Approximation of GBS Type q-Jakimovski-Leviatan-Beta Integral Operators in Bögel Space

Abstract

In the present article, we introduce the bivariate variant of Beta integral type operators based on Appell polynomials via q-calculus. We study the local and global type approximation properties for these new operators. Next, we introduce the GBS form for these new operators and then study the degree of approximation by means of modulus of smoothness, mixed modulus of smoothness and Lipschitz class of Bögel continuous functions.

1. Introduction and Preliminaries

In 1950, very famous mathematician, Szász, introduced the operators known as Szász positive linear operators [1]. Suppose $v\in [0,\infty )$ and the class of all continuous functions on $[0,\infty )$ is $C[0,\infty )$, then, for all $f\in C[0,\infty )$, the Szász operators are defined as:

$$S_r(f;v)=e^{-rv}\sum _{k=0}^{\infty }\frac{{\left(rv\right)}^k}{k!}f\left(\frac{k}{r}\right).$$

(1)

Later, the development of Szász operators introduced by the identity of Appell polynomials [2] and an advance technique of Appell polynomials were been addressed by Jakimovski and Leviatan in 1969 [3] by:

$$X\left(v\right)e^{vx}=\sum _{k=0}^{\infty }{\beta }_k\left(x\right)v^k,$$

(2)

where ${\beta }_k\left(x\right)={\sum }_{i=0}^k{\alpha }_i\frac{x^{m-i}}{(m-i)!}$ for $m\in \mathbb{N}$ and $X\left(v\right)={\sum }_{k=0}^{\infty }{\alpha }_k{v}^k,X\left(1\right)\ne 0$. Let $E[0,\infty )$ denote the set of functions defined by $[0,\infty )$ such that $\left|f\left(x\right)\right|\le \kappa e^{\gamma x}$, where $\kappa ,\gamma$ are positive constants. For further development, we prefer to see the Jakimovski–Leviatan types and other related articles [4,5,6,7,8,9,10].

The family of q-Appell polynomials introduced by Al-Salam (see [11,12]) by using the generating functions $X_q\left(t\right)={\displaystyle \sum _{m=0}^{\infty }}{X}_{m,q}\frac{t^m}{{\left[m\right]}_q!},X_q\left(1\right)\ne 0$, such that:

$$X_{m,q}\left(x\right)=\sum _{k=0}^m{\left[\begin{array}{c}m\\ k\end{array}\right]}_q{A}_{m-k,q}{x}^k,(m\in \mathbb{N}),$$

(3)

where the q-differential $D_{q,x}\left(X_{m,q}\left(x\right)\right)={\left[m\right]}_q{X}_{m-1,q}\left(x\right),m=1,2,\cdots$ and notice that $D_{q,x}\left(X_{1,q}\left(x\right)\right)={\left[1\right]}_q{X}_{0,q}\left(x\right)=X_{0,q}$, with $X_{0,q}$ is a positive constant. In addition, the generating function satisfies the equality $X_q\left(t\right)e_q\left(tx\right)={\displaystyle \sum _{m=0}^{\infty }}{X}_{m,q}\left(x\right)\frac{t^m}{{\left[m\right]}_q!},0<q<1$. Here, we note some basic formulae of q-calculus, thus, the q-integer is given by:

$${\left[\eta \right]}_q=\left\{\begin{array}{c}\frac{1-q^{\eta }}{1-q},q\ne 1\hfill \\ \eta ,q=1\hfill \end{array}\right.for\eta \in \mathbb{N}and{\left[0\right]}_q=0.$$

(4)

For $\left|q\right|<1,$ the q-factorial ${\left[\eta \right]}_q!$ is defined by:

$${\left[\eta \right]}_q!=\left\{\begin{array}{cc}1,\hfill & (\eta =0)\hfill \\ {\displaystyle \prod _{k=1}^{\eta }}{\left[k\right]}_q,\hfill & (\eta \in \mathbb{N}).\hfill \end{array}\right.$$

(5)

In the standard approach, the exponential functions for q-calculus:

$$e_q\left(x\right)=\sum _{k=0}^{\infty }\frac{x^k}{{\left[k\right]}_q!},$$

(6)

$$E_q\left(x\right)=\sum _{k=0}^{\infty }\frac{q^{\frac{k(k-1)}{2}}{x}^k}{{\left[k\right]}_q!},$$

(7)

while the q-Jackson improper integral of the function f is given as:

$${\int }_0^{\infty /A}f\left(x\right)d_{q}x=(1-q)\sum _{m\in \mathbb{N}}f\left(\frac{q^m}{A}\right)\frac{q^m}{A},A\in \mathbb{R}-\left\{0\right\},$$

(8)

where $\mathbb{R}$ is the set of real numbers.

The q-beta function $B_q(l,m)$ is defined as follows:

$$B_q(l,m){\int }_0^1{t}^{l-1}{(1-qt)}_q^{m-1}{d}_{q}t(l>0;m>0)$$

(9)

and for $l>1;m>0$:

$$B_q(l,m)=\mathcal{K}(A,l){\int }_0^{\infty /A}\frac{t^{l-1}}{{(1+t)}_q^{l+m}}{d}_{q}t=\frac{{[l-1]}_q}{{\left[m\right]}_q}{B}_q(l-1,m+1)$$

(10)

with the explicit form of:

$$\mathcal{K}(A,l+1)=q^l\mathcal{K}(A,l).$$

$$\mathcal{K}(A,l)=\frac{1}{A+1}{A}^l{\left(1+\frac{1}{A}\right)}_q^l{\left(1+A\right)}_q^{1-l},$$

(11)

while the Gamma function in the $q-$ form is defined as follows:

$${\Gamma }_q\left(x\right)={\int }_0^{1/1-q}{t}^{x-1}{E}_q(-qt)d_{q}t(x>0)$$

(12)

satisfying ${\Gamma }_q(x+1)={\left[x\right]}_q{\Gamma }_q\left(x\right)$, ${\Gamma }_q\left(1\right)=1$, and

$${\Gamma }_q\left(x\right)={\int }_0^{\infty /1-q}{t}^{x-1}{E}_q(-qt)d_{q}t(x>0)$$

(13)

and a relation of the Beta and Gamma functions given as:

$$B_q(l,m)=\frac{{\Gamma }_q\left(l\right){\Gamma }_q\left(m\right)}{{\Gamma }_q(l+m)}.$$

(14)

In the present context, we consider the recent development of the published article [13] and then construct the operators in terms of two variables supposing $0\le v_1,v_2<\infty$. Finally, we study the approximation results in terms bivariate functions by modulus and mixed-modulus of continuity and Lipschitz maximal functions. Next, we also construct the GBS-type polynomial functions and obtain the approximation in terms of Bögel continuous functions. For more related concepts on these classes of functions, we prefer to see the recent published article by Nasiruzzaman et al. [14,15,16,17,18,19]. In addition, there are various operators in several functional spaces given by the authors: Mohiuddine et al. [20,21,22,23], Mursaleen et al. [24,25], Acar et al. [26], Kajla et al. [27], Özger et al. [28], and Rao et al. [29,30].

2. Operators and Their Associated Moments

Take ${\mathcal{M}}^2=\{(v_1,v_2):0\le v_1<\infty ,0\le v_2<\infty \}$, and $C\left({\mathcal{M}}^2\right)$ is the class of all continuous functions on ${\mathcal{M}}^2$ and satisfies the norm by ${\left|\right|g\left|\right|}_{C\left({\mathcal{M}}^2\right)}={sup}_{(v_1,v_2)\in {\mathcal{M}}^2}\left|f(v_1,v_2)\right|$. Suppose $f\in C\left({\mathcal{M}}^2\right)$. Then, for all $i=1,2$ such that $m_i\in \mathbb{N}$, $X_{r_i,q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right)\ge 0$ and $X_{q_i}\left(1\right)\ne 0,$ we define:

$${\mathcal{J}}_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)=\sum _{r_1,r_2=0}^{\infty }{\mathcal{U}}_{m_1,q_1}^{r_1}\left(v_1\right){\mathcal{V}}_{m_2,q_2}^{r_2}\left(v_2\right)\underset{0}{\overset{\infty /A_1}{\int }}\underset{0}{\overset{\infty /A_2}{\int }}{\mathcal{T}}_{m_1,m_2}^{q_1,q_2}(t,s)f\left(q_1^{r_1}t,q_2^{r_2}s\right)d_{q_1}t{d}_{q_2}s,$$

(15)

where

$${\mathcal{U}}_{m_1,r_1}^{q_1}\left(v_1\right)=\frac{e_{q_1}(-{\left[m_1\right]}_{q_1}{v}_1)}{X_{q_1}\left(1\right)}\frac{R_{r_1,q_1}\left({\left[m_1\right]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}\frac{{\mathcal{K}}_1(A_1,r_1+1)}{B_{q_1}(r_1+1,m_1)},$$

(16)

$${\mathcal{V}}_{m_2,r_2}^{q_2}\left(v_2\right)=\frac{e_{q_2}(-{\left[m_2\right]}_{q_2}{v}_2)}{X_{q_2}\left(1\right)}\frac{R_{r_2,q_2}\left({\left[m_2\right]}_{q_2}{v}_2\right)}{{\left[r_2\right]}_{q_2}!}\frac{{\mathcal{K}}_2(A_2,r_2+1)}{B_{q_2}(r_2+1,m_2)}$$

(17)

and

$${\mathcal{T}}_{m_1,m_2}^{q_1,q_2}(t,s)=\frac{t^{r_1}}{{(1+t)}_{q_1}^{r_1+m_1+1}}\frac{s^{r_2}}{{(1+s)}_{q_2}^{r_2+m_2+1}}.$$

(18)

It is noticed that for all $i=1,2,$ and $r_i>1,m_i>0$ we have:

$$B_{q_i}(r_i,m_i)={\mathcal{K}}_i(A_i,r_i){\int }_0^{\infty /A_i}\frac{u^{r_i-1}}{{(1+u_i)}_{q_i}^{r_i+m_i}}{d}_{q_i}{u}_i=\frac{{[r_i-1]}_{q_i}}{{\left[m_i\right]}_{q_i}}{B}_{q_i}(r_i-1,m_i+1),$$

(19)

with

$${\mathcal{K}}_i(A_i,r_i+1)=q_i^{r_i}{\mathcal{K}}_i(A_i,r_i),$$

$${\mathcal{K}}_i(A_i,r_i)=q_i^{\frac{r_i(r_i-1)}{2}},{\mathcal{K}}_i(A_i,0)=1.$$

Lemma 1.

Operators $J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)$ defined by (15) have the equality:

$$J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)={\mathcal{C}}_{m_1,r_1}^{q_1}\left({\mathcal{D}}_{m_2,r_2}^{q_2}(f;v_1,v_2)\right)={\mathcal{D}}_{m_2,r_2}^{q_2}\left({\mathcal{C}}_{m_1,r_1}^{q_1}(f;v_1,v_2)\right)$$

(20)

where

$${\mathcal{C}}_{m_1,r_1}^{q_1}(f;v_1,v_2)=\sum _{r_1=0}^{\infty }{\mathcal{U}}_{m_1,r_1}^{q_1}\left(v_1\right)\underset{0}{\overset{\infty /A_1}{\int }}\frac{t^{r_1}}{{(1+t)}_{q_1}^{r_1+m_1+1}}f\left(q_1^{r_1}t,q_2^{r_2}s\right)d_{q_1}t$$

(21)

$${\mathcal{D}}_{m_2,r_2}^{q_2}(f;v_1,v_2)=\sum _{r_2=0}^{\infty }{\mathcal{V}}_{m_2,r_2}^{q_2}\left(v_2\right)\underset{0}{\overset{\infty /A_2}{\int }}\frac{s^{r_2}}{{(1+s)}_{q_2}^{r_2+m_2+1}}f\left(q_1^{r_1}t,q_2^{r_2}s\right)d_{q_2}s.$$

(22)

Proof. 

We easily see that:

$$\begin{array}{ccc}\hfill {\mathcal{C}}_{m_1,r_1}^{q_1}\left({\mathcal{D}}_{m_2,r_2}^{q_2}(f;v_1,v_2)\right)& =& {\mathcal{C}}_{m_1,r_1}^{q_1}\left(\sum _{r_2=0}^{\infty }{\mathcal{V}}_{m_2,r_2}^{q_2}\left(v_2\right)\underset{0}{\overset{\infty /A_2}{\int }}\frac{s^{r_2}}{{(1+s)}_{q_2}^{r_2+m_2+1}}f\left(q_1^{r_1}t,q_2^{r_2}s\right)d_{q_2}s\right)\hfill \\ & =& \sum _{r_2=0}^{\infty }{\mathcal{V}}_{m_2,r_2}^{q_2}\left(v_2\right){\mathcal{C}}_{m_1,r_1}^{q_1}\left(\underset{0}{\overset{\infty /A_2}{\int }}\frac{s^{r_2}}{{(1+s)}_{q_2}^{r_2+m_2+1}}f\left(q_1^{r_1}t,q_2^{r_2}s\right)\right)d_{q_2}s\hfill \\ & =& \sum _{r_1,r_2=0}^{\infty }{\mathcal{U}}_{m_1,q_1}^{r_1}\left(v_1\right){\mathcal{V}}_{m_2,q_2}^{r_2}\left(v_2\right)\underset{0}{\overset{\infty /A_1}{\int }}\underset{0}{\overset{\infty /A_2}{\int }}{\mathcal{T}}_{m_1,m_2}^{q_1,q_2}(t,s)f\left(q_1^{r_1}t,q_2^{r_2}s\right)d_{q_1}t{d}_{q_2}s\hfill \\ & =& J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2).\hfill \end{array}$$

On the other hand, we obtain ${\mathcal{D}}_{m_2,r_2}^{q_2}\left({\mathcal{C}}_{m_1,r_1}^{q_1}(f;v_1,v_2)\right)=J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)$. □

Lemma 2.

