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Bivariate Bernstein–Schurer–Stancu type GBS operators in \((p,q)\)-analogue
Advances in Difference Equations volume 2020, Article number: 76 (2020)
Abstract
The purpose of this paper is to construct a \((p,q)\)-analogue of Bernstein–Schurer–Stancu type GBS (generalized Boolean sum) operators for approximating B-continuous and B-differentiable functions. We also establish uniform convergence theorem and estimate the degree of approximation of B-continuous and B-differentiable functions.
1 Introduction
Badea et al. [5] introduced the following operators known as GBS operators associated with L.
Let \(I_{1},I_{2}\subseteq \mathbb{R}\) be nonempty intervals, and let \(L:\mathbb{R}^{I_{1}\times I_{2}}\rightarrow \mathbb{R}^{I_{1}\times I_{2}}\) be a positive linear bivariate operator, where \(\mathbb{R} ^{I_{1}\times I_{2}}=\{f|f:I_{1}\times I_{2}\rightarrow \mathbb{R}\}\). If \(f(\circ ,\ast )\in \mathbb{R}^{I_{1}\times I_{2}}\), then the bivariate operators \(U:\mathbb{R}^{I_{1}\times I_{2}}\rightarrow \mathbb{R}^{I_{1}\times I_{2}}\) are defined by
In 1934, Karl Bögel [13] introduced the notion of B-continuity and B-differentiability. B-continuity by means of bivariate mixed difference operator \(\Delta _{2}: \mathbb{R}^{I_{1} \times I_{2}}\rightarrow \mathbb{R}^{I_{1}\times I_{2}}\) is defined in [12].
We now recall some definitions and results based on B-continuity as follows.
Definition 1.1
A function \(f\in \mathbb{R}^{I_{1}\times I_{2}}\) is B-continuous if, for each \((x,y)\in I_{1}\times I_{2}\),
where \(\Delta _{u,v} [f:x,y ]\) is the mixed difference defined by
If the function f is B-continuous at any point \((x,y)\in I_{1} \times I_{2}\), then it is B-continuous on the interval \(I_{1}\times I_{2}\).
For any \((x,y),(u,v)\in I_{1}\times I_{2}\), if there exists \(M>0\) such that
holds, then f is B-bounded.
Definition 1.2
A function \(f\in \mathbb{R}^{I_{1}\times I_{2}}\) is said to be uniformly B-continuous if, for any \(\epsilon >0\), there exists \(\delta ( \epsilon )>0\) such that, for every \((x,y),(u,v)\in I_{1}\times I_{2}\) with \(|x-u|<\delta (\epsilon )\), \(|y-v|<\delta (\epsilon )\), we have
If \(f\in C_{b}(I_{1}\times I_{2})\) and \(I_{1}\times I_{2}\subseteq \mathbb{R}\) are compact intervals of \(\mathbb{R}\), then f is uniform B-continuous on \(I_{1}\times I_{2}\), where \(C_{b}(I_{1}\times I_{2})\) is the set of B-continuous functions. For more information, we refer to [35].
Badea et al. [6] proved the following Korovkin type theorem to approximate bivariate function in the space of Bögel-continuous (B-continuous) functions.
Theorem 1.3
Let\(\{L_{m,n}\}\)be a sequence of positive linear operators which maps\(\mathbb{R}^{I_{1}\times I_{2}}\)to\(\mathbb{R}^{I_{1}\times I_{2}}\)such that, for all\((x,y)\in I_{1}\times I_{2}\),
- (i)
\(L_{m,n}(e_{0 0};x,y)=L(1,x,y)=1\),
- (ii)
\(L_{m,n}(e_{1 0};x,y)=L(u,x,y)=x+u_{m,n}(x,y)\),
- (iii)
\(L_{m,n}(e_{0 1};x,y)=L(v,x,y)=y+v_{m,n}(x,y)\),
- (iv)
\(L_{m,n}(e_{0 2}+e_{2 0};x,y)=L(u^{2}+v^{2},x,y)=x^{2}+y ^{2}+w_{m,n}(x,y)\),
where\(u_{m,n}(x,y)\), \(v_{m,n}(x,y)\), and\(w_{m,n}(x,y)\)converge uniformly to zero as\(m,n\rightarrow \infty \). Then the sequence\(\{U_{m,n}f\}\)converges uniformly tofon\(I_{1}\times I_{2}\)for any\(f\in C_{b}(I_{1}\times I_{2})\), where\(I_{1}\), \(I_{2}\)are compact intervals of\(\mathbb{R}\); and\(U_{m,n}\)is a GBS operator associated with\(L_{m,n}\).
