Open Access

Bivariate tensor product \((p, q)\)-analogue of Kantorovich-type Bernstein-Stancu-Schurer operators

Journal of Inequalities and Applications20172017:284

https://doi.org/10.1186/s13660-017-1559-9

Received: 18 July 2017

Accepted: 8 November 2017

Published: 14 November 2017

Abstract

In this paper, we construct a bivariate tensor product generalization of Kantorovich-type Bernstein-Stancu-Schurer operators based on the concept of \((p, q)\)-integers. We obtain moments and central moments of these operators, give the rate of convergence by using the complete modulus of continuity for the bivariate case and estimate a convergence theorem for the Lipschitz continuous functions. We also give some graphs and numerical examples to illustrate the convergence properties of these operators to certain functions.

Keywords

\((p, q)\)-integersBernstein-Stancu-Schurer operatorsmodulus of continuityLipschitz continuous functionsbivariate tensor product

MSC

41A1041A2541A36

1 Introduction

In recent years, \((p, q)\)-integers have been introduced to linear positive operators to construct new approximation processes. A sequence of \((p, q)\)-analogue of Bernstein operators was first introduced by Mursaleen [1, 2]. Besides, \((p, q)\)-analogues of Szász-Mirakyan operators [3], Baskakov-Kantorovich operators [4], Bleimann-Butzer-Hahn operators [5] and Kantorovich-type Bernstein-Stancu-Schurer operators [6] were also considered. For further developments, one can also refer to [712]. These operators are double parameters corresponding to p and q versus single parameter q-type operators [1316]. The aim of these generalizations is to provide appropriate and powerful tools to application areas such as numerical analysis, CAGD and solutions of differential equations (see, e.g., [17]).

Motivated by all the above results, in 2016, Cai et al. [6] introduced a new kind of Kantorovich-type Bernstein-Stancu-Schurer operators based on \((p, q)\)-integers as follows:
$$ K_{n,p,q}^{\alpha,\beta,l}(f;x)=\bigl([n+1]_{p,q}+\beta\bigr)\sum _{k=0}^{n+l}\frac {b_{n+l,k}(p,q;x)}{[k+1]_{p,q}-[k]_{p,q}} \int_{\frac{[k]_{p,q}+\alpha }{[n+1]_{p,q}+\beta}}^{\frac{[k+1]_{p,q}+\alpha}{[n+1]_{p,q}+\beta }}f(t)\, d_{p,q}t, $$
(1)
where \(b_{n+l,k}(p,q;x)=\bigl [ {\scriptsize\begin{matrix}{} n+l\cr k \end{matrix}} \bigr ]_{p,q}x^{k}(1-x)_{p,q}^{n+l-k} \) for \(f\in C(I)\), \(I=[0,1+l]\), \(l\in\mathbb{N}\), \(0\leq\alpha\leq\beta\), \(0< q< p\leq1\) and \(n\in\mathbb{N}\). They got some approximation properties, since convergence properties of bivariate operators are important in approximation theory, and it seems there has been no papers mentioning the bivariate forms of above operators (1). Hence, we will propose the bivariate case in the following. Before doing this, in [6] (Lemma 2.1), they got \(K_{n,p,q}^{\alpha,\beta,l}(1;x)=1\), that is, the operators reproduce constant functions. However, this conclusion is incorrect. In fact, \(\sum_{k=0}^{n+l}b_{n+l,k}(p,q;x)\neq1\). Hence, we re-introduce the revised operators as
$$ K_{n,p,q}^{\alpha,\beta,l}(f;x)=\bigl([n+1]_{p,q}+\beta\bigr)\sum _{k=0}^{n+l}\frac {\widetilde{b_{n+l,k}}(p,q;x)}{[k+1]_{p,q}-[k]_{p,q}} \int_{\frac {[k]_{p,q}+\alpha}{[n+1]_{p,q}+\beta}}^{\frac{[k+1]_{p,q}+\alpha }{[n+1]_{p,q}+\beta}}f(t)\, d_{p,q}t, $$
(2)
where
$$ \widetilde{b_{n+l,k}}(p,q;x)=\frac{1}{p^{\frac{(n+l)(n+l-1)}{2}}}\left [ \textstyle\begin{array}{@{}c@{}} n+l\\ k \end{array}\displaystyle \right ]_{p,q}p^{\frac{k(k-1)}{2}}x^{k}(1-x)_{p,q}^{n+l-k}. $$
(3)
From [2], we know \(\sum_{k=0}^{n+l}\widetilde {b_{n+l,k}}(p,q;x)=1\), and this ensures the operators reproduce constant functions.
On this basis, let \(C(I^{2})\) denote the space of all real-valued continuous functions on \(I^{2}\) endowed with the norm \(\|f\| _{I^{2}}=\sup_{(x,y)\in I^{2}}|f(x,y)|\). For \(f\in C(I^{2})\), \(I^{2}=I\times I=[0,1+l]\times[0,1+l]\), \(l\in\mathbb{N}\), \(0\leq\alpha\leq\beta\), \(0< q_{n_{1}}, q_{n_{2}}< p_{n_{1}}, p_{n_{2}}\leq1\) and \(n_{1}, n_{2}\in\mathbb {N}\). We propose the bivariate tensor product \((p, q)\)-analogue of Kantorovich-type Bernstein-Stancu-Schurer operators as follows:
$$\begin{aligned}& {K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}}(f;x,y) \\& \quad = \bigl([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta \bigr) \bigl([n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta \bigr) \\& \qquad {}\times\sum_{k_{1}=0}^{n_{1}+l}\sum _{k_{2}=0}^{n_{2}+l}\frac {b_{n_{1}+l,n_{2}+l,k_{1},k_{2}}^{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}(x,y)}{ ([k_{1}+1]_{p_{n_{1}},q_{n_{1}}}-[k_{1}]_{p_{n_{1}},q_{n_{1}}} ) ([k_{2}+1]_{p_{n_{2}},q_{n_{2}}}-[k_{2}]_{p_{n_{2}},q_{n_{2}}} )} \\& \qquad {}\times \int_{\frac{[k_{1}]_{p_{n_{1}},q_{n_{1}}}+\alpha }{[n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta}}^{\frac {[k_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\alpha}{[n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta}} \int _{\frac{[k_{2}]_{p_{n_{2}},q_{n_{2}}}+\alpha}{[n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta }}^{\frac{[k_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\alpha }{[n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta }}f(t,s)\, d_{p_{n_{1}},q_{n_{1}}}t\, d_{p_{n_{2}},q_{n_{2}}}s, \end{aligned}$$
(4)
where
$$\begin{aligned}& b_{n_{1}+l,n_{2}+l,k_{1},k_{2}}^{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}(x,y) \\& \quad =\frac{1}{p_{n_{1}}^{\frac{(n_{1}+l)(n_{1}+l-1)}{2}}p_{n_{2}}^{\frac {(n_{2}+l)(n_{2}+l-1)}{2}}}\left [ \textstyle\begin{array}{@{}c@{}} n_{1}+l\\ k_{1} \end{array}\displaystyle \right ]_{p_{n_{1}},q_{n_{1}}}\left [ \textstyle\begin{array}{@{}c@{}} n_{2}+l\\ k_{2} \end{array}\displaystyle \right ]_{p_{n_{2}},q_{n_{2}}} \\& \qquad {}\times p_{n_{1}}^{\frac{k_{1}(k_{1}-1)}{2}}p_{n_{2}}^{\frac {k_{2}(k_{2}-1)}{2}}x^{k_{1}}y^{k_{2}}(1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-k_{1}}(1-y)_{p_{n_{2}},q_{n_{2}}}^{n_{2}+l-k_{2}} \end{aligned}$$
(5)
for \(x, y\in[0,1]\).

