# Bivariate tensor product \((p, q)\) analogue of Kantorovich type Bernstein Stancu Schurer operators

N/A
N/A
Protected

Share "Bivariate tensor product \((p, q)\) analogue of Kantorovich type Bernstein Stancu Schurer operators"

Copied!
14
0
0

Full text

(1)

## operators

1

2,3

### and Guorong Zhou

4*

*Correspondence:

goonchow@xmut.edu.cn

4School of Applied Mathematics,

Xiamen University of Technology, Xiamen, 361024, China Full list of author information is available at the end of the article

Abstract

In this paper, we construct a bivariate tensor product generalization of Kantorovich-type Bernstein-Stancu-Schurer operators based on the concept of (p,q)-integers. We obtain moments and central moments of these operators, give the rate of convergence by using the complete modulus of continuity for the bivariate case and estimate a convergence theorem for the Lipschitz continuous functions. We also give some graphs and numerical examples to illustrate the convergence properties of these operators to certain functions.

MSC: 41A10; 41A25; 41A36

Keywords: (p,q)-integers; Bernstein-Stancu-Schurer operators; modulus of continuity; Lipschitz continuous functions; bivariate tensor product

1 Introduction

In recent years, (p,q)-integers have been introduced to linear positive operators to con-struct new approximation processes. A sequence of (p,q)-analogue of Bernstein opera-tors was ﬁrst introduced by Mursaleen [, ]. Besides, (p,q)-analogues of Szász-Mirakyan operators [], Baskakov-Kantorovich operators [], Bleimann-Butzer-Hahn operators [] and Kantorovich-type Bernstein-Stancu-Schurer operators [] were also considered. For further developments, one can also refer to [–]. These operators are double parame-ters corresponding topandqversus single parameterq-type operators [–]. The aim of these generalizations is to provide appropriate and powerful tools to application areas such as numerical analysis, CAGD and solutions of diﬀerential equations (see, e.g., []).

Motivated by all the above results, in , Cai et al. [] introduced a new kind of Kantorovich-type Bernstein-Stancu-Schurer operators based on (p,q)-integers as follows:

Knα,,pβ,q,l(f;x) =[n+ ]p,q+β

n+l k=

bn+l,k(p,q;x)

[k+ ]p,q– [k]p,q

[k+]p,q+α

[n+]p,q+β

[k]p,q+α

[n+]p,q+β

f(t)dp,qt, ()

wherebn+l,k(p,q;x) =

n+l

k

p,qx

k( –x)n+lk

p,q forfC(I), I= [,  +l], l∈N, ≤αβ,

 <q<p≤ and n∈N. They got some approximation properties, since convergence properties of bivariate operators are important in approximation theory, and it seems

(2)

there has been no papers mentioning the bivariate forms of above operators (). Hence, we will propose the bivariate case in the following. Before doing this, in [] (Lemma .), they gotKnα,,,,ql(;x) = , that is, the operators reproduce constant functions. However, this

conclusion is incorrect. In fact,nk+=l bn+l,k(p,q;x)= . Hence, we re-introduce the revised

operators as

Knα,,pβ,,ql(f;x) =[n+ ]p,q+β

n+l k=

bn+l,k(p,q;x)

[k+ ]p,q– [k]p,q

[k+]p,q+α

[n+]p,q+β

[k]p,q+α

[n+]p,q+β

f(t)dp,qt, ()

where

bn+l,k(p,q;x) =

p(n+l)(n+l–) n+l

k

p,q

pk(k–)xk( –x)n+lk

p,q . ()

From [], we knownk=+lbn+l,k(p,q;x) = , and this ensures the operators reproduce

con-stant functions.

On this basis, letC(I) denote the space of all real-valued continuous functions onI en-dowed with the normfI=sup(x,y)I|f(x,y)|. ForfC(I),I=I×I= [,  +l]×[,  + l],l∈N, ≤αβ,  <qn,qn <pn,pn ≤ andn,n∈N. We propose the bivariate tensor product (p,q)-analogue of Kantorovich-type Bernstein-Stancu-Schurer operators as follows:

Kn,n,α,β,l

pn,qn,pn,qn(f;x,y)

=[n+ ]pn,qn+β

[n+ ]pn,qn +β

× n+l

k=

n+l

k=

bpn,qn,pn,qn

n+l,n+l,k,k (x,y)

([k+ ]pn,qn– [k]pn,qn)([k+ ]pn,qn– [k]pn,qn)

×

[k+]pn ,qn+α [n+]pn ,qn+β [k]pn ,qn+α [n+]pn ,qn+β

[k+]pn ,qn+α [n+]pn ,qn+β [k]pn ,qn+α [n+]pn ,qn+β

f(t,s)dpn,qnt dpn,qns, ()

where

bpn,qn,pn,qn

n+l,n+l,k,k (x,y)

= 

p

(n+l)(n+l–) 

np

(n+l)(n+l–) 

n

n+l k

pn,qn

n+l k

pn,qn

×p

k(k–)

np

k(k–)

nx

kyk( –x)n+lk

pn,qn( –y)pnn+,lqnk ()

forx,y∈[, ].

