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1. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
1
Some Fixed Point and Common Fixed Point Theorems of Integral
type on 2-Banach Spaces
1
Pravin B. Prajapati, 2
Ramakant Bhardwaj, Piyush Bhatnagar3
1
S.P.B.Patel Engineering College, Linch
1
The Research Scholar of Sai nath University, Ranchi (Jharkhand)
2
Truba Institute of Engineering & Information Technology, Bhopal (M.P)
3
Department of Mathematics, Govt. M.L.B.College Bhopal
E-mail: rkbhardwaj100@gmail.com
Abstract: In the present paper we prove some fixed point and common fixed point theorems in 2-Banach spaces
for new rational expression. Which generalize the well-known results.
Keywords: Banach Space, 2-Banach Spaces, Fixed point, Common Fixed point.
2. INTRODUCTION
Fixed point theory plays basic role in application of various branches of mathematics from elementary calculus
and linear algebra to topology and analysis. Fixed point theory is not restricted to mathematics and this theory
has many application in other disciplines.
The study of non-contraction mapping concerning the existence of fixed points draws attention of various
authors in non-linear analysis. It is well known that the differential and integral equations that arise in physical
problems are generally non-linear, therefore the fixed point methods especially Banach’s contraction principle
provides a powerful tool for obtaining the solutions of these equations which were very difficult to solve by any
other methods. Recently Verma [13] described about the application of Banach’s contraction principle [4].
Ghalar [8] introduced the concept of 2-Banach spaces. Recently Badshah and Gupta [5], Yadava, Rajput and
Bhardwaj [14], Yadava, Rajput, Choudhary and Bhardwaj [15] also worked for Banach and 2-Banch spaces for
non-contraction mappings. In present paper we prove some fixed point and common fixed point theorems for
non-contraction mappings, in 2-Banach spaces motivated by above, before starting the main result
first we write some definitions .
Definition (2.A), 2-Banach Spaces: In a paper Gahler [8] define a linear 2-normed space to be
pair where is a linear space and non- negative, real valued function defined on
such that a,b,c
(i) = 0 if and only if a and b are Linearly dependent
(ii) =
(iii) = , is real
(iv)
Hence is called a 2- norm.
Definition (2.B):
A sequence in a linear 2 – normed space L ,is called a convergent sequence if there is , x , such
that = 0 for all y .

2. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
2
Definition (2.C):
A sequence in a linear 2 – normed space L,is called a Cauchy sequence if there exists y, z , such
that y and z are linearly independent and
= 0
Definition (2.D): A linear 2-normed space in which every Cauchy sequence is convergent is called 2-Banach
spaces.
Theorem (2.E) (Banach’s contraction principle) Let (X, d) be a complete metric space, c∈(0,1) and f: X→X be a
mapping such that for each x, y ∈X,
d(fx, fy) ≤ cd (x, y) Then f has a unique fixed point a ∈X, such that for each
x∈ X,
After the classical result, Kannan [11] gave a subsequently new contractive mapping to prove the fixed point
theorem. Since then a number of mathematicians have been worked on fixed point theory dealing with mappings
Satisfying various type of contractive conditions.
In 2002, A. Branciari [3] analysed the existence of fixed point for mapping f defined on a complete metric
space (X,d) satisfying a general contractive condition of integral type.
Theorem (2.F) (Branciari) Let (X,d) be a complete metric space ,c and let f : X be a mapping
such that for each x, y ∈X,
≤ c where [0,+ [0,+ is a Lebesgue integrable
mapping which is summable on each compact subset of [0,+ , non-negative ,and such that for each
, , then f has a unique fixed point a ∈X, such that for each x ∈ X,
After the paper of Branciari, a lot of research works have been carried out on generalizing contractive condition
of integral type for different contractive mappings satisfying various known properties. A fine work has been
done by Rhoades [5] extending the result of Branciari by replacing the condition [1.2] by the following
≤ .
Theorem (2.G):
Let T be a mapping of a 2 – Banach spaces into itself. If T satisfies the following conditions:
(1) , where is identity mapping
(2) +
+ +
Where x , a is real with 8 Then T has unique fixed
point.
Our main result is modified the above result in integral type mapping.

3. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
3
3. MAIN RESULTS
Theorem 3.1
Let T be a mappings of a 2- Banach space X into itself. T satisfy the following conditions :
(1) , where is identity mapping,
(2) +
+
+ +
+
For every x, y ∈X, ∈ [0,1] with x and
4. Also [0,+ [0,+ is a Lebesgue
integrable mapping which is summable on each compact subset of [0,+ , non- negative ,and
such that for each , , Then T has unique fixed point .
Proof : Suppose x is any point in 2- Banach space X.
Taking y = x, z = T(y)
= =
+
+ +
+ +
+
+ +

4. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
4
+ +
+
+ +
+ +
+ +
+ +
+
+ +
+ +
+
+
+
----------------------------------- (3.1.1)
Now for

5. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
5
= =
+ +
+ +
+
+
+ +
+ +
+ +
+ +
+
+
+
--------------------------------------------------- (3.1.2)
Now
= +
+
+

6. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
6
+
+
+
On the other hand
=
=
= 2
So
2
+
k
as ( 4)
Where k = 1
Let R = ( T+I ) , then
=

7. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
7
= =
By the definition of R we claim that is a Cauchy sequence in X , is converges to so
element in X . So = . Hence T( =
So is a fixed point of T.
Uniqueness:
If possible let is another fixed point of T . Then
=
+
+ +
+
+
+ +
+
Which is contradiction as 4
so . Hence fixed point in unique.
Theorem 3.2
Let T and G be two expansion mappings of a 2- Banach space X into itself. T and G satisfy the following
conditions:
(1) T and G commute
(2) and , where is identity mapping,

8. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
8
(3) +
+ +
+ +
For every x, y ∈X, ∈ [0,1] with x and and
1. Also [0,+ [0,+ is a Lebesgue integrable mapping which is
summable on each compact subset of [0,+ , non- negative ,and such that for each ,
Then there exists a unique common fixed point of T and G such that T( and G(
.
Proof:-
Suppose x is point in 2- Banach space X ,it is clear that
+
+
+
+
+
+ +
+

9. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
9
+ +
Taking G(x) = p , G(y) = q , where p q
+ +
+
+ +
Taking TG = R we get
+
+ +
+ +
It is clear by theorem (1.1) ; that TG = R has at least one fixed point say in K that is R( =
TG( =
And so T.(TG) = T( or T(
G ( = T (
Now
=
=

10. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
10
+
+ +
+
+
= ( + )
So = ( 1)
That is is the fixed point of T.
But =G ( ) so G ( ) =
Hence is the fixed point of T and G.
Uniqueness:
If possible let is another common fixed point of T and G.
=
=
+
+
+ +
+

11. Innovative Systems Design and Engineering www.iiste.org
ISSN 2222-1727 (Paper) ISSN 2222-2871 (Online)
Vol.5, No.7, 2014
11
( + )
But 1
So , so common fixed point in unique.
REFERENCES:
[1] A.S.Saluja , Alkesh Kumar Dhakde “ Some Fixed Point Theorems in 2-
Banach Spaces “American Journal of Engineering Research Vol.2 (2013)
pp-122-127.
[2] Ahmad, and Shakily, M. “Some fixed point theorems in Banach spaces”
Nonlinear Funct.Anal. & Appl. 11 (2006) 343-349.
[3] A. Branciari, A fixed point theorem for mappings satisfying a general
Contractive condition of integral type, Int.J.Math.Math.Sci, 29(2002), no.9,
531 - 536.
[4] Banach, S. “Surles operation dans les ensembles abstraits et leur
Application aux equations integrals” Fund. Math.3 (1922) 133-181.
[5] Badshah, V.H. and Gupta, O.P. “Fixed point theorems in Banach and
2-Banach spaces” Jnanabha 35(2005) 73-78.
[6] B.E Rhoades, Two fixed point theorems for mappings satisfying a general
Contractive condition of integral Type, International Journal of Mathematics
and Mathematical Sciences, 63, (2003), 4007 - 4013.
[7] Goebel, K. and Zlotkiewics, E. “Some fixed point theorems in Banach
Spaces” Colloq Math 23(1971) 103-106.
[8] Gahlar, S. “2-metrche raume and ihre topologiscche structure”
Math.Nadh.26 (1963-64) 115-148.
[9] Isekey, K. “fixed point theorem in Banach space” Math.Sem.Notes, Kobe
University 2 (1974) 111-115.
[10] Jong, S.J.Viscosity approximation methods for a family of finite non
Expansive in Banach spaces” nonlinear Analysis 64 (2006) 2536-2552.
[11] R. Kannan, Some results on fixed points, Bull.Calcutta Math. Soc.
60(1968), 71-76.
[12] Sharma, P.L. and Rajput S.S. “Fixed point theorem in Banach space”
Vikram Mathematical Journal 4 (1983) 35-38.
[13] Verma, B.P. “Application of Banach fixed point theorem to solve non
linear equations and its generalization” Jnanabha 36 (2006) 21-23.
[14] Yadava, R.N., Rajput, S.S. and Bhardwaj, R.K. “Some fixed point and
Common fixed point theorems in Banach spaces” Acta Ciencia Indica 33
No 2 (2007) 453-460.
[15] Yadava, R.N., Rajput, S.S, Choudhary , S. and Bhardwaj , R.K. “Some
fixed point and common fixed point theorems for non-contraction mapping
on 2-Banach spaces” Acta Ciencia Indica 33 No 3 (2007) 737-744.

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