1) The document presents a fixed point and common fixed point theorem for four self-mappings (E, F, G, H) in a complete metric space that satisfy certain contractive conditions.
2) It establishes that if the mappings satisfy a contractive modulus condition and the pairs (G, E) and (H, F) are weakly compatible, then the mappings have a unique common fixed point.
3) The proof involves showing that a sequence defined using the mappings converges to a fixed point, and that this fixed point is unique.
Accurate Numerical Method for Singular Initial-Value Problemsaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular initial value problems. The second-order singular initial value problem under consideration is transferred into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods is confirmed by solving three model examples, and the obtained approximate solutions are compared with the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical method for solving the singular initial value problems.
ABSTRACT: In this paper, we construct new classes of derivative-free of tenth-order iterative methods for solving nonlinear equations. The new methods of tenth-order convergence derived by combining of theSteffensen's method, the Kung and Traub’s of optimal fourth-order and the Al-Subaihi's method. Several examples to compare of other existing methods and the results of new iterative methods are given the encouraging results and have definite practical utility.
Accurate Numerical Method for Singular Initial-Value Problemsaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular initial value problems. The second-order singular initial value problem under consideration is transferred into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods is confirmed by solving three model examples, and the obtained approximate solutions are compared with the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical method for solving the singular initial value problems.
ABSTRACT: In this paper, we construct new classes of derivative-free of tenth-order iterative methods for solving nonlinear equations. The new methods of tenth-order convergence derived by combining of theSteffensen's method, the Kung and Traub’s of optimal fourth-order and the Al-Subaihi's method. Several examples to compare of other existing methods and the results of new iterative methods are given the encouraging results and have definite practical utility.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
We consider an elliptic BVP.
How to compute a part of the solution? For instance, solution on the interface, solution in s subdomain in a point without computing the whole solution and with O(n log n) complexity/storage.
ملزمة الرياضيات للصف السادس التطبيقي الفصل الاول الاعداد المركبة 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
In this paper based on recently introduced approach we formulated some recommendations to optimize
manufacture drift bipolar transistor to decrease their dimensions and to decrease local overheats during
functioning. The approach based on manufacture a heterostructure, doping required parts of the heterostructure
by dopant diffusion or by ion implantation and optimization of annealing of dopant and/or radiation
defects. The optimization gives us possibility to increase homogeneity of distributions of concentrations
of dopants in emitter and collector and specific inhomogenous of concentration of dopant in base and at the
same time to increase sharpness of p-n-junctions, which have been manufactured framework the transistor.
We obtain dependences of optimal annealing time on several parameters. We also introduced an analytical
approach to model nonlinear physical processes (such as mass- and heat transport) in inhomogenous media
with time-varying parameters.
Numerical solution of fuzzy differential equations by Milne’s predictor-corre...mathsjournal
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of fuzzy dot Ideals of BH-algebras and investigate some of their results.
On optimization ofON OPTIMIZATION OF DOPING OF A HETEROSTRUCTURE DURING MANUF...ijcsitcejournal
We introduce an approach of manufacturing of a p-i-n-heterodiodes. The approach based on using a δ-
doped heterostructure, doping by diffusion or ion implantation of several areas of the heterostructure. After
the doping the dopant and/or radiation defects have been annealed. We introduce an approach to optimize
annealing of the dopant and/or radiation defects. We determine several conditions to manufacture more
compact p-i-n-heterodiodes
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
We consider an elliptic BVP.
How to compute a part of the solution? For instance, solution on the interface, solution in s subdomain in a point without computing the whole solution and with O(n log n) complexity/storage.
ملزمة الرياضيات للصف السادس التطبيقي الفصل الاول الاعداد المركبة 2022anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
In this paper based on recently introduced approach we formulated some recommendations to optimize
manufacture drift bipolar transistor to decrease their dimensions and to decrease local overheats during
functioning. The approach based on manufacture a heterostructure, doping required parts of the heterostructure
by dopant diffusion or by ion implantation and optimization of annealing of dopant and/or radiation
defects. The optimization gives us possibility to increase homogeneity of distributions of concentrations
of dopants in emitter and collector and specific inhomogenous of concentration of dopant in base and at the
same time to increase sharpness of p-n-junctions, which have been manufactured framework the transistor.
