Approximation by (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-Lupaş–Schurer–Kantorovich operators

In the current paper, we examine the (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-analogue of Kantorovich type Lupaş–Schurer operators with the help of (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-Jackson integral. Then, we estimate the rate of convergence for the constructed operators by using the modulus of continuity in terms of a Lipschitz class function and by means of Peetre’s K-functionals based on Korovkin theorem. Moreover, we illustrate the approximation of the (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-Lupaş–Schurer–Kantorovich operators to appointed functions by the help of Matlab algorithm and then show the comparison of the convergence of these operators with Lupaş–Schurer operators based on (p,q)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(p,q)$\end{document}-integers.


Introduction
In 1962, Bernstein-Schurer operators were identified in the paper of Schurer [25]. In 1987, Lupaş [16] initiated the q-generalization of Bernstein operators in rational form.
Some other q-Bernstein polynomial was defined by Phillips [22] in 1997. The development q-calculus applications established a precedent in the field of approximation theory.
First of all, we introduce some important notations and definitions for the (p, q)-calculus, which is a generalization of q-oscillator algebras. For 0 < q < p ≤ 1 and m ≥ 0, the (p, q)-number of m is denoted by [m] p,q and is defined by [m] p,q := p m-1 + p m-2 q + · · · + pq m-2 + q m-1 = The formula for the (p, q)-binomial expansion is defined by The (p, q)-Jackson integrals are defined by For detailed information about the theory of (p, q)-integers, we refer to [11] and [24].

Construction of the operator
Definition 1 For any 0 < q < p ≤ 1, we construct a (p, q)-analogue of Kantorovich type Lupaş-Schurer operator by where m ∈ N, f ∈ C[0, s + 1], s > 0 is a fixed natural number and After some calculations we obtain In the following lemma, we present some equalities for the (p, q)-analogue of Lupaş-Schurer-Kantorovich operators.
m,s (·; ·) be given by Eq. (4). Then we have Proof (i) From the definition of the operators in (4), we can easily prove the first claim as follows:   [2] p,q p s × m+s l=0 m+s l p,q p (m+s-l)(m+s-l-1) 2 q l(l-1) Thus, (6) is obtained.
(iii) For the third identity involving K Firstly, we calculate B1 as Now by using the equality we acquire Secondly, we work out B2 as follows: Thirdly, we deal with B3 as As a consequence, K If we reorganize, we obtain (v) Similarly, we write the second central moment K We now plug-in into equation (18) expressions (5), (6) and (7). Then we get We can easily see that K m,s (f ; x) are linear positive operators.
Before mentioning local approximation properties, we will give two lemmas as follows.

Lemma 2 If f is a monotone increasing function, then the constructed operators K (p,q)
m,s (f ; x) are linear and positive.
m,s (f ; x) satisfy the following Hölder inequality:

Local approximation properties
Let f be a continuous function on C[0, s + 1]. The modulus of continuity of f is denoted by w(f , σ ) and given as Then we know from the properties of modulus of continuity that for each σ > 0, we have And also, for f ∈ C[0, s + 1] we have lim σ →0 + w(f , σ ) = 0. First of all, we begin by giving the rate of convergence of the operators K and K x) is as given by (19).
Proof By the positivity and linearity of the operators K After that we apply (21) and obtain Then, taking supremum of the last equation, we have Thus, we achieve This result completes the proof of the theorem.
In what follows, by using Lipschitz functions, we will give the rate of convergence of the operators K (p,q) m,s (f ; x). We remember that if the inequality is satisfied, then f belongs to the class Lip M (α).
Theorem 3 Denote p := (p m ) and q := (q m ) satisfying 0 < q m < p m ≤ 1. Then, for every f ∈ Lip M (α), we have where σ m (x) is the same as in (22).
Proof Let f belong to the class Lip M (α) for some 0 < α ≤ 1. Using the monotonicity of the operators K (p,q) m,s (f ; x) and (24), we obtain Taking p = 2 α , q = 2 2-α and applying Hölder inequality yields By choosing σ m (x) as in Theorem 2, we complete the proof as desired.
Finally, in the light of Peetre-K functionals, we obtain the rate of convergence of the constructed operators K (p,q) m,s (f ; x). We recall the properties of Peetre-K functionals, which are defined as Here C 2 [0, s + 1] defines the space of the functions f such that f , f , f ∈ C[0, s + 1]. The norm in this space is given by Also we consider the second modulus of smoothness of f ∈ C[0, s + 1], namely We know from [7] that for M > 0 Before giving the main theorem, we present an auxiliary lemma, which will be used in the proof of the theorem.
Lemma 4 For any f ∈ C[0, s + 1], we have Proof and Proof Define an auxiliary operator K * m,s as follows: Taylor's expansion for a function g ∈ C 2 [0, s + 1] can be written as follows: Then applying operator K * m,s to both sides of (30), we get Using (29) and (28), we obtain and ([2] Let us employ (32) and (33) when taking the absolute value of (31). We obtain Accordingly, , where Finally, for all g ∈ C 2 [0, s + 1], taking the infimum of (35), we get Consequently, using the property of Peetre-K functional, we obtain This completes the proof.

Graphical illustrations
In this section, we illustrate an approximation of the operators K

Algorithm 2
Initially, we discuss the error estimates of the Kantorovich type Lupaş-Schurer operators based on (p, q)-integers for different values of x and m in Table 1 by using Algorithm 1.

Conclusion
In this paper, we constructed a new kind of Lupaş operators based on (p, q)-integers to provide a better error estimation. Firstly, we investigated some local approximation results by the help of the well-known Korovkin theorem. Also, we calculated the rate of convergence of the constructed operators employing the modulus of continuity, by using Lipschitz functions and then with the help of Peetre's K-functional. Additionally, we presented a table of error estimates of the (p, q)-Lupaş-Schurer-Kantorovich operators for a certain function. Finally, we compared the convergence of the new operator to that of the (p, q)-analogue of Lupaş-Schurer operator.

Funding
The authors have not received any research funding for this manuscript.