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J. Nonlinear Sci. Appl. 9 (2016), 3090–3097
Research Article
On approximation properties of certain
multidimensional nonlinear integrals
Sevgi Esen Almalia,∗, Akif D. Gadjievb
a
Faculty of Sciences and Art, Depatment of Mathematics, Kirikkale University, Kirikkale,Turkey.
b
National Academy of Sciences of Azerbaijan, 9 F. Agaev Street, 1141, Baku, Azerbaijan.
Communicated by J. Brzdek
Abstract
We prove theorems on convergence of multidimensional nonlinear integrals in Lebesgue points of generated function, and show that the main results are applicable to a wide class of exponentially nonlinear
integral operators, which may be constructed by using well known positive kernels in approximation theory.
c
2016
All rights reserved.
Keywords: Nonlinear integral operators, positive kernels, Lebesgue points, approximation, exponentially
nonlinear integrals.
2010 MSC: 42A35, 42A63.
1. Introduction
It is well known that the solution of boundary value problems for elliptic and parabolic differential
equations can be expressed as integral operators with positive kernels. The most famous of these are the
kernels of Gauss–Weierstrass, Poisson, Abel–Poisson, etc. Sequences or families of integral operators with
positive kernels play an important role in approximation of functions and serve as the main model in the
creation of the so called theory of approximation of positive linear operators [2, 6].
Approximative properties of the sequences of linear integral operators with positive kernels generated
interest to study approximation by nonlinear integrals [3]–[5], [8]–[12]. Related to that, presumably it would
be interesting to find a connection between the approximation by nonlinear integrals and boundary value
problems for certain nonlinear differential operators.
∗
Corresponding author
Email addresses: sevgi_esen@hotmail.com (Sevgi Esen Almali), akif_gadjiev@mail.us (Akif D. Gadjiev)
Received 2015-09-30
S. E. Almali, A. D. Gadjiev, J. Nonlinear Sci. Appl. 9 (2016), 3090–3097
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On the other hand, given that the classical theorems on the convergence of sequences of operators are
only valid for linear operators, approximation by using nonlinear integrals is of interest in terms of the
convergence of sequences (or families) of these integrals and founding its limit values. The results of the
papers [1, 7], on limiting properties of multidimensional integrals, called Riesz and Bessel potentials, can
be considered as regarding this. The main goal of our work is to find appropriate expressions of limits of a
family of certain nonlinear integrals.
We will study nonlinear multidimensional integrals of the form
Z
λn
Lλ (u, x) =
K(λ |t − x| , u(t))dt,
wn−1 Rn
1
2
where x ∈ Rn , |t − x| = (t1 − x1 )2 + (t2 − x2 )2 + · · · + (tn − xn )2
and wn−1 is the area of unit sphere
S n−1 in Rn .
The kernel-function K(ξ, u), depending on two variables ξ ≥ 0 and u, |u| ≤ M satisfy some properties
which will be given in Section 2. We want to find the limit of Lλ (u, x0 ) as λ → ∞ where x0 is the Lebesgue
point of the function u(x), x ∈ Rn . In Section 3 we will show that the kernel-function K(ξ, u), satisfying
all our conditions, may be easily obtained by using classical positive kernels of approximation theory, such
as Gauss–Weierstrass, Abel–Poisson, Jacson and other kernels. The main theorem, proved in Part 2, allow
us to give a definition of a general subclass of nonlinear integrals Lλ (u, x0 ) exponentially depending of the
function u(x), x ∈ Rn .
2. Main result
First, we recall the well known definition of Lebesgue points in Rn . Let S n−1 be the unit sphere in Rn
and let for x ∈ Rn ,
Z
f (r, u(x)) =
|u(x + rθ) − u(x)| dθ, 0 < r < ∞.
S n−1
Definition 2.1. The point x ∈ Rn is called the Lebesgue point of function u ∈ L1 (Rn ) if
Z
1
lim
|u(x − t) − u(x)| dt = 0.
r→0 r n |t|≤r
From this definition, we can find an δ > 0 such that
Z
1
|u(x − t) − u(x)| dt < δ
rn |t|≤r
provided r ≤ δ.
Let us define a new function by
Z
F (ρ) =
ρ
rn−1 f (r, u(x0 ))dr.
(2.1)
0
Definition 2.1 gives that for any ε > 0 there exist a positive number δ = δ(ε) such that
F (ρ) < ερn , for all ρ ≤ δ,
(2.2)
dF (ρ) = ρn−1 f (ρ, u(x0 ))dρ.
