Blending type approximation by GBS$GBS$ operators of ...

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Cai and Zhou Journal of Inequalities and Applications (2018) 2018:268 https://doi.org/10.1186/s13660-018-1862-0 RESEARCH Open Access Blending type approximation by GBS operators of bivariate tensor product of λ-Bernstein–Kantorovich type Qing-Bo Cai 1* and Guorong Zhou 2 * Correspondence: [email protected] 1 School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou, China Full list of author information is available at the end of the article Abstract In this paper, we introduce a family of GBS operators of bivariate tensor product of λ-Bernstein–Kantorovich type. We estimate the rate of convergence of such operators for B-continuous and B-differentiable functions by using the mixed modulus of smoothness, establish the Voronovskaja type asymptotic formula for the bivariate λ-Bernstein–Kantorovich operators, as well as give some examples and their graphs to show the effect of convergence. MSC: 41A10; 41A25; 41A36 Keywords: B-continuous functions; B-differentiable functions; GBS operators; Bernstein–Kantorovich operators; Mixed modulus of smoothness 1 Introduction In 1912, Bernstein [1] constructed a sequence of polynomials to prove the Weierstrass approximation theorem as follows: B n (f ; x)= n k=0 f k n b n,k (x), (1) for any continuous function f C[0, 1], where x [0, 1], n = 1, 2, . . . , and Bernstein basis functions b n,k (x) are defined by b n,k (x)= n k x k (1 – x) nk . (2) The polynomials in (1), called Bernstein polynomials, possess many remarkable proper- ties. Recently, Cai et al. [2] proposed a new type λ-Bernstein operators with parameter λ [–1, 1], they obtained some approximation properties and gave some graphs and nu- merical examples to show that these operators converge to continuous functions f . These © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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Cai and Zhou Journal of Inequalities and Applications (2018) 2018:268 https://doi.org/10.1186/s13660-018-1862-0

R E S E A R C H Open Access

Blending type approximation by GBSoperators of bivariate tensor product ofλ-Bernstein–Kantorovich typeQing-Bo Cai1* and Guorong Zhou2

*Correspondence: [email protected] of Mathematics andComputer Science, QuanzhouNormal University, Quanzhou, ChinaFull list of author information isavailable at the end of the article

AbstractIn this paper, we introduce a family of GBS operators of bivariate tensor product ofλ-Bernstein–Kantorovich type. We estimate the rate of convergence of suchoperators for B-continuous and B-differentiable functions by using the mixedmodulus of smoothness, establish the Voronovskaja type asymptotic formula for thebivariate λ-Bernstein–Kantorovich operators, as well as give some examples and theirgraphs to show the effect of convergence.

MSC: 41A10; 41A25; 41A36

Keywords: B-continuous functions; B-differentiable functions; GBS operators;Bernstein–Kantorovich operators; Mixed modulus of smoothness

1 IntroductionIn 1912, Bernstein [1] constructed a sequence of polynomials to prove the Weierstrassapproximation theorem as follows:

Bn(f ; x) =n∑

k=0

f(

kn

)bn,k(x), (1)

for any continuous function f ∈ C[0, 1], where x ∈ [0, 1], n = 1, 2, . . . , and Bernstein basisfunctions bn,k(x) are defined by

bn,k(x) =

(nk

)xk(1 – x)n–k . (2)

The polynomials in (1), called Bernstein polynomials, possess many remarkable proper-ties.

Recently, Cai et al. [2] proposed a new type λ-Bernstein operators with parameterλ ∈ [–1, 1], they obtained some approximation properties and gave some graphs and nu-merical examples to show that these operators converge to continuous functions f . These

© The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in anymedium, pro-vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, andindicate if changes were made.