The paper is organized as follows. The following section contains some basic definitions regarding \((p, q)\)-integers and \((p, q)\)-calculus. In Section 3, we estimate the moments and central moments of the revised operators (2) and then deduce the corresponding results of a bivariate case. In Section 4, we give the rate of convergence by using the modulus of continuity and estimate a convergent theorem for the Lipschitz continuous functions. In Section 5, we give some graphs and numerical examples to illustrate the convergence properties of operators (4) to certain functions.

2 Some notations

We mention some definitions based on \((p, q)\)-integers, details can be found in [1822]. For any fixed real number \(0< q< p\leq 1\) and each nonnegative integer k, we denote \((p, q)\)-integers by \([k]_{p,q}\), where
$$ [k]_{p,q}=\frac{p^{k}-q^{k}}{p-q}. $$
Also \((p, q)\)-factorial and \((p, q)\)-binomial coefficients are defined as follows:
$$\begin{aligned}& [k]_{p,q}!= \textstyle\begin{cases} [k]_{p,q}[k-1]_{p,q}\cdots[1]_{p,q}, &k=1,2,\ldots, \\ 1, &k=0, \end{cases}\displaystyle \\& \left [ \textstyle\begin{array}{@{}c@{}} n\\ k \end{array}\displaystyle \right ]_{p,q}= \frac{[n]_{p,q}!}{[k]_{p,q}![n-k]_{p,q}!}\quad (n\geq k\geq0). \end{aligned}$$
The \((p, q)\)-Binomial expansion is defined by
$$ (x+y)_{p,q}^{n}= \textstyle\begin{cases} 1, &n=0, \\ (x+y)(px+qy)\cdots (p^{n-1}x+q^{n-1}y ), &n=1,2,\ldots. \end{cases} $$
The definite \((p, q)\)-integrals are defined by
$$\begin{aligned}& \int_{0}^{a}f(x)\, d_{p,q}x=(p-q)a\sum _{k=0}^{\infty}\frac {q^{k}}{p^{k+1}}f \biggl( \frac{q^{k}}{p^{k+1}}a \biggr)\quad \mbox{and} \\& \int_{0}^{a_{1}} \int _{0}^{a_{2}}f(x,y)\, d_{p_{n_{1}},q_{n_{1}}}x\, d_{p_{n_{2}},q_{n_{2}}}y \\& \quad = (p_{n_{1}}-q_{n_{1}} ) (p_{n_{2}}-q_{n_{2}} )a_{1}a_{2} \sum_{k_{1}=0}^{\infty} \sum_{k_{2}=0}^{\infty}\frac {q_{n_{1}}^{k_{1}}}{p_{n_{1}}^{k_{1}+1}} \frac {q_{n_{2}}^{k_{2}}}{p_{n_{2}}^{k_{2}+1}}f \biggl(\frac {q_{n_{1}}^{k_{1}}}{p_{n_{1}}^{k_{1}+1}}a_{1}, \frac {q_{n_{2}}^{k_{2}}}{p_{n_{2}}^{k_{2}+1}}a_{2} \biggr). \end{aligned}$$
When \(p=1\), all the definitions of \((p, q)\)-calculus above are reduced to q-calculus.

3 Auxiliary results

In order to obtain the convergence properties, we need the following lemmas.

Lemma 3.1

For the \((p, q)\)-analogue of Kantorovich-type Bernstein-Stancu-Schurer operators (2), we have
$$\begin{aligned}& {K_{n,p,q}^{\alpha,\beta,l}}(1;x)=1, \end{aligned}$$
(6)
$$\begin{aligned}& {K_{n,p,q}^{\alpha,\beta,l}}(t;x)=\frac {(1+q)[n+l]_{p,q}(px+1-x)_{p,q}^{n+l-1}}{[2]_{p,q} ([n+1]_{p,q}+\beta )p^{n+l-1}}x +\frac{(px+1-x)_{p,q}^{n+l}+2\alpha}{[2]_{p,q} ([n+1]_{p,q}+\beta )} , \end{aligned}$$
(7)
$$\begin{aligned}& {K_{n,p,q}^{\alpha,\beta,l}} \bigl(t^{2};x \bigr) \\& \quad = \frac{ (q+q^{2}+q^{3} )[n+l]_{p,q}[n+l-1]_{p,q} (p^{2}x+1-x )_{p,q}^{n+l-2}}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}p^{2n+2l-4}}x^{2} \\& \qquad{} +\frac{ (1+q+q^{2}+p+2pq )[n+l]_{p,q} (p^{2}x+1-x )_{p,q}^{n+l-1}x}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}p^{n+l-2}} \\& \qquad {}+\frac{3\alpha(1+q)[n+l]_{p,q}(px+1-x)_{p,q}^{n+l-1}x}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}p^{n+l-1}}+\frac{ (p^{2}x+1-x )_{p,q}^{n+l}}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}} \\& \qquad {}+\frac{3\alpha (px+1-x )_{p,q}^{n+l}+3{\alpha }^{2}}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}}. \end{aligned}$$
(8)

