Chauhan et al. Journal of Inequalities and Applications (2017) 2017:50 DOI 10.1186/s13660-017-1298-y
RESEARCH
Open Access
A new kind of Bernstein-Schurer-StancuKantorovich-type operators based on q-integers Ruchi Chauhan1* , Nurhayat Ispir2 and PN Agrawal1 *
Correspondence:
[email protected] 1 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India Full list of author information is available at the end of the article
Abstract Agrawal et al. (Boll. Unione Mat. Ital. 8:169-180, 2015) introduced a Stancu-type Kantorovich modification of the operators proposed by Ren and Zeng (Bull. Korean Math. Soc. 50(4):1145-1156, 2013) and studied a basic convergence theorem by using the Bohman-Korovokin criterion, the rate of convergence involving the modulus of continuity, and the Lipschitz function. The concern of this paper is to obtain Voronoskaja-type asymptotic result by calculating an estimate of fourth order central moment for these operators and discuss the rate of convergence for the bivariate case by using the complete and partial moduli of continuity and the degree of approximation by means of a Lipschitz-type function and the Peetre K-functional. Also, we consider the associated GBS (generalized Boolean sum) operators and estimate the rate of convergence for these operators with the help of a mixed modulus of smoothness. Furthermore, we show the rate of convergence of these operators (univariate case) to certain functions with the help of the illustrations using Maple algorithms and in the bivariate case, the rate of convergence of these operators is compared with the associated GBS operators by illustrative graphics. MSC: 41A25; 26A15; 41A28 Keywords: q-Bernstein-Schurer-Kantorovich; rate of convergence; modulus of continuity; GBS operators; mixed modulus of continuity
1 Introduction Following [], for any fixed real number q > , satisfying the condition < q < , the qinteger [k]q , for k ∈ N and q-factorial [k]q ! are defined as [k]q =
(–qk ) , (–q)
if q = ,
k,
if q = ,
and [k]q ! =
[k]q [k – ]q · · · , if k ≥ , , if k = ,
© The Author(s) 2017. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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respectively. For any integers n, k satisfying ≤ k ≤ n, the q-binomial coefficient is given by [n]q ! n . = k q [n – k]q ![k]q ! The q-analogue of ( – x)n is given by ( – x)nq =
n– j j= ( – q x),
n = , , . . . , n = .
,
The q-integration in the interval [, a] is defined by
a
f (t) dq t = a( – q)
∞ f aqn qn ,
< q < ,
n=
provided the series converges. Let I = [, + p] and p ∈ N ∪ {}. For f ∈ C(I), the space of all continuous functions on I endowed with the norm f = supx∈[,+p] |f (x)| and < q < , Ren and Zeng [] defined the following new version of the q-Bernstein-Schurer operator which preserves the linear functions: ∗ p p˜ n,k (q, x)f Bn f (t); q, x = n+p
k=
[k]q , [n]q
(.)
where p˜ ∗n,k (q, x) =
n+p–k n+p [n]q n+p k [n + p]q x – x . n+p [n]q k q [n + p]q q
Later, Acu [] proposed a q-Durrmeyer modification of the operators (.) as
Dn,p (f ; q, x) =
n+p [n + p + ]q [n]q ∗ p˜ n,k (q, x) [n + p]q
[n+p]q [n]q
f (t)b˜ n,k (q, qt) dq t p
(.)
k=
and discussed the rate of convergence in terms of the modulus of continuity, a Lipschitz class function, and a Voronovskaja-type result. Subsequently, for α, β ∈ R such that ≤ α ≤ β and f ∈ C(I), Agrawal et al. [] introduced a Stancu-type Kantorovich modification of the operators (.), defined as (α,β) Kn,p (f ; q, x) =
n+p k=
p˜ ∗n,k (q, x)
f
[k]q + qk t + α dq t, [n + ]q + β
(.)
and discussed the basic convergence theorem, the rate of convergence involving modulus of continuity and Lipschitz function. Significant contributions have been made by researchers in this area of approximation theory (cf. [] and the references their in). The purpose of this paper is to discuss the Voronoskaja asymptotic result by calculating an estimate of the fourth order central moment for the operators (.) and construct
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the bivariate case of these operators. We obtain the rate of approximation of the bivariate operators by using the complete and partial moduli of continuity and the degree of approximation with the aid of a Lipschitz-type space and the Peetre K -functional. Lastly, we consider the associated GBS (generalized Boolean sum) operators and study the approximation of Bögel continuous and Bögel differentiable functions by means of the mixed modulus of smoothness. Lemma ([]) For the operators given by (.), the following equalities hold: (α,β)
(i) Kn,p (; q, x) = ; q[n]q x+ ; []q ([n+]q +β) [n]q [n+p–]q (α,β) Kn,p (t ; q, x) = [] [] ([n+] +β) { [n+p]q ([]q q q q q []q )}[n]q x + []q α + α[]q + ( + qα )[]q }. (α,β)
(ii) Kn,p (t; q, x) = (iii)
α [n+]q +β
+
+ q )x + {(α + )q[]q + q ( +
(α,β)
Lemma ([]) For m ∈ N ∪ {}, the mth order central moment of Kn,p (f ; q, x) defined as α,β μ∗n,m,q (x) = Kn,p ((t – x)m ; q, x), we have (–[]q )q[n]q x–(β+)[]q x+ (i) μ∗n,,q (x) = + [n+]αq +β ; []q ([n+]q +β) [n]q [n+p–]q ([]q q +q ) q[n] (ii) μ∗n,,q (x) = { [n+p] – []q [n+]qq +β + }x q ([n+]q +β) []q []q [] α +α[] +(+qα )[] α – []q [n+] }x + q([n+] +β)q [] [] q . [n+]q +β q +β q q q
+{
{(α+)[]q q+q (+[]q )}[n]q ([n+]q +β) []q []q
–
In the following we obtain an estimate of the fourth order central moment of the operators defined by (.). By the definition of the Jackson integral and the inequality (a + b) ≤ (a + b ), where a > , b > , and Lemma . in [], we have
(α,β) Kn,p
(t – x) ; qn , x = [n + ]qn
n+p
p˜ ∗n,k (qn , x)
k=
=
n+p
p˜ ∗n,k (qn , x)( – qn )
n+p
p˜ ∗n,k (qn , x)
k=
+ ( – qn )
n+p
n
[n + ]qn + β
[k]qn + α –x [n + ]qn + β
p∗n,k (qn , x)
k=
+
n+p
∞ j=
k=
[k]qn + α [n]qn + β
p˜ ∗n,k (qn , x)
k=
+
∞ j [k]qn + qk qn + α
p˜ ∗n,k (qn , x)
n+p
≤
[k]qn + qnk t + α –x [n + ]qn + β
j=
k=
≤
dqn t
–x
× qnj
qnk [n + ]qn + β
qnn [n + ]qn + β
[k]qn + α –x [n]qn + β
qnj
n+p qnk ∗ ˜ p (q , x) n n,k + qn + qn + qn + qn [n + ]qn k=
n+p
=
k=
p∗n,k (qn , x)
[k]qn + α [n]qn + β
qnn [n + ]qn + β
Chauhan et al. Journal of Inequalities and Applications (2017) 2017:50
p
+ Bn
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[k]qn + α –x [n]qn + β
; q, x
n+p qnk ∗ p˜ n,k (qn , x) + + qn + qn + qn + qn [n + ]qn k=
+ M + / ≤ + M + = . [n]qn [n]qn [n]qn [n]qn
(.)
