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Bivariate Bernstein–Schurer–Stancu type GBS operators in \((p,q)\)-analogue

Abstract

The purpose of this paper is to construct a \((p,q)\)-analogue of Bernstein–Schurer–Stancu type GBS (generalized Boolean sum) operators for approximating B-continuous and B-differentiable functions. We also establish uniform convergence theorem and estimate the degree of approximation of B-continuous and B-differentiable functions.

Introduction

Badea et al. [5] introduced the following operators known as GBS operators associated with L.

Let \(I_{1},I_{2}\subseteq \mathbb{R}\) be nonempty intervals, and let \(L:\mathbb{R}^{I_{1}\times I_{2}}\rightarrow \mathbb{R}^{I_{1}\times I_{2}}\) be a positive linear bivariate operator, where \(\mathbb{R} ^{I_{1}\times I_{2}}=\{f|f:I_{1}\times I_{2}\rightarrow \mathbb{R}\}\). If \(f(\circ ,\ast )\in \mathbb{R}^{I_{1}\times I_{2}}\), then the bivariate operators \(U:\mathbb{R}^{I_{1}\times I_{2}}\rightarrow \mathbb{R}^{I_{1}\times I_{2}}\) are defined by

$$ Uf(x,y)=L \bigl(f(\circ ,y)+f(x,\ast )-f(\circ ,\ast ) \bigr) (x,y), \quad \text{for } (x,y)\in I_{1}\times I_{2}. $$
(1.1)

In 1934, Karl Bögel [13] introduced the notion of B-continuity and B-differentiability. B-continuity by means of bivariate mixed difference operator \(\Delta _{2}: \mathbb{R}^{I_{1} \times I_{2}}\rightarrow \mathbb{R}^{I_{1}\times I_{2}}\) is defined in [12].

We now recall some definitions and results based on B-continuity as follows.

Definition 1.1

A function \(f\in \mathbb{R}^{I_{1}\times I_{2}}\) is B-continuous if, for each \((x,y)\in I_{1}\times I_{2}\),

$$ \lim_{(u,v)\rightarrow (x,y)}\Delta _{u,v} [f:x,y ]=0, $$
(1.2)

where \(\Delta _{u,v} [f:x,y ]\) is the mixed difference defined by

$$ \Delta _{u,v} [f:x,y ]=f(u,v)-f(u,y)-f(x,v)+f(x,y). $$

If the function f is B-continuous at any point \((x,y)\in I_{1} \times I_{2}\), then it is B-continuous on the interval \(I_{1}\times I_{2}\).

For any \((x,y),(u,v)\in I_{1}\times I_{2}\), if there exists \(M>0\) such that

$$ \bigl\vert \Delta _{u,v} [f:x,y ] \bigr\vert \leq M $$

holds, then f is B-bounded.

Definition 1.2

A function \(f\in \mathbb{R}^{I_{1}\times I_{2}}\) is said to be uniformly B-continuous if, for any \(\epsilon >0\), there exists \(\delta ( \epsilon )>0\) such that, for every \((x,y),(u,v)\in I_{1}\times I_{2}\) with \(|x-u|<\delta (\epsilon )\), \(|y-v|<\delta (\epsilon )\), we have

$$ \bigl\vert \Delta _{u,v} [f:x,y ] \bigr\vert < \epsilon . $$
(1.3)

If \(f\in C_{b}(I_{1}\times I_{2})\) and \(I_{1}\times I_{2}\subseteq \mathbb{R}\) are compact intervals of \(\mathbb{R}\), then f is uniform B-continuous on \(I_{1}\times I_{2}\), where \(C_{b}(I_{1}\times I_{2})\) is the set of B-continuous functions. For more information, we refer to [35].

Badea et al. [6] proved the following Korovkin type theorem to approximate bivariate function in the space of Bögel-continuous (B-continuous) functions.

Theorem 1.3

Let\(\{L_{m,n}\}\)be a sequence of positive linear operators which maps\(\mathbb{R}^{I_{1}\times I_{2}}\)to\(\mathbb{R}^{I_{1}\times I_{2}}\)such that, for all\((x,y)\in I_{1}\times I_{2}\),

  1. (i)

    \(L_{m,n}(e_{0 0};x,y)=L(1,x,y)=1\),

  2. (ii)

    \(L_{m,n}(e_{1 0};x,y)=L(u,x,y)=x+u_{m,n}(x,y)\),

  3. (iii)

    \(L_{m,n}(e_{0 1};x,y)=L(v,x,y)=y+v_{m,n}(x,y)\),

  4. (iv)

    \(L_{m,n}(e_{0 2}+e_{2 0};x,y)=L(u^{2}+v^{2},x,y)=x^{2}+y ^{2}+w_{m,n}(x,y)\),

where\(u_{m,n}(x,y)\), \(v_{m,n}(x,y)\), and\(w_{m,n}(x,y)\)converge uniformly to zero as\(m,n\rightarrow \infty \). Then the sequence\(\{U_{m,n}f\}\)converges uniformly tofon\(I_{1}\times I_{2}\)for any\(f\in C_{b}(I_{1}\times I_{2})\), where\(I_{1}\), \(I_{2}\)are compact intervals of\(\mathbb{R}\); and\(U_{m,n}\)is a GBS operator associated with\(L_{m,n}\).

The mixed modulus of continuity is an important tool to approximate degree of B-continuous functions introduced by Marchaud [25]. Let \(f\in \mathbb{R}^{I_{1}\times I_{2}}\) and \(I_{1}\), \(I _{2}\) be compact intervals of \(\mathbb{R}\). Then \(\omega _{ \mathrm{mixed}}:[0,\infty )\times [0,\infty )\rightarrow [0,\infty )\) is defined by

$$ \omega _{\mathrm{mixed}}(\delta _{1},\delta _{2})=\sup \bigl\{ \bigl\vert \Delta _{u,v} [f:x,y ] \bigr\vert : \vert u-x \vert < \delta _{1}, \vert v-y \vert < \delta _{2} \bigr\} , $$
(1.4)

for any \(\delta _{1},\delta _{2}\in (0,\infty )\times (0,\infty )\) and \((x,y),(u,v)\in I_{1}\times I_{2}\).

Badea et al. [5] proved the following Shisha–Mond type theorem (introduced by Mahmedov [24]) to evaluate the degree of approximation of Bögel-continuous (continuous in Bögel sense) functions using GBS operators.

Theorem 1.4

Let\(L:C_{b}(I_{1}\times I_{2})\rightarrow C_{b}(I_{1}\times I_{2})\)be a positive linear operator and\(Uf(x,y)\)be the associated GBS operator. Then the following inequality holds for any\(f\in C_{b}(I_{1}\times I _{2})\), \((x,y)\in I_{1}\times I_{2}\), and\(\delta _{1},\delta _{2} \geq 0\):

$$ \begin{aligned} \bigl\vert f(x,y)-Uf(x,y) \bigr\vert &\leq \bigl\vert f(x,y) \bigr\vert \bigl\vert 1-L(e_{0 0};x,y) \bigr\vert + \bigl\{ L(e_{0 0};x,y) \\ &\quad {}+\delta ^{-1}_{1}\sqrt{L \bigl((e_{1 0}-x)^{2};x,y \bigr)}+\delta ^{-1}_{2}\sqrt{L \bigl((e _{0 1}-y)^{2};x,y \bigr)} \\ &\quad {}+(\delta _{1}\delta _{2})^{-1} \sqrt{L \bigl((e_{1 0}-x)^{2};x,y \bigr)L \bigl((e_{0 1}-y)^{2};x,y \bigr)} \bigr\} \omega _{\mathrm{mixed}}(\delta _{1},\delta _{2}). \end{aligned} $$

Note that \(\omega _{\mathrm{mixed}}(\delta _{1},\delta _{2})\) is a B-continuous function and \(\omega _{\mathrm{mixed}}(0,0)=0\). By the inequality defined in Theorem 1.4 and the properties of \(\omega _{\mathrm{mixed}}(\delta _{1},\delta _{2})\), it is possible to obtain the uniform convergence for the sequence introduced by GBS operators.

In [10], Bărbosu defined Schurer–Stancu type GBS operators

$$ \tilde{U}_{m,n,r_{1},r_{2}}^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}:C[0,1+r_{1}]\times C[0,1+r_{2}]\rightarrow C[0,1]\times C[0,1] $$

as follows:

$$ \begin{aligned} &\tilde{U}_{m,n,r_{1},r_{2}}^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}f(x,y) \\ &\quad =\sum_{k=0}^{m+r_{1}} \sum _{l=0}^{n+r_{2}}\tilde{r}_{m,k} \tilde{r}_{n,l} \biggl\lbrace f \biggl(\frac{k+\alpha _{1}}{m+\beta _{1}},y \biggr)+f \biggl(x,\frac{l+\alpha _{2}}{n+\beta _{2}} \biggr)-f \biggl(\frac{k+ \alpha _{1}}{m+\beta _{1}},\frac{l+\alpha _{2}}{n+\beta _{2}} \biggr) \biggr\rbrace , \end{aligned} $$

where \(r_{1}\), \(r_{2}\) are nonnegative integers and \(\alpha _{i}\), \(\beta _{i}\) are real parameters with \(0\leq \alpha _{i}\leq \beta _{i}\) (\(i=1,2\)). For \(\alpha _{i}=\beta _{i}=0\) (\(i=1,2\)), the above operators reduce to the first GBS operators which were introduced by Dobrescu and Matei [15]. For detailed study, one can refer to [7, 8], and [19].

q-Bernstein–Schurer–Stancu GBS operators

Quantum calculus (q-calculus) plays an important role in approximation theory. First of all, the q-calculus was applied by Lupaş on Bernstein polynomials. Then, focusing on bivariate case, Bărbosu [9] introduced the generalized bivariate Stancu operators, and many researchers have worked on different operators: Örkcü [32] established the q-Szász–Mirakjan–Kantorovich bivariate operators; Mursaleen and Ahasan [28] introduced the Dunkl generalization of Stancu type q-Szász–Mirakjan–Kantorovich operators; Ostrovska [33] determined the relation between the theory of q-Bernstein polynomials and limit q-Bernstein operators. For detailed study, we refer to [3, 6, 11, 14, 20, 34], and [37].

Agrawal et al. [4] introduced the q-Bernstein–Schurer–Stancu operators

$$ S^{\alpha ,\beta }_{n,r}:C[0,1+r]\rightarrow C[0,1] $$

as follows:

$$ S^{\alpha ,\beta }_{n,r}(f;q;x)=\sum_{k=0}^{n+r} \begin{bmatrix} n+r \\ k \end{bmatrix}x^{k}\prod _{j=0}^{n+r-k-1} \bigl(1-q^{j}x \bigr)f \biggl( \frac{[k]_{q}+ \alpha }{[n]_{q}+\beta } \biggr), \quad q\in (0,1), x\in [0,1+r]. $$

Recently Bărbosu et al. [12] introduced Bernstein–Schurer–Stancu type GBS operators based on q-integers.

