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Fixed Point Theorems of Integral Type Contraction in
Normal Cone Metric Space
R. Krishnakumar
1, R. Livingston
2Department of Mathematics,Urumu Dhanalakshmi College, Tiruchirappalli-620017,Tamilnadu, . Department of mathematics,Urumu Dhanalakshmi College, Tiruchirappalli-620017, Tamilnadu, .
Email: [email protected]1
Abstract- In this paper we extent a some fixed point theorems of integral type contraction function in complete normal cone metric space and discussed corollaries.
Index Terms- Complete Cone Metric Space, Normal Cone Metric Space, Integral Function,.
1. INTRODUCTION
There exist a number of generalizations of metric spaces, and one of them is the cone metric spaces. The notion of cone metric space is introduced by Huang and Zhang [2] and also they discussed some properties of the convergence of sequences and proved the fixed point theorems of a contraction mappings cone metric spaces. In 2010, Farshid Khojasteh [9] et al, are initiate on integral type contraction on some fixed point theorem. Further, many researchers work in this field of fixed point theory like us [10,11,12]. In this paper we discuss existence and unique fixed point in complete normal cone metric spaces, which are the generalization of some existing theorems.
2. PERMAILERIES
DEFINITION 2.1[1]: A subset of is called a cone if and only if;
1. is closed, nonempty and
2. for all and nonnegative real numbers
3. .
Given a cone , we define a partial ordering with respect to by if and only if . We will write to indicate that but , while will stand for , where
denotes the interior of . The cone is called normal if there is a number such that
implies for all . The least positive number satisfying the above is called the normal constant. The cone is called regular if every increasing sequence which is bounded from above is convergent. That is, if is sequence such that
for some , then there is such that as . Equivalently the cone is regular if and only if every decreasing sequence which is bounded from below is convergent. It is well known that a regular cone is a normal cone. Suppose is a Banach space, is a cone in E with and is partial ordering with respect to .
Example 2.2[12]:Let be given. Consider the real vector space with
{ [ ]}
With supremum norm and the cone
in . The cone is regular and so normal.
DEFINITION 2.3[1]:Suppose that is a real Banach
space, then is a cone in with , and is partial ordering with respect to . Let be a nonempty set, a function is called a cone metric on if it satisfies the following conditions with
1. , and if and only if
2.
3.
Then is called a cone metric space (CMS)
Example 2.4[1]:Let
and such that
| | | |
Where is a constant. Then is a cone metric space.
DEFINITION 2.5[1]: Let be a CMS and
be a sequence in X. Then converges to in X whenever for every with , there is a natural number such that for all . It is denoted by or . DEFINITION 2.6[1]: Let be a CMS and
be a sequence in X. Then is a Cauchy sequence whenever for every with , there is a natural number such that
for all .
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converges to , then . That is the limit of is unique.
DEFINITION 2.8[4]: Let be a cone metric space, if every Cauchy sequence is convergent in X, then X is called a complete cone metric space.
LEMMA 2.9[5]:Let be a cone metric space, P
be a normal cone with normal constant K. Let be a sequence in X. Then is a Cauchy sequence if and only if .
DEFINITION 2.10[9]: The set
is called a partition for
if and only if the sets are pairwise disjoint and ⋃
DEFINITION 2.11[9]: For each partition of
and each increasing function , we define cone lower summation and cone upper summation as
∑ ‖ ‖
∑
‖ ‖
Respectively
DEFINITION 2.12[9]: Suppose that is a normal cone in . is called on integrable function on with respect to cone or to simplicity, Cone integrable function, if and only if for all partition of
,
, where must be unique. We show the common value by ∫ to simplicity ∫
DEFINITION 2.13[9]: The function is
called sub additive cone integrable function if and only if for all
∫
∫ ∫
EXAMPLE 2.14[9]: Let
| | and
for all . Then for all
∫
∫ ∫
Since ,then . Therefore
This shows that is an example of sub additive cone inegrable function.
3. MAIN RESULT
THEOREM 3.1: Let
(
X
,
d
)
be a complete cone metric space andP
be a normal cone with constantK
. Suppose the mapping
T
:
X
X
satisfies the following conditions:∫
{
∫
( )
∫ }
For all where . The function
is defined as for each ∫ .Then the followings are
(i) T has unique fixed point in .
(ii) is converges to a fixed point, for all .
Proof: (i) Let be arbitrary and choose a sequence such that .
