Some approximation properties of Urysohn type nonlinear operators

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DOI: 10.24193/subbmath.2019.2.05

Some approximation properties of Urysohn type nonlinear operators

Harun Karsli

Abstract. The central issue of this paper is to continue the investigation of convergence properties of Urysohn type operators. By using Urysohn type operators we will extend the theory of interpolation to functionals and operators. In details, the present paper centers around Urysohn type nonlinear counterpart of the two dimensional Stancu operators defined on a triangle. We construct our nonlinear operators by defining a nonlinear forms of the kernel functions. Afterwards, we investigate the convergence problem for these operators.

Mathematics Subject Classification (2010):41A25, 41A35, 47G10, 47H30.

Keywords:Urysohn integral operators, Stancu operator, two dimensional nonlinear Stancu operators, Urysohn type nonlinear Stancu operators.

1. Introduction

In functional analysis, the superposition problem is known as the problem of representing a functionf as the composition of “simpler and more easily calculated”

functions. In 1885, Weierstrass gave a positive answer to this problem with his famous theorem, which states that every continuous function defined on a closed interval [a, b] can be uniformly approximated by a sequence of polynomials. Since that time many researchers try to find an explicit form of such polynomials to give a simple proof of this theorem. A well-known and most celebrated proof of the Weierstrass approximation theorem for f ∈C[0,1] is due to Bernstein, in which he defined the following polynomials

(B_nf) (x) =

n

X

k=0

f k

n

p_n,k(x), n≥1, (1.1)

This paper has been presented at the fourth edition of the International Conference on Numerical Analysis and Approximation Theory (NAAT 2018), Cluj-Napoca, Romania, September 6-9, 2018.

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where pn,k(x) = n

k

x^k(1−x)^n−k is the Binomial distribution, and proved that Bnf converges uniformly to anyf ∈C[0,1] (see [7]). Further investigations are obtained by Lorentz in [19]. Since Bernstein operators are the prototype of many positive linear operators used in the theory of approximation, a great number of generaliza- tions of these operators are given.

For the same functions, Stancu defined another positive linear operator as follows (P_n^αf) (x) =

n

X

k=0

f k

n

p^α_n,k(x), n≥1,

whereαis a non-negative parameter, which may depend only on the natural number nandp^α_n,k(x) called Markov-Polya distribution (see [23]).

The special caseα= 0 yields the Bernstein operator, while the Szasz-Mirakyan operator is shown to be a limiting case ofP_n^α. Whenα= 1/nwe obtain the Lupa¸s and Lupa¸s [20] operators corresponding to the equally spaced pointsk/n(k= 0,1, ..., n).

Up to the work of the famous polish mathematician Julian Musielak in 1981, see [22], the theory of approximation was strongly related with the linearity of the considered operators. Based on the idea developed in [22] and afterwards the works of C. Bardaro, G. Vinti and their research group on nonlinear operators, the approximation problem was proved by using nonlinear operators in some function spaces (see the fundamental book due to Bardaro, Musielak and Vinti [5]). For the approximation by linear and nonlinear operators, please see also the papers [3]-[2] and the monographs [10] and [26].

In view of the approach due to Musielak [22] and the techniques introduced by Bardaro-Mantellini in [4], Karsli-Tiryaki and Altin [18] considered the following nonlinear Bernstein operators;

(N B_nf)(x) =

n

X

k=0

P_n,k

x, f k

n

, 0≤x≤1, n∈N, (1.2) acting on bounded functionsf on an interval [0,1],wherePn,k satisfy some suitable assumptions. For further results we refer the papers [17], [16] and [18].

To generalize and extend the superposition or approximation problem for the functionals and operators, very recently in [13] and [14] Karsli defined and investigated the Urysohn type nonlinear Bernstein operators as;

(N BnF)x(t) =

1

Z

0

" _n X

k=0

Pk,n

x(s), f

t, s,k

n #

ds, 0≤x(s)≤1, wherePk,nsatisfy some suitable assumptions.

As a continuation of the above studies, in [14] the author also obtained Voronovskaya-type theorems for these operators.