For all $i=1,2,$ if we take $R_{r_i,q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right)\ge 0$ and $X_{q_i}\left(1\right)\ne 0,$ then for each $m_i\in \mathbb{N},$ we have:

$$\begin{array}{ccc}\hfill \sum _{r_i=0}^{\infty }\frac{X_{r_i,q_i}\left({\left[m_1\right]}_{q_i}{v}_i\right)}{{\left[r_i\right]}_{q_i}!}& =& X_{q_i}\left(1\right)e_{q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right),\hfill \\ \hfill \sum _{r_i=0}^{\infty }{r}_i\frac{X_{r_i,q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right)}{{\left[r_i\right]}_{q_i}!}& =& \left[{\left[m_i\right]}_{q_i}{X}_{q_i}\left(1\right)v_i+X_{q_i}^{\prime }\left(1\right)\right]e_{q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right),\hfill \\ \hfill \sum _{r_i=0}^{\infty }{r}_i^2\frac{X_{r_i,q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right)}{{\left[r_i\right]}_{q_i}!}& =& \left[{\left[m_i\right]}_{q_i}^2{X}_{q_i}\left(1\right)v_i^2+2{\left[m_i\right]}_{q_i}{X}_{q_i}^{\prime }\left(1\right)v_i+X_{q_i}^{″}\left(1\right)\right]e_{q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right),\hfill \\ \hfill \sum _{r_i=0}^{\infty }{r}_i^3\frac{X_{r_i,q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right)}{{\left[r_i\right]}_{q_i}!}& =& \left[{\left[m_i\right]}_{q_i}^3{X}_{q_i}\left(1\right)v_i^3+3{\left[m_i\right]}_{q_i}^2{X}_{q_i}^{\prime }\left(1\right)v_i^2+3{\left[m_i\right]}_{q_i}{X}_{q_i}^{″}\left(1\right)v_i+X_{q_i}^{‴}\left(1\right)\right]e_{q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right),\hfill \\ \hfill \sum _{r_i=0}^{\infty }{r}_i^4\frac{X_{r_i,q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right)}{{\left[r_i\right]}_{q_i}!}\end{array}$$

$$=\left[{\left[m_i\right]}_{q_i}^4{X}_{q_i}\left(1\right)v_i^4+4{\left[m_i\right]}_{q_i}^3{X}_{q_i}^{\prime }\left(1\right)v_i^3+6{\left[m_i\right]}_{q_i}^2{X}_{q_i}^{″}\left(1\right)v_i^2+4{\left[m_i\right]}_{q_i}{X}_{q_i}^{‴}\left(1\right)v_i+X_{q_i}^{\left(4\right)}\left(1\right)\right]e_{q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right).$$

Lemma 3.

Let $i,j=\{0,1,2,3,4\}$ and $f(t,s)={\lambda }_{i,j}$ be the test functions such that ${\lambda }_{i,j}=t^i{s}^j$. Then, for every $m_1,m_2\in \mathbb{N}\setminus \{1,2,3,4\},$ operators $J_{m_1,m_2}^{q_1,q_2}(.;.)$ have the following equalities:

$$\begin{array}{ccc}\hfill \left(1\right)J_{m_1,m_2}^{q_1,q_2}({\lambda }_{0,0};v_1,v_2)& =& {\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{0,0};v_1,v_2)\hfill \\ & =& {\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{0,0};v_1,v_2)=1;\hfill \\ \hfill \left(2\right)J_{m_1,m_2}^{q_1,q_2}({\lambda }_{1,0};v_1,v_2)& =& {\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{1,0};v_1,v_2)\hfill \\ & =& \frac{1}{q_1{[m_1-1]}_{q_1}}+\frac{1}{{[m_1-1]}_{q_1}}\left({\left[m_1\right]}_{q_1}{v}_1+\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}\right);\hfill \\ \hfill \left(3\right)J_{m_1,m_2}^{q_1,q_2}({\lambda }_{0,1};v_1,v_2)& =& {\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{0,1};v_1,v_2)\hfill \\ & =& \frac{1}{q_2{[m_2-1]}_{q_2}}+\frac{1}{{[m_2-1]}_{q_2}}\left({\left[m_2\right]}_{q_2}{v}_2+\frac{X_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}\right);\hfill \\ \hfill \left(4\right)J_{m_1,m_2}^{q_1,q_2}({\lambda }_{2,0};v_1,v_2)& =& {\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{2,0};v_1,v_2)\hfill \\ & =& \frac{(1+q_1)}{q_1^3{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}}+\frac{(1+2q_1)}{q_1^2{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}}\left({\left[m_1\right]}_{q_1}{v}_1+\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}\right)\hfill \\ & +& \frac{1}{{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}}\left({\left[m_1\right]}_{q_1}^2{v}_1^2+\frac{2{\left[m_1\right]}_{q_1}{X}_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}{v}_1+\frac{X_{q_1}^{″}\left(1\right)}{X_{q_1}\left(1\right)}\right);\hfill \\ \hfill \left(5\right)J_{m_1,m_2}^{q_1,q_2}({\lambda }_{0,2};v_1,v_2)& =& {\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{0,2};v_1,v_2)\hfill \\ & =& \frac{(1+q_2)}{q_2^3{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}}+\frac{(1+2q_2)}{q_2^2{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}}\left({\left[m_2\right]}_{q_2}{v}_2+\frac{X_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}\right)\hfill \\ & +& \frac{1}{{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}}\left({\left[m_2\right]}_{q_2}^2{v}_2^2+\frac{2{\left[m_2\right]}_{q_2}{X}_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}{v}_2+\frac{X_{q_2}^{″}\left(1\right)}{X_{q_2}\left(1\right)}\right);\hfill \\ \hfill \left(6\right)J_{m_1,m_2}^{q_1,q_2}({\lambda }_{3,0};v_1,v_2)& =& {\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{3,0};v_1,v_2)\hfill \\ & =& \frac{1+2q_1+2q_1^2+q_1^3}{q_1^4{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}{[m_1-3]}_{q_1}}\hfill \\ & & +\frac{(1+3q_1+4q_1^2+3q_1^3)}{q_1^3{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}{[m_1-3]}_{q_1}}\left(({\left[m_1\right]}_{q_1}{v}_1+\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}\right)\hfill \\ & +& \frac{(1+2q_1+3q_1^2)}{q_1{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}{[m_1-3]}_{q_1}}({\left[m_1\right]}_{q_1}^2{v}_1^2+\frac{2{\left[m_1\right]}_{q_1}{X}_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}{v}_1\hfill \\ & & +\frac{X_{q_1}^{″}\left(1\right)}{X_{q_1}\left(1\right)})+\frac{q_1^2}{{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}{[m_1-3]}_{q_1}}({\left[m_1\right]}_{q_1}^3{v}_1^3\hfill \\ & & +3{\left[m_1\right]}_{q_1}^2\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}{v}_1^2+3{\left[m_1\right]}_{q_1}\frac{X_{q_1}^{″}\left(1\right)}{X_{q_1}\left(1\right)}{v}_1+\frac{X_{q_1}^{‴}\left(1\right)}{X_{q_1}\left(1\right)});\hfill \\ \hfill \left(7\right)J_{m_1,m_2}^{q_1,q_2}({\lambda }_{0,3};v_1,v_2)& =& {\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{0,3};v_1,v_2)\hfill \\ & =& \frac{1+2q_2+2q_2^2+q_2^3}{q_2^4{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}{[m_2-3]}_{q_2}}\hfill \\ & & +\frac{(1+3q_2+4q_2^2+3q_2^3)}{q_2^3{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}{[m_2-3]}_{q_2}}\left(({\left[m_2\right]}_{q_2}{v}_2+\frac{X_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}\right)\hfill \\ & +& \frac{(1+2q_2+3q_2^2)}{q_2{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}{[m_2-3]}_{q_2}}({\left[m_2\right]}_{q_2}^2{v}_2^2+\frac{2{\left[m_2\right]}_{q_2}{X}_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}{v}_2\hfill \\ & & +\frac{X_{q_2}^{″}\left(1\right)}{X_{q_2}\left(1\right)})+\frac{q_2^2}{{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}{[m_2-3]}_{q_2}}({\left[m_2\right]}_{q_2}^3{v}_2^3\hfill \\ & & +3{\left[m_2\right]}_{q_2}^2\frac{X_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}{v}_2^2+3{\left[m_2\right]}_{q_2}\frac{X_{q_2}^{″}\left(1\right)}{X_{q_2}\left(1\right)}{v}_2+\frac{X_{q_2}^{‴}\left(1\right)}{X_{q_2}\left(1\right)});\hfill \\ \hfill \left(8\right)J_{m_1,m_2}^{q_1,q_2}({\lambda }_{4,0};v_1,v_2)& =& {\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{4,0};v_1,v_2)\hfill \\ & =& \frac{1+3q_1+5q_1^2+6q_1^3+5q_1^4+3q_1^5+q_1^6}{q_1^5{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}{[m_1-3]}_{q_1}{[m_1-4]}_{q_1}}\hfill \\ & +& \frac{(1+5q_1+10q_1^2+13q_1^3+12q_1^4+7q_1^5+2q_1^6)}{q_1^3{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}{[m_1-3]}_{q_1}{[m_1-4]}_{q_1}}\left({\left[m_1\right]}_{q_1}{v}_1+\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}\right)\hfill \\ & +& \frac{(1+3q_1+7q_1^2+9q_1^3+9q_1^4+6q_1^5)}{q_1{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}{[m_1-3]}_{q_1}{[m_1-4]}_{q_1}}({\left[m_1\right]}_{q_1}^2{v}_1^2\hfill \\ & & +\frac{2{\left[m_1\right]}_{q_1}{X}_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}{v}_1+\frac{X_{q_1}^{″}\left(1\right)}{X_{q_1}\left(1\right)})\hfill \\ & +& \frac{(q_1^2+2q_1^3+2q_1^4+2q_1^5+q_1^6+2q_1^7)}{{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}{[m_1-3]}_{q_1}{[m_1-4]}_{q_1}}({\left[m_1\right]}_{q_1}^3{v}_1^3\hfill \\ & & +3{\left[m_1\right]}_{q_1}^2\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}{v}_1^2+3{\left[m_1\right]}_{q_1}\frac{X_{q_1}^{″}\left(1\right)}{X_{q_1}\left(1\right)}{v}_1+\frac{X_{q_1}^{‴}\left(1\right)}{X_{q_1}\left(1\right)})\hfill \\ & +& \frac{q_1^6}{{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}{[m_1-3]}_{q_1}{[m_1-4]}_{q_1}}({\left[m_1\right]}_{q_1}^4{v}_1^4+4{\left[m_1\right]}_{q_1}^3\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}{v}_1^3\hfill \\ & & +6{\left[m_1\right]}_{q_1}^2\frac{X_{q_1}^{″}\left(1\right)}{X_{q_1}\left(1\right)}{v}_1^2+4{\left[m_1\right]}_{q_1}\frac{X_{q_1}^{‴}\left(1\right)}{X_{q_1}\left(1\right)}{v}_1+\frac{X_{q_1}^{\left(4\right)}\left(1\right)}{X_{q_1}\left(1\right)});\hfill \\ \hfill \left(9\right)J_{m_1,m_2}^{q_1,q_2}({\lambda }_{0,4};v_1,v_2)& =& {\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{0,4};v_1,v_2)\hfill \\ & =& \frac{1+3q_2+5q_2^2+6q_2^3+5q_2^4+3q_2^5+q_2^6}{q_2^5{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}{[m_2-3]}_{q_2}{[m_2-4]}_{q_2}}\hfill \\ & +& \frac{(1+5q_2+10q_2^2+13q_2^3+12q_2^4+7q_2^5+2q_2^6)}{q_2^3{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}{[m_2-3]}_{q_2}{[m_2-4]}_{q_2}}\left({\left[m_2\right]}_{q_2}{v}_2+\frac{X_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}\right)\hfill \\ & +& \frac{(1+3q_2+7q_2^2+9q_2^3+9q_2^4+6q_2^5)}{q_2{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}{[m_2-3]}_{q_2}{[m_2-4]}_{q_2}}({\left[m_2\right]}_{q_2}^2{v}_2^2\hfill \\ & & +\frac{2{\left[m_2\right]}_{q_2}{X}_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}{v}_2+\frac{X_{q_2}^{″}\left(1\right)}{X_{q_2}\left(1\right)})\hfill \\ & +& \frac{(q_2^2+2q_2^3+2q_2^4+2q_2^5+q_2^6+2q_2^7)}{{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}{[m_2-3]}_{q_2}{[m_2-4]}_{q_2}}({\left[m_2\right]}_{q_2}^3{v}_2^3\hfill \\ & & +3{\left[m_2\right]}_{q_2}^2\frac{X_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}{v}_2^2+3{\left[m_2\right]}_{q_2}\frac{X_{q_2}^{″}\left(1\right)}{X_{q_2}\left(1\right)}{v}_2+\frac{X_{q_2}^{‴}\left(1\right)}{X_{q_2}\left(1\right)})\hfill \\ & +& \frac{q_2^6}{{[m_2-1]}_{q_2}{[m_2-2]}_{q_2}{[m_2-3]}_{q_2}{[m_2-4]}_{q_2}}({\left[m_2\right]}_{q_2}^4{v}_2^4+4{\left[m_2\right]}_{q_2}^3\frac{X_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}{v}_2^3\hfill \\ & & +6{\left[m_2\right]}_{q_2}^2\frac{X_{q_2}^{″}\left(1\right)}{X_{q_2}\left(1\right)}{v}_2^2+4{\left[m_2\right]}_{q_2}\frac{X_{q_2}^{‴}\left(1\right)}{X_{q_2}\left(1\right)}{v}_2+\frac{X_{q_2}^{\left(4\right)}\left(1\right)}{X_{q_2}\left(1\right)}).\hfill \end{array}$$

Proof. 

