# Nonlinear Bivariate Bernstein–Chlodowsky Operators of Maximum Product Type

Authors: Özge Özalp Güller; Ecem Acar; Sevilay Kırcı Serenbay
Journal: Journal of Mathematics
 (2022)

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## Abstract

The positive nonlinear operators with maximum and product were introduced by Bede. In this study, nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators is defined and the approximation properties are investigated with the help of new definitions. In this paper, it was aimed that the order of approximation obtained with the nonlinear maximum product type of operator sequences would be better than the degree of approximation of the known linear operator sequences.


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## Body

## 1. Introduction

The main topic in the classical approximation theory is approximating a continuous functiong:α,β⟶ℝ with more elementary functions such as polynomials, trigonometric functions, and so on. The well-known Korovkin’s theorem, which gives a simple proof of Weierstrass theorem, is based on the approximation of functions by positive and linear operators. In the last century, researchers’ attention has turned to the fact that the operations that allow us to construct linear approximation operators do not have to be just addition and multiplication operations, and the approximation operators do not have to be linear. The maximum product type of operators (given in the study by Gal in [1]) were defined for many operators such as Bernstein, Bleimann-Butzer-Hahn, Favard-Szász-Mirakyan, Meyer-König and Zeller, Baskakov [2–8] and their approaches and some shape preserving properties have been studied. In [9], the authors obtained nonlinear Shepard type operators by replacing the operations sum and product by other pairs. Then, the maximum product type Bernstein operators were introduced as follows:(1)BnMfx=∨k=0npn,kxfk/n∨k=0npn,kx,where(2)pn,kx=nkxk1−xn−k,f:0,1⟶ℝ+,(see [9]). In these studies, it was aimed that the order of approximation obtained with the nonlinear maximum product type of operator sequences would be better than the order of approximation of the known linear operator sequences. In this study, the nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators is defined and the approximation properties are investigated with the help of new definitions.


## 2. Preliminaries

In [10], the authors defined the maximum product kind of Bernstein–Chlodowsky operators as(3)CnMfx=∨k=0nhn,kxfbnk/n∨k=0nhn,kx,x∈0,bn,where(4)hn,kx=nkxbnk1−xbnn−k,f:0,bn⟶ℝ+,and bn is a sequence of positive real numbers such that limn⟶∞bn=∞ and limn⟶∞bn/n=0.In addition, in the book published in 2016, Bede et al. [11] gave the definition of the maximum product bivariate Bernstein operators and examined various approximation properties. In this study, we will give the definitions of the nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators and examine various approach features.Definition 1.
(see [11]). Suppose that h:I⟶ℝ; for I=0,1×0,1,(i)
Let for ally,y+l,z∈0,1 and for l>0, φy+l,z−φy,z≥0, (resp., ≤0). Then, the function φ is increasing with respect to y on I (resp., decreasing).(ii)
Let for ally,z,z+l∈0,1 and for l>0, φy,z+l−φy,z≥0, (resp., ≤0). Then, the function φ is increasing with respect to z on I (resp., decreasing).(iii)
Let for ally,y+t,z,z+l∈0,1 and for t,l≥0, Δ2φy,z=φy+t,z+l−φy,z+l−φy+t,z+φy,z≥0 (resp., ≤0). Then, the function φ is upper bidimensional monotone on I (resp., lower bidimensional monotone).(iv)
Let (i)–(iii) hold for all simultaneously≥0 (resp., ≤0). Then, the function φ is totally upper monotone on I (resp., lower bidimensional monotone).(v)
Let, for any fixedz∈0,1,φ be quasiconvex as function of y, i.e., for all y1, y2, z∈0,1 and μ∈0,1, φμy1+1−μy2,z≤maxφy1,z,φy2,z. Then, the continuous function φ is called quasiconvex with respect to y on I.(vi)
Let, for any fixedy∈0,1, φ be quasiconvex as function of z, i.e., for all y, z1, z2∈0,1 and η∈0,1, φy,ηz1+1−ηz2≤maxφy,z1,φy,z2. Then, the continuous function φ is called quasiconvex with respect to z on I.(vii)
Let for ally1, y2, z1, z2∈0,1 and ρ, η∈0,1,(5)φρy1+1−ρy2,ηz1+1−ηz2≤maxφy1,z1,φy1,z2,φy2,z1,φy2,z2.
Then, the continuous functionφ is called bidimensional quasiconvex on I.
Note that if the continuous functionφ is bidimensional quasiconvex and satisfies (v) and (vi), then φ is called totally quasiconvex on I.
There are a few more items (see [11]).Remark 1.
(see [11]). Let φ have partial derivative. The conditions (i)–(iv) in Definition 1 can be expressed as follows:(i)
For ally,z∈0,1, ∂φy,z/∂y≥0 (or ≤0).(ii)
For ally,z∈0,1, ∂φy,z/∂z≥0 (or ≤0).(iii)
For ally,z∈0,1, ∂2φy,z/∂y∂z≥0 (or ≤0).(iv)
is represented by conditions (i)–(iii).


