Approximation of A Class of Non-linear Integral Operators
Sevgi Esen Almalı
Kırıkkale Üniversitesi Fen-Edebiyat Fakültesi Matematik Bölümü, Yahşihan, Kırıkkale- =903183574242
[email protected] Recieved: 11th October 2016
Accepted: 28th March 2017 DOI: http://dx.doi.org/10.18466/cbujos.77466
Abstract
In this study,we investigate the problem of pointwise convergence at lebesgue points of f funtions for the family of non-linear integral operators
1
=
, ( , ) )
(
= ) , (
m b
a
m
m t K x t dt
f x
f
L
where is a real parameter, K,m(x,t) is non-negative kernels and f is the function in L1(a,b). We consider two cases where (a,b) is a finite interval and when is the whole real axis.
Keywords — Approximation, nonlinear integral operators, lebesque point
1 Introduction
In [1], the concept of singularity was extended to cover the case of nonlinear integral operators,
, x , )) ( , (
= )
(s K t x f t dt G
f T
G
the assumption of linearity of the operators being replaced by an assumption of a Lipschitz condition for Kλwith respect to the second variable. Later, Swiderski and Wachnicki [2] investigated the problem of convergence of above the same operators to f as
w,s
w0,s0
where s0 is an accumulation point of the locally compact abelian group G in Lp(,) and Lp( R).In [3] Karsli examined both the pointwise convergence and the rate of pointwise convergence of above operators on a generalized Lebesque point to fL1(a,b) as (x,)(x0,0). And in [4] it is studied the rate of convergence of nonlinear integral operators for functions of bounded variation at a point x, which has a discontinuity of
the first kinds as 0. In [5] they obtained estimates, convergence results and rate of approximation for functions belonging to BV-spaces for a family of nonlinear integral operators of the convolution type
R s w dt t s f t K s
f
Tw )( ) =
w( , ( )) , > 0, (
in the periodic case. In paper [6], they obtained pointwise convergence and rate of pointwise convergence results at Lebesgue points for a family of nonlinear integral operators of the form
( , )= ( , , ( )) , > 0, > 0,
0
x z z dz f z x K x f T
with
K is a family of kernel satisfying a Lipschitz condition.Karsli [7] stated some approximation theorems about pointwise convergence and its rate for a class of non-convolution type nonlinear integral
operators.
Soon, in [8], Almali and Gadjiev proved convergence of exponentially nonlinear integrals in Lebesgue points of generated function, having many applications in approximation theory [9,10].
The aim of the article is to obtain pointwise convergence results for a family of non-linear operators of the form
(1) )
, ( ) (
= ) , (
1
=
, mb
a
m m t K x t dt f
x f
L
where K,m(x,t) is a family of kernels depending on ..We study convergence of the family (1) at every Lebesque point of the function f in the spaces of L1(a,b) and L1(,) with
1
=
, ( , ) )
(
m b
a
m
m t K x t dt
f is convergence.
Now we give the following definition
Definition 1 (Class A): We take a family ( K) of functions K,m(x,t):RXR R. We will say that the function K(x,t) belongs the class A, if the following conditions are satisfied:
a) K,m(x,t) is a non-negative function defined for all t and x on (a,b) and .
b) As function of t , K,m(x,t) is non-decreasing on
a,x and non-increasing on
x,b for any fixedx and .
c)For any fixed x, m m
b
a
C dt t x K, ( , ) =
.d) m
m
C1
=
is convergence.
e) For , lim ( , )= 0.
1
=
,
m
m x y K
x
y
2. Main Result
We are going to prove the family of non-linear integral operators (1) with the positive kernel convergence to the functions f L1(a,b)
Theorem 1. Suppose that f L1(a,b) and f is bounded on (a,b). If non-negative the kernel K,m belongs to Class A, then, for the operator L(f,x) which is defined in (1)
m=1
0 m m λ 0
λ
) (x f C
= ) x (f, lim L
holds at every x0 Lebesque point of f function with Cmfm(x0)
1
= m
is convergence.Proof. For integral (1), from c), we can write
1
=
0
0) ( )
, (
m
m m f x C
x f L
1
=
0 ,
0)] ( , ) (
) ( [
=
m b
a
m m
m t f x K x t dt
f
and in view of a)
1
=
0
0) ( )
, (
m
m m f x C
x f L
) , (
) , ( ) ( ) (
0 1
=
0 , 0
x I
dt t x K x f t f
m b
a
m m
m
Now we consider I(x0,). For any fixed >0, we can write I(x0,) as follow.
dt t x K x f t f x
I m m m
m
b
x x
x x
x x
a
) , ( ) ( ) (
= ) ,
( 0 , 0
1
= 0
0
0 0
0 0
0
) 2 ( )
, , ( ) , , ( ) , , ( ) , , (
=I1 x0 m I2 x0 m I3 x0 m I4 x0 m
Firstly we shall calculate I1(x0,,m), that’s
. ) , ( ) ( ) (
= ) , , (
1
= 0
0 , 0 0
1
m x
a
m m
m t f x K x t dt
f m
x I
By the condition b), we have
0
1
=
0 0 , 0
1( , , ) ( , ) ( )
x
a m
m
m x x f t dt
K m
x I
0
0) (
x
a
m x dt
f
and
. ) ( ) ( )
,
( 0
) , 1( 1
=
0 0
,
K x x f fm x b ab a L m
m
m
(3)
In the same way, we can estimate I4(x0,,m) .From property b)
b
x m
m
m x x f t dt
K m
x I
0 1
=
0 0 , 0
4( , , ) ( , ) ( )
b
x
m x dt
f
0
0) (
. ) ( ) ( )
, (
1
=
) 0 , 1( 0
0
,
m
m b a L m
m x x f f x b a
K (4)
On the other hand, since x0 is a Lebesque point of ,
f for every > 0, there exists a > 0 such that
h x
x
h dt x f t f 0
0
0) <
( )
( (5)
and
h dt x f t f x
h x
<
) ( )
( 0
0
0
(6)
for all 0<h. Now let’s define a new function as follows,
t
x
du x f u f t F
0
0) . ( ) (
= ) (
Then from (5), for tx0 we have ).