Proof

Since \(\sum_{k=0}^{n+l}{b_{n+l,k}}(p,q;x)=1\), (6) is easily obtained. Using (2) and \([k+1]_{p,q}=p^{k}+q[k]_{p,q}\), we have
$$\begin{aligned}& {K_{n,p,q}^{\alpha,\beta,l}}(t;x) \\& \quad = \bigl([n+1]_{p,q}+\beta\bigr)\sum_{k=0}^{n+l} \frac {b_{n+l,k}(p,q;x)}{[k+1]_{p,q}-[k]_{p,q}} \int_{\frac{[k]_{p,q}+\alpha }{[n+1]_{p,q}+\beta}}^{\frac{[k+1]_{p,q}+\alpha}{[n+1]_{p,q}+\beta }}t\, d_{p,q}t \\& \quad = \bigl([n+1]_{p,q}+\beta\bigr)\sum_{k=0}^{n+l} \frac {b_{n+l,k}(p,q;x)}{[k+1]_{p,q}-[k]_{p,q}}\frac{([k+1]_{p,q}+\alpha )^{2}-([k]_{p,q}+\alpha)^{2}}{[2]_{p,q}([n+1]_{p,q}+\beta)^{2}} \\& \quad = \frac{1}{[2]_{p,q}([n+1]_{p,q}+\beta)}\sum_{k=0}^{n+l}b_{n+l,k}(p,q;x) \bigl([k+1]_{p,q}+[k]_{p,q}+2\alpha\bigr) \\& \quad = \frac{(1+q)[n+l]_{p,q}x}{[2]_{p,q} ([n+1]_{p,q}+\beta )p^{n+l-1}p^{\frac{(n+l-1)(n+l-2)}{2}}}\sum_{k=0}^{n+l-1} \left [ \textstyle\begin{array}{@{}c@{}} n+l-1 \\ k \end{array}\displaystyle \right ]_{p,q}p^{\frac{k(k-1)}{2}} \\& \qquad {}\times(px)^{k}(1-x)_{p,q}^{n+l-k-1}+ \frac{(px+1-x)_{p,q}^{n+l}+2\alpha }{[2]_{p,q} ([n+1]_{p,q}+\beta )} \\& \quad = \frac{(1+q)[n+l]_{p,q}(px+1-x)_{p,q}^{n+l-1}}{[2]_{p,q} ([n+1]_{p,q}+\beta )p^{n+l-1}}x+\frac{(px+1-x)_{p,q}^{n+l}+2\alpha }{[2]_{p,q} ([n+1]_{p,q}+\beta )}. \end{aligned}$$
Thus, (7) is proved. Finally, from (2), we get
$$\begin{aligned}& {K_{n,p,q}^{\alpha,\beta}} \bigl(t^{2};x \bigr) \\& \quad = \bigl([n+1]_{p,q}+\beta\bigr)\sum_{k=0}^{n+l} \frac {b_{n+l,k}(p,q;x)}{[k+1]_{p,q}-[k]_{p,q}} \int_{\frac{[k]_{p,q}+\alpha }{[n+1]_{p,q}+\beta}}^{\frac{[k+1]_{p,q}+\alpha}{[n+1]_{p,q}+\beta }}t^{2}\, d_{p,q}t \\& \quad = \frac{ ([n+1]_{p,q}+\beta )}{[3]_{p,q}}\sum_{k=0}^{n+l} \frac {b_{n+l,k}(p,q;x)}{[k+1]_{p,q}-[k]_{p,q}}\frac{([k+1]_{p,q}+\alpha )^{3}-([k]_{p,q}+\alpha)^{3}}{([n+1]_{p,q}+\beta)^{3}} \\& \quad = \frac{1}{[3]_{p,q}([n+1]_{p,q}+\beta)^{2}}\sum_{k=0}^{n+l} \bigl([k+1]_{p,q}^{2}+[k]_{p,q}^{2}+[k+1]_{p,q}[k]_{p,q} \\& \qquad {} +3\alpha[k+1]_{p,q}+3\alpha[k]_{p,q}+3{ \alpha}^{2}\bigr)b_{n+l,k}(p,q;x). \end{aligned}$$
Since \([k+1]_{p,q}=p^{k}+q[k]_{p,q}\), by some computations, we get
$$\begin{aligned} \begin{aligned} &[k+1]_{p,q}^{2}+[k]_{p,q}^{2}+[k+1]_{p,q}[k]_{p,q}+3 \alpha [k+1]_{p,q}+3\alpha[k]_{p,q}+3{\alpha}^{2} \\ &\quad = \bigl(q+q^{2}+q^{3} \bigr)[k]_{p,q}[k-1]_{p,q}+ \biggl(1+2q+\frac {1+q+q^{2}}{p} \biggr)p^{k}[k]_{p,q} \\ &\qquad {}+3\alpha (1+q)[k]_{p,q}+p^{2k}+3\alpha p^{k}+3{\alpha}^{2}. \end{aligned} \end{aligned}$$
So, we can obtain
$$\begin{aligned}& {K_{n,p,q}^{\alpha,\beta,l}} \bigl(t^{2};x \bigr) \\& \quad = \frac{ (q+q^{2}+q^{3} )[n+l]_{p,q}[n+l-1]_{p,q} (p^{2}x+1-x )_{p,q}^{n+l-2}}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}p^{2n+2l-4}}x^{2} \\& \qquad {} +\frac{ (1+q+q^{2}+p+2pq )[n+l]_{p,q} (p^{2}x+1-x )_{p,q}^{n+l-1}x}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}p^{n+l-2}} \\& \qquad {} +\frac{3\alpha(1+q)[n+l]_{p,q}(px+1-x)_{p,q}^{n+l-1}x}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}p^{n+l-1}}+\frac{ (p^{2}x+1-x )_{p,q}^{n+l}}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}} \\& \qquad {} +\frac{3\alpha (px+1-x )_{p,q}^{n+l}+3{\alpha }^{2}}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}}. \end{aligned}$$
Thus, (8) is proved. □

Lemma 3.2

Using Lemma 3.1 and easy computations, we have
$$\begin{aligned}& K_{n,p,q}^{\alpha,\beta,l}(t-x;x) \\& \quad = \biggl(\frac{(1+q)[n+l]_{p,q}(px+1-x)_{p,q}^{n+l-1}}{[2]_{p,q} ([n+1]_{p,q}+\beta )p^{n+l-1}}-1 \biggr)x+\frac {(px+1-x)_{p,q}^{n+l}+2\alpha}{[2]_{p,q} ([n+1]_{p,q}+\beta )}, \end{aligned}$$
(9)
$$\begin{aligned}& K_{n,p,q}^{\alpha,\beta,l} \bigl((t-x)^{2};x \bigr) \\& \quad = \biggl(\frac{ (q+q^{2}+q^{3} )[n+l]_{p,q}[n+l-1]_{p,q} (p^{2}x+1-x )_{p,q}^{n+l-2}}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}p^{2n+2l-4}}+1 \\& \qquad {} -\frac{2(1+q)[n+l]_{p,q} (px+1-x )_{p,q}^{n+l-1}}{[2]_{p,q} ([n+l]_{p,q}+\beta )p^{n+l-1}} \biggr)x^{2}+\frac{ (p^{2}x+1-x )_{p,q}^{n+l}}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}} \\& \qquad {}+\frac{ (1+q+q^{2}+p+2pq )[n+l]_{p,q} (p^{2}x+1-x )_{p,q}^{n+l-1}x}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}p^{n+l-2}} \\& \qquad {}+\frac{3\alpha(1+q)[n+l]_{p,q}(px+1-x)_{p,q}^{n+l-1}x}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}p^{n+l-1}}+\frac{3\alpha (px+1-x)_{p,q}^{n+l}+3{\alpha}^{2}}{[3]_{p,q} ([n+1]_{p,q}+\beta )^{2}} \\& \qquad {}-\frac{2(px+1-x)_{p,q}^{n+l}x+4\alpha x}{[2]_{p,q} ([n+1]_{p,q}+\beta )}. \end{aligned}$$
(10)