## 3. Construction of the OperatorsCα,βM and HαM

In this section, we introduce two kinds of nonlinear maximum product type bivariate Bernstein–Chlodowsky operators.Definition 2.
Letφ:0,cα×0,dβ⟶ℝ+, where cα, dβ are considered a sequence of positive real numbers such that limα⟶∞cα=∞, limβ⟶∞dβ=∞ and limα⟶∞cα/α=0, limβ⟶∞dβ/β=0, for α,β∈ℕ, be a continuous function. The nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators is defined as follows:(6)Cα,βMφx,y=∨i=0α∨l=0βsα,ixsβ,lyφi/αcα,l/βdβ∨i=0αsα,ix⋅∨l=0βsβ,ly,with(7)sα,ix=αixcαi1−xcαα−i,sβ,ly=βlydβl1−ydββ−l,for all x,y∈0,cα×0,dβ, α, β∈ℕ.Definition 3.
Letφ:Δcα⟶ℝ+, where cα is a sequence of positive real numbers such thatlimα⟶∞cα=∞,, and limα⟶∞cα/α=0, forα∈ℕ, be a continuous function. The nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators in the triangular domain Δcα=x,y;x,y≥0,x+y≤cα is defined as follows:(8)HαMφx,y=∨i=0α∨l=0α−isα,i,lx,yφi/αcα,l/αcα∨i=0α∨l=0α−isα,i,lx,y,with(9)sα,i,lx,y=αiα−ilxcαiycαl1−xcα−ycαα−i−l,for all x,y∈Δcα, α∈ℕ.
In the following sections, we give some properties of theCα,βM and HαM operators, as well as an error estimate for the nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators in terms of modulus of continuity.