For $i,j=0,$ we take:

$$\begin{array}{ccc}\hfill J_{m_1,m_2}^{q_1,q_2}({\lambda }_{0,0};v_1,v_2)& =& \sum _{r_1,r_2=0}^{\infty }{\mathcal{U}}_{m_1,q_1}^{r_1}\left(v_1\right){\mathcal{V}}_{m_2,q_2}^{r_2}\left(v_2\right)\underset{0}{\overset{\infty /A_1}{\int }}\underset{0}{\overset{\infty /A_2}{\int }}{\mathcal{T}}_{m_1,m_2}^{q_1,q_2}(t,s)d_{q_1}t{d}_{q_2}s\hfill \\ & =& \sum _{r_1,r_2=0}^{\infty }{\mathcal{U}}_{m_1,q_1}^{r_1}\left(v_1\right){\mathcal{V}}_{m_2,q_2}^{r_2}\left(v_2\right)\underset{0}{\overset{\infty /A_1}{\int }}\frac{t^{r_1}}{{(1+t)}_{q_1}^{r_1+m_1+1}}{d}_{q_1}t\underset{0}{\overset{\infty /A_2}{\int }}\frac{s^{r_2}}{{(1+s)}_{q_2}^{r_2+m_2+1}}{d}_{q_2}s\hfill \\ & =& \frac{e_{q_1}(-{\left[m_1\right]}_{q_1}{v}_1)}{X_{q_1}\left(1\right)}\sum _{r_1=0}^{\infty }\frac{X_{r_1,q_1}\left({\left[m_1\right]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}\frac{B_{q_1}(r_1+1,m_1)}{B_{q_1}(r_1+1,m_1)}\hfill \\ & & \times \frac{e_{q_2}(-{\left[m_2\right]}_{q_2}{v}_2)}{X_{q_2}\left(1\right)}\sum _{r_2=0}^{\infty }\frac{X_{r_2,q_2}\left({\left[m_2\right]}_{q_2}{v}_2\right)}{{\left[r_2\right]}_{q_2}!}\frac{B_{q_2}(r_2+1,m_2)}{B_{q_2}(r_2+1,m_2)}\hfill \\ & =& 1.\hfill \end{array}$$

For $i=1,j=0,$ we obtain:

$$\begin{array}{ccc}\hfill J_{m_1,m_2}^{q_1,q_2}({\lambda }_{1,0};v_1,v_2)& =& \sum _{r_1,r_2=0}^{\infty }{\mathcal{U}}_{m_1,q_1}^{r_1}\left(v_1\right){\mathcal{V}}_{m_2,q_2}^{r_2}\left(v_2\right)\underset{0}{\overset{\infty /A_1}{\int }}\frac{{\left(q_{1}t\right)}^{r_1}}{{(1+t)}_{q_1}^{r_1+m_1+1}}{d}_{q_1}t\underset{0}{\overset{\infty /A_2}{\int }}\frac{s^{r_2}}{{(1+s)}_{q_2}^{r_2+m_2+1}}{d}_{q_2}s\hfill \\ & =& \frac{e_{q_1}(-{\left[m_1\right]}_{q_1}{v}_1)}{X_{q_1}\left(1\right)}\sum _{r_1=0}^{\infty }\frac{X_{r_1,q_1}\left({\left[m_1\right]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}{q}_1^{r_1}\frac{{\mathcal{K}}_1(A_1,r_1+1)}{B_{q_1}(r_1+1,m_1)}\underset{0}{\overset{\infty /A_1}{\int }}\frac{t^{r_1+1}}{{(1+t)}_{q_1}^{r_1+m_1+1}}{d}_{q_1}t\hfill \\ & & \times \frac{e_{q_2}(-{\left[m_2\right]}_{q_2}{v}_2)}{X_{q_2}\left(1\right)}\sum _{r_2=0}^{\infty }\frac{X_{r_2,q_2}\left({\left[m_2\right]}_{q_2}{v}_2\right)}{{\left[r_2\right]}_{q_2}!}\frac{B_{q_2}(r_2+1,m_2)}{B_{q_2}(r_2+1,m_2)}\hfill \\ & =& \frac{e_{q_1}(-{\left[m_1\right]}_{q_1}{v}_1)}{X_{q_1}\left(1\right)}\sum _{r_1=0}^{\infty }\frac{X_{r_1,q_1}\left({\left[m_1\right]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}{q}_1^{r_1}\frac{{\mathcal{K}}_1(A_1,r_1+1)}{B_{q_1}(r_1+1,m_1)}\frac{B_{q_1}(r_1+2,m_1-1)}{{\mathcal{K}}_1(A_1,r_1+2)}\hfill \\ & =& \frac{1}{q_1{[m_1-1]}_{q_1}}\frac{e_{q_1}(-{\left[m_1\right]}_{q_1}{v}_1)}{X_{q_1}\left(1\right)}\sum _{r_1=0}^{\infty }\frac{X_{r_1,q_1}\left({\left[m_1\right]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}{[r_1+1]}_{q_1}.\hfill \end{array}$$

By applying:

$${[r_1+1]}_{q_1}=1+q_1{\left[r_1\right]}_{q_1},$$

(23)

$$\begin{array}{ccc}\hfill J_{m_1,m_2}^{q_1,q_2}({\lambda }_{1,0};v_1,v_2)& =& \frac{1}{q_1{[m_1-1]}_{q_1}}\frac{e_{q_1}(-{\left[m_1\right]}_{q_1}{v}_1)}{X_{q_1}\left(1\right)}\sum _{r_1=0}^{\infty }\frac{X_{r_1,q_1}\left({\left[m_1\right]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}\hfill \\ & +& \frac{1}{{[m_1-1]}_{q_1}}\frac{e_{q_1}(-{\left[m_1\right]}_{q_1}{v}_1)}{X_{q_1}\left(1\right)}\sum _{r_1=0}^{\infty }\frac{X_{r_1,q_1}\left({\left[m_1\right]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}{\left[r_1\right]}_{q_1}\hfill \\ & =& \frac{1}{q_1{[m_1-1]}_{q_1}}+\frac{1}{{[m_1-1]}_{q_1}}\left({\left[m_1\right]}_{q_1}{v}_1+\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}\right)\hfill \end{array}$$

For $i=2,j=0,$ we have:

$$\begin{array}{ccc}\hfill J_{m_1,m_2}^{q_1,q_2}({\lambda }_{2,0};v_1,v_2)& =& \frac{e_q(-{\left[m\right]}_{q}x)}{X_q\left(1\right)}\sum _{r_1=0}^{\infty }\frac{X_{r_1,q_1}\left({[m-1]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}{q}_1^{2r_1}\frac{{\mathcal{K}}_1(A_1,r_1+1)}{B_{q_1}(r_1+1,m_1)}\underset{0}{\overset{\infty /A_1}{\int }}\frac{t^{r_1+2}}{{(1+t)}_{q_1}^{r_1+m_1+1}}{d}_{q_1}t\hfill \\ & & \times \frac{e_{q_2}(-{\left[m_2\right]}_{q_2}{v}_2)}{X_{q_2}\left(1\right)}\sum _{r_2=0}^{\infty }\frac{X_{r_2,q_2}\left({\left[m_2\right]}_{q_2}{v}_2\right)}{{\left[r_2\right]}_{q_2}!}\frac{B_{q_2}(r_2+1,m_2)}{B_{q_2}(r_2+1,m_2)}\hfill \\ & =& \frac{e_{q_1}(-{\left[m_1\right]}_{q_1}{v}_1)}{X_{q_1}\left(1\right)}\sum _{r_1=0}^{\infty }\frac{X_{r_1,q_1}\left({\left[m_1\right]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}{q}_1^{2r_1}\frac{{\mathcal{K}}_1(A_1,r_1+1)}{{\mathcal{K}}_1(A_1,r_1+3)}\frac{B_{q_1}(r_1+3,m_1-2)}{B_{q_1}(r_1+1,m_1)}\hfill \\ & =& \frac{1}{q_1^3{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}}\frac{e_{q_1}(-{\left[m_1\right]}_{q_1}{v}_1)}{X_{q_1}\left(1\right)}\sum _{r_1=0}^{\infty }\frac{X_{r_1,q_1}\left({\left[m_1\right]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}{[r_1+2]}_{q_1}{[r_1+1]}_{q_1}.\hfill \end{array}$$

We use the equality (23) and ${[r_1+2]}_{q_1}=1+q_1+q_1^2{\left[r_1\right]}_{q_1}$:

$$\begin{array}{ccc}\hfill J_{m_1,m_2}^{q_1,q_2}({\lambda }_{2,0};v_1,v_2)& =& \frac{1}{q_1^3{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}}\frac{e_{q_1}(-{\left[m_1\right]}_{q_1}{v}_1)}{X_{q_1}\left(1\right)}\sum _{r_1=0}^{\infty }\frac{X_{r_1,q_1}\left({\left[m_1\right]}_{q_1}{v}_1\right)}{{\left[r_1\right]}_{q_1}!}\hfill \\ & & \left((1+q_1)+q_1(1+2q_1){\left[r_1\right]}_{q_1}+q_1^3{\left[r_1\right]}_{q_1}^2\right)\hfill \\ & =& \frac{(1+q_1)}{q_1^3{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}}+\frac{(1+2q_1)}{q_1^2{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}}\left(({\left[m_1\right]}_{q_1}{v}_1+\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}\right)\hfill \\ & +& \frac{1}{{[m_1-1]}_{q_1}{[m_1-2]}_{q_1}}\left({\left[m_1\right]}_{q_1}^2{v}_1^2+\frac{2{\left[m_1\right]}_{q_1}{X}_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}{v}_1+\frac{X_{q_1}^{″}\left(1\right)}{X_{q_1}\left(1\right)}\right).\hfill \end{array}$$

Similarly, we obtain the other identities. This completes the proof. □

Lemma 4.

Take ${\Phi }_{v_1,v_2}^{c,d}(t,s)={(t-v_1)}^c{(s-v_2)}^d$ for all $c,d=\{0,1,2,3,4\}$. Then, for any $i=1,2,$$R_{r_i,q_i}\left({\left[m_i\right]}_{q_i}{v}_i\right)\ge 0$ and $X_{q_i}\left(1\right)\ne 0,$ suppose ${\left({\delta }_{m_1,m_2}^{c,d}\right)}^2=J_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{c,d}(t,s);v_1,v_2\right)$.

Then, for all $i=1,2,$ with $m_i\in \mathbb{N}\setminus \{1,2\}$, we obtain:

$${\left({\delta }_{m_1,m_2}^{c,d}\right)}^2=\left\{\begin{array}{c}{\displaystyle \left(\frac{{\left[m_i\right]}_{q_i}^2}{{[m_i-1]}_{q_i}{[m_i-2]}_{q_i}}+1-\frac{2{\left[m_i\right]}_{q_i}}{{[m_i-1]}_{q_i}}\right)v_i^2}\hfill \\ +{\displaystyle \frac{1}{{[m_i-1]}_{q_i}}\left(\frac{(1+2q_i){\left[m_i\right]}_{q_i}}{q_i^2{[m_i-2]}_{q_i}}+\frac{2{\left[m_i\right]}_{q_i}}{{[m_i-2]}_{q_i}}\frac{X_{q_i}^{\prime }\left(1\right)}{X_{q_i}\left(1\right)}-\frac{2X_{q_i}^{\prime }\left(1\right)}{X_{q_i}\left(1\right)}-\frac{2}{q_i}\right)v_i}\hfill \\ +{\displaystyle \frac{1}{q_i^2{[m_i-1]}_{q_i}{[m_i-2]}_{q_i}}\left(\frac{(1+q_i)}{q_i}+(1+2q_i)\frac{X_{q_i}^{\prime }\left(1\right)}{X_{q_i}\left(1\right)}\right)}\hfill \\ {\displaystyle {\displaystyle fori=1c=2,d=0\mathrm{and}i=2,d=2,c=0}}\hfill \\ {\displaystyle \left(\frac{{\left[m_i\right]}_{q_i}}{{[m_i-1]}_{q_i}}-1\right)v_i+\frac{1}{{[m_i-1]}_{q_i}}\left(\frac{1}{q_i}+\frac{X_{q_i}^{\prime }\left(1\right)}{X_{q_i}\left(1\right)}\right)}\hfill \\ {\displaystyle {\displaystyle fori=1,c=1,d=0\mathrm{and}i=2,c=0,d=1}}.\hfill \end{array}\right.$$

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Remark 1.

Let $m_1,m_2\in \mathbb{N},$ then, for any ${\delta }_{m_1},{\delta }_{m_2}>0,$ we take:

$${\delta }_{m_1,m_2}^{c,d}=\left\{\begin{array}{c}{\displaystyle \sqrt{J_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{c,d}(t,s);v_1,v_2\right)}={\delta }_{m_1}\left(suppose\right)}\hfill \\ {\displaystyle {\displaystyle forc=2\mathrm{and}d=0.}}\hfill \\ {\displaystyle \sqrt{J_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{c,d}(t,s);v_1,v_2\right)}={\delta }_{m_2}\left(suppose\right)}\hfill \\ {\displaystyle {\displaystyle ford=2\mathrm{and}c=0.}}\hfill \end{array}\right.$$

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Lemma 5.

