Journal of Applied Mathematics & Bioinformatics, vol.2, no.2, 2012, 1-10
ISSN: 1792-6602 (print), 1792-6939 (online)
Scienpress Ltd, 2012
Application on Inner Product Space with
Fixed Point Theorem in Probabilistic
Rajesh Shrivasatava1 , B. Krishna Reddy2 , K. Shashidhra Reddy3 ,
M. Vijaya Kumar4 and P. Devidas5
Abstract
In this present paper we obtain a fixed point theorem in complete
probabilistic - Inner product space. To study the existence and uniqueness of solution for linear valterra integral equation.
Mathematics Subject Classification : 46S50
Keywords: Fixed point, valterra Integral Equation Probabilistic Inner Product Space
1
Introduction
This paper is to obtain a new fixed point theorem in probabilistic ∆-inner
product space, where ∆ is a t-norm of h-type, to study the existence and
1
2
3
4
5
e-mail:
e-mail:
e-mail:
e-mail:
e-mail:
rajeshraja0101@redeffmail.com
bkrbkr 07@gmail.com
shashidharkotta9@yahoo.co.in
greengirl ias2001@yahoo.com
devidasjalpath@gmail.com
Article Info: Received : January 12, 2012. Revised : March 23, 2012
Published online : September 15, 2012
2
Application on Inner Product Space with Fixed Point Theorem
uniqueness of solution for linear Valterra integral equation in complete ∆PIP-space. Throughout this paper, let R = (−∞, +∞), R+ = (0, +∞), D
denotes the set of all distribution functions.
2
Preliminaries
Definition 2.1. A mapping F : R → R+ is called a distribution function if
it is nondecreasing and left-continuous with inf F (t) = 0, sup F (t) = 1.
t∈R
t∈R
Definition 2.2. A probabilistic ∆-inner product space (briefly, a ∆-PIPspace) is a triplet (X, F, ∆), where X is a real linear space, ∆ is a t-norm and F
is a mapping of X × X → D(Fx,y will denote the distribution function F (x, y)
and Fx,y (t) will represent the value of Fx,y at t ∈ R) satisfying the following
conditions:
(P I − 1 − ∆)fx,x (0) = 0
(P I − 2 − ∆)fx,y = Fy,x
(P I − 3 − ∆)fx,x = H(t)∀t > 0 ⇔ x = 0
where

0 t ≤ 0
H(t) =
1 t > 0
(P I − 4 − ∆)
Fax,y


t


fx,y ( a ),
= H(t),



1 − Fx,y ( t +),
a
a>0
a=0
a<0
(P I − 5 − ∆)
Fx+y,z (t) = sup∆(Fx,z (s), Fy,z (s), Fy,z (r), t ∈ R, S + r = t, S ∈ R, r ∈ R,
Definition 2.3. A t-norm ∆ is h-type if the family of functions {∆m (t)}∞
m=1
is equi-continuous at t = 1, where
∆1 (t) = ∆(t, t), ∆m (t) = ∆(t, ∆m−1 (t)), t ∈ [0, 1], m = 2, 3, . . .
3
R. Shrivasatava et al
Definition 2.4. Let (X, F, ∆) be a ∆-PIP-space.
1. A sequence xn ⊂ X is said to converge to x ∈ X if ∀ > 0, ∀α ∈
(0, 1], ∃N, whenn ≥ N, Fxn −x,xn −x (ε) > 1 − α.
2. A sequence xn ⊂ X is called a Cauchy sequence if ∀ > 0, ∀α ∈ (0, 1], ∃N ,
when m, n ≥ N, Fxn −xm ,xn −xm (ε) > 1 − α.
3
Main Results
Theorem 3.1. Let (X, F, ∆) be a complete ∆ PIP space and ∆ be a t-norm
of h-type. Let T : (X, F, ∆) → (X, F, ∆) be a linear mapping satisfying the
following condition.
!
!
t
t
Fxy
+ Fy.z
k (α, β)
k (β.γ)
!
FTx,y,z (t) ≥
(1)
t
Fx,y
k (α.β)
For all x, y, z ∈ X, t ≥ 0, α, γ ∈ (0, +∞) and k(α.γ) : (0, +∞) × (0, +∞) →
(0.1) is a function. Then T has exactly one fixed point x∗ ∈ X. Further more
for any x0 ∈ X The iterative sequence {T n x0 } T converges to X∗ .