## 4. Approximation and Shape Preserving Properties forCα,βM

In this section, we give an error estimate in terms of continuity modulus for the nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators. Additionally, we investigate the shape preserving properties for these operators.Let us now give the following lemma which is necessary.Lemma 1.
Letsα,ix and sβ,ly be defined as in Definition 2. Then, we have(10)∨i=0αsα,ix⋅∨l=0βsβ,ly=sα,rx⋅sβ,ty,for all x,y∈rcα/α+1,r+1cα/α+1×tdβ/β+1,t+1dβ/β+1, r=0,1,2,…,α, t=0,1,2,…,β.Proof.
Firstly, we claim that forα,β∈ℕ and 0≤r<r+1≤α, 0≤t<t+1≤β, we have(11)0≤sα,r+1x≤sα,rx,0≤sβ,t+1x≤sβ,tx,if and only if(12)x∈0,r+1cαα+1,y∈0,t+1dββ+1.
We can easily write the following inequalities:(13)0≤αr+1xcαr+11−xcαα−r+1≤αrxcαr1−xcαα−r,0≤βt+1ydβt+11−ydββ−t+1≤βtydβt1−ydββ−t,and after simplification,(14)0≤xcααr+1+αr≤αr,0≤ydββt+1+βt≤βt.
From the equalityαi+1+αi=α+1i+1, we get 0≤x≤r+1cα/α+1, 0≤y≤t+1dβ/β+1. Also, denoting(15)Bi,α,rx=sα,ixsα,rx=αiαrx/cα1−x/cαi−r,(16)Bl,β,ty=sβ,lysβ,ty=βlβty/dβ1−y/dβl−t,(17)Bi,α,r,l,β,tx,y=Bi,α,rx⋅Bl,β,ty,write the following formula which is useful in proving approximate results:(18)Cα,βMφx,y=∨i=0α∨l=0βBi,α,r,l,β,tx,yφiαcα,lβdβ,for all x,y∈rcα/α+1,r+1cα/α+1×tdβ/β+1,t+1dβ/β+1, r=0,1,2,…,α, t=0,1,2,…,β.
Note that the nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators is tensor product of the univariate type of these maximum operators. It easily can be followed:(19)Cα,βMφ=Cα,xMCβ,yMφ.
Now, we give an error estimate for nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators in terms of the modulus of continuity, where(20)ω1φ;ν,η=supφx,y−φz,t;x,y,z,t∈0,cα×0,dβ,x−z≤ν,y−t≤η.Theorem 1.
Letφ:0,cα×0,dβ⟶ℝ+ be a continuous function. Suppose that Cα,βMφ is nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators defined in (6). Then, the following holds:(21)Cα,βMφx,y−φx,y≤18ω1φ;cαα+1,dββ+1,for all x,y∈0,cα×0,dβ and α,β∈ℕ.Proof.
Consider the following inequality for the positive numbersϕp, ψp, and p∈0,1,2,…,j:(22)maxp∈0,1,…,jϕp−maxp∈0,1,…,jψp≤maxp∈0,1,…,jϕp−ψp.
We surely obtain(23)Cα,βMφx,y−φx,y=∨i=0α∨l=0βsα,ixsβ,lyφi/αcα,l/βdβ∨i=0αsα,ix⋅∨l=0βsβ,ly−∨i=0α∨l=0βsα,ixsβ,lyφx,y∨i=0αsα,ix⋅∨l=0βsβ,ly≤∨i=0α∨l=0βsα,ixsβ,lyφi/αcα,l/βdβ−φx,y∨i=0αsα,ix⋅∨l=0βsβ,ly.
In addition,ξ≥0, η≥0, ρ≥0, and τ≥0, if the following known modulus of continuity of properties is used:(24)ω1φ;ρξ,τη≤1+ρ+τω1φ;ξ,η,ω1φ;nξ,nη≤nω1φ;ξ,η,for all α∈ℕ (see, [12]); then, we have(25)Cα,βMφx,y−φx,y≤∨i=0α∨l=0βsα,ixsβ,lyω1φ;i/αcα−x,l/βdβ−y∨i=0αsα,ix⋅∨l=0βsβ,ly≤∨i=0α∨l=0βsα,ixsβ,ly1+i/αcα−x/ξ+l/βdβ−y/ηω1φ;ξ,η∨i=0αsα,ix⋅∨l=0βsβ,ly=ω1φ;ξ,η1+1ξ∨i=0αsα,ixi/αcα−x∨i=0αsα,ix+1η∨l=0βsβ,lyl/βdβ−y∨l=0βsβ,ly.