For any $v_1,v_2\in {\mathcal{M}}^2$ and sufficiently large $m_1,m_2\in \mathbb{N}\setminus \{1,2,3,4\},$ the operators $J_{m_1,m_2}^{q_1,q_2}(.;.)$ verify the following inequalities:

$$\begin{array}{ccc}\hfill \left(1\right)J_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{2,0}(t,s);v_1,v_2\right)& =& O\left(\frac{1}{{\left[m_1\right]}_{q_1}^2}\right){(v_1+1)}^2\le M_1{(v_1+1)}^{2}as{m}_1,m_2\to \infty ;\hfill \\ \hfill \left(2\right)J_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{0,2}(t,s);v_1,v_2\right)& =& O\left(\frac{1}{{\left[m_2\right]}_{q_2}^2}\right){(v_2+1)}^2\le M_2{(v_2+1)}^{2}as{m}_1,m_2\to \infty ;\hfill \\ \hfill \left(3\right)J_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{4,0}(t,s);v_1,v_2\right)& =& O\left(\frac{1}{{\left[m_1\right]}_{q_1}^4}\right){(v_1+1)}^4\le M_3{(v_1+1)}^{4}as{m}_1,m_2\to \infty ;\hfill \\ \hfill \left(4\right)J_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{0,4}(t,s);v_1,v_2\right)& =& O\left(\frac{1}{{\left[m_2\right]}_{q_2}^4}\right){(v_2+1)}^4\le M_4{(v_2+1)}^{2}as{m}_1,m_2\to \infty .\hfill \end{array}$$

3. Approximation in Weighted Space and Degree of Convergence

For any weight function $\psi$ given, such that $\psi (v_1,v_2)=1+v_1^2+v_2^2$, which satisfies the properties $B_{\psi }\left({\mathcal{M}}^2\right)=\{f:\mid f(v_1,v_2)\mid \le M_f\psi (v_1,v_2),M_f>0\},$ where $B_{\psi }\left({\mathcal{M}}^2\right)$ is defined for the class of all bounded functions on ${\mathcal{M}}^2=[0,\infty )\times [0,\infty )$. In addition, take $C^{\left(m\right)}\left({\mathcal{M}}^2\right)$ as the m-times continuously differentiable functions on ${\mathcal{M}}^2=\{(v_1,v_2)\in {\mathcal{M}}^2:v_1,v_2\in [0,\infty )\}$. The norm on $B_{\psi }$ is equipped by ${\Vert f\Vert }_{\psi }={sup}_{v_1,v_2\in {\mathcal{M}}^2}\frac{\mid f(v_1,v_2)\mid }{\psi (v_1,v_2)}$. Moreover, some classified functions are defined as follows:

$$C_{\psi }^m\left({\mathcal{M}}^2\right)=\{f:f\in C_{\psi }\left({\mathcal{M}}^2\right);\mathrm{such}\mathrm{that}\underset{(v_1,v_2)\to \infty }{lim}\frac{f(v_1,v_2)}{\psi (v_1,v_2)}=k_f<\infty \},$$

(26)

$$C_{\psi }^0\left({\mathcal{M}}^2\right)=\{f:f\in C_{\psi }^m\left({\mathcal{M}}^2\right);\mathrm{such}\mathrm{that}\underset{(v_1,v_2)\to \infty }{lim}\frac{f(v_1,v_2)}{\psi (v_1,v_2)}=0\},$$

(27)

$$C_{\psi }\left({\mathcal{M}}^2\right)=\{f:f\in B_{\psi }\cap C_{\psi }\left({\mathcal{M}}^2\right)\}.$$

(28)

For all $f\in C_{\psi }^0\left({\mathcal{M}}^2\right)$ and ${\delta }_{m_1},{\delta }_{m_2}>0,$ the weighted modulus of continuity of function f, suppose ${\omega }_{\psi }(f;{\delta }_{m_1},{\delta }_{m_2})$ be defined by:

$${\omega }_{\psi }(f;{\delta }_{m_1},{\delta }_{m_2})=\underset{(v_1,v_2)\in [0,\infty )}{sup}\underset{0\le \mid {\phi }_1\mid \le {\delta }_{m_1},0\le \mid {\phi }_2\mid \le {\delta }_{m_2}}{sup}\frac{\mid f(v_1+{\phi }_1,v_2+{\phi }_2)-f(v_1,v_2)\mid }{\psi (v_1,v_2)\psi ({\phi }_1,{\phi }_2)},$$

(29)

for any ${\mu }_1,{\mu }_2>0$, one has:

$${\omega }_{\psi }(f;{\mu }_1{\delta }_{m_1},{\mu }_2{\delta }_{m_2})\le 4(1+{\mu }_1)(1+{\mu }_2)(1+{\delta }_{m_1}^2)(1+{\delta }_{m_2}^2){\omega }_{\psi }(f;{\delta }_{m_1},{\delta }_{m_2}),$$

(30)

$$\begin{array}{ccc}\hfill \mid f(t,s)-f(v_1,v_2)\mid & \le & \psi (v_1,v_2)\psi \left(\mid t-v_1\mid ,\mid s-v_2\mid \right){\omega }_{\psi }\left(f;\mid t-v_1\mid ,\mid s-v_2\mid \right)\hfill \\ & \le & (1+v_1^2+v_2^2)(1+{(t-v_1)}^2)(1+{(s-v_2)}^2){\omega }_{\psi }\left(f;\mid t-v_1\mid ,\mid s-v_2\mid \right).\hfill \end{array}$$

Theorem 1.

Let $f\in C_{\psi }^0\left({\mathcal{M}}^2\right)$. Then, for sufficiently large $m_1,m_2\in \mathbb{N},$ the operators $J_{m_1,m_2}^{q_1,q_2}$ satisfy the inequality:

$$\begin{array}{ccc}\hfill \frac{\mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid }{4(1+v_1^2+v_2^2)}& \le & {\mathsf{\Psi }}_{v_1,v_2}\left(1+O\left({\left[m_1\right]}_{q_1}^{-2}\right)\right)\left(1+O\left({\left[m_2\right]}_{q_2}^{-2}\right)\right)\hfill \\ & \times & {\omega }_{\psi }\left(f;O\left({\left[m_1\right]}_{q_1}^{-1}\right),O\left({\left[m_2\right]}_{q_2}^{-1}\right)\right),\hfill \end{array}$$

where ${\mathsf{\Psi }}_{v_1,v_2}=(1+(v_1+1)+M_1{(v_1+1)}^2+M_1{(v_1+1)}^3)(1+(v_2+1)+M_2{(v_2+1)}^2+M_2{(v_2+1)}^3)$$M_1,M_2>0$ and ${\delta }_{m_i}=O\left(\frac{1}{{\left[m_i\right]}_{q_i}}\right),$ for $i=1,2$.

Proof. 

For all ${\delta }_{m_1},{\delta }_{m_2}>0$, we have:

$$\begin{array}{ccc}\hfill \mid f(t,s)-f(v_1,v_2)\mid & \le & 4(1+v_1^2+v_2^2)\left(1+{(t-v_1)}^2\right)\left(1+{(s-v_2)}^2\right)\hfill \\ & \times & \left(1+\frac{\mid t-v_1\mid }{{\delta }_{m_1}}\right)\left(1+\frac{\mid s-v_2\mid }{{\delta }_{m_2}}\right)(1+{\delta }_{m_1}^2)(1+{\delta }_{m_2}^2){\omega }_{\psi }\left(f;{\delta }_{m_1},{\delta }_{m_2}\right)\hfill \\ & =& 4(1+v_1^2+v_2^2)(1+{\delta }_{m_1}^2)(1+{\delta }_{m_2}^2)\hfill \\ & \times & \left(1+\frac{\mid t-v_1\mid }{{\delta }_{m_1}}+{(t-v_1)}^2+\frac{1}{{\delta }_{m_1}}\mid t-v_1\mid {(t-v_1)}^2\right)\hfill \\ & \times & \left(1+\frac{\mid s-v_2\mid }{{\delta }_{m_2}}+{(s-v_2)}^2+\frac{\mid s-v_2\mid }{{\delta }_{m_2}}{(s-v_2)}^2\right){\omega }_{\psi }\left(f;{\delta }_{m_1},{\delta }_{m_2}\right).\hfill \end{array}$$

On applying the positive linear operator $J_{m_1,m_2}^{q_1,q_2}$ and taking into account the well-known Cauchy–Schwarz inequality:

$$\begin{array}{ccc}\hfill \frac{\mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid }{4(1+v_1^2+v_2^2)}& \le & J_{m_1,m_2}^{q_1,q_2}\left(\right(1+\frac{\mid t-v_1\mid }{{\delta }_{m_1}}+{(t-v_1)}^2\hfill \\ & & +\frac{1}{{\delta }_{m_1}}\mid t-v_1\mid {(t-v_1)}^2;v_1,v_2)\hfill \\ & \times & J_{m_1,m_2}^{q_1,q_2}\left(1+\frac{\mid s-v_2\mid }{{\delta }_{m_2}}+{(s-v_2)}^2+\frac{\mid s-v_2\mid }{{\delta }_{m_2}}{(s-v_2)}^2;v_1,v_2\right)\hfill \\ & \times & (1+{\delta }_{m_1}^2)(1+{\delta }_{m_2}^2){\omega }_{\psi }\left(f;{\delta }_{m_1},{\delta }_{m_2}\right)\hfill \\ & =& (1+v_1^2+v_2^2)(1+{\delta }_{m_1}^2)(1+{\delta }_{m_2}^2){\omega }_{\psi }\left(f;{\delta }_{m_1},{\delta }_{m_2}\right)\hfill \\ & \times & (1+\frac{1}{{\delta }_{m_1}}{J}_{m_1,m_2}^{q_1,q_2}(\mid t-v_1\mid ;v_1,v_2)+J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^2;v_1,v_2)\hfill \\ & +& \frac{1}{{\delta }_{m_1}}{J}_{m_1,m_2}^{q_1,q_2}(\mid t-v_1\mid {(t-v_1)}^2;v_1,v_2)\hfill \\ & \times & (1+\frac{1}{{\delta }_{m_2}}{J}_{m_1,m_2}^{q_1,q_2}(\mid s-v_2\mid ;v_1,v_2)+J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^2;v_1,v_2)\hfill \\ & +& \frac{1}{{\delta }_{m_2}}{J}_{m_1,m_2}^{q_1,q_2}(\mid s-v_2\mid {(s-v_2)}^2;v_1,v_2);\hfill \\ \hfill \frac{\mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid }{4(1+v_1^2+v_2^2)}& \le & (1+{\delta }_{m_1}^2)(1+{\delta }_{m_2}^2){\omega }_{\psi }\left(f;{\delta }_{m_1},{\delta }_{m_2}\right)\hfill \\ & \times & [1+\frac{1}{{\delta }_{m_1}}\sqrt{J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^2;v_1,v_2)}+J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^2;v_1,v_2)\hfill \\ & +& \frac{1}{{\delta }_{m_1}}\sqrt{J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^2;v_1,v_2)}\sqrt{J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^4;v_1,v_2)}]\hfill \\ & \times & [1+\frac{1}{{\delta }_{m_2}}\sqrt{J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^2;v_1,v_2)}+J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^2;v_1,v_2)\hfill \\ & +& \frac{1}{{\delta }_{m_2}}\sqrt{J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^2;v_1,v_2)}\sqrt{J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^4;v_1,v_2)}].\hfill \end{array}$$

Take into account Lemma 5 and for all $i=1,2,$ if we choose ${\delta }_{m_i}=O\left(\frac{1}{{\left[m_i\right]}_{q_i}}\right),$ then:

$$\begin{array}{ccc}\hfill \frac{\mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid }{4(1+v_1^2+v_2^2)}& \le & (1+{\delta }_{m_1}^2)(1+{\delta }_{m_2}^2){\omega }_{\psi }\left(f;{\delta }_{m_1},{\delta }_{m_2}\right)\hfill \\ & \times & [1+\frac{1}{{\delta }_{m_1}}O\left(\frac{1}{{\left[m_1\right]}_{q_1}}\right)(v_1+1)+O\left(\frac{1}{{\left[m_1\right]}_{q_1}^2}\right){(v_1+1)}^2\hfill \\ & +& \frac{1}{{\delta }_{m_1}}O\left(\frac{1}{{\left[m_1\right]}_{q_1}}\right)(v_1+1)O\left(\frac{1}{{\left[m_1\right]}_{q_1}^2}\right){(v_1+1)}^2]\hfill \\ & \times & [1+\frac{1}{{\delta }_{m_2}}O\left(\frac{1}{{\left[m_2\right]}_{q_2}}\right)(v_2+1)+O\left(\frac{1}{{\left[m_2\right]}_{q_2}^2}\right){(v_2+1)}^2\hfill \\ & +& \frac{1}{{\delta }_{m_2}}O\left(\frac{1}{{\left[m_2\right]}_{q_2}}\right)(v_2+1)O\left(\frac{1}{{\left[m_2\right]}_{q_2}^2}\right){(v_2+1)}^2]\hfill \\ & \le & 4(1+O\left(\frac{1}{{\left[m_1\right]}_{q_1}^2}\right))(1+O\left(\frac{1}{{\left[m_2\right]}_{q_2}^2}\right))\hfill \\ & \times & {\omega }_{\psi }(f;O\left(\frac{1}{{\left[m_1\right]}_{q_1}}\right),O\left(\frac{1}{{\left[m_2\right]}_{q_2}}\right))\hfill \\ & \times & [1+(v_1+1)+M_1{(v_1+1)}^2+M_1{(v_1+1)}^3]\hfill \\ & \times & [1+(v_2+1)+M_2{(v_2+1)}^2+M_2{(v_2+1)}^3].\hfill \end{array}$$