The mixed modulus of continuity is an important tool to approximate degree of B-continuous functions introduced by Marchaud [25]. Let \(f\in \mathbb{R}^{I_{1}\times I_{2}}\) and \(I_{1}\), \(I _{2}\) be compact intervals of \(\mathbb{R}\). Then \(\omega _{ \mathrm{mixed}}:[0,\infty )\times [0,\infty )\rightarrow [0,\infty )\) is defined by
for any \(\delta _{1},\delta _{2}\in (0,\infty )\times (0,\infty )\) and \((x,y),(u,v)\in I_{1}\times I_{2}\).
Badea et al. [5] proved the following Shisha–Mond type theorem (introduced by Mahmedov [24]) to evaluate the degree of approximation of Bögel-continuous (continuous in Bögel sense) functions using GBS operators.
Theorem 1.4
Let\(L:C_{b}(I_{1}\times I_{2})\rightarrow C_{b}(I_{1}\times I_{2})\)be a positive linear operator and\(Uf(x,y)\)be the associated GBS operator. Then the following inequality holds for any\(f\in C_{b}(I_{1}\times I _{2})\), \((x,y)\in I_{1}\times I_{2}\), and\(\delta _{1},\delta _{2} \geq 0\):
Note that \(\omega _{\mathrm{mixed}}(\delta _{1},\delta _{2})\) is a B-continuous function and \(\omega _{\mathrm{mixed}}(0,0)=0\). By the inequality defined in Theorem 1.4 and the properties of \(\omega _{\mathrm{mixed}}(\delta _{1},\delta _{2})\), it is possible to obtain the uniform convergence for the sequence introduced by GBS operators.
In [10], Bărbosu defined Schurer–Stancu type GBS operators
as follows:
where \(r_{1}\), \(r_{2}\) are nonnegative integers and \(\alpha _{i}\), \(\beta _{i}\) are real parameters with \(0\leq \alpha _{i}\leq \beta _{i}\) (\(i=1,2\)). For \(\alpha _{i}=\beta _{i}=0\) (\(i=1,2\)), the above operators reduce to the first GBS operators which were introduced by Dobrescu and Matei [15]. For detailed study, one can refer to [7, 8], and [19].
2 q-Bernstein–Schurer–Stancu GBS operators
Quantum calculus (q-calculus) plays an important role in approximation theory. First of all, the q-calculus was applied by Lupaş on Bernstein polynomials. Then, focusing on bivariate case, Bărbosu [9] introduced the generalized bivariate Stancu operators, and many researchers have worked on different operators: Örkcü [32] established the q-Szász–Mirakjan–Kantorovich bivariate operators; Mursaleen and Ahasan [28] introduced the Dunkl generalization of Stancu type q-Szász–Mirakjan–Kantorovich operators; Ostrovska [33] determined the relation between the theory of q-Bernstein polynomials and limit q-Bernstein operators. For detailed study, we refer to [3, 6, 11, 14, 20, 34], and [37].
Agrawal et al. [4] introduced the q-Bernstein–Schurer–Stancu operators
as follows:
Recently Bărbosu et al. [12] introduced Bernstein–Schurer–Stancu type GBS operators based on q-integers.
For any \((x,y)\in I=[0,1+r_{1}]\times [0,1+r_{2}]\), the operators
associated with \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}}\) are defined as follows:
where
and
The operators (2.1) satisfy the following properties as proved in [12].