Now, considering the univariate case (see [10] in Theorem 1) and for α,β∈ℕ, we can write(26)Cα,βMφx,y−φx,y≤3ω1φ;6cαα+1,6dββ+1≤18ω1φ;cαα+1,dββ+1,by choosing ξ=6cα/α+1, η=6dβ/β+1, for all x,y∈0,cα×0,dβ.Theorem 2.
Letφ:0,cα×0,dβ⟶ℝ+ be continuous on 0,cα×0,dβ. Suppose that for all y∈0,dβ fixed, φ is a concave function with respect to x, and for all x∈0,cα fixed, a concave function with respect to y. Then, for all x,y∈0,cα×0,dβ and α,β∈ℕ, we have(27)Cα,βMφx,y−φx,y≤4ω1φ;cαα,dββ.Proof.
See the proof in the appendix.Theorem 3.
Let the continuous functionφ:0,cα×0,dβ⟶ℝ+ be strictly positive on 0,cα×0,dβ. Then, for all x,y∈0,cα×0,dβ,α,β∈ℕ, we have(28)Cα,βMφx,y−φx,y≤8ω1φ;cαα,dββ+1cφαω1,xφ;cαα2+βω1,yφ;dββ2,where cφ=minφx,y;x,y∈0,cα×0,dβ and, for all x+l,y,x,y∈0,cα×0,dβ, ω1,xφ;ν≤supφx+l,y−φx,y;0≤l≤ν, and, for all x,y+l,x,y∈0,cα×0,dβ,ω1,yφ;ν≤supφx,y+l−φx,y;0≤l≤ν.Proof.
We get(29)Cα,βMφx,y−φx,y=∨i=0α∨l=0βsα,ixsβ,lyφi/αcα,l/βdβ∨i=0αsα,ix⋅∨l=0βsβ,ly−∨i=0α∨l=0βsα,ixsβ,lyφx,y∨i=0αsα,ix⋅∨l=0βsβ,ly≤∨i=0αsα,ix∨r=0αsα,rx⋅∨l=0βsβ,ly∨t=0βsβ,tyφiαcα,lβdβ−∨i=0αsα,ix∨r=0αsα,rx⋅∨l=0βsβ,ly∨t=0βsβ,tyφiαcα,y+∨i=0αsα,ix∨r=0αsα,rx⋅∨l=0βsβ,ly∨t=0βsβ,tyφiαcα,y−∨i=0αsα,ix∨r=0αsα,rx⋅∨l=0βsβ,ly∨t=0βsβ,tyφx,y,and the following inequality is easily obtained.(30)Cα,βMφx,y−φx,y≤∨i=0αsα,ix∨r=0αsα,rx.∨l=0βsβ,ly∨t=0βsβ,tyφiαcα,jβdβ−∨l=0βsβ,ly∨t=0βsβ,tyφiαcα,y+∨l=0βsβ,ly∨t=0βsβ,ty∨i=0αsα,ix∨r=0αsα,rxφiαcα,y−∨i=0αsα,ix∨r=0αsα,rxφx,y.
We also know that the following equation is satisfied for the nonlinear maximum product type of univariate Bernstein–Chlodowsky operators. That is,(31)CαMφx−φx≤αω1φ,cα/αcφ+4ω1φ,cαα,(see Theorem 2.2.18 given in [11]), which implies(32)Cα,βMφx,y−φx,y≤α⋅ω1,xφ;cα/αcφ+4ω1,xφ;cαα+β⋅ω1,yφ;dβ/βcφ+4ω1,yφ;dββ.
Fromω1,xφ;cα/α+ω1,yφ;dβ/β≤2ω1φ;cα/α,dβ/β (see [13], p. 81), for the last inequality, we easily can write(33)Cα,βMφx,y−φx,y≤8ω1φ;cαα,dββ+1cφαω1,xφ;cαα2+βω1,yφ;dββ2.
In the following example, we graphically show the rate of convergence of the operatorsCα,βM to certain function. We also compare the convergence of the nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators and the classical linear bivariate Bernstein–Chlodowsky operators (see [14]) to functions.Remark 2.
The classical linear bivariate Bernstein–Chlodowsky operators have the following form:(34)Cα,βφ;x,y=∑i=0α∑j=0βφiαcα,jβdβαixcαi1−xcαα−i×βjydβj1−ydββ−j,where cα, dβ are considered a sequence of positive real numbers such that limα⟶∞cα=∞, limβ⟶∞dβ=∞ and limα⟶∞cα/α=0, limβ⟶∞dβ/β=0, for α,β∈ℕ (see [14]).Example 1.
Letφx=x−32+y−32, cα=α1/3, and dβ=β1/3.
In the graph in Figure1, we illustrate the convergence of the operators Cα,βM (blue) and Cα,β (yellow) to φx,y (red) for α=5 and β=5.