This completes the proof. □

Lemma 6

([31,32]). For the positive sequence of operators ${\left\{L_{m_1,m_2}\right\}}_{m_1,m_2\ge 1},$ which act as $C_{\psi }\to B_{\psi }$, then there exists some positive K such that:

$$\Vert X_{m_1,m_2}(\psi ;v_1,v_2){\Vert }_{\psi }\le K.$$

Theorem 2

([31,32]). For the positive sequence of operators ${\left\{X_{m_1,m_2}\right\}}_{m_1,m_2\ge 1}$ acting as $C_{\psi }\to B_{\psi }$ defined earlier, the following conditions are satisfied:

$$\begin{array}{ccc}\hfill \left(1\right)\underset{m_1,m_2\to \infty }{lim}{\Vert X_{m_1,m_2}(1;v_1,v_2)-1\Vert }_{\psi }& =& 0;\hfill \\ \hfill \left(2\right)\underset{m_1,m_2\to \infty }{lim}{\Vert X_{m_1,m_2}(t;v_1,v_2)-v_1\Vert }_{\psi }& =& 0;\hfill \\ \hfill \left(3\right)\underset{m_1,m_2\to \infty }{lim}{\Vert X_{m_1,m_2}(s;v_1,v_2)-v_2\Vert }_{\psi }& =& 0;\hfill \\ \hfill \left(4\right)\underset{m_1,m_2\to \infty }{lim}{\Vert X_{m_1,m_2}((t^2+s^2);v_1,v_2)-(v_1^2+v_2^2)\Vert }_{\psi }& =& 0.\hfill \end{array}$$

Then, for all $f\in C_{\psi }^0,$:

$$\underset{m_1,m_2\to \infty }{lim}{\Vert X_{m_1,m_2}f-f\Vert }_{\psi }=0$$

and there exists another function $f\in C_{\psi }\setminus C_{\psi }^0,$ such that:

$$\underset{m_1,m_2\to \infty }{lim}{\Vert X_{m_1,m_2}f-f\Vert }_{\psi }\ge 1.$$

Theorem 3.

If $f\in C_{\psi }^0\left({\mathcal{M}}^2\right),$ then we have:

$$\underset{m_1,m_2\to \infty }{lim}{\Vert J_{m_1,m_2}^{q_1,q_2}\left(f\right)-f\Vert }_{\psi }=0.$$

Proof. 

$$\begin{array}{ccc}\hfill \Vert J_{m_1,m_2}^{q_1,q_2}(\psi ;v_1,v_2){\Vert }_{\psi }& =& \underset{(v_1,v_2)\in {\mathcal{M}}^2}{sup}\frac{\mid J_{m_1,m_2}^{q_1,q_2}(1+v_1^2+v_2^2;v_1,v_2)\mid }{1+v_1^2+v_2^2}\hfill \\ & =& 1+\underset{(v_1,v_2)\in {\mathcal{M}}^2}{sup}\left[\frac{1}{1+v_1^2+v_2^2}\right|\left(J_{m_1,m_2}^{q_1,q_2}(v_1^2;v_1,v_2)+J_{m_1,m_2}^{q_1,q_2}(v_2^2;v_1,v_2)\right)\left|\right]\hfill \\ & =& 1+\left|\frac{{\left[m_1\right]}_{q_1}^2}{{[m_1-1]}_{q_1}{[m_2-1]}_{q_1}}\right|\underset{(v_1,v_2)\in {\mathcal{M}}^2}{sup}\frac{v_1^2}{1+v_1^2+v_2^2}\hfill \\ & +& \left|\frac{(1+2q_1)+2q_1^2\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}}{q_1^2{[m_1-1]}_{q_1}{[m_2-1]}_{q_1}}\right|\underset{(v_1,v_2)\in {\mathcal{M}}^2}{sup}\frac{v_1}{1+v_1^2+v_2^2}\hfill \\ & +& \left|\frac{(1+2q_1)+q_1(1+2q_1)\frac{X_{q_1}^{\prime }\left(1\right)}{X_{q_1}\left(1\right)}+q_1^3\frac{X_{q_1}^{″}\left(1\right)}{X_{q_1}\left(1\right)}}{q_1^3{[m_1-1]}_{q_1}{[m_2-1]}_{q_1}}\right|\underset{(v_1,v_2)\in {\mathcal{M}}^2}{sup}\frac{1}{1+v_1^2+v_2^2}\hfill \\ & +& \left|\frac{{\left[m_2\right]}_{q_2}^2}{{[m_1-1]}_{q_2}{[m_2-1]}_{q_2}}\right|\underset{(v_1,v_2)\in {\mathcal{M}}^2}{sup}\frac{v_2^2}{1+v_1^2+v_2^2}\hfill \\ & +& \left|\frac{(1+2q_2)+2q_2^2\frac{X_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}}{q_2^2{[m_1-1]}_{q_2}{[m_2-1]}_{q_2}}\right|\underset{(v_1,v_2)\in {\mathcal{M}}^2}{sup}\frac{v_2}{1+v_1^2+v_2^2}\hfill \\ & +& \left|\frac{(1+2q_2)+q_2(1+2q_2)\frac{X_{q_2}^{\prime }\left(1\right)}{X_{q_2}\left(1\right)}+q_2^3\frac{X_{q_2}^{″}\left(1\right)}{X_{q_2}\left(1\right)}}{q_2^3{[m_1-1]}_{q_2}{[m_2-1]}_{q_2}}\right|\underset{(v_1,v_2)\in {\mathcal{M}}^2}{sup}\frac{1}{1+v_1^2+v_2^2}.\hfill \end{array}$$

Now, for all $m_1,m_2\in \mathbb{N}\setminus \{1,2\},$ there exists a positive constant K such that:

$$\Vert J_{m_1,m_2}^{q_1,q_2}(\psi ;v_1,v_2){\Vert }_{\psi }\le K.$$

Therefore, in order to prove Theorem 3, it is sufficient to take from the Lemma 3 and Theorem 2. Thus, we aimed to prove of Theorem 3. □

Here, we obtain the approximations of our new operators by the use of the second-order modulus of continuity. For this purpose, if any $f\in C\left({\mathcal{M}}^2\right)$ and $\delta >0$, then the second-order modulus of continuity in terms of partial continuity is given by:

$$\omega \left(f;{\delta }_{m_1},{\delta }_{m_2}\right)=sup\{\mid f(t,s)-f(v_1,v_2)\mid :(t,s),(v_1,v_2)\in {\mathcal{M}}^2\}$$

(31)

with $\mid t-v_1\mid \le {\delta }_{m_1},\mid s-v_2\mid \le {\delta }_{m_2}$ with the partial modulus of continuity defined as:

$${\omega }_1\left(f;{\delta }_{m_1}\right)=\underset{0\le v_1,v_2\le 1}{sup}\underset{\mid x_1-x_2\mid \le {\delta }_{m_1}}{sup}\{\mid f(x_1,v_2)-f(x_2,v_2)\mid \},$$

(32)

$${\omega }_2\left(f;{\delta }_{m_2}\right)=\underset{0\le v_1,v_2\le 1}{sup}\underset{\mid y_1-y_2\mid \le {\delta }_{m_2}}{sup}\{\mid f(v_1,y_1)-f(v_1,y_2)\mid \}.$$

(33)

While for any function $\phi$ and ${\delta }^{\ast }>0$, one has taken the modulus of continuity of order one ${\omega }^{\ast }(f;{\delta }^{\ast })$, and it is defined such that ${lim}_{{\delta }^{\ast }\to 0+}{\omega }^{\ast }(\phi ;{\delta }^{\ast })=0$ and:

$${\omega }^{\ast }(\phi ;{\delta }^{\ast })=\underset{\mid {\nu }_1-{\nu }_2\mid \le {\delta }^{\ast }}{sup}\mid \phi \left({\nu }_1\right)-\phi \left({\nu }_2\right)\mid ;{\nu }_1,{\nu }_2\in [0,\infty ),$$

(34)

$$\mid \phi \left({\nu }_1\right)-\phi \left({\nu }_2\right)\mid \le \left(1+\frac{\mid {\nu }_1-{\nu }_2\mid }{{\delta }^{\ast }}\right){\omega }^{\ast }(\phi ;{\delta }^{\ast }).$$

(35)

Theorem 4.

If $f\in C\left({\mathcal{M}}^2\right)$, then we obtain the inequality for the operators $J_{m_1,m_2}^{q_1,q_2}$:

$$\mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid \le 2\left({\omega }_1(f;{\delta }_{m_1,m_2}^{2,0})+{\omega }_2(f;{\delta }_{m_1,m_2}^{0,2})\right).$$

Proof. 

For proof of Theorem 4, we use the Cauchy–Schwarz inequality, therefore, we obtain here:

$$\begin{array}{ccc}\hfill \mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid & \le & J_{m_1,m_2}^{q_1,q_2}\left(\mid f(t,s)-f(v_1,v_2)\mid ;v_1,v_2\right)\hfill \\ & \le & J_{m_1,m_2}^{q_1,q_2}\left(\mid f(t,s)-f(v_1,s)\mid ;v_1,v_2\right)\hfill \\ & +& J_{m_1,m_2}^{q_1,q_2}\left(\mid f(v_1,s)-f(v_1,v_2)\mid ;v_1,v_2\right)\hfill \\ & \le & J_{m_1,m_2}^{q_1,q_2}\left({\omega }_1(f;\mid t-v_1\mid );v_1,v_2\right)+J_{m_1,m_2}^{q_1,q_2}\left({\omega }_2(f;\mid s-v_2\mid );v_1,v_2\right)\hfill \\ & \le & {\omega }_1(f;{\delta }_{m_1})\left(1+{\delta }_{m_1}^{-1}{J}_{m_1,m_2}^{q_1,q_2}(\mid t-v_1\mid ;v_1,v_2)\right)\hfill \\ & +& {\omega }_2(f;{\delta }_{m_2})\left(1+{\delta }_{m_2}^{-1}{J}_{m_1,m_2}^{q_1,q_2}(\mid s-v_2\mid ;v_1,v_2)\right)\hfill \\ & \le & {\omega }_1(f;{\delta }_{m_1})\left(1+\frac{1}{{\delta }_{m_1}}\sqrt{J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^2;v_1,v_2)}\right)\hfill \\ & +& {\omega }_2(f;{\delta }_{m_2})\left(1+\frac{1}{{\delta }_{m_2}}\sqrt{J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^2;v_1,v_2)}\right),\hfill \end{array}$$

where ${\delta }_{m_1}={\delta }_{m_1,m_2}^{2,0}=\sqrt{J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^2;v_1,v_2)}$ and ${\delta }_{m_2}$ = ${\delta }_{m_1,m_2}^{0,2}=\sqrt{J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^2;v_1,v_2)},$ which then led to results. □

Next, we obtain the convergence by use of class of the Lipschitz maximal in bivariate form for any function $f\in C\left({\mathcal{M}}^2\right)$. For any positive constant K and $0\le {\rho }_1,{\rho }_2\le 1$ the Lipschitz maximal function on any set $\mathcal{I}\times \mathcal{I}\subset {\mathcal{M}}^2$ is given by:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{{\rho }_1,{\rho }_2}(\mathcal{I}\times \mathcal{I})& =& \{f:sup{(1+t)}^{{\rho }_1}{(1+s)}^{{\rho }_2}\left(f_{{\rho }_1,{\rho }_2}(t,s)-f_{{\rho }_1,{\rho }_2}(v_1,v_2)\right)\hfill \\ & \le & K\frac{1}{{(1+v_1)}^{{\rho }_1}}\frac{1}{{(1+v_2)}^{{\rho }_2}}\}.\hfill \end{array}$$

$$f_{{\rho }_1,{\rho }_2}(t,s)-f_{{\rho }_1,{\rho }_2}(v_1,v_2)=\frac{\mid f(t,s)-f(v_1,v_2)\mid }{\mid t-v_1{\mid }^{{\rho }_1}\mid s-v_2{\mid }^{{\rho }_2}};(t,s),(v_1,v_2)\in {\mathcal{M}}^2.$$

(36)

Theorem 5.

For all $f\in {\mathcal{L}}_{{\rho }_1,{\rho }_2}(\mathcal{I}\times \mathcal{I}),$ if $0\le {\rho }_1,{\rho }_2\le 1,$ then we obtain:

$$\begin{array}{ccc}\hfill \mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid & \le & K\left\{\right({\left(d(v_1,\mathcal{I})\right)}^{{\rho }_1}+{\left({\delta }_{m_1,m_2}^{2,0}\right)}^{\frac{{\rho }_1}{2}}\left)\right({\left(d(v_2,\mathcal{I})\right)}^{{\rho }_2}+{\left({\delta }_{m_1,m_2}^{0,2}\right)}^{\frac{{\rho }_2}{2}})\hfill \\ & +& {\left(d(v_1,\mathcal{I})\right)}^{{\rho }_1}{\left(d(v_2,\mathcal{I})\right)}^{{\rho }_2}\},\hfill \end{array}$$

where ${\delta }_{m_1,m_2}^{2,0}$ and ${\delta }_{m_1,m_2}^{0,2}$ are defined by Theorem 4 and $K>0$.

Proof. 