Lemma 2.1
Let\(e_{i,j}:I\rightarrow I\), where\(I=[0,1+r_{1}]\times [0,1+r_{2}]\)is the test functions defined by\(e_{i,j}(x,y)=x^{i}y^{j}\) (i, jare nonnegative integers). Then the following equalities hold:
- (i)
\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{0,0};q_{1},q_{2};x,y)=e_{0,0}(x,y)\),
- (ii)
\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{1,0};q_{1},q_{2};x,y)=\frac{[m+r_{1}]_{q_{1}}x+\alpha _{1}}{[m]_{q _{1}}+\beta _{1}}\),
- (iii)
\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}}(e_{0,1};q_{1},q_{2};x,y)=\frac{[n+r_{2}]_{q_{2}}y+\alpha _{2}}{[n]_{q_{2}}+\beta _{2}}\),
- (iv)
\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{2,0};q_{1},q_{2};x,y)=\frac{ ([m+r_{1}]_{q_{1}}^{2}x ^{2}+[m+r_{1}]_{q_{1}}x(1-x)+2\alpha _{1}[m+r_{1}]_{q_{1}}x+\alpha ^{2} _{1} )}{([m]_{q_{1}}+\beta _{1})^{2}}\),
- (v)
\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{0,2};q_{1},q_{2};x,y)=\frac{ ([n+r_{2}]_{q_{2}}^{2}y ^{2}+[n+r_{2}]_{q_{2}}y(1-y)+2\alpha _{2}[n+r_{2}]_{q_{2}}y+\alpha ^{2} _{2} )}{([n]_{q_{2}}+\beta _{2})^{2}}\).
3 \((p,q)\)-Bernstein–Schurer–Stancu type GBS operators
In 2015 Mursaleen et al. [29] used \((p,q)\)-calculus in approximation theory and defined first \((p,q)\)-analogue of Bernstein polynomials. Later on this idea was used to generalize several operators, e.g., [1, 2, 16–18, 21, 26, 27, 30, 31]; for its applications, see [22] and [23].
We now recall some notations on \((p,q)\)-calculus.
For any \(p > 0\) and \(q > 0\), the \((p,q)\) integers \([k]_{p,q}\) are defined as follows:
\(k = 0,1,2,3,4,\ldots \) .
Also,
and
For \(y=0\), formula (3.5) becomes
For \(x=0\), formula (3.6) becomes
Now we define some useful notations which are used in this paper. For any nonnegative integer k, we have
For \(p=1\) in (3.1)–(3.10) all these reduce to q-analogues.
Now first of all, we construct a \((p,q)\)-analogue of Bernstein–Schurer–Stancu operators as follows:
for any \(x\in [0,1+r]\) and \(0< q< p\leq 1\), where r is a nonnegative integer.
Lemma 3.1
The operators (3.11) satisfy the following properties for the test functions\(e_{i}=x^{i}\) (\(i=0,1,2,3,4\)):
- (i)
\(S^{\alpha ,\beta }_{n,r}(e_{0};p,q;x)=1\),
- (ii)
\(S^{\alpha ,\beta }_{n,r}(e_{1};p,q;x)=\frac{[n+r]_{p,q}x+ \alpha }{[n]_{p,q}+\beta }\),
- (iii)
\(S^{\alpha ,\beta }_{n,r}(e_{2};p,q;x)=\frac{[n+r]_{p,q} ^{2}x^{2}+p^{n+r-1}[n+r]_{p,q}x(1-x)+2\alpha [n+r]_{p,q}x+\alpha ^{2}}{([n]_{p,q}+ \beta )^{2}}\),
- (iv)
\(S^{\alpha ,\beta }_{n,r}(e_{3};p,q;x) =\frac{[n+r]_{p,q}p ^{n+r} ( p^{-2}-[n+r-1]_{p,q}+3\alpha +3\alpha ^{2} )x}{([n]_{p,q}+ \beta )^{3}}+ \frac{[n+r]_{p,q}[n+r-1]_{p,q} \lbrace p^{2}q+p^{n+r-3}[2]_{p,q}+p ^{n+r}-p^{n+r-1}[2]_{p,q}+p^{n+r-2}q[2]_{p,q}+3\alpha q \rbrace x ^{2}}{([n]_{p,q}+\beta )^{3}}+\frac{[n+r]_{p,q}[n+r-1]_{p,q}[n+r-2]_{p,q}\{p^{2}q^{2}-pq^{2}+q ^{3}\}x^{3}+\alpha ^{3}}{([n]_{p,q}+\beta )^{3}}\).
