Let \(C(I_{1}\times I_{2})\), where \(I_{1}=[0,1+p_{1}]\) and \(I_{2}=[0,1+p_{2}]\), denote the space of all real valued continuous functions on \(I_{1}\times I_{2}\) endowed with the norm
$$\|f\|_{C(I_{1}\times I_{2})}= \sup_{(x,y)\in I_{1}\times I_{2}}\bigl\vert f(x,y)\bigr\vert . $$
For \(f\in C(I_{1}\times I_{2})\), \(0< q_{1}\), \(q_{2}<1 \) and \(J=[0,1]\), the bivariate generalization of the operators given by (1.3) is defined as
$$\begin{aligned}& \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}\bigl(f(t,s);q_{1},q_{2},x,y \bigr) \\& \quad =\sum_{k_{1}=0}^{n_{1}+p_{1}}\sum _{k_{2}=0}^{n_{2}+p_{2}} \tilde{p}_{n_{1},n_{2},k_{1},k_{2}}^{*}(q_{1},q_{2};x,y) \\& \qquad{} \times \int _{0}^{1} \int_{0}^{1}f\bigl(\Psi_{n_{1},k_{1},q_{1}}^{\alpha_{1},\beta_{1}}(t), \Psi _{n_{2},k_{2},q_{2}}^{\alpha_{2},\beta_{2}}(s)\bigr)\, d_{q_{1}}t\, d_{q_{2}}s, \end{aligned}$$
(3.1)
where
$$\begin{aligned}& \tilde{p}_{n_{1},n_{2},k_{1},k_{2}}^{*}(q_{1},q_{2},x,y) \\& \quad =\frac{[n_{1}]_{q_{1}}^{n_{1}+p_{1}}}{[n_{1}+p_{1}]_{q_{1}}^{n_{1}+p_{1}}}{n_{1}+p_{1} \brack k_{1}}_{q_{1}} x^{k_{1}} \biggl(\frac{[n_{1}+p_{1}]_{q_{1}}}{[n_{1}]_{q_{1}}}-x \biggr)_{q_{1}}^{n_{1}+p_{1}-k_{1}} \\& \qquad {}\times\frac {[n_{2}]_{q_{2}}^{n_{2}+p_{2}}}{[n_{2}+p_{2}]_{q_{2}}^{n_{2}+p_{2}}}{n_{2}+p_{2} \brack k_{2}}_{q_{2}} y^{k_{2}} \biggl(\frac{[n_{2}+p_{2}]_{q_{2}}}{[n_{2}]_{q_{2}}}-y \biggr)_{q_{2}}^{n_{2}+p_{2}-k_{2}}, \quad x,y\in J\quad \mbox{and} \\& \Psi_{n_{1},k_{1},q_{1}}^{\alpha_{1},\beta_{1}}(t)=\frac {[k_{1}]_{q_{1}}+q_{1}^{k_{1}}t+\alpha_{1}}{[n_{1}+1]+\beta_{1}},\qquad \Psi_{n_{2},k_{2},q_{2}}^{\alpha_{2},\beta_{2}}(s)=\frac {[k_{2}]_{q_{2}}+q_{2}^{k_{2}}s+\alpha_{2}}{[n_{2}+1]+\beta_{2}}. \end{aligned}$$
Lemma 3
let
\(e_{ij}(t,s)=t^{i}s^{j}\), \((t,s)\in(I_{1}\times I_{2})\), \((i,j)\in N^{0}\times N^{0}\)
with
\(i+j\leq2\)
be the two dimensional test functions. Then the following equalities hold for the operators (3.1):
-
(i)
\(\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(e_{00};q_{1},q_{2},x,y)=1\);
-
(ii)
\(\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(e_{10};q_{1},q_{2},x,y)=\frac{\alpha_{1}}{[n_{1}+1]_{q_{1}}+\beta _{1}}+\frac{2q_{1}[n_{1}]_{q_{1}}x+1}{[2]_{q_{1}}([n_{1}+1]_{q_{1}}+\beta_{1})}\);
-
(iii)
\(\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(e_{01};q_{1},q_{2},x,y)=\frac{\alpha_{2}}{[n_{2}+1]_{q_{2}}+\beta _{2}}+\frac{2q_{2}[n_{2}]_{q_{2}}y+1}{[2]_{q_{2}}([n_{2}+1]_{q_{2}}+\beta_{2})}\);
-
(iv)
\(\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(e_{20};q_{1},q_{2},x,y)=\frac {1}{[2]_{q_{1}}[3]_{q_{1}}([n_{1}+1]_{q_{1}}+\beta_{1})^{2}} \{\frac {[n_{1}]_{q_{1}}^{2}[n_{1}+p_{1}-1]_{q_{1}}}{[n_{1}+p_{1}]_{q_{1}}}([3]_{q_{1}}q_{1}^{2}+3q_{1}^{4})x^{2} +\{(4\alpha_{1}+3)q_{1}[3]_{q_{1}}+q_{1}^{2}(1+[2]_{q_{1}})\}[n_{1}]_{q_{1}}x +[4]_{q_{1}}\alpha_{1}^{2}+2\alpha_{1}[3]_{q_{1}}+(1+q_{1}\alpha_{1}^{2})[2]_{q_{1}} \}\).
-
(v)
\(\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(e_{02};q_{1},q_{2},x,y)=\frac {1}{[2]_{q_{2}}[3]_{q_{2}}([n_{2}+1]_{q_{2}}+\beta_{2})^{2}} \{\frac {[n_{2}]_{q_{2}}^{2}[n_{2}+p_{2}-1]_{q_{2}}}{[n_{2}+p_{2}]_{q_{2}}}([3]_{q_{2}}q_{2}^{2}+3q_{2}^{4})y^{2} +\{(4\alpha_{2}+3)q_{2}[3]_{q_{2}}+q_{2}^{2}(1+[2]_{q_{2}})\}[n_{2}]_{q_{2}}y +[4]_{q_{2}}\alpha_{2}^{2}+2\alpha_{2}[3]_{q_{2}}+(1+q_{2}\alpha_{2}^{2})[2]_{q_{2}} \}\).
Proof
We have \(\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(t^{i}s^{j};q_{1},q_{2},x,y)=\mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta _{1})}(t^{i};q_{1},x) \mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta _{2})}(s^{j};q_{2},y)\), for \(0\leq i,j\leq2\).