In the following, let (qn )n , < qn < be a sequence satisfying limn→∞ qn = and limn→∞ qnn = a ( ≤ a < ).
2 Voronovskaja-type theorem Let C [, + p] denote the space of twice continuously differentiable functions on [, + p]. Theorem For any f ∈ C [, + p],
(α,β) lim [n]qn Kn,p (f ; qn , x) – f (x) n→∞
=
–x(a + + β) +α+ f (x) – x f (x)
uniformly in [, ]. Proof Using Taylor’s expansion for f , we obtain f (t) = f (x) + f (x)(t – x) +
f (x)(t – x) + ξ (t, x)(t – x) ,
(.)
where the function ξ (t, x) is the Peano form of the remainder, ξ (t, x) ∈ C[, + p], and limt→x ξ (t, x) = . (α,β) By linearity of the operators Kn,p (; qn , x) and using Lemma , we get
(α,β) (f ; qn , x) – f (x) lim [n]qn Kn,p n→∞
–x(a + + β) +α+ f (x) – x f (x) = (α,β) ξ (t, x)(t – x) ; qn , x + lim [n]qn Kn,p
n→∞
(.)
uniformly in [, ]. For the last term of the right side, using the Cauchy-Schwarz inequality, we are led to
(α,β) (α,β) (α,β) [n]qn Kn,p ξ (t, x)(t – x) ; qn , x ≤ [n]qn Kn,p ξ (t, x); qn , x Kn,p (t – x) ; qn , x . We observe that ξ (t, x) ∈ C[, + p] and ξ (x, x) = , hence, by Theorem (α,β) ξ (t, x); qn , x = ξ (x, x) = , lim Kn,p
n→∞
uniformly with respect to x ∈ [, ].
(α,β) Further using (.), limn→∞ [n]qn Kn,p ((t – x) ; qn , x) is finite. Hence, (α,β) ξ (t, x)(t – x) ; qn , x = lim [n]qn Kn,p
n→∞
uniformly in x ∈ [, ]. Finally, consideration of (.) and (.) completes the proof.
(.)
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(α ,β )
Figure 1 The convergence of Kn,p (f ; q, x) to f (x).
(α ,β )
Figure 2 The convergence of Kn,p (f ; q, x) to f (x).
In the following examples, we illustrate the rate of convergence of the operators given by (.) to certain functions. Example Let qn = (n – )/n. For α = ., β = ., p = with n = and , the conver(α,β) gence of Kn,p (f ; q, x) given by (.) to f (x) = x + sin(πx/) is shown in Figure . It is observed that the approximation becomes better on increasing the value of n. Example Let f (x) = arctan(x ), p = ., n = , q = . and n = , q = .. For (α,β) α = β = , α = ., β = . and α = , β = the convergence of Kn,p (f ; q, x) to f (x) is shown in Figures and respectively. It is observed that the approximation becomes better when the values of α, β ∈ [, ) and the convergence is better in a small interval for larger values of α, β.
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(α ,β )
Figure 3 The convergence of Kn,p (f ; q, x) to f (x).
3 Construction of the bivariate operators Let C(I × I ), where I = [, + p ] and I = [, + p ], denote the space of all real valued continuous functions on I × I endowed with the norm f C(I ×I ) =
sup f (x, y) .
(x,y)∈I ×I
For f ∈ C(I × I ), < q , q < and J = [, ], the bivariate generalization of the operators given by (.) is defined as Kn(α,n,α,p,β ,p,β ) f (t, s); q , q , x, y
n +p n +p
=
p˜ ∗n ,n ,k ,k (q , q ; x, y)
k = k =
×
α ,β α ,β f n,k,q (t), n,k,q (s) dq t dq s,
(.)