We obtain dependences of optimal annealing time on several parameters. We also introduced an analytical
approach to model nonlinear physical processes (such as mass- and heat transport) in inhomogenous media
with time-varying parameters.
Numerical solution of fuzzy differential equations by Milne’s predictor-corre...mathsjournal
The study of this paper suggests on dependency problem in fuzzy computational method by using the numerical solution of Fuzzy differential equations(FDEs) in Milne’s predictor-corrector method. This method is adopted to solve the dependency problem in fuzzy computation. We solve some fuzzy initial value problems to illustrate the theory.
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
Research on information and communication technologies have been developed rapidly since it can be applied easily to several areas like computer science, medical science, economics, environments, engineering, among other. Applications of soft set theory, especially in information systems have been found paramount importance. Recently, Mukherjee and Das defined some new operations in fuzzy soft multi set theory and show that the De-Morgan’s type of results hold in fuzzy soft multi set theory with respect to these newly defined operations. In this paper, we extend their work and study some more basic properties of their defined operations. Also, we define some basic supporting tools in information system also application of fuzzy soft multi sets in information system are presented and discussed. Here we define the notion of fuzzy multi-valued information system in fuzzy soft multi set theory and show that every fuzzy soft multi set is a fuzzy multi valued information system.
In this paper we introduce the notions of Fuzzy Ideals in BH-algebras and the notion
of fuzzy dot Ideals of BH-algebras and investigate some of their results.
On optimization ofON OPTIMIZATION OF DOPING OF A HETEROSTRUCTURE DURING MANUF...ijcsitcejournal
We introduce an approach of manufacturing of a p-i-n-heterodiodes. The approach based on using a δ-
doped heterostructure, doping by diffusion or ion implantation of several areas of the heterostructure. After
the doping the dopant and/or radiation defects have been annealed. We introduce an approach to optimize
annealing of the dopant and/or radiation defects. We determine several conditions to manufacture more
compact p-i-n-heterodiodes
In this paper, an accurate numerical method is presented to find the numerical solution of the singularinitial value problems. The second-order singular initial value problem under consideration is transferredinto a first-order system of initial value problems, and then it can be solved by using the fifth-order RungeKutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methodsis confirmed by solving three model examples, and the obtained approximate solutions are compared withthe existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numericalmethod for solving the singular initial value problems.
ACCURATE NUMERICAL METHOD FOR SINGULAR INITIAL-VALUE PROBLEMSaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular
initial value problems. The second-order singular initial value problem under consideration is transferred
into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge
Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods
is confirmed by solving three model examples, and the obtained approximate solutions are compared with
the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical
method for solving the singular initial value problems.
ACCURATE NUMERICAL METHOD FOR SINGULAR INITIAL-VALUE PROBLEMSaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular initial value problems. The second-order singular initial value problem under consideration is transferred into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods is confirmed by solving three model examples, and the obtained approximate solutions are compared with the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical method for solving the singular initial value problems.
Accurate Numerical Method for Singular Initial-Value Problemsaciijournal
In this paper, an accurate numerical method is presented to find the numerical solution of the singular
initial value problems. The second-order singular initial value problem under consideration is transferred
into a first-order system of initial value problems, and then it can be solved by using the fifth-order Runge
Kutta method. The stability and convergence analysis is studied. The effectiveness of the proposed methods
is confirmed by solving three model examples, and the obtained approximate solutions are compared with
the existing methods in the literature. Thus, the fifth-order Runge-Kutta method is an accurate numerical
method for solving the singular initial value problems.
Using Cauchy Inequality to Find Function ExtremumsYogeshIJTSRD
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Mx/G(a,b)/1 With Modified Vacation, Variant Arrival Rate With Restricted Admi...IJRES Journal
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Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
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Neuro-symbolic is not enough, we need neuro-*semantic*Frank van Harmelen
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JMeter webinar - integration with InfluxDB and GrafanaRTTS
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Length: 30 minutes
Session Overview
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During this webinar, we will cover the following topics while demonstrating the integrations of JMeter, InfluxDB and Grafana:
- What out-of-the-box solutions are available for real-time monitoring JMeter tests?
