Int. Journal of Math. Analysis, Vol. 4, 2010, no. 38, 1857 - 1864
Some Applications of Integro-Substitution Operators in Integral Equations Kuldip Raj, Sandeep Bhougal and Sunil K. Sharma School of Mathematics Shri Mata Vaishno Devi University Katra-182320, J&K, India
[email protected] Abstract In this paper we shall establish connection between Integro-substitution operators and integral equations.
Mathematics Subject Classification : Primary 47B20, Secondary 46B38 Keywords: Integro-Substitution Operator, Integral Operator, Lp -space
1.Introduction and Preliminaries p Let (X, s, μ) be a σ-finite measure space. For 1 ≤ p < ∞, let L (X, C) = {f |f : X → C is measurable and |f |p dμ < ∞}. Then Lp (X, C) is a Banach
space under the norm
X
||f || = (
|f |p dμ)1/p
X
Let T : X → X be a non singular measurable transformation that is if μ(E) = 0 implies that μ(T −1(E)) = 0 . If T is non singular, then the measure μT −1 is absloutely continuous with respect to the measure μ. Hence by RadonNikodym derivative theorem, there exists a positive measureable function f0 such that −1 f0 dμ for every E ∈ s. μ(T (E)) = X
The function f0 is called the Radon Nikodym derivative of the measure −1 . μT with respect to measure μ and we denote it by dμT dμ −1
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K. Raj, S. Bhougal and S. K. Sharma
Suppose (X, s, μ) is a measure space and k : X × X → C is a measurable function. For 1 ≤ p < ∞, if the operator Tk : Lp (X, C) → Lp (X, C) defined by
(Tk f )(x) =
k(x, y)f (y)dμ(y) X
is bounded, we call it an integral operator induced by k. If we compose an integral operator with a substitution operator CT , we get another operator called Integro substitution operator. We define the operators LT k : Lp (X, C) → Lq (X, C) and RkT : Lp (X, C) → (Lq X, C) respectively by k(T (x), y)f (y)dμ(y) (LT k f )(x) = X
and (RkT f )(x) =
k(x, y)f (T (y))dμ(y). X
For f0 =
dμT −1 . dμ
We set kf0 (x, y) = f0 (x)k(x, y). The study of integral operators have been carried out by several mathematician in sevaral direction. A few of them are Halmos and Sunder (3), Whitely [4], Steponov [6], Sinnamon [2], Bloom and Kerman [5], A. Gupta and B.S. Komal [1]. In this paper we shall establish connection between Integro-substitution operators and integral equations. Some cases for which the integral equations of the type b g(x) = f (x) + λ k(x, y)g(T (y))dμ(y) a
have unique solutions are discussed. The operator RkT : L2 ([a, b], C) → L2 ([a, b], C) where the kernel function k(x, y) is the characteristic function of the set {(x, y) : 0 ≤ |y| ≤ |x| ≤ 1}
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Applications of integro-substitution operators
is infact a quasinilpotent operator. In this special case RkT will be denoted by RT .
2. Applications of Integro-Substitution Operators in Integral Equations Theorem 2.1: Suppose kT ∈ L2 ([a, b] × [a, b], C), where kT (x, y) = E(k(x, .))oT −1 (y)f0 (y). Then the equation
b
k(x, y)g(T (y))dμ(y)
g(x) = f (x) + λ
...(1)
a
has unique solution in L2 ([a, b], C) for sufficiently small value of λ. Proof: For every g ∈ L2 ([a, b], C), consider the operator (RkT g)(x) = f (x) + λ
b
k(x, y)g(T (y))dμ(y). a
We first show that this operator takes every function g in L2 ([a, b], C) to a function in L2 ([a, b], C). Since f ∈ L2 ([a, b], C), it sufficies to show that the operator, b
(RkT g)(x) = λ
k(x, y)g(T (y))dμ(y) takes every function g ∈ L2 ([a, b], C)
a
into a function in L2 ([a, b], C). Consider
b
(| a
b
k(x, y)g(T (y))dμ(y)|2)dμ(x) a b b = | E(kx )oT −1 (y)f0 (y)g(y)dμ(y)|2dμ(x) b a b ab −1 2 ( |f0 (y)E(kx )oT (y)| dμ(y) |g(y)|2dμ(y)dμ(x) ≤ a a a b b = ||g||2 |f0 (y)E(kx )oT −1 (y)|2dμ(y)dμ(x) a
= ||g||2.M Now
a
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K. Raj, S. Bhougal and S. K. Sharma
||RkT g − RkT h|| b |(RkT g)(x) − (RkT h)(x)|2 dμ(y))1/2 = ( a b b 2 = (|λ| | k(x, y)(g(T (y)) − h(T (y))|2dμ(y)dμ(x))1/2 a a b b ≤ |λ|( ( |k(x, y)||g(T (y) − h(T (y))|)2dμ(y)dμ(x))1/2 a a b b b 2 −1 2 = |λ| ( E|k(x, .)| oT (y)f0 (y)( |g(y) − h(y)|2dμ(y)dμ(x))1/2 a a a b b = |λ|||g − h||2 f02 (y)E(|k(x, .)|)oT −1(y)dμ(y) a a b b < ||g − h|| if |λ| |k(x, .)|oT −1(y)f02 (y)dμ(y)dμ(x) < 1. a
a
By Banach contraction principle there exists unique g0 ∈ L2 ([a, b], C) such that b k(x, y)g(T (y))dμ(y). Hence the given RkT g0 = g0 , that is g0 (x) = f (x) + λ equation (1) has a unique solution.
a
Theorem 2.2 : Let kT ∈ L2 ([0, 1] × [0, 1], C) and T −1 [0, x] ⊃ [0, x] for each x ∈ [0, 1]. Then the equation s g(s) = f (s) + λ k(s, t)g(T (t))dt 0
has unique solution in L2 ([0, 1], C) for arbitrary 1 λ ∈ C. s Proof : Let p(s) = |k(s, t)|2 dt, q(t) = |k(s, t)|2 dt and 0 t 1 f0 (z)p(z)dz. Define the mapping A : L2 ([0, 1], C) → L2 ([0, 1], C) by r(s) = 0
s
k(s, t)g(T (t))dt
(Ag)(s) = f (s) + λ 0
we first prove that An is a contraction for some n ∈ N. Now Ag = f + λRkT g so that An g = f + λRkT f + λ2 Rk2T f + − − − + λn RknT f.