where p˜ ∗n ,n ,k ,k (q , q , x, y) n +p –k
n +p [n ]q n + p k [n + p ]q x –x = n +p k [n ]q [n + p ]q q q
n +p –k n +p [n ]q n + p k [n + p ]q × y – y , n +p k [n ]q [n + p ]q q q α ,β
n,k,q (t) =
[k ]q + qk t + α , [n + ] + β
α ,β
n,k,q (s) =
x, y ∈ J
and
[k ]q + qk s + α . [n + ] + β
Lemma let eij (t, s) = t i sj , (t, s) ∈ (I × I ), (i, j) ∈ N × N with i + j ≤ be the two dimensional test functions. Then the following equalities hold for the operators (.):
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(α ,α ,β ,β )
(i) Kn,n,p ,p (e ; q , q , x, y) = ; (α ,α ,β ,β ) (ii) Kn,n,p ,p (e ; q , q , x, y) = [n +]αq (iii) (iv) (v)
q [n ]q x+ ; []q ([n +]q +β ) q [n ]q y+ (α ,α ,β ,β ) Kn ,n ,p ,p (e ; q , q , x, y) = + []q ([n +]q +β ) ; [n ]q [n +p –]q (α ,α ,β ,β ) Kn ,n ,p ,p (e ; q , q , x, y) = [] [] ([n +] +β ) { [n +p ]q ([]q q + q )x + q q q {(α + )q []q + q ( + []q )}[n ]q x + []q α + α []q + ( + q α )[]q }. [n ] [n +p –]q Kn(α,n,α,p,β ,p,β ) (e ; q , q , x, y) = [] [] ([n +] +β ) { q[n +p ]q ([]q q + q )y + q q q {(α + )q []q + q ( + []q )}[n ]q y + []q α + α []q + ( + q α )[]q }. +β α [n +]q +β
(α ,α ,β ,β )
+
(α ,β )
(α ,β )
Proof We have Kn,n,p ,p (t i sj ; q , q , x, y) = Kn,p (t i ; q , x)Kn,p (sj ; q , y), for ≤ i, j ≤ . By using Lemma , the proof of the lemma is straightforward. Hence the details are omitted. For f ∈ C(I × I ) and δ > , the first order complete modulus of continuity for the bivariate case is defined as follows: ω(f ; δ , δ ) = sup f (t, s) – f (x, y) : |t – x| ≤ δ , |s – y| ≤ δ , where δ , δ > . Further ω(f ; δ , δ ) satisfies the following properties: (a) ω(f ; δ , δ ) → if δ → and δ → , )( + |s–y| ). (b) |f (t, s) – f (x, y)| ≤ ω(f ; δ , δ )( + |t–x| δ δ Now, we give an estimate of the rate of convergence of the bivariate operators. In the n following, let < qni < be sequences in (, ) such that qni → and qnii → ai ( ≤ ai < ), (α ,β ) (α ,β ) as ni → ∞ for i = , . Further, let δn (x) = Kn ,p ((t – x) ; qn , x) and δn (y) = Kn,p ((s – y) ; qn , y). Theorem For f ∈ C(I × I ) and all (x, y) ∈ J , we have
(α ,α ,β ,β )
K (f ; qn , qn , x, y) – f (x, y) ≤ ω f ; δn (x), δn (y) . n ,n ,p ,p (α ,α ,β ,β )
Proof Since Kn,n,p ,p (f ; qn , qn , x, y) is a linear positive operator, by the property (b) of bivariate modulus of continuity, Lemma , and the Cauchy-Schwarz inequality (α ,α ,β ,β ) K f (t, s); qn , qn , x, y – f (x, y) n ,n ,p ,p ≤ Kn(α,n,α,p,β ,p,β ) f (t, s) – f (x, y) ; qn , qn , x, y
(α ,β ) (α ,β ) ≤ ω f ; δn (x), δn (y) Kn ,p (; qn , x) + Kn ,p |t – x|; qn , x δn (x) Kn(α,p,β ) |s – y|; qn , y × Kn(α,p,β ) (; qn , y) + δn (y)
Kn(α,p,β ) (t – x) ; qn , x ≤ ω f ; δn (x), δn (y) + δn (x)
× + Kn(α,p,β ) (s – y) ; qn , y , δn (y) we get the desired result.
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Theorem If f (x, y) has continuous partial derivatives
∂f ∂x
and
∂f , ∂y
then the inequality
(α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p
≤ M λn (x) + ω fx , δn (x) + δn (x)
+ M λn (y) + ω fy , δn (y) + δn (y) , where M , M are the positive constants such that ∂f ≤ M , ∂x
∂f ≤ M ∂y
( ≤ x ≤ a, ≤ y ≤ b)
and ( – []qn )qn [n ]qn – (β + )[]qn ( + []qn α ) x + λn (x) = ; []qn ([n + ]qn + β ) [n + ]qn + β ( – []qn )qn [n ]qn – (β + )[]qn ( + []qn α ) y + . λn (y) = []qn ([n + ]qn + β ) []qn ([n + ]qn + β ) Proof From the mean value theorem we have f (t, s) – f (x, y) = f (t, y) – f (x, y) + f (t, s) – f (t, y) ∂f (ξ , y) ∂f (x, ξ ) + (s – y) ∂x ∂y ∂f (x, y) ∂f (x, y) ∂f (ξ , y) ∂f (x, y) + (t – x) – + (s – y) = (t – x) ∂x ∂x ∂x ∂y ∂f (x, ξ ) ∂f (x, y) – , + (s – y) ∂y ∂y = (t – x)
(.)
where x < ξ < t and y < ξ < s. Since ∂f (ξ , y) ∂f (x, y) ≤ ω f ; |t – x| ≤ + |t – x| ω f , δn and – x x ∂x ∂x δn ∂f (x, ξ ) ∂f (x, y) |s – y| ω fy , δn ∂y – ∂y ≤ ω fy ; |s – y| ≤ + δ n (α ,α ,β ,β )
for some δn , δn > , on applying the operator Kn,n,p ,p (·; qn , qn , x, y) on both sides of (.), we have (α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p ≤ M Kn(α,p,β ) (e – x; qn , x)
n +p n +p
+
p˜ ∗n ,n ,k ,k (qn , qn ; x, y)
k = k =
×
α ,β |n ,k ,qn (t) – x| ω f , δn (t) – x + dqn t dqn s x n ,k ,qn δn
α ,β
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+ M Kn(α,p,β,q)n (e – y; y)
n +p n +p
+
p˜ ∗n ,n ,k ,k (qn , qn ; x, y)
k = k =
×
α ,β |n ,k ,qn (s) – y| ω f (s) – y , δ + dqn t dqn s. n y n ,k ,qn δn
α ,β
Now applying the Cauchy-Schwarz inequality (α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p ≤ M Kn(α,p,β ) (e ; qn , x) n +p n +p p˜ ∗n ,n ,k ,k (qn , qn ; x, y) + ω fx , δn k = k =
×
+
×
α ,β n,k,qn (t) – x dqn t dqn s
n +p n +p ω(fx , δn ) p˜ ∗n ,n ,k ,k (qn , qn ; x, y) δn
k = k =
α ,β n,k,qn (t) – x dqn t dqn s
+ M Kn(α,p,β ) (e ; qn , y) n +p n +p p˜ ∗n ,n ,k ,k (qn , qn ; x, y) + ω fy , δn k = k =
×
+
×
α ,β n,k,qn (s) – y dqn t dqn s
+p n +p ω(fy , δn ) n
δn
p˜ ∗n ,n ,k ,k (qn , qn ; x, y)
k = k =
α ,β n,k,qn (s) – y dqn t dqn s
= M λn (x) + ω fx , δn ( + δn ) + M λn (y) + ω fy , δn ( + δn ), on choosing δn = δn (x) and δn = δn (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 < ξ ≤ and < ξ ≤ , we define the Lipschitz class LipM (ξ , ξ ) for the bivariate case as follows: f (t, s) – f (x, y) ≤ M|t – x|ξ |s – y|ξ , where (t, s), (x, y) ∈ (I × I ) are arbitrary.