By using Lemma 1, the proof of the lemma is straightforward. Hence the details are omitted. □
For \(f\in C(I_{1}\times I_{2})\) and \(\delta>0 \), the first order complete modulus of continuity for the bivariate case is defined as follows:
$$ \omega(f;\delta_{1},\delta_{2})=\sup \bigl\{ \bigl\vert f(t,s)-f(x,y)\bigr\vert : |t-x|\leq \delta_{1},|s-y|\leq \delta_{2} \bigr\} , $$
where \(\delta_{1},\delta_{2}>0\). Further \(\omega(f;\delta_{1},\delta_{2})\) satisfies the following properties:
-
(a)
\(\omega(f;\delta_{1},\delta_{2})\rightarrow0 \) if \(\delta_{1}\rightarrow 0 \) and \(\delta_{2}\rightarrow0\),
-
(b)
\(|f(t,s)-f(x,y)|\leq\omega(f;\delta_{1},\delta_{2}) (1+\frac {|t-x|}{\delta_{1}} ) (1+\frac{|s-y|}{\delta_{2}} )\).
Now, we give an estimate of the rate of convergence of the bivariate operators. In the following, let \(0< q_{n_{i}}<1\) be sequences in \((0,1)\) such that \(q_{n_{i}}\rightarrow1\) and \(q_{n_{i}}^{n_{i}}\rightarrow a_{i}\) (\(0\leq a_{i}<1\)), as \(n_{i}\rightarrow\infty\) for \(i=1,2\). Further, let \(\delta_{n_{1}}(x)= \mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta _{1})}((t-x)^{2};q_{n_{1}},x)\) and \(\delta_{n_{2}}(y)= \mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta _{2})}((s-y)^{2};q_{n_{2}},y)\).
Theorem 2
For
\(f\in C(I_{1}\times I_{2})\)
and all
\((x,y)\in J^{2}\), we have
$$ \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \leq4\omega\bigl(f;\sqrt{\delta _{n_{1}}(x)},\sqrt{ \delta_{n_{2}}(y)}\bigr). $$
Proof
Since \(\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)\) is a linear positive operator, by the property (b) of bivariate modulus of continuity, Lemma 1, and the Cauchy-Schwarz inequality
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}\bigl(f(t,s);q_{n_{1}},q_{n_{2}},x,y \bigr)-f(x,y)\bigr\vert \\& \quad \leq \bigl(\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}\bigl\vert f(t,s)-f(x,y)\bigr\vert ;q_{n_{1}},q_{n_{2}},x,y\bigr) \\& \quad \leq \omega\bigl(f;\sqrt{\delta_{n_{1}}(x)},\sqrt{ \delta_{n_{2}}(y)}\bigr) \biggl(\mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta_{1})}(1;q_{n_{1}},x)+ \frac{1}{\sqrt {\delta_{n_{1}}(x)}}\mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta _{1})}\bigl(\vert t-x \vert ;q_{n_{1}},x\bigr) \biggr) \\& \qquad {}\times \biggl(\mathcal {K}_{n_{2},p_{2}}^{(\alpha_{2},\beta_{2})}(1;q_{n_{2}},y)+ \frac{1}{\sqrt{\delta _{n_{2}}(y)}} \mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta _{2})}\bigl(\vert s-y \vert ;q_{n_{2}},y\bigr) \biggr) \\& \quad \leq \omega\bigl(f;\sqrt{\delta_{n_{1}}(x)},\sqrt{ \delta_{n_{2}}(y)}\bigr) \biggl(1+\frac{1}{\sqrt{\delta_{n_{1}}(x)}} \sqrt{\bigl( \mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta _{1})}(t-x)^{2};q_{n_{1}},x \bigr)} \biggr) \\& \qquad {}\times \biggl(1+\frac{1}{\sqrt{\delta _{n_{2}}(y)}}\sqrt{\mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta _{2})} \bigl((s-y)^{2};q_{n_{2}},y\bigr)} \biggr), \end{aligned}$$
we get the desired result. □
Theorem 3
If
\(f(x,y)\)
has continuous partial derivatives
\(\frac{\partial f}{\partial x}\)
and
\(\frac{\partial f}{\partial y}\), then the inequality
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad \leq M_{1} \lambda_{n_{1}}(x) +\omega \bigl(f'_{x},\delta_{n_{1}}(x)\bigr) \bigl(1+ \sqrt{\delta_{n_{1}}(x)}\bigr) \\& \qquad {}+M_{2} \lambda_{n_{2}}(y)+\omega\bigl(f'_{y}, \delta_{n_{2}}(y)\bigr) \bigl(1+\sqrt{\delta_{n_{2}}(y)}\bigr), \end{aligned}$$
where
\(M_{1}\), \(M_{2}\)
are the positive constants such that
$$ \biggl\vert \frac{\partial f}{\partial x}\biggr\vert \leq M_{1},\qquad \biggl\vert \frac{\partial f}{\partial y}\biggr\vert \leq M_{2} \quad (0\leq x \leq a, 0\leq y\leq b) $$
and
$$\begin{aligned}& \lambda_{n_{1}}(x)= \biggl\vert \frac{(2-[2]_{q_{n_{1}}})q_{n_{1}}[n_{1}]_{q_{n_{1}}} -(\beta_{1}+1)[2]_{q_{n_{1}}}}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta _{1})} \biggr\vert x+ \frac{(1+[2]_{q_{n_{1}}}\alpha_{1})}{[n_{1}+1]_{q_{n_{1}}}+\beta _{1}} ; \\& \lambda_{n_{2}}(y)= \biggl\vert \frac {(2-[2]_{q_{n_{2}}})q_{n_{2}}[n_{2}]_{q_{n_{2}}}-(\beta _{2}+1)[2]_{q_{n_{2}}}}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta_{2})} \biggr\vert y + \frac{(1+[2]_{q_{n_{2}}}\alpha_{2})}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta_{2})} . \end{aligned}$$
Proof
From the mean value theorem we have