(3)

of continuity and estimate a convergent theorem for the Lipschitz continuous functions. In Section , we give some graphs and numerical examples to illustrate the convergence properties of operators () to certain functions.

2 Some notations

We mention some deﬁnitions based on (p,q)-integers, details can be found in [–]. For any ﬁxed real number  <q<p≤ and each nonnegative integerk, we denote (p, q)-integers by [k]p,q, where

[k]p,q=

pkqk

pq .

Also (p,q)-factorial and (p,q)-binomial coeﬃcients are deﬁned as follows:

[k]p,q! =

⎧ ⎨ ⎩

[k]p,q[k– ]p,q· · ·[]p,q, k= , , . . . ,

, k= ,

n k

p,q

= [n]p,q! [k]p,q![n–k]p,q!

(n≥k≥).

The (p,q)-Binomial expansion is deﬁned by

(x+y)np,q=

⎧ ⎨ ⎩

, n= ,

(x+y)(px+qy)· · ·(pn–x+qn–y), n= , , . . . .

The deﬁnite (p,q)-integrals are deﬁned by

a

f(x)dp,qx= (p–q)a

k= qk

pk+f

qk

pk+a

and

a 

a 

f(x,y)dpn,qnx dpn,qny

= (pn–qn)(pn–qn)aa

k=

k= qk

n

pkn+ qk

n

pk+

nf

qk

n

pkn+ a, q

k

n

pk+

na

.

Whenp= , all the deﬁnitions of (p,q)-calculus above are reduced toq-calculus.

3 Auxiliary results

In order to obtain the convergence properties, we need the following lemmas.

Lemma . For the(p,q)-analogue of Kantorovich-type Bernstein-Stancu-Schurer

opera-tors(),we have

Knα,,pβ,,ql(;x) = , ()

,β,l n,p,q(t;x) =

( +q)[n+l]p,q(px+  –x)np+,ql–

[]p,q([n+ ]p,q+β)pn+l–

x+(px+  –x)

n+l p,q+ α

[]p,q([n+ ]p,q+β)

(4)

Knα,p,β,,qlt;x =(q+q

+q)[n+l]

p,q[n+l– ]p,q(px+  –x)np+,ql–

[]p,q([n+ ]p,q+β)pn+l–

x

+( +q+q

+p+ pq)[n+l]

p,q(px+  –x)np+,ql–x

[]p,q([n+ ]p,q+β)pn+l–

+α( +q)[n+l]p,q(px+  –x)

n+l–

p,q x

[]p,q([n+ ]p,q+β)pn+l–

+ (p

x+  –x)n+l p,q

[]p,q([n+ ]p,q+β)

+α(px+  –x)

n+l p,q+ α

[]p,q([n+ ]p,q+β)

. ()

Proof Sincenk+=l bn+l,k(p,q;x) = , () is easily obtained. Using () and [k+ ]p,q=pk+

q[k]p,q, we have

,β,l n,p,q(t;x)

=[n+ ]p,q+β

n+l k=

bn+l,k(p,q;x)

[k+ ]p,q– [k]p,q

[k+]p,q+α

[n+]p,q+β

[k]p,q+α

[n+]p,q+β

t dp,qt

=[n+ ]p,q+β

n+l k=

bn+l,k(p,q;x)

[k+ ]p,q– [k]p,q

([k+ ]p,q+α)– ([k]p,q+α)

[]p,q([n+ ]p,q+β)

= 

[]p,q([n+ ]p,q+β) n+l

k=

bn+l,k(p,q;x)

[k+ ]p,q+ [k]p,q+ α

= ( +q)[n+l]p,qx []p,q([n+ ]p,q+β)pn+l–p

(n+l–)(n+l–) 

n+l–

k=

n+l–  k

p,q

pk(k–)

×(px)k( –x)np,+qlk–+ (px+  –x)

n+l p,q + α

[]p,q([n+ ]p,q+β)

=( +q)[n+l]p,q(px+  –x)

n+l–

p,q

[]p,q([n+ ]p,q+β)pn+l–

x+ (px+  –x)

n+l p,q+ α

[]p,q([n+ ]p,q+β)

.

