Blending type approximation by GBS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$GBS$\end{document} operators of bivariate tensor product of λ-Bernstein–Kantorovich type

In this paper, we introduce a family of GBS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$GBS$\end{document} 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.

In [3], Cai introduced the λ-Bernstein-Kantorovich operators as where b n,k (λ, x) (k = 0, 1, . . . , n) are defined in (2) and λ ∈ [-1, 1]. He established a global approximation theorem in terms of second order modulus of continuity, obtained a direct approximation theorem by means of the Ditzian-Totik modulus of smoothness and derived an asymptotically estimate on the rate of convergence for certain absolutely continuous functions. Very recently, Acu et al. provided a quantitative Voronovskaja type theorem, a Grüss-Voronovskaja type theorem, and also gave some numerical examples of the operators 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-continuous functions was established by Badea et al. in [6,7]. In 2013, Miclăuş [8] studied the approximation by the GBS operators of Bernstein-Stancu type. In 2016, Agrawal et al. [9] considered the bivariate generalization of Lupaş-Durrmeyer type operators based on Pólya distribution 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 on q-integers, studied the uniform convergence theorem and the degree of approximation of these operators. Very recently, Kajla and Miclăuş [11] introduced the GBS operators of generalized Bernstein-Durrmeyer type and estimated the degree of approximation in terms of the mixed modulus of smoothness.
Motivated by the above research, the aims of this paper are to propose the bivariate tensor product of λ-Bernstein-Kantorovich operators and the GBS operators of bivariate tensor product of λ-Bernstein-Kantorovich type. We use the mixed modulus of smoothness 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 establish a Voronovskaja type asymptotic formula for the bivariate λ-Bernstein-Kantorovich operators. In order to show the effect of convergence, we also give some examples and graphs.
This paper is mainly organized as follows: In Sect. 2, we introduce the bivariate tensor product of λ-Bernstein-Kantorovich operators K λ 1 ,λ 2 m,n (f ; x, y) and the GBS operators UK λ 1 ,λ 2 m,n (f ; x, y). In Sect. 3, some lemmas are given to prove the main results. In Sect. 4, the rate of convergence for B-continuous and B-differentiable functions of GBS operators UK λ 1 ,λ 2 m,n (f ; x, y) is proved. In Sect. 5, we investigate the Voronovskaja type asymptotic formula for bivariate operators K λ 1 ,λ 2 m,n (f ; x, y).
The GBS operators of the bivariate tensor product of λ-Bernstein-Kantorovich type are defined as for f ∈ C b (I 2 ). Obviously, the operators UK λ 1 ,λ 2 m,n (f ; x, y) are positive linear operators.

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

Rate of convergence
We first introduce the definitions of B-continuity and B-differentiability, details can be found in [12] and [13]. Let X and Y be compact real intervals. A function f : function at (x 0 , y 0 ) ∈ X × Y if the following limit exists and is finite: , y), (x 0 , y 0 )) (xx 0 )(yy 0 ) .
The limit is named the B-differential of f at the point (x 0 , y 0 ) and denoted by D B f (x 0 , y 0 ).
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 the following function sets: Then the mixed modulus of smoothness ω mixed (f ; ·, ·) is defined by for any δ 1 , δ 2 ≥ 0.
Let L : C b (X × Y ) → B(X × Y ) be a linear positive operator. The operator UL : C b (X × Y ) → B(X × Y ) defined for any function f ∈ C b (X × Y ) and any (x, y) ∈ X × Y by UL(f (t, s); x, y) = L(f (t, y) + f (x, s)f (t, s); x, y) is called the GBS operator associated to the operator L.
In the sequel, we will consider functions e ij : X × Y → R, e ij (x, y) = x i y j for any (x, y) ∈ X × Y , and i, j ∈ N. In order to estimate the rate of convergence of UK λ 1 ,λ 2 m,n (f ; x, y), we need the following two theorems.
be a linear positive operator and UL : First, we will use B-continuous functions to estimate the rate of convergence of UK λ 1 ,λ 2 m,n (f ; x, y) to f ∈ C b (I 2 ) by using the mixed modulus of smoothness. We have , (x, y) ∈ I 2 and m, n > 1, we have the following inequality: Proof Applying Theorem 4.1 and using Lemma 3.4, we get Therefore, (8) can be obtained from the above inequality by choosing δ 1 = 1 √ m+1 and δ 2 = 1 √ n+1 .
Next, we will give the rate of convergence to the B-differentiable functions for UK λ 1 ,λ 2 m,n (f ; x, y).

Theorem 4.4 Let f ∈ D b (I 2 )
, D B f ∈ B(I 2 ), (x, y) ∈ I 2 and m, n > 1, we have the following inequality: where C and M are positive constants.

Conclusion
In this paper, we deduce the rate of convergence of GBS operators of bivariate tensor product of λ-Bernstein-Kantorovich type for B-continuous and B-differentiable functions by using the mixed modulus of smoothness, as well as obtain the Voronovskaja type asymptotic formula for bivariate λ-Bernstein-Kantorovich operators.