$$\begin{aligned} f(t,s)-f(x,y) =&f(t,y)-f(x,y)+f(t,s)-f(t,y) \\ =&(t-x)\frac{\partial f(\xi_{1},y)}{\partial x}+(s-y)\frac{\partial f(x,\xi_{2})}{\partial y} \\ =&(t-x)\frac{\partial f(x,y)}{\partial x}+(t-x) \biggl(\frac{\partial f(\xi_{1},y)}{\partial x}-\frac{\partial f(x,y)}{\partial x} \biggr)+(s-y)\frac{\partial f(x,y)}{\partial y} \\ &{}+(s-y) \biggl(\frac {\partial f(x,\xi_{2})}{\partial y}-\frac{\partial f(x,y)}{\partial y} \biggr), \end{aligned}$$
(3.2)
where \(x <\xi< t\) and \(y<\xi_{2}<s\). Since
$$\begin{aligned}& \biggl\vert \frac{\partial f(\xi_{1},y)}{\partial x}-\frac{\partial f(x,y)}{\partial x}\biggr\vert \leq\omega \bigl(f'_{x};\vert t-x\vert \bigr)\leq \biggl(1+ \frac {\vert t-x\vert }{\delta_{n_{1}}} \biggr)\omega\bigl(f'_{x}, \delta_{n_{1}}\bigr)\quad \mbox{and} \\& \biggl\vert \frac{\partial f(x,\xi_{2})}{\partial y}-\frac{\partial f(x,y)}{\partial y}\biggr\vert \leq\omega \bigl(f'_{y};\vert s-y\vert \bigr)\leq \biggl(1+ \frac {\vert s-y\vert }{\delta_{n_{2}}} \biggr)\omega\bigl(f'_{y}, \delta_{n_{2}}\bigr) \end{aligned}$$
for some \(\delta_{n_{1}},\delta_{n_{2}}>0\), on applying the operator \(\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(\cdot;q_{n_{1}},q_{n_{2}},x,y)\) on both sides of (3.2), we have
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad \leq M_{1}\bigl\vert \mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta _{1})}(e_{10}-x;q_{n_{1}},x) \bigr\vert \\& \qquad {}+\sum_{k_{1}=0}^{n_{1}+p_{1}}\sum _{k_{2}=0}^{n_{2}+p_{2}} \tilde{p}_{n_{1},n_{2},k_{1},k_{2}}^{*}(q_{n_{1}},q_{n_{2}};x,y) \\& \qquad {}\times \int_{0}^{1} \int _{0}^{1}\bigl\vert \Psi_{n_{1},k_{1},q_{n_{1}}}^{\alpha_{1},\beta_{1}}(t)-x \bigr\vert \omega \bigl(f'_{x},\delta_{n_{1}} \bigr) \biggl(\frac{\vert \Psi_{n_{1},k_{1},q_{n_{1}}}^{\alpha_{1},\beta_{1}}(t)-x\vert }{\delta _{n_{1}}}+1 \biggr)\, d_{q_{n_{1}}}t\, d_{q_{n_{2}}}s \\& \qquad {}+M_{2}\bigl\vert \mathcal{K}_{n_{2},p_{2},q_{n_{2}}}^{(\alpha_{2},\beta_{2})}(e_{01}-y;y) \bigr\vert \\& \qquad {}+\sum_{k_{1}=0}^{n_{1}+p_{1}}\sum _{k_{2}=0}^{n_{2}+p_{2}} \tilde{p}_{n_{1},n_{2},k_{1},k_{2}}^{*}(q_{n_{1}},q_{n_{2}};x,y) \\& \qquad {}\times \int_{0}^{1} \int _{0}^{1}\bigl\vert \Psi_{n_{2},k_{2},q_{n_{2}}}^{\alpha_{2},\beta_{2}}(s)-y \bigr\vert \omega \bigl(f'_{y},\delta_{n_{2}} \bigr) \biggl(\frac{\vert \Psi_{n_{2},k_{2},q_{n_{2}}}^{\alpha_{2},\beta_{2}}(s)-y\vert }{\delta _{n_{2}}}+1 \biggr)\, d_{q_{n_{1}}}t\, d_{q_{n_{2}}}s. \end{aligned}$$
Now applying the Cauchy-Schwarz inequality
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad \leq M_{1}\bigl\vert \mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta _{1})}(e_{10};q_{n_{1}},x) \bigr\vert \\& \qquad {}+ \omega\bigl(f'_{x},\delta_{n_{1}} \bigr) \Biggl\{ \sum_{k_{1}=0}^{n_{1}+p_{1}}\sum _{k_{2}=0}^{n_{2}+p_{2}} \tilde{p}_{n_{1},n_{2},k_{1},k_{2}}^{*}(q_{n_{1}},q_{n_{2}};x,y) \\& \qquad {}\times \int_{0}^{1} \int _{0}^{1}\bigl(\Psi_{n_{1},k_{1},q_{n_{1}}}^{\alpha_{1},\beta_{1}}(t)-x \bigr)^{2}\,d_{q_{n_{1}}}t \,d_{q_{n_{2}}}s \Biggr\} ^{\frac{1}{2}} \\& \qquad {}+\frac{\omega(f'_{x},\delta_{n_{1}})}{ \delta_{n_{1}}}\sum_{k_{1}=0}^{n_{1}+p_{1}} \sum_{k_{2}=0}^{n_{2}+p_{2}} \tilde{p}_{n_{1},n_{2},k_{1},k_{2}}^{*}(q_{n_{1}},q_{n_{2}};x,y) \\& \qquad {}\times \int_{0}^{1} \int _{0}^{1}\bigl(\Psi_{n_{1},k_{1},q_{n_{1}}}^{\alpha_{1},\beta_{1}}(t)-x \bigr)^{2}\,d_{q_{n_{1}}}t \,d_{q_{n_{2}}}s \\& \qquad {}+M_{2}\bigl\vert \mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta_{2})}(e_{01};q_{n_{2}},y) \bigr\vert \\& \qquad {}+\omega\bigl(f'_{y},\delta_{n_{2}} \bigr) \Biggl\{ \sum_{k_{1}=0}^{n_{1}+p_{1}}\sum _{k_{2}=0}^{n_{2}+p_{2}} \tilde{p}_{n_{1},n_{2},k_{1},k_{2}}^{*}(q_{n_{1}},q_{n_{2}};x,y) \\& \qquad {}\times \int_{0}^{1} \int _{0}^{1}\bigl(\Psi_{n_{2},k_{2},q_{n_{2}}}^{\alpha_{2},\beta_{2}}(s)-y \bigr)^{2}\,d_{q_{n_{1}}}t \,d_{q_{n_{2}}}s \Biggr\} ^{\frac{1}{2}} \\& \qquad {}+\frac{\omega(f'_{y},\delta_{n_{2}})}{ \delta_{n_{2}}}\sum_{k_{1}=0}^{n_{1}+p_{1}} \sum_{k_{2}=0}^{n_{2}+p_{2}} \tilde{p}_{n_{1},n_{2},k_{1},k_{2}}^{*}(q_{n_{1}},q_{n_{2}};x,y) \\& \qquad {}\times \int_{0}^{1} \int _{0}^{1}\bigl(\Psi_{n_{2},k_{2},q_{n_{2}}}^{\alpha_{2},\beta_{2}}(s)-y \bigr)^{2}\,d_{q_{n_{1}}}t \,d_{q_{n_{2}}}s \\& \quad = M_{1} \lambda_{n_{1}}(x)+\omega\bigl(f'_{x}, \delta_{n_{1}}\bigr) (1+\sqrt{\delta _{n_{1}}})+M_{2} \lambda_{n_{2}}(y)+\omega\bigl(f'_{y}, \delta_{n_{2}}\bigr) (1+\sqrt{\delta _{n_{2}}}), \end{aligned}$$
on choosing \(\delta_{n_{1}}=\delta_{n_{1}}(x)\) and \(\delta_{n_{2}}=\delta_{n_{2}}(y)\), we obtain the required result. □
3.1 Degree of approximation
In our next result, we study the degree of approximation for the bivariate operators by means of the Lipschitz class.