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operators, which they called λ-Bernstein operators, are defined as follows:

Bn,λ(f ; x) =n∑

k=0

bn,k(λ; x)f(

kn

), (3)

where

⎧⎪⎪⎨

⎪⎪⎩

bn,0(λ; x) = bn,0(x) – λn+1 bn+1,1(x),

bn,i(λ; x) = bn,i(x) + λ( n–2i+1n2–1 bn+1,i(x) – n–2i–1

n2–1 bn+1,i+1(x)),

bn,n(λ; x) = bn,n(x) – λn+1 bn+1,n(x),

(4)

1 ≤ i ≤ n – 1, bn,k(x) (k = 0, 1, . . . , n) are defined in (2) and λ ∈ [–1, 1].In [3], Cai introduced the λ-Bernstein–Kantorovich operators as

Kn,λ(f ; x) = (n + 1)n∑

k=0

bn,k(λ; x)∫ k+1

n+1

kn+1

f (t) dt, (5)

where bn,k(λ, x) (k = 0, 1, . . . , n) are defined in (2) and λ ∈ [–1, 1]. He established a globalapproximation theorem in terms of second order modulus of continuity, obtained a di-rect approximation theorem by means of the Ditzian–Totik modulus of smoothness andderived an asymptotically estimate on the rate of convergence for certain absolutely con-tinuous functions. Very recently, Acu et al. provided a quantitative Voronovskaja type the-orem, a Grüss–Voronovskaja type theorem, and also gave some numerical examples of theoperators defined in (5) in [4].

As we know, the generalized Boolean sum operators (abbreviated by GBS operators)were first studied by Dobrescu and Matei in [5]. The Korovkin theorem for B-continuousfunctions was established by Badea et al. in [6, 7]. In 2013, Miclăuş [8] studied the approx-imation by the GBS operators of Bernstein–Stancu type. In 2016, Agrawal et al. [9] con-sidered the bivariate generalization of Lupaş–Durrmeyer type operators based on Pólyadistribution and studied the degree of approximation for the associated GBS operators.In 2017, Bărbosu et al. [10] introduced the GBS operators of Durrmeyer type based onq-integers, studied the uniform convergence theorem and the degree of approximationof these operators. Very recently, Kajla and Miclăuş [11] introduced the GBS operatorsof generalized Bernstein–Durrmeyer type and estimated the degree of approximation interms of the mixed modulus of smoothness.

Motivated by the above research, the aims of this paper are to propose the bivariatetensor product of λ-Bernstein–Kantorovich operators and the GBS operators of bivariatetensor product of λ-Bernstein–Kantorovich type. We use the mixed modulus of smooth-ness to estimate the rate of convergence of GBS operators of bivariate tensor product of λ-Bernstein–Kantorovich type for B-continuous and B-differentiable functions, and estab-lish a Voronovskaja type asymptotic formula for the bivariate λ-Bernstein–Kantorovichoperators. In order to show the effect of convergence, we also give some examples andgraphs.

This paper is mainly organized as follows: In Sect. 2, we introduce the bivariate ten-sor product of λ-Bernstein–Kantorovich operators Kλ1,λ2

m,n (f ; x, y) and the GBS operatorsUKλ1,λ2

m,n (f ; x, y). In Sect. 3, some lemmas are given to prove the main results. In Sect. 4,

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the rate of convergence for B-continuous and B-differentiable functions of GBS opera-tors UKλ1,λ2

m,n (f ; x, y) is proved. In Sect. 5, we investigate the Voronovskaja type asymptoticformula for bivariate operators Kλ1,λ2

m,n (f ; x, y).

2 Construction of operatorsFor f ∈ C(I2), I2 = [0, 1] × [0, 1], λ1,λ2 ∈ [–1, 1], we introduce the bivariate tensor productof λ-Bernstein–Kantorovich operators as

Kλ1,λ2m,n (f ; x, y)

= (m + 1)(n + 1)m∑

i=0

n∑

j=0

bm,i(λ1; x)bn,j(λ2; y)∫ i+1

m+1

im+1

∫ j+1n+1

jn+1

f (t, s) dt ds, (6)

where bm,i(λ1; x) (i = 0, 1, . . . , n) and bn,j(λ2; y) (j = 0, 1, . . . , n) are defined in (4), λ1,λ2 ∈[–1, 1]. Obviously, when λ1 = λ2 = 0, B0,0

m,n(f ; x, y) reduce to the bivariate tensor product ofclassical Bernstein–Kantorovich operators.