Thus, () is proved. Finally, from (), we get

,β

n,p,q

t;x

=[n+ ]p,q+β

n+l k=

bn+l,k(p,q;x)

[k+ ]p,q– [k]p,q

[k+]p,q+α

[n+]p,q+β

[k]p,q+α

[n+]p,q+β

tdp,qt

=([n+ ]p,q+β) []p,q

n+l

k=

bn+l,k(p,q;x)

[k+ ]p,q– [k]p,q

([k+ ]p,q+α)– ([k]p,q+α)

([n+ ]p,q+β)

= 

[]p,q([n+ ]p,q+β) n+l

k=

[k+ ]

p,q+ [k]p,q+ [k+ ]p,q[k]p,q

+ α[k+ ]p,q+ α[k]p,q+ α

(5)

Since [k+ ]p,q=pk+q[k]p,q, by some computations, we get

[k+ ]p,q+ [k]p,q+ [k+ ]p,q[k]p,q+ α[k+ ]p,q+ α[k]p,q+ α

=q+q+q[k]p,q[k– ]p,q+

 + q+ +q+q

p

pk[k]p,q

+ α( +q)[k]p,q+pk+ αpk+ α.

So, we can obtain

Knα,p,β,,qlt;x =(q+q

+q)[n+l]

p,q[n+l– ]p,q(px+  –x)np+,ql–

[]p,q([n+ ]p,q+β)pn+l–

x

+( +q+q

+p+ pq)[n+l]

p,q(px+  –x)np+,ql–x

[]p,q([n+ ]p,q+β)pn+l–

+α( +q)[n+l]p,q(px+  –x)

n+l–

p,q x

[]p,q([n+ ]p,q+β)pn+l–

+ (p

x+  –x)n+l p,q

[]p,q([n+ ]p,q+β)

+α(px+  –x)

n+l p,q+ α

[]p,q([n+ ]p,q+β)

.

Thus, () is proved.

Lemma . Using Lemma.and easy computations,we have

Knα,,pβ,,ql(t–x;x)

=

( +q)[n+l]

p,q(px+  –x)np+,ql–

[]p,q([n+ ]p,q+β)pn+l–

– 

x+ (px+  –x)

n+l p,q + α

[]p,q([n+ ]p,q+β)

, ()

Knα,,pβ,,ql(t–x);x =

(q+q+q)[n+l]

p,q[n+l– ]p,q(px+  –x)np+,ql–

[]p,q([n+ ]p,q+β)pn+l–

+ 

–( +q)[n+l]p,q(px+  –x)

n+l–

p,q

[]p,q([n+l]p,q+β)pn+l–

x+ (p

x+  –x)n+l p,q

[]p,q([n+ ]p,q+β)

+( +q+q

+p+ pq)[n+l]

p,q(px+  –x)np+,ql–x

[]p,q([n+ ]p,q+β)pn+l–

+α( +q)[n+l]p,q(px+  –x)

n+l–

p,q x

[]p,q([n+ ]p,q+β)pn+l–

+α(px+  –x)

n+l p,q + α

[]p,q([n+ ]p,q+β)

–(px+  –x)

n+l p,qx+ αx

[]p,q([n+ ]p,q+β)

. ()

Lemma . Let ei,j(x,y) =xiyj,i,j∈N, i+j≤, (x,y)Ibe the two-dimensional test

(6)

Bernstein-Stancu-Schurer operators deﬁned in()satisﬁes the following equalities:

Kn,n,α,β,l

pn,qn,pn,qn(e,;x,y) = , ()

Kn,n,α,β,l

pn,qn,pn,qn(e,;x,y)

= (pnx+  –x)

n+l pn,qn+ α

[]pn,qn([n+ ]pn,qn+β)

+( +qn)[n+l]pn,qn(pnx+  –x)

n+l–

pn,qn

[]pn,qn([n+ ]pn,qn+β)pnn+ l–

x, ()

Kn,n,α,β,l

pn,qn,pn,qn(e,;x,y)

= (pny+  –y)

n+l pn,qn+ α

[]pn,qn([n+ ]pn,qn +β)

+( +qn)[n+l]pn,qn(pny+  –y)

n+l–

pn,qn

[]pn,qn([n+ ]pn,qn +β)pnn+ l–

y, ()

Kn,n,α,β,l

pn,qn,pn,qn(e,;x,y)

=

(p

nx+  –x)

n+l pn,qn+ α

[]pn,qn([n+ ]pn,qn+β)