- (v)
\(S^{\alpha ,\beta }_{n,r}(e_{4};p,q;x)=\frac{[n+r]_{p,q}\{4 \alpha ^{3} p^{3(n+r-1)}+4\alpha p^{2(n+r-1)}+6\alpha ^{2}p^{n+r-1}\}x}{([n]_{p,q}+ \beta )^{4}}+\frac{[n+r]_{p,q}[n+r-1]_{p,q}\{6\alpha ^{2}q+p^{n+r-1}+2qp^{2(n+r-1)}+q[2]_{p,q}p ^{2(n+r)-3}+4\alpha q(p^{n+r-1}+[2]_{p,q}p^{n+r-2})\}x^{2}}{([n]_{p,q}+ \beta )^{4}}+\frac{[n+r]_{p,q}[n+r-1]_{p,q}[n+r-2]_{p,q}\{4\alpha pq^{2}(1-p)+4 \alpha q^{3}+pq+q([2]_{p,q}+q)p^{n+r}-qp^{n+r+1}+2q^{3}p^{n+r-1}\}x ^{3}}{([n]_{p,q}+\beta )^{4}}+\frac{[n+r]_{p,q}[n+r-1]_{p,q}[n+r-2]_{p,q}[n+r-3]_{p,q}p^{4}q^{4}x ^{4}+\alpha ^{4}}{([n]_{p,q}+\beta )^{4}}\).
Proof
After solving, we get
Finally, we have
By using (3.8), (3.9), and (3.10), we obtain \((v)\). □
Rao and Wafi [36] introduced a \((p,q)\)-analogue of Bivariate–Schurer–Stancu operators in the following form:
Let \(I_{1}\times I_{2}=[0,1+r_{1}]\times [0,1+r_{2}]\), \(0< q_{1}< p_{1} \leq 1\), \(0< q_{2}< p_{2}\leq 1\), and \(m,n\in \mathbb{N}\times \mathbb{N}\). Then, for any \(f\in C(I_{1}\times I_{2})\), the operators \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}:C(I _{1}\times I_{2})\rightarrow C([0,1]\times [0,1])\) are defined by
where
In the following, we define a \((p,q)\)-analogue of the bivariate Schurer–Stancu operators as follows:
Let \(I_{1}\times I_{2}=[0,1+r_{1}]\times [0,1+r_{2}]\), \(0< q_{1}< p_{1} \leq 1\), \(0< q_{2}< p_{2}\leq 1\), and \(m,n\in \mathbb{N}\times \mathbb{N}\). Then, for any \(f\in C(I_{1}\times I_{2})\), the operators \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}:C(I _{1}\times I_{2})\rightarrow C([0,1]\times [0,1])\) are defined by
where
The operators (3.16) satisfy the following properties.
Lemma 3.2
Let\(e_{i,j}(x,y)=x^{i}y^{j}\), \(0\leq i\), \(j\leq 2\), be two-dimensional test functions. Then
- (i)
\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{0,0};p_{1},p_{2},q_{1},q_{2};x,y)=1\),
- (ii)
\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{1,0};p_{1},p_{2},q_{1},q_{2};x,y)=\frac{[m+r_{1}]_{p_{1},q _{1}}x+\alpha _{1}}{[m]_{p_{1},q_{1}}+\beta _{1}}\),
- (iii)
\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r _{1},r_{2}}(e_{0,1};p_{1},p_{2},q_{1},q_{2};x,y)=\frac{[n+r_{2}]_{p _{2},q_{2}}y+\alpha _{2}}{[n]_{p_{2},q_{2}}+\beta _{2}}\),
- (iv)
\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{2,0};p_{1},p_{2},q_{1},q_{2};x,y)=\frac{[m+r_{1}]_{p_{1},q _{1}}(p_{1}^{m+r_{1}-1}+2\alpha _{1})x}{([m]_{p_{1},q_{1}}+\beta _{1})^{2}}+\frac{q _{1}[m+r_{1}]_{p_{1},q_{1}}[m+r_{1}-1]_{p_{1},q_{1}}x^{2}}{([m]_{p _{1},q_{1}} +\beta _{1})^{2}} +\frac{\alpha _{1}^{2}}{([m]_{p_{1},q_{1}}+\beta _{1})^{2}}\),
- (v)
\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}(e_{0,2};p_{1},p_{2},q_{1},q_{2};x,y)=\frac{[n+r_{2}]_{p_{2},q _{2}}(p_{2}^{n+r_{2}-1}+2\alpha _{2})y}{([n]_{p_{2},q_{2}}+\beta _{2})^{2}}+\frac{q _{2}[n+r_{2}]_{p_{2},q_{2}}[n+r_{2}-1]_{p_{2},q_{2}}y^{2}}{([n]_{p _{2},q_{2}}+\beta _{2})^{2}} +\frac{\alpha _{2}^{2}}{([n]_{p_{2},q_{2}}+\beta _{2})^{2}}\).