Take into account $\mid v_1-x_0\mid =d(v_1,\mathcal{I})$ and $\mid v_2-y_0\mid =d(v_2,E)$. For any $(v_1,v_2)\in {\mathcal{M}}^2$ and $(x_0,y_0)\in \mathcal{I}\times \mathcal{I}$, we let $d(v_1,\mathcal{I})=inf\{\mid v_1-v_2\mid :v_2\in \mathcal{I}\}$. Thus, we can write here:

$$\mid f(t,s)-f(v_1,v_2)\mid \le K\mid f(t,s)-f(x_0,y_0)\mid +\mid f(x_0,y_0)-f(v_1,v_2)\mid .$$

(37)

Applying $J_{m_1,m_2}^{q_1,q_2}$, we obtain:

$$\begin{array}{ccc}\hfill \mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid & \le & J_{m_1,m_2}^{q_1,q_2}\left(\mid f(v_1,v_2)-f(x_0,y_0)\mid +\mid f(x_0,y_0)-f(v_1,v_2)\mid \right)\hfill \\ & \le & K{J}_{m_1,m_2}^{q_1,q_2}\left(\mid t-x_0{\mid }^{{\rho }_1}\mid s-y_0{\mid }^{{\rho }_2};v_1,v_2\right)\hfill \\ & +& K\mid v_1-x_0{\mid }^{{\rho }_1}\mid v_2-y_0{\mid }^{{\rho }_2}.\hfill \end{array}$$

For all $U,V\ge 0$ and $\rho \in [0,1],$ we know the inequality ${(U+V)}^{\rho }\le U^{\rho }+V^{\rho }$. Then, it is easy to obtain:

$$\mid t-x_0{\mid }^{{\rho }_1}\le \mid t-v_1{\mid }^{{\rho }_1}+\mid v_1-x_0{\mid }^{{\rho }_1},$$

$$\mid s-y_0{\mid }^{{\rho }_2}\le \mid s-v_2{\mid }^{{\rho }_2}+\mid v_2-y_0{\mid }^{{\rho }_2}.$$

Therefore:

$$\begin{array}{ccc}\hfill \mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid & \le & K{J}_{m_1,m_2}^{q_1,q_2}\left(\mid t-v_1{\mid }^{{\rho }_1}\mid s-v_2{\mid }^{{\rho }_2};v_1,v_2\right)\hfill \\ & +& K\mid v_1-x_0{\mid }^{{\rho }_1}{J}_{m_1,m_2}^{q_1,q_2}\left(\mid s-v_2{\mid }^{{\rho }_2};v_1,v_2\right)\hfill \\ & +& K\mid v_2-y_0{\mid }^{{\rho }_2}{J}_{m_1,m_2}^{q_1,q_2}\left(\mid t-v_1{\mid }^{{\rho }_1};v_1,v_2\right)\hfill \\ & +& 2K\mid v_1-x_0{\mid }^{{\rho }_1}\mid v_2-y_0{\mid }^{{\rho }_2}{J}_{m_1,m_2}^{q_1,q_2}\left({\lambda }_{0,0};v_1,v_2\right),\hfill \end{array}$$

by the use of the Hölder inequality, we obtain the equality:

$$\begin{array}{ccc}\hfill J_{m_1,m_2}^{q_1,q_2}\left(\mid t-v_1{\mid }^{{\rho }_1}\mid s-v_2{\mid }^{{\rho }_2};v_1,v_2\right)& =& {\mathcal{C}}_{m_1,r_1}^{q_1}\left(\mid t-v_1{\mid }^{{\rho }_1};v_1,v_2\right){\mathcal{D}}_{m_2,r_2}^{q_2}\left(\mid s-v_2{\mid }^{{\rho }_2};v_1,v_2\right)\hfill \\ & \le & {\left(J_{m_1,m_2}^{q_1,q_2}(\mid t-v_1{\mid }^2;v_1,v_2)\right)}^{\frac{{\rho }_1}{2}}{\left(J_{m_1,m_2}^{q_1,q_2}({\lambda }_{0,0};v_1,v_2)\right)}^{\frac{2-{\rho }_1}{2}}\hfill \\ & \times & {\left(J_{m_1,m_2}^{q_1,q_2}(\mid s-v_2{\mid }^2;v_1,v_2)\right)}^{\frac{{\rho }_2}{2}}{\left(J_{m_1,m_2}^{q_1,q_2}({\lambda }_{0,0};v_1,v_2)\right)}^{\frac{2-{\rho }_2}{2}}.\hfill \end{array}$$

Thus, it is easy to obtain:

$$\begin{array}{ccc}\hfill \mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid & \le & K{\left({\delta }_{m_1,m_2}^{2,0}\right)}^{\frac{{\rho }_1}{2}}{\left({\delta }_{m_1,m_2}^{0,2}\right)}^{\frac{{\rho }_2}{2}}+2K{\left(d(v_1,\mathcal{I})\right)}^{{\rho }_1}{\left(d(v_2,\mathcal{I})\right)}^{{\rho }_2}\hfill \\ & +& K{\left(d(v_1,\mathcal{I})\right)}^{{\rho }_1}{\left({\delta }_{m_1,m_2}^{2,0}\right)}^{\frac{{\rho }_2}{2}}+K{\left(d(v_2,\mathcal{I})\right)}^{{\rho }_2}{\left({\delta }_{m_1,m_2}^{0,2}\right)}^{\frac{{\rho }_1}{2}}.\hfill \end{array}$$

This completes the proof. □

Theorem 6.

Suppose $f\in C^{\prime }\left({\mathcal{M}}^2\right)$. Then, for all $(v_1,v_2)\in {\mathcal{M}}^2$, the operators $J_{m_1,m_2}^{q_1,q_2}$ have the equality:

$$\mid J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)-f(v_1,v_2)\mid \le \Vert f_{v_1}{\Vert }_{C\left({\mathcal{M}}^2\right)}{\left({\delta }_{m_1,m_2}^{2,0}\right)}^{\frac{1}{2}}+{\Vert f_{v_2}\Vert }_{C\left({\mathcal{M}}^2\right)}{\left({\delta }_{m_1,m_2}^{0,2}\right)}^{\frac{1}{2}},$$

where ${\delta }_{m_1,m_2}^{2,0}$ and ${\delta }_{m_1,m_2}^{0,2}$ are given by Theorem 4.

Proof. 

For any $f\in C^{\prime }\left({\mathcal{M}}^2\right)$ and fixed $(v_1,v_2)\in {\mathcal{M}}^2,$ we take:

$$f(t,s)-f(v_1,v_2)={\int }_{v_1}^t{f}_{\varrho }(\varrho ,s)\mathrm{d}\varrho +{\int }_{v_2}^s{f}_{\mu }(v_1,\mu )\mathrm{d}\mu .$$

(38)

Apply operators $J_{m_1,m_2}^{q_1,q_2}$ on both sides, then:

$$J_{m_1,m_2}^{q_1,q_2}\left(f(t,s);v_1,v_2\right)-f(v_1,v_2)=J_{m_1,m_2}^{q_1,q_2}\left({\int }_{v_1}^t{f}_{\varrho }(\varrho ,s)\mathrm{d}\varrho ;v_1,v_2\right)+J_{m_1,m_2}^{q_1,q_2}\left({\int }_{v_2}^s{f}_{\mu }(v_1,\mu )\mathrm{d}\mu ;v_1,v_2\right).$$

(39)

The sup-norm on ${\mathcal{M}}^2$ gives us:

$$\mid {\int }_{v_1}^t{f}_{\varrho }(\varrho ,s)\mathrm{d}\varrho \mid \le {\int }_{v_1}^t\mid f_{\varrho }(\varrho ,s)\mathrm{d}\varrho \mid \le {\Vert f_{v_1}\Vert }_{C\left({\mathcal{M}}^2\right)}\mid t-v_1\mid$$

(40)

$$\mid {\int }_{v_2}^s{f}_{\mu }(v_1,\mu )\mathrm{d}\mu \mid \le {\int }_{v_2}^s\mid f_{\mu }(v_1,\mu )\mathrm{d}\mu \mid \le {\Vert f_{v_2}\Vert }_{C\left({\mathcal{M}}^2\right)}\mid s-v_2\mid .$$

(41)

Taking into account (39)–(41), it is easy to obtain:

$$\begin{array}{ccc}\hfill \mid J_{m_1,m_2}^{q_1,q_2}\left(f(v_1,v_2);v_1,v_2\right)-f(v_1,v_2)\mid & \le & J_{m_1,m_2}^{q_1,q_2}\left(\left|{\int }_{v_1}^t{f}_{\varrho }(\varrho ,s)\mathrm{d}\varrho \right|;v_1,v_2\right)\hfill \\ & +& J_{m_1,m_2}^{q_1,q_2}\left(\left|{\int }_{v_2}^s{f}_{\mu }(v_1,\mu )\mathrm{d}\mu \right|;v_1,v_2\right)\hfill \\ & \le & \Vert f_{v_1}{\Vert }_{C\left({\mathcal{M}}^2\right)}{J}_{m_1,m_2}^{q_1,q_2}\left(\mid t-v_1\mid ;v_1,v_2\right)\hfill \\ & +& \Vert f_{v_2}{\Vert }_{C\left({\mathcal{M}}^2\right)}{J}_{m_1,m_2}^{q_1,q_2}\left(\mid s-v_2\mid ;v_1,v_2\right)\hfill \\ & \le & \Vert f_{v_1}{\Vert }_{C\left({\mathcal{M}}^2\right)}{\left(J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^2;v_1,v_2)J_{m_1,m_2}^{q_1,q_2}(1;v_1,v_2)\right)}^{\frac{1}{2}}\hfill \\ & +& \Vert f_{v_2}{\Vert }_{C\left({\mathcal{M}}^2\right)}{\left(J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^2;v_1,v_2)J_{m_1,m_2}^{q_1,q_2}(1;v_1,v_2)\right)}^{\frac{1}{2}}\hfill \\ & =& \Vert f_{v_1}{\Vert }_{C\left({\mathcal{M}}^2\right)}{\left({\delta }_{m_1,m_2}^{2,0}\right)}^{\frac{1}{2}}+{\Vert f_{v_2}\Vert }_{C\left({\mathcal{M}}^2\right)}{\left({\delta }_{m_1,m_2}^{0,2}\right)}^{\frac{1}{2}}.\hfill \end{array}$$

□

Theorem 7.

For an arbitrary function $f\in C\left({\mathcal{M}}^2\right),$ we define the auxiliary operators as:

$$\begin{array}{ccc}\hfill S_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)& =& J_{m_1,m_2}^{q_1,q_2}(f;v_1,v_2)+f(v_1,v_2)-f\left({\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{1,0};v_1,v_2),{\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{1,0};v_1,v_2)\right).\hfill \end{array}$$

Then, for all $f\in C^{\prime }\left({\mathcal{M}}^2\right)$, the operators $S_{m_1,m_2}^{q_1,q_2}$ verify the inequality:

$$\begin{array}{ccc}\hfill S_{m_1,m_2}^{q_1,q_2}(f;t,s)-f(v_1,v_2)& \le & \{{\delta }_{m_1,m_2}^{2,0}+{\delta }_{m_1,m_2}^{0,2}+{\left({\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{1,0};v_1,v_2)-v_1\right)}^2\hfill \\ & +& {\left({\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{1,0};v_1,v_2)-v_2\right)}^2\}{\Vert f\Vert }_{C^2\left({\mathcal{M}}^2\right)}.\hfill \end{array}$$

where ${\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{1,0};v_1,v_2)$ and ${\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{0,1};v_1,v_2)$ are defined by Lemma 3.

Proof. 