Lemma 3.3

Let \(e_{i,j}(x,y)=x^{i}y^{j}\), \(i,j\in\mathbb{N}\), \(i+j\leq2\), \((x,y)\in I^{2}\) be the two-dimensional test functions. Using Lemma 3.1, the bivariate \((p, q)\)-analogue of Kantorovich-type Bernstein-Stancu-Schurer operators defined in (4) satisfies the following equalities:
$$\begin{aligned}& K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(e_{0,0};x,y)=1, \end{aligned}$$
(11)
$$\begin{aligned}& K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(e_{1,0};x,y) \\ & \quad =\frac{(p_{n_{1}}x+1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l}+2\alpha }{[2]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )}+\frac {(1+q_{n_{1}})[n_{1}+l]_{p_{n_{1}},q_{n_{1}}}(p_{n_{1}}x+1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-1}}{[2]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )p_{n_{1}}^{n_{1}+l-1}}x, \end{aligned}$$
(12)
$$\begin{aligned}& K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(e_{0,1};x,y) \\ & \quad =\frac{(p_{n_{2}}y+1-y)_{p_{n_{2}},q_{n_{2}}}^{n_{2}+l}+2\alpha }{[2]_{p_{n_{2}},q_{n_{2}}} ([n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta )}+\frac {(1+q_{n_{2}})[n_{2}+l]_{p_{n_{2}},q_{n_{2}}}(p_{n_{2}}y+1-y)_{p_{n_{2}},q_{n_{2}}}^{n_{2}+l-1}}{[2]_{p_{n_{2}},q_{n_{2}}} ([n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta )p_{n_{2}}^{n_{2}+l-1}}y, \end{aligned}$$
(13)
$$\begin{aligned}& K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(e_{1,1};x,y) \\ & \quad =\biggl(\frac {(p_{n_{1}}x+1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l}+2\alpha }{[2]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )}+\frac {(1+q_{n_{1}})[n_{1}+l]_{p_{n_{1}},q_{n_{1}}}(p_{n_{1}}x+1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-1}}{[2]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )p_{n_{1}}^{n_{1}+l-1}}x\biggr) \\ & \qquad {}\times\biggl(\frac {(1+q_{n_{2}})[n_{2}+l]_{p_{n_{2}},q_{n_{2}}}(p_{n_{2}}y+1-y)_{p_{n_{2}},q_{n_{2}}}^{n_{2}+l-1}}{[2]_{p_{n_{2}},q_{n_{2}}} ([n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta )p_{n_{2}}^{n_{2}+l-1}}y \\ & \qquad {}+\frac{(p_{n_{2}}y+1-y)_{p_{n_{2}},q_{n_{2}}}^{n_{2}+l}+2\alpha }{[2]_{p_{n_{2}},q_{n_{2}}} ([n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta )}\biggr), \end{aligned}$$
(14)
$$\begin{aligned}& K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(e_{2,0};x,y) \\ & \quad = \frac{ (q_{n_{1}}+q_{n_{1}}^{2}+q_{n_{1}}^{3} )[n_{1}+l]_{p_{n_{1}},q_{n_{1}}}[n_{1}+l-1]_{p_{n_{1}},q_{n_{1}}} (p_{n_{1}}^{2}x+1-x )_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-2}}{[3]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )^{2}p_{n_{1}}^{2n_{1}+2l-4}}x^{2} \\ & \qquad {}+\frac{ (1+q_{n_{1}}+q_{n_{1}}^{2}+p_{n_{1}}+2p_{n_{1}}q_{n_{1}} )[n_{1}+l]_{p_{n_{1}},q_{n_{1}}} (p_{n_{1}}^{2}x+1-x )_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-1}x}{[3]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )^{2}p_{n_{1}}^{n_{1}+l-2}} \\ & \qquad {}+\frac{3\alpha (1+q_{n_{1}})[n_{1}+l]_{p_{n_{1}},q_{n_{1}}}(p_{n_{1}}x+1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-1}x}{[3]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )^{2}p_{n_{1}}^{n_{1}+l-1}} \\ & \qquad {}+\frac{ (p_{n_{1}}^{2}x+1-x )_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l}}{[3]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )^{2}}+\frac{3\alpha (p_{n_{1}}x+1-x )_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l}+3{\alpha }^{2}}{[3]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )^{2}}, \end{aligned}$$
(15)
$$\begin{aligned}& K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(e_{0,2};x,y) \\ & \quad = \frac{ (q_{n_{2}}+q_{n_{2}}^{2}+q_{n_{2}}^{3} )[n_{2}+l]_{p_{n_{2}},q_{n_{2}}}[n_{2}+l-1]_{p_{n_{2}},q_{n_{2}}} (p_{n_{2}}^{2}y+1-y )_{p_{n_{2}},q_{n_{2}}}^{n_{2}+l-2}}{[3]_{p_{n_{2}},q_{n_{2}}} ([n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta )^{2}p_{n_{2}}^{2n_{2}+2l-4}}y^{2} \\ & \qquad {}+\frac{ (1+q_{n_{2}}+q_{n_{2}}^{2}+p_{n_{2}}+2p_{n_{2}}q_{n_{2}} )[n_{2}+l]_{p_{n_{2}},q_{n_{2}}} (p_{n_{2}}^{2}y+1-y )_{p_{n_{2}},q_{n_{2}}}^{n_{2}+l-1}y}{[3]_{p_{n_{2}},q_{n_{2}}} ([n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta )^{2}p_{n_{2}}^{n_{2}+l-2}} \\ & \qquad {}+\frac{3\alpha (1+q_{n_{2}})[n_{2}+l]_{p_{n_{2}},q_{n_{2}}}(p_{n_{2}}y+1-y)_{p_{n_{2}},q_{n_{2}}}^{n_{2}+l-1}y}{[3]_{p_{n_{2}},q_{n_{2}}} ([n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta )^{2}p_{n_{2}}^{n_{2}+l-1}} \\ & \qquad {}+\frac{ (p_{n_{2}}^{2}y+1-y )_{p_{n_{2}},q_{n_{2}}}^{n_{2}+l}}{[3]_{p_{n_{2}},q_{n_{2}}} ([n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta )^{2}}+\frac{3\alpha (p_{n_{2}}y+1-y )_{p_{n_{2}},q_{n_{2}}}^{n_{2}+l}+3{\alpha }^{2}}{[3]_{p_{n_{2}},q_{n_{2}}} ([n_{2}+1]_{p_{n_{2}},q_{n_{2}}}+\beta )^{2}}. \end{aligned}$$
(16)

