Journal of Function Spaces
Volume 2019 (2019), Article ID 7587401, 11 pages
https://doi.org/10.1155/2019/7587401

Research Article

Bivariate Chlodowsky-Stancu Variant of (p,q)-Bernstein-Schurer Operators

Tuba Vedi-Dilek¹ and Eser Gemikonakli²

¹Institute of Applied Sciences, University of Kyrenia, Girne, Mersin 10, Turkey
²Department of Computer Engineering, University of Kyrenia, Girne, Mersin 10, Turkey

Correspondence should be addressed to Tuba Vedi-Dilek; tuba.vedi@kyrenia.edu.tr and Eser Gemikonakli; eser.gemikonakli@kyrenia.edu.tr

Received 21 January 2019; Accepted 6 March 2019; Published 24 April 2019

Guest Editor: Vita Leonessa

Copyright © 2019 Tuba Vedi-Dilek and Eser Gemikonakli. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

In this study, it is proposed to define bivariate Chlodowsky variant of -Bernstein-Stancu-Schurer operators. Therefore, Korovkin-type approximation theorems and the error of approximation by using full modulus of continuity are presented. Beside this, we introduce a generalization of the bivariate Chlodowsky variant of -Bernstein-Stancu-Schurer operators and investigate its approximation in more general weighted space. Moreover, the numerical results are discussed in order to validate the accuracy of the bivariate Chlodowsky variant of -Bernstein-Schurer operators.

1. Introduction

In this area, many operators about -integer and -integer [1–25] were studied. It may not be possible to find the exact solution for the models developed for the engineering fields because of its mathematical intractability. Therefore, these operators can be used effectively to find an algorithm for approximating solutions [26].

In 2014, Vedi and Ozarslan introduced the Chlodowsky variant of -Bernstein-Schurer-Stancu operators in [27] as

where is increasing sequence of real numbers satisfying , , , , and and investigated its approximation properties.

In 2017, Vedi and Ozarslan defined the two dimensional Chlodowsky variant of -Bernstein-Schurer-Stancu operators in [28] by

where and are increasing sequences of real numbers satisfying and , , , and investigated its approximation properties on the rectangular unbounded domain.

Moreover, Gemikonakli and Vedi-Dilek [29] introduced the Chlodowsky variant of Bernstein-Schurer operators based on -integers as

where , , and .

Let us discuss some well-known basic definitions of -calculus. For , the -numbers are given as [30]

For each the -factorial is represented by

and -binomial coefficients are defined as

where

Then, Chlodowsky variant of -Bernstein-Stancu-Schurer operators was constructed in [31] as

where , , with , , and has the same properties as the operator .

Lemma 1. Let be defined in [31].
Then the first few moments of the operators are

The bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on -integers is defined and then the first few moments of the operator are provided in Section 2. Next, in Section 3, some Korovkin-type theorems are studied. And following this, the order of convergence of the bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on -integers by means of the first modulus of continuity is obtained in Section 4. Moreover, in Section 5, we study the generalization of the bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on -integers and seek its approximation properties in more general weighted space. Finally, in Section 6, numerical results for the operators constructed are provided in detail.

2. Construction of the Operators

Let denote

and and are increasing sequences of real numbers satisfying

For , we construct the bivariate Chlodowsky variant of Bernstein-Stancu-Schurer operators based on -integers as

where , , and , .

Lemma 2. Let be given in (11).
Then, the first few moments of the operators are

3. Korovkin-Type Approximation Theorems

In this section, Korovkin-type approximation theorems are given for the bivariate Chlodowsky variant of Bernstein–Stancu-Schurer operators based on -integers. For fixed consider the space , which consists of all continuous functions satisfying the condition

Obviously, is a linear normed space with the following norm:

Theorem 3. Let the numbers and be any fixed positive real numbers.
Let with and and be increasing sequences of positive real numbers that satisfy the following properties:

For all , we obtain

Proof. Using Lemma 2, we have

And again using Lemma 2 we get

Finally, from the above equality, we obtain

Therefore, from the hypothesis of the theorem, we get

when and .
Hence, the proof is completed by the two dimensional Korovkin theorem.

Theorem 4 (see [13]). There exists a sequence of positive operators , acting from to , satisfying the conditions

and there exists a function for which

where

Now, consider the following operator:

Theorem 5. Let for any

where , , , and have the same conditions as in Theorem 3.

Proof. For all , there exist sufficiently large positive real numbers and such that

when and .
Let , be sufficiently large so that

By Theorem 3, and for the proof of the second term we have

Finally, since , the term is bounded. Furthermore, because of the fact that

in case Lemma 2 is used, the term is bounded for sufficiently large and . Hence, we get by (25) that

Since is arbitrary, then . This completes the proof.

Now, consider the subspace of which is defined by

Theorem 6. Let the sequences , , , and satisfy the same properties as in Theorem 3. Then for all , we obtain

Proof. For all , observe that

Therefore, for all , we can find sufficiently large numbers and such that

for and and there exist natural numbers and such that

for all and .
Hence, for large and , we have

By Theorem 3, it is sufficient to show that as .
Using (33) and (34), we get

where

By Lemma 2, it is clear that there exist independent of and such that

Therefore, for and we have

This completes the proof.

4. Order of Convergence

In this section, we compute the rate of convergence of the operators in terms of the full modulus of continuity and partial modulus of continuities.