For any \((x,y)\in I=[0,1+r_{1}]\times [0,1+r_{2}]\), the operators

$$ U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}:C_{b} \bigl([0,1+r _{1}] \times [0,1+r_{2}] \bigr)\rightarrow C_{b} \bigl([0,1] \times [0,1] \bigr) $$

associated with \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}}\) are defined as follows:

$$\begin{aligned} U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(f;q _{1},q_{2};x,y) =&\sum_{k_{1}=0}^{m+r_{1}}\sum _{k_{2}=0}^{n+r_{2}} \begin{bmatrix} m+r_{1} \\ k_{1} \end{bmatrix} \begin{bmatrix} n+r_{2} \\ k_{2} \end{bmatrix}\prod_{s=0}^{m+r_{1}-k_{1}-1} \bigl(1-q^{s}_{1}x \bigr) \\ &{}\times \prod_{t=0}^{n+r_{2}-k_{2}-1} \bigl(1-q^{t}_{2}y \bigr)x^{k_{1}}y^{k_{2}} \{f_{k_{1}}+f_{k_{2}}-f_{k_{1}}f_{k_{2}}\}, \end{aligned}$$
(2.1)

where

$$ f_{k_{1}}(y)=f \biggl(\frac{[k_{1}]_{q_{1}}+\alpha _{1}}{[m]_{q_{1}}+ \beta _{1}},y \biggr), \qquad f_{k_{2}}(x)=f \biggl(x,\frac{[k_{2}]_{q_{2}}+\alpha _{2}}{[n]_{q_{2}}+ \beta _{2}} \biggr) $$

and

$$ f_{k_{1}}f_{k_{2}}(x,y)=f \biggl(\frac{[k_{1}]_{q_{1}}+\alpha _{1}}{[m]_{q _{1}}+\beta _{1}}, \frac{[k_{2}]_{q_{2}}+\alpha _{2}}{[n]_{q_{2}}+\beta _{2}} \biggr). $$

The operators (2.1) satisfy the following properties as proved in [12].

Lemma 2.1

Let\(e_{i,j}:I\rightarrow I\), where\(I=[0,1+r_{1}]\times [0,1+r_{2}]\)is the test functions defined by\(e_{i,j}(x,y)=x^{i}y^{j}\) (i, jare nonnegative integers). Then the following equalities hold:

  1. (i)

    \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{0,0};q_{1},q_{2};x,y)=e_{0,0}(x,y)\),

  2. (ii)

    \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{1,0};q_{1},q_{2};x,y)=\frac{[m+r_{1}]_{q_{1}}x+\alpha _{1}}{[m]_{q _{1}}+\beta _{1}}\),

  3. (iii)

    \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}}(e_{0,1};q_{1},q_{2};x,y)=\frac{[n+r_{2}]_{q_{2}}y+\alpha _{2}}{[n]_{q_{2}}+\beta _{2}}\),

  4. (iv)

    \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{2,0};q_{1},q_{2};x,y)=\frac{ ([m+r_{1}]_{q_{1}}^{2}x ^{2}+[m+r_{1}]_{q_{1}}x(1-x)+2\alpha _{1}[m+r_{1}]_{q_{1}}x+\alpha ^{2} _{1} )}{([m]_{q_{1}}+\beta _{1})^{2}}\),

  5. (v)

    \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{0,2};q_{1},q_{2};x,y)=\frac{ ([n+r_{2}]_{q_{2}}^{2}y ^{2}+[n+r_{2}]_{q_{2}}y(1-y)+2\alpha _{2}[n+r_{2}]_{q_{2}}y+\alpha ^{2} _{2} )}{([n]_{q_{2}}+\beta _{2})^{2}}\).

\((p,q)\)-Bernstein–Schurer–Stancu type GBS operators

In 2015 Mursaleen et al. [29] used \((p,q)\)-calculus in approximation theory and defined first \((p,q)\)-analogue of Bernstein polynomials. Later on this idea was used to generalize several operators, e.g., [1, 2, 1618, 21, 26, 27, 30, 31]; for its applications, see [22] and [23].

We now recall some notations on \((p,q)\)-calculus.

For any \(p > 0\) and \(q > 0\), the \((p,q)\) integers \([k]_{p,q}\) are defined as follows:

$$ [k]_{p,q} = p^{k-1} + p^{k-2}q + p^{k-3}q^{2} + \cdots + pq^{k-2} + q ^{k-1} = \textstyle\begin{cases} \frac{p^{k} - q^{k}}{p - q}, & \text{when } p \neq q \neq 1 \\ {k} p^{k-1}, & \text{when } p = q \neq 1 \\ [k ] _{q}, & \text{when } p=1 \\ {k}, & \text{when } p = q = 1 \end{cases} $$
(3.1)

\(k = 0,1,2,3,4,\ldots \) .

Also,

$$ [k]_{p,q}! =\prod_{j=1}^{k}[J]_{p,q}=[k]_{p,q}[k-1]_{p,q} \cdots [1]_{p,q}, \quad k = 1,2,3,\ldots , $$
(3.2)

and

$$\begin{aligned}& \begin{bmatrix} n \\ k \end{bmatrix}_{p,q} = \frac{[n]_{p,q}!}{[k]_{p,q}![n-k]_{p,q}!},\quad \text{for } k = 1,2,3,\ldots , \end{aligned}$$
(3.3)
$$\begin{aligned}& (ax + by)^{n}_{p,q} = \sum_{i= 0}^{n}p^{\frac{(n-i)(n-i-1)}{2}} q^{\frac{i(i - 1)}{2}} \begin{bmatrix} n \\ i \end{bmatrix}_{p,q} (ax)^{n-i} (by)^{i}, \end{aligned}$$
(3.4)
$$\begin{aligned}& (x + y)^{n}_{p,q} =\prod _{i=0}^{n-1} \bigl(p^{i}x+q^{i}y \bigr)=(x + y) (px + qy) \bigl(p ^{2}x + q^{2}y \bigr)\cdots \bigl(p^{n-1}x + q^{n-1}y \bigr). \end{aligned}$$
(3.5)

For \(y=0\), formula (3.5) becomes

$$\begin{aligned}& (x + 0)^{n}_{p,q}=p^{\frac{n(n-1)}{2}}x^{n}, \\& (1 - x)^{n}_{p,q} = (1 - x) (p - qx) \bigl(p^{2} - q^{2}x \bigr)\cdots \bigl(p^{n-1} - q ^{n-1}x \bigr). \end{aligned}$$
(3.6)

For \(x=0\), formula (3.6) becomes

$$ (1- 0)^{n}_{p,q}=p^{\frac{n(n-1)}{2}}. $$

Now we define some useful notations which are used in this paper. For any nonnegative integer k, we have

$$\begin{aligned}& [n+k]_{p,q}=p^{n}[k]_{p,q}+q^{k}[n]_{p,q}, \end{aligned}$$
(3.7)
$$\begin{aligned}& [k]^{2}_{p,q}=p^{k-1}[k]_{p,q}+q[k]_{p,q}[k-1]_{p,q}, \end{aligned}$$
(3.8)
$$\begin{aligned}& [k]^{3}_{p,q}=p^{k-1}q^{k-1}[k]_{p,q}+p[k]^{2}_{p,q}[k-1]_{p,q}+q^{k}[k]_{p,q}[k-1]_{p,q}, \end{aligned}$$
(3.9)
$$\begin{aligned}& \begin{aligned}[b] [k]^{4}_{p,q}&=p^{2k-2}q^{k-1}[k]_{p,q}+p[k]^{3}_{p,q}[k-1]_{p,q} \\ &\quad {}+q ^{k}[k]^{2}_{p,q}[k-1]_{p,q}+p^{k-1}q^{k}[k]_{p,q}[k-1]_{p,q}. \end{aligned} \end{aligned}$$
(3.10)

For \(p=1\) in (3.1)–(3.10) all these reduce to q-analogues.

Now first of all, we construct a \((p,q)\)-analogue of Bernstein–Schurer–Stancu operators as follows:

$$ \begin{aligned}[b] S^{\alpha ,\beta }_{n,r}(f;p,q;x)&= \frac{1}{p^{\frac{(n+r)(n+r-1)}{2}}}\sum_{k=0}^{n+r} \begin{bmatrix} n+r \\ k \end{bmatrix}p^{\frac{k(k-1)}{2}}x^{k} \\ &\quad {}\times\prod_{j=0}^{n+r-k-1} \bigl(p^{j}-q^{j}x \bigr)f \biggl(\frac{p^{n+r-k}[k]+\alpha }{[n]+\beta } \biggr) \end{aligned} $$
(3.11)

for any \(x\in [0,1+r]\) and \(0< q< p\leq 1\), where r is a nonnegative integer.

Lemma 3.1

The operators (3.11) satisfy the following properties for the test functions\(e_{i}=x^{i}\) (\(i=0,1,2,3,4\)):

  1. (i)

    \(S^{\alpha ,\beta }_{n,r}(e_{0};p,q;x)=1\),

  2. (ii)

    \(S^{\alpha ,\beta }_{n,r}(e_{1};p,q;x)=\frac{[n+r]_{p,q}x+ \alpha }{[n]_{p,q}+\beta }\),

  3. (iii)

    \(S^{\alpha ,\beta }_{n,r}(e_{2};p,q;x)=\frac{[n+r]_{p,q} ^{2}x^{2}+p^{n+r-1}[n+r]_{p,q}x(1-x)+2\alpha [n+r]_{p,q}x+\alpha ^{2}}{([n]_{p,q}+ \beta )^{2}}\),

  4. (iv)

    \(S^{\alpha ,\beta }_{n,r}(e_{3};p,q;x) =\frac{[n+r]_{p,q}p ^{n+r} ( p^{-2}-[n+r-1]_{p,q}+3\alpha +3\alpha ^{2} )x}{([n]_{p,q}+ \beta )^{3}}+ \frac{[n+r]_{p,q}[n+r-1]_{p,q} \lbrace p^{2}q+p^{n+r-3}[2]_{p,q}+p ^{n+r}-p^{n+r-1}[2]_{p,q}+p^{n+r-2}q[2]_{p,q}+3\alpha q \rbrace x ^{2}}{([n]_{p,q}+\beta )^{3}}+\frac{[n+r]_{p,q}[n+r-1]_{p,q}[n+r-2]_{p,q}\{p^{2}q^{2}-pq^{2}+q ^{3}\}x^{3}+\alpha ^{3}}{([n]_{p,q}+\beta )^{3}}\).