Proof. Firstly we prove that any x0 ∈ X. The sequence {xm }∞
m=0 is a τ cauchy sequence where
m
{xm }∞
m=0 = {x0 , x1 = T x0 ................xm = T x0 ...........}
Let us consider
Fxy
FTx,y,z (t) ≥
t
k (α.β)
Fxy
!
t
k (β.γ)
+ Fy.z
t
k (α.β)
!
!
4
Application on Inner Product Space with Fixed Point Theorem
Fx,z
t
k (α.γ)
!
Fxy
t
k (α.β)
!
Fx,z
t
k (α.γ)
!
Fxy
t
k (α.β)
!
Fxyz (t) ≥
or
Fx,z (t) ≥
By (∆ - PI – 5) we have
Fx0 −T m x0 .z
t
k (α.γ)
!
= Fx0 −T x0 +T m x0 ,z
≥ ∆(Fx0 −T x0 ,z
t (1 − k (α, γ))
k (α, γ)
!
≥ ∆(Fx0 −T x0 ,z
t (1 − k (α, γ))
k (α, γ)
!
t
k (α.γ)
, FT x0 −T m x0 ,z
, FT x0 −T m−1 x0 ,z
!
tk (α, γ)
k (α, γ)
!
t
k (α, γ)
)
!
)
!
t
)
, FT x0 −T x0 +T x0 −T m−1 x0 ,z
= ∆(Fx0 −T x0 ,z
k (α, γ)
!
!
t (1 − k (α, γ))
t (1 − k (α, γ))
,
, ∆ (Fx0 −T x0 ,z
≥ ∆(Fx0 −T x0 ,z
k (α, γ)
k (α, γ)
!
tkα, β
FT x0 −T m−1 x0 ,z
)
k (α, γ)
!
!
t (1 − k (α, β))
t (1 − k (α, γ))
≥ ∆(Fx0 −T x0 ,z
, ∆ (Fx0 −T x0 ,y
,
k (α, γ)
k (α, γ)
!
t
FT x0 −T m−2 x0 ,z
)
k (α, γ)
!
!
t (1 − k (α, γ))
t (1 − k (α, γ))
≥ ....... ≥ ∆ Fx0 −T x0 ,z
, ∆ (Fx0 −T x0 ,y
,
k (α, γ)
k (α, γ)
!
!
!
t (1 − k (α, γ))
t
∆ (......., ∆ (Fx0 −T x0 ,z
, (Fx0 −T x0 ,y
.......
k (α, γ)
k (α, γ)
t (1 − k (α, γ))
k (α, γ)
!
5
R. Shrivasatava et al
Because of k (α, γ) ∈ (0, 1), therefore we get
t (1 − k (α, γ))
t
≤
k (α, γ)
k (α, γ)
By the property of a distribution function, we have
!
!
t (1 − k (α, γ))
t
≥ Fx0 −T x0 ,z
Fx0 −T x0 ,z
k (α, γ)
k (α, γ)
By the property of t-norm, we obtain
!
t
Fx0 −T m x0 ,z
k (α, γ)
≥
Fx0 −T x0 ,z
t (1 − k (α, γ))
k (α, γ)
∆ (......, ∆ (Fx0 −T x0 ,z
= ∆m−1
Fx0 −T x0 ,z
!
, ∆ Fx0 −T x0 ,z
t (1 − k (α, γ))
k (α, γ)
t (1 − k (α, γ))
k (α, γ)
!
t (1 − k (α, γ))
k (α, γ)
, Fx0 −T x0 ,z
!!
t (1 − k (α, γ))
k (α, γ)
!!
So for any positive integer m, n we have
FT n x0 −T m+n x0 ,T n x0 −T m+n x0
t
k (α, γ)
!
t
≥ Fx0 −T m x0 ,T n x0 −T m+n x0
k n+1 (α, γ)
!
t
≥ Fx0 −T m x0 ,x0 −T m x0
k 2n+1 (α, γ)
≥ ∆m−1
≥ ∆m−1
≥ ∆2m−2
Fx0 −T m x0 ,x0 −T m x0
!
t (1 − k (α, γ))
k 2n+1 (α, γ)
!!
t (1 − k (α, γ))2
∆m−1 Fx0 −T x0 ,x0 −T x0
k 2n+1 (α, γ)
!!
t (1 − k (α, γ))2
.