Lemma 3.4

Using Lemmas 3.2 and 3.3, the following equalities hold:
$$\begin{aligned}& K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(t-x;x,y) \\& \quad = \biggl(\frac {(1+q_{n_{1}})[n_{1}+l]_{p_{n_{1}},q_{n_{1}}}(p_{n_{1}}x+1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-1}}{[2]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )p_{n_{1}}^{n_{1}+l-1}}-1 \biggr)x \\& \qquad {}+\frac{(p_{n_{1}}x+1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l}+2\alpha }{[2]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )}:=A_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x), \end{aligned}$$
(17)
$$\begin{aligned}& K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(s-y;x,y)=A_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y), \end{aligned}$$
(18)
$$\begin{aligned}& K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \bigl((t-x)^{2};x,y \bigr) \\& \quad = \biggl(\frac{ (q_{n_{1}}+q_{n_{1}}^{2}+q_{n_{1}}^{3} )[n_{1}+l]_{p_{n_{1}},q_{n_{1}}}[n_{1}+l-1]_{p_{n_{1}},q_{n_{1}}} (p_{n_{1}}^{2}x+1-x )_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-2}}{[3]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )^{2}p_{n_{1}}^{2n_{1}+2l-4}} \\& \qquad {}+1-\frac{2(1+q_{n_{1}})[n_{1}+l]_{p_{n_{1}},q_{n_{1}}} (p_{n_{1}}x+1-x )_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-1}}{[2]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+l]_{p_{n_{1}},q_{n_{1}}}+\beta )p_{n_{1}}^{n_{1}+l-1}} \biggr)x^{2} \\& \qquad {} +\frac{ (1+q_{n_{1}}+q_{n_{1}}^{2}+p_{n_{1}}+2p_{n_{1}}q_{n_{1}} )[n_{1}+l]_{p_{n_{1}},q_{n_{1}}} (p_{n_{1}}^{2}x+1-x )_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-1}x}{[3]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )^{2}p_{n_{1}}^{n_{1}+l-2}} \\& \qquad {}+\frac{3\alpha (1+q_{n_{1}})[n_{1}+l]_{p_{n_{1}},q_{n_{1}}}(p_{n_{1}}x+1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l-1}x}{[3]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )^{2}p_{n_{1}}^{n_{1}+l-1}} \\& \qquad {}+\frac{ (p_{n_{1}}^{2}x+1-x )_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l}}{[3]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )^{2}}+\frac{3\alpha (p_{n_{1}}x+1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l}+3{\alpha }^{2}}{[3]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )^{2}} \\& \qquad {}-\frac{2(p_{n_{1}}x+1-x)_{p_{n_{1}},q_{n_{1}}}^{n_{1}+l}x+4\alpha x}{[2]_{p_{n_{1}},q_{n_{1}}} ([n_{1}+1]_{p_{n_{1}},q_{n_{1}}}+\beta )}:=B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x), \end{aligned}$$
(19)
$$\begin{aligned}& K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \bigl((s-y)^{2};x,y \bigr)=B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y). \end{aligned}$$
(20)

Lemma 3.5

(see Theorem 2.1 of [23])

For \(0< q_{n}< p_{n}\leq1\), set \(q_{n}:=1-\alpha_{n}\), \(p_{n}:=1-\beta _{n}\) such that \(0\leq\beta_{n}<\alpha_{n}<1\), \(\alpha_{n}\rightarrow0\), \(\beta _{n}\rightarrow0\) as \(n\rightarrow\infty\). The following statements are true:
  1. (A)

    If \(\lim_{n\rightarrow\infty}e^{n(\beta_{n}-\alpha_{n})}=1\) and \(e^{n\beta_{n}}/n\rightarrow0\), then \([n]_{p_{n},q_{n}}\rightarrow\infty\).

     
  2. (B)

    If \(\varlimsup_{n\rightarrow\infty}e^{n(\beta_{n}-\alpha _{n})}<1\) and \(e^{n\beta_{n}}(\alpha_{n}-\beta_{n})\rightarrow0\), then \([n]_{p_{n},q_{n}}\rightarrow\infty\).

     
  3. (C)

    If \(\varliminf_{n\rightarrow\infty}e^{n(\beta_{n}-\alpha _{n})}<1\), \(\varlimsup_{n\rightarrow\infty}e^{n(\beta_{n}-\alpha_{n})}=1\) and \(\max \{e^{n\beta_{n}}/n,e^{n\beta_{n}}(\alpha_{n}-\beta_{n}) \} \rightarrow0\), then \([n]_{p_{n},q_{n}}\rightarrow\infty\).

     

Remark 3.6

Let sequences \(\{p_{n_{1}}\}\), \(\{q_{n_{1}}\}\), \(\{p_{n_{2}}\}\), \(\{ q_{n_{2}}\}\) (\(0< q_{n_{1}}, q_{n_{2}}< p_{n_{1}}, p_{n_{2}}\leq1\)) satisfy the conditions of Lemma 3.5(A), (B) or (C). We have \([n_{1}]_{p_{n_{1}},q_{n_{1}}}\rightarrow\infty\), \([n_{2}]_{p_{n_{2}},q_{n_{2}}}\rightarrow\infty\). From Lemmas 3.3 and 3.4, the following statements are true.
$$\begin{aligned}& \lim_{n_{1},n_{2}\rightarrow\infty }{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} (e_{1,0};x,y )}=x, \\& \lim_{n_{1},n_{2}\rightarrow\infty }{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} (e_{0,1};x,y )}=y, \\& \lim_{n_{1},n_{2}\rightarrow\infty }{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} (e_{2,0}+e_{0,2};x,y )}=x^{2}+y^{2}, \\& \lim_{n_{1},n_{2}\rightarrow\infty }{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \bigl((t-x)^{2};x,y \bigr)}=\lim_{n_{1}\rightarrow\infty }B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x)=0, \\& \lim_{n_{1},n_{2}\rightarrow\infty }{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \bigl((s-y)^{2};x,y \bigr)}=\lim_{n_{2}\rightarrow\infty }B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)=0. \end{aligned}$$

4 Convergence properties

In order to ensure the convergence of operators defined in (4), in the sequel, let \(\{p_{n_{1}}\}\), \(\{q_{n_{1}}\}\), \(\{p_{n_{2}}\}\), \(\{q_{n_{2}}\} \), \(0< q_{n_{1}}, q_{n_{2}}< p_{n_{1}}, p_{n_{2}}\leq1\) be sequences satisfying Lemma 3.5(A), (B) or (C).