  5. (v)

    \(S^{\alpha ,\beta }_{n,r}(e_{4};p,q;x)=\frac{[n+r]_{p,q}\{4 \alpha ^{3} p^{3(n+r-1)}+4\alpha p^{2(n+r-1)}+6\alpha ^{2}p^{n+r-1}\}x}{([n]_{p,q}+ \beta )^{4}}+\frac{[n+r]_{p,q}[n+r-1]_{p,q}\{6\alpha ^{2}q+p^{n+r-1}+2qp^{2(n+r-1)}+q[2]_{p,q}p ^{2(n+r)-3}+4\alpha q(p^{n+r-1}+[2]_{p,q}p^{n+r-2})\}x^{2}}{([n]_{p,q}+ \beta )^{4}}+\frac{[n+r]_{p,q}[n+r-1]_{p,q}[n+r-2]_{p,q}\{4\alpha pq^{2}(1-p)+4 \alpha q^{3}+pq+q([2]_{p,q}+q)p^{n+r}-qp^{n+r+1}+2q^{3}p^{n+r-1}\}x ^{3}}{([n]_{p,q}+\beta )^{4}}+\frac{[n+r]_{p,q}[n+r-1]_{p,q}[n+r-2]_{p,q}[n+r-3]_{p,q}p^{4}q^{4}x ^{4}+\alpha ^{4}}{([n]_{p,q}+\beta )^{4}}\).

Proof

$$\begin{aligned}& S^{\alpha ,\beta }_{n,r}(e_{0};p,q;x)=\frac{1}{p^{ \frac{(n+r)(n+r-1)}{2}}}\sum _{k=0}^{n+r} \begin{bmatrix} n+r \\ k \end{bmatrix}p^{\frac{k(k-1)}{2}}x^{k}\prod _{j=0}^{n+r-k-1} \bigl(p^{j}-q^{j}x \bigr)=1, \end{aligned}$$
(3.12)
$$\begin{aligned}& \begin{aligned} S^{\alpha ,\beta }_{n,r}(e_{1};p,q;x) &=\frac{1}{p^{ \frac{(n+r)(n+r-1)}{2}}}\sum_{k=0}^{n+r} \frac{[n+r]_{p,q}!}{[k]_{p,q}![n+r-k]_{p,q}!}p^{\frac{k(k-1)}{2}}x ^{k} \biggl( \frac{p^{n+r-k}[k]_{p,q}+\alpha }{[n]_{p,q}+\beta } \biggr) \\ &\quad {}\times \prod_{j=0}^{n+r-k-1} \bigl(p^{j}-q^{j}x \bigr) \\ &=\frac{1}{p^{\frac{(n+r)(n+r-1)}{2}}}\sum_{k=1}^{n+r} \frac{[n+r]_{p,q}!}{[k-1]_{p,q}![n+r-k]_{p,q}!}p^{\frac{k(k-1)}{2}}x ^{k} \frac{p^{n+r-k}}{[n]_{p,q}+\beta } \\ &\quad {}\times \prod_{j=0}^{n+r-k-1} \bigl(p^{j}-q^{j}x \bigr)+\frac{\alpha }{[n]_{p,q}+ \beta } \\ &=\frac{[n+r]_{p,q}}{p^{\frac{(n+r)(n+r-3)}{2}}}\sum_{k=0}^{n+r} \frac{[n+r-1]_{p,q}!}{[k]_{p,q}![n+r-k-1]_{p,q}!}p^{ \frac{k^{2}-k-2}{2}}x^{k+1} \frac{1}{[n]_{p,q}+\beta } \\ &\quad {}\times \prod_{j=0}^{n+r-k-1} \bigl(p^{j}-q^{j}x \bigr)+\frac{\alpha }{[n]_{p,q}+ \beta } \\ &=\frac{[n+r]_{p,q}x}{[n]_{p,q}+\beta } +\frac{\alpha }{[n]_{p,q}+ \beta }, \end{aligned} \\& S^{\alpha ,\beta }_{n,r}(e_{2};p,q;x) = \frac{1}{p^{ \frac{(n+r)(n+r-1)}{2}}}\sum _{k=0}^{n+r} \frac{[n+r]_{p,q}!}{[k]_{p,q}![n+r-k]_{p,q}!}p^{\frac{k(k-1)}{2}}x ^{k}\prod_{j=0}^{n+r-k-1} \bigl(p^{j}-q^{j}x \bigr) \\& \hphantom{S^{\alpha ,\beta }_{n,r}(e_{2};p,q;x) }\quad {}\times \biggl(\frac{p^{n+r-k}[k]_{p,q}+\alpha }{[n]_{p,q}+\beta } \biggr) ^{2} \\& \hphantom{S^{\alpha ,\beta }_{n,r}(e_{2};p,q;x) }=\frac{1}{p^{\frac{(n+r)(n+r-1)}{2}}}\sum_{k=0}^{n+r} \frac{[n+r]_{p,q}!}{[k]_{p,q}![n+r-k]_{p,q}!}p^{\frac{k(k-1)}{2}}x ^{k}\prod _{j=0}^{n+r-k-1} \bigl(p^{j}-q^{j}x \bigr) \\& \hphantom{S^{\alpha ,\beta }_{n,r}(e_{2};p,q;x) }\quad {}\times \frac{p^{2(n+r-k)}[k]_{p,q}^{2}+\alpha ^{2}+2\alpha p^{n+r-k}[k]_{p,q}}{([n]_{p,q}+ \beta )^{2}} \\& \hphantom{S^{\alpha ,\beta }_{n,r}(e_{2};p,q;x) }=\frac{[n+r]_{p,q}^{2}x^{2}+p^{n+r-1}[n+r]_{p,q}x(1-x)+2\alpha [n+r]_{p,q}x+ \alpha ^{2}}{([n]_{p,q}+\beta )^{2}}, \end{aligned}$$
(3.13)
$$\begin{aligned}& \begin{aligned}[b] S^{\alpha ,\beta }_{n,r}(e_{3};p,q;x) &= \frac{1}{p^{ \frac{(n+r)(n+r-1)}{2}}}\sum_{k=0}^{n+r} \frac{[n+r]_{p,q}!}{[k]_{p,q}![n+r-k]_{p,q}!}p^{\frac{k(k-1)}{2}}x ^{k}\prod _{j=0}^{n+r-k-1} \bigl(p^{j}-q^{j}x \bigr) \\ &\quad {}\times \frac{p^{3(n+r-k)}[k]^{3}_{p,q}+\alpha ^{3}+3\alpha p^{2(n+r-k)}[k]^{2} _{p,q}+3\alpha ^{2}p^{n+r-k}[k]_{p,q}}{([n]_{p,q}+\beta )^{3}}. \end{aligned} \end{aligned}$$
(3.14)

After solving, we get

$$\begin{aligned} =&\frac{[n+r]_{p,q}p^{n+r} ( p^{-2}-[n+r-1]_{p,q}+3\alpha +3 \alpha ^{2} )x}{([n]_{p,q}+\beta )^{3}} \\ &{}+ \bigl([n+r]_{p,q}[n+r-1]_{p,q} \bigl\lbrace p^{2}q+p^{n+r-3}[2]_{p,q}+p ^{n+r} \\ &{}-p^{n+r-1}[2]_{p,q}+p^{n+r-2}q[2]_{p,q}+3 \alpha q \bigr\rbrace x ^{2} \bigr) \\ &{}/ \bigl( \bigl([n]_{p,q}+\beta \bigr)^{3} \bigr) \\ &{}+\frac{[n+r]_{p,q}[n+r-1]_{p,q}[n+r-2]_{p,q}\{p^{2}q^{2}-pq^{2}+q ^{3}\}x^{3}+\alpha ^{3}}{([n]_{p,q}+\beta )^{3}}. \end{aligned}$$

Finally, we have

$$\begin{aligned} &S^{\alpha ,\beta }_{n,r}(e_{4};p,q;x) \\ &\quad =\frac{1}{p^{ \frac{(n+r)(n+r-1)}{2}}} \sum_{k=0}^{n+r} \frac{[n+r]_{p,q}!}{[k]_{p,q}![n+r-k]_{p,q}!}p^{\frac{k(k-1)}{2}}x ^{k}\prod _{j=0}^{n+r-k-1} \bigl(p^{j}-q^{j}x \bigr) \\ &\qquad {}\times \frac{p^{4(n+r-k)}[k]^{4}_{p,q}+\alpha ^{4}+6\alpha ^{2} p^{2(n+r-k)}[k]^{2} _{p,q}+4\alpha p^{3(n+r-k)}[k]^{3}_{p,q}+4\alpha ^{3}p^{(n+r-k)}[k]_{p,q}}{([n]_{p,q}+ \beta )^{4}}. \end{aligned}$$

By using (3.8), (3.9), and (3.10), we obtain \((v)\). □

Rao and Wafi [36] introduced a \((p,q)\)-analogue of Bivariate–Schurer–Stancu operators in the following form:

Let \(I_{1}\times I_{2}=[0,1+r_{1}]\times [0,1+r_{2}]\), \(0< q_{1}< p_{1} \leq 1\), \(0< q_{2}< p_{2}\leq 1\), and \(m,n\in \mathbb{N}\times \mathbb{N}\). Then, for any \(f\in C(I_{1}\times I_{2})\), the operators \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}:C(I _{1}\times I_{2})\rightarrow C([0,1]\times [0,1])\) are defined by

$$\begin{aligned} S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(f;p _{1},p_{2},q_{1},q_{2};x,y) =&\sum_{k_{1}=0}^{m+r_{1}}\sum _{k_{2}=0} ^{n+r_{2}}s^{p_{1},q_{1}}_{m,r_{1},k_{1}}(x)s^{p_{2},q_{2}}_{n,r_{2},k _{2}}(y) \\ &{}\times f \biggl(\frac{p^{m-k_{1}}[k_{1}]_{p_{1},q_{1}}+\alpha _{1}}{[m]_{p _{1},q_{1}}+\beta _{1}},\frac{p^{n-k_{2}}[k_{2}]_{p_{2},q_{2}}+\alpha _{2}}{[n]_{p_{2},q_{2}}+\beta _{2}} \biggr), \end{aligned}$$
(3.15)

where

$$ \begin{gathered} s^{p_{1},q_{1}}_{m,r_{1},k_{1}}(x)=\frac{1}{p_{1}^{\frac{(m+r_{1})(m+r _{1}-1)}{2}}} \begin{bmatrix} m+r_{1} \\ k_{1} \end{bmatrix}_{p_{1},q_{1}}p_{1}^{\frac{k_{1}(k_{1}-1)}{2}}x^{k_{1}} \prod_{s=0}^{m+r_{1}-k_{1}-1} \bigl(p_{1}^{s}-q^{s}_{1}x \bigr), \\ s^{p_{2},q_{2}}_{n,r_{2},k_{2}}(y)=\frac{1}{p_{2}^{\frac{(n+r_{2})(n+r _{2}-1)}{2}}} \begin{bmatrix} n+r_{2} \\ k_{2} \end{bmatrix}_{p_{2},q_{2}}p_{2}^{\frac{k_{2}(k_{2}-1)}{2}}y^{k_{2}} \prod_{s=0}^{n+r_{2}-k_{2}-1} \bigl(p_{2}^{s}-q^{s}_{2}y \bigr). \end{gathered} $$