For \(0<\xi_{1}\leq1 \) and \(0<\xi_{2}\leq1 \), we define the Lipschitz class \(\operatorname{Lip}_{M}(\xi_{1},\xi_{2})\) for the bivariate case as follows:
$$ \bigl\vert f(t,s)-f(x,y)\bigr\vert \leq M|t-x|^{\xi_{1}}|s-y|^{\xi_{2}}, $$
where \((t,s), (x,y)\in(I_{1}\times I_{2})\) are arbitrary.
Theorem 4
Let
\(f\in \operatorname{Lip}_{M}(\xi_{1},\xi_{2}) \). Then, for all
\((x,y)\in J^{2}\), we have
$$ \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \leq M \bigl(\delta_{n_{1}}(x)\bigr)^{\frac{\xi _{1}}{2}}\bigl( \delta_{n_{2}}(y)\bigr)^{\frac{\xi_{2}}{2}}. $$
Proof
By our hypothesis, we can write
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad \leq \mathcal {K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}\bigl(\bigl\vert f(t,s)-f(x,y)\bigr\vert ;q_{n_{1}},q_{n_{2}},x,y\bigr) \\& \quad \leq M \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta _{1},\beta_{2})}\bigl(\vert t-x\vert ^{\xi_{1}}\vert s-y\vert ^{\xi_{2}};q_{n_{1}},q_{n_{2}},x,y \bigr) \\& \quad = M \bigl(\mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta_{1})}\vert t-x\vert ^{\xi _{1}};q_{n_{1}},x\bigr)\mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta_{2})} \bigl(\vert s-y\vert ^{\xi _{2}};q_{n_{2}},y\bigr). \end{aligned}$$
Now, applying the Hölder’s inequality with \(u_{1}=\frac{2}{\xi_{1}}\), \(v_{1}=\frac{2}{2-\xi_{1}}\) and \(u_{2}=\frac{2}{\xi_{2}}\) and \(v_{2}=\frac{2}{2-\xi_{2}}\), respectively, we have
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x) \bigr\vert \\& \quad \leq M {\bigl(\mathcal {K}_{n_{1},p_{1}}^{(\alpha_{1},\beta_{1})}(t-x)^{2};q_{n_{1}},x \bigr)}^{\frac{\xi _{1}}{2}}{\mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta_{1})}(1;q_{n_{1}},x)}^{\frac {2-\xi_{1}}{2}} \\& \qquad {}\times{\mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta _{2})}\bigl((s-y)^{2};q_{n_{2}},y \bigr)}^{\frac{\xi_{2}}{2}}{\mathcal{K}_{n_{2},p_{2}}^{\alpha _{2},\beta_{2}}(1;q_{n_{2}},y)}^{\frac{2-\xi_{2}}{2}} \\& \quad = M \bigl(\delta_{n_{1}}(x)\bigr)^{\frac{\xi_{1}}{2}}\bigl( \delta_{n_{2}}(y)\bigr)^{\frac{\xi_{2}}{2}}. \end{aligned}$$
Hence, the proof is completed. □
Let \(C^{1}(I_{1}\times I_{2})\) denote the space of all continuous functions on \(I_{1}\times I_{2}\) such that their first partial derivatives are continuous on \(I_{1}\times I_{2}\).
Theorem 5
For
\(f\in C^{1}(I_{1}\times I_{2})\)
and
\((x,y)\in J^{2} \)
we have
$$ \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \leq\bigl\Vert f'_{x}\bigr\Vert _{C(I_{1}\times I_{2})}\sqrt{\delta_{n_{1}}(x)}+\bigl\Vert f'_{y}\bigr\Vert _{C(I_{1}\times I_{2})}\sqrt{ \delta_{n_{2}}(y)}. $$
Proof
Let \((x,y)\in J^{2} \) be a fixed point. Then by our hypothesis
$$ f(t,s)-f(x,y)= \int_{x}^{t}f'_{u}(u,s)\, d_{q} u+ \int_{y}^{s}f'_{v}(x,v) \, d_{q} v. $$
Now, operating by \(\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta_{2})}(\cdot;q_{n_{1}},q_{n_{2}},x,y) \) on both sides of the above equation, we are led to
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad \leq \mathcal {K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta_{2})} \biggl(\biggl\vert \int _{t}^{x}\bigl\vert f'_{u}(u,s) \bigr\vert \, d_{q} u\biggr\vert ;q_{n_{1}},q_{n_{2}},x,y \biggr) \\& \qquad {}+\mathcal {K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})} \biggl(\biggl\vert \int _{y}^{s}\bigl\vert f'_{v}(x,v) \bigr\vert \, d_{q} v\biggr\vert ;q_{n_{1}},q_{n_{2}},x,y \biggr). \end{aligned}$$
Since \(|\int_{t}^{x}|f'_{u}(u,s)|\, d_{q} u|\leq\| f'_{x}\| _{C(I_{1}\times I_{2})}|t-x|\) and \(|\int_{y}^{s}|f'_{v}(x,v)|\, d_{q} v|\leq \| f'_{y}\|_{C(I_{1}\times I_{2})}|s-y|\), we have
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad \leq\bigl\Vert f'_{x}\bigr\Vert _{C(I_{1}\times I_{2})}\mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1}, \beta_{1})}\bigl(\vert t-x \vert ;q_{n_{1}},x\bigr)+\bigl\Vert f'_{y} \bigr\Vert _{C(I_{1}\times I_{2})}\mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta _{2})}\bigl( \vert s-y\vert ;q_{n_{2}},y\bigr). \end{aligned}$$
Applying the Cauchy-Schwarz inequality and Lemma 1, we have
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad \leq \bigl\Vert f'_{x}\bigr\Vert _{C(I_{1}\times I_{2})}\sqrt{\mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1}, \beta_{1})} \bigl((t-x)^{2};q_{n_{1}},x\bigr)} \sqrt{ \mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta_{1})}(1;q_{n_{1}},x)} \\& \qquad {}+\bigl\Vert f'_{y}\bigr\Vert _{C(I_{1}\times I_{2})} \sqrt{\mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta _{2})}\bigl((s-y)^{2};q_{n_{2}},y \bigr)}\sqrt{\mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta _{2})}(1;q_{n_{2}},y)} \\& \quad = \bigl\Vert f'_{x}\bigr\Vert _{C(I_{1}\times I_{2})} \sqrt{\delta _{n_{1}}(x)}+\bigl\Vert f'_{y} \bigr\Vert _{C(I_{1}\times I_{2})}\sqrt{\delta_{n_{2}}(y)}. \end{aligned}$$
This completes the proof of the theorem. □
For \(f\in C(I_{1}\times I_{2})\) and \(\delta>0\), the partial moduli of continuity with respect to x and y are given by
$$ \bar{\omega}_{1}(f;\delta)=\sup \bigl\{ \bigl\vert f(x_{1},y)-f(x_{2},y)\bigr\vert :y\in I_{2} \mbox{ and } \vert x_{1}-x_{2}\vert \leq\delta \bigr\} $$
and
$$ \bar{\omega}_{2}(f;\delta)=\sup \bigl\{ \bigl\vert f(x,y_{1})-f(x,y_{2})\bigr\vert :x\in I_{1} \mbox{ and } \vert y_{1}-y_{2}\vert \leq\delta \bigr\} . $$
Clearly, both moduli of continuity satisfy the properties of the usual modulus of continuity.