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Theorem Let f ∈ LipM (ξ , ξ ). Then, for all (x, y) ∈ J , we have (α ,α ,β ,β ) ξ ξ K (f ; qn , qn , x, y) – f (x, y) ≤ M δn (x) δn (y) . n ,n ,p ,p Proof By our hypothesis, we can write (α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p ≤ Kn(α,n,α,p,β ,p,β ) f (t, s) – f (x, y) ; qn , qn , x, y ≤ MKn(α,n,α,p,β ,p,β ) |t – x|ξ |s – y|ξ ; qn , qn , x, y = M Kn(α,p,β ) |t – x|ξ ; qn , x Kn(α,p,β ) |s – y|ξ ; qn , y . Now, applying the Hölder’s inequality with u = respectively, we have
, ξ
v =
–ξ
and u =
ξ
and v =
, –ξ
(α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x) n ,n ,p ,p –ξ ξ ≤ M Kn(α,p,β ) (t – x) ; qn , x Kn(α,p,β ) (; qn , x) –ξ ξ × Kn(α,p,β ) (s – y) ; qn , y Knα,p,β (; qn , y)
ξ ξ = M δn (x) δn (y) .
Hence, the proof is completed.
Let C (I × I ) denote the space of all continuous functions on I × I such that their first partial derivatives are continuous on I × I . Theorem For f ∈ C (I × I ) and (x, y) ∈ J we have (α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) ≤ f n ,n ,p ,p x C(I
×I )
δn (x) + fy C(I
×I )
δn (y).
Proof Let (x, y) ∈ J be a fixed point. Then by our hypothesis
t
f (t, s) – f (x, y) = x
fu (u, s) dq u +
y
s
fv (x, v) dq v.
(α ,α ,β ,β )
Now, operating by Kn,n,p ,p (·; qn , qn , x, y) on both sides of the above equation, we are led to (α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p x (α ,α ,β ,β ) fu (u, s) dq u ; qn , qn , x, y ≤ Kn ,n ,p ,p t
s + Kn(α,n,α,p,β ,p,β ) fv (x, v) dq v ; qn , qn , x, y . y
Chauhan et al. Journal of Inequalities and Applications (2017) 2017:50
Since |
x t
Page 11 of 24
|fu (u, s)| dq u| ≤ fx C(I ×I ) |t – x| and |
s y
|fv (x, v)| dq v| ≤ fy C(I ×I ) |s – y|, we have
(α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p K(α ,β ) |t – x|; qn , x + f ≤ f x C(I ×I )
n ,p
(α ,β ) y C(I ×I ) Kn ,p
|s – y|; qn , y .
Applying the Cauchy-Schwarz inequality and Lemma , we have (α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p
(α ,β ) (α ,β ) ≤ fx C(I ×I ) Kn,p (t – x) ; qn , x Kn,p (; qn , x)
(α ,β ) (α ,β ) + fy C(I ×I ) Kn ,p (s – y) ; qn , y Kn,p (; qn , y)
= fx C(I ×I ) δn (x) + fy C(I ×I ) δn (y).
This completes the proof of the theorem.
For f ∈ C(I × I ) and δ > , the partial moduli of continuity with respect to x and y are given by ω¯ (f ; δ) = sup f (x , y) – f (x , y) : y ∈ I and |x – x | ≤ δ and ω¯ (f ; δ) = sup f (x, y ) – f (x, y ) : x ∈ I and |y – y | ≤ δ . Clearly, both moduli of continuity satisfy the properties of the usual modulus of continuity. Theorem If f ∈ C(I × I ) and (x, y) ∈ J , then we have
|Kn(α,n,α,p,β ,p,β ) (f ; qn , qn , x, y) – f (x, y)| ≤ ω¯ f ; δn (x) + ω¯ f ; δn (y) . Proof Using the definition of partial moduli of continuity, Lemma , and the CauchySchwarz inequality, we have (α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p ≤ Kn(α,n,α,p,β ,p,β ) f (t, s) – f (x, y) ; qn , qn , x, y ≤ Kn(α,n,α,p,β ,p,β ) f (t, s) – f (t, y) ; qn , qn , x, y + Kn(α,n,α,p,β ,p,β ) f (t, y) – f (x, y) ; qn , qn , x, y
(α ,β ) (α ,β ) Kn ,p |t – x|; qn , x ≤ ω¯ f ; δn (x) Kn ,p (; qn , x) + δn (x)
(α ,β ) (α ,β ) Kn ,p |s – y|; qn , y + ω¯ f ; δn (y) Kn ,p (; qn , y) + δn (y)
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(α ,β ) ≤ ω¯ f ; δn (x) + Kn ,p (t – x) ; qn , x δn (x)
(α ,β ) + ω¯ f ; δn (y) + Kn ,p (s – y) ; qn , y , δn (y)
from which the required result is straightforward.
Let C (I × I ) be the space of all functions f ∈ C(I × I ) such that second partial derivatives of f belong to C(I × I ). The norm on the space C (I × I ) is defined as f C (I ×I ) = f +
i i ∂ f ∂ f ∂xi + ∂yi . i=
The Peetre K -functional of the function f ∈ C(I × I ) is defined as
K(f ; δ) =
inf
g∈C (I ×I )
f – gC(I ×I ) + δgC (I ×I ) ,
δ > .
Also by [], it follows that √ K(f ; δ) ≤ M ω˜ (f ; δ) + min(, δ)f C(I ×I )
(.)
holds for all δ > . √ The constant M in the above inequality is independent of δ and f and ω˜ (f ; δ) is the second order modulus of continuity. Theorem For the function f ∈ C(I × I ), we have the following inequality: (α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p
(p ,p ) ≤ M ω˜ f ; An,n (qn , qn , x, y) + min , An(p,n,p ) (qn , qn , x, y) f C(I ×I )
(p ,p ) + ω f ; Bn,n (qn , qn , x, y) , where An(p,n,p ) (qn , qn , x, y) = δn (x) + δn (y) + +
α [n + ]qn
qn [n ]qn x + α –x + [n + ]qn + β []qn ([n + ]qn + β ) qn [n ]qn y + –y + + β []qn ([n + ]qn + β )
and
Bn(p,n,p ) (qn , qn , x, y)
qn [n ]qn x + α = + –x [n + ]qn + β []qn ([n + ]qn + β ) qn [n ]qn y + α –y , + + [n + ]qn + β []qn ([n + ]qn + β ) (p ,p )
and the constant M (> ), is independent of f and An,n (qn , qn , x, y).