In the following, we define a \((p,q)\)-analogue of the bivariate Schurer–Stancu operators as follows:

Let \(I_{1}\times I_{2}=[0,1+r_{1}]\times [0,1+r_{2}]\), \(0< q_{1}< p_{1} \leq 1\), \(0< q_{2}< p_{2}\leq 1\), and \(m,n\in \mathbb{N}\times \mathbb{N}\). Then, for any \(f\in C(I_{1}\times I_{2})\), the operators \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}:C(I _{1}\times I_{2})\rightarrow C([0,1]\times [0,1])\) are defined by

$$ \begin{aligned}[b] &S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(f;p _{1},p_{2},q_{1},q_{2};x,y) \\ &\quad =\sum_{k_{1}=0}^{m+r_{1}}\sum _{k_{2}=0} ^{n+r_{2}}s^{p_{1},q_{1}}_{m,r_{1},k_{1}}(x)s^{p_{2},q_{2}}_{n,r_{2},k _{2}}(y) \\ &\qquad {}\times f \biggl(\frac{p^{m+r_{1}-k_{1}}[k_{1}]_{p_{1},q_{1}}+\alpha _{1}}{[m]_{p_{1},q_{1}}+\beta _{1}},\frac{p^{n+r_{2}-k_{2}}[k_{2}]_{p _{2},q_{2}}+\alpha _{2}}{[n]_{p_{2},q_{2}}+\beta _{2}} \biggr), \end{aligned} $$
(3.16)

where

$$\begin{aligned}& s^{p_{1},q_{1}}_{m,r_{1},k_{1}}(x)=\frac{1}{p_{1}^{\frac{(m+r_{1})(m+r _{1}-1)}{2}}} \begin{bmatrix} m+r_{1} \\ k_{1} \end{bmatrix}_{p_{1},q_{1}}p_{1}^{\frac{k_{1}(k_{1}-1)}{2}}x^{k_{1}} \prod_{s=0}^{m+r_{1}-k_{1}-1} \bigl(p_{1}^{s}-q^{s}_{1}x \bigr), \\& s^{p_{2},q_{2}}_{n,r_{2},k_{2}}(y)=\frac{1}{p_{2}^{\frac{(n+r_{2})(n+r _{2}-1)}{2}}} \begin{bmatrix} n+r_{2} \\ k_{2} \end{bmatrix}_{p_{2},q_{2}}p_{2}^{\frac{k_{2}(k_{2}-1)}{2}}y^{k_{2}} \prod_{s=0}^{n+r_{2}-k_{2}-1} \bigl(p_{2}^{s}-q^{s}_{2}y \bigr). \end{aligned}$$

The operators (3.16) satisfy the following properties.

Lemma 3.2

Let\(e_{i,j}(x,y)=x^{i}y^{j}\), \(0\leq i\), \(j\leq 2\), be two-dimensional test functions. Then

  1. (i)

    \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{0,0};p_{1},p_{2},q_{1},q_{2};x,y)=1\),

  2. (ii)

    \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{1,0};p_{1},p_{2},q_{1},q_{2};x,y)=\frac{[m+r_{1}]_{p_{1},q _{1}}x+\alpha _{1}}{[m]_{p_{1},q_{1}}+\beta _{1}}\),

  3. (iii)

    \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}}(e_{0,1};p_{1},p_{2},q_{1},q_{2};x,y)=\frac{[n+r_{2}]_{p _{2},q_{2}}y+\alpha _{2}}{[n]_{p_{2},q_{2}}+\beta _{2}}\),

  4. (iv)

    \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{2,0};p_{1},p_{2},q_{1},q_{2};x,y)=\frac{[m+r_{1}]_{p_{1},q _{1}}(p_{1}^{m+r_{1}-1}+2\alpha _{1})x}{([m]_{p_{1},q_{1}}+\beta _{1})^{2}}+\frac{q _{1}[m+r_{1}]_{p_{1},q_{1}}[m+r_{1}-1]_{p_{1},q_{1}}x^{2}}{([m]_{p _{1},q_{1}} +\beta _{1})^{2}} +\frac{\alpha _{1}^{2}}{([m]_{p_{1},q_{1}}+\beta _{1})^{2}}\),

  5. (v)

    \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{0,2};p_{1},p_{2},q_{1},q_{2};x,y)=\frac{[n+r_{2}]_{p_{2},q _{2}}(p_{2}^{n+r_{2}-1}+2\alpha _{2})y}{([n]_{p_{2},q_{2}}+\beta _{2})^{2}}+\frac{q _{2}[n+r_{2}]_{p_{2},q_{2}}[n+r_{2}-1]_{p_{2},q_{2}}y^{2}}{([n]_{p _{2},q_{2}}+\beta _{2})^{2}} +\frac{\alpha _{2}^{2}}{([n]_{p_{2},q_{2}}+\beta _{2})^{2}}\).

Proof

From Lemma 3.1, \((i)\) follows immediately.

$$ \begin{aligned} &S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{0,0};p _{1},p_{2},q_{1},q_{2};x,y) \\ &\quad =S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{0};p_{1},q_{1};x,y)S^{\alpha _{1},\beta _{1}, \alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{0};p_{2},q_{2};x,y)=1. \end{aligned} $$

Now for \((ii)\), again from Lemma 3.1, we have

$$\begin{aligned} S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{1,0};p _{1},p_{2},q_{1},q_{2};x,y) =&S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{1};p_{1},q_{1};x,y)S^{\alpha _{1},\beta _{1}, \alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{0};p_{2},q_{2};x,y) \\ =&\frac{[m+r_{1}]_{p_{1},q_{1}}x+\alpha _{1}}{[m]_{p_{1},q_{1}}+\beta _{1}}. \end{aligned}$$

Similarly, we obtain \((iii)\).

Further, for \((iv)\), we have

$$\begin{aligned} S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{2,0};p _{1},p_{2},q_{1},q_{2};x,y) =&S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{2};p_{1},q_{1};x,y)S^{\alpha _{1},\beta _{1}, \alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{0};p_{2},q_{2};x,y) \\ =&\frac{[m+r_{1}]_{p_{1},q_{1}}(p_{1}^{m+r_{1}-1}+2\alpha _{1})x}{([m]_{p _{1},q_{1}}+\beta _{1})^{2}}+\frac{\alpha _{1}^{2}}{([m]_{p_{1},q_{1}}+ \beta _{1})^{2}} \\ &{}+\frac{q_{1}[m+r_{1}]_{p_{1},q_{1}}[m+r_{1}-1]_{p_{1},q_{1}}x^{2}}{([m]_{p _{1},q_{1}}+\beta _{1})^{2}}. \end{aligned}$$

In a similar way, we get (v). □

Now, motivated by q-Bernstein–Schurer–Stancu type GBS operators (2.1), we construct \((p,q)\)-Bernstein–Schurer–Stancu type GBS operators as follows.

For any \((x,y)\in I=[0,1+r_{1}]\times [0,1+r_{2}]\), the \((p,q)\)-Bernstein–Schurer–Stancu type GBS operators \(U^{\alpha _{1}, \beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}:C_{b}([0,1+r_{1}] \times [0,1+r_{2}])\rightarrow C_{b}([0,1]\times [0,1])\) associated with \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\) are defined by

$$\begin{aligned} U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(f;p _{1},p_{2},q_{1},q_{2};x,y) =&\frac{1}{p_{1}^{ \frac{(m+r_{1})(m+r_{1}-1)}{2}}}\frac{1}{p_{2}^{\frac{(n+r_{2})(n+r _{2}-1)}{2}}}\sum_{k_{1}=0}^{m+r_{1}} \sum_{k_{2}=0}^{n+r_{2}} \begin{bmatrix} m+r_{1} \\ k_{1} \end{bmatrix} \\ &{}\times \begin{bmatrix} n+r_{2} \\ k_{2} \end{bmatrix}p_{1}^{\frac{k_{1}(k_{1}-1)}{2}}p_{2}^{ \frac{k_{2}(k_{2}-1)}{2}} \prod_{s=0}^{m+r_{1}-k_{1}-1} \bigl(p_{1}^{s}-q ^{s}_{1}x \bigr) \\ &{}\times \prod_{t=0}^{n+r_{2}-k_{2}-1} \bigl(p_{2}^{t}-q^{t}_{2}y \bigr)x^{k_{1}}y ^{k_{2}}\{f_{k_{1}}+f_{k_{2}}-f_{k_{1}}f_{k_{2}} \}, \end{aligned}$$
(3.17)

where

$$\begin{aligned}& f_{k_{1}}(y)=f \biggl(\frac{p_{1}^{m+r_{1}-k_{1}}[k_{1}]_{p_{1},q_{1}}+ \alpha _{1}}{[m]_{p_{1},q_{1}}+\beta _{1}},y \biggr), \qquad f_{k_{2}}(x)=f \biggl(x,\frac{p_{2}^{n+r_{2}-k_{2}}[k_{2}]_{p_{2},q _{2}}+\alpha _{2}}{[n]_{p_{2},q_{2}}+\beta _{2}} \biggr), \\& f_{k_{1}}f_{k_{2}}(x,y)=f \biggl(\frac{p_{1}^{m+r_{1}-k_{1}}[k_{1}]_{p _{1},q_{1}}+\alpha _{1}}{[m]_{p_{1},q_{1}}+\beta _{1}}, \frac{p_{2}^{n+r _{2}-k_{2}}[k_{2}]_{p_{2},q_{2}}+\alpha _{2}}{[n]_{p_{2},q_{2}}+\beta _{2}} \biggr). \end{aligned}$$