Theorem 6
If
\(f\in C(I_{1}\times I_{2})\)
and
\((x,y)\in J^{2}\), then we have
$$ |\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y)| \leq2\bigl\{ \bigl(\bar{\omega}_{1}\bigl(f;\sqrt{\delta _{n_{1}}(x)}\bigr)\bigr)+\bigl(\bar{\omega}_{2}\bigl(f;\sqrt{ \delta_{n_{2}}(y)}\bigr)\bigr)\bigr\} . $$
Proof
Using the definition of partial moduli of continuity, Lemma 1, and the Cauchy-Schwarz inequality, we have
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad \leq \mathcal {K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}\bigl(\bigl\vert f(t,s)-f(x,y)\bigr\vert ;q_{n_{1}},q_{n_{2}},x,y\bigr) \\& \quad \leq \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}\bigl(\bigl\vert f(t,s)-f(t,y)\bigr\vert ;q_{n_{1}},q_{n_{2}},x,y\bigr) \\& \qquad {}+\mathcal {K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}\bigl(\bigl\vert f(t,y)-f(x,y)\bigr\vert ;q_{n_{1}},q_{n_{2}},x,y\bigr) \\& \quad \leq \bar{\omega}_{1}\bigl(f;\sqrt{\delta_{n_{1}}(x)} \bigr) \biggl(\mathcal {K}_{n_{1},p_{1}}^{(\alpha_{1},\beta_{1})}(1;q_{n_{1}},x) + \frac{1}{\sqrt{\delta_{n_{1}}(x)}}\mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta _{1})}\bigl(\vert t-x \vert ;q_{n_{1}},x\bigr) \biggr) \\& \qquad {}+\bar{\omega}_{2}\bigl(f;\sqrt{\delta _{n_{2}}(y)} \bigr) \biggl(\mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta_{2})}(1;q_{n_{2}},y)+ \frac{1}{\sqrt{\delta_{n_{2}}(y)}}\mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta _{2})}\bigl(\vert s-y \vert ;q_{n_{2}},y\bigr) \biggr) \\& \quad \leq \bar{\omega}_{1}\bigl(f;\sqrt{\delta_{n_{1}}(x)} \bigr) \biggl(1+\frac{1}{\sqrt {\delta_{n_{1}}(x)}}\sqrt{\mathcal{K}_{n_{1},p_{1}}^{(\alpha_{1},\beta _{1})} \bigl((t-x)^{2};q_{n_{1}},x\bigr)} \biggr) \\& \qquad {}+ \bar{\omega}_{2}\bigl(f;\sqrt{\delta_{n_{2}}(y)} \bigr) \biggl(1+\frac{1}{\sqrt{\delta _{n_{2}}(y)}}\sqrt{\mathcal{K}_{n_{2},p_{2}}^{(\alpha_{2},\beta _{2})} \bigl((s-y)^{2};q_{n_{2}},y\bigr)} \biggr), \end{aligned}$$
from which the required result is straightforward. □
Let \(C^{2}(I_{1}\times I_{2})\) be the space of all functions \(f\in C(I_{1}\times I_{2})\) such that second partial derivatives of f belong to \(C(I_{1}\times I_{2})\). The norm on the space \(C^{2}(I_{1}\times I_{2}) \) is defined as
$$ \Vert f\Vert _{C^{2}(I_{1}\times I_{2})}=\Vert f\Vert +\sum _{i=1}^{2}\biggl(\biggl\Vert \frac {\partial^{i}f}{\partial x^{i}} \biggr\Vert +\biggl\Vert \frac{\partial^{i}f}{\partial y^{i}}\biggr\Vert \biggr). $$
The Peetre K-functional of the function \(f\in C(I_{1}\times I_{2})\) is defined as
$$ \mathcal{K}(f;\delta)=\inf_{g\in{C^{2}(I_{1}\times I_{2})}}\bigl\{ \Vert f-g\Vert _{C(I_{1}\times I_{2})}+\delta \Vert g\Vert _{C^{2}(I_{1}\times I_{2})}\bigr\} ,\quad \delta>0. $$
Also by [6], it follows that
$$ \mathcal{K}(f;\delta)\leq M \bigl\{ \tilde{\omega}_{2}(f; \sqrt{\delta })+\min(1,\delta)\|f\|_{C(I_{1}\times I_{2})} \bigr\} $$
(3.3)
holds for all \(\delta>0\).
The constant M in the above inequality is independent of δ and f and \(\tilde{\omega}_{2}(f;\sqrt{\delta})\) is the second order modulus of continuity.
Theorem 7
For the function
\(f\in C(I_{1}\times I_{2})\), we have the following inequality:
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta _{1},\beta_{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad \leq M \Bigl\{ \tilde{\omega}_{2}\Bigl(f;\sqrt {A_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y)}\Bigr) +\min\bigl\{ 1,A_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y) \bigr\} \|f\|_{C(I_{1}\times I_{2})} \Bigr\} \\& \qquad {}+\omega \Bigl(f;\sqrt {B_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y)} \Bigr), \end{aligned}$$
where
$$\begin{aligned}& A_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y) \\& \quad = \biggl\{ \delta ^{2}_{n_{1}}(x)+\delta^{2}_{n_{2}}(y)+ \biggl(\frac{\alpha_{1}}{[n_{1}+1]_{q_{n_{1}}}+\beta_{1}}+\frac {2q_{n_{1}}[n_{1}]_{q_{n_{1}}}x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta _{1})}-x \biggr)^{2} \\& \qquad {}+ \biggl(\frac{\alpha_{2}}{[n_{2}+1]_{q_{n_{2}}}+\beta_{2}}+\frac {2q_{n_{2}}[n_{2}]_{q_{n_{2}}}y+1}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta _{2})}-y \biggr)^{2} \biggr\} \end{aligned}$$
and
$$\begin{aligned} B_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y) =& \biggl(\frac{\alpha _{1}}{[n_{1}+1]_{q_{n_{1}}}+\beta_{1}}+\frac{2q_{n_{1}}[n_{1}]_{q_{n_{1}}} x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta_{1})}-x \biggr)^{2} \\ &{}+ \biggl(\frac{\alpha_{2}}{[n_{2}+1]_{q_{n_{2}}}+\beta_{2}}+\frac {2q_{n_{2}}[n_{2}]_{q_{n_{2}}}y+1}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta _{2})}-y \biggr)^{2}, \end{aligned}$$
and the constant
M (>0), is independent of
f
and
\(A_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y)\).