- What are the benefits of integrating InfluxDB and Grafana into the load testing stack?
- Which features are provided by Grafana?
- Demonstration of InfluxDB and Grafana using a practice web application
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Fixed point and common fixed point theorems in complete metric spaces
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
193
Fixed Point and Common Fixed Point Theorems in Complete
Metric Spaces
Pushpraj Singh Choudhary*
, Renu Praveen Pathak** and M.K. Sahu***,Ramakant Bhardwaj****
*Dept. of Mathematics & Computer Science Govt. Model Science College (Auto.) Jabalpur.
**Department of Mathematics, Late G.N. Sapkal College of Engineering,Anjaneri Hills, Nasik (M.H.), India
***Department of Mathematics,Govt PG College Gadarwara.
****Truba Institute of Engineering and Information Technology,Bhopal.
pr_singh_choudhary@yahoo.co.in, pathak_renu@yahoo.co.in
Abstract
In this paper we established a fixed point and a unique common fixed point theorems in four pair of weakly
compatible self-mappings in complete metric spaces satisfy weakly compatibility of contractive modulus.
Keywords : Fixed point, Common Fixed point, Complete metric space, Contractive modulus, Weakly
compatible maps.
Mathematical Subject Classification : 2000 MSC No. 47H10, 54H25
Introduction
The concept of the commutativity has generalised in several ways .In 1998, Jungck Rhoades [3] introduced the
notion of weakly compatible and showed that compatible maps are weakly compatible but conversely. Brian
Fisher [1] proved an important common fixed point theorem. Seesa S [9] has introduced the concept of weakly
commuting and Gerald Jungck [2] initiated the concept of compatibility. It can be easily verified that when the
two mappings are commuting then they are compatible but not conversely. Thus the study of common fixed
point of mappings satisfying contractive type condition have been a very active field of research activity during
the last three decades.
In 1922 the polished mathematician, Banach, proved a theorem ensures, under appropriate conditions, the
existence and uniqueness of a fixed point. His result is called Banach fixed point theorem or the Banach
contraction principle. This theorem provides a technique for solving a variety of a applied problems in a
mathematical science and engineering. Many authors have extended, generalized and improved Banach fixed
point theorem in a different ways. Jungck [2] introduced more generalised commuting mappings, called
compatible mappings, which are more general than commuting and weakly commuting mappings.
The main purpose of this paper is to present fixed point results for two pair of four self maps satisfying a new
contractive modulus condition by using the concept of weakly compatible maps in complete metric spaces.
Preliminaries
The definition of complete metric spaces and other results that will be needed are:
Definition 1: Let f and g two self-maps on a set X. Maps f and g are said to be commuting if
= .
Definition 2: Let f and g two self-maps on a set X. If = , ℎ called coincidence
point of f and g.
Definition 3: Let f and g two self-maps defined on a set X, then f and g are said to be compatible if they
commute at coincidence points. That is, if = , ℎ = .
Definition 4: Let f and g be two weakly compatible self-maps defined on a set X, If f and g have a unique point
of coincidence, that is, = = , ℎ .
Definition 5: A sequence in a metric space , ! ,
denoted by lim →& = , lim →& , = 0.
Definition 6: A sequence in a metric space , ( ℎ)
lim
*→&
, + = 0 , > .
Definition 7: A metric space , ( every Cauchy sequence in X is convergent.
Definition 8: A function -: /0, ∞ → /0, ∞ !
-: /0, ∞ → /0, ∞ - < > 0 .
Definition 9: A real valued function φ defined on RX ⊆ is said to be upper semi continuous
lim
→&
- ≤ - , ! ) ℎ → → ∞ .
Hence it is clear that every continuous function is upper semi continuous but converse may not true.
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
194
MAIN RESULT
In this section we established a common fixed point theorem for two pairs of weakly compatible mappings in
complete metric spaces using a contractive modulus.
Theorem 1: Let (X,d) be a complete metric space. Suppose that the mapping E, F, G and H are four self maps
of X satisfying the following condition:
(i) 3 ⊆ 5 6 ⊆ 7
(ii) 869, 3:; ≤ - < , )
Where - is a upper semi continuous, contractive modulous and
< , ) = max
?