For the linear forms of the Urysohn Bernstein and Urysohn Stancu operators we refer to the reader [11] and [21].

Moreover, in [15], Karsli considered a sequence N BF = (N BnF) of operators, which represents the Urysohn type nonlinear form of the two dimensional Bernstein

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operators defined by P.L. Butzer on the squareS= [0,1]×[0,1] (see [8], [9]), having the form:

(N B_nF) (x(t), y(t)) =

1

Z

0 1

Z

0

" _n X

k=0 n

X

i=0

P_k,i,n

x(s), y(z), f

t, s, z,k n, i

n #

dsdz,

0≤x(s), y(z)≤1, n∈N,

acting on bounded functionsf on [0,1]⁵,whereP_k,i,n satisfy some suitable assumptions.

The central issue of this paper is to give a positive answer to the superposition problem for functionals and operators by introducing the Urysohn nonlinear operators of the two dimensional Stancu operators (P_n^αf) (x, y) defined on the triangle

4:={(s, z) :s, z≥0, s+z≤1}.

Afterwards, we investigate the convergence problem for these nonlinear operators.

This paper is organized as follows: in Section 2, we construct the operators and further we present a basic lemma together with some definitions, which will be used in the sequel. Section 3 deals with the main convergence results for these operators.

2. Preliminaries and auxiliary results

This section is devoted to collecting some definitions and results which will be needed further on.

Now, we consider the following two dimensional Urysohn integral operator over the triangle4:={(s, z) :s, z≥0, s+z≤1},

F(x(t), y(t)) = Z Z

4

f(t, s, z, x(s), y(z))dsdz, t, s, z∈[0,1]

with unknown kernelf. If this representation exists, thenf(t, s, z, x(.), y(.)) is called the two dimensional Green’s function, which is strongly related to the functionsxand y (see [25] and [26]).

In view of the above relations, we assume that the two dimensional continuous interpolation conditions hold:

F(xi(t), yj(t)) = Z Z

4

f(t, s, z, xi(s), yj(z))dsdz, t∈[0,1], (2.1) where

xi(s) = i

nH(s−ξ);ξ∈[0; 1]

yj(z) = j

nH(z−ς);ς ∈[0; 1]

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andi, j= 0,1,2, ...n. By a straightforward calculation we have

∂²F _nⁱH(s−ξ),_n^jH(z−ς)

∂ξ∂ς = f(t, ξ, ς,i

n, j

n)−f(t, ξ, ς, i n,0) +f(t, ξ, ς,0,0)−f(t, ξ, ς,0, j

n).

Say

F₁

t, ξ, ς, i n, j

n

:= ∂²F _nⁱH(s−ξ),_n^jH(z−ς)

∂ξ∂ς .

According to the above definition, it is possible to construct an approximation operator in order to generalize and extend of the theory of interpolation of functions to operators.

For a bounded function defined on the triangle4:={(x, y) :x, y≥0, x+y≤1}, two dimensional Stancu polynomials is given by:

(P_n^αf) (x, y) =

n

X

k=0 n−k

X

j=0

p^α_n,k,j(x, y)f k

n, j n

,

whereαis a non-negative parameter, which may depend only on the natural number nand

p^α_n,k,j(x, y) = n

k

n−k j

∗

k−1

Q

l₁=0

(x+l₁α)

j−1

Q

l₂=0

(y+l₂α)

n−k−j−1

Q

l₃=0

(1−x−y+l₃α)

n−1

Q

l₄=0

(1 +l₄α) is the two dimensional Markov-Polya distribution ([24]).

Finally, let us now consider a sequenceN P^αF = (N P_n^αF) of operators, which represents Urysohn type nonlinear counterpart of the two dimensional Stancu operators defined on the triangle4:={(s, z) :s, z≥0, s+z≤1}, having the form:

(N P_n^αF) (x(t), y(t))

= Z Z

4

" _n X

k=0 n−k

X

i=0

P_k,i,n^α

x(s), y(z), f

t, s, z,k n, i

n #

dsdz, (2.2)

0≤x(s), y(z) andx(s) +y(z)≤1, n∈N,

acting on bounded functionsf on [0,1]⁵,whereP_k,i,n^α satisfy some suitable assumptions. In particular, we will putDom N P^αF = T

n∈N

Dom N P_n^αF,whereDom N P_n^αF is the set of all functionsf : [0,1]⁵→Rfor which the operator is well defined.