Let and . Then the definition of the modulus of continuity of is given by

It is known that [32] for any we know that

Also, for any we have

Theorem 7. For any , the following inequalities

are satisfied, where

Proof. We directly have

By linearity and positivity of the operators, we get

Using Lemma 1 and Cauchy-Schwartz inequality, we have

Then, we get

where we choose as in (46).
In the same way, we obtain

where is given in (45). Combining (50) and (51), we obtain (43).
Now, by using linearity and the monotonicity of the operators, and taking into account (40), we have

Using (41) and the Cauchy-Schwartz inequality, we get (44).

5. Generalization of the Bivariate Chlodowsky Variant of Bernstein-Schurer Operators Based on -Integers

In this section, we introduce generalization of bivariate Chlodowsky variant of Bernstein-Schurer operators based on -integers. The generalized operators help us to approximate continuous functions defined on more general weighted spaces. Note that this kind of generalization was considered earlier for the Chlodowsky-Bernstein polynomials [12].

For , consider any continuous function and define

Let us consider the generalization of the as follows:

where and and have the same properties of bivariate Chlodowsky variant of Bernstein-Schurer operators based on -integers.

Theorem 8. For all continuous functions satisfying , , and , we have

where

Proof. Clearly,

thus

Since , then for and is continuous function on . Furthermore, from , we have

Thus, from Theorem 6 we get the result.

Finally, note that taking , then the operators reduce .

6. Numerical Results and Discussions

In this section, numerical results are presented in order to demonstrate the effectiveness and validate the accuracy of to with different values of parameters. The parameters used in the mathematical calculations are given in Table 1.

Table 1: parameters used for numerical analysis.

Parameters
Values

n
20

m
5

0.989

0.99

0.989

0.99

s
1

Variable

Figures 1 and 2 are given for as a function of x and y for different and values considering two different multivariable functions and accordingly.

Figure 1: Convergence of to multivariable function .

Figure 2: Convergence of to multivariable function .

Figure 1 demonstrates the convergence of to the multivariable function . For different x and y values, various and values are taken into account. According to the results obtained, the behaves like the function for smaller and values. Therefore, the error of approximation of to the function is minimised for and , where s=1, n=20, and m=5 for the sequences and as provided in Table 1.

Figure 2 shows the convergence to the function but this time for different and values, while keeping all the other parameters constant. Unlike in Figure 1, increasing and values upto and respectively provided better approximate results. It is evident that, similar to Figure 1, and values have a significant effect on the convergence of to the multivariable function , which provide better approximate results.

Comparative results are given in Tables 2 and 3, for the errors of the approximation of   to the functions and .

Table 2: Errors of approximation to , ().

x
y

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

0
0.0015
0.0039
0.0075
0.0112
0.0149
0.0186
0.0222
0.0258
0.0294
0.0329
0.0364

0.1
0.0044
0.0015
0.0022
0.0032
0.0041
0.0049
0.0056
0.0063
0.0069
0.0074
0.0078

0.2
0.0086
0.0023
0.0034
0.0049
0.0063
0.0075
0.0086
0.0096
0.0104
0.011
0.0116

0.3
0.0128
0.0033
0.0046
0.0066
0.0083
0.0099
0.0114
0.0128
0.0141
0.0153
0.0164

0.4
0.0169
0.0038
0.0051
0.0072
0.0092
0.0109
0.0127
0.0145
0.0163
0.0183
0.0206

0.5
0.0208
0.004
0.0048
0.0066
0.0082
0.0099
0.0117
0.0138
0.0164
0.0198
0.0243

0.6
0.0245
0.0035
0.0033
0.0042
0.0051
0.0061
0.0076
0.01
0.0137
0.0193
0.0274

0.7
0.028
0.0025
0.0006
0.0002
0.0009
0.0012
0.0004
0.0021
0.0074
0.0164
0.0303

0.8
0.0313
0.0007
0.0038
0.0071
0.0104
0.0128
0.0133
0.0105
0.0031
0.0108
0.0332

0.9
0.0342
0.002
0.0101
0.0171
0.0239
0.0294
0.0317
0.0288
0.0185
0.0023
0.0366

1
0.0368
0.0057
0.0184
0.0303
0.0421
0.0517
0.0566
0.0537
0.0393
0.0093
0.0407

Table 3: Errors of approximation to , .

x
y

0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1

0
0
0
0.0001
0.0002
0.0004
0.0005
0.0007
0.001
0.0013
0.0017
0.0021

0.1
0.0001
0.0002
0.0004
0.0007
0.0012
0.0016
0.0022
0.0029
0.0037
0.0048
0.0066

0.2
0.0003
0.0005
0.001
0.0017
0.0025
0.0033
0.0043
0.0055
0.0069
0.0093
0.0134

0.3
0.0006
0.001
0.0019
0.0029
0.004
0.0052
0.0065
0.0079
0.0099
0.0136
0.0209

0.4
0.001
0.0017
0.003
0.0044
0.0058
0.0072
0.0086
0.0102
0.0127
0.0177
0.0289

0.5
0.0015
0.0024
0.0043
0.0061
0.0078
0.0093
0.0108
0.0124
0.0151
0.0217
0.0374

0.6
0.0021
0.0034
0.0058
0.0081
0.01
0.0116
0.0129
0.0143
0.0173
0.0256
0.0466

0.7
0.0028
0.0045
0.0075
0.0103
0.0124
0.014
0.0151
0.0162
0.0192
0.0292
0.0563

0.8
0.0036
0.0058
0.0094
0.0127
0.0151
0.0166
0.0173
0.0179
0.0209
0.0327
0.0665

0.9
0.0046
0.0072
0.0116
0.0154
0.018
0.0193
0.0195
0.0194
0.0222
0.0361
0.0773

1
0.0056
0.0087
0.014
0.0183
0.0211
0.0221
0.0217
0.0207
0.0233
0.0392
0.0887

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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