+( +qn)[n+l]pn,qn(pnx+  –x)

n+l–

pn,qn

[]pn,qn([n+ ]pn,qn+β)pnn+ l– x

×

( +q

n)[n+l]pn,qn(pny+  –y)

n+l–

pn,qn

[]pn,qn([n+ ]pn,qn+β)pnn+ l– y

+ (pny+  –y)

n+l pn,qn + α

[]pn,qn([n+ ]pn,qn +β)

, ()

Kn,n,α,β,l

pn,qn,pn,qn(e,;x,y)

=(qn+q

n+q

n)[n+l]pn,qn[n+l– ]pn,qn(p 

nx+  –x)

n+l–

pn,qn

[]pn,qn([n+ ]pn,qn+β)pnn+l–

x

+( +qn+q

n+pn+ pnqn)[n+l]pn,qn(p 

nx+  –x)

n+l–

pn,qnx

[]pn,qn([n+ ]pn,qn+β)pnn+ l–

+α( +qn)[n+l]pn,qn(pnx+  –x)

n+l–

pn,qnx

[]pn,qn([n+ ]pn,qn+β)pnn+ l–

+ (p 

nx+  –x)

n+l pn,qn

[]pn,qn([n+ ]pn,qn+β)

+ α(pnx+  –x)

n+l pn,qn+ α

[]pn,qn([n+ ]pn,qn+β)

, ()

Kn,n,α,β,l

pn,qn,pn,qn(e,;x,y)

=(qn+q

n+q

n)[n+l]pn,qn[n+l– ]pn,qn(p 

ny+  –y)

n+l–

pn,qn

[]pn,qn([n+ ]pn,qn +β)pnn+l–

y

+( +qn+q

n+pn+ pnqn)[n+l]pn,qn(p 

ny+  –y)

n+l–

pn,qny

[]pn,qn([n+ ]pn,qn +β)pnn+ l–

+α( +qn)[n+l]pn,qn(pny+  –y)

n+l–

pn,qny

[]pn,qn([n+ ]pn,qn +β)pnn+ l–

+ (p 

ny+  –y)

n+l pn,qn

[]pn,qn([n+ ]pn,qn +β)

+ α(pny+  –y)

n+l

pn,qn + α

[]pn,qn([n+ ]pn,qn +β)

(7)

Lemma . Using Lemmas.and.,the following equalities hold:

Kn,n,α,β,l

pn,qn,pn,qn(t–x;x,y)

=

( +q

n)[n+l]pn,qn(pnx+  –x)

n+l–

pn,qn

[]pn,qn([n+ ]pn,qn+β)pnn+ l– – 

x

+ (pnx+  –x)

n+l pn,qn+ α

[]pn,qn([n+ ]pn,qn+β)

:=n,,βpn,l

,qn(x), ()

Kn,n,α,β,l

pn,qn,pn,qn(s–y;x,y) =A

α,β,l

n,pn,qn(y), ()

Kn,n,α,β,l

pn,qn,pn,qn

(t–x);x,y

=

(q

n+q

n+q

n)[n+l]pn,qn[n+l– ]pn,qn(p 

nx+  –x)

n+l–

pn,qn

[]pn,qn([n+ ]pn,qn+β)pnn+l–

+  –( +qn)[n+l]pn,qn(pnx+  –x)

n+l–

pn,qn

[]pn,qn([n+l]pn,qn+β)pnn+ l–

x

+( +qn+q

n+pn+ pnqn)[n+l]pn,qn(p 

nx+  –x)

n+l–

pn,qnx

[]pn,qn([n+ ]pn,qn+β)pnn+ l–

+α( +qn)[n+l]pn,qn(pnx+  –x)

n+l–

pn,qnx

[]pn,qn([n+ ]pn,qn+β)pnn+ l–

+ (p 

nx+  –x)

n+l pn,qn

[]pn,qn([n+ ]pn,qn+β)

+ α(pnx+  –x)

n+l pn,qn+ α

[]pn,qn([n+ ]pn,qn+β)

–(pnx+  –x)

n+l

pn,qnx+ αx

[]pn,qn([n+ ]pn,qn+β)

:=n,,βpn,l

,qn(x), ()

Kn,n,α,β,l

pn,qn,pn,qn

(s–y);x,y=n,,βpn,l

,qn(y). ()

Lemma .(see Theorem . of []) For <qn<pn≤,set qn:=  –αn,pn:=  –βnsuch

that≤βn<αn< ,αn→,βn→as n→ ∞.The following statements are true: (A) Iflimn→∞en(βnαn)= andenβn/n,then[n]

pn,qn→ ∞. (B) Iflimn→∞en(βnαn)< andenβn(αnβn),then[n]

pn,qn→ ∞. (C) Iflimn→∞en(βnαn)< ,lim

n→∞en(βnαn)= andmax{enβn/n,enβn(αnβn)} →,

then[n]pn,qn→ ∞.