@
A
@
B 859, 7:;, 59, 69 , 87:, 3:;,
C
D
E 859, 3:; + 87:, 69;G ,
C
H
I
J8KL,MN;OJ KL,PL OJ8MN,PL;
COJ8KL,MN;J KL,PL J8MN,PL;
Q ,
R
D
I
J8MN,SN;OJ8MN,PL;OJ8MN,KL;
COJ8MN,SN;J8MN,PL;J8MN,KL;
Q
T
@
U
@
V
(iii) The pair (G, E) and (H, F) are weakly compatible then E, F, G& H have a unique common fixed point.
Proof : Suppose ( ) ℎ ) ℎ ℎ
) = 6 = 7 OC
) OC = 3 OC = 5 OD
By (ii) we have
) , ) OC = 6 , 3 OC
( )( )1,n nx x +≤ φ λ
Where < , OC =
?
@
A
@
B
5 , 7 OC , 5 , 6 , 7 OC, 3 OC ,
C
D
/ 5 , 3 OC + 7 OC, 6 W ,
C
H
X
J K9Y,M9YZ[ OJ K9Y,P9Y OJ M9YZ[,P9Y
COJ K9Y,M9YZ[ J K9Y,P9Y J M9YZ[,P9Y
,
R
D
X
J M9YZ[,S9YZ[ OJ M9YZ[,P9Y OJ M9YZ[,K9Y
COJ M9YZ[,S9YZ[ J M9YZ[,P9Y J M9YZ[,K9Y
T
@
U
@
V
=
?
@
@
A
@
@
B
3 ]C, 6 , 3 ]C, 6 , 6 , 3 OC ,
1
2
/ 3 ]C, 3 OC + 6 , 6 W ,
1
4
a
3 ]C, 6 + 3 ]C, 6 + 6 , 6
1 + 3 ]C, 6 3 ]C, 6 6 , 6
b ,
3
2
a
6 , 3 OC + 6 , 6 + 6 , 3 ]C
1 + 6 , 3 OC 6 , 6 6 , 3 ]C
b
T
@
@
U
@
@
V
≤ max d 3 ]C, 6 , 6 , 3 OC ,
1
2
/ 3 ]C, 3 OC W
1
4
e
3 ]C, 6
1
f
3
2
e
3 ]C, 3 OC
1
fg
≤ max I ) ]C, ) , ) , ) OC ,
1
2
/ ) ]C, ) OC W,
1
4
) ]C, ) ,
3
2
/ ) ]C, ) OC WQ
≤ max ) ]C, ) , ) , ) OC
Since - is a contractive modulus; < , OC = ) , ) OC , hℎ
) , ) OC ≤ - ) ]C, ) 1
Since - is an upper semi continuous contractive modulus, Equation (1) implies that the sequence ) , ) OC
is monotonic decreasing and continuous.
Hence there exists a real number, say ≥ 0 ℎ ℎ lim →& ) , ) OC =
∴ → ∞ , 5 1 ℎ
≤ -
Which is possible only If r = 0 because - is a contractive modulus, Thus
lim
→&
) , ) OC = 0
Now we show that ) is a Cauchy sequence.