(2.3)
(see [11]). Moreover, from (2.1) we have
S. E. Almali, A. D. Gadjiev, J. Nonlinear Sci. Appl. 9 (2016), 3090–3097
Consider now the following nonlinear multidimensional integral
Z
λn
Lλ (u, x) =
K(λ |t − x| , u(t))dt,
wn−1 Rn
3092
(2.4)
where, as we noted above, wn−1 is the area of unit sphere S n−1 in Rn and λ is a positive parameter.
Note that for partial derivatives of the function K(ξ, u) with respect to the variable u, we will write for
brevity
∂ m K(ξ, u)
Kum (ξ, 0) =
|u=0 , m = 1, 2, 3, . . . .
∂um
We assume that the kernel function K(ξ, u) satisfy the following conditions:
(a) For any ξ ≥ 0, K(ξ, u) is an infinitely differentiable function of variables u and ξ and K(ξ, 0) = 0.
(m)
(b) For any natural m, Ku (ξ, 0) is nonnegative monotonically decreasing function of ξ on semiaxis [0, ∞).
(c)
Kum (ξ, 0) ≤ Ku(m−1) (ξ, 0),
(d)
Z
m = 1, 2, 3, . . . .
∞
Kum (ξ, 0)ξ n−1 dξ = cm < ∞,
m = 1, 2, . . . .
(2.5)
0
From the conditions(a)-(d), it follows that the kernel K(ξ, u) may be expanded in Maclaurin series by
the powers of function u. Moreover, (c) gives that the first partial derivative Ku1 (ξ, 0) is a majorant function
for other derivatives, that is
Ku(m.) (ξ, 0) ≤ Ku1 (ξ, 0), m = 1, 2, . . . .
From (d) we can infer the inequality
Z p
Z
(m)
n−1
(m)
Ku (ξ, 0)ξ
dξ ≥ Ku (p, 0)
p
2
=
p
ξ n−1 dξ
p
2
Ku(m) (p, 0)pn
1 1
1− n
, m = 1, 2, . . . ,
2
n
for any positive number.
Using (d) we obtain
lim pn Ku(m) (p, 0) = 0,
p→∞
and from this it follows that for any m = 1, 2, . . . ,
lim Ku(m) (p, 0) = 0.
p→∞
The properties (d) gives also
Z
lim
p→∞ p
(2.6)
∞
Ku(m) (ξ, 0)ξ n−1 dξ = 0.
(2.7)
We will prove now the main theorem on convergence of the family Lλ (u, x) as λ → ∞ at a fixed point
x0 , being the Lebesgue point of the function u(x).
Theorem 2.2. Let u(x) be a bounded function in Rn , belonging to L1 (Rn ) and let the kernel K(ξ, u) satisfy
conditions (a)–(d). Then at each Lebesgue point x0 of the function u(x) we have
∞
X
cm m
lim Lλ (u, x0 ) =
u (x0 ),
λ→∞
m!
m=1
where cm are defined in (2.5).
S. E. Almali, A. D. Gadjiev, J. Nonlinear Sci. Appl. 9 (2016), 3090–3097
3093
Proof. First we write the Maclurin expansion of the function K(ξ, u) by the powers of u. We have [1]
∞
X
1 (m)
K (λ |t − x0 | , 0)um (t).
m! u
K(λ |t − x| , u(t)) =
m=1
By condition (c) and boundedness of the function u(t), the series in right-hand side is majorating and
therefore
Z
∞
λn X 1
K (m) (λ |t − x0 | , 0)um (t)dt.
Lλ (u, x0 ) =
wn−1
m! Rn u
m=1
Firstly, let us transform t − x0 = p and then take p = t
Z
∞
λn X 1
Lλ (u, x0 ) =
K (m) (λ |t| , 0)um (t + x0 )dt.
wn−1
m! Rn u
m=1
From binomial expansion of um (x0 + t) we can write
Z
∞
m
X
λn X 1
k
Lλ (u, x0 ) =
Ku(m) (λ |t| , 0)
Cm
(u (x0 + t) − u(x0 ))m−k uk (x0 )dt,
wn−1
m! Rn
m=1
k=0
k are binomial coefficients. From this we obtain
where Cm
Z
∞
m−1
∞
X
X
λn X 1
cm m
k
(m)
Cm
(u (x0 + t) − u(x0 ))m−k uk (x0 )dt.
u (x0 ) =
Ku (λ |t| , 0)
Lλ (u, x0 ) −
m!
wn−1
m! Rn
m=1
m=1
k=0
Since u(x) is bounded, say, |u(x)| ≤ M < ∞, we have
m−1
m−1
X
X k
m−k k
k
Cm
(2M )m−1−k M k
C
(u
(x
+
t)
−
u(x
))
u
(x
)
≤
|u
(x
+
t)
−
u(x
)|
0
0
0 0
0
m
k=0
k=0
≤ (4M )m |u (x0 + t) − u(x0 )| .
