5, " , )<
⋮
E"3, "5, Y c
H ≥ "3, "5, )< ð "5, " , )< ð … … ð "395, "3, )<

Metric Space
And hence,
E"5, " , Y c
Hð E" , "~, Y c
Hð … … ð E"3, "5, Y c
H ≥
"5, " , )<
3
ð " , "~, )<
3
ð … … ð "3, "5, )<
3
By induction
D"5, " , Yk
c
I ð D" , "~, Yk
c
I ð … … ð D"3, "5, Yk
c
I ≥
Q D"5, " , Yk95
c
IR
3
ð Q D" , "~, Yk95
c
IR
3
ð … … ð Q D"3, "5, Yk95
c
IR
3
⋮
≥ "5, " , )<
3d
ð " , "~, )<
3d
ð … … ð "3, "5, )<
3d
But, ∑ Y3
c
.
345 < ) and hence
"5, " , ) ð " , "~, ) ð … … ð "3, "5, )
≥ D"5, " , ∑ Yk
c
.
k43c
I ð D" , "~, ∑ Yk
c
.
k43c
I ð … … ð D"3, "5, ∑ Yk
c
.
k43c
I
≥ D"5, " , Y3c
c
I ð D" , "~, Y3c
c
I ð … … ð D"3, "5, Y3c
c
I
≥ "5, " , )<
3:c
ð " , "~, )<
3:c
ð … … ð "3, "5, )<
3:c
≥ 1 − g ð 1 − g ð … … ð 1 − g ≥ 1 − ʎ
⟹ "5 " "~ … "3
And hence, by (3) & (4)
M5 DM E… … EM3 F
H … . . HI ℛ5 Dℛ E… … Eℛ3 F, G,…, :
H … . . HI "5
H … . . HI ℛ5 Dℛ E… … Eℛ3 G, K,…, F
H … . . HI "
⋮ (5)
M5 DM E… … EM3 :
H… . . HI ℛ5 Dℛ E… … Eℛ3 :, F,…, :LF
H … . . HI "3
Finally, we shall prove the uniqueness suppose that
"5̀ , "̀ , … … , "3̀ ∈ ‹ Satisfy (5)
E"5, "5̀ , Y H
Dℛ5 Dℛ E… … Eℛ3 F, G,…, :
H… . . HI , ℛ5 Dℛ E… … Eℛ3 F̀ , G̀ ,…, :̀ H… . . HI , Y I
≥ DM5 DM E… … EM3 F
H … . . HI , M5 DM E… … EM3 F̀ H … . . HI , )I ð … … ð
H … . . HI , M5 DM E… … EM3 :̀ H … . . HI , )I

Zena Hussein Maibed
"5, "5̀ , ) ð … … ð "3, "3̀ , ) > "5, "5̀ , )
By lemma (1.3), we get"5 "5̀ . As the same way, we get
" "̀ , … 678 "3 "3̀
Corollary (2.7)
Let Ε, , ð be a fuzzy metric space .Under the same assumptions of theorem (2.5) but
Qℛ5 Dℛ E… … Eℛ3 F, G,…, :
H… . . HI , ℛ5 Dℛ E… … Eℛ3 F, G,…, :
H … . . HI , •)R
≥ QM5 DM E… … EM3 F
H … . . HI , M5 DM E… … EM3 F
H… . . HI , )R ð
H… . . HI , M5 DM E… … EM3 G
H… . . HI , )R ð … … ð
H… . . HI , M5 DM E… … EM3 :
H… . . HI , )R
Where • ∈ 0,1 , ) > 0 and "`, #` ∈ Ε ∀ a 1,2, … … , 7.Then there exists a unique Œ. 7 – tupled
common fixed point of compose the mappings in • 678 Ž.
Corollary (2.8)
Let Ε, , ð be a fuzzy metric space .Under the same assumptions of theorem (2.5) but
Qℛ5 Dℛ E… … Eℛ3 F, G,…, :
H… . . HI , ℛ5 Dℛ E… … Eℛ3 F, G,…, :
H … . . HI , Y ) R
H… . . HI , M5 DM E… … EM3 F
H… . . HI , )R
zF
ð
H … . . HI , M5 DM E… … EM3 G
H… . . HI , )R
zG
ð … … ð
H … . . HI , M5 DM E… … EM3 :
H… . . HI , )R
z:
Where ∑ 6`
3
`45 1, ) > 0 and "`, #` ∈ Ε , ∀ a 1,2, … … , 7.Then there exists a unique
Œ. 7 – tupled common fixed point of compose the mappings in• 678 Ž.
Corollary (2.9)
Let Ε, , ð be a fuzzy metric space .Under the same assumptions of theorem(2.5) but
Qℛ5 Dℛ E… … Eℛ3 F, G,…, :
H… . . HI , ℛ5 Dℛ E… … Eℛ3 F, G,…, :
H … . . HI , •)R
H… . . HI , M5 DM E… … EM3 F
H… . . HI , )R
zF
ð
H … . . HI , M5 DM E… … EM3 G
H… . . HI , )R
zG
ð … … ð
H … . . HI , M5 DM E… … EM3 :
H… . . HI , )R
z:
Where ∑ 6`
3
`45 1 , • ∈ 0,1 , and "`, #` ∈ Ε ∀ a 1,2, … … , 7.Then there exists a
unique Œ. 7 – tupled common fixed point of compose the mappingsin• 678 Ž.

Metric Space
Corollary (2.10):
Let Ε, , ð be a fuzzy metric space .Under the same assumptions of theorem(2.5) but
Qℛ5 Dℛ E… … Eℛ3 F, G,…, :
H… . . HI , ℛ5 Dℛ E… … Eℛ3 F, G,…, :
H … . . HI , •)R
≥ "5, #5, ) ð " , # , ) ð … … ð "3, #3, )
Where • ∈ 0,1 , ) > 0 and "`, #` ∈ Ε ∀ a 1,2, … … , 7.Then there exists a unique Œ. 7 – tupled
fixed point of compose the mappings in •.
CONCLUSION
We have introduced the new concepts of generalized tupled common fixed point and weakly
compatible mapping and established some common fixed point theorems in Fuzzy metric
space. In the end, we must say that, this paper is just a beginning of a new structure and we
have studied many of ideas , it will be necessary to carry out more theoretical research to
establish a general framework for the practical application.
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Ijciet 10 01_025

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