Let If possible we assume that ) is not a cauchy sequence , Then there exist an ∈ > 0 and subsequence
l l ℎ ℎ l < l < lOC
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
195
( ) ( )1
, and , im n m nd y y d y y −
≥ ∈ <∈ (2)
So that ( ) ( ) ( ) ( )1 1 1
, , , ,i i i i i i im n m n n n n nd y y d y y d y y d y y− − −
∈≤ ≤ + <∈ +
Therefore ( ) ( ) ( )( )lim , , ,i i i i i im n m n m n
n
d y y d Gx Hx x x
→∞
= ≤ φ λ
i.e. ( )( ),i im nx x∈≤ φ λ (3)
Where,
( )
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
, , , , , ,
1
, ,
2
, , ,1, max
4 1 , , ,
, , ,3
2 1 , ,
i i i i i i
i i i i
i i i i i i
i i
i i i i i i
i i i i i i
i i i
m n m m n n
m n n m
m n m m n m
m n
m n m m n m
n n n m n m
n n n
d Ex Fx d Ex Gx d Fx Hx
d Ex Hx d Fx Gx
d Ex Fx d Ex Gx d Fx Gxx x
d Ex Fx d Ex Gx d Fx Gx
d Fx Hx d Fx Gx d Fx Ex
d Fx Hx d Fx G
+
+ +λ =
+ +
+ +
+ ( ) ( ),i i im n mx d Fx Ex
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
1 1 1 1
1 1
1 1 1 1
1 1 1 1
1 1
, , , , , ,
1
, , ,
2
, , ,1max
4 1 , , ,
, ,3
2
i i i i i i
i i i i
i i i i i i
i i i i i i
i i i i i
m n m m n n
m n n m
m n m m n m
m n m m n m
n n n m n
d Hx Gx d Hx Gx d Gx Hx
d Hx Hx d Gx Gx
d Hx Gx d Hx Gx d Gx Gx
d Hx Gx d Hx Gx d Gx Gx
d Gx Hx d Gx Gx d Gx
− − − −
− −
− − − −
− − − −
− −
+
+ +=
+
+ + ( )
( ) ( ) ( )
1 1
1 1 1 1
,
1 , , ,
i
i i i i i i
m
n n n m n m
Hx
d Gx Hx d Gx Gx d Gx Hx
− −
− − − −
+
( ) ( ) ( )
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( )
1 1 1 1
1 1
1 1 1 1
1 1 1 1
1 1 1 1
1 1
, , , , , ,
1
, , ,
2
, , ,1max
4 1 , , ,
, , ,3
2 1 , ,
i i i i i i
i i i i
i i i i i i
i i i i i i
i i i i i i
i i i
m n m m n n
m n n m
m n m m n m
m n m m n m
n n n m n m
n n n
d y y d y y d y y
d y y d y y
d y y d y y d G y
d y y d y y d y y
d y y d y y d y y
d y y d y
− − − −
− −
− − − −
− − − −
− − − −
− −
+
+ +=
+
+ +
+ ( ) ( )1 1
,i i im n my d y y− −
By taking limit as → ∞ ,
lim
l→&
< +l, l = I∈ ,0,0,
1
2
∈ +∈ ,
1
4
m
∈ +0 + 0
1 + 0
n ,
3
2
m
0 + 0+∈
1 + 0
nQ
( ) ( )1
, and , im n m nd y y d y y −
≥ ∈ <∈
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196
Thus we have ,lim*→& < +, =∈
Therefore from (3) ∈ ≤ - 5
This is a contraction because 0 <∈ - ! .
Thus ) ℎ) ,
, ℎ o ℎ ℎ lim →& ) = o.
Thus lim
→&
6 = lim
→&
7 OC = o lim
→&
3 OC = lim
→&
5 OD = o
i. e. lim
→&
6 = lim
→&
7 OC = lim
→&
3 OC = lim
→&
5 OD = o
since 6 ⊆ 7 , ℎ ∈ ℎ ℎ o = 5
Then by (ii), we have
6 , o ≤ 6 , 3 OC + 3 OC, o
≤ -8< , OC ; + 3 OC, o
Where, < , OC =
?
@
A
@
B
5 , 7 OC , 5 , 6 , 7 OC, 3 OC ,
C
D
/ 5 , 3 OC + 7 OC, 6 W,
C
H
X
J Kp,M9YZ[ OJ Kp,Pp OJ M9YZ[,Pp
COJ Kp,M9YZ[ J Kp,Pp J M9YZ[,Pp
,
R
D
X
J M9YZ[,S9YZ[ OJ M9YZ[,Pp OJ M9YZ[,Kp
COJ M9YZ[,S9YZ[ J M9YZ[,Pp J M9YZ[,Kp
T
@
U
@
V
=
?
@
@
A
@
@
B
o, 6 , o, 6 , 6 , 3 OC ,
1
2
/ o, 3 OC + 6 , 6 W,
1
4
a
o, 6 + o, 6 + 6 , 6
1 + o, 6 o, 6 6 , 6
b ,
3
2
a
6 , 3 OC + 6 , 6 + 6 , o
1 + 6 , 3 OC 6 , 6 6 , o
b
T
@
@
U
@
@
V
Taking the limit as → ∞ )
< , OC =
?