Proof
We define the auxiliary operators as follows:
$$\begin{aligned}& \mathcal{L}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y) \\& \quad = \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha _{1},\alpha_{2}, \beta_{1},\beta_{2})}(f;q_{n_{1}},q_{n_{2}},x,y) \\& \qquad {}-f \biggl(\frac{\alpha_{1}}{[n_{1}+1]_{q_{n_{1}}}+\beta_{1}}+\frac {2q_{n_{1}}[n_{1}]_{q_{n_{1}}}x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta _{1})}, \\& \qquad \frac{\alpha_{2}}{[n_{2}+1]_{q_{n_{2}}}+\beta_{2}}+\frac {2q_{n_{2}}[n_{2}]_{q_{n_{2}}}y+1}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta _{2})} \biggr)+f(x,y). \end{aligned}$$
(3.4)
Considering Lemma 3, one has \(\mathcal{L}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha _{1},\alpha_{2}, \beta_{1},\beta_{2})}(1;q_{n_{1}},q_{n_{2}},x,y)=1\), \(\mathcal{L}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}((t-x);q_{n_{1}},q_{n_{2}}, x,y)=0\), and \(\mathcal {L}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}((s-y);q_{n_{1}},q_{n_{2}},x,y)=0\).
Let \(g\in C^{2}(I_{1}\times I_{2})\) and \((x,y)\in J^{2} \). Using Taylor’s theorem, we may write
$$\begin{aligned} g(t,s)-g(x,y) =&g(t,y)-g(x,y)+g(t,s)-g(t,y) \\ =&\frac{\partial g(x,y)}{\partial x}(t-x)+ \int^{t}_{x}(t-u)\frac{\partial ^{2} g(u,y)}{\partial u^{2}}\, du \\ &{}+\frac{\partial g(x,y)}{\partial y}(s-y)+ \int^{s}_{y}(s-v)\frac{\partial ^{2} g(x,v)}{\partial v^{2}}\, dv. \end{aligned}$$
Applying the operator \(\mathcal{L}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha _{2},\beta_{1},\beta_{2})}(\cdot;q_{n_{1}},q_{n_{2}},x,y)\) on the above equation and using (3.4), we are led to
$$\begin{aligned}& \mathcal{L}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(g;q_{n_{1}},q_{n_{2}},x,y)-g(x,y) \\& \quad = \mathcal{L}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})} \biggl( \int^{t}_{x}(t-u)\frac{\partial^{2} g(u,y)}{\partial u^{2}} \,du;q_{n_{1}},q_{n_{2}},x,y \biggr) \\& \qquad {}+\mathcal{L}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})} \biggl( \int^{s}_{y}(s-v)\frac{\partial^{2} g(x,v)}{\partial v^{2}} \,dv;q_{n_{1}},q_{n_{2}},x,y \biggr) \\& \quad = \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})} \biggl( \int^{t}_{x}(t-u)\frac{\partial^{2} g(u,y)}{\partial u^{2}} \,du;q_{n_{1}},q_{n_{2}},x,y \biggr) \\& \qquad {}- \int_{x}^{\frac{\alpha_{1}}{[n_{1}+1]_{q_{1}}+\beta_{1}}+\frac {2q_{n_{1}}[n_{1}]_{q_{n_{1}}}x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta _{1})}} \biggl(\frac{\alpha_{1}}{[n_{1}+1]_{q_{n_{1}}}+\beta_{1}}+ \frac {2q_{n_{1}}[n_{1}]_{q_{n_{1}}}x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta _{1})}-u \biggr) \\& \qquad {}\times\frac{\partial^{2} g(u,y)}{\partial u^{2}}\,du+ \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})} \biggl( \int^{s}_{y}(s-v)\frac{\partial^{2} g(x,v)}{\partial v^{2}} \,dv;q_{n_{1}},q_{n_{2}},x,y \biggr) \\& \qquad {}- \int_{y}^{\frac{\alpha_{2}}{[n_{2}+1]_{q_{n_{2}}}+\beta_{2}}+\frac {2q_{n_{2}}[n_{2}]_{q_{n_{2}}}y+1}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta _{2})}} \biggl(\frac{\alpha_{2}}{[n_{2}+1]_{q_{n_{2}}}+\beta_{2}}+ \frac {2q_{n_{2}}[n_{2}]_{q_{n_{2}}}y+1}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta _{2})}-v \biggr) \\& \qquad {}\times\frac{\partial^{2} g(x,v)}{\partial v^{2}}\,dv. \end{aligned}$$
Hence,
$$\begin{aligned}& \bigl\vert \mathcal{L}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(g;q_{n_{1}},q_{n_{2}},x,y)-g(x,y) \bigr\vert \\& \quad \leq \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})} \biggl( \biggl\vert \int^{t}_{x} \bigl\vert (t-u) \bigr\vert \biggl\vert \frac{\partial^{2} g(u,y)}{\partial u^{2}} \biggr\vert \, du \biggr\vert ;q_{n_{1}},q_{n_{2}},x,y \biggr) \\& \qquad {} + \biggl\vert \int_{x}^{\frac{\alpha_{1}}{[n_{1}+1]_{q_{n_{1}}}+\beta_{1}}+\frac {2q_{n_{1}}[n_{1}]_{q_{n_{1}}}x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta _{1})}} \biggl\vert \biggl( \frac{\alpha_{1}}{[n_{1}+1]_{q_{n_{1}}}+\beta_{1}}+\frac {2q_{n_{1}}[n_{1}]_{q_{n_{1}}}x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta _{1})}-u \biggr) \biggr\vert \\& \qquad {}\times \biggl\vert \frac{\partial^{2} g(u,y)}{\partial u^{2}} \biggr\vert \, du \biggr\vert +\mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta_{2})} \biggl( \biggl\vert \int^{s}_{y} \bigl\vert (s-v) \bigr\vert \biggl\vert \frac {\partial^{2} g(x,v)}{\partial v^{2}} \biggr\vert \, dv \biggr\vert ;q_{n_{1}},q_{n_{2}},x,y \biggr) \\& \qquad {}+ \biggl\vert \int_{y}^{\frac{\alpha_{2}}{[n_{2}+1]_{q_{n_{2}}}+\beta _{2}}+\frac {2q_{n_{2}}[n_{2}]_{q_{n_{2}}}y+1}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta _{2})}} \biggl\vert \biggl( \frac{\alpha_{2}}{[n_{2}+1]_{q_{n_{2}}}+\beta_{2}}+\frac {2q_{n_{2}}[n_{2}]_{q_{n_{2}}}y+1}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta _{2})}-v \biggr) \biggr\vert \\& \qquad {}\times\biggl\vert \frac{\partial^{2} g(x,v)}{\partial v^{2}} \biggr\vert \, dv \biggr\vert \\& \quad = A_{n_{1},n_{2}}^{p_{1},p_{2}}(q_{n_{1}},q_{n_{2}},x,y) \|g\|_{C^{2}(I_{1}\times I_{2})}. \end{aligned}$$