Lemma 3.3

Let\(\psi _{x}, \psi _{y}:I\rightarrow \mathbb{R}\)be defined as

$$ \psi _{x}(u,v)= \vert u-x \vert ,\qquad \psi _{y}(u,v)= \vert v-y \vert , \quad \textit{for any } (u,v)\in I \textit{ and } (x,y)\in I, $$

where\(I=[0,1+r_{1}]\times [0,1+r_{2}]\). Then the following equalities hold:

$$ \begin{aligned}[b] &S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{x};p_{1},p_{2},q_{1},q_{2};x,y \bigr) \\ &\quad =\frac{(((p^{r_{1}}_{1}-1)[m]_{p_{1},q_{1}}+q^{m}_{1}[r_{1}]_{p_{1},q _{1}}-\beta _{1})x+\alpha _{1})^{2}+p^{m+r_{1}-1}_{1}[m+r_{1}]_{p_{1},q _{1}}x(1-x)}{([m]_{p_{1},q_{1}}+\beta _{1})^{2}} \end{aligned} $$
(3.18)

and

$$\begin{aligned} &S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{y};p_{1},p_{2},q_{1},q_{2};x,y \bigr) \\ &\quad =\frac{(((p^{r_{2}}_{2}-1)[n]_{p_{2},q_{2}}+q^{n}_{2}[r_{2}]_{p_{2},q _{2}}-\beta _{2})y+\alpha _{2})^{2}+p^{n+r_{2}-1}_{2}[n+r_{2}]_{p_{2},q _{2}}y(1-y)}{([n]_{p_{2},q_{2}}+\beta _{2})^{2}}. \end{aligned}$$
(3.19)

Similarly, we have

$$\begin{aligned} &S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{4}_{x };p_{1},p_{2},q_{1},q_{2};x,y \bigr) \\ &\quad =\frac{1}{([m]_{p_{1},q_{1}}+\beta _{1})^{4}} [4\alpha _{1} \biggl\{ [m+r _{1}]_{p_{1},q_{1}}p^{m+r_{1}-1}_{1} \biggl(\alpha _{1}^{2} p^{2(m+r_{1}-1)} _{1}+p^{m+r_{1}-1}_{1}+ \frac{3}{2}\alpha _{1} \biggr) \\ &\qquad {}-\alpha _{1}^{2} \bigl([m]_{p_{1},q_{1}}+ \beta _{1} \bigr) \biggr\} x+ \bigl\{ [m+r_{1}]_{p_{1},q _{1}}[m+r_{1}-1]_{p_{1},q_{1}} \bigl(6\alpha _{1}^{2}q_{1} \\ &\qquad {}+p^{m+r_{1}-1}_{1} \bigl(1+2q _{1}p^{m+r_{1}-1}_{1} \bigr)+q_{1}[2]_{p_{1},q_{1}}p^{2(m+r_{1})-3}_{1}+4 \alpha _{1} q_{1}p^{m+r _{1}-1}_{1} \bigl(1+[2]_{p_{1},q_{1}}p^{-1}_{1} \bigr) \bigr) \\ &\qquad {}-4 \bigl([m]_{p_{1},q_{1}}+\beta _{1} \bigr)[m+r_{1}]_{p_{1},q_{1}} p_{1}^{m+r_{1}} \bigl(p_{1}^{-2}-[m+r_{1}-1]_{p_{1},q_{1}}+3 \alpha _{1}(1+\alpha _{1}) \bigr) \\ &\qquad {}+6\alpha _{1}^{2} \bigl([m]_{p_{1},q_{1}}+\beta _{1} \bigr)^{2} \bigr\} x^{2}+ \bigl\{ [m+r_{1}]_{p_{1},q_{1}} q_{1}[m+r_{1}-1]_{p_{1},q_{1}}[m+r_{1}-2]_{p_{1},q_{1}} \\ &\qquad {}\times \bigl(4 \alpha _{1} p_{1}q_{1}(1-p_{1})+q_{1}^{2} \bigl(4\alpha _{1}-2p_{1}^{m+r_{1}-1} \bigr)+p _{1} \bigl(1-p_{1}^{m+r_{1}} \bigr)+p_{1}^{m+r_{1}} \bigl([2]_{p_{1},q_{1}}+q_{1} \bigr) \bigr) \\ &\qquad {}-4[m+r_{1}]_{p_{1},q_{1}}[m+r _{1}-1]_{p_{1},q_{1}} \bigl([m]_{p_{1},q_{1}}+\beta _{1} \bigr) \bigl(q_{1} \bigl(p_{1}^{2}+3 \alpha _{1} \bigr) \\ &\qquad {}+p_{1}^{m+r_{1}}+[2]_{p_{1},q_{1}}p^{m+r_{1}-3}_{1} \bigl(1-p^{2}_{1}+q _{1}p_{1} \bigr) \bigr)+ 12\alpha _{1} \bigl([m]_{p_{1},q_{1}}+\beta _{1} \bigr)^{2}[m+r_{1}]_{p _{1},q_{1}} \\ &\qquad {}+6p_{1}^{m+r_{1}-1}[m+r_{1}]_{p_{1},q_{1}} \bigl([m]_{p_{1},q_{1}}+\beta _{1} \bigr)^{2}-4\alpha _{1} \bigl([m]_{p_{1},q_{1}}+\beta _{1} \bigr)^{3} \bigr\} x^{3} \\ &\qquad {}+[m+r _{1}]_{p_{1},q_{1}} [m+r_{1}-1]_{p_{1},q_{1}}[m+r_{1}-2]_{p_{1},q_{1}}[m+r_{1}-3]_{p _{1},q_{1}}p_{1}^{4}q_{1}^{4}x^{4}+ \alpha _{1}^{4} \\ &\qquad {}+ \bigl([m]_{p_{1},q_{1}}+ \beta _{1} \bigr)^{4}x^{4} + \bigl\{ -6p^{m+r_{1}-1}[m+r_{1}]_{p_{1},q_{1}} \bigl([m]_{p_{1},q_{1}}+\beta _{1} \bigr)^{2} \\ &\qquad {}-4 \bigl([m]_{p_{1},q_{1}}+\beta _{1} \bigr)[m+r_{1}]_{p_{1},q_{1}}[m+r _{1}-1]_{p_{1},q_{1}}[m+r_{1}-2]_{p_{1},q_{1}}q^{2}_{1} \bigl(p^{2}_{1}-p_{1}+q_{1} \bigr) \\ &\qquad {}-4 \bigl([m]_{p _{1},q_{1}}+\beta _{1} \bigr)^{3}[m+r_{1}]_{p_{1},q_{1}} \bigr\} x^{4} ]. \end{aligned}$$
(3.20)

Finally, we obtain

$$\begin{aligned} &S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{4}_{y };p_{1},p_{2},q_{1},q_{2};x,y \bigr) \\ &\quad =\frac{1}{([n]_{p_{2},q_{2}}+\beta _{2})^{4}} [4\alpha _{2} \biggl\{ [n+r _{2}]_{p_{2},q_{2}}p^{n+r_{2}-1}_{2} \biggl(\alpha _{2}^{2} p^{2(n+r_{2}-1)} _{2}+p^{n+r_{2}-1}_{2}+ \frac{3}{2}\alpha _{2} \biggr) \\ &\qquad {}-\alpha _{2}^{2} \bigl([n]_{p,q}+ \beta _{2} \bigr) \biggr\} y + \bigl\{ [n+r_{2}]_{p_{2},q_{2}}[n+r_{2}-1]_{p_{2},q_{2}} \bigl(6\alpha _{2}^{2}q _{2} \\ &\qquad {}+p^{n+r_{2}-1}_{2} \bigl(1+2q_{2}p^{n+r_{2}-1}_{2} \bigr)+q_{2}[2]_{p,q}p^{2(n+r _{2})-3}_{2}+4\alpha _{2} q_{2}p^{n+r_{2}-1}_{2} \bigl(1+[2]_{p_{2},q_{2}}p^{-1}_{2} \bigr) \bigr) \\ &\qquad {}-4 \bigl([n]_{p _{2},q_{2}}+\beta _{2} \bigr)[n+r_{2}]_{p_{2},q_{2}}p_{2}^{n+r_{2}} \bigl(p_{2} ^{-2}-[n+r_{2}-1]_{p_{2},q_{2}}+3\alpha _{2}(1+\alpha _{2}) \bigr) \\ &\qquad {}+6 \alpha _{2}^{2} \bigl([n]_{p_{2},q_{2}}+ \beta _{2} \bigr)^{2} \bigr\} y^{2}+ \bigl\{ [n+r_{2}]_{p_{2},q_{2}}[n+r_{2}-1]_{p_{2},q_{2}}[n+r _{2}-2]_{p_{2},q_{2}} \\ &\qquad {}\times q_{2} \bigl(4\alpha _{2} p_{2}q_{2}(1-p_{2})+q_{2}^{2} \bigl(4\alpha _{2}-2p _{2}^{n+r_{2}-1} \bigr)+p_{2} \bigl(1-p_{2}^{n+r_{2}} \bigr)+p_{2}^{n+r_{2}} \bigl([2]_{p_{2},q _{2}}+q_{2} \bigr) \bigr) \\ &\qquad {}-4[n+r_{2}]_{p_{2},q_{2}}[n+r_{2}-1]_{p_{2},q_{2}} \bigl([n]_{p_{2},q_{2}}+ \beta _{2} \bigr) \bigl(q_{2} \bigl(p_{2}^{2}+3\alpha _{2} \bigr)+p_{2}^{n+r_{2}} \\ &\qquad {}+[2]_{p_{2},q _{2}}p^{n+r_{2}-3}_{2} \bigl(1-p^{2}_{2}+q_{2}p_{2} \bigr) \bigr)+ 12\alpha _{2} \bigl([n]_{p_{2},q_{2}}+\beta _{2} \bigr)^{2}[n+r_{2}]_{p_{2},q _{2}} \\ &\qquad {}+6p_{2}^{n+r_{2}-1}[n+r_{2}]_{p_{2},q_{2}} \bigl([n]_{p_{2},q_{2}}+ \beta _{2} \bigr)^{2}-4\alpha \bigl([n]_{p_{2},q_{2}}+\beta _{2} \bigr)^{3} \bigr\} y^{3} \\ &\qquad {}+[n+r_{2}]_{p_{2},q_{2}}[n+r_{2}-1]_{p_{2},q_{2}}[n+r_{2}-2]_{p_{2},q _{2}}[n+r_{2}-3]_{p_{2},q_{2}}p_{2}^{4}q_{2}^{4}y^{4} \\ &\qquad {}+ \alpha _{2}^{4}+ \bigl([n]_{p _{2},q_{2}}+ \beta _{2} \bigr)^{4}y^{4}+ \bigl\{ -6p^{n+r_{2}-1}[n+r_{2}]_{p,q} \bigl([n]_{p_{2},q_{2}}+ \beta _{2} \bigr)^{2} \\ &\qquad {}-4 \bigl([n]_{p _{2},q_{2}}+\beta _{2} \bigr)[n+r_{2}]_{p,q}[n+r_{2}-1]_{p,q}[n+r_{2}-2]_{p,q} \\ &\qquad {}\times q^{2}_{2} \bigl(p^{2}_{2}-p_{2}+q_{2} \bigr)-4 \bigl([n]_{p_{2},q_{2}}+\beta _{2} \bigr)^{3}[n+r_{2}]_{p_{2},q_{2}} \bigr\} y^{4} ]. \end{aligned}$$
(3.21)

Proof

By definition of \(\psi _{x}\), \(\psi _{y}\) and the linearity of \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\), we obtain the results (3.18), (3.19), (3.20), and (3.21). □

Theorem 3.4

Let\(p_{1}=p_{1}^{m}\), \(p_{2}=p_{2}^{n}\)and\(q_{1}=q_{1}^{m}\), \(q_{2}=q_{2}^{n}\)such that

$$ \lim_{m\rightarrow \infty }p_{1}^{m}=\lim _{m\rightarrow \infty }q_{1}^{m}=\lim_{n\rightarrow \infty }p_{2}^{n}= \lim_{n\rightarrow \infty }q_{2}^{n}=1 \quad \textit{with } \lim_{r_{1}\rightarrow \infty }p_{1}^{r_{1}}=\lim _{r_{2}\rightarrow \infty }p_{2}^{r_{2}}=1. $$

Then the sequence\(\{U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}} _{m,n,r_{1},r_{2}}\}\)converges uniformly toffor any\(f\in C_{b}(I)\)on the interval\([0,1]\times [0,1]\).