Proof
From Lemma 3.1, \((i)\) follows immediately.
Now for \((ii)\), again from Lemma 3.1, we have
Similarly, we obtain \((iii)\).
Further, for \((iv)\), we have
In a similar way, we get (v). □
Now, motivated by q-Bernstein–Schurer–Stancu type GBS operators (2.1), we construct \((p,q)\)-Bernstein–Schurer–Stancu type GBS operators as follows.
For any \((x,y)\in I=[0,1+r_{1}]\times [0,1+r_{2}]\), the \((p,q)\)-Bernstein–Schurer–Stancu type GBS operators \(U^{\alpha _{1}, \beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}:C_{b}([0,1+r_{1}] \times [0,1+r_{2}])\rightarrow C_{b}([0,1]\times [0,1])\) associated with \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\) are defined by
where
Lemma 3.3
Let\(\psi _{x}, \psi _{y}:I\rightarrow \mathbb{R}\)be defined as
where\(I=[0,1+r_{1}]\times [0,1+r_{2}]\). Then the following equalities hold:
and
Similarly, we have
Finally, we obtain
Proof
By definition of \(\psi _{x}\), \(\psi _{y}\) and the linearity of \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\), we obtain the results (3.18), (3.19), (3.20), and (3.21). □
Theorem 3.4
Let\(p_{1}=p_{1}^{m}\), \(p_{2}=p_{2}^{n}\)and\(q_{1}=q_{1}^{m}\), \(q_{2}=q_{2}^{n}\)such that
Then the sequence\(\{U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}} _{m,n,r_{1},r_{2}}\}\)converges uniformly toffor any\(f\in C_{b}(I)\)on the interval\([0,1]\times [0,1]\).
Proof
The new operators are linear and positive in view of linearity and positivity of q-Bernstein–Schurer–Stancu operators on \([0,1] \times [0,1]\). Now we have to prove that the \((p,q)\)-Bernstein–Schurer–Stancu operators satisfy the hypotheses of Theorem 1.3. By Lemma 3.2\((i)\), we have
That is, condition \((i)\) of Theorem 1.3 verified.
From the second condition \((ii)\) of Lemma 3.2, we have
i.e., condition \((ii)\) of Theorem 1.3 is verified with
In a similar way, we get
where
From statements \((iv)\) and \((v)\), again by applying Lemma 3.2, we obtain
where
From (3.22), (3.23), (3.24), and the hypotheses of Theorem 3.4, it follows that
uniformly on \([0,1]\times [0,1]\).
The desired uniform convergence is verified as a consequence of Theorem 1.3. □
In the next result, we express the degree of approximation of B-continuous functions f by using \((p,q)\)-GBS operators.
Theorem 3.5
For any\(f\in C_{b}(I)\)and\((x,y)\in [0,1]\times [0,1]\), the following estimation holds true:
where
and
Proof
By applying Theorem 1.3, since \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\) is a constant reproducing operator, we have
for any \(\delta _{1}, \delta _{2}\geq 0\).
Since, for any \((x,y)\in [0,1]\times [0,1]\),
Then
where \(\delta _{1}\), \(\delta _{2}\) are defined in (3.26) and (3.27), respectively. Thus, from (3.28), (3.29), and (3.30), we obtain the desired result. □
Remark 3.6
Suppose \(p_{1}=p^{m}_{1}\), \(q_{1}=q^{m}_{1}\) and \(p_{2}=p^{n}_{2}\), \(q_{2}=q^{n}_{2}\) such that
Then
It directly follows from (3.25) that the sequence \(\lbrace U ^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}f \rbrace _{m,n\in \mathbb{N}}\) converges uniformly to f for any \(f\in C_{b}(I)\) on \([0,1]\times [0,1]\).