Remark . Let sequences{pn},{qn},{pn},{qn}( <qn,qn<pn,pn ≤) satisfy the conditions of Lemma .(A), (B) or (C). We have [n]pn,qn → ∞, [n]pn,qn → ∞. From

Lemmas . and ., the following statements are true.

lim

n,n→∞K

n,n,α,β,l

pn,qn,pn,qn(e,;x,y) =x,

lim

n,n→∞

Kn,n,α,β,l

pn,qn,pn,qn(e,;x,y) =y,

lim

n,n→∞

Kn,n,α,β,l

(8)

lim

n,n→∞

Kn,n,α,β,l

pn,qn,pn,qn

(t–x);x,y= lim

n→∞ n,,βpn,l

,qn(x) = ,

lim

n,n→∞

Kn,n,α,β,l pn,qn,pn,qn

(s–y);x,y= lim

n→∞ ,β,l

n,pn,qn(y) = .

4 Convergence properties

In order to ensure the convergence of operators deﬁned in (), in the sequel, let{pn},{qn},

{pn},{qn},  <qn,qn<pn,pn≤ be sequences satisfying Lemma .(A), (B) or (C).

Theorem . For fC(I),we have

lim

n,n→∞

Kn,n,α,β,l

pn,qn,pn,qn(f;·,·) –fI= .

Proof Using (), Remark . and a bivariate-type Korovkin theorem (see []), we obtain

Theorem . easily.

ForfC(I), the complete modulus of continuity for the bivariate case is deﬁned as

ω(f;δ,δ)

=supf(t,s) –f(x,y): (t,s), (x,y)I,|tx| ≤δ,|sy| ≤δ

,

whereδ,δ> . Furthermore,ω(f;δ,δ) satisﬁes the following properties:

(i) ω(f;δ,δ)→, ifδ,δ→;

(ii) f(t,s) –f(x,y)ω(f;δ,δ)

 +|tx|

δ

|sy|

δ

.

The partial modulus of continuity with respect toxandyis deﬁned as

ω()(f;δ) =supf(x,y) –f(x,y):yIand|x–x| ≤δ

,

ω()(f;δ) =supf(x,y) –f(x,y):xIand|yy| ≤δ.

Details of the modulus of continuity for the bivariate case can be found in []. We also use the notation

CI=fCI:fx,fyCI.

Now, we give the estimate of the rate of convergence of operators deﬁned in ().

Theorem . For fC(I),we have

Kn,n,α,β,l

pn,qn,pn,qn(f;x,y) –f(x,y)≤ω

f;

Bαn,,βpn,l,qn(x),

Bαn,,βpn,l,qn(y)

, ()

(9)

Proof From Lemmas . and ., using the property (ii) of the complete modulus of con-tinuity for the bivariate case above and the Cauchy-Schwarz inequality, we get

Kpnn,n,α,β,l

,qn,pn,qn(f;x,y) –f(x,y)

Kpnn,n,α,β,l

,qn,pn,qnf(t,s) –f(x,y);x,y

ω

f;

Bαn,,βpn,l,qn(x),

Bαn,,βpn,l,qn(y)

×

 +

Kpnn,n,qn,α,,pnβ,l,qn((t–x);x,y)

Bαn,,βpn,l,qn(x)

×

 +

Kpnn,n,qn,α,,pnβ,l,qn((s–y);x,y)

Bαn,,βpn,l,qn(y)

.

Theorem . is proved.

Theorem . For fC(I),under the conditions of Lemma.,we have

Kn,n,α,β,l

pn,qn,pn,qn(f;x,y) –f(x,y) ≤

f;ω()

f;

Bαn,,βpn,l,qn(x)

+ω()

f;

Bαn,β,pn,l,qn(y)

,

where Bαn,,βpn,l,qn(x)and Bαn,β,pn,l,qn(y)are deﬁned in()and().

Proof Using the deﬁnition of partial modulus of continuity above and the Cauchy-Schwarz inequality, we have

Kn,n,α,β,l

pn,qn,pn,qn(f;x,y) –f(x,y)Kpnn,n,α,β,l

,qn,pn,qnf(t,s) –f(x,y);x,y

Kpnn,n,α,β,l ,qn,pn,qn

f(t,s) –f(t,y);x,y

+Kpnn,n,α,β,l

,qn,pn,qnf(t,y) –f(x,y);x,y

Kpnn,n,α,β,l ,qn,pn,qn

ω()f;|sy|;x,y +Kn,n,α,β,l

pn,qn,pn,qn

ω()f;|tx|;x,y

ω()

f;

Bαn,,βpn,l,qn(y)

 +

Kpnn,n,qn,α,,βpn,l,qn((s–y);x,y)

Bαn,,βpn,l,qn(y)

+ω()

f;

Bαn,,βpn,l,qn(x)

 +

Kpnn,n,qn,α,,βpn,l,qn((t–x);x,y)

Bαn,,βpn,l,qn(x)

.