In the graph in Figure2, we show the operator Cα,βM approximation to φx,y (yellow) for the values α,β=3 (blue) and α,β=5 (red).Figure 1
Cα,βM (blue) and Cα,β (yellow) to φx,y (red) for α=5 and β=5.Figure 2
Cα,βM approximation to φx,y (yellow) for α,β=3 (blue), α,β=5 (red).Example 2.
Letφx=x−32+y−32, cα=α1/3, and dβ=β1/3. The comparison of the values of the operators Cα,βM, Cα,β and the function φx,y at different x,y points is shown in Table 1 for values α=5 and β=5.
As a result, from graph in Figure1, it is clearly seen that for the corresponding functions, the nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators approximates much better than the classical linear bivariate Bernstein–Chlodowsky operators.
Taking into account that the nonlinear maximum product type of univariate Bernstein–Chlodowsky operators preserves the monotonicity and quasiconvexity, we expect these maximum operators to preserve the types of monotonicity in Definition1 in the bivariate case. In this case, we present the following theorem.Table 1
φ, Cα,βM, and Cα,β values for x,y.
(0, 0)(0, 5)(5, 0)(5, 6)(6, 5)(6, 6)φ4.2426406863.6055512753.6055512753.6055512753.6055512754.242640686Cα,βM4.2426406873.6055512753.7239437933.7239437932.9778779984.242640686Cα,β0002.85092.027003.2730Theorem 4.
Letφ:0,cα×0,dβ⟶ℝ+ be continuous on 0,cα×0,dβ. For all α, β∈ℕ,(i)
For all fixed, arbitraryy, if φ is increasing with respect to x (or decreasing), then Cα,βM is increasing (or decreasing).(ii)
For all fixed, arbitraryx, if φ is increasing with respect to y (or decreasing), then Cα,βM is increasing (or decreasing).(iii)
Letφ be simultaneously monotonous with respect to x and y. If φ has the same monotonicity (resp., opposite monotonicity), then Cα,βM is upper bidimensional monotone on 0,cα×0,dβ (resp., lower bidimensional monotone).(iv)
Letφ be simultaneously monotonous with respect to x and y. If φ has the same monotonicity (resp., opposite monotonicity), then Cα,βM is totally upper monotone on 0,cα×0,dβ (resp., lower bidimensional monotone).(v)
If the continuous functionφ is quasiconvex on 0,cα×0,dβ with respect to x, then Cα,βM is quasiconvex and continuous on 0,cα×0,dβ.(vi)
If the continuous functionφ is quasiconvex on 0,cα×0,dβ with respect to y, then Cα,βM is quasiconvex and continuous on 0,cα×0,dβ.(vii)
If the continuous functionφ is simultaneously quasiconvex with respect to x and quasiconvex with respect to y on 0,cα×0,dβ, then Cα,βM is bidimensional quasiconvex and continuous on 0,cα×0,dβ.(viii)
If the continuous functionφ is simultaneously quasiconvex with respect to x and quasiconvex with respect to y on 0,cα×0,dβ, then Cα,βM is totally quasiconvex and continuous on 0,cα×0,dβ.Proof.
See the proof in the appendix.


