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A Common Unique Random Fixed Point Theorem in Hilbert Space using Integral Type Mappings RAMAKANT BHARDWAJ Mr. PIYUSH M. PATEL* Associate Professor in Truba Institute of Engineering & Information Technology, Bhopal (M.P) Research Scholar of CMJ University, Shillong (Meghalaya), pmpatel551986@gmail.com drrkbhardwaj100@gmail.com *Corresponding author ABSTRACT The object of this paper is to obtain a common random fixed point theorem for two continuous random operators defined on a non empty closed subset of a separable Hilbert space for integral type mapping. Key wards: common fixed point, rational expression, hilbert space random variable 1. INTRODUCTION AND PRELIMINARIES Impact of fixed point theory in different branches of mathematics and its applications is immense. The first result on fixed points for Contractive type mapping was the much celebrated Banach’s contraction principle by S. Banach [9] in 1922. In the general setting of complete metric space, this theorem runs as the follows, Theorem 1.1 (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, for each x ∈ X,

d (fx,fy) ≤ cd (x,y)

Then f has a unique fixed point a ∈X, such that

limn→∞ f x = a . n

After the classical result, Kannan [7] 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 [1] analyzed 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 1.2 (Branciari) Let

(X,d) be a complete metric space, c ∈ (0,1) and let f: X→X be a mapping such

that for each x, y ∈ X,

d ( x, y ) d (fx,fy) ξ (t ) dt ≤ ∫0 ξ (t ) dt ∫0

Where

ξ :[0, +∞) → [0, +∞)

is

a

Lebesgue

is summable on each compact subset of [0, +∞) , non

integrable mapping which

∈ negative, and such that for each ϵ > o, ∫ ξ (t ) dt , then f has a unique fixed point a ∈ X such that for each x∈X, 0 n limn→∞ f x = a . After the paper of Branciari, a lot of a research works have been carried out on generalizing

contractive conditions of integral type for different contractive mappings satisfying various known properties. A fine work has been done by Rhoades [2] extending the result of Branciari by replacing the condition [1.2] by the following

max  d ( x, y ),d ( x, fx),d ( y, fy ), d ( x, fy )+ d ( y , fx)  2   

d ( fx, fy )

ξ (t ) dt ≤ ∫

ξ (t )dt

0 0 The aim of this paper is to generalize some mixed type of contractive conditions to the mapping and then a pair of mappings satisfying general contractive mappings such as Kannan type [7], Chatrterjee type [8], Zamfirescu type [11], etc. In 2010, the study of random fixed points has attracted much attention. Some of the recent literatures in random fixed point may be noted in Rhoades [3], and Binayak S. Choudhary [4].In this paper, we construct a sequence of measurable functions and consider its convergence to the common unique random fixed point of four continuous random operators defined on a non-empty closed subset of a separable Hilbert space. For the purpose of obtaining the random fixed point of the four continuous random operators. We have used a rational inequality (from B. Fisher [5] and S.S. Pagey [10]) and the parallelogram law. Throughout this paper, (Ω, Σ) denotes a measurable space consisting of a set Ω and sigma algebra Σ of subsets of Ω, H stands for a separable Hilbert space and C is a nonempty closed subset of H.

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Definition 1.3. A function f: Ω → C is said to be measurable if

f −1 ( B ∩ C) ∈∑ for every Borel subset B of H. Definition 1.4. A function F : Ω × C → C is said to be a random operator if F (., x) : Ω → C is measurable for every x ∈ C Definition 1.5. A measurable g : Ω → C is said to be a random fixed point of the random operator F : Ω × C → C if F (t , g (t )) = g (t ) for all t ∈ Ω Definition 1.6. A random operator F: Ω×C → C is said to be continuous if for fixed t ∈ Ω , F (t ,.) : C → C is continuous. Condition (A). Two mappings, T, S: C → C, where C is a non-empty closed subset of a Hilbert space H, is said to satisfy condition (A) if 2 2 1 + x − Sx    2 1+ x − y

y −Ty

S x − Ty

≤ a∫

0

w ith a , b ≥ 0 , a + b <

ξ (t ) d t + b ∫

x − Sx

2

+ y −Ty

0

2

ξ (t ) d t

1 2

where x , y ∈ C ξ : R + → R + is a lebesgue- integrable mapping which is ∈ subset of R + , non negative, and such that for each

∈ >o,

ξ (t )d t

0

summable on each compact

.

2. MAIN RESULTS Theorem 2.1. Let C be a non-empty closed subset of a separable Hilbert space H . Let S and T be two continuous random operators defined on C such that for, t ∈ Ω T (t ,.), S (t ,.) : C → C satisfy condition (A)

S and T have unique common random fixed point. Proof: We define a sequence of functions { g n } as g 0 : Ω → C is arbitrary measurable function for t ∈ Ω , and

then

n = 0, 1, 2, 3, . . .

g 2 n +1 (t ) = S (t , g 2 n (t )) , g 2 n + 2 (t ) = T (t , g 2 n +1 (t ))

If g 2 n (t ) = g 2 n +1 (t ) = g 2 n + 2 (t ) for

------------------------ (1)

t ∈ Ω for some n then we see that { g 2n } is a random fixed point of S

and T. Therefore, we suppose that no two

consecutive terms of sequence { g n } are equal.