(10)

Theorem . For fC(I),using Lemma.,we have

Kpnn,n,α,β,l

,qn,pn,qn(f;x,y) –f(x,y)

fxI

Bαn,,βpn,l,qn(x) +fyI

Bαn,,βpn,l,qn(y),

where Bαn,,βpn,l,qn(x)and B

α,β,l

n,pn,qn(y)are deﬁned in()and().

Proof Sincef(t,s) –f(x,y) =xtfu(u,s)du+ysfv(x,v)dv. Applying the operators deﬁned in () on both sides above, we have

Kpnn,n,α,β,l

,qn,pn,qn(f;x,y) –f(x,y)

Kpnn,n,α,β,l ,qn,pn,qn

xtfu(u,s)du;x,y

+Kpnn,n,α,β,l ,qn,pn,qn

s

y

fv(x,v)dv;x,y

.

Due to|xtfu(u,s)du| ≤ fxI|tx|and|

s

yfv(x,v)dv| ≤ fyI|sy|, we have

Kpnn,n,α,β,l

,qn,pn,qn(f;x,y) –f(x,y)

fxIKn,n,α,β,l pn,qn,pn,qn

|tx|;x,y+fyIKn,n,α,β,l pn,qn,pn,qn

|sy|;x,y.

Using the Cauchy-Schwarz inequality and Lemma ., we obtain

Kpnn,n,α,β,l

,qn,pn,qn(f;x,y) –f(x,y)

fxI

Kpnn,n,qn,α,,βpn,l,qn

(t–x);x,yKn,n,α,β,l

pn,qn,pn,qn(;x,y)

+fyI

Kpnn,n,qn,α,,βpn,l,qn

(s–y);x,yKn,n,α,β,l

pn,qn,pn,qn(;x,y)

fxI

Bαn,,βpn,l,qn(x) +fyI

Bαn,,βpn,l,qn(y).

Theorem . is proved.

Finally, we study the rate of convergence ofKpnn,n,qn,α,,βpn,l,qn(f;x,y) by means of functions

of Lipschitz classLipM(θ,θ) if

f(t,s) –f(x,y)M|tx|θ|sy|θ, (t,s), (x,y)I.

Theorem . Let f ∈LipM(θ,θ),under the conditions of Lemma.,we have

Kn,n,α,β,l

pn,qn,pn,qn(f;x,y) –f(x,y)M

,β,l n,pn,qn(x)

θ Bα,β,l

n,pn,qn(y)

θ ,

(11)

Proof Sincef ∈LipM(θ,θ), we have

Kpnn,n,α,β,l

,qn,pn,qn(f;x,y) –f(x,y)

Kpnn,n,α,β,l

,qn,pn,qnf(t,s) –f(x,y);x,y

MKn,n,α,β,l pn,qn,pn,qn

|tx|θ;x,yKn,n,α,β,l pn,qn,pn,qn

|sy|θ;x,y.

Using Hölder’s inequality for the last formula, respectively, we get

Kn,n,α,β,l

pn,qn,pn,qn(f;x,y) –f(x,y)MKpnn,n,α,β,l

,qn,pn,qn

(t–x);x,y

θ

Kn,n,α,β,l

pn,qn,pn,qn(;x,y)

–θ 

×Kpnn,n,α,β,l ,qn,pn,qn

(s–y);x,y

θ

Kn,n,α,β,l

pn,qn,pn,qn(;x,y)

–θ 

=MBα,β,l n,pn,qn(x)

θ Bα,β,l

n,pn,qn(y)

θ .

Theorem . is proved.

5 Graphical and numerical examples analysis

In this section, we give several graphs and numerical examples to show the convergence ofKpnn,n,qn,α,,βpn,l,qn(f;x,y) tof(x,y) with diﬀerent values of parameters which satisfy the

con-clusions of Lemma ..

Letf(x,y) =xy, the graphs ofKn,n,α,β,l

pn,qn,pn,qn(f;x,y) with diﬀerent values ofqn,qn,pn, pn andn,nare shown in Figures ,  and . In Tables ,  and , we give the errors of the approximation ofKn,n,α,β,l

[image:11.595.118.477.479.699.2]

pn,qn,pn,qn(f;x,y) tof(x,y) with diﬀerent parameters.