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Proof We define the auxiliary operators as follows: ,α ,β ,β ) L(α n ,n ,p ,p (f ; qn , qn , x, y)
= Kn(α,n,α,p,β ,p,β ) (f ; qn , qn , x, y) qn [n ]qn x + α –f , + [n + ]qn + β []qn ([n + ]qn + β ) qn [n ]qn y + α + + f (x, y). [n + ]qn + β []qn ([n + ]qn + β ) (α ,α ,β ,β )
(.) (α ,α ,β ,β )
Considering Lemma , one has Ln,n,p ,p (; qn , qn , x, y) = , Ln,n,p ,p ((t – x); qn , qn , (α ,α ,β ,β ) x, y) = , and Ln,n,p ,p ((s – y); qn , qn , x, y) = . Let g ∈ C (I × I ) and (x, y) ∈ J . Using Taylor’s theorem, we may write g(t, s) – g(x, y) = g(t, y) – g(x, y) + g(t, s) – g(t, y) t ∂ g(u, y) ∂g(x, y) (t – x) + (t – u) du = ∂x ∂u x s ∂g(x, y) ∂ g(x, v) + (s – y) + (s – v) dv. ∂y ∂v y (α ,α ,β ,β )
Applying the operator Ln,n,p ,p (·; qn , qn , x, y) on the above equation and using (.), we are led to ,α ,β ,β ) L(α n ,n ,p ,p (g; qn , qn , x, y) – g(x, y) t ∂ g(u, y) (α ,α ,β ,β ) (t – u) du; qn , qn , x, y = Ln ,n ,p ,p ∂u x s ∂ g(x, v) ,α ,β ,β ) (s – v) dv; q , q , x, y + L(α n n n ,n ,p ,p ∂v y t ∂ g(u, y) (t – u) du; q , q , x, y = Kn(α,n,α,p,β ,p,β ) n n ∂u x
qn [n ]qn x + α –u + – [n + ]qn + β []qn ([n + ]qn + β ) x s ∂ g(x, v) ∂ g(u, y) (α ,α ,β ,β ) du + Kn ,n ,p ,p (s – v) dv; qn , qn , x, y × ∂u ∂v y
qn [n ]qn x+ α [n +]q +β + []qn ([n +]qn +β )
qn [n ]qn y+ α [n +]qn +β + []qn ([n +]qn +β )
– y
×
qn [n ]qn y + α –v + [n + ]qn + β []qn ([n + ]qn + β )
∂ g(x, v) dv. ∂v
Hence, (α ,α ,β ,β ) L (g; qn , qn , x, y) – g(x, y) n ,n ,p ,p t ∂ g(u, y) du ; qn , qn , x, y ≤ Kn(α,n,α,p,β ,p,β ) (t – u) ∂u x
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qn [n ]qn x + α – u + [n + ] + β []qn ([n + ]qn + β ) qn x s ∂ g(u, y) du + K(α ,α ,β ,β ) (s – v) ∂ g(x, v) dv ; qn , qn , x, y × n ,n ,p ,p ∂v ∂u y
+
+
qn [n ]qn x+ α [n +]qn +β + []qn ([n +]qn +β )
qn [n ]qn y+ α [n +]qn +β + []qn ([n +]qn +β )
y
∂ g(x, v) dv × ∂v
qn [n ]qn y + α [n + ] + β + [] ([n + ] + β ) – v qn qn qn
= Apn ,p ,n (qn , qn , x, y)gC (I ×I ) .
(.)
Also, (α ,α ,β ,β ) L (f ; qn , qn , x, y) n ,n ,p ,p
(α ,α ,β ,β ) ≤ Kn ,n ,p ,p (f ; qn , qn , x, y) + f α [n + ]qn
qn [n ]qn x + α , + [n + ]qn + β []qn ([n + ]qn + β ) qn [n ]qn y + + f (x, y) + + β []qn ([n + ]qn + β )
≤ f C(I ×I ) .
(.)
Hence, considering (.), (.), and (.) (in that order), (α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p n ,n ,p ,p = L(α ,α ,β ,β ) (f ; qn , qn , x, y) – f (x, y) + f
qn [n ]qn x + α , + [n + ]qn + β []qn ([n + ]qn + β ) qn [n ]qn y + α + – f (x, y) [n + ]qn + β []qn ([n + ]qn + β ) n ,n ,p ,p n ,n ,p ,p ≤ L(α ,α ,β ,β ) (f – g; qn , qn , x, y) + L(α ,α ,β ,β ) (g; qn , qn , x, y) – g(x, y) + g(x, y) – f (x, y) qn [n ]qn x + α , + + f [n + ]qn + β []qn ([n + ]qn + β ) qn [n ]qn y + α – f (x, y) + [n + ]qn + β []qn ([n + ]qn + β ) n ,n ,p ,p ≤ f – gC(I ×I ) + K (g; qn , qn , x, y) – g(x, y) + f
(α ,α ,β ,β )
qn [n ]qn x + α , + [n + ]qn + β []qn ([n + ]qn + β ) q [n ]q y + α – f (x, y) + [n + ]q + β []q ([n + ]q + β ) ≤ f – gC(I ×I ) + Apn ,p ,n (qn , qn , x, y)gC (I ×I )
(p ,p ) + ω f ; Bn,n (qn , qn , x, y) .
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Now, taking the infimum on the right hand side all over g ∈ C (I × I ) and using (.) (α ,α ,β ,β ) K (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p
(p ,p ) ,p ) (q , q , x, y) + ω f ; ≤ K f ; A(p B (q , q , x, y) ,n n n n n n n ,n
p ,p ,p ) ≤ M ω˜ f ; An ,n (qn , qn , x, y) + min , A(p n ,n (qn , qn , x, y) f C(I ×I )
(p ,p ) + ω f ; Bn,n (qn , qn , x, y) . Thus, we get the desired result.