The GBS operators of the bivariate tensor product of λ-Bernstein–Kantorovich type aredefined as

UKλ1,λ2m,n

(f (t, s); x, y

)

= Kλ1,λ2m,n

(f (x, s) + f (t, y) – f (t, s); x, y

)

= (m + 1)(n + 1)m∑

i=0

n∑

j=0

bm,i(λ1; x)bn,j(λ2; y)∫ i+1

m+1

im+1

∫ j+1n+1

jn+1

[f (x, s) + f (t, y)

– f (t, s)]

ds dt, (7)

for f ∈ Cb(I2). Obviously, the operators UKλ1,λ2m,n (f ; x, y) are positive linear operators.

3 Auxiliary resultsIn order to obtain the main results, we need the following lemmas.

Lemma 3.1 ([4]) For λ-Bernstein–Kantorovich operators Kn,λ(f ; x) and n > 1, we have thefollowing equalities:

Kn,λ(1; x) = 1;

Kn,λ(t; x) = x +1 – 2x

2(n + 1)+

1 – 2x + xn+1 – (1 – x)n+1

n2 – 1λ;

Kn,λ(t2; x

)= x2 –

9nx2 – 6nx + 3x2 – 13(n + 1)2 +

2(–2x2n + xn+1n + xn + xn+1 – x)λ(n – 1)(n + 1)2 ;

Kn,λ(t3; x

)= x3 –

24n2x3 – 18n2x2 + 4nx3 + 18nx2 + 4x3 – 14nx – 14(n + 1)3

2(n + 1)3(n – 1)[–12n2x3 + 6n2x2 + 12x3n + 6xn+1n2 – 30x2n

+ 12xn+1n + 6xn + 7xn+1 – (1 – x)n+1 – 8x + 1];

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Kn,λ(t4; x

)=

15(n + 1)4

(5n5x4 – 30n3x4 + 40n3x3 + 55n2x4 – 120n2x3 – 30nx4

+ 75n2x2 + 80nx3 – 75nx2 + 30nx + 1)

+2λ

(n – 1)(n + 1)4

(–4n3x4

+ 2n3x3 + 12n2x4 – 24n2x3 – 8x4n + 2xn+1n3 + 6n2x2 + 22x3n

+ 6xn+1n2 – 24x2n + 3xn + 3xn+1 – 3x).

Lemma 3.2 ([4]) For λ-Bernstein–Kantorovich operators Kn,λ(f ; x) and n > 1, we have

Kn,λ(t – x; x) =1 – 2x

2(n + 1)+

1 – 2x + xn+1 – (1 – x)n+1

n2 – 1λ;

Kn,λ((t – x)2; x

)=

x(1 – x)n + 1

+1 – 6x + 6x2

3(n + 1)2 +2λ[xn+1(1 – x) + x(1 – x)n+1]

n2 – 1

–4x(1 – x)λ

(n + 1)2(n – 1).

Lemma 3.3 (See [4, Lemma 2.4]) We have

limn→∞ nKn,λ(t – x; x) =

12

– x;

limn→∞ nKn,λ

((t – x)2; x

)= x(1 – x), x ∈ (0, 1),

limn→∞ n2Kn,λ

((t – x)4; x

)= O(1), x ∈ (0, 1).

Lemma 3.4 For the bivariate tensor product of λ-Bernstein–Kantorovich operatorsKλ1,λ2

m,n (f ; x, y), we have the following inequalities:

Kλ1,λ2m,n

((t – x)2; x, y

) ≤ 2m + 1

;

Kλ1,λ2m,n

((s – y)2; x, y

) ≤ 2n + 1

;

Kλ1,λ2m,n

((t – x)2(s – y)2; x, y

) ≤ 4(m + 1)(n + 1)

;

Kλ1,λ2m,n

((t – x)4(s – y)2; x, y

) ≤ C(m + 1)2(n + 1)

;

Kλ1,λ2m,n

((t – x)2(s – y)4; x, y

) ≤ C(m + 1)(n + 1)2 ,

where C is a positive constant.

4 Rate of convergenceWe first introduce the definitions of B-continuity and B-differentiability, details can befound in [12] and [13]. Let X and Y be compact real intervals. A function f : X × Y → R

is called a B-continuous function at (x0, y0) ∈ X × Y if

lim(x,y)→(x0,y0)

�f((x, y), (x0, y0)

)= 0,

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where �f ((x, y), (x0, y0)) = f (x, y) – f (x0, y) – f (x, y0) + f (x0, y0) denotes the mixed differenceof f . A function f : X × Y → R is a B-differentiable function at (x0, y0) ∈ X × Y if thefollowing limit exists and is finite:

lim(x,y)→(x0,y0)

�f ((x, y), (x0, y0))(x – x0)(y – y0)

.