LetX be the set of all bounded Lebesgue measurable functions f : [0,1]⁵→R⁺0 = [0,∞).

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Let Ψ be the class of all functions ψ : R⁺0 → R⁺0 such that the function ψ is continuous and concave withψ(0) = 0, ψ(u)>0 for u >0.

We now introduce a sequence of functions. Let n P_k,i,n^α o

n∈N

be a sequence of functionsP_k,i,n^α : [0,1] x [0,1] xR→Rdefined by

P_k,i,n^α (t, l, u) =p^α_k,n(t)p^α_i,n(l)Hn(u) (2.3) for everyt, l∈[0,1], u∈R, whereHn:R→Ris such that Hn(0) = 0 and p^α_k,n(•) is the Markov-Polya basis.

Throughout the paper we assume thatµ:N→R⁺ is an increasing and continuous function such that lim

n→∞µ(n) =∞.

Assume that the following conditions hold:

a)Hn:R→Ris such that

|Hn(u)−Hn(v)| ≤ψ(|u−v|), (2.4) holds for every u, v ∈R, for every n∈N.That is,H_n satisfies a (L−Ψ) Lipschitz condition.

b)Denoting by rn(u) :=Hn(u)−u,u∈Rand n∈N,such that for nsufficiently large

sup

u

|rn(u)|= sup

u

|Hn(u)−u| ≤ 1

µ(n), (2.5)

holds.

The symbol [a] will denote the greatest integer not greater thana.

Following our announced aim, in this part we recall results regarding the uni- variate and linear case of the celebrated Stancu operators.

Lemma 2.1. [23]For(P_n^αt^s)(x), s= 0,1,2, one has (P_n^α1)(x) = 1

(P_n^αt)(x) = x

(P_n^αt²)(x) = x²+(1 +αn)x(1−x) n(1 +α) . By direct calculation, we find the following equalities:

(P_n^α(t−x)²)(x) =x(1−x) (1 +αn)

n(1 +α) , (P_n^α(t−x))(x) = 0. Moreover, for the second order central moment one has

(P_n^α(t−x)²)(x)≤ 1 +αn 4n(1 +α). Definition 2.2. Letf ∈C

[a, b]⁵

andδ >0 be given. Then the complete modulus of continuity is given by:

ω(f;δ) =√ sup

(u₁−u₂)²+(v₁−v₂)²≤δ

|f(t, s, z, u1, v1)−f(t, s, z, u2, v2)|. (2.6)

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Further on, the partial modulus of continuity with respect to forth and fifth variables are defined by

ω1(f;δ) = sup

t,s,z,v₁

sup

|u1−u2|≤δ

|f(t, s, z, u1, v1)−f(t, s, z, u2, v1)|

! ,

and

ω2(f;δ) = sup

t,s,z,u1

sup

|v₁−v₂|≤δ

|f(t, s, z, u1, v1)−f(t, s, z, u1, v2)|

! , respectively. Note thatω(f;δ) has the following properties;

(i) Letλ∈R⁺, then

ω(f;λδ)≤(λ+ 1)ω(f;δ), (ii) lim

δ→0⁺ω(f;δ) = 0,

(iii)|f(t, s, z, u1, v1)−f(t, s, z, u2, v2)|

≤ω(f;δ)



1 + q

(u1−u2)²+ (v1−v2)² δ



.

The same properties also hold for partial moduli of continuity.

Now, we are ready to state some convergence results of the operators defined on the triangle.

3. Main theorems

Theorem 3.1. LetF be the Urysohn integral operator with0≤x(s), y(z)and x(s) +y(z)≤1.

Then(N P_n^αF)converges to F uniformly in x, y∈C[0,1]. That is

n→∞lim k(N P_n^αF) (x(t), y(t))−F(x(t), y(t))k_C(4)= 0.