Figure 1 The ﬁgures ofKpnn1,n2,α,β,l

1 ,qn1 ,pn2 ,qn2(f;x,y) (the upper one) forn1=n2= 20,pn1=pn2= 1 – 1/10 8,

(12)
[image:12.595.119.478.81.300.2]

Figure 2 The ﬁgures ofKpnn1,n2,α,β,l

1 ,qn1 ,pn2 ,qn2(f;x,y) (the upper one) forn1=n2= 20,pn1=pn2= 1 – 1/10 14,

qn1=qn2= 0.9999,l= 1,α= 3,β= 2 andf(x,y) =x2y2(the below one).

Figure 3 The ﬁgures ofKpnn1,n2,α,β,l

1 ,qn1 ,pn2 ,qn2(f;x,y) (the upper one) forn1=n2= 50,pn1=pn2= 1 – 1/10 14,

[image:12.595.117.479.251.569.2]

qn1=qn2= 0.9999,l= 1,α= 3,β= 2 andf(x,y) =x2y2(the below one).

Table 1 The errors of the approximation ofKpnn1,n2,α,β,l

1,qn1,pn2,qn2(f;x,y) forpn1=pn2= 1–1/10 15,

qn1=qn2= 0.9999,l= 1,α= 3,β= 2 and different values ofn1,n2

n1= n2 f (x, y) – Kn1,n2,

α,β,l

pn1 ,qn1 ,pn2 ,qn2(f ; x, y)

5 0.801911

10 0.406691

15 0.259663

20 0.188202

30 0.150588

[image:12.595.219.371.652.732.2]
(13)
[image:13.595.214.375.118.196.2]

Table 2 The errors of the approximation ofKpnn1,n2,α,β,l

1,qn1,pn2,qn2(f;x,y) forn1=n2= 10,

pn1=pn2= 1–1/1015,l= 1,α= 3,β= 2 and different values ofq

n1,qn2

qn1= qn2 f (x, y) – Kpn1 ,qn1 ,pn2 ,qn2n1,n2,α,β,l (f ; x, y)

0.99 2.923910

0.995 1.194710

0.999 0.643543

0.9995 0.594722 0.9999 0.406691 0.99995 0.130489

Table 3 The errors of the approximation ofKpnn1,n2,α,β,l

1,qn1,pn2,qn2(f;x,y) forqn1=qn2= 0.9999,l= 1,

α= 3,β= 2 and different values ofpn1,pn2andn1,n2

n1= n2 pn1= pn2 f (x, y) – Kpn1 ,qn1 ,pn2 ,qn2n1,n2,α,β,l (f ; x, y)

10 1–1/1010 0.406691

15 1–1/1011 0.259663

20 1–1/1012 0.188202

25 1–1/1013 0.150589

30 1–1/1014 0.131836

35 1–1/1015 0.125673

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11601266 and 11626201), the Natural Science Foundation of Fujian Province of China (Grant No. 2016J05017) and the Program for New Century Excellent Talents in Fujian Province University. We also thank Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing and Information Processing of Fujian Province University.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

QBC, XWX and GZ carried out the molecular genetic studies, participated in the sequence alignment and drafted the manuscript. QBC, XWX and GZ carried out the immunoassays. QBC, XWX and GZ participated in the sequence alignment. QBC, XWX and GZ participated in the design of the study and performed the statistical analysis. QBC, XWX and GZ conceived of the study and participated in its design and coordination and helped to draft the manuscript. All authors read and approved the ﬁnal manuscript.

Author details

1School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou, 362000, China.2School of

Mathematical Sciences, Xiamen University, Xiamen, 361005, China.3Computer Sciences, Rice University, Houston, USA. 4School of Applied Mathematics, Xiamen University of Technology, Xiamen, 361024, China.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 18 July 2017 Accepted: 8 November 2017 References

1. Mursaleen, M, Ansari, KJ, Khan, A: On (p,q)-analogue of Bernstein operators. Appl. Math. Comput.266, 874-882 (2015) 2. Mursaleen, M, Ansari, KJ, Khan, A: Erratum to ‘On (p,q)-analogue of Bernstein operators [Appl. Math. Comput. 266

(2015) 874-882]’. Appl. Math. Comput.278, 70-71 (2016)

3. Acar, T: (p,q)-Generalization of Szász-Mirakyan operators. Math. Methods Appl. Sci.39(10), 2685-2695 (2016) 4. Acar, T, Aral, A, Mohiuddine, SA: On Kantorovich modiﬁcation of (p,q)-Baskakov operators. J. Inequal. Appl. (2016).