Proof

The new operators are linear and positive in view of linearity and positivity of q-Bernstein–Schurer–Stancu operators on \([0,1] \times [0,1]\). Now we have to prove that the \((p,q)\)-Bernstein–Schurer–Stancu operators satisfy the hypotheses of Theorem 1.3. By Lemma 3.2\((i)\), we have

$$ S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{0,0};p _{1},p_{2},q_{1},q_{2};x,y)=1. $$

That is, condition \((i)\) of Theorem 1.3 verified.

From the second condition \((ii)\) of Lemma 3.2, we have

$$ \begin{aligned}[b] &S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{1,0};p _{1},p_{2},q_{1},q_{2};x,y) \\ &\quad =x+ \frac{((p^{r_{1}}_{1}-1)[m]_{p_{1},q _{1}}+q^{m}_{1}[r_{1}]_{p_{1},q_{1}}-\beta _{1})x+\alpha _{1}}{[m]_{p _{1},q_{1}}+\beta _{1}}, \end{aligned} $$
(3.22)

i.e., condition \((ii)\) of Theorem 1.3 is verified with

$$ u_{m,n}(x,y)=\frac{((p^{r_{1}}_{1}-1)[m]_{p_{1},q_{1}}+q^{m}_{1}[r _{1}]_{p_{1},q_{1}}-\beta _{1})x+\alpha _{1}}{[m]_{p_{1},q_{1}}+\beta _{1}}. $$

In a similar way, we get

$$ \begin{aligned}[b] &S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{0,1};p _{1},p_{2},q_{1},q_{2};x,y) \\ &\quad =y+ \frac{((p^{r_{2}}_{2}-1)[n]_{p_{2},q _{2}}+q^{n}_{2}[r_{2}]_{p_{2},q_{2}}-\beta _{2})y+\alpha _{2}}{[n]_{p _{2},q_{2}}+\beta _{2}}, \end{aligned} $$
(3.23)

where

$$ v_{m,n}(x,y)=\frac{((p^{r_{2}}_{2}-1)[n]_{p_{2},q_{2}}+q^{n}_{2}[r _{2}]_{p_{2},q_{2}}-\beta _{2})y+\alpha _{2}}{[n]_{p_{2},q_{2}}+\beta _{2}}. $$

From statements \((iv)\) and \((v)\), again by applying Lemma 3.2, we obtain

$$ S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(e_{2,0}+e _{0,2};p_{1},p_{2},q_{1},q_{2};x,y)=x^{2}+y^{2}+ \omega _{m,n}(x,y), $$
(3.24)

where

$$\begin{aligned} &\omega _{m,n}(x,y) \\ &\quad =\bigl(\bigl([m+r_{1}]_{p_{1},q_{1}}^{2}- \bigl([m]_{p_{1},q_{1}}+\beta _{1}\bigr)^{2}\bigr)x ^{2}+p^{m+r_{1}-1}_{1}[m+r_{1}]_{p_{1},q_{1}}x(1-x) \\ &\qquad {}+2\alpha _{1}x[m+r _{1}]_{p_{1},q_{1}}+\alpha ^{2}_{1}\bigr)/\bigl(\bigl([m]_{p_{1},q_{1}}+\beta _{1}\bigr)^{2}\bigr) \\ &\qquad {}+\bigl(\bigl([n+r_{2}]_{p_{2},q_{2}}^{2}- \bigl([n]_{p_{2},q_{2}}+\beta _{2}\bigr)^{2}\bigr)y ^{2}+p^{n+r_{2}-1}_{2}[n+r_{2}]_{p_{2},q_{2}}y(1-y) \\ &\qquad {}+2\alpha _{2}y[n+r _{2}]_{p_{2},q_{2}}+\alpha ^{2}_{2}\bigr)/\bigl(\bigl([n]_{p_{2},q_{2}}+\beta _{2}\bigr)^{2}\bigr). \end{aligned}$$

From (3.22), (3.23), (3.24), and the hypotheses of Theorem 3.4, it follows that

$$ \lim_{m,n \rightarrow \infty }u_{m,n}(x,y)=\lim_{m,n\rightarrow \infty }v_{m,n}(x,y)= \lim_{m,n\rightarrow \infty }\omega _{m,n}(x,y)=0 $$

uniformly on \([0,1]\times [0,1]\).

The desired uniform convergence is verified as a consequence of Theorem 1.3. □

In the next result, we express the degree of approximation of B-continuous functions f by using \((p,q)\)-GBS operators.

Theorem 3.5

For any\(f\in C_{b}(I)\)and\((x,y)\in [0,1]\times [0,1]\), the following estimation holds true:

$$ \bigl\vert f(x,y)-U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(f;p_{1},p_{2},q_{1},q_{2};x,y) \bigr\vert \leq \frac{9}{4} \omega _{\mathrm{mixed}}(\delta _{1},\delta _{2}), $$
(3.25)

where

$$ \begin{aligned}[b] \delta _{1}&=\frac{1}{[m]_{p_{1},q_{1}}+\beta _{1}} \\ &\quad {}\times\sqrt{4\max_{x\in [0,1]} \bigl( \bigl( \bigl(p^{r_{1}}_{1}-1 \bigr)[m]_{p_{1},q_{1}}+q^{m}_{1}[r _{1}]_{p_{1},q_{1}}- \beta _{1} \bigr)x+\alpha _{1} \bigr)^{2}+p^{m+r_{1}-1}_{1}[m+r _{1}]_{p_{1},q_{1}}} \end{aligned} $$
(3.26)

and

$$ \begin{aligned}[b] \delta _{2}&=\frac{1}{[n]_{p_{2},q_{2}}+\beta _{2}} \\ &\quad {}\times\sqrt{4\max_{y\in [0,1]} \bigl( \bigl( \bigl(p^{r_{2}}_{2}-1 \bigr)[n]_{p_{2},q_{2}}+q^{n}_{2}[r _{2}]_{p_{2},q_{2}}- \beta _{2} \bigr)y+\alpha _{2} \bigr)^{2}+p^{n+r_{2}-1}_{2}[n+r _{2}]_{p_{2},q_{2}}}. \end{aligned} $$
(3.27)

Proof

By applying Theorem 1.3, since \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\) is a constant reproducing operator, we have

$$\begin{aligned} & \bigl\vert f(x,y)-U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(f;p_{1},p_{2},q_{1},q_{2};x,y) \bigr\vert \\ &\quad \leq \Bigl\{ 1+\delta ^{-1}_{1}\sqrt {S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{x};p_{1},p_{2},q_{1},q _{2};x,y \bigr)} \\ &\qquad {}+\delta ^{-1}_{2}\sqrt {S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{y};p_{1},p_{2},q_{1},q_{2};x,y \bigr)} \\ &\qquad {}+\delta ^{-1}_{1}\delta ^{-1}_{2} \sqrt{S^{\alpha _{1},\beta _{1}, \alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{x};p_{1},p_{2},q _{1},q_{2};x,y \bigr)S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}} \bigl( \psi ^{2}_{y};p_{1},p_{2},q_{1},q_{2};x,y \bigr)} \Bigr\} \\ &\qquad {}\times \omega _{\mathrm{mixed}}(\delta _{1},\delta _{2}) \end{aligned}$$
(3.28)

for any \(\delta _{1}, \delta _{2}\geq 0\).

Since, for any \((x,y)\in [0,1]\times [0,1]\),

$$ x(1-x)\leq \frac{1}{4} \quad \text{and} \quad y(1-y)\leq \frac{1}{4}. $$

Then

$$\begin{aligned}& S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{x};p_{1},p_{2},q_{1},q_{2};x,y \bigr)\leq \frac{1}{2}\delta _{1}, \end{aligned}$$
(3.29)
$$\begin{aligned}& S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{y};p_{1},p_{2},q_{1},q_{2};x,y \bigr)\leq \frac{1}{2}\delta _{2}, \end{aligned}$$
(3.30)

where \(\delta _{1}\), \(\delta _{2}\) are defined in (3.26) and (3.27), respectively. Thus, from (3.28), (3.29), and (3.30), we obtain the desired result. □

Remark 3.6

Suppose \(p_{1}=p^{m}_{1}\), \(q_{1}=q^{m}_{1}\) and \(p_{2}=p^{n}_{2}\), \(q_{2}=q^{n}_{2}\) such that

$$ \lim_{m\rightarrow \infty }p^{m}_{1}=\lim _{m\rightarrow \infty }q^{m}_{1}=1,\qquad \lim _{n\rightarrow \infty }p^{n}_{2}=\lim_{n\rightarrow \infty }p^{n}_{2}=1 \quad \text{and} \lim_{r_{1}\rightarrow \infty }p_{1}^{r_{1}}= \lim_{r_{2}\rightarrow \infty }p_{2}^{r_{2}}=1. $$

Then

$$ \lim_{m\rightarrow \infty }\delta _{1}=0 \quad \text{and} \quad \lim_{n\rightarrow \infty }\delta _{2}=0. $$

It directly follows from (3.25) that the sequence \(\lbrace U ^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}f \rbrace _{m,n\in \mathbb{N}}\) converges uniformly to f for any \(f\in C_{b}(I)\) on \([0,1]\times [0,1]\).