Proof. Owing to the definition of the operator given by (2.2), by considering (2.1), (2.3), (2.4) and (2.5), we have

|(N P_n^αF) (x(t), y(t))−F(x(t), y(t))|

=

Z Z

4

" _n X

k=0 n−k

X

i=0

P_k,i,n^α

x(s), y(z), f

t, s, z,k n, i

n #

dsdz−F(x(t), y(t))

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≤ Z Z

4 n

X

k=0 n−k

X

i=0

p^α_k,n(x(s))p^α_i,n(y(z))·

·

H_n

f

t, s, z,k n, i

n

−H_n(f(t, s, z, x(s), y(z)))

dsdz

+ Z Z

4 n

X

k=0 n−k

X

i=0

· |Hn(f(t, s, z, x(s), y(z)))−f(t, s, z, x(s), y(z))|dsdz :=I1+I2.

Owing to the assumptionb),one has I2 =

Z Z

4 n

X

k=0 n−k

X

i=0

· |H_n(f(t, s, z, x(s), y(z)))−f(t, s, z, x(s), y(z))|dsdz

≤ Z Z

4 n

X

k=0 n−k

X

i=0

p^α_k,n(x(s))p^α_i,n(y(z)) 1 µ(n)dsdz

= 1

µ(n), which tends to zero asn→ ∞.

Using the definition of the functionF1(t, s, z, x(s), y(z)),by concavity of the function ψ,and using Jensen inequality, we obtain

I₁≤ Z Z

4 n

X

k=0 n−k

X

i=0

p^α_k,n(x(s))p^α_i,n(y(z))

×ψ

f

t, s, z,k

n, i n

−f(t, s, z, x(s), y(z))

dsdz

≤ψ





 Z Z

4 n

X

k=0 n−k

X

i=0

p^α_k,n(x(s))p^α_i,n(y(z)) f

t, s, z, k n, i

n

−f(t, s, z, x(s), y(z))

dsdz







≤ψ (

RR

4 n

P

k=0 n−k

P

i=0

p^α_k,n(x(s))p^α_i,n(y(z))×

F1(t, s, z, x(s), y(z))−F1 t, s, z,^k_n,_nⁱ dsdz

+ Z Z

4

f(t, s, z, x(s),0)−

n

X

k=0

p^α_k,n(x(s))f

t, s, z, k n,0

dsdz

+ Z Z

4

f(t, s, z,0, y(z))−

n

X

i=0

p^α_i,n(y(z))f

t, s, z,0, i n

dsdz







≤I_1,1+I_1,2+I_1,3.

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Let us divide the first term into four parts as;

I_1,1 = ψ





RR

4 n

P

k=0 n−k

P

i=0

·

F1(t, s, z, x(s), y(z))−F1 t, s, z,_n^k,_nⁱ dsdz





: ≤I1,1,1+I1,1,2+I1,1,3+I1,1,4, where

I_1,1,1=ψ



 RR

4

P

|n^k−x(s)|^<δ1

P

|nⁱ−y(z)|^<δ2

·



,

I1,1,2=ψ



 RR

4

P

|n^k−x(s)|^<δ1

P

|nⁱ−y(z)|^≥δ2

·



,

I1,1,3=ψ



 RR

4

P

|_n^k−x(s)|≥δ1

P

|_nⁱ−y(z)|^<δ2

·



,

and

I_1,1,4=ψ



 RR

4

P

|_n^k^−x(s)|^≥δ1

P

|_nⁱ^−y(z)|^≥δ2

·

F₁(t, s, z, x(s), y(z))−F₁ t, s, z,_n^k,_nⁱ dsdz



.

Sincex, y∈C[0,1], then there exist δ1, δ2>0 such that

F1(t, s, z, x(s), y(z))−F1

t, s, z,k

n, i n

<

holds true when

^k_n−x(s)

< δ₁and

_nⁱ −y(z)

< δ₂.So one can easily obtain I1,1,1< ψ().

As to the other terms

F1(t, s, z, x(s), y(z))−F1

t, s, z,k

n, i n

≤2M

holds true for someM >0, when

^k_n−x(s)

≥δ1or

_nⁱ −y(z) ≥δ2.

