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

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R E S E A R C H

Open Access

Bivariate tensor product

(

p

,

q

)-analogue of

Kantorovich-type Bernstein-Stancu-Schurer

operators

Qing-Bo Cai

1

, Xiao-Wei Xu

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

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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∈[, ].

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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)

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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

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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

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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)

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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

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Figure

Figure 1 The figures of Kn1,n2,α,β,lpn1 ,qn1 ,pn2 ,qn2 (f;x,y) (the upper one) for n1 = n2 = 20, pn1 = pn2 = 1 – 1/108,qn1 = qn2 = 0.999, l = 1, α = 3, β = 2 and f(x,y) = x2y2 (the below one).
Figure 2 The figures of Kn1,n2,α,β,lpn1 ,qn1 ,pn
Table 2 The errors of the approximation of Kpn1,n2,α,β,lpn1,qn1,pn2,qn2 (f;x,y) for n1 = n2 = 10,n1 = pn2 = 1–1/1015, l = 1, α = 3, β = 2 and different values of qn1, qn2

References

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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