@
@
A
@
@
B o, o , o, 6 , o, o ,
1
2
/ o, o + o, 6 W,
,
1
4
a
o, o + o, 6 + o, 6
1 + o, o o, 6 o, 6
b ,
3
2
a
o, o + o, 6 + o, o
1 + o, o o, 6 o, o
b
T
@
@
U
@
@
V
= 6 , o
Thus as → ∞, 6 , o ≤ -8 6 , o ; + o, o = -8 6 , o ;
q 6 ≠ o ℎ 6 , o > 0 ℎ - ! -8 6 , o ; < 6 , o .
hℎ 6 , o < 6 , o ℎ ℎ .
hℎ 6 = o , 5 = 6 = o.
s 5 6 .
s ℎ 6 5 t ) ,
65 = 56 , . 6o = 5o .
u 6 ⊆ 7 , ℎ ! ∈ ℎ ℎ o = 7!.
hℎ ) , ℎ !
o, 3! = 6!, 3!
≤ - 8< , ! ;
ℎ , < , ! =
?
@@
A
@@
B 5 , 7! , 5 , 6 , 7!, 3! ,
1
2
/ 5 , 3! + 7!, 6 W,
1
4
a
5 , 7! + 5 , 6 + 7!, 6
1 + 5 , 7! 5 , 6 7!, 6
b ,
3
2
a
7!, 3! + 7!, 6 + 7!, 5
1 + 7!, 3! 7!, 6 7!, 5
b
T
@@
U
@@
V
5. Mathematical Theory and Modeling www.iiste.org
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Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
197
=
?
@
A
@
B o, o , o, o , o, 3! ,
C
D
/ o, 3! + o, o W,
C
H
X
J v,v OJ v,v OJ v,v
COJ v,v J v,v J v,v
,
R
D
X
J v,Sw OJ v,v OJ v,v
COJ v,Sw J v,v J v,v
T
@
U
@
V
= o, 3!
hℎ o, 3! ≤ - o, 3!
q 3! ≠ o ℎ o, 3! > 0 ℎ - ! ,
- o, 3! < o, 3! , ℎ ℎ .
Therefore 3! = 7! = o.
So ! 7 3.
Since the pair of maps F and H are weakly compatible,
73! = 37! , . 7o = 3o.
Now we show that z is a fixed point of G.
hℎ ) , ℎ ! 6o, o = 6o, 3!
≤ - 8< o, ! ;
Where,
< o, ! =
?
@@
A
@@
B 5o, 7! , 5o, 6o , 7!, 3! ,
1
2
/ 5o, 3! + 7!, 6o W,
1
4
a
5o, 7! + 5o, 6o + 7!, 6o
1 + 5o, 7! 5o, 6o 7!, 6o
b ,
3
2
a
7!, 3! + 7!, 6o + 7!, 5o
1 + 7!, 3! 7!, 6o 7!, 5o
b
T
@@
U
@@
V
=
?
@
A
@
B 6o, o , 6o, 6o , o, o ,
C
D
/ 6o, o + o, 6o W,
C
H
X
J Pv,v OJ Pv,Pv OJ v,Pv
COJ Pv,v J Pv,Pv J v,Pv
,
R
D
X
J v,v OJ v,Pv OJ v,Pv
COJ v,v J v,Pv J v,Pv
T
@
U
@
V
= 6o, o
hℎ 6o, o ≤ - 8 6o, o ;
q 6o ≠ o ℎ 6o, o > 0 ℎ - !
- 8 6o, o ; < 6o, o .
Therefore 6o = o
Hence 6o = 5o = o
By (ii), we have,
o, 3o = 5o, 3o
( )( ),z z≤ φ λ
Where,
< o, o =
?
@@
A
@@
B 5o, 7o , 5o, 6o , 7o, 3o ,
1
2
/ 5o, 3o + 7o, 6o W,
1
4
a
5o, 7o + 5o, 6o + 7o, 6o
1 + 5o, 7o 5o, 6o 7o, 6o
b ,
3
2
a
7o, 3o + 7o, 6o + 7o, 5o
1 + 7o, 3o 7o, 6o 7o, 5o
b
T
@@
U
@@
V
=
?