Theorem Let f ∈ C (I × I ). Then for every (x, y) ∈ J , (α ,α ,β ,β ) f (t, s); q , x, y – f (x, y) lim [n]qn Kn,n,p n ,p
[n]qn →∞
–x(a + + β ) –y(a + + β ) + α + + fy (x, y) + α + x( – x) y( – y) fxx (x, y) + fyy (x, y) +
= fx (x, y)
uniformly in (x, y) ∈ J . Proof By Taylor’s formula for f , we have f (t, s) = f (x, y) + fx (x, y)(t – x) + fy (x, y)(s – y) fxx (x, y)(t – x) + fxy (x, y)(t – x)(s – y) + fyy (x, y)(s – y) + ξ (t, s, x, y) (t – x) + (s – y) , +
where ξ (t, s, x, y) → as (t, s) → (x, y) and ξ (t, s, x, y) ∈ C (I × I ). Now, applying the op(α ,α ,β ,β ) (·; qn , x, y) on the above equation, we get erator Kn,n,p ,p (α ,α ,β ,β ) f (t, s); qn , x, y Kn,n,p ,p (α ,β ) (α ,β ) (t – x); qn , x + fy (x, y)Kn,p (s – y); qn , y = f (x, y) + fx (x, y)Kn,p (α ,β ) (α ,β ) (t – x) ; qn , x + fxy (x, y)Kn,p (t – x); qn , x fxx (x, y)Kn,p α ,β α ,β (s – y); qn , y + fyy Kn,p (s – y) ; qn , y × Kn,p (α ,α ,β ,β ) ξ (t, s, x, y) (t – x) + (s – y) ; qn , x, y . + Kn,n,p ,p
+
Hence, using Lemma , (α ,α ,β ,β ) f (t, s); q , x, y – f (x, y) lim [n]qn Kn,n,p n ,p
[n]qn →∞
= fx (x, y)
–x(a + + β ) –y(a + + β ) + α + + fy (x, y) + α +
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x( – x) y( – y) + fxx (x, y) + fyy (x, y) (α ,α ,β ,β ) ξ (t, s, x, y) (t – x) + (s – y) ; x, y + lim [n]qn Kn,n,p ,p [n]qn →∞
uniformly in (x, y) ∈ J . Applying the Cauchy-Schwarz inequality (α ,α ,β ,β ) K ξ (t, s) (t – x) + (s – y) ; qn , x, y n,n,p ,p
(α ,α ,β ,β ) α ,α ,β ,β ξ (t – x) + (s – y) ; qn , x, y . (t, s); q , x, y Kn,n,p ,p ≤ Kn,n,p n ,p Since, by Theorem and in view of Lemma , (α ,α ,β ,β ) lim Kn,n,p ξ (t, s); x, y = ξ (x, y) = , ,p
[n]qn →∞
, [n]qn (α ,α ,β ,β ) (s – y) Kn,p ; q , y = O n [n]qn (α ,α ,β ,β ) (t – x) ; qn , x = O Kn,n,p ,p
and
uniformly in (x, y) ∈ J , it follows that (α ,α ,β ,β ) ξ (t, s) (t – x) + (s – y) ; q , x, y lim [n]qn Kn,n,p = n ,p
n→∞
uniformly in (x, y) ∈ J , the desired result is obtained.
In the following example, the rate of convergence of the bivariate operators given by (.) to a certain function is shown by illustrative graphics. We observe that when the val(α ,α ,β ,β ) ues of q and q increase, the approximation of f by the operator Kn,n,p p (f ; q , q , x, y) becomes better. Example Let n = n = , α = ., β = ., α = ., β = ., p = p = . For q = ., q = . (green) and q = ., q = . (pink), the convergence of the operators Kn(α,n,α,p,β p,β ) (f ; q , q , x, y) given by (.) to f (x, y) = sin(x + y)/( + xy) (yellow) is illustrated in Figure .
4 Construction of GBS operator of q-Bernstein-Schurer-Kantorovich type In [] and [], Bögel proposed the concepts of B-continuous and B-differentiable functions. Later, Dobrescu and Matei [] discussed the approximation of B-continuous functions on a bounded interval by a generalized Boolean sum of bivariate generalization of Bernstein polynomials. Subsequently, Badea and Cottin [] established Korovkin theorems for GBS operators. Pop [] studied the GBS operators associated to a certain class of linear and positive operators defined by an infinite sum and discussed the approximation of B-continuous and B-differentiable functions by these operators. Recently, Sidharth et al. [] proposed the GBS operators of q-Bernstein-Schurer-Kantorovich type and studied the rate of convergence of these operators by means of the mixed modulus of smoothness. Agrawal and Ispir [] introduced the bivariate generalization of Chlodowsky-SzaszCharlier-type operators and obtained the degree of approximation for the associated GBS
Chauhan et al. Journal of Inequalities and Applications (2017) 2017:50
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(α ,α ,β ,β )
1 2 (f ; q , Figure 4 The convergence of Kn11,n2 2,p1 ,p 1 2 q2 , x, y) to f (x, y).
operators. In this section, we give some basic definitions and notations, for further details, one can see []. Let X and Y be compact subsets of R. A function f : X ×Y −→ R is called a B-continuous (Bögel continuous) function at (x , y ) ∈ X × Y if lim
(x,y)→(x ,y )
f (x , y ); (x, y) = ,
where f [(x , y ); (x, y)] denotes the mixed difference defined by f (x , y ); (x, y) = f (x, y) – f (x, y ) – f (x , y) + f (x , y ).
(.)
The function f : X × Y → R is called B-bounded on X × Y if there exists M > such that | f [(t, s); (x, y)]| ≤ M, for every (x, y), (t, s) ∈ (X × Y ). Since X × Y is a compact subset of R , each B-continuous function is a B-bounded function on X × Y → R. Throughout this paper, Bb (X × Y ) denotes all B-bounded functions on X × Y → R, equipped with the norm f B = sup(x,y),(t,s)∈X×Y | f [(t, s); (x, y)]|. We denote by Cb (X × Y ), the space of all B-continuous functions on X × Y . B(X × Y ), C(X × Y ) denote the space of all bounded functions and the space of all continuous (in the usual sense) functions on X × Y endowed with the sup-norm · ∞ . It is well known that C(X × Y ) ⊂ Cb (X × Y ) ([], p.). A function f : X × Y −→ R is called a B-differentiable (Bögel differentiable) function at (x , y ) ∈ X × Y if the limit lim
(x,y)→(x ,y )
f [(x , y ); (x, y)] (x – x )(y – y )
exists and is finite. The limit is said to be the B-differential of f at the point (x , y ) and is denoted by DB (f ; x , y ) and the space of all B-differentiable functions is denoted by Db (X × Y ). The mixed modulus of smoothness of f ∈ Cb (I × I ) is defined as ωmixed (f ; δ , δ ) := sup f (t, s); (x, y) : |x – t| < δ , |y – s| < δ
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for all (x, y), (t, s) ∈ (I × I ) and for any (δ , δ ) ∈ (, ∞) × (, ∞) with ωmixed : [, ∞) × [, ∞) → R. The basic properties of ωmixed were obtained by Badea et al. in [] and [], which are similar to the properties of the usual modulus of continuity. (α ,α ,β ,β ) We define the GBS operator of the operator Kn,n,p ,p given by (.), for any f ∈ Cb (I × I ) and m, n ∈ N, by ,α ,β ,β ) f (t, s); q , q , x, y Tn(α ,n n n ,p ,p := Kn(α,n,α,p,β ,p,β ) f (t, y) + f (x, s) – f (t, s); qn , qn , x, y
(.)