Let \(\mathrm{Lip}_{M}(\gamma _{1},\gamma _{2})\)\((\gamma _{1},\gamma _{2} \in (0,1])\) be a Lipschitz class defined by
Theorem 3.7
Let\(f\in \mathrm{Lip}_{M}(\gamma _{1},\gamma _{2})\). Then, for any\(M>0\)and\(\gamma _{1},\gamma _{2}\in (0,1]\), we have
Proof
By the definition of the operators \(U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\) and linearity of the operators \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\), we obtain
By the hypothesis, we get
By using Hölder’s inequality with \(s_{1}=\frac{2}{\gamma _{1}}\), \(t_{1}=\frac{2}{2-\gamma _{1}}\) and \(s_{2}=\frac{2}{\gamma _{2}}\), \(t_{2}=\frac{2}{2-\gamma _{2}}\), we have
Considering Lemma 3.2, we get the degree of local approximation for B-continuous functions \(f\in \mathrm{Lip}_{M}(\gamma _{1},\gamma _{2})\). □
4 Degree of approximation of B-differentiable functions
In this section, we consider the case of B-differentiable functions. Note that a function \(f:\mathbb{R}\rightarrow \mathbb{R}\) is called B-differentiable if, for each \((x,y)\in \mathbb{R}\), the following equality holds:
where \(D_{B}f\) is a B-derivative of f.
The mixed modulus of smoothness \(\omega _{\mathrm{mixed}}:[0,\infty ) \times [0,\infty )\rightarrow [0,\infty )\) is defined as follows.
Definition 4.1
For any \(\delta _{1},\delta _{2}\in [0,\infty )\times [0,\infty )\) and for all \((x,y),(u,v)\in I_{1}\times I_{2}\),
In [5], Badea et al. proved the following Shisha–Mond type theorem for B-differentiable functions by using GBS operators. In what follows, we try to prove this result by using \((p,q)\)-GBS operators.
Theorem 4.2
Let\(L:C_{b}(I_{1}\times I_{2})\rightarrow C_{b}(I_{1}\times I_{2})\)be a positive linear operator and\(Uf(x,y)\)be the associated GBS operator. Then the following inequality holds for any\(f\in C_{b}(I_{1}\times I _{2})\), \((x,y)\in I_{1}\times I_{2}\), and\(\delta _{1},\delta _{2} \geq 0\):
Theorem 4.3
Let\(U^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\)be a GBS operators associated with\(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r_{2}}\)andfhas a boundedB-derivative\(D_{B}f\). Then the following inequality holds:
Proof
Since \(f\in C_{b}(I)\), we have
where
Since \(D_{B}f\in B(I)\), we can write
Since the mixed modulus of smoothness is nondecreasing, we have
Now substituting in inequality (4.3), by linearity of the operators \(S^{\alpha _{1},\beta _{1},\alpha _{2},\beta _{2}}_{m,n,r_{1},r _{2}}\) and by the Cauchy–Schwarz inequality, we obtain
By using the following equality for \((x,y),(u,v)\in I_{1}\times I_{2}\):
\(i,j=1,2\), from (3.29), (3.30), and Lemma 3.3, for any \((x,y)\in [0,1]\times [0,1]\), we get the result. □
Remark 4.4
For \(p=1\), all the above results reduce to q-analogues and for \(p=q=1\) these results further reduce to the classical ones.
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Authors are thankful to the learned referees for their valuable comments which improved the presentation of the paper.
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The author (K.J. Ansari) extends his appreciation to the Deanship of Scientific Research at King Khalid University for funding this work through research groups program under grant number G.R.P-93-41.
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Mursaleen, M., Ahasan, M. & Ansari, K.J. Bivariate Bernstein–Schurer–Stancu type GBS operators in \((p,q)\)-analogue. Adv Differ Equ 2020, 76 (2020). https://doi.org/10.1186/s13662-020-02547-7
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DOI: https://doi.org/10.1186/s13662-020-02547-7