@
A
@
B o, 3o , o, o , 3o, 3o ,
C
D
/ o, 3o + 3o, o W,
C
H
X
J v,Sv OJ v,v OJ Sv,v
COJ v,Sv J v,v J Sv,v
,
R
D
X
J Sv,Sv OJ Sv,v OJ Sv,v
COJ Sv,Sv J Sv,v J Sv,v
T
@
U
@
V
= o, 3o
hℎ o, 3o ≤ - 8 o, 3o ;
If o ≠ 3o ℎ o, 3o > 0 ℎ - ! ,
- 8 o, 3o ; < o, 3o
Therefore o, 3o < o, 3o , ℎ ℎ .
Henceo = 3o
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198
Therefore 3o = 7o = o.
Therefore 6o = 5o = 3o = 7o = o , . o 5, 7, 6 3.
Uniqueness : For uniqueness, Let we assume that o , o ≠
5, 7, 6 3.
By (ii), we have,
o, = 6o, 3
( )( ),z w≤ φ λ
Where,
o, =
?
@
A
@
B 5o, 7 , 5o, 6o , 7 , 3 ,
C
D
/ 5o, 3 + 7 , 6o W,
C
H
X
J Kv,Mx OJ Kv,Pv OJ Mx,Pv
COJ Kv,Mx J Kv,Pv J Mx,Pv
,
R
D
X
J Mx,Sx OJ Mx,Pv OJ Mx,Kv
COJ Mx,Sx J Mx,Pv J Mx,Kv
T
@
U
@
V
=
?
@
A
@
B o, , o, o , , ,
C
D
/ o, + , o W,
C
H
X
J v,x OJ v,v OJ x,v
COJ v,x J v,v J x,v
,
R
D
X
J x,x OJ x,v OJ x,v
COJ x,x J x,v J x,v
T
@
U
@
V
= o,
Thus o, ≤ - 8 o, ;
Since o ≠ ℎ o, > 0 ℎ - ! ,
- 8 o, ; < o,
o, < o, ℎ ℎ .
Therefore o =
Thus z is the unique common fixed point of E, F, G & H .
Hence the theorem.
Corollary 1 : Let , ℎ ℎ 5, 6 3
, s ) ℎ
3 ⊆ 5 6 ⊆ 5
6 , 3) ≤ -8< , ) ;
Where - , !
< , ) =
?
@
A
@
B 859, 5:;, 59, 69 , 85:, 3:;,
C
D
E 859, 3:; + 85:, 69;G ,
C
H
I
J8KL,KN;OJ KL,PL OJ8KN,PL;
COJ8KL,KN;J KL,PL J8KN,PL;
Q ,
R
D
I
J8KN,SN;OJ8KN,PL;OJ8KN,KL;
COJ8KN,SN;J8KN,PL;J8KN,KL;
Q
T
@
U
@
V
hℎ 6, 5 3, 5 t )
Then E, G and H have a unique common fixed point.
Proof : By taking E = F in main theorem we get the proof.
Corollary 2 : Let (X,d) be a complete metric space, suppose that the mappings E and G are self maps of X,
satisfying the following conditions :
6 ⊆ 5
6 , 6) ≤ -8< , ) ;
Where - , !
< , ) =
?
@
@
A
@
@
B 859, 5:;, 59, 69 , 85:, 6:;,
1
2
E 859, 6:; + 85:, 69;G ,
1
4
e
859, 5:; + 59, 69 + 85:, 69;
1 + 859, 5:; 59, 69 85:, 69;
f ,
3
2
e
85:, 6:; + 85:, 69; + 85:, 59;
1 + 85:, 6:; 85:, 69; 85:, 59;
f
T
@
@
U
@
@
V
hℎ 6, 5 t )
Then E and G have a unique common fixed point.
Proof : By taking E = F and G = H in theorem 3.1 we get the proof.
7. Mathematical Theory and Modeling www.iiste.org
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199
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[7] M. Imdad, Santosh Kumar and M. S. Khan, Remarks on fixed point theorems satisfying Implicit
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[8] S. Rezapour and R. Hamlbarani, Some notes on the paper : Cone Metric spaces and Fixed point
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[9] S. Sessa, On a weak Commutativity condition of mappings in a fixed point considerations, Publ. Inst.
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