∫
∫
{
∫
(
)
∫
}
∫
(
)
Take
∫
(
)
We have
∫
∫
∫
∫
.Observe that is non increasing, with positive terms. So, and .It follows that
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Thus, it is verified that
Now, for all and , we have
∫
{
∫
∫
∫
}
{
[
]
∫
} ∑
‖∫
‖
∑ ∫
‖∫
‖
∑ ‖ ∫
‖
‖∫
‖
∑ ‖ ‖
‖∫
‖
Where, and K is normal constant of P. ‖ ‖ ‖ ‖‖ ‖
Now, and ∑ is a finite, and
∑
, as .
Hence, is convergent by D’ Alembert’s ratio test, therefore is a Cauchy sequence. There is,
such that as .
∫
( )
∫
( )
∫
∫
( ( ) ( )
)
∫
{
∫
( ( ) ( )
( ) ) ( )
∫
( )
} as
Therefore, ‖∫ ( ) ‖ . Thus .
UNIQUENESS
Suppose and are two fixed points of T.
∫
( )
∫
{
(
) ∫
( )
}
Therefore, ‖∫ ( ) ‖ . Thus . Hence, is an unique fixed point of T.
Now,
∫
( )
∫
( ( ) )
∫
( )
∫
( ( ) )
∫
( )
Hence, converges to a fixed point, for all .
COROLLARY 3.2: Let be a complete cone
metric space and P be a normal cone with normal constant K. suppose the mapping satisf ies the following conditions:
∫
∫
( )
For all . Where . The function is defined as for each ∫ . Then the followings are
(i) T has unique fixed point in X. (ii) Converges to a fixed point, for
all .
Proof: The proof of the corollary immediately by taking in the above theorem.
THEOREM 3.3:Let be a complete metric
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(ii) Converges to a fixed point, for all
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∫
( )
∫
( ( ) )
∫
( )
Hence, converges to a fixed point, for all .
COROLLARY 3.4:Let be a complete metric
space and let T be a mapping from X into itself. Suppose that T satisfies the following conditions:
∫
∫
( )
For all , where . The function is defined as for each ∫ . Then the followings are
(i) T has unique fixed point in X. (ii) Converges to a fixed point, for
all .
Proof: The proof of the corollary follows immediately by taking in the above theorem.
REFERENCES
[1] Huang L. G., Zhang, Cone metric spaces and fixed point theorems of contractive mappings. J. Math. Anal. ppl., 332 (2007), 1468-1476.
[2] Kannan. R., Some results on fixed points-II. The American Mathematical Monthly, 76(4) (1969), 405-408.
[3] Kirk W. A., Contraction Mappings and Extensions, in Handbook of Metric Fixed Point Theory, 1-34, Kluwer Academic, Dordrecht, The Netherlands, (2001).
[4] Krishnakumar R. and Marudai M., Cone Convex Metric Space and Fixed Point Theorems, Int. Journal of Math. Analysis, 6(22) (2012), 1087- 1093.
[5] Krishnakumar R. and Marudai M., Generalization of a Fixed Point Theorem in Cone Metric Spaces, Int. Journal of Math. Analysis, 5(11) (2011), 507- 512.
[6] Krishnakumar R. and Dhamodharan D., Some Fixed Point Theorems in Cone Banach Spaces Using Operator, International Journal of Mathematics And its Applications, 4(2-B) (2016), 105-112.
[7] Krishnakumar R. and Dhamodharan D., Metric Space and Fixed Point theorems, International J. of Pure &Engg. Mathematics, 2(II) (2014), 75-84. [8] Subrahmaniyam P. V., Remarks On Some Fixed Point Theorems Related to Banach’s Contraction Principle, Journal of Mathematics and Physical Sciences, 8(1974), 445-457.
[9] Farshid Khojasteh, Zahra Goodarzi and
Abdolrahman Razani, Some Fixed Point Theorems of Integral Type Contraction In Cone Metric
Spaces, Fixed point theory & Applications, 2010:189684. https://doi.org/10.1155/2010/189684. [10] Rahim Shah, Akbar Zada, Ishfaq Khan, Some Fixed Point Theorems of Integral Type
Contraction In Cone b – Metric Spaces, Turkish Journal of Analysis and Number Theory, Vol. 3, No. 6, 2015, Pp165-169.
[11] R. Krishnakumar, D. Dhamodharan, Some Fixed Point Theorems For Contraction of Rational Expression On Cone Metric Space, International Journal of Mathematics Trends and Technology, Vol. 47, No. 5, 2017, 318-322.