Let \(\mathrm{Lip}_{M}(\gamma _{1},\gamma _{2})\)\((\gamma _{1},\gamma _{2} \in (0,1])\) be a Lipschitz class defined by

$$ \begin{aligned} &\mathrm{Lip}_{M}(\gamma _{1},\gamma _{2}) \\ &\quad = \bigl\{ f\in C_{b}(I):\Delta _{u,v} [f:x,y ]\leq M \vert u-x \vert ^{\gamma _{1}} \vert v-y \vert ^{\gamma _{2}}, (u,v),(x,y)\in [0,1]\times [0,1] \bigr\} . \end{aligned} $$

Theorem 3.7

Let\(f\in \mathrm{Lip}_{M}(\gamma _{1},\gamma _{2})\). Then, for any\(M>0\)and\(\gamma _{1},\gamma _{2}\in (0,1]\), we have

$$ \bigl\vert f(x,y)-U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(f;p_{1},p_{2},q_{1},q_{2};x,y) \bigr\vert \leq M\delta _{1}^{\frac{\gamma _{1}}{2}}\delta _{2}^{\frac{\gamma _{2}}{2}}. $$
(3.31)

Proof

By the definition of the operators \(U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\) and linearity of the operators \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\), we obtain

$$\begin{aligned} U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}(f;p _{1},p_{2},q_{1},q_{2};x,y) =&S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(f(x,v)+f(u,y)-f(u,v);p_{1},p_{2},q_{1},q _{2};x,y \bigr) \\ =&S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(f(x,y)-\Delta _{u,v} [f:x,y ];p_{1},p_{2},q_{1},q _{2};x,y \bigr) \\ =&f(x,y)S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{00};p_{1},p_{2},q_{1},q_{2};x,y) \\ &{}-S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\Delta _{u,v} [f:x,y ];p_{1},p_{2},q_{1},q_{2};x,y \bigr). \end{aligned}$$

By the hypothesis, we get

$$\begin{aligned} & \bigl\vert f(x,y)-U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(f;p_{1},p_{2},q_{1},q_{2};x,y) \bigr\vert \\ &\quad \leq S^{\alpha _{1},\beta _{1}, \alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\Delta _{u,v} [f:x,y ];p _{1},p_{2},q_{1},q_{2};x,y \bigr) \\ &\quad \leq M S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}} \bigl( \vert u-x \vert ^{\gamma _{1}} \vert v-y \vert ^{\gamma _{2}};p_{1},p_{2},q_{1},q_{2};x,y \bigr) \\ &\quad =M S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl( \vert u-x \vert ^{ \gamma _{1}};p_{1},q_{1};x \bigr) \\ &\qquad {}\times S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}} \bigl( \vert v-y \vert ^{\gamma _{2}};p_{2},q_{2};y \bigr). \end{aligned}$$

By using Hölder’s inequality with \(s_{1}=\frac{2}{\gamma _{1}}\), \(t_{1}=\frac{2}{2-\gamma _{1}}\) and \(s_{2}=\frac{2}{\gamma _{2}}\), \(t_{2}=\frac{2}{2-\gamma _{2}}\), we have

$$\begin{aligned} \bigl\vert f(x,y)-U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(f;p_{1},p_{2},q_{1},q_{2};x,y) \bigr\vert \leq &M \bigl( S^{\alpha _{1}, \beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl((u-x)^{2};p _{1},q_{1};x \bigr) \bigr)^{\frac{\gamma _{1}}{2}} \\ &{}\times \bigl(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}} (e_{0};p_{1},q_{1};x ) \bigr)^{\frac{2-\gamma _{1}}{2}} \\ &{}\times \bigl( S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}} \bigl((v-y)^{2};p_{1},q_{1};y \bigr) \bigr)^{\frac{ \gamma _{2}}{2}} \\ &{}\times \bigl(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}} (e_{0};p_{1},q_{1};y ) \bigr)^{\frac{2-\gamma _{2}}{2}}. \end{aligned}$$

Considering Lemma 3.2, we get the degree of local approximation for B-continuous functions \(f\in \mathrm{Lip}_{M}(\gamma _{1},\gamma _{2})\). □

Degree of approximation of B-differentiable functions

In this section, we consider the case of B-differentiable functions. Note that a function \(f:\mathbb{R}\rightarrow \mathbb{R}\) is called B-differentiable if, for each \((x,y)\in \mathbb{R}\), the following equality holds:

$$ \lim_{(u,v)\rightarrow (x,y)}\frac{\Delta _{u,v} [f:x,y ]}{(u-x)(v-y )}=D_{B}f< \infty , $$
(4.1)

where \(D_{B}f\) is a B-derivative of f.

The mixed modulus of smoothness \(\omega _{\mathrm{mixed}}:[0,\infty ) \times [0,\infty )\rightarrow [0,\infty )\) is defined as follows.

Definition 4.1

For any \(\delta _{1},\delta _{2}\in [0,\infty )\times [0,\infty )\) and for all \((x,y),(u,v)\in I_{1}\times I_{2}\),

$$ \omega _{\mathrm{mixed}}(\delta _{1},\delta _{2})=\sup \bigl\{ \bigl\vert \Delta _{u,v} [f:x,y ] \bigr\vert : \vert u-x \vert \leq \delta _{1}, \vert v-y \vert \leq \delta _{2} \bigr\} . $$
(4.2)

In [5], Badea et al. proved the following Shisha–Mond type theorem for B-differentiable functions by using GBS operators. In what follows, we try to prove this result by using \((p,q)\)-GBS operators.

Theorem 4.2

Let\(L:C_{b}(I_{1}\times I_{2})\rightarrow C_{b}(I_{1}\times I_{2})\)be a positive linear operator and\(Uf(x,y)\)be the associated GBS operator. Then the following inequality holds for any\(f\in C_{b}(I_{1}\times I _{2})\), \((x,y)\in I_{1}\times I_{2}\), and\(\delta _{1},\delta _{2} \geq 0\):

$$\begin{aligned} \bigl\vert f(x,y)-Uf(x,y) \bigr\vert \leq & \bigl\vert f(x,y) \bigr\vert \bigl\vert 1-L(e_{0 0};x,y) \bigr\vert \\ &{}+3 \Vert D_{B}f \Vert _{\infty }\sqrt{L \bigl((e_{1 0}-x)^{2};x,y \bigr)L \bigl((e _{0 1}-y)^{2};x,y \bigr)} \\ &{}+ \bigl\{ \sqrt{L \bigl((e_{1 0}-x)^{2};x,y \bigr)L \bigl((e_{0 1}-y)^{2};x,y \bigr)} \\ &{}+\delta ^{-1}_{1} \sqrt{L \bigl((e_{1 0}-x)^{4};x,y \bigr)L \bigl((e_{0 1}-y)^{2};x,y \bigr)} \\ &{}+\delta ^{-1}_{2} \sqrt{L \bigl((e_{1 0}-x)^{2};x,y \bigr)L \bigl((e_{0 1}-y)^{4};x,y \bigr)} \\ &{}+(\delta _{1}\delta _{2})^{-1}L \bigl((e_{1 0}-x)^{2};x,y \bigr)L \bigl((e_{0 1}-y)^{2};x,y \bigr) \bigr\} \omega _{\mathrm{mixed}}(D_{B}f;\delta _{1}, \delta _{2}). \end{aligned}$$

Theorem 4.3

Let\(U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\)be a GBS operators associated with\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\)andfhas a boundedB-derivative\(D_{B}f\). Then the following inequality holds:

$$\begin{aligned} & \bigl\vert f(x,y)-U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(f;p_{1},p_{2},q_{1},q_{2};x,y) \bigr\vert \\ &\quad \leq \frac{M}{\sqrt{([m]_{p _{1},q_{1}}+\beta _{1})([n]_{p_{2},q_{2}}+\beta _{2})}} \\ &\qquad {}\times \bigl\lbrace \Vert D_{B}f \Vert _{\infty }+ \omega _{\mathrm{mixed}} \bigl( D_{B}f; \bigl([m]_{p_{1},q_{1}}+\beta _{1} \bigr)^{-1/2}, \bigl([n]_{p _{2},q_{2}}+\beta _{2} \bigr)^{-1/2} \bigr) \bigr\rbrace . \end{aligned}$$

Proof

Since \(f\in C_{b}(I)\), we have

$$ \Delta _{u,v} [f:x,y ]=(u-x) (v-y)D_{B}f(\lambda ,\mu ), \quad \text{with } x< \lambda < u; y< \mu < v, $$

where

$$ D_{B}f(\lambda ,\mu )=\Delta _{u,v}D_{B}f( \lambda ,\mu )+D_{B}f(\lambda ,y)+D_{B}f(x,\mu )-D_{B}f(x,y). $$

Since \(D_{B}f\in B(I)\), we can write

$$\begin{aligned} & \bigl\vert S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\Delta _{u,v} [f:x,y ];p_{1},p_{2},q_{1},q_{2},x,y \bigr) \bigr\vert \\ &\quad \leq S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}} \bigl( \vert u-x \vert \vert v-y \vert \omega _{\mathrm{mixed}} \bigl( D_{B}f; \vert \lambda -x \vert , \vert \mu -y \vert \bigr);p_{1},p_{2},q_{1},q_{2};x,y \bigr) \\ &\qquad {}+3\|D_{B}f\| _{\infty }S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl( \vert u-x \vert \vert v-y \vert ;p_{1},p_{2},q_{1},q_{2};x,y \bigr). \end{aligned}$$
(4.3)

Since the mixed modulus of smoothness is nondecreasing, we have

$$\begin{aligned} \omega _{\mathrm{mixed}} \bigl( D_{B}f; \vert \lambda -x \vert , \vert \mu -y \vert \bigr) \leq &\omega _{\mathrm{mixed}} \bigl( D_{B}f; \vert u-x \vert , \vert v-y \vert \bigr) \\ \leq & \bigl(1-\delta _{1}^{-1} \vert u-x \vert \bigr) \bigl(1-\delta _{2}^{-1} \vert v-y \vert \bigr) \omega _{\mathrm{mixed}} ( D_{B}f;\delta _{1},\delta _{2} ). \end{aligned}$$