By the use of Lemma 3, it is easy to obtain $S_{m_1,m_2}^{q_1,q_2}(1;v_1,v_2)=1,$$S_{m_1,m_2}^{q_1,q_2}(t-v_1;v_1,v_2)=0$ and $S_{m_1,m_2}^{q_1,q_2}(s-v_2;v_1,v_2)=0$. In addition, for any $f\in C^{\prime }\left({\mathcal{M}}^2\right)$ from the Taylor series expansions, we obtain:

$$\begin{array}{ccc}\hfill f(t,s)-f(v_1,v_2)& =& \frac{\partial f(v_1,v_2)}{\partial v_1}(t-v_1)+{\int }_{v_1}^t(t-\chi )\frac{{\partial }^{2}f(\chi ,v_2)}{\partial {\chi }^2}\mathrm{d}\chi \hfill \\ & +& \frac{\partial f(v_1,v_2)}{\partial v_2}(s-v_2)+{\int }_{v_2}^s(s-\sigma )\frac{{\partial }^{2}f(v_1,\sigma )}{\partial {\sigma }^2}\mathrm{d}\sigma .\hfill \end{array}$$

On applying $S_{m_1,m_2}^{q_1,q_2},$ we see that:

$S_{m_1,m_2}^{q_1,q_2}\left(f(t,s);v_1,v_2\right)-S_{m_1,m_2}^{q_1,q_2}\left(f(v_1,v_2\right)$

$$\begin{array}{ccc}& =& S_{m_1,m_2}^{q_1,q_2}\left({\int }_{v_1}^t(t-\chi )\frac{{\partial }^{2}f(\chi ,v_2)}{\partial {\chi }^2}\mathrm{d}\chi ;v_1,v_2\right)+S_{m_1,m_2}^{q_1,q_2}\left({\int }_{v_2}^s(s-\sigma )\frac{{\partial }^{2}f(v_1,\sigma )}{\partial {\sigma }^2}\mathrm{d}\sigma ;v_1,v_2\right)\hfill \\ & =& J_{m_1,m_2}^{q_1,q_2}\left({\int }_{v_1}^t(t-\chi )\frac{{\partial }^{2}f(\chi ,v_2)}{\partial {\chi }^2}\mathrm{d}\chi ;v_1,v_2\right)+J_{m_1,m_2}^{q_1,q_2}\left({\int }_{v_2}^s(s-\sigma )\frac{{\partial }^{2}f(v_1,\sigma )}{\partial {\sigma }^2}\mathrm{d}\sigma ;v_1,v_2\right)\hfill \\ & -& {\int }_{v_1}^{{\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{1,0};v_1,v_2)}\left({\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{1,0};v_1,v_2)-\chi \right)\frac{{\partial }^{2}f(\chi ,v_2)}{\partial {\chi }^2}\mathrm{d}\chi \hfill \\ & -& {\int }_{v_2}^{{\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{1,0};v_1,v_2)}\left({\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{1,0};v_1,v_2)-\sigma \right)\frac{{\partial }^{2}f(v_1,\sigma )}{\partial {\sigma }^2}\mathrm{d}\sigma .\hfill \end{array}$$

From this hypothesis, we easily obtain:

$$\left|{\int }_{v_1}^t(t-\chi )\frac{{\partial }^{2}f(\chi ,v_2)}{\partial {\chi }^2}\mathrm{d}\chi \right|\le {\int }_{v_1}^t\left|(t-\chi )\frac{{\partial }^{2}f(\chi ,v_2)}{\partial {\chi }^2}\mathrm{d}\chi \right|\le {\Vert f\Vert }_{C^2\left({\mathcal{M}}^2\right)}{(t-v_1)}^2,$$

$$\left|{\int }_{v_2}^s(s-\sigma )\frac{{\partial }^{2}f(v_1,\sigma )}{\partial {\sigma }^2}\mathrm{d}\sigma \right|\le {\int }_{v_2}^s\left|(s-\sigma )\frac{{\partial }^{2}f(v_1,\sigma )}{\partial {\sigma }^2}\mathrm{d}\sigma \right|\le {\Vert f\Vert }_{C^2\left({\mathcal{M}}^2\right)}{(s-v_2)}^2,$$

$$\begin{array}{ccc}& & \left|{\int }_{v_1}^{{\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{1,0};v_1,v_2)}\left({\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{1,0};v_1,v_2)-\chi \right)\frac{{\partial }^{2}f(\chi ,v_2)}{\partial {\chi }^2}\mathrm{d}\chi \right|\hfill \\ & \le & {\Vert f\Vert }_{C^2\left({\mathcal{M}}^2\right)}{\left({\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{1,0};v_1,v_2)-v_1\right)}^2\hfill \end{array}$$

$$\begin{array}{ccc}& & \left|{\int }_{v_2}^{{\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{1,0};v_1,v_2)}\left({\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{1,0};v_1,v_2)-\sigma \right)\frac{{\partial }^{2}f(v_1,\sigma )}{\partial {\sigma }^2}\mathrm{d}\sigma \right|\hfill \\ & \le & {\Vert f\Vert }_{C^2\left({\mathcal{M}}^2\right)}{\left({\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{1,0};v_1,v_2)-v_2\right)}^2.\hfill \end{array}$$

Thus:

$$\begin{array}{ccc}\hfill \mid S_{m_1,m_2}^{q_1,q_2}\left(f;t,s\right)-f(v_1,v_2)\mid & \le & \{J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^2;v_1,v_2)+J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^2;v_1,v_2)\hfill \\ & +& {\left({\mathcal{C}}_{m_1,r_1}^{q_1}({\lambda }_{1,0};v_1,v_2)-v_1\right)}^2\hfill \\ & +& {\left({\mathcal{D}}_{m_2,r_2}^{q_2}({\lambda }_{1,0};v_1,v_2)-v_2\right)}^2\}{\Vert f\Vert }_{C^2\left({\mathcal{M}}^2\right)},\hfill \end{array}$$

which completes the desired proof of Theorem 7.

□

4. GBS-Type Approximation in Bögel Space

Suppose ${\mathcal{M}}_1\times {\mathcal{M}}_2=\{(v_1,v_2):0\le v_1,v_2<\infty \}$. Take any function $f:{\mathcal{M}}_1\times {\mathcal{M}}_2\to \mathbb{R}$ for real compact intervals of ${\mathcal{M}}_1\times {\mathcal{M}}_2$. For all $(t,s),(v_1,v_2)\in {\mathcal{M}}_1\times {\mathcal{M}}_2$ suppose ${\Delta }_{(t,s)}^{\ast }f(v_1,v_2)$ denotes the bivariate mixed difference operators defined as follows:

$${\Delta }_{(t,s)}^{\ast }f(v_1,v_2)=f(t,s)-f(t,v_2)-f(v_1,s)+f(v_1,v_2).$$

(42)

If at any point $(v_1,v_2)\in {\mathcal{M}}_1\times {\mathcal{M}}_2$, the function $f:{\mathcal{M}}_1\times {\mathcal{M}}_2\to \mathbb{R}$ defined on ${\mathcal{M}}_1\times {\mathcal{M}}_2,$ then ${lim}_{(t,s)\to (v_1,v_2)}{\Delta }_{(t,s)}^{\ast }f(v_1,v_2)=0$.

If set of all the space of all Bögel-continuous (B-continuous) denoted by $C_B({\mathcal{M}}_1\times {\mathcal{M}}_2)$ on $(v_1,v_2)\in {\mathcal{M}}_1\times {\mathcal{M}}_2$ and be defined such that $C_B({\mathcal{M}}_1\times {\mathcal{M}}_2)=\{f,suchthatf:{\mathcal{M}}_1\times {\mathcal{M}}_2\to \mathbb{R}isf,B-boundedon{\mathcal{M}}_1\times {\mathcal{M}}_2\}$. Next, the Bögel-differentiable function on $(v_1,v_2)\in {\mathcal{M}}_1\times {\mathcal{M}}_2$ be $f:{\mathcal{M}}_1\times {\mathcal{M}}_2\to \mathbb{R}$ and limit exists finite defined by:

$$\underset{(t,s)\to (v_1,v_2),t\ne v_1,s\ne v_2}{lim}\frac{1}{(t-v_1)(s-v_2)}\left({\Delta }_{(t,s)}^{\ast }\right)=D_{B}f(v_1,v_2)<\infty .$$

(43)

Let the classes of all Bögel-differentiable function denoted by $D_{\psi }f(v_1,v_2)$ and be $D_{\psi }({\mathcal{M}}_1\times {\mathcal{M}}_2)=\{f,suchthatf:{\mathcal{M}}_1\times {\mathcal{M}}_2\to \mathbb{R}isf,$ B—$differentiableon{\mathcal{M}}_1\times {\mathcal{M}}_2\}$. Suppose the function f is B-bounded on D and be $f:{\mathcal{M}}_1\times {\mathcal{M}}_2\to \mathbb{R},$ then for all $(t,s),(v_1,v_2)\in {\mathcal{M}}_1\times {\mathcal{M}}_2$ there exists positive constant M such that $\mid {\Delta }_{(t,s)}^{\ast }f(v_1,v_2)\mid \le M$. The classes of all B-continuous function is called a B-bounded on ${\mathcal{M}}_1\times {\mathcal{M}}_2,$ when ${\mathcal{M}}_1\times {\mathcal{M}}_2$ is compact subset. Let $B_{\psi }({\mathcal{M}}_1\times {\mathcal{M}}_2)$ denote the classes of all B-bounded function defined on ${\mathcal{M}}_1\times {\mathcal{M}}_2$, which equipped the norm on B as ${\Vert f\Vert }_B={sup}_{(t,s),(v_1,v_2)\in {\mathcal{M}}_1\times {\mathcal{M}}_2}\mid {\Delta }_{(t,s)}^{\ast }f(v_1,v_2)\mid$. As we know to approximate the degree for a set of all B-continuous function on positive linear operators, it is essential to use the properties of mixed-modulus of continuity. So, we let for all $(t,s)$ and $(v_1,v_2)\in {\mathcal{M}}_1\times {\mathcal{M}}_2$ and $f\in B_{\psi }({\mathcal{M}}_1\times {\mathcal{M}}_2)$, the mixed-modulus of continuity of function f such that ${\omega }_B:[0,\infty )\times [0,\infty )\to {\mathbb{R}}^2$ and be defined such as:

$${\omega }_B(f;{\delta }_1,{\delta }_2)=sup\{{\Delta }_{(t,s)}^{\ast }f(v_1,v_2):\mid t-v_1\mid \le {\delta }_1,\mid s-v_2\mid \le {\delta }_2\}.$$

(44)

For any ${\mathcal{M}}_1\times {\mathcal{M}}_2=[0,\infty )\times [0,\infty )$, we suppose the classes of all B-continuous function defined on ${\mathcal{M}}_1\times {\mathcal{M}}_2$ denoted by $C_{\psi }({\mathcal{M}}_1\times {\mathcal{M}}_2)$. Moreover, let set of all ordinary continuous function defined on ${\mathcal{M}}_1\times {\mathcal{M}}_2$ be $C({\mathcal{M}}_1\times {\mathcal{M}}_2)$. For further details on space of Bögel functions and GBS-type operators related to this paper, we propose the article [33,34,35,36,37,38].

Let $(v_1,v_2)\in {\mathcal{M}}_1\times {\mathcal{M}}_2$ and $m_1,m_2\in \mathbb{N}$. Then, for all $f\in C({\mathcal{M}}_1\times {\mathcal{M}}_2)$, we define the GBS-type new positive linear operators $K_{m_1,m_2}^{q_1,q_2}$ for the operators $J_{m_1,m_2}^{q_1,q_2}$. Thus, we suppose:

$$K_{m_1,m_2}^{q_1,q_2}\left(f(t,s);v_1,v_2\right)=J_{m_1,m_2}^{q_1,q_2}\left(f(t,v_2)+f(v_1,s)-f(t,s);v_1,v_2\right).$$

(45)

More precisely, the generalized GBS operator for the bivariate function is defined as follows:

$$K_{m_1,m_2}^{q_1,q_2}\left(f(t,s);v_1,v_2\right)=\sum _{r_1,r_2=0}^{\infty }{\mathcal{U}}_{m_1,q_1}^{r_1}\left(v_1\right){\mathcal{V}}_{m_2,q_2}^{r_2}\left(v_2\right)\underset{0}{\overset{\infty /A_1}{\int }}\underset{0}{\overset{\infty /A_2}{\int }}{Q}_{v_1,v_2}(t,s)f\left(q_1^{r_1}t,q_2^{r_2}s\right)d_{q_1}t{d}_{q_2}s,$$

(46)

where $Q_{v_1,v_2}(t,s)={\mathcal{T}}_{m_1,m_2}^{q_1,q_2}(t,s)P_{v_1,v_2}(t,s)$ and $P_{v_1,v_2}(t,s)=\left(f(t,v_2)+f(v_1,s)-f(t,s)\right)$.

Theorem 8.

For all $f\in C_{\psi }({\mathcal{M}}_1\times {\mathcal{M}}_2)$, the operators $K_{m_1,m_2}^{q_1,q_2}$ verify that:

$$\mid K_{m_1,m_2}^{q_1,q_2}\left(f(t,s);v_1,v_2\right)-f(v_1,v_2)\mid \le 4{\omega }_B\left(f;{\delta }_{m_1,m_2}^{2,0},{\delta }_{m_1,m_2}^{0,2}\right),$$

where ${\delta }_{m_1,m_2}^{2,0}$ and ${\delta }_{m_1,m_2}^{0,2}$ are defined by Theorem 4.

Proof. 

Let $(t,s)$ and $(v_1,v_2)\in {\mathcal{M}}_1\times {\mathcal{M}}_2.$ For all $m_1,m_2\in \mathbb{N}$ and ${\delta }_{m_1},{\delta }_{m_2}>0,$ it follows that:

$$\begin{array}{ccc}\hfill \mid {\Delta }_{(v_1,v_2)}^{\ast }f(t,s)\mid & \le & {\omega }_B\left(f;\mid t-v_1\mid \mid s-v_2\mid \right)\hfill \\ & \le & \left(1+\frac{t-v_1}{{\delta }_{m_1}}\right)\left(1+\frac{s-v_2}{{\delta }_{m_2}}\right){\omega }_B\left(f;{\delta }_{m_1},{\delta }_{m_2}\right).\hfill \end{array}$$

From (45) and the well-known Cauchy–Schwarz inequality, we easily conclude that:

$$\begin{array}{ccc}\hfill \mid K_{m_1,m_2}^{q_1,q_2}\left(f(t,s);v_1,v_2\right)-f(v_1,v_2)\mid & \le & J_{m_1,m_2}^{q_1,q_2}\left(\mid {\Delta }_{(v_1,v_2)}^{\ast }f(t,s)\mid ;v_1,v_2\right)\hfill \\ & \le & (J_{m_1,m_2}^{q_1,q_2}({\lambda }_{0,0};v_1,v_2)+\frac{1}{{\delta }_{m_1}}{\left(J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^2;v_1,v_2)\right)}^{\frac{1}{2}}\hfill \\ & +& \frac{1}{{\delta }_{m_2}}{\left(J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^2;v_1,v_2)\right)}^{\frac{1}{2}}\hfill \\ & +& \frac{1}{{\delta }_{m_1}}{\left(J_{m_1,m_2}^{q_1,q_2}({(t-v_1)}^2;v_1,v_2)\right)}^{\frac{1}{2}}\hfill \\ & \times & \frac{1}{{\delta }_{m_2}}{\left(J_{m_1,m_2}^{q_1,q_2}({(s-v_2)}^2;v_1,v_2)\right)}^{\frac{1}{2}}){\omega }_B\left(f;{\delta }_{m_1},{\delta }_{m_2}\right).\hfill \end{array}$$

In the view of Theorem 4, we easily obtain our results.