for all (x, y) ∈ J . Hence for any f ∈ Cb (I × I ), the GBS operator of the q-Bernstein-Schurer-Kantorovich type is ,α ,β ,β ) : C (I × I ) − Tn(α ,n b → C(I × I ) ,p ,p
given by ,α ,β ,β ) (f ; q , q , x, y) Tn(α ,n n n ,p ,p
n +p n +p
=
p˜ ∗n ,n ,k ,k (qn , qn ; x, y)
k = k =
[k ]qn + qnk s + α , y + f x, [n + ]qn + β [n + ]qn + β [k ]qn + qnk t + α [k ]qn + qnk s + α –f , dqn t dqn s, [n + ]qn + β [n + ]qn + β ×
f
[k ]qn + qnk t + α
where p˜ ∗n ,n ,k ,k (qn , qn , x, y)
n +p
=
[n ]qn
n +p
[n + p ]qn ×
n + p k
n +p [n ]qn n +p [n + p ]qn
x qn
n + p k
k
[n + p ]qn [n ]qn
k
y qn
n +p –k –x
[n + p ]qn [n ]qn
qn
n +p –k –y . qn
(α ,α ,β ,β )
is linear and preserves linear functions. Clearly, the operator Tn ,n ,p ,p
Theorem For every f ∈ Cb (I × I ), at each point (x, y) ∈ J , the operator (.) verifies the following inequality:
(α ,α ,β ,β )
T (f ; qn , qn , x, y) – f (x, y) ≤ ωmixed f ; δn (x), δn (y) . n ,n ,p ,p Proof By the property ωmixed (f ; λ δ , λ δ ) ≤ ( + λ )( + λ )ωmixed (f , δ , δ );
λ , λ > ,
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we can write f (t, s); (x, y) ≤ ωmixed f ; |t – x|, |s – y| |s – y| |t – x| + ωmixed (f ; δ , δ ) ≤ + δ δ
(.)
for every (t, s) ∈ (I × I ), (x, y) ∈ J and any δ , δ > . From (.) and the definition of the mixed difference f [(t, s); (x, y)], on applying Lemma and the inequality (.), we get (α ,α ,β ,β ) T (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p ≤ Kn(α,n,α,p,β ,p,β ) f (t, s); (x, y) ; qn , qn , x, y ≤ Kn(α,p,β ) (; qn , x) + Kn(α,p,β ) |t – x|; qn , x δn + Kn(α,p,β ) |s – y|; qn , y + Kn(α,p,β ) |t – x|; qn , x δn δn δn × Kn(α,p,β ) |s – y|; qn , y ωmixed (f ; δn , δn ). Now, applying the Cauchy-Schwarz inequality (α ,α ,β ,β ) T (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p
≤ Kn(α,p,β ) (e ; qn , x) + Kn(α,p,β ) (t – x) ; qn , x δn
Kn(α,p,β ) (s – y) ; qn , y + Kn(α,p,β ) (t – x) ; qn , x + δn δn δn
(α ,β ) × Kn,p (s – y) ; qn , y ωmixed (f , δn , δn ) = ωmixed (f ; δn , δn ), on choosing δn = δn (x) and δn = δn (y). This completes the proof.
Next, let us define the Lipschitz class for B-continuous functions. For f ∈ Cb (I × I ), the Lipschitz class LipM (ξ , η) with ξ , η ∈ (, ] is defined by LipM (ξ , η) = f ∈ Cb (I × I ) : f (t, s); (x, y) ≤ M|t – x|ξ |s – y|η , for (t, s), (x, y) ∈ I × I . (α ,β ,α ,β )
In our next result, we determine the degree of approximation for the operators Tn ,n ,p ,p by means of the class LipM (ξ , η) of the class of Bögel continuous functions.
Theorem For f ∈ LipM (ξ , η), we have (α ,α ,β ,β ) ξ η T (f ; qn , qn , x, y) – f (x, y) ≤ M δn (x) δn (y) n ,n ,p ,p for M > , ξ , η ∈ (, ].
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Proof From (.), (.), and by our hypothesis, we may write (α ,α ,β ,β ) T (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p ≤ Kn(α,n,α,p,β ,p,β ) f (t, s); (x, y) ; x, y ≤ MKn(α,n,α,p,β ,p,β ) |t – x|ξ |s – y|η ; x, y (α ,β ) = MKn(α,p,β ) |t – x|ξ ; x K n,p |s – y|η ; y . Applying Hölder’s inequality with p = /ξ , q = /( – ξ ) and p = /η, q = /( – η), we are led to (α ,β ,α ,β ) T (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p ξ / (α ,β ) ≤ M Kn(α,p,β ) (t – x) ; x Kn,p (e ; x)(–ξ )/ η/ (α ,β ) × Kn(α,p,β ) (s – y) ; y Kn,p (e ; y)(–η)/ . In view of Lemma , the desired result is immediate. Theorem For f ∈ Db (I × I ) with DB f ∈ B(I × I ) and each (x, y) ∈ J , we have (α ,β ,α ,β ) T (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p ≤
M –/ DB f ∞ + ωmixed DB f ; [n ]–/ . qn , [n ]qn / [n ]/ [n ] qn qn
Proof By our hypothesis, using the relations f (t, s); (x, y) = (t – x)(s – y)DB f (ξ , η),
where x < ξ < t; y < η < s,
and DB f (ξ , η) = DB f (ξ , η) + DB f (ξ , y) + DB f (x, η) – DB f (x, y), we obtain (α ,α ,β ,β ) K f (t, s); (x, y) ; qn , qn , x, y n ,n ,p ,p = Kn(α,n,α,p,β ,p,β ) (t – x)(s – y)DB f (ξ , η); qn , qn , x, y ≤ Kn(α,n,α,p,β ,p,β ) |t – x||s – y| DB f (ξ , η) ; x, y + Kn(α,n,α,p,β ,p,β ) |t – x||s – y| DB f (ξ , y) + DB f (x, η) + DB f (x, y) ; qn , qn , x, y ≤ Kn(α,n,α,p,β ,p,β ) |t – x||s – y|ωmixed DB f ; |ξ – x|, |η – y| ; x, y + DB f ∞ Kn(α,n,α,p,β ,p,β ) |t – x||s – y|; qn , qn , x, y .