Fx0 −T x0 ,x0 −T x0
k 2n+1 (α, γ)
!!!
Note that ∆ is a t – norm of h-type, the family of functions ∆m (t)∞
m=1 is equicontinuous at t = 1, and the distribution function F is nondecreaseing with
!!
.....
!
6
Application on Inner Product Space with Fixed Point Theorem
supt∈R F(t) = 1, then we have
lim FT n x0 −T m+n x0 ,T n x0 −T m+n x0
1→∞
≥ lim ∆2m−2
n→∞
t
k (α, γ)
!
t (1 − k (α, γ))
k 2n+1 (α, γ)
Fx0 −T x0 ,x0 −T x0
2
!!
By (∆ - P1-3), we have lim (T n x0 − T n+m x0 ) = 0. so {T m x0 }∞
m=0 is a Cauchy
n→∞
sequence in X. By the completeness of X, let xm → x∗ ∈ X(m → ∞).
Secondly, we prove that X∗ is a fixed point of T. Because of
!
t
Fxi −T xi ,x∗ −T x∗ (t) ≥ Fxi−1−T xi−1 ,x∗ −T x∗
k (α, γ)
≥ ..... ≥ Fx0 −T x0 ,x∗ −T x∗
t
i
k (α, γ)
!
Then we have
lim Fxi −T xi ,x∗ −T x∗ (t) ≥ lim Fx0 −T x0 ,x∗ −T x∗
i→∞
Fx∗ −T x∗ ,x∗ −T xi
i→∞
t
k (α, γ)
!
≥ ∆ Fx∗ −xi ,x∗ −T x∗
t
k i (α, γ)
!
t (1 − k (α, γ))
k (α, γ)
=1
!
, Fxi −T xi ,x∗ −T x∗ (t)
Because xi → x∗ (when i → ∞), ∆(. . . ) is equi-continous at (3.1) and
Fθ,x∗ −T x∗ (t) = 1, we have
!
t
= 1, ∀t > 0.
lim Fx∗ −T xi ,x∗ −T x∗
i→∞
k (α, γ)
Hence
Fx∗ −T x∗ ,x∗ −T x∗
t
k (α, γ)
!
≥ ∆ Fx∗ −T xi ,x∗ −T x∗
t (1 − k (α, γ))
k (α, γ)
!
≥ ∆ Fx∗ −T xi ,x∗ −T x∗
t (1 − k (α, γ))
k (α, γ)
!
, FT xi −T x∗ ,x∗ −T x∗ (t)
, FT xi −x∗ ,x∗ −T x∗
!
t
k (α, γ)
!!
!
7
R. Shrivasatava et al
So Fx∗ −T x∗ ,x∗ −T x∗
t
k (α, γ)
!
→ 1 (i → ∞, ∀t > 0) . By (∆ − P I − 3), we
have x∗ = T x∗ .
If there exists a point y ∗ ∈ X such that y∗ = T y∗ , then
Fx∗ −y∗ ,x∗ −y∗ (t) = FT x∗ −T y∗ ,T x∗ −T y∗ (t) ≥ Fx∗ −y∗ ,x∗ −y∗
t
k 2 (α, γ)
!
In the same way, we obtain
Fx∗ −y∗ ,x∗ −y∗ (t) ≥ Fx∗ −y∗ ,x∗ −y∗
t
k 2 (α, γ)
!
t
k 2n (α, γ)
≥ ..... ≥ fx∗ −y∗ ,x∗ −y∗
!
So Fx∗ −y∗ ,x∗ −y∗ (t) → 1 (n → ∞, ∀t > 0). By ( ∆ - PI - 3), we have x∗ = y∗ .
Therefore x* is the unique fixed point in X.