The limit is named the B-differential of f at the point (x0, y0) and denoted by DBf (x0, y0).The function f : X × Y → R is B-bounded on X × Y if there exists a k > 0 such that

|�f ((x, y), (t, s))| ≤ K for any (x, y), (t, s) ∈ X × Y .Let B(X ×Y ), C(X ×Y ) denote the spaces of all bounded functions and of all continuous

functions on X × Y endowed with the sup-norm ‖ · ‖∞, respectively. We also define thefollowing function sets:

Bb(X × Y ) = {f : X × Y →R|f is B-bounded on X × Y }

with the norm ‖f ‖B = sup(x,y),(t,s)∈X×Y |�f ((x, y), (t, s))|,

Cb(X × Y ) = {f : X × Y →R|f is B-continuous on X × Y },

and Db(X × Y ) = {f : X × Y → R|f is B-differentiable on X × Y }. It is known that C(X ×Y ) ⊂ Cb(X × Y ).

Let f ∈ Bb(X × Y ). Then the mixed modulus of smoothness ωmixed(f ; ·, ·) is defined by

ωmixed(f ; δ1, δ2) = sup{∣∣�f

((x, y), (t, s)

)∣∣ : |x – t| ≤ δ1, |y – s| ≤ δ2}

,

for any δ1, δ2 ≥ 0.Let L : Cb(X × Y ) → B(X × Y ) be a linear positive operator. The operator UL : Cb(X ×

Y ) → B(X × Y ) defined for any function f ∈ Cb(X × Y ) and any (x, y) ∈ X × Y byUL(f (t, s); x, y) = L(f (t, y) + f (x, s) – f (t, s); x, y) is called the GBS operator associated to theoperator L.

In the sequel, we will consider functions eij : X × Y → R, eij(x, y) = xiyj for any (x, y) ∈X × Y , and i, j ∈N. In order to estimate the rate of convergence of UKλ1,λ2

m,n (f ; x, y), we needthe following two theorems.

Theorem 4.1 ([7]) Let L : Cb(X × Y ) → B(X × Y ) be a linear positive operator and UL :Cb(X × Y ) → B(X × Y ) the associated GBS operator. Then for any f ∈ Cb(X × Y ), any(x, y) ∈ (X × Y ) and δ1, δ2 > 0, we have

∣∣UL(f (t, s); x, y

)– f (x, y)

∣∣

≤ ∣∣f (x, y)∣∣∣∣1 – L(e00; x, y)

∣∣ +[L(e00; x, y) + δ–1

1

√L((t – x)2; x, y

)

+ δ–12

√L((s – y)2; x, y

)+ δ–1

1 δ–12

√L((t – x)2(s – y)2; x, y

)]ωmixed(f ; δ1, δ2).

Theorem 4.2 ([14]) Let L : Cb(X × Y ) → B(X × Y ) be a linear positive operator and UL :Cb(X × Y ) → B(X × Y ) the associated GBS operator. Then for any f ∈ Db(X × Y ) with

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DBf ∈ B(X × Y ), any (x, y) ∈ (X × Y ) and δ1, δ2 > 0, we have

∣∣UL(f (t, s); x, y

)– f (x, y)

∣∣

≤ ∣∣f (x, y)∣∣∣∣1 – L(e00; x, y)

∣∣ + 3‖DBf ‖∞√

L((t – x)2(s – y)2; x, y

)

+[√

L((t – x)2(s – y)2; x, y

)+ δ–1

1

√L((t – x)4(s – y)2; x, y

)

+ δ–12

√L((t – x)2(s – y)4; x, y

)+ δ–1

1 δ–12 L

((t – x)2(s – y)2; x, y

)]

× ωmixed(DBf ; δ1, δ2).