□

If we let $x=(t,s),y=(v_1,v_2)\in {\mathcal{M}}_1\times {\mathcal{M}}_2,$ then the Lipschitz function in terms of B-continuous functions is defined by:

$$Li{p}_M^{\xi }=\{f\in C({\mathcal{M}}_1\times {\mathcal{M}}_2):\mid {\Delta }_{(v_1,v_2)}^{\ast }f(x,y)\mid \le M{\Vert x-y\Vert }^{\xi }\},$$

(47)

where M is a positive constant, $0<\xi \le 1,$ and the Euclidean norm $\Vert x-y\Vert$ = $\sqrt{{(t-v_1)}^2+{(s-v_2)}^2}$.

Theorem 9.

For all $f\in Li{p}_M^{\xi },$ the operators $K_{m_1,m_2}^{q_1,q_2}$ satisfy:

$$\mid K_{m_1,m_2}^{q_1,q_2}\left(f(x,y);v_1,v_2\right)-f(v_1,v_2)\mid \le M{\left({\delta }_{m_1,m_2}^{2,0}+{\delta }_{m_1,m_2}^{0,2}\right)}^{\frac{\xi }{2}},$$

where ${\delta }_{m_1,m_2}^{2,0}$ and ${\delta }_{m_1,m_2}^{0,2}$ are defined by Theorem 4.

Proof. 

We easily see that:

$$\begin{array}{ccc}\hfill K_{m_1,m_2}^{q_1,q_2}\left(f(x,y);v_1,v_2\right)& =& J_{m_1,m_2}^{q_1,q_2}\left(f(v_1,y)+f(x,v_2)-f(x,s);v_1,v_2\right)\hfill \\ & =& J_{m_1,m_2}^{q_1,q_2}\left(f(v_1,v_2)-{\Delta }_{(v_1,v_2)}^{\ast }f(x,s);v_1,v_2\right)\hfill \\ & =& f(v_1,v_2)-J_{m_1,m_2}^{q_1,q_2}\left({\Delta }_{(v_1,v_2)}^{\ast }f(x,s);v_1,v_2\right).\hfill \end{array}$$

Therefore:

$$\begin{array}{ccc}\hfill \mid K_{m_1,m_2}^{q_1,q_2}\left(f(x,y);v_1,v_2\right)-f(v_1,v_2)\mid & \le & J_{m_1,m_2}^{q_1,q_2}\left(\mid {\Delta }_{(v_1,v_2)}^{\ast }f(x,y)\mid ;v_1,v_2\right)\hfill \\ & \le & M{J}_{m_1,m_2}^{q_1,q_2}\left({\Vert x-y\Vert }^{\xi };v_1,v_2\right)\hfill \\ & \le & M(J_{m_1,m_2}^{q_1,q_2}\left({\Vert x-y\Vert }^2;v_1,v_2\right){)}^{\frac{\xi }{2}}\hfill \\ & \le & M(J_{m_1,m_2}^{q_1,q_2}\left({(t-v_1)}^2;v_1,v_2\right)+J_{m_1,m_2}^{q_1,q_2}\left({(s-v_2)}^2;v_1,v_2\right){)}^{\frac{\xi }{2}}.\hfill \end{array}$$

□

Theorem 10.

For all $f\in D_{\psi }({\mathcal{M}}_1\times {\mathcal{M}}_2)$ and $D_{B}f\in B({\mathcal{M}}_1\times {\mathcal{M}}_2),$ we obtain:

$$\begin{array}{ccc}\hfill \mid K_{m_1,m_2}^{q_1,q_2}\left(f;v_1,v_2\right)-f(v_1,v_2)\mid & \le & C\{1+3\Vert D_B{f\Vert }_{\infty }\}(v_1+1)(v_2+1)\hfill \\ & +& \{1+\sqrt{M_2}(v_1+1)+\sqrt{M_1}(v_2+1)\}(v_1+1)(v_2+1)\hfill \\ & \times & {\varpi }_{mixed}(D_{B}f;O\left(\frac{1}{{\left[m_1\right]}_{q_1}}\right),O\left(\frac{1}{{\left[m_2\right]}_{q_2}}\right)),\hfill \end{array}$$

where C is any positive constant.

Proof. 

Suppose $\rho \in (v_1,t),\xi \in (v_2,s)$ and

$${\Delta }_{(v_1,v_2)}^{\ast }f(t,s)=(t-v_1)(s-v_2)D_{B}f(\rho ,\xi ),$$

$$D_{B}f(\rho ,\xi )={\Delta }_{(v_1,v_2)}^{\ast }{D}_{B}f(\rho ,\xi )+D_{B}f(\rho ,y)+D_{B}f(x,\xi )-D_{B}f(v_1,v_2).$$

For all $D_{B}f\in B({\mathcal{M}}_1\times {\mathcal{M}}_2)$, it follows that:

$$\begin{array}{ccc}\hfill \mid K_{m_1,m_2}^{q_1,q_2}\left({\Delta }_{(v_1,v_2)}^{\ast }f(t,s);v_1,v_2\right)\mid & =& \mid K_{m_1,m_2}^{q_1,q_2}\left((t-v_1)(s-v_2)D_{B}f(\rho ,\xi );v_1,v_2\right)\mid \hfill \\ & \le & K_{m_1,m_2}^{q_1,q_2}\left(\mid t-v_1\mid \mid s-v_2\mid \mid {\Delta }_{(v_1,v_2)}^{\ast }{D}_{B}f(\rho ,\xi )\mid ;v_1,v_2\right)\hfill \\ & +& K_{m_1,m_2}^{q_1,q_2}(\mid t-v_1\mid \mid s-v_2\mid (\mid D_{B}f(\rho ,v_2)\mid \hfill \\ & +& \mid D_{B}f(v_1,\xi )\mid +\mid D_{B}f(v_1,v_2)\mid );v_1,v_2)\hfill \\ & \le & K_{m_1,m_2}^{q_1,q_2}(\mid t-v_1\mid \mid s-v_2\mid \hfill \\ & \times & {\varpi }_{mixed}\left(D_{B}f;\mid \rho -v_1\mid ,\mid \xi -v_2\mid \right);v_1,v_2)\hfill \\ & +& 3\Vert D_B{f\Vert }_{\infty }{K}_{m_1,m_2}^{q_1,q_2}\left(\mid t-v_1\mid \mid s-v_2\mid ;v_1,v_2\right).\hfill \end{array}$$

Here, ${\varpi }_{mixed}$ is the mixed-modulus of continuity, and it follows that:

${\varpi }_{mixed}\left(D_{B}f;\mid \rho -v_1\mid ,\mid \xi -v_2\mid \right)$

$$\begin{array}{ccc}& \le & {\varpi }_{mixed}\left(D_{B}f;\mid t-v_1\mid ,\mid s-v_2\mid \right)\hfill \\ & \le & \left(1+{\delta }_{m_1}^{-1}\mid t-v_1\mid \right)\left(1+{\delta }_{m_2}^{-1}\mid s-v_2\mid \right){\varpi }_{mixed}\left(D_{B}f;\mid {\delta }_{m_1},{\delta }_{m_2}\right).\hfill \end{array}$$

Therefore, it is obvious that:

$$\begin{array}{ccc}\hfill \mid K_{m_1,m_2}^{\ast }\left(f;v_1,v_2\right)-f(v_1,v_2)\mid & =& \mid {\Delta }_{(v_1,v_2)}^{\ast }f(t,s);v_1,v_2\mid \hfill \\ & \le & 3\Vert D_B{f\Vert }_{\infty }(K_{m_1,m_2}^{q_1,q_2}\left({(t-v_1)}^2{(s-v_2)}^2;v_1,v_2\right){)}^{\frac{1}{2}}\hfill \\ & +& (K_{m_1,m_2}^{q_1,q_2}\left(\mid t-v_1\mid \mid s-v_2\mid ;v_1,v_2\right)\hfill \\ & +& {\delta }_{m_1}^{-1}{K}_{m_1,m_2}^{q_1,q_2}\left({(t-v_1)}^2\mid s-v_2\mid ;v_1,v_2\right))\hfill \\ & +& {\delta }_{m_2}^{-1}{K}_{m_1,m_2}^{q_1,q_2}\left(\mid t-v_1\mid {(s-v_2)}^2;v_1,v_2\right)\hfill \\ & +& {\delta }_{m_1}^{-1}{\delta }_{m_2}^{-1}{K}_{m_1,m_2}^{q_1,q_2}\left({(t-v_1)}^2{(s-v_2)}^2;v_1,v_2\right){\varpi }_{mixed}\left(D_{B}f;{\delta }_{m_1},{\delta }_{m_2}\right);\hfill \\ \hfill \mid K_{m_1,m_2}^{\ast }\left(f;v_1,v_2\right)-f(v_1,v_2)\mid & =& \mid {\Delta }_{(v_1,v_2)}^{\ast }f(t,s);v_1,v_2\mid \hfill \\ & \le & 3\Vert D_B{f\Vert }_{\infty }(K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{2,2};v_1,v_2\right){)}^{\frac{1}{2}}\hfill \\ & +& \left\{\right(K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{2,2};v_1,v_2\right){)}^{\frac{1}{2}}\hfill \\ & +& {\delta }_{m_1}^{-1}(K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{4,2};v_1,v_2\right){)}^{\frac{1}{2}}+{\delta }_{m_2}^{-1}(K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{2,4};v_1,v_2\right){)}^{\frac{1}{2}}\hfill \\ & +& {\delta }_{m_1}^{-1}{\delta }_{m_2}^{-1}{K}_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{2,2};v_1,v_2\right)\}{\varpi }_{mixed}\left(D_{B}f;{\delta }_{m_1},{\delta }_{m_2}\right)\hfill \end{array}$$

which follows that:

$$\begin{array}{ccc}\hfill \mid K_{m_1,m_2}^{\ast }\left(f;v_1,v_2\right)-f(v_1,v_2)\mid & =& 3\Vert D_B{f\Vert }_{\infty }(K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{2,0};v_1,v_2\right)K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{0,2};v_1,v_2\right){)}^{\frac{1}{2}}\hfill \\ & +& \left\{\right(K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{2,0};v_1,v_2\right)K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{0,2};v_1,v_2\right){)}^{\frac{1}{2}}\hfill \\ & +& {\delta }_{m_1}^{-1}(K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{4,0};v_1,v_2\right)K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{0,2};v_1,v_2\right){)}^{\frac{1}{2}}\hfill \\ & +& {\delta }_{m_2}^{-1}(K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{2,0};v_1,v_2\right)K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{0,4};v_1,v_2\right){)}^{\frac{1}{2}}\hfill \\ & +& {\delta }_{m_1}^{-1}{\delta }_{m_2}^{-1}{K}_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{2,0};v_1,v_2\right)K_{m_1,m_2}^{q_1,q_2}\left({\Phi }_{v_1,v_2}^{0,2};v_1,v_2\right)\}\hfill \\ & \times & {\varpi }_{mixed}\left(D_{B}f;{\delta }_{m_1},{\delta }_{m_2}\right).\hfill \end{array}$$

From Lemma 5, we demonstrate:

$$\begin{array}{ccc}\hfill \mid K_{m_1,m_2}^{\ast }\left(f;v_1,v_2\right)-f(v_1,v_2)\mid & \le & 3\Vert D_B{f\Vert }_{\infty }\left(\sqrt{M_1{M}_2}(v_1+1)(v_2+1)\right)\hfill \\ & +& \left\{\right(\sqrt{M_1{M}_2}(v_1+1)(v_2+1))\hfill \\ & +& {\delta }_{m_1}^{-1}\left(\sqrt{M_2}O\left(\frac{1}{{\left[m_1\right]}_{q_1}}\right){(v_1+1)}^2(v_2+1)\right)\hfill \\ & +& {\delta }_{m_2}^{-1}\left(\sqrt{M_1}O\left(\frac{1}{{\left[m_2\right]}_{q_2}}\right){(v_2+1)}^2(v_1+1)\right)\hfill \\ & +& {\delta }_{m_1}^{-1}{\delta }_{m_2}^{-1}\left(O\left(\frac{1}{{\left[m_1\right]}_{q_1}}\right)O\left(\frac{1}{{\left[m_2\right]}_{q_2}}\right)(v_1+1)(v_2+1)\right)\}\hfill \\ & \times & {\varpi }_{mixed}\left(D_{B}f;{\delta }_{m_1},{\delta }_{m_2}\right),\hfill \end{array}$$

where, in the view of Theorem 1, we have ${\delta }_{m_i}=O\left(\frac{1}{{\left[m_i\right]}_{q_i}}\right)$ for $i=1,2$, which completes the proof of Theorem 10. □

5. Conclusions

In our investigation, we introduced the bivariate form of the previous operators [13] and then obtained the GBS-type approximation in the Bögel continuous function. These types of constructions of operators give an extended form of Jakimovski–Leviatan-Beta type $q$—integral operators given by [13]. We studied the bivariate properties of our new operators by the use of the modulus of continuity and the mixed-modulus of continuity and then obtained the degree of approximations for Lipschitz-class maximal functions. Moreover, we are mentioning here that the convergence at a certain point by GBS-type operators defined by (46) is more generalized than the bivariate operator (15) and the operator defined by [13]. Finally, in our future work, we will study the statistical approximation of these operators, as well as the approximation properties, by using power series methods, including some shape-preserving properties of these types operators.