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In view of Lemma 2.1, we obtain

I_1,1,2=ψ



 RR

4

P

|^kn−x(s)|^<δ1

P

|nⁱ−y(z)|^≥δ2

·





≤ ψ





2M Z Z

4

X

|^kn−x(s)|^<δ1

X

|nⁱ−y(z)|^≥δ2

i−ny(z) δ2

2

p^α_k,n(x(s))p^α_i,n(y(z))dsdz







≤ ψ





2M Z Z

4

X

|^k_n−x(s)|^<δ1

X

|_nⁱ−y(z)|≥δ2

i−ny(z) δ2

²







≤ ψ 2M

δ₂²

1 +αn 4n(1 +α)

.

Similarly one has

I1,1,3≤ψ 2M

δ²

1 +αn 4n(1 +α)

,

and

I1,1,4≤ψ 2M δ₁²δ₂²

1 +αn 4n(1 +α)

²! .

Collecting these estimates we have

|(N P_n^αF) (x(t), y(t))−F(x(t), y(t))|

≤ ψ() +ψ 2M

δ₁²

1 +αn 4n(1 +α)

+ψ

2M δ₂²

1 +αn 4n(1 +α)

+ψ 2M

δ₁²δ₂²

1 +αn 4n(1 +α)

²!

+ 1

µ(n). That is

n→∞lim k(N P_n^αF) (x(t), y(t))−F(x(t), y(t))k_C([0,1]2)= 0.

This completes the proof.

Theorem 3.2. LetF be the Urysohn integral operator withx, y∈C[0,1]and0≤x(s), y(z)≤1.Then

|(N P_n^αF) (x(t), y(t))−F(x(t), y(t))| ≤2ψ(ω(f;δ)) + 1 µ(n) holds true, where δ=q _1+αn

2n(1+α).

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Proof.Clearly one has

|(N P_n^αF) (x(t), y(t))−F(x(t), y(t))|

≤ Z Z

4 n

X

k=0 n−k

X

i=0

·

Hn

f

t, s, z,k

n, i n

−Hn(f(t, s, z, x(s), y(z)))

dsdz

+ 1

µ(n)

:=I_n,1(x) + 1

µ(n), (3.1)

say. Sincex, y∈C[0,1] we can rewrite (3.1) as follows I_n,1(x)≤

Z Z

4 n

X

k=0 n−k

X

i=0

·ψ

f

t, s, z,k

n, i n

−f(t, s, z, x(s), y(z))

dsdz

≤ Z Z

4 n

X

k=0 n−k

X

i=0

p^α_k,n(x(s))p^α_i,n(y(z))ψ(ω(f;δ))dsdz

≤ψ





 Z Z

4 n

X

k=0 n−k

X

i=0

p^α_k,n(x(s))p^α_i,n(y(z))ω(f;δ)dsdz







≤ψ







RR

4 n

P

k=0 n−k

P

i=0

·

q(^k_n−x(s))²⁺(_nⁱ−y(z))²

δ + 1

!

ω(f;δ)dsdz







=ψ







ω(f;δ)RR

4 n

P

k=0 n−k

P

i=0

·

q(n^k−x(s))²⁺(nⁱ−y(z))²

δ dsdz







+ψ





ω(f;δ) Z Z

4 n

X

k=0 n−k

X

i=0







≤ψ





 ω(f;δ)

δ Z Z

4







n

P

k=0 n−k

P

i=0

·h

k

n−x(s)²

+ _nⁱ −y(z)²i







1/2

dsdz







+ψ(ω(f;δ))

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≤ψ ω(f;δ) δ

1 +αn 2n(1 +α)

^1/2!

+ψ(ω(f;δ)).

Taking into account that ω(f;δ) is the modulus of continuity defined as (2.6). If we choose

δ=

s 1 +αn 2n(1 +α), then one can obtain the desired estimate, namely,

|(N P_n^αF) (x(t), y(t))−F(x(t), y(t))| ≤2ψ(ω(f;δ)) + 1 µ(n). Thus the proof is now complete.