Therefore, using the property (c), we can write
Z
∞
∞
X
cm m
λn X (4M )m
u (x0 ) ≤
|u (x0 + t) − u(x0 )| Ku1 (λ |t| , 0)dt.
Lλ (u, x0 ) −
wn−1
m
m!
n
R
m=1
m=1
So, for any fixed δ, which will be chosen below, we have
Z
Z ∞
X
cm m
λn
u (x0 ) ≤
+
|u (x0 + t) − u(x0 )| Ku1 (λ |t| , 0)dt
Lλ (u, x0 ) −
wn−1
m
t<δ
t≥δ
m=1
=
1
wn−1
e4M
(2.8)
− 1 {λn Iλı + λn Iλıı } .
From (b), for λn Iλıı , we immediately obtain
Z ∞Z
Z
n ıı
n 1
n−1
n
λ Iλ ≤ λ Ku (λδ, 0)
|u(x0 + ρθ| ρ
dθdρ + |u(x0 )| λ
Ku1 (λ |t| , 0)dt
δ
S n−1
t≥δ
Z ∞
≤ λn kukL1 (Rn ) Ku1 (λδ, 0) + λn u(x0 )wn−1
Ku1 (λρ, 0) ρn−1 dρ.
δ
The first term in right-hand side tends to zero as λ → ∞ by (2.6), and the second term by (2.7). Therefore
lim λn Iλıı = 0.
λ→∞
(2.9)
S. E. Almali, A. D. Gadjiev, J. Nonlinear Sci. Appl. 9 (2016), 3090–3097
3094
Consider λn Iλı . As we noted above, assuming that x0 is a Lebesgue point of the function u, we obtain,
by Definition 2.1, that for ε > 0 there exist a number δ > 0 such that the function F (ρ), defined in (2.1)
satisfies the inequality (2.2). Fixing this δ > 0, we can write
λn Iλı
=λ
n
Z
δ
Z
S n−1
0
|u(x0 + ρθ) − u(x0 )| Ku1 (λρ, 0) ρn−1 dθdρ,
or, using (2.3) and integrating by parts
λn Iλı
=λ
n
Z
δ
Ku1 (λρ, 0) dF (ρ)
0
= λn Ku1 (λρ, 0) F (δ) + λn
δ
Z
F (ρ)d −Ku1 (λρ, 0) .
0
Using (2.2), we have
λn Iλı
δ
ρn d −Ku1 (λρ, 0)
< ε (λδ)
+ ελ
0
Z δ
Ku1 (λρ, 0) ρn−1 dρ
= nελn
0
Z δ
= nε
Ku1 (ρ, 0) ρn−1 dρ.
n
Ku1 (λρ, 0)
n
Z
0
By (2.5), the limit of the integral in the right-hand side is finite and therefore
lim λn Iλı = 0,
λ→∞
(2.10)
since ε is an arbitrary small number. Using (2.9) and (2.10) in (2.8), proves the theorem.
We give the following simple example of an application of Theorem 2.2. Lets consider the kernel-function
i
1 h
2
exp e−ξ u − 1 .
K(ξ, u) = √
2π
(2.11)
Obviously K(ξ, 0) = 0. We have
1
2
Ku(m) (ξ, 0) = √ e−mξ ,
2π
and all the conditions of Theorem 2.2 are satisfied. Therefore we use the following well known formula,
Z ∞
Γ n2
1
n−1 −mρ2
dρ =
ρ
e
.
2 m n+1
2
0
3. Exponentially nonlinear integrals
The example of kernel K(ξ, u), given in formula (2.11), shows that Theorem 2.2 covers the subclass of
nonlinear integrals which may be described by the following definition.
Definition 3.1. We will say that a nonlinear integral operator contains the exponential nonlinearity if its
kernel-function K(ξ, u) has the form
K(ξ, u) = eA(ξ)u − 1,
(3.1)
where A(ξ) is a given function, defined for ξ ≥ 0.