Finally, we prove that the sequence {Tn x0 } T-converges to x∗ for any x0 ∈
X. Because of
Fx∗ −T n x0 ,x∗ −T n x0 (l) = FT x∗ −T n x0 ,x∗ −T n−1 x0 (t)
≥ Fx∗ −T n−1 x0 ,x∗ −T n−1 x0
t
k 2 (α, γ)
!
t
= FT x∗ −T n−1 x0 ,T x∗ −T n−1 x0
k 2 (α, γ)
!
t
≥ Fx∗ −T n−2 x0 ,x∗ −T n−2 x0
k 4 (α, γ)
!
t
≥ ...... ≥ Fx∗ −x0 ,x∗ −x0
k 2n (α, γ)
!
We have
lim Fx∗ −T n x0 ,x∗ −T n x0 (t) ≥ lim Fx∗ −x0 ,x∗ −x0
n→∞
n→∞
t
2n
k (α, γ)
!
=1
8
Application on Inner Product Space with Fixed Point Theorem
Similarly lim Fx∗ −T n x0 −T n x0 (t) ≥ lim Fx∗ −x0 ,x∗ −x0
n→∞
n→∞
lim Fx∗ −x0 ,x∗ −x0
n→∞
∴ lim Fx∗ −T n x0 x0 −T n x0 (t) ≥
n→∞
lim Fx∗ −x0 ,x∗ −x0
n→∞
!
t
=1
k 2n (α, γ)
!
t
k 2n (α, γ)
1
! = =1
1
t
k 2n (α, γ)
∴ So T n x0 → x0 (n → ∞) This completes the proof.
4
Application
Theorem 3.1, we utilize this theorem to study the existence and uniqueness
of solution of linear Valterra integral equation in complete ∆ - PIP – space.
Let [a, b] be a fixed real interval. We define linear operation in L2 [a, b],
(x + y)(t) = x(t) + y(t)), (αx(t) = αx(t)
Then L2 [a, b] is a linear space. We lead inner product into
Z c
2
x (t) y (t) dt.
L [a, b], (x, y) =
a
Hence (x, y) is a finite number, (., .) satisfies all conditions of inner product and
L2 [a, b] is a inner product space by (., .) Because L2 [a, b] is infinite dimension
and completeness. Define a space (L2 [a, b,], F, ∆), where
F : L3[a, b, c] × L3[a, b, c] → D, F x, y, z(t) = H(t − −(x − y − z)).
Then (L2 [a, b], F, ∆) is a ∆ - PIP – space. In fact, let {xn} be a Cauchy
sequence in (L2 [a, b], F, ∆). Then for any > 0, λ ∈ (0, 1], ∃N, when m, n
≥ N, we have Fxm −xn ,xm −xn (ε) > 1 − λ because of
Fxm −xn ,xm −xn (ε) = H (ε − (xm − xn , xm − xn ) φ(µ))
c
Z
(xm − xn ) (t) (xm − xn ) (t) dtφ(µ)
xm − xn = H ε −
a
Z b
2
[(xm − xn ) (t)] dtφ(µ) > 1 − λ,
= H ε−
a
9
R. Shrivasatava et al
We have
Z
c
[(xm − xn ) (t)]2 dt → 0,
a
And then xm − xn → 0, So xn is a Cauchy sequence in L2 [a, b]. By the completeness of L2 [a, b] we have xn → x∗ ∈ L2 [a, b]. Hence x∗ ∈ L2 [a, b], F, ∆). So
(L2 [a, b], F, ∆) is a complete ∆ - PIP – space.
Theorem 4.1. Let (L2 [a, b], F, ∆) be a complete ∆ - PIP – space. Then
the following conditions are satisfied:
i)
Rt
a
(x(s) − z(s), t)ds ≤ x(t) − z(t), ∀x(·) ∈ L2 [a, b]
ii) Let T be a linear mapping and defined as follows
Z t
k (t, s) x (s) ds
(T x) (t) = f (t) +
a
Where f ∈ L2 [a, b] is a given function k (t, s) is a continuous function
defined on a ≤ t ≤ b ≤ c, a ≤ s ≤ t, λ is a constant, we denote
max
a≤t≤b,a≤s≤t
k (t, s) = M.
Then when λM ∈ (0, 1), T has a unique fixed point in L2 [a, b]. Furthermore,
for any x0 ∈ L2 [a, b], the iterative sequence {T n x0 } T - converges to the fixed
point.
10
Application on Inner Product Space with Fixed Point Theorem
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