( )
(t t x0
F
Also, since f is bounded, there exists M >0 such that
M x f t f x f t
f m( ) m( 0) ( ) ( 0)
is satisfied. Therefore, we can estimate I3(x0,,m)
as follows.
1
= 0
0
0 , 0 0
3( , , ) ( ) ( ) ( , )
m x
x
m x t dt K
x f t f M
m x I
. ) ( ) , ( 1
= 0
0
0
,
m
x
x
m x t dF t K
M
We apply integration by part, then we obtain the following result.
1
=
0 0 , 0 0 0
3( , , ) ( , ) ( , )
m
m x x
K x x F M m x
I
0
0
0 , ( , ) )
( x
x
m x t K
d t F
Since K,m is decreasing on
x ,0 b
, it is clear that K,m is increasing. Hence its differential is positive. Therefore, we can wirte
1
=
0 0
, 0
3( , , ) ( , )
m
m x x
K M
m x
I
x
x
m x t
K d x
t ) ( , )
( 0 , 0
Integration by parts again, we have the following inequality
1
= 0
0
0 , 0
3( , , ) ( , )
m x
x
m x t dt K
M m x I
. ) , (
1
=
0
,
m b
a
m x t dt K
M
(7)
Now, we can use similar method for evaluation ).
, , ( 0
2 x m
I Let
. ) ( ) (
= )
(
x
t
dy x f y f t G
Then, the statement
dt x f t f t
dG( ) = ( ) ( 0)
is satisfied. For x0 t , by using (6), it can be written as follows
t x t
G( ) 0 Hence, we get
1
= 0
0
0 , 0 0
2( , , ) ( ) ( ) ( , ) .
m x
x
m x t dt K
x f t f M
m x I
Then, we shall write
. ) ( ) , ( )
, , (
1
=
0 , 0
0 0
2
m
m x
x
t dG t x K M
m x
I
By integration of parts, we have
1
=
,
2( , , ) ( ( , )
m
m x x
K x G M m x
I
x
x
m t K x t d
t G
( , ))
( )
( ,
From (6), we obtain
1
=
0 0 , 0
2( , , ) ( , )
m
m x x K
M m x
I
0
0
0 ,
0 ) ( ( , ))
(
x
x
m
t K x t
d t x
By using integration of parts again, we find . ) , ( )
, , (
1
=
0 , 0
2
m b
a
m x t dt K
M m
x
I (8)
Combined (7) and (8), we get
) , , ( ) , ,
( 0 3 0
2 x m I x m
I
. ) , ( 2
1
=
0
,
m b
a
m x t dt K
M
(9)
From condition d), (9) tends to 0 as . Finally,from (3), (4) and (9),the terms on right hand side of these inequalitys tend to 0 as . That’s
. ) (
= ) , lim (
1
=
0
0
m
m m f x C
x f L
Thus, the proof is completed.
In this theorem, specially interval (a,b) may be expanded interval (,). In this case, we can give the following theorem.
Theorem 2 Let f L1(,) and f is bounded. If non-negative the kernel K,m belongs to Class A and
satisfies also the following properties ,
0
= ) , lim (
1
=
,
m
x
m t x dt K
(10)
and
0,
= ) , lim (
1
=
,
m x
m t x dt K
(11)
then the statement
=1
) (
= ) , lim (
m
m m f x C
x f L
is satisfied at almost every x R with )
(
1
=
x f Cm m
m
is convergence.Proof. We can write, for a fixed > 0 )
( )
, (
1
x f C x
f
L m m
m
1
=
, ( , ) )
( ) (
m
m m
m t f x K x t dt
f
1
=
, ( , ) )
( ) (
=
m
m m
m
x x
x x
x x
dt t x K x f t
f
) , , ( ) , , ( ) , , ( ) , , (
= A1 x m A2 x m A3 x m A4 x m
) , ,
2(x m
A
and A3(x,,m) integrals are calculated as the proof in Theorem1. For proof, it is sufficent to show that A1(x,
,m) and) , ,
4(x m
A
tend to zero as .Firstly, we consider A1(x,
,m). Since f is bounded and by the property b), this integration is written in the form
1
=
,
1( , , ) ( ) ( ) ( , )
m x
m x t dt K
x f t f M
m x A
1
=
, ( , ) ( )
m
x
m x x f t dt
K M
. ) , ( )
(
1
=
,
m x
m x t dt K
x f M
1
= , 1
= ) , , 1(
) , ( )
(
) , (
m x
m m L m
dt t x K x
f M
x x K M f
In addition to, we obtain the inequality
1
=
,
4( , , ) ( ) ( ) ( , )
m x
m x t dt K
x f t f M
m x A
1
= ) , , 1(
) , (
m
L M K m x x
f
. ) , ( )
(
1
,
m x
m x t dt K
x f M
According to the conditions d), (10) and (11),we find that A1(x,
,m) A4(x,
,m) 0 as . This completes the proof.3 References
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