S. E. Almali, A. D. Gadjiev, J. Nonlinear Sci. Appl. 9 (2016), 3090–3097
3095
Definition 3.2. Nonlinear integral, having exponential nonlinearity, will be called exponentially nonlinear
integral.
As an example, consider the exponentially nonlinear integral
Z h
i
λn
Bλ (u, x) =
eA(λ|t−x|)u(t) − 1 dt.
wn−1 Rn
(3.2)
Following Theorem 2.2, we can see that the following statement holds.
Theorem 3.3. Let a nonnegative function A(ξ) satisfy the following conditions:
(a1 ) A(ξ) < 1 for all ξ ≥ 0,
(b1 ) A(ξ) is monotonically decreasing function in semiaxis [0, ∞),
R∞
(c1 ) 0 A(ξ)ξ n−1 dξ < ∞.
Then at each Lebesgue point x0 of a bounded function u ∈ L1 (Rn ), we have
lim Bλ (u, x0 ) =
λ→∞
where dm =
R∞
0
∞
X
dm m
u (x0 ),
m!
(3.3)
m=1
Am (ξ)ξ n−1 dξ.
(m)
Proof. It is clear that for function (3.1) we have K(ξ, 0) = 0, Ku (ξ, 0) = Am (ξ), m = 1, 2, . . . , and by
(a1 )
Ku(m) (ξ, 0) < Ku(m−1) (ξ, 0),
and (b1 ) implies (b). Therefore, the conditions (a)–(c) of Theorem 2.2 are satisfied. Moreover, using (c1 ),
we see that all conditions (a)–(d) of Theorem 2.2 are satisfied, which gives the proof.
Remark 3.4. It is clear that Theorem 3.3 is a corollary of Theorem 2.2. We give this statement in the form
of special Theorem since it covers many applicable cases.
Now we consider the case of exponential nonlinear integrals where the function A(ξ) does not satisfy the
condition of monotony.
Theorem 3.5. Let A(ξ) be a nonnegative function such that there exist a monotonically decreasing majorant
function D (ξ) on [0, ∞) such that
Am (ξ) ≤ D (ξ) , m = 1, 2, . . . ,
and
Z
(3.4)
∞
ξ n−1 D(ξ)dξ < ∞.
(3.5)
0
Then (3.3) holds at every Lebesgue point x0 of bounded function u ∈ L1 (Rn ).
Proof. As in the proof of Theorem 2.2 we can write the inequality (2.8) for integral (3.2) in the following
form
∞
X
dm m
λn
B
(u,
x
)
−
u
(x
)
e4M − 1 {λn Iλı + λn Iλıı } ,
λ
0
0 ≤
wn−1
m!
m=1
where
λn Iλı
Z
|u (x0 + t) − u(x0 )| D(λ |t|)dt,
=
t<δ
S. E. Almali, A. D. Gadjiev, J. Nonlinear Sci. Appl. 9 (2016), 3090–3097
and
λn Iλıı =
3096
Z
|u (x0 + t) − u(x0 )| D(λ |t|)dt.
t≥δ
Since D (ξ) is monotonically decreasing, we obtain that
lim λn Iλı = lim λn Iλıı = 0.
λ→∞
λ→∞
By the same way as above (2.9) and (2.10). The proof is complete.
Consider the following Fejer-type kernel
F (t) =
sin t
t
2n
,
and corresponding exponentially nonlinear integral (3.2)
Z h
i
λn
Fλ (u, x) =
eF (λ|t−x|)u(t) − 1 dt.
wn−1 Rn
We have
λn
Fλ (u, x) =
wn−1
∞ X
sin (λ |t − x|) 2nm
Z
Rn
λ |t − x|
m=1
um (t)dt.
Obviously for any natural number n and m
sin (λ |t − x|)
λ |t − x|
2nm
<
sin (λ |t − x|)
λ |t − x|
Therefore, we have
F (λ |t − x|) < 2n
<
2n
1 + λ2 |t − x|2
2n
1 + λ2 |t − x|2
n .
n ,
which means that the majorant function D (ξ) has the form
D (ξ) =
2n
,
(1 + ξ 2 )n
ξ ∈ [0, ∞) .
It is easy to see that D (ξ) is a monotonically decreasing function on [0, ∞) and
Z
∞
ξ
0
n−1
Z
D (ξ) dξ ≤
1
n
2 dξ + 2
0
n
Z
∞
1
dξ
ξ n+1
< ∞.
This show that all the conditions of Theorem 3.5 are satisfied.
Acknowledgements
The authors would like to thank the referee for valuable comments.
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