(3.5)
Also,
$$\begin{aligned}& \bigl\vert \mathcal{L}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y) \bigr\vert \\& \quad \leq \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y) \bigr\vert +\biggl\vert f\biggl(\frac{\alpha _{1}}{[n_{1}+1]_{q_{n_{1}}}+\beta_{1}}+\frac {2q_{n_{1}}[n_{1}]_{q_{n_{1}}}x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta _{1})}, \\& \qquad {}\frac{\alpha_{2}}{[n_{2}+1]_{q_{n_{2}}}+\beta_{2}}+\frac {2q_{n_{2}}[n_{2}]_{q_{n_{2}}}y+1}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta _{2})}\biggr)\biggr\vert +\bigl\vert f(x,y)\bigr\vert \\& \quad \leq 3\|f\|_{C(I_{1}\times I_{2})}. \end{aligned}$$
(3.6)
Hence, considering (3.4), (3.6), and (3.5) (in that order),
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta _{1},\beta_{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad = \biggl\vert \mathcal{L}^{n_{1},n_{2},p_{1},p_{2}}_{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y)+f \biggl(\frac{\alpha _{1}}{[n_{1}+1]_{q_{n_{1}}}+\beta_{1}}+\frac {2q_{n_{1}}[n_{1}]_{q_{n_{1}}}x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta _{1})}, \\& \qquad \frac{\alpha_{2}}{[n_{2}+1]_{q_{n_{2}}}+\beta_{2}}+\frac {2q_{n_{2}}[n_{2}]_{q_{n_{2}}}y+1}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta _{2})} \biggr)-f(x,y) \biggr\vert \\& \quad \leq \bigl\vert \mathcal{L}^{n_{1},n_{2},p_{1},p_{2}}_{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(f-g;q_{n_{1}},q_{n_{2}},x,y) \bigr\vert +\bigl\vert \mathcal{L}^{n_{1},n_{2},p_{1},p_{2}}_{(\alpha _{1},\alpha_{2},\beta_{1},\beta _{2})}(g;q_{n_{1}},q_{n_{2}},x,y)-g(x,y) \bigr\vert \\& \qquad {}+\bigl\vert g(x,y)-f(x,y)\bigr\vert \\& \qquad {}+ \biggl\vert f \biggl(\frac{\alpha_{1}}{[n_{1}+1]_{q_{n_{1}}}+\beta_{1}}+\frac {2q_{n_{1}}[n_{1}]_{q_{n_{1}}}x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta _{1})}, \\& \qquad \frac{\alpha_{2}}{[n_{2}+1]_{q_{n_{2}}}+\beta_{2}}+\frac {2q_{n_{2}}[n_{2}]_{q_{n_{2}}}y+1}{[2]_{q_{n_{2}}}([n_{2}+1]_{q_{n_{2}}}+\beta _{2})} \biggr)-f(x,y) \biggr\vert \\& \quad \leq 4\|f-g\|_{C(I_{1}\times I_{2})}+\bigl\vert \mathcal {K}^{n_{1},n_{2},p_{1},p_{2}}_{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(g;q_{n_{1}},q_{n_{2}},x,y)-g(x,y) \bigr\vert \\& \qquad {} + \biggl\vert f \biggl(\frac{\alpha_{1}}{[n_{1}+1]_{q_{n_{1}}}+\beta_{1}}+\frac {2q_{n_{1}}[n_{1}]_{q_{n_{1}}}x+1}{[2]_{q_{n_{1}}}([n_{1}+1]_{q_{n_{1}}}+\beta_{1})}, \\& \qquad \frac{\alpha_{2}}{[n_{2}+1]_{q_{2}}+\beta_{2}}+\frac {2q_{2}[n_{2}]_{q_{2}}y+1}{[2]_{q_{2}}([n_{2}+1]_{q_{2}}+\beta_{2})} \biggr)-f(x,y) \biggr\vert \\& \quad \leq \bigl(4\|f-g\|_{C(I_{1}\times I_{2})}+A_{n_{1},n_{2}}^{p_{1},p_{2}}(q_{n_{1}},q_{n_{2}},x,y) \|g\|_{C^{2}(I_{1}\times I_{2})} \bigr) \\& \qquad {}+\omega \Bigl(f;\sqrt {B_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y)} \Bigr). \end{aligned}$$
Now, taking the infimum on the right hand side all over \(g\in C^{2}(I_{1}\times I_{2})\) and using (3.3)
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n_{1},n_{2},p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}(f;q_{n_{1}},q_{n_{2}},x,y)-f(x,y) \bigr\vert \\& \quad \leq 4\mathcal {K}\bigl(f;A_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y) \bigr) +\omega \Bigl(f;\sqrt{B_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y)} \Bigr) \\& \quad \leq M \Bigl\{ \tilde{\omega}_{2} \Bigl(f;\sqrt {A_{n_{1},n_{2}}^{p_{1},p_{2}}(q_{n_{1}},q_{n_{2}},x,y)} \Bigr) + \min\bigl\{ 1,A_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y) \bigr\} \|f\|_{C(I_{1}\times I_{2})} \Bigr\} \\& \qquad {}+\omega \Bigl(f;\sqrt {B_{n_{1},n_{2}}^{(p_{1},p_{2})}(q_{n_{1}},q_{n_{2}},x,y)} \Bigr). \end{aligned}$$
Thus, we get the desired result. □
Theorem 8
Let
\(f\in C^{2}(I_{1}\times I_{2})\). Then for every
\((x,y)\in J^{2}\),
$$\begin{aligned}& \lim_{[n]_{q_{n}}\rightarrow\infty}[n]_{q_{n}}\bigl\{ \mathcal {K}_{n,n,p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}\bigl(f(t,s);q_{n},x,y\bigr)-f(x,y) \bigr\} \\& \quad = f_{x}(x,y) \biggl(\frac{-x(a+1+2\beta _{1})}{2}+\alpha_{1}+ \frac{1}{2} \biggr) +f_{y}(x,y) \biggl(\frac{-y(a+1+2\beta _{2})}{2}+\alpha_{2}+ \frac{1}{2} \biggr) \\& \qquad {}+\frac{1}{2}\biggl\{ f_{xx}(x,y)\frac {x(1-x)}{2}+f_{yy}(x,y) \frac{y(1-y)}{2}\biggr\} \end{aligned}$$
uniformly in
\((x,y)\in J^{2}\).