Theorem 4.1

For \(f\in C(I^{2})\), we have
$$ \lim_{n_{1},n_{2}\rightarrow\infty} \bigl\Vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f; \cdot ,\cdot)-f \bigr\Vert _{I^{2}}=0. $$

Proof

Using (6), Remark 3.6 and a bivariate-type Korovkin theorem (see [24]), we obtain Theorem 4.1 easily. □

For \(f\in C(I^{2})\), the complete modulus of continuity for the bivariate case is defined as
$$\begin{aligned}& \omega(f;\delta_{1},\delta_{2}) \\& \quad = \sup\bigl\{ \bigl\vert f(t,s)-f(x,y) \bigr\vert : (t,s), (x,y) \in I^{2}, |t-x|\leq\delta_{1}, |s-y|\leq\delta_{2} \bigr\} , \end{aligned}$$
where \(\delta_{1}, \delta_{2}>0\). Furthermore, \(\omega(f;\delta_{1},\delta _{2})\) satisfies the following properties:
$$\begin{aligned} (\mathrm{i})&\quad \omega(f;\delta_{1},\delta_{2}) \rightarrow0,\quad \mbox{if }\delta_{1}, \delta _{2} \rightarrow0; \\ (\mathrm{ii})&\quad \bigl|f(t,s)-f(x,y)\bigr|\leq\omega(f;\delta_{1},\delta) \biggl(1+\frac {|t-x|}{\delta_{1}} \biggr) \biggl(\frac{|s-y|}{\delta_{2}} \biggr). \end{aligned}$$
The partial modulus of continuity with respect to x and y is defined as
$$\begin{aligned}& \omega^{(1)}(f;\delta)=\sup\bigl\{ \bigl\vert f(x_{1},y)-f(x_{2},y) \bigr\vert : y\in I\mbox{ and }\vert x_{1}-x_{2} \vert \leq \delta\bigr\} , \\& \omega^{(2)}(f;\delta)=\sup\bigl\{ \bigl\vert f(x,y_{1})-f(x,y_{2}) \bigr\vert : x\in I\mbox{ and }\vert y_{1}-y_{2} \vert \leq \delta\bigr\} . \end{aligned}$$
Details of the modulus of continuity for the bivariate case can be found in [25]. We also use the notation
$$ C^{1}\bigl(I^{2}\bigr)=\bigl\{ f\in C\bigl(I^{2} \bigr): f_{x}', f_{y}'\in C \bigl(I^{2}\bigr)\bigr\} . $$

Now, we give the estimate of the rate of convergence of operators defined in (4).

Theorem 4.2

For \(f\in C(I^{2})\), we have
$$ \bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \leq4 \omega \Bigl(f;\sqrt {B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x)}, \sqrt {B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)} \Bigr), $$
(21)
where \(B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x)\) and \(B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)\) are defined in (19) and (20).

Proof

From Lemmas 3.3 and 3.4, using the property (ii) of the complete modulus of continuity for the bivariate case above and the Cauchy-Schwarz inequality, we get
$$\begin{aligned}& \bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \\& \quad \leq K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}\bigl( \bigl\vert f(t,s)-f(x,y) \bigr\vert ;x,y\bigr) \\& \quad \leq \omega \Bigl(f;\sqrt{B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta ,l}(x)}, \sqrt{B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)} \Bigr) \\& \qquad {} \times \biggl(1+\sqrt{\frac {K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} ((t-x)^{2};x,y )}{B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x)}} \biggr) \\& \qquad {} \times \biggl(1+\sqrt{\frac {K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} ((s-y)^{2};x,y )}{B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)}} \biggr). \end{aligned}$$
Theorem 4.2 is proved. □

Theorem 4.3

For \(f\in C(I^{2})\), under the conditions of Lemma 3.4, we have
$$\begin{aligned}& \bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \\& \quad \leq 2 \Bigl(f;\omega^{(1)} \Bigl(f;\sqrt{B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha ,\beta,l}(x)} \Bigr)+\omega^{(2)} \Bigl(f;\sqrt {B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)} \Bigr) \Bigr), \end{aligned}$$
where \(B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x)\) and \(B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)\) are defined in (19) and (20).

Proof

Using the definition of partial modulus of continuity above and the Cauchy-Schwarz inequality, we have
$$\begin{aligned}& \bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \\& \quad \leq K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}\bigl( \bigl\vert f(t,s)-f(x,y) \bigr\vert ;x,y\bigr) \\& \quad \leq K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}\bigl( \bigl\vert f(t,s)-f(t,y) \bigr\vert ;x,y\bigr) \\& \qquad {}+K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}\bigl( \bigl\vert f(t,y)-f(x,y) \bigr\vert ;x,y\bigr) \\& \quad \leq K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \bigl(\omega^{(2)}\bigl(f; \vert s-y \vert \bigr);x,y \bigr) \\& \qquad {}+K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \bigl(\omega^{(1)}\bigl(f; \vert t-x \vert \bigr);x,y \bigr) \\& \quad \leq \omega^{(2)} \Bigl(f;\sqrt{B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta ,l}(y)} \Bigr) \biggl(1+\sqrt{\frac {K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} ((s-y)^{2};x,y )}{B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)}} \biggr) \\& \qquad {}+\omega^{(1)} \Bigl(f;\sqrt{B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta ,l}(x)} \Bigr) \biggl(1+\sqrt{\frac {K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} ((t-x)^{2};x,y )}{B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x)}} \biggr). \end{aligned}$$
Theorem 4.3 is proved. □

Theorem 4.4

For \(f\in C^{1}(I^{2})\), using Lemma 3.4, we have
$$\begin{aligned}& \bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \\& \quad \leq \bigl\Vert f_{x}' \bigr\Vert _{I}\sqrt{B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x)} + \bigl\Vert f_{y}' \bigr\Vert _{I}\sqrt {B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)}, \end{aligned}$$
where \(B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x)\) and \(B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)\) are defined in (19) and (20).

Proof

Since \(f(t,s)-f(x,y)=\int_{x}^{t}f_{u}'(u,s)\,du+\int_{y}^{s}f_{v}'(x,v)\,dv\). Applying the operators defined in (4) on both sides above, we have
$$\begin{aligned}& \bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \\& \quad \leq K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \biggl( \biggl\vert \int_{x}^{t}f_{u}'(u,s) \,du \biggr\vert ;x,y \biggr) \\& \qquad {} +K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \biggl( \biggl\vert \int_{y}^{s}f_{v}'(x,v) \,dv \biggr\vert ;x,y \biggr). \end{aligned}$$
Due to \(\vert \int_{x}^{t}f_{u}'(u,s)\,du \vert \leq\|f_{x}'\|_{I}|t-x|\) and \(\vert \int_{y}^{s}f_{v}'(x,v)\,dv \vert \leq\|f_{y}'\|_{I}|s-y|\), we have
$$\begin{aligned} &\bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \\ &\quad \leq \bigl\Vert f_{x}' \bigr\Vert _{I}K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha ,\beta,l}\bigl( \vert t-x \vert ;x,y \bigr)+ \bigl\Vert f_{y}' \bigr\Vert _{I}K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}\bigl( \vert s-y \vert ;x,y \bigr). \end{aligned}$$
Using the Cauchy-Schwarz inequality and Lemma 3.3, we obtain
$$\begin{aligned}& \bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \\& \quad \leq \bigl\Vert f_{x}' \bigr\Vert _{I}\sqrt {K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \bigl((t-x)^{2};x,y \bigr)}\sqrt {K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(1;x,y)} \\& \qquad {} + \bigl\Vert f_{y}' \bigr\Vert _{I}\sqrt {K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \bigl((s-y)^{2};x,y \bigr)}\sqrt {K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(1;x,y)} \\& \quad \leq \bigl\Vert f_{x}' \bigr\Vert _{I}\sqrt{B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta ,l}(x)}+ \bigl\Vert f_{y}' \bigr\Vert _{I}\sqrt {B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)}. \end{aligned}$$
Theorem 4.4 is proved. □
Finally, we study the rate of convergence of \(K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)\) by means of functions of Lipschitz class \(\operatorname{Lip}_{M}(\theta_{1},\theta_{2})\) if
$$ \bigl\vert f(t,s)-f(x,y) \bigr\vert \leq M|t-x|^{\theta_{1}}|s-y|^{\theta_{2}}, \quad (t,s), (x,y)\in I^{2}. $$

Theorem 4.5

Let \(f\in \operatorname{Lip}_{M}(\theta_{1},\theta_{2})\), under the conditions of Lemma 3.4, we have
$$ \bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \leq M \bigl(B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha ,\beta,l}(x) \bigr)^{\frac{\theta_{1}}{2}} \bigl(B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y) \bigr)^{\frac{\theta_{2}}{2}}, $$
where \(B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x)\) and \(B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y)\) are defined in (19) and (20).