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Hence taking into account and applying the Cauchy-Schwarz inequality we obtain ωmixed
|t – x| |s – y| DB f ; |ξ – x|, |η – y| ≤ + + ωmixed (DB f ; δn , δn ). δn δn
We have (α ,α ,β ,β ) T (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p
(α ,α ,β ,β ) ≤ DB f ∞ Kn,n,p ,p (t – x) (s – y) ; qn , qn , x, y
Kn(α,n,α,p,β ,p,β ) (t – x) (s – y) ; qn , qn , x, y +
(α ,α ,β ,β ) – + δn Kn,n,p ,p (t – x) (s – y) ; qn , qn , x, y
(α ,α ,β ,β ) + δn– Kn,n,p ,p (t – x) (s – y) ; qn , qn , x, y + δn– δn– Kn(α,n,α,p,β ,p,β ) (t – x) (s – y) ; qn , qn , x, y ωmixed (DB f ; δn , δn ).
(.)
From Lemma , we observe that for (t, s) ∈ (I × I ), (x, y) ∈ J and i, j = , , ,α ,β ,β ) (t – x)i (s – y)j ; q , q , x, y Kn(α ,n n n ,p ,p ,α ,β ,β ) (t – x)i ; q , x, y K (α ,α ,β ,β ) (s – y)j ; q , x, y . = Kn(α ,n n n n ,n ,p ,p ,p ,p = Kn(α ,p ,β ) (t – x)i ; qn , x Kn(α ,p,β ) (s – y)j ; qn , y ≤
M M i [n ]qn [n ]jqn
for some constants M , M > . Now, let δn = / and δn = [n ]qn
[n ]/ qn
,
(α ,α ,β ,β ) T (f ; qn , qn , x, y) – f (x, y) n ,n ,p ,p O = DB ∞ O [n ]/ [n ]/ qn qn –/ O ωmixed DB f ; [n ]–/ +O qn , [n ]qn . / / [n ]qn [n ]qn
(.)
Thus, we obtain the required result.
Now, we illustrate the rate of convergence of the GBS operators (.) to certain functions by graphics. It is observed that when the values of q and q increase, the convergence of (α ,α ,β ,β ) the GBS operator Tn ,n ,p p (f ; q , q , x, y) to the function f (x, y) becomes better. Example Let n = n = , α = , β = , α = , β = , p = p = . For q = ., q = (α ,α ,β ,β ) . and q = ., q = ., the convergence of the GBS operators Tn ,n ,p p (f ; q , q , x, y) (turquoise, orange) to f (x, y) = cos(x )/( + y) (yellow) is shown in Figure .
Chauhan et al. Journal of Inequalities and Applications (2017) 2017:50
(α ,α ,β ,β )
1 2 (f ; q , q , x, Figure 5 Convergence of Tn11,n2 2,p1 ,p 1 2 2 y) to f (x, y).
Figure 6 The comparison of rate of convergence (α ,α ,β ,β )
(α ,α ,β ,β )
1 2 (f ; q , q , x, y) and T 1 2 1 2 (f ; of Kn11,n2 2,p1 ,p 1 2 n1 ,n2 ,p1 ,p2 2 q1 , q2 , x, y) to f (x, y).
Figure 7 The comparison of rate of convergence (α ,α ,β ,β )
(α ,α ,β ,β )
1 2 (f ; q , q , x, y) and T 1 2 1 2 (f ; of Kn11,n2 2,p1 ,p 1 2 n1 ,n2 ,p1 ,p2 2 q1 , q2 , x, y) to f (x, y).
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Figure 8 The comparison of rate of convergence (α ,α ,β ,β )
(α ,α ,β ,β )
1 2 (f ; q , q , x, y) and T 1 2 1 2 (f ; of Kn11,n2 2,p1 ,p 1 2 n1 ,n2 ,p1 ,p2 2 q1 , q2 , x, y) to f (x, y).
(α ,α ,β ,β )
Lastly, we compare the convergence of the operators Kn,n,p p (f ; q , q , x, y) given by (α ,α ,β ,β ) (.) and its GBS operators Tn ,n ,p p (f ; q , q , x, y) to some functions. Example For n , n = , α = , β = , α = , β = , p = p = and q = ., q = ., (α ,α ,β ,β ) the comparison of convergence of the operators Kn,n,p p (f ; q , q , x, y) (green) and (α ,α ,β ,β ) Tn ,n ,p p (f ; q , q , x, y) (gray) to the functions f (x, y) = arctan(x + y ), f (x, y) = sin(x )/( + y ), f (x, y) = sin(x )/( + y ) is illustrated, respectively, in Figures , , and . We ob(α ,α ,β ,β ) serve that the rate of convergence of Tn ,n ,p p (f ; q , q , x, y) is better than the operator (α ,α ,β ,β ) Kn ,n ,p p (f ; q , q , x, y). Competing interests The authors declare that they have no competing interest. Authors’ contributions All authors contributed equally and significantly in writing this article. All the authors read and approved the final manuscript. Author details 1 Department of Mathematics, Indian Institute of Technology Roorkee, Roorkee, 247667, India. 2 Department of Mathematics, Faculty of Sciences, Gazi University, Ankara, 06500, Turkey. Acknowledgements The authors are extremely grateful to the reviewers for a critical reading of the manuscript and making valuable comments and suggestions leading to an overall improvement of the paper. The authors are thankful to the editor for sending the reports timely. The first author is thankful to the Ministry of Human Resource and Development, India, for financial support to carry out the above research work. Received: 7 November 2016 Accepted: 15 January 2017 References 1. Agrawal, PN, Goyal, M, Kajla, A: On q-Bernstein-Schurer-Kantorovich type operators. Boll. Unione Mat. Ital. 8, 169-180 (2015) 2. Ren, MY, Zeng, XM: On statistical approximation properties of modified q-Bernstein-Schurer operators. Bull. Korean Math. Soc. 50(4), 1145-1156 (2013) 3. Kac, V, Cheung, P: Quantum Calculus. Springer, New York (2002) 4. Acu, AM: Stancu-Schurer-Kantorovich operators based on q-integers. Appl. Math. Comput. 259, 896-907 (2015) 5. Aral, A, Gupta, V, Agarwal, RP: Applications of q-Calculus in Operator Theory. Springer, New York (2013) 6. Ditzian, Z, Totik, V: Moduli of Smoothness. Springer, New York (1987) 7. Bögel, K: Mehrdimensionale differemtiation, Von functionen mehrerer Veränderlicher. J. Reine Angew. Math. 170, 197-217 (1934) 8. Bögel, K: Mehrdimensionale differemtiation, integration and beschränkte variation. J. Reine Angew. Math. 173, 5-29 (1935)
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