Proof
By Taylor’s formula for f, we have
$$\begin{aligned} f(t,s) =&f(x,y)+f_{x}(x,y) (t-x)+f_{y}(x,y) (s-y) \\ &{}+ \frac{1}{2}\bigl\{ f_{xx}(x,y) (t-x)^{2} +2f_{xy}(x,y) (t-x) (s-y)+f_{yy}(x,y) (s-y)^{2}\bigr\} \\ &{}+\xi (t,s,x,y)\sqrt{(t-x)^{4}+(s-y)^{4}}, \end{aligned}$$
where \(\xi(t,s,x,y)\rightarrow0\) as \((t,s)\rightarrow(x,y)\) and \(\xi (t,s,x,y)\in C^{2}(I_{1}\times I_{2})\). Now, applying the operator \(\mathcal{K}_{n,n,p_{1},p_{2}}^{(\alpha_{1},\alpha _{2}, \beta_{1},\beta_{2})}(\cdot;q_{n},x,y)\) on the above equation, we get
$$\begin{aligned}& \mathcal{K}_{n,n,p_{1},p_{2}}^{(\alpha_{1},\alpha_{2}, \beta_{1},\beta _{2})}\bigl(f(t,s);q_{n},x,y \bigr) \\& \quad = f(x,y)+f_{x}(x,y)\mathcal{K}_{n,p_{1}}^{(\alpha _{1},\beta_{1})} \bigl((t-x);q_{n},x\bigr)+f_{y}(x,y)\mathcal{K}_{n,p_{2}}^{(\alpha_{2},\beta _{2})} \bigl((s-y);q_{n},y\bigr) \\& \qquad {}+\frac{1}{2}\bigl\{ f_{xx}(x,y)\mathcal {K}_{n,p_{1}}^{(\alpha_{1},\beta_{1})}\bigl((t-x)^{2};q_{n},x \bigr)+2f_{xy}(x,y)\mathcal {K}_{n,p_{1}}^{(\alpha_{1},\beta_{1})} \bigl((t-x);q_{n},x\bigr) \\& \qquad {}\times\mathcal {K}_{n,p_{2}}^{\alpha_{2},\beta_{2}}\bigl((s-y);q_{n},y \bigr)+f_{yy}\mathcal {K}_{n,p_{2}}^{\alpha_{2},\beta_{2}} \bigl((s-y)^{2};q_{n},y\bigr)\bigr\} \\& \qquad {}+\mathcal {K}_{n,n,p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})}\bigl(\xi (t,s,x,y) \sqrt{(t-x)^{4}+(s-y)^{4}};q_{n},x,y\bigr). \end{aligned}$$
Hence, using Lemma 2,
$$\begin{aligned}& \lim_{[n]_{q_{n}}\rightarrow\infty}[n]_{q_{n}}\bigl\{ \mathcal {K}_{n,n,p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}\bigl(f(t,s);q_{n},x,y\bigr)-f(x,y) \bigr\} \\& \quad = f_{x}(x,y) \biggl(\frac{-x(a+1+2\beta _{1})}{2}+\alpha_{1}+ \frac{1}{2} \biggr) +f_{y}(x,y) \biggl(\frac{-y(a+1+2\beta _{2})}{2}+\alpha_{2}+ \frac{1}{2} \biggr) \\& \qquad {}+\frac{1}{2}\biggl\{ f_{xx}(x,y)\frac {x(1-x)}{2}+f_{yy}(x,y) \frac{y(1-y)}{2}\biggr\} \\& \qquad {}+\lim_{[n]_{q_{n}}\rightarrow\infty}[n]_{q_{n}}\mathcal {K}_{n,n,p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})} \bigl(\xi (t,s,x,y)\sqrt{(t-x)^{4}+(s-y)^{4}};x,y \bigr) \end{aligned}$$
uniformly in \((x,y)\in J^{2}\).
Applying the Cauchy-Schwarz inequality
$$\begin{aligned}& \bigl\vert \mathcal{K}_{n,n,p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})}\bigl(\xi (t,s) \sqrt{(t-x)^{4}+(s-y)^{4}};q_{n},x,y\bigr)\bigr\vert \\& \quad \leq\sqrt{\mathcal{K}_{n,n,p_{1},p_{2}}^{\alpha_{1},\alpha_{2},\beta_{1},\beta _{2}}\bigl( \xi^{2}(t,s);q_{n},x,y\bigr)}\sqrt{ \mathcal{K}_{n,n,p_{1},p_{2}}^{(\alpha _{1},\alpha_{2},\beta_{1},\beta_{2})}\bigl((t-x)^{4}+(s-y)^{4};q_{n},x,y \bigr)}. \end{aligned}$$
Since, by Theorem 2 and in view of Lemma 2,
$$\begin{aligned}& \lim_{[n]_{q_{n}}\rightarrow\infty}\mathcal{K}_{n,n,p_{1},p_{2}}^{(\alpha _{1},\alpha_{2},\beta_{1},\beta_{2})}\bigl( \xi^{2}(t,s);x,y\bigr)=\xi^{2}(x,y)=0, \\& \mathcal{K}_{n,n,p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}\bigl((t-x)^{4};q_{n},x \bigr)=O \biggl(\frac{1}{[n]_{q_{n}}^{2}} \biggr),\quad \mbox{and} \\& \mathcal{K}_{n,p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta _{2})}\bigl((s-y)^{4};q_{n},y \bigr)=O \biggl(\frac{1}{[n]_{q_{n}}^{2}} \biggr) \end{aligned}$$
uniformly in \((x,y)\in J^{2}\), it follows that
$$\lim_{n\rightarrow\infty}[n]_{q_{n}}\bigl\{ \mathcal {K}_{n,n,p_{1},p_{2}}^{(\alpha_{1},\alpha_{2},\beta_{1},\beta_{2})}\bigl(\xi(t,s)\sqrt {(t-x)^{4}+(s-y)^{4}};q_{n},x,y \bigr)\bigr\} =0 $$
uniformly in \((x,y)\in J^{2}\), the desired result is obtained. □
In the following example, the rate of convergence of the bivariate operators given by (3.1) to a certain function is shown by illustrative graphics. We observe that when the values of \(q_{1}\) and \(q_{2}\) increase, the approximation of f by the operator \(\mathcal {K}_{n_{1},n_{2},p_{1}p_{2}}^{ ( \alpha_{1},\alpha_{2},\beta _{1},\beta _{2} ) } ( f;q_{1},q_{2},x,y ) \) becomes better.
Example 3
Let \(n_{1}=n_{2}=5\), \(\alpha_{1}=0.5\), \(\beta_{1}=0.6\), \(\alpha _{2}=0.7\), \(\beta_{2}=0.8\), \(p_{1}=p_{2}=1\). For \(q_{1}=0.45\), \(q_{2}=0.50\) (green) and \(q_{1}=0.85\), \(q_{2}=0.90\) (pink), the convergence of the operators \(\mathcal {K}_{n_{1},n_{2},p_{1}p_{2}}^{ ( \alpha_{1},\alpha_{2},\beta _{1},\beta _{2} ) } ( f;q_{1},q_{2},x,y ) \) given by (3.1) to \(f ( x,y ) =\sin ( x+y ) /(1+xy)\) (yellow) is illustrated in Figure 4.