Now substituting in inequality (4.3), by linearity of the operators \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}\) and by the Cauchy–Schwarz inequality, we obtain

$$\begin{aligned} & \bigl\vert f(x,y)-U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(f;p_{1},p_{2},q_{1},q_{2};x,y) \bigr\vert \\ &\quad = \bigl\vert S^{\alpha _{1},\beta _{1}, \alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\Delta _{u,v}(f;x,y) \bigr) \bigr\vert \\ &\quad \leq 3 \Vert D_{B}f \Vert _{\infty }\sqrt {S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{x};p_{1},p_{2},q _{1},q_{2};x,y \bigr)S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}} \bigl( \psi ^{2}_{y};p_{1},p_{2},q_{1},q_{2};x,y \bigr)} \\ &\qquad {}+ \Bigl[\sqrt{S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}} \bigl(\psi ^{2}_{x};p_{1},p_{2},q_{1},q_{2};x,y \bigr)S^{\alpha _{1}, \beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{y};p_{1},p _{2},q_{1},q_{2};x,y \bigr)} \\ &\qquad {}+\delta _{1}^{-1}\sqrt{S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{4}_{x};p_{1},p_{2},q_{1},q_{2};x,y \bigr)S ^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{y};p_{1},p_{2},q_{1},q_{2};x,y \bigr)} \\ &\qquad {}+\delta _{2}^{-1}\sqrt{S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{x};p_{1},p_{2},q_{1},q_{2};x,y \bigr)S ^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{4}_{y};p_{1},p_{2},q_{1},q_{2};x,y \bigr)} \\ &\qquad {}+(\delta _{1}\delta _{2})^{-1}S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl(\psi ^{2}_{x}\psi ^{2}_{y};p_{1},p_{2},q_{1},q _{2};x,y \bigr) \Bigr]\omega _{\mathrm{mixed}}(D_{B}f;\delta _{1},\delta _{2}). \end{aligned}$$

By using the following equality for \((x,y),(u,v)\in I_{1}\times I_{2}\):

$$\begin{aligned} S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl((u-x)^{2i}(v-y)^{2j};x,y \bigr) =&S ^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}} \bigl((u-x)^{2i};x,y \bigr) \\ &{}\times S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}} \bigl((v-y)^{2j};x,y \bigr), \end{aligned}$$

\(i,j=1,2\), from (3.29), (3.30), and Lemma 3.3, for any \((x,y)\in [0,1]\times [0,1]\), we get the result. □

Remark 4.4

For \(p=1\), all the above results reduce to q-analogues and for \(p=q=1\) these results further reduce to the classical ones.

References

  1. 1.

    Acar, T.: \((p,q)\)-Generalization of Szász–Mirakyan operators. Math. Methods Appl. Sci. 39(10), 2685–2695 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  2. 2.

    Acar, T., Mohiuddine, S.A., Mursaleen, M.: Approximation by \((p, q)\)-Baskakov–Durrmeyer–Stancu operators. Complex Anal. Oper. Theory 12(6), 1453–1468 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  3. 3.

    Agratini, O.: Bivariate positive operators in polynomial weighted spaces. Abstr. Appl. Anal. 2013, Article ID 850760 (2013)

    MathSciNet  MATH  Article  Google Scholar 

  4. 4.

    Agrawal, P.N., Gupta, V., Kumar, A.S.: On q-analogue of Bernstein–Schurer–Stancu. Appl. Math. Comput. 219(14), 7754–7764 (2013)

    MathSciNet  MATH  Google Scholar 

  5. 5.

    Badea, C., Badea, I., Cottin, C., Gonska, H.H.: Notes on the degree of approximation of B-continuous and B-differentiable functions. J. Approx. Theory Appl. 4, 95–108 (1988)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Badea, C., Badea, I., Gonska, H.H.: A test function theorem and approximation by pseudopolynomials. Bull. Aust. Math. Soc. 34, 53–64 (1986)

    MathSciNet  MATH  Article  Google Scholar 

  7. 7.

    Badea, C., Cottin, C.: Korovkin type theorems for generalized Boolean sum operators. In: Colloquia Mathematica Societatis “Janos Bolyai”. Approximation Theorie, vol. 58, pp. 51–67. Kecsemet (1990)

    Google Scholar 

  8. 8.

    Badea, I.: Modulul de continuitate în sens Bögelşi unele aplicaţii în aproximarea printr-un operator Bernstein. Stud. Univ. Babeş–Bolyai, Math. 4(2), 69–78 (1973)

    Google Scholar 

  9. 9.

    Bărbosu, D.: Bivariate operators of Schurer–Stancu type. An. Ştiinţ. Univ. ‘Ovidius’ Constanţa 11(1), 1–8 (2003)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Bărbosu, D.: GBS operators of Schurer–Stancu type. An. Univ. Craiova, Ser. Mat. Inform. 31(1), 1–7 (2003)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Bărbosu, D.: Approximation properties of a bivariate Stancu type operator. Rev. Anal. Numér. Théor. Approx. 34(1), 17–21 (2005)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Bărbosu, D., Muraru, C.V.: Approximating B-continuous functions using GBS operators of Bernstein–Schurer–Stancu type based on q-integers. Appl. Math. Comput. 259, 80–87 (2015)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Bögel, K.: Mehrdimensionale differentiation von funktionen mehrerer veränderlicher. J. Reine Angew. Math. 170, 197–217 (1934)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Dirik, F., Demirici, K.: Approximation in statistical sense to n-variate B-continuous functions by positive linear operators. Math. Slovaca 60(6), 877–886 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  15. 15.

    Dobrescu, E., Matei, I.: Aproximarea prin polinoame de tip Bernstein a funcţiilor bidimensional continue. Anal. Univ. Timiş., Ser. Sştiinţe Mat.-Fiz. IV, 85–90 (1966)

    Google Scholar 

  16. 16.

    Jebreen, H.B., Mursaleen, M., Naaz, A.: Approximation by quaternion \((p,q)\)-Bernstein polynomials and Voronovskaja type result on compact disk. Adv. Differ. Equ. 2018, 448 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  17. 17.

    Kadak, U.: On weighted statistical convergence based on \((p,q)\)-integers and related approximation theorems for functions of two variables. J. Math. Anal. Appl. 443(2), 752–764 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  18. 18.

    Kadak, U., Mishra, V.N., Pandey, S.: Chlodowsky type generalization of \((p, q)\)-Szász operators involving Brenke type polynomials. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 112(4), 1443–1462 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  19. 19.

    Kajla, A., Miclăuş, D.: Blending type approximation by GBS operators of generalized Bernstein–Durrmeyer type. Results Math. 73, 1 (2018)

    MathSciNet  MATH  Article  Google Scholar 

  20. 20.

    Karakus, S., Demirici, K.: Statistical r-approximation to Bögel type continuous functions. Studia Sci. Math. Hung. 48(4), 475–488 (2011)

    MATH  Google Scholar 

  21. 21.

    Khan, A., Sharma, V.: Statistical approximation by \((p, q)\)-analogue of Bernstein–Stancu operators. Azerb. J. Math. 8(2), 100–121 (2018)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Khan, K., Lobiyal, D.K.: Bézier curves based on Lupaş \((p,q)\)-analogue of Bernstein functions in CAGD. J. Comput. Appl. Math. 317, 458–477 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  23. 23.

    Khan, K., Lobiyal, D.K., Kilicman, A.: Bézier curves and surfaces based on modified Bernstein polynomials. Azerb. J. Math. 9(1), 3–21 (2019)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Mahmedov, R.G.: On the order of approximation of functions by linear positive operators. Dokl. Akad. Nauk SSSR 128, 674–676 (1959) (Russian)

    MathSciNet  Google Scholar 

  25. 25.

    Marchaud, A.: Sur les Dérivées et sur les Différences des functions de variables Réelles. J. Math. Pures Appl. 6, 337–425 (1972)

    MATH  Google Scholar 

  26. 26.

    Mishra, V.N., Mursaleen, M., Pandey, S., Alotaibi, A.: Approximation properties of Chlodowsky variant of \((p,q)\)-Bernstein–Stancu–Schurer operators. J. Inequal. Appl. 2017, 176 (2017)

    MathSciNet  MATH  Article  Google Scholar 

  27. 27.

    Mishra, V.N., Pandey, S.: Certain modifications of \((p, q)\)-Szász–Mirakyan operator. Azerb. J. Math. 9(2), 81–95 (2019)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Mursaleen, M., Ahasan, M.: The Dunkl generalization of Stancu type q-Szász–Mirakjan–Kantorovich operators and some approximation results. Carpath. J. Math. 34(3), 363–370 (2018)

    MATH  Google Scholar 

  29. 29.

    Mursaleen, M., Ansari, K.J., Khan, A.: On \((p,q)\)-analogue of Bernstein operators. Appl. Math. Comput. 264, 392–402 (2015) [Erratum: Appl. Math. Comput., 278 (2016) 70–71]

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Mursaleen, M., Nasiruzzaman, M., Khan, A.: Some approximation results on Bleimann–Butzer–Hahn operators defined by \((p, q)\)-integers. Filomat 30(3), 639–648 (2016)

    MathSciNet  MATH  Article  Google Scholar 

  31. 31.

    Mursaleen, M., Nasiruzzaman, M., Nurgali, A.: Some approximation results on Bernstein–Schurer operators defined by \((p,q)\)-integers. J. Inequal. Appl. 2015, 249 (2015)

    MathSciNet  MATH  Article  Google Scholar 

  32. 32.

    Örkcü, M.: q-Szász–Mirakjan–Kantorovich operators of functions of two variables in polynomial weighted spaces. Abstr. Appl. Anal., 2013 (2013)

  33. 33.

    Ostrovska, S., Özban, A.Y.: On the q-Bernstein polynomials of rational function with real poles. J. Math. Anal. Appl. 413, 547–556 (2014)

    MathSciNet  MATH  Article  Google Scholar 

  34. 34.

    Phillips, G.M.: A survey of results on the q-Bernstein polynomials. IMA J. Numer. Anal. 30(1), 277–288 (2010)

    MathSciNet  MATH  Article  Google Scholar 

  35. 35.

    Pop, O.T.: Aproximation of B-continuous and B-differentiable functions defined by finite sum. Facta Univ., Ser. Math. Inform. 22(1), 33–41 (2007)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Rao, N., Wafi, A.: Bivariate–Schurer–Stancu operators based on \((p,q)\)-integers. Filomat 32(4), 1251–1258 (2018)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Sidharth, M., Ispir, N., Agrawal, P.N.: Approximation of B-continuous and B-differentiable functions by GBS operators of q-Bernstein–Schurer–Stancu type. Turk. J. Math. 40, 1298–1315 (2016)

    MathSciNet  MATH  Article  Google Scholar 

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Acknowledgements

Authors are thankful to the learned referees for their valuable comments which improved the presentation of the paper.

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The author (K.J. Ansari) extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant number G.R.P-93-41.

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Mursaleen, M., Ahasan, M. & Ansari, K.J. Bivariate Bernstein–Schurer–Stancu type GBS operators in \((p,q)\)-analogue. Adv Differ Equ 2020, 76 (2020). https://doi.org/10.1186/s13662-020-02547-7

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MSC

  • 41A10
  • 41A25
  • 41A36

Keywords

  • \((p,q)\)-integers
  • Bernstein–Schurer–Stancu type operators
  • GBS operators
  • B-continuous functions
  • B-differentiable functions
  • Mixed modulus of continuity and smoothness