First, we will use B-continuous functions to estimate the rate of convergence ofUKλ1,λ2

m,n (f ; x, y) to f ∈ Cb(I2) by using the mixed modulus of smoothness. We have

Theorem 4.3 For f ∈ Cb(I2), (x, y) ∈ I2 and m, n > 1, we have the following inequality:

∣∣UKλ1,λ2m,n (f ; x, y) – f (x, y)

∣∣ ≤ (3 + 2√

2)ωmixed

(f ;

1√m + 1

,1√

n + 1

). (8)

Proof Applying Theorem 4.1 and using Lemma 3.4, we get

∣∣UKλ1,λ2m,n (f ; x, y) – f (x, y)

∣∣

≤[

1 +1δ1

√2

m + 1+

1δ2

√2

n + 1+

2δ1δ2

√(m + 1)(n + 1)

]ωmixed(f ; δ1, δ2).

Therefore, (8) can be obtained from the above inequality by choosing δ1 = 1√m+1 and δ2 =

1√n+1 . �

Next, we will give the rate of convergence to the B-differentiable functions forUKλ1,λ2

m,n (f ; x, y).

Theorem 4.4 Let f ∈ Db(I2), DBf ∈ B(I2), (x, y) ∈ I2 and m, n > 1, we have the followinginequality:

∣∣UKλ1,λ2m,n (f ; x, y) – f (x, y)

∣∣

≤ M√(m + 1)(n + 1)

[‖DBf ‖∞ + ωmixed

(DBf ;

1√m + 1

,1√

n + 1

)], (9)

where C and M are positive constants.

Proof Using Theorem 4.2 and Lemma 3.4, we have

∣∣UKλ1,λ2m,n (f ; x, y) – f (x, y)

∣∣

≤ 6‖DBf ‖∞√(m + 1)(n + 1)

+[

2√(m + 1)(n + 1)

+1

δ1(m + 1)

√C

n + 1

+1

δ2(n + 1)

√C

m + 1+

4δ1δ2(m + 1)(n + 1)

]ωmixed(DBf ; δ1, δ2).

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Figure 1 Graphs of UK1,110,10(f (s, t); x, y) and f (x, y)

Hence, taking δ1 = 1√m+1 , δ2 = 1√

n+1 and using the above inequality, we get the desired result(9). �

Example 4.5 Let f (x, y) = xy + x2, x, y ∈ [0, 1], the graphs of f (x, y) and UK1,110,10(f (s, t); x, y)

are shown in Fig. 1. Figure 2 shows the partially enlarged graphs of f (x, y) andUK1,1

10,10(f (s, t); x, y).

5 Voronovskaja type asymptotic formulas for Kλ1,λ2m,n (f ; x, y)

In this section, we will give a Voronovskaja type asymptotic formula for Kλ1,λ2m,n (f ; x, y).

Theorem 5.1 Consider an f ∈ C(I2). Then for any x, y ∈ (0, 1) and λ1,λ2 ∈ [–1, 1], we have

limn→∞ n

[Kλ1,λ2

n,n (f ; x, y) – f (x, y)]

=f ′x(x, y)

2(1 – 2x) +

f ′y (x, y)

2(1 – 2y) +

12[f ′′x2 (x, y)x(1 – x) + f ′′

y2 (x, y)y(1 – y)].

Proof For (x, y), (t, s) ∈ I2, by Taylor’s expansion, we have

f (t, s)

= f (x, y) + f ′x(x, y)(t – x) + f ′

y (x, y)(s – y) +12[f ′′x2 (x, y)(t – x)2 + 2f ′′

xy(x, y)

× (t – x)(s – y) + f ′′y2 (x, y)(s – y)2] + ρ(t, s; x, y)

√(t – x)4 + (s – y)4, (10)

where ρ(t, s; x, y) ∈ C(I2) and lim(t,s)→(x,y) ρ(t, s; x, y) = 0.

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Figure 2 Partially enlarged graphs of UK1,110,10(f (s, t); x, y) and f (x, y)

Applying Kλ1,λ2n,n (f ; x) to (10), we obtain

Kλ1,λ2n,n (f ; x, y) = f (x, y) + f ′

x(x, y)Kn,λ1 (t – x; x) + f ′y (x, y)Kn,λ2 (s – y; y)

+12[f ′′x2 (x, y)Kn,λ1

((t – x)2; x

)+ f ′′

y2 (x, y)Kn,λ2

((s – y)2; y

)

+ 2f ′′xy(x, y)Kλ1,λ2

n,n((t – x)(s – y); x, y

)]

+ Kλ1,λ2n,n

(ρ(t, s; x, y)

√(t – x)4 + (s – y)4; x, y

).