Common random fixed point theorem: Now Consider for

155

t ∈Ω

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g 2 n +1 ( t ) − g 2 n + 2 ( t )

2

ξ (λ ) d λ ≤

0

S ( t , g 2 n ( t )) − T ( t , g 2 n +1 ( t )

0

+b

=a

0

2  g (t )− S (t , g (t ) 2 + g 2n 2 n +1 ( t ) − T ( t , g 2 n +1 ( t )   2 n 

0 2 2 g 2 n +1 ( t ) − g 2 n + 2 ( t ) 1+ g 2 n ( t ) − g 2 n +1 ( t )    2 1+ g 2 n ( t ) − g 2 n +1 ( t )

+b

0

2 2  g (t )− g 2 n +1 ( t ) + g 2 n +1 ( t ) − g 2 n + 2 ( t )   2 n 

0

= (a + b)

[

⇒ 1 − (a + b)

] ∫0

ξ (λ )d λ ≤ b

b

ξ (λ ) d λ ≤

1 − (a + b)

ξ (λ ) d λ ≤ k

0

1 2

   1 − (a + b) 

where k = 

b

p

0

g 2 n ( t ) − g 2 n +1 ( t )

2

g 2 n+1 ( t ) − g 2 n+2 ( t )

g 2 n ( t ) − g 2 n +1 ( t )

0

g 2 n ( t ) − g 2 n+1 ( t )

ξ (λ ) d λ

ξ (λ ) d λ

2

In Integral

∫ ∫

g n ( t ) − g n +1 ( t ) 0 g n ( t ) − g n +1 ( t ) 0

ξ (λ )d λ ≤ k

ξ (λ )d λ ≤ k

n

g n −1 ( t ) − g n ( t ) 0

ξ (λ )d λ

g n −1 ( t ) − g n ( t ) 0

ξ (λ )d λ

fo r t ∈ Ω − − − − − − − − − − − − − − ( 2 ) Now, we shall prove that for t ∈ Ω ,

gn (t ) is a Cauchy sequence.

for Common random fixed point theorem every positive integer p, We have,

156

ξ (λ )d λ

2

1

ξ (λ )d λ

2

2

0

ξ (λ ) d λ

ξ (λ )d λ

0

2

0

ξ (λ ) d λ

ξ (λ ) d λ

2 2  g ( t ) − g 2 n + 2 ( t ) + b g 2 n ( t ) − g 2 n +1 ( t )   2 n +1 

2

g 2 n +1 ( t ) − g 2 n + 2 ( t )

g 2 n +1 ( t ) − g 2 n + 2 ( t )

ξ (λ ) d λ

2 2 g 2 n +1 ( t ) − T ( t , g 2 n +1 ( t ) 1+ g 2 n ( t ) − S ( t , g 2 n ( t )    2 1+ g 2 n ( t ) − g 2 n +1 ( t )

≤a

2

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g n ( t ) − g n +1 ( t ) + g n +1 ( t ) − g n + 2 ( t ) + g n + 2 ( t ) + ... + g n + p −1 ( t ) − g n + p ( t )

0

g n ( t ) − g n +1 ( t ) + g n +1 ( t ) − g n + 2 ( t ) + ... + g n + p −1 ( t ) − g n + p ( t )

 k n + k n +1 + ... + k n + p −1  ∫   0

0

= k

k

1+ k + k 2 + ... + k p −1  ∫   0

n

n

(1 − k ) ∫

g0 ( t ) − g1 ( t )

0

Lim n→∞ ∫0

g 0 ( t ) − g1 ( t )

g 0 ( t ) − g1 ( t )

ξ (λ )d λ

ξ (λ )d λ

ξ (λ )d λ

ξ (λ )d λ

ξ (λ ) d λ

g n ( t ) − gn+ p ( t )

=0

ξ (λ ) d λ

for t ∈ Ω

It follows that for, { g n (t )} is a Cauchy sequence and hence is convergent in closed subset C of Hilbert space H. {g n ( t )}

lim n→∞

ξ (λ )d λ = g (t )

For t ∈ Ω ------------------------------------- (3)

0

Existence of random fixed point: Consider for

∫ =

T ( t , g ( t )) − g ( t )) 0

≤ 2

2

ξ (λ )d λ

T ( t , g ( t )) − g 2 n +1 ( t ) + g 2 n +1 ( t ) − g ( t ))

2

0

T ( t , g ( t )) − g 2 n +1 ( t ) 0

t ∈Ω

2

ξ (λ )d λ + 2

ξ (λ )d λ

g 2 n +1 ( t ) − g ( t ))

2

ξ (λ )d λ

0

[By parallelogram law x + y

2

≤ x

2

+ y

2

]

{ g2n+1 ( t )} and { g ( t )} are subsequences of { g ( t )} as n → ∞ , { g ( t )} → { g ( t )} and { g ( t )} → { g ( t )} As

2n+2

n

2 n +1

2 n +1

157

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Therefore,

g ( t )− T ( t , g (t ) )