Theorem 3.3. LetF be the Urysohn integral operator withx, y∈C[0,1],and0≤x(s), y(z)≤1.Then

|(N P_n^αF) (x(t), y(t))−F(x(t), y(t))|

≤2

"

ψ ω1 f;

1 +αn 4n(1 +α)

^1/2!!

+ψ ω2 f;

1 +αn 4n(1 +α)

^1/2!!#

+ 1

µ(n) holds true.

Proof.In view of the definition of the considered operator, one has

|(N P_n^αF) (x(t), y(t))−F(x(t), y(t))|

≤ Z Z

4 n

X

k=0 n−k

X

i=0

·

H_n

f

t, s, z,k n, i

n

−H_n(f(t, s, z, x(s), y(z)))

dsdz + 1

µ(n)

= Z Z

4 n

X

k=0 n−k

X

i=0

·

Hn f t, s, z,_n^k,_nⁱ

−Hn f t, s, z, x(s),_nⁱ +Hn f t, s, z, x(s),_nⁱ

−Hn(f(t, s, z, x(s), y(z)))

dsdz + 1

µ(n)

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≤ Z Z

4 n

X

k=0 n−k

X

i=0

·

H_n

f

t, s, z,k n, i

n

−H_n

f

t, s, z, x(s), i n

dsdz

+ Z Z

4 n

X

k=0 n−k

X

i=0

p^α_k,n(x(s))p^α_i,n(y(z))

Hn

f

t, s, z, x(s), i n

−Hn(f(t, s, z, x(s), y(z)))

dsdz

+ 1

µ(n)

: =I_n,1(x) +I_n,2(x) + 1 µ(n),

say. Sincex, y∈C[0,1] we can rewrite (3.1) as follows: By concavity of the function ψ,and using Jensen inequality, we obtain

In,1(x) = Z Z

4 n

X

k=0 n−k

X

i=0

·

H_n

f

t, s, z,k n, i

n

−H_n

f

t, s, z, x(s), i n

dsdz

≤ Z Z

4 n

X

k=0 n−k

X

i=0

p^α_k,n(x(s))p^α_i,n(y(z))ψ

ω1

f;

k n−x(s)

dsdz

≤ ψ





 Z Z

4 n

X

k=0 n−k

X

i=0

p^α_k,n(x(s))p^α_i,n(y(z))ω₁

f; k n−x(s)

dsdz







Sinceψis non decreasing, then one has

In,1(x) ≤ ψ





 RR

4 n

P

k=0 n−k

P

i=0

·

q(^kn−x(s))²

δ₁ + 1

!

ω1(f;δ)dsdz







≤ ψ ω1(f;δ) δ

1 +αn 4n(1 +α)

1/2!

+ψ(ω1(f;δ)). Similarly

I_n,1(x)≤ψ ω₂(f;δ) δ

1 +αn 4n(1 +α)

^1/2!

+ψ(ω₂(f;δ)).

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If we chooseδ=h

1+αn 4n(1+α)

i^1/2

,so we get the desired estimate.

Acknowledgments.The author is thankful to the referee for his/her valuable remarks and suggestions leading to a better presentation of this paper.

References

[1] Altomare, F., Campiti, M.,Korovkin-Type Approximation Theory and its Applications, De Gruyter Studies in Mathematics, 17, Walter de Gruyter and Co., Berlin, 1994.

[2] Bardaro, C., Karsli, H., Vinti, G.,Nonlinear integral operators with homogeneous kernels:

pointwise approximation theorems, Appl. Anal.,90(2011), no 3-4, 463–474.

[3] Bardaro, C., Mantellini, I., On the reconstruction of functions by means of nonlinear discrete operators, J. Concr. Appl. Math.,1(2003), no. 4, 273–285.

[4] Bardaro, C., Mantellini, I.,Approximation properties in abstract modular spaces for a class of general sampling-type operators, Appl. Anal.,85(2006), no. 4, 383–413.

[5] Bardaro, C., Musielak, J., Vinti, G.,Nonlinear Integral Operators and Applications, De Gruyter Series in Nonlinear Analysis and Applications, Vol. 9, xii + 201 pp., 2003.