Taking the limit on both sides of the above equality, we have

limn→∞ n

[Kλ1,λ2

n,n (f ; x, y) – f (x, y)]

= f ′x(x, y) lim

n→∞ nKn,λ1 (t – x; x) + f ′y (x, y) lim

n→∞ nKn,λ2 (s – y; y)

+12

[f ′′x2 (x, y) lim

n→∞ nKn,λ1

((t – x)2; x

)

+ f ′′y2 (x, y) lim

n→∞ nKn,λ2

((s – y)2; y

)

+ 2f ′′xy(x, y) lim

n→∞ nKλ1,λ2n,n

((t – x)(s – y); x, y

)]

+ limn→∞ nKλ1,λ2

n,n(ρ(t, s; x, y)

√(t – x)4 + (s – y)4; x, y

). (11)

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Using Lemma 3.2, we have

limn→∞ nKλ1,λ2

n,n((t – x)(s – y); x, y

)= lim

n→∞ n[Kn,λ1 (t – x; x)Kn,λ2 (s – y; y)

]= 0. (12)

By Cauchy–Schwarz inequality, we have

nKλ1,λ2n,n

(ρ(t, s; x, y)

√(t – x)4 + (s – y)4; x, y

)

≤√

Kλ1,λ2n,n

(ρ2(t, s; x, y); x, y

)√n2Kλ1,λ2

n,n((t – x)4 + (s – y)4; x, y

)

≤√

Kλ1,λ2n,n

(ρ2(t, s; x, y); x, y

)

×√

n2Kn,λ1

((t – x)4; x

)+ n2Kn,λ2

((s – y)4; y

).

Since lim(t,s)→(x,y) ρ(t, s; x, y) = 0, using Lemma 3.3, we obtain

limn→∞ nKλ1,λ2

n,n(ρ(t, s; x, y)

√(t – x)4 + (s – y)4; x, y

)= 0. (13)

Therefore, by (11), (12), (13) and Lemma 3.3, we have

limn→∞ n

[Kλ1,λ2

n,n (f ; x, y) – f (x, y)]

=f ′x(x, y)

2(1 – 2x) +

f ′y (x, y)

2(1 – 2y) +

12[f ′′x2 (x, y)x(1 – x) + f ′′

y2 (x, y)y(1 – y)].

Thus we have obtained the desired result. �

Example 5.2 Consider the function f (x, y) = xy + x2, x, y ∈ [0, 1]. The graphs of f (x, y)and K1,1

20,20(f ; x, y) are shown in Fig. 3. We also give the graphs of K1,110,10(f ; x, y) and

Figure 3 Graphs of K1,120,20(f ; x, y) and f (x, y)

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Figure 4 Graphs of K1,110,10(f ; x, y) and UK1,110,10(f (s, t); x, y)

UK1,110,10(f (s, t); x, y) in Fig. 4 to compare the bivariate λ-Bernstein–Kantorovich operators

with GBS operators.

6 ConclusionIn this paper, we deduce the rate of convergence of GBS operators of bivariate tensor prod-uct of λ-Bernstein–Kantorovich type for B-continuous and B-differentiable functions byusing the mixed modulus of smoothness, as well as obtain the Voronovskaja type asymp-totic formula for bivariate λ-Bernstein–Kantorovich operators.

AcknowledgementsWe thank Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing andInformation Processing of Fujian Province University.

FundingThis work is supported by the National Natural Science Foundation of China (Grant No. 11601266), the Natural ScienceFoundation of Fujian Province of China (Grant No. 2016J05017) and the Program for New Century Excellent Talents inFujian Province University.

Availability of data and materialsAll data generated or analyzed during this study are included in this published article.

Competing interestsThe authors declare that they have no competing interests.

Authors’ contributionsThe authors wrote the whole manuscript. All authors read and approved the final manuscript.

Author details1School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou, China. 2School of AppliedMathematics, Xiamen University of Technology, Xiamen, China.

Publisher’s NoteSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 6 April 2018 Accepted: 20 September 2018

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