ξ (λ )d λ ≤ 2

0

∫ ∫

+2b

 g ( t )− g ( t ) 

2

) ∫0

[

+ g (t )− T ( t , g ( t ) )

g ( t )− T ( t , g ( t ) )

⇒ 1 − 2(a + b)

] ∫0

g ( t ) − T ( t , g ( t ))

0

ξ (λ )d λ

2

 

ξ (λ )d λ

ξ (λ )d λ

2

ξ (λ )d λ

g ( t ) − T ( t , g ( t ))

2

ξ (λ )d λ ≤ 0

2

ξ ( λ ) d λ = 0 ( as 2 ( a + b ) p 1)

0

T ( t , g ( t ))

2

2

0

(

0

0

≤ 2 a +b

1 + g ( t ) − g ( t ) 2    1 + g ( t ) − g ( t ) 2   

g ( t )− T ( t , g ( t ) )

+2 a

g (t )− g ( t )

2

ξ (λ )d λ =

g (t ) 0

ξ (λ )d λ ∀t ∈ Ω

− − − − − − − − − − − − − − − − − − − − − (4) Common unique random fixed point theorem In an exactly similar way we can prove that for all t ∈ Ω

S ( t , g ( t ))

0

ξ (λ ) d λ =

g (t )

0

ξ (λ ) d λ

--------------------- (5)

Again, if A: Ω × C → C is a continuous random operator on a non-empty subset C of a separable Hilbert space H, then for any measurable function f : Ω → C, the function h(t) = A(t, f(t)) is also measurable [3]. It follows from the construction of { g n } (by (1)) and the above

consideration that { g n } is a sequence of

measurable functions. From (3), it follows that g is also a measurable function. This fact along with (4) and (5) shows that g: Ω → C is a common random fixed point of S and T. UniquenessLet h: Ω → C be another random fixed point common to S and T, that is, for t ∈ Ω

T ( t , h ( t ))

0

,∫

ξ (λ ) d λ =

S ( t , h ( t ))

0

h (t )

0

ξ (λ ) d λ =

ξ (λ ) d λ

h (t )

0

,∫

S ( t , h ( t ))

0

ξ (λ ) d λ =

h (t )

0

ξ (λ ) d λ

Then for t ∈ Ω ,

158

ξ (λ ) d λ

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g (t )− h (t )) 0

2

ξ (λ )d λ = 2

+b

+b

+ h (t )− T ( t , h ( t ) )

2

0 1 + g (t )− , g ( t )  1 + g (t )− h (t ) 2   

2

)

2

0  g (t )− , g ( t ) 

+ h (t )− h ( t )

2

2

 

0

g

g (t )− , h ( t ) 0 g (t 0

)

ξ (λ )d λ  

ξ (λ )d λ

 

ξ (λ )d λ

ξ (λ )d λ

2

ξ (λ )d λ ≤ 0

ξ (λ )d λ

(t ) =

2

ξ (λ )d λ

 

2

0

h (t )− , h (t

= a

2

 g (t )− S ( t , g ( t ) ) 

2

0

 1 + g ( t )− S ( t , g ( t ) )  1 + g (t )− h (t ) 2   

h (t )− T ( t , h (t ))

= a

S ( t , g ( t )) − T ( t , h ( t ))

=

h (t ) 0

ξ (λ )d λ ∀ t ∈ Ω

h (t ) ∀ t ∈ Ω

This completes proof of theorem (2.1). ACKNOWLEDGEMENTS: The authors are thankful to Prof. B.E.Rhoades [Indiana University, Bloomington, USA] for providing us necessary Literature of fixed point theory. References: [1] A. Branciari, A fixed point theorem for mappings satisfying a general con-tractive condition of integral type, Int.J.Math.Math.Sci, 29(2002), no.9, 531 - 536. [2] 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. [3] B.E. Rhoades, Iteration to obtain random solutions and fixed points of operators in uniformly convex Banach spaces, Soochow Journal of mathematics, 27(4) (2001), 401 – 404 [4] Binayak S. Choudhary, A common unique fixed point theorem for two random operators in Hilbert space, IJMMS 32(3)(2002), 177 - 182. [5] B. Fisher, Common fixed point and constant mapping satisfying a rational inequality, Math. Sem. Kobe Univ, 6(1978), 29 - 35. [6] C.J. Himmelberg, Measurable relations, Fund Math, 87 (1975), 53 - 72. [7] R. Kannan, Some results on fixed points, Bull.Calcutta Math. Soc. 60(1968), 71-76[8] S.K.Chatterjea, Fixed point theorems, C.R.Acad.Bulgare Sci. 25(1972), 727- 730. [9] S. Banach, Sur les oprations dans les ensembles abstraits et leur application aux quations intgrals, Fund. Math.3, (1922)133181 (French). [10] S.S. Pagey, Shalu Srivastava and Smita Nair, Common fixed point theorem for rational inequality in a quasi 2-metric space, Jour. Pure Math., 22(2005), 99 - 104. [11] T.Zamfrescu, Fixed point theorems in metric spaces, Arch.Math.(Basel) 23(1972), 292-298.

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