Proof

Since \(f\in \operatorname{Lip}_{M}(\theta_{1},\theta_{2})\), we have
$$\begin{aligned}& \bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \\& \quad \leq K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}\bigl( \bigl\vert f(t,s)-f(x,y) \bigr\vert ;x,y\bigr) \\& \quad \leq MK_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \bigl( \vert t-x \vert ^{\theta_{1}};x,y \bigr)K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l} \bigl( \vert s-y \vert ^{\theta_{2}};x,y \bigr). \end{aligned}$$
Using Hölder’s inequality for the last formula, respectively, we get
$$\begin{aligned}& \bigl\vert K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)-f(x,y) \bigr\vert \\& \quad \leq M \bigl(K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}\bigl((t-x)^{2};x,y \bigr) \bigr)^{\frac{\theta_{1}}{2}} \bigl(K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(1;x,y) \bigr)^{\frac{2-\theta_{1}}{2}} \\& \qquad {} \times \bigl(K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}\bigl((s-y)^{2};x,y \bigr) \bigr)^{\frac{\theta_{2}}{2}} \bigl(K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(1;x,y) \bigr)^{\frac{2-\theta_{2}}{2}} \\& \quad = M \bigl(B_{n_{1},p_{n_{1}},q_{n_{1}}}^{\alpha,\beta,l}(x) \bigr)^{\frac {\theta_{1}}{2}} \bigl(B_{n_{2},p_{n_{2}},q_{n_{2}}}^{\alpha,\beta,l}(y) \bigr)^{\frac{\theta_{2}}{2}}. \end{aligned}$$
Theorem 4.5 is proved. □

5 Graphical and numerical examples analysis

In this section, we give several graphs and numerical examples to show the convergence of \(K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)\) to \(f(x,y)\) with different values of parameters which satisfy the conclusions of Lemma 3.5.

Let \(f(x,y) = x^{2}y^{2}\), the graphs of \(K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)\) with different values of \(q_{n_{1}}\), \(q_{n_{2}}\), \(p_{n_{1}}\), \(p_{n_{2}}\) and \(n_{1}\), \(n_{2}\) are shown in Figures 1, 2 and 3. In Tables 1, 2 and 3, we give the errors of the approximation of \(K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)\) to \(f(x,y)\) with different parameters.
Figure 1

The figures of \(\pmb{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)}\) (the upper one) for \(\pmb{n_{1} = n_{2} = 20}\) , \(\pmb{p_{n_{1}} = p_{n_{2}} = 1 - 1/10^{8}}\) , \(\pmb{q_{n_{1}} = q_{n_{2}} = 0.999}\) , \(\pmb{l=1}\) , \(\pmb{\alpha=3}\) , \(\pmb{\beta=2}\) and \(\pmb{f(x,y)=x^{2}y^{2}}\) (the below one).

Figure 2

The figures of \(\pmb{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)}\) (the upper one) for \(\pmb{n_{1} = n_{2} = 20}\) , \(\pmb{p_{n_{1}} = p_{n_{2}} = 1 - 1/10^{14}}\) , \(\pmb{q_{n_{1}} = q_{n_{2}} = 0.9999}\) , \(\pmb{l=1}\) , \(\pmb{\alpha=3}\) , \(\pmb{\beta=2}\) and \(\pmb{f(x,y)=x^{2}y^{2}}\) (the below one).

Figure 3

The figures of \(\pmb{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)}\) (the upper one) for \(\pmb{n_{1} = n_{2} = 50}\) , \(\pmb{p_{n_{1}} = p_{n_{2}} = 1 - 1/10^{14}}\) , \(\pmb{q_{n_{1}} = q_{n_{2}} = 0.9999}\) , \(\pmb{l=1}\) , \(\pmb{\alpha=3}\) , \(\pmb{\beta=2}\) and \(\pmb{f(x,y)=x^{2}y^{2}}\) (the below one).

Table 1

The errors of the approximation of \(\pmb{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)}\) for \(\pmb{p_{n_{1}} = p_{n_{2}} = 1{-}1/10^{15}}\) , \(\pmb{q_{n_{1}} = q_{n_{2}} = 0.9999}\) , \(\pmb{l=1}\) , \(\pmb{\alpha=3}\) , \(\pmb{\beta=2}\) and different values of \(\pmb{n_{1}}\) , \(\pmb{n_{2}}\)

\(\boldsymbol{n_{1} = n_{2}}\)

\(\boldsymbol{\|f(x,y) - K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)\| _{\infty}}\)

5

0.801911

10

0.406691

15

0.259663

20

0.188202

30

0.150588

35

0.131835

Table 2

The errors of the approximation of \(\pmb{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)}\) for \(\pmb{n_{1} = n_{2} = 10}\) , \(\pmb{p_{n_{1}} = p_{n_{2}} = 1{-}1/10^{15}}\) , \(\pmb{l=1}\) , \(\pmb{\alpha=3}\) , \(\pmb{\beta=2}\) and different values of \(\pmb{q_{n_{1}}}\) , \(\pmb{q_{n_{2}}}\)

\(\boldsymbol{q_{n_{1}} = q_{n_{2}}}\)

\(\boldsymbol{\|f(x,y) - K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)\| _{\infty}}\)

0.99

2.923910

0.995

1.194710

0.999

0.643543

0.9995

0.594722

0.9999

0.406691

0.99995

0.130489

Table 3

The errors of the approximation of \(\pmb{K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta,l}(f;x,y)}\) for \(\pmb{q_{n_{1}} = q_{n_{2}} = 0.9999}\) , \(\pmb{l=1}\) , \(\pmb{\alpha=3}\) , \(\pmb{\beta=2}\) and different values of \(\pmb{p_{n_{1}}}\) , \(\pmb{p_{n_{2}}}\) and \(\pmb{n_{1}}\) , \(\pmb{n_{2}}\)

\(\boldsymbol{n_{1} =n_{2}}\)

\(\boldsymbol{p_{n_{1}}=p_{n_{2}}}\)

\(\boldsymbol{\|f(x,y) - K_{p_{n_{1}},q_{n_{1}},p_{n_{2}},q_{n_{2}}}^{n_{1},n_{2},\alpha,\beta ,l}(f;x,y)\|_{\infty}}\)

10

1 − 1/1010

0.406691

15

1 − 1/1011

0.259663

20

1 − 1/1012

0.188202

25

1 − 1/1013

0.150589

30

1 − 1/1014

0.131836

35

1 − 1/1015

0.125673

Declarations

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11601266 and 11626201), the Natural Science Foundation of Fujian Province of China (Grant No. 2016J05017) and the Program for New Century Excellent Talents in Fujian Province University. We also thank Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing and Information Processing of Fujian Province University.