Applications of integro-substitution operators
and
(RknT g)(s)
s
s
= 0
kTn (s, t)g(T (t))dt, where kT1 (s, t) = k.
k(s, z)k(T (z), t)dz kT2 (s, t) = 0 ........ ........ kTm (s, t) = k(s, z)kTm−1 (T (z), t)dz n To calculate ||RkT ||, consider
|kT2 (s, t)|
2
s
= | k(s, z)k(T (z), t)|2 dz s s0 2 |k(s, z)| dz |k(T (z), t)|2 dz ≤ 0 0 ≤ p(s) |k(T (z), t)|2 dz −1 T ([0,s]) s ≤ p(s) f0 (z)|k(z, t)|2 dz 0
≤ p(s)q(t)
|kT3 (s, t)|
2
s
= | k(s, z)kT2 (T (z), t)dz|2 0s s 2 ≤ |k(s, z)| dz |kT2 (T (z), t)|2 dz 0 0 s |k(s, z)|2 dz kT2 (T (z), t)|2 dz ≤ −1 0 T ([0,s]) s f0 (z)|kT2 (z, t)|2 dz ≤ p(s) 0 s f0 (z)p(z)q(t)dz ≤ p(s) t s f0 (z)p(z)dz = p(s)q(t) t
= p(s)q(t)[r(s) − r(t)]
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K. Raj, S. Bhougal and S. K. Sharma
|kT4 (s, t)|
2
s
= | k(s, z)kT3 (T (z), t)|2 dz s 0s 2 |k(s, z)| dz |(T (z), t)|2 dz ≤ 0 0 s f0 (y)|kT3 (z, t)|2 dz = p(s) t s f0 (z)p(z)q(t)(r(s) − r(t))dz ≤ p(s) t
= p(s)q(t)
(r(s) − r(t))2 2
Similarly we can prove that |kTn (s, t)|2 ≤ p(s)q(t) Now |(A g)(s) − (A h)(s)| n
n
2
[r(s) − r(t)]n−2 , for n ≥ 2 (n − 2)!
s
kTn (s, t)[g(T (t) − h(T (t))]|2 dt 0 s s 2n n 2 |kT (s, t)| dt |g(T (t)) − h(T (t))|2 dt ≤ |λ| 0 s 0 s [r(s) − r(t)]n−2 2n dt = |λ| p(s)q(t) f0 (t)|g(t) − h(t)|2 dt (n − 2)! 0 0 n−2 [r(s) − r(t)] M.M1 ||g − h||2 ≤ |λ|2n p(s) (n − 2)! = |λ
n
Integrating with respect to s, we get ||A g − A h|| n
n
2
1 − h||2 MM1 ≤ |λ| (r(s))n−2 p(s)ds (n − 2)! 0 1 2 2n ||g − h|| n−1 ≤ |λ| M M1 p(s)ds (n − 2)! 0 ||g − h||2 n ≤ |λ|2n M M1 , for n ≥ 2 (n − 2)! 2n ||g
2n0
n0
M1 < 1, so that An0 is a conTherefore, we can choose n0 such that |λ| (n0M −2)! traction. Hence A has a unique fixed point and that fixed point is a unique solution.
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Applications of integro-substitution operators
Theorem 2.3 : The integro substitution operator RT is a quasinilpotent operator i.e σ(RT ) = {0}. Proof : For f ∈ L2 ([a, b], C), consider (RT f )(x) = RT (RT f )(x) =
−x
(f oT )(x)dx and
x
x
−x 0
(RT f )(y)dy
−x
(RT f )(y)dy +
x
(RT f )(y)dy x −x (RT f )(y)dy + (RT f )(y)dy = − x0 0 x (RT f )(t)dt + (RT f )(y)dy = −
=
0
0
0
= 0 Thus RT is a quasinilpotent operator of index 2. Hence σ(RT ) = {0}.
References: 1. A. Gupta and B.S. Komal, Composite integral operator on L2 (μ), Pitman Lecture notes in Mathematics series, 377(1997), 92-99.
2. G. Sinnamon, Weighted Hardy and opial type inequalities, J. Math. Anal. Appl. 160, (1991), 434-435. 3. P. R. Halmos and V. S. Sunder, Bounded integral operators on L2 -spaces, Springer Verlag, New York, 1978.
4. R. Whitely, Normal and quasinormal composition operators, Proc. Amer. Math. Soc. 70(1978), 114-118.
5. S. Bloom and R. Kerman, Weighted norm inequalities for operators of Hardy type, Proc. Amer. Math. Soc. 113, (1991), 135-141.
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K. Raj, S. Bhougal and S. K. Sharma
6. V.D. Stepanov, On the boundedness and compactness of a class of convolution operators, Soviet math. Dokl. 41, (1990), 446-470.
Received: April, 2010