## 5. The Approximation Properties forHαM

In this section, we will examine the approximation properties of the nonlinear maximum product type bivariate Bernstein–Chlodowsky operatorsHαMφ in the triangular domain Δcα.Throughout the study,Δcα=x,y;x,y≥0,x+y≤cα.Firstly, we need some auxiliary results.Theorem 5.
LetC+Δcα=φ:Δcα⟶0,∞;fis continuous onΔcα.(i)
Ifφ∈C+Δcα, then for all α∈ℕ, HαMφ∈C+Δcα.(ii)
HαMλφ=λHαMφ for λ≥0, for all α∈ℕ and φ∈C+Δcα.(iii)
For allφ,ϕ∈C+Δcα, α∈ℕ, and x,y∈Δcα, HαMφ+ϕx,y≤HαMφx,y+HαMϕx,y.(iv)
Ifφ,ϕ∈C+Δcα, for all x,y∈Δcα, α∈ℕ, satisfy φx,y≤ϕx,y, then HαMφx,y≤HαMϕx,y.(v)
Denotinge0x,y=1, for all x,y∈Δcα, α∈ℕ, we have HαMe0x,y=e0x,y.(vi)
HαMφx,y−HαMϕx,y≤HαMφ−ϕx,y, for all φ,ϕ∈C+Δcα, α∈ℕ, and x,y∈Δcα.Proof.
The proof of properties from (i) to (vi) is obvious.Lemma 2.
For allx,y∈Δcα, denoting(35)sα,it=αitcαi1−tcαn−i,we have(36)∨i=0α∨l=0α−isα,i,lx,y=∨i=0αsα,ix⋅∨l=0α−isα−i,lycα−x,with x<cα for α∈ℕ.Proof.
Forx≠cα, we can write(37)∨i=0α∨l=0α−iαiα−ilxcαiycαl1−xcα−ycαα−i−l=∨i=0ααixcαi1−xcαα−i∨l=0α−iα−ilycαl1−x/cα−y/cαα−i−l1−x/cαα−i=∨i=0αsα,ix⋅∨l=0α−isα−i,lycα−x.
Let us now give the following corollary.Corollary 1.
We have(38)HαMφx,y−φx,y≤ω1f;νΔcα1+1νHαMχxx,y+1νHαMχyx,y,for all φ∈C+Δcα, x,y∈Δcα and α∈ℕ, where χxt=t−x and χyr=r−y, t,r are variable and x,y are supposed fixed, and(39)ω1φ;νΔcα=supφt,r−φx,y;t,r,x,y∈Δcα,t−x2+r−y2≤ν.Proof.
From (vi) of Theorem5, we can write(40)HαMφx,y−φx,y=HαMφt,rx,y−HαMφx,yx,y≤HαMφt,r−φx,yx,y.
Additionally, we clearly have(41)φt,r−φx,y≤ω1φ;t,r−x,yΔcα≤ω1φ;νΔcα1+1νt−x2+r−y2≤ω1φ;νΔcα1+1νt−x+1νr−y,for all t,r,x,y∈Δcα. Thus, the proof of the above inequalities is completed.Remark 3.
From Corollary1, the properties of HαM are controlled by(42)HαMχxx,y=∨i=0α∨l=0α−isα,i,lx,yi/αcα−x∨i=0α∨l=0α−isα,i,lx,y,HαMχyx,y=∨i=0α∨l=0α−isα,i,lx,yj/αcα−y∨i=0α∨l=0α−isα,i,lx,y,for all x,y∈Δcα, α∈ℕ, where(43)sα,i,lx,y=αiα−ilxcαiycαl1−xcα−ycαα−i−l.
Now, we give an error estimate in terms of modulus of continuity for nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators.Theorem 6.
Let, for allx,y∈Δcα and α∈ℕ, φ∈C+Δcα. If HαM are the nonlinear maximum product type bivariate Bernstein–Chlodowsky operators defined in (8), then the following holds:(44)HαMφx,y−φx,y≤18ω1φ;cαα+1,cαα+1.Proof.
By the proof of Lemma2, we get(45)∨i=0α∨l=0α−isα,i,lx,yiαcα−x=∨i=0αsα,ixiαcα−x⋅∨l=0α−isα−i,lycα−x,which immediately implies(46)HαMχxx,y=CαMχxx.
Additionally, since we have(47)αl⋅α−li=αi⋅α−il,we can clearly write(48)∨l=0α∨i=0α−lsα,l,ix,y=∨l=0αsα,ly∨i=0α−lsα−l,iycα−x,which immediately implies(49)HαMχyx,y=CαMχyy.
Using Corollary1 and from (2.2) in [11], we get(50)HαMφx,y−φx,y≤18ω1φ;cαα+1Δcα,with the property of modulus of continuity ω1φ;νΔcα≤ω1φ;ν,ν (see [13], p. 81).


## 6. Conclusions

Approaching functions have been an active study subject for more than a century with their useful structure and applications in many disciplines (physics, engineering sciences, computer technologies, and so on). For example, since the topic of approximating multivariate functions is interesting for problems involving large dimensions, it has an important place in related topics such as kernel methods in machine learning (see, e.g., [15–17]). In addition to these, the approach of continuous functions through sequences of linear positive operators, which have many applications in disciplines such as engineering and physics apart from mathematics, has also been an important research topic. In 1912, Bernstein [18] gave the definition, which was referred to by his name, for the proof of the Weierstrass approximation theorem. Some generalizations and modifications of Bernstein operators have been extensively studied in the literature. Bernstein–Chlodowsky [19] operators, which are generalizations of Bernstein operators over an unlimited set, are also one of the important generalizations, and many studies have been done involving Bernstein–Chlodowsky operators (see, e.g., [14, 20–23]). In approximation theory, the well-known Korovkin type theorems are based on linear positive approximation operators. However, in recent years, nonlinear positive operators have been introduced instead of linear positive operators. In the last century, researchers’ attention has turned to the fact that the operations that allow us to construct linear approximation operators do not have to be just addition and multiplication operations, and the approximation operators do not have to be linear. The maximum product type of operators that emerged as a solution to the open problems was defined for many operators and their approaches have been studied. In this study, nonlinear maximum product type of bivariate Bernstein–Chlodowsky operators is defined in this direction and the approximation properties are investigated with the help of new definitions.



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