Authors’ contributions

QBC, XWX and GZ carried out the molecular genetic studies, participated in the sequence alignment and drafted the manuscript. QBC, XWX and GZ carried out the immunoassays. QBC, XWX and GZ participated in the sequence alignment. QBC, XWX and GZ participated in the design of the study and performed the statistical analysis. QBC, XWX and GZ conceived of the study and participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access 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, provided 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.

Authors’ Affiliations

(1)
School of Mathematics and Computer Science, Quanzhou Normal University
(2)
School of Mathematical Sciences, Xiamen University
(3)
Computer Sciences, Rice University
(4)
School of Applied Mathematics, Xiamen University of Technology

References

  1. Mursaleen, M, Ansari, KJ, Khan, A: On \((p, q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266, 874-882 (2015) MathSciNetGoogle Scholar
  2. Mursaleen, M, Ansari, KJ, Khan, A: Erratum to ‘On \((p, q)\)-analogue of Bernstein operators [Appl. Math. Comput. 266 (2015) 874-882]’. Appl. Math. Comput. 278, 70-71 (2016) MathSciNetGoogle Scholar
  3. Acar, T: \((p, q)\)-Generalization of Szász-Mirakyan operators. Math. Methods Appl. Sci. 39(10), 2685-2695 (2016) MathSciNetView ArticleMATHGoogle Scholar
  4. Acar, T, Aral, A, Mohiuddine, SA: On Kantorovich modification of \((p, q)\)-Baskakov operators. J. Inequal. Appl. (2016). doi:10.1186/s13660-016-1045-9 MathSciNetMATHGoogle Scholar
  5. Mursaleen, M, Nasiruzzaman, M, Khan, A, Ansari, KJ: Some approximation results on Bleimann-Butzer-Hahn operators defined by \((p, q)\)-integers. Filomat 30(3), 639-648 (2016) MathSciNetView ArticleMATHGoogle Scholar
  6. Cai, QB, Zhou, GR: On \((p, q)\)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators. Appl. Math. Comput. 276, 12-20 (2016) MathSciNetGoogle Scholar
  7. Mursaleen, M, Ansari, KJ, Khan, A: Some approximation results by \((p, q)\)-analogue of Bernstein-Stancu operators. Appl. Math. Comput. 264, 392-402 (2015) MathSciNetGoogle Scholar
  8. Mursaleen, M, Alotaibi, A, Ansari, KJ: On a Kantorovich variant of \((p, q)\)-Szász-Mirakjan operators. J. Funct. Spaces 2016, Article ID 1035253 (2016) MATHGoogle Scholar
  9. Mursaleen, M: Some approximation results on Bleimann-Butzer-Hahn operators defined by \((p, q)\)-integers. Filomat 30(3), 639-648 (2016) MathSciNetView ArticleMATHGoogle Scholar
  10. Mursaleen, M, Ansari, KJ, Khan, A: Some approximation results for Bernstein-Kantorovich operators based on \((p, q)\)-calculus. UPB Sci. Bull., Ser. A 78(4), 129-142 (2016) MathSciNetGoogle Scholar
  11. Acar, T, Agrawal, P, Kumar, A: On a modification of \((p, q)\)-Szász-Mirakyan operators. Complex Anal. Oper. Theory (2016). doi:10.1007/s11785-016-0613-9 Google Scholar
  12. Ilarslan, H, Acar, T: Approximation by bivariate \((p, q)\)-Baskakov-Kantorovich operators. Georgian Math. J. (2016). doi:10.1515/gmj-2016-0057 Google Scholar
  13. Phillips, GM: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4, 511-518 (1997) MathSciNetMATHGoogle Scholar
  14. Gupta, V, Kim, T: On the rate of approximation by q modified beta operators. J. Math. Anal. Appl. 377, 471-480 (2011) MathSciNetView ArticleMATHGoogle Scholar
  15. Gupta, V, Aral, A: Convergence of the q analogue of Szász-beta operators. Appl. Math. Comput. 216, 374-380 (2010) MathSciNetMATHGoogle Scholar
  16. Acar, T, Aral, A: On pointwise convergence of q-Bernstein operators and their q-derivatives. Numer. Funct. Anal. Optim. 36(3), 287-304 (2015) MathSciNetView ArticleMATHGoogle Scholar
  17. Khan, K, Lobiyal, DK: Bézier curves based on Lupas \((p, q)\)-analogue of Bernstein functions in CAGD. J. Comput. Appl. Math. 317, 458-477 (2017) MathSciNetView ArticleMATHGoogle Scholar
  18. Hounkonnou, MN, Désiré, J, Kyemba, B: \(R(p, q)\)-Calculus: differentiation and integration. SUT J. Math. 49, 145-167 (2013) MathSciNetMATHGoogle Scholar
  19. Jagannathan, R, Rao, KS: Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series. In: Proceedings of the International Conference on Number Theory and Mathematical Physics, pp. 20-21 (2005) Google Scholar
  20. Katriel, J, Kibler, M: Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers. J. Phys. A, Math. Gen. 24, 2683-2691 (1992) MathSciNetView ArticleMATHGoogle Scholar
  21. Sadjang, PN: On the fundamental theorem of \((p, q)\)-calculus and some \((p, q)\)-Taylor formulas. arXiv:1309.3934v1 (2015)
  22. Sahai, V, Yadav, S: Representations of two parameter quantum algebras and \(p, q\)-special functions. J. Math. Anal. Appl. 335, 268-279 (2007) MathSciNetView ArticleMATHGoogle Scholar
  23. Cai, QB, Xu, XW: A basic problem of \((p, q)\)-Bernstein operators. J. Inequal. Appl. 2017, 140 (2017). doi:10.1186/s13660-017-1413-0 MathSciNetView ArticleMATHGoogle Scholar
  24. Altomare, F, Campiti, M: Korovkin-Type Approximation Theory and Its Applications. De Gruyter Studies in Mathematics, vol. 17. de Gruyter, Berlin (1994) View ArticleMATHGoogle Scholar
  25. Anastassiou, GA, Gal, SG: Approximation Theory: Moduli of Continuity and Global Smoothness Preservation. Birkhäuser, Boston (2000) View ArticleMATHGoogle Scholar

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