[6] Bardaro, C., Vinti, G., Urysohn integral operators with homogeneous kernel: approximation properties in modular spaces, Comment. Math. (Prace Mat.), 42(2002), no. 2, 145–182.

[7] Bernstein S.N., Demonstration du Théoreme de Weierstrass fondée sur le calcul des probabilités, Comm. Soc. Math. Kharkow,13(1912/13), 1-2.

[8] Butzer, P.L.,On Bernstein Polynomials, Ph.D. Thesis, University of Toronto, 1951.

[9] Butzer, P.L., On two dimensional Bernstein polynomials, Canad. J. Math., 5(1953), 107–113.

[10] Butzer, P.L., Nessel, R.J.,Fourier Analysis and Approximation, Vol. 1, Academic Press, New York, London, 1971.

[11] Demkiv, I.I., On Approximation of the Urysohn operator by Bernstein type operator polynomials, Visn. L’viv. Univ., Ser. Prykl. Mat. Inform., (2000), no. 2, 26-30.

[12] Karsli, H.,Approximation by Urysohn type Meyer-K¨onig and Zeller operators to Urysohn integral operators, Results Math., 72(2017), no. 3, 1571–1583.

[13] Karsli, H., Approximation Results for Urysohn Type Nonlinear Bernstein Operators, Advances in Summability and Approximation Theory, Book Chapter, Springer Nature Singapore Pte Ltd., 223-241, 2018.

[14] Karsli, H.,Voronovskaya-type theorems for Urysohn type nonlinear Bernstein operators, Mathematical Methods in the Applied Sciences, (2018), accepted.

[15] Karsli, H.,Approximation results for Urysohn type two dimensional nonlinear Bernstein operators, Const. Math. Anal.,1(2018), no. 1, 45-57.

[16] Karsli, H., Altin, H.E.,A Voronovskaya-type theorem for a certain nonlinear Bernstein operators, Stud. Univ. Babe¸s-Bolyai Math.,60(2015), no. 2, 249–258.

[17] Karsli, H., Tiryaki, I.U., Altin, H.E., On convergence of certain nonlinear Bernstein operators, Filomat,30(2016), no. 1, 141–155.

[18] Karsli, H., Tiryaki I.U., Altin H.E.,Some approximation properties of a certain nonlinear Bernstein operators, Filomat,28(6)(2014), 1295-1305.

[19] Lorentz, G.G.,Bernstein Polynomials, University of Toronto Press, Toronto, 1953.

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[20] Lupa¸s, L., Lupa¸s, A.,Polynomials of binomial type and approximation operators, Studia Univ. Babe¸s-Bolyai, Mathematica,32(1987), 61-69.

[21] Makarov, V.L., Demkiv, I.I.,Approximation of the Urysohn operator by operator polynomials of Stancu type, Ukrainian Math Journal,64(2012), no. 3, 356-386.

[22] Musielak, J.,On some approximation problems in modular spaces, In: Constructive Func- tion Theory 1981, (Proc. Int. Conf., Varna, June 1-5, 1981), 455-461, Publ. House Bul- garian Acad. Sci., Sofia 1983.

[23] Stancu, D.D.,Approximation of functions by a new class of linear polynomial operators, Rev. Roumaine Math. Pures Appl.,13(1968), 1173–1194.

[24] Stancu, D.D., A new class of uniform approximating polynomial operators in two and several variables, Proceedings of the Conference on the Constructive Theory of Functions (Approximation Theory) (Budapest, 1969), 443-455, Akademiai Kiado, Budapest, 1972.

[25] Urysohn, P.,On a type of nonlinear integral equation, Mat. Sb.,31(1924), 236–255.

[26] Zabreiko, P.P., Koshelev, A.I., Krasnosel’skii, M.A., Mikhlin, S.G., Rakovscik, L.S., Stet- senko, V.Ja.,Integral Equations: A Reference Text, Noordhoff Int. Publ., Leyden, 1975.

Harun Karsli

Bolu Abant Izzet Baysal University Faculty of Science and Arts Department of Mathematics

14030 Golkoy Bolu, Turkey e-mail:karsli [email protected]