doi:10.1186/s13660-016-1045-9

5. Mursaleen, M, Nasiruzzaman, M, Khan, A, Ansari, KJ: Some approximation results on Bleimann-Butzer-Hahn operators deﬁned by (p,q)-integers. Filomat30(3), 639-648 (2016)

6. Cai, QB, Zhou, GR: On (p,q)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators. Appl. Math. Comput.

276, 12-20 (2016)

7. Mursaleen, M, Ansari, KJ, Khan, A: Some approximation results by (p,q)-analogue of Bernstein-Stancu operators. Appl. Math. Comput.264, 392-402 (2015)

[image:13.595.197.396.249.328.2]
(14)

9. Mursaleen, M: Some approximation results on Bleimann-Butzer-Hahn operators deﬁned by (p,q)-integers. Filomat

30(3), 639-648 (2016)

10. Mursaleen, M, Ansari, KJ, Khan, A: Some approximation results for Bernstein-Kantorovich operators based on (p,q)-calculus. UPB Sci. Bull., Ser. A78(4), 129-142 (2016)

11. Acar, T, Agrawal, P, Kumar, A: On a modiﬁcation of (p,q)-Szász-Mirakyan operators. Complex Anal. Oper. Theory (2016). doi:10.1007/s11785-016-0613-9

12. Ilarslan, H, Acar, T: Approximation by bivariate (p,q)-Baskakov-Kantorovich operators. Georgian Math. J. (2016). doi:10.1515/gmj-2016-0057

13. Phillips, GM: Bernstein polynomials based on theq-integers. Ann. Numer. Math.4, 511-518 (1997)

14. Gupta, V, Kim, T: On the rate of approximation byqmodiﬁed beta operators. J. Math. Anal. Appl.377, 471-480 (2011) 15. Gupta, V, Aral, A: Convergence of theqanalogue of Szász-beta operators. Appl. Math. Comput.216, 374-380 (2010) 16. Acar, T, Aral, A: On pointwise convergence ofq-Bernstein operators and theirq-derivatives. Numer. Funct. Anal.

Optim.36(3), 287-304 (2015)

17. Khan, K, Lobiyal, DK: Bézier curves based on Lupas (p,q)-analogue of Bernstein functions in CAGD. J. Comput. Appl. Math.317, 458-477 (2017)

18. Hounkonnou, MN, Désiré, J, Kyemba, B:R(p,q)-Calculus: diﬀerentiation and integration. SUT J. Math.49, 145-167 (2013)

19. Jagannathan, R, Rao, KS: Two-parameter quantum algebras, twin-basic numbers, and associated generalized hypergeometric series. In: Proceedings of the International Conference on Number Theory and Mathematical Physics, pp. 20-21 (2005)

20. Katriel, J, Kibler, M: Normal ordering for deformed boson operators and operator-valued deformed Stirling numbers. J. Phys. A, Math. Gen.24, 2683-2691 (1992)

21. Sadjang, PN: On the fundamental theorem of (p,q)-calculus and some (p,q)-Taylor formulas. arXiv:1309.3934v1 (2015) 22. Sahai, V, Yadav, S: Representations of two parameter quantum algebras andp,q-special functions. J. Math. Anal. Appl.

335, 268-279 (2007)

23. Cai, QB, Xu, XW: A basic problem of (p,q)-Bernstein operators. J. Inequal. Appl.2017, 140 (2017). doi:10.1186/s13660-017-1413-0

24. Altomare, F, Campiti, M: Korovkin-Type Approximation Theory and Its Applications. De Gruyter Studies in Mathematics, vol. 17. de Gruyter, Berlin (1994)

Figure

References

Related documents

gifted education programmein terms of student number per class, teachers’ lesson delivery and teachers’ assessment process at Federal Government Academy(Centre for

Os resultados contribuem para que os profissionais de saúde, com destaque para os que atuam na Atenção Básica, reflitam sobre a inserção das PIC de forma efetiva e eficaz

In this work, the surface acidity and redox properties of various transition metals: Mo, Ce, Mn, Cu, V, Co, Zr supported on WTi-PILC are investigated then

Model 6: Blood pressure, waist circumference, occupation, physical activity, marital status, education, age and social class, alcohol consumption;.. Model 7 : Blood

stage of lung development, most Fgf10 -expressing cells gave rise to lipofibroblasts that are believed to represent a stem cell niche that signals to adjacent type 2

Samples are rejected and canceled when they are considered inadequate (saturated, wet and wrong material for examination), samples with insufficient blood

As a result of geophysical research studies along the profile I-I using VES method the sections of apparent resistivity and geoelectrical section based on the values of

Preparation of working standard drug solution: The standard Dorzolamide HCl (100 mg) was weighed accurately and transferred to 100 ml volumetric flask containing few