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.

1 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$(x,y)\in I_{1}\times I_{2}$,

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

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

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

• (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].

2 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:

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

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

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

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

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

3 $(p,q)$-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, 16–18, 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$e_{i}=x^{i}$ ($i=0,1,2,3,4$):

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

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

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

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

• (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

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

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

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

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

• (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 $f\in \mathrm{Lip}_{M}(\gamma _{1},\gamma _{2})$. □

4 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.