- Review
- Open access
- Published:
On the convergence of Lupaş \((p,q)\)-Bernstein operators via contraction principle
Journal of Inequalities and Applications volume 2019, Article number: 34 (2019)
Abstract
The present paper deals with the limit behavior of iterates of Lupaş q- and \((p,q)\)-Bernstein operators. We obtain the convergence for Lupaş q- and \((p,q)\)-Bernstein operators by using the contraction principle.
1 Introduction and preliminaries
For any \(f\in \mathcal{C}[0,1]\), the sequence of operators \(B_{n}:C[0,1] \rightarrow \mathcal{C}[0,1]\) defined by
is known as Bernstein polynomials [6].
Lupaş [18] defined the first q-analogue of Bernstein operators (rational) for \(q>0\) as follows:
where
Later Phillips [32] proposed another q-analog of Bernstein operators.
For \(q={1}\), both are reduced to the original ones. However, for \(q\neq {1}\), there are considerable differences between them. The convergence properties for iterates of q-Bernstein polynomials have been investigated in [5, 29, 30, 33, 37], and [38].
For elementary properties of q-analogs of Bernstein polynomials, we refer to [12, 18, 19, 31].
The q-integer \([k]_{q}\) for \(k\in \mathbb{N}\) and a fixed real number \(q>{0}\) is defined by
Set \([0]_{q}={0}\). The q-factorial coefficients are defined by
and the q-binomial coefficients by
with and for \(k>n\).
Also,
For any function f, the divided differences are denoted by \(\Delta _{q}^{0}f_{i}=f_{i}\) for \(i=0,1,2,\ldots,{n-1}\) and, recursively, by \(\Delta _{q}^{k+1}f_{i} = \Delta _{q}^{k}f_{i+1}- {q^{k}}\Delta _{q}^{k}f_{i}\) for \(k\geqq {0}\), where \(f_{i}\) denotes \(f(\frac{[i]}{[m]})\). It is established by induction that
Note that (1.2) may be written in the q-difference form
We may deduce that
Also, we can see that these operators verify for the test functions \(e_{j}(x)=x^{j}\), \(j=0,1,2\).
In 1993, Rus [34] introduced and developed the theory of (weakly) Picard operators, which is one of the most strong tools of fixed point theory with several applications to operator equations and inclusions. Berinde [8] showed that an almost contraction is more general than most of the contractions in the literature.
We now recall some basic features from fixed point theory (see [34]).
Definition 1.1
The operator S on a metric space \((X,d)\) is called a weakly Picard operator (WPO) if the sequence of iterates \((S^{m}(x))_{m}\geqq 1\) converges to a fixed point of S for all \(x\in X\).
We denote, as usual, \(S^{0}=I_{X}\), \(S^{m+1}=S\circ S^{m}\), \(m\in \mathbb{N}\).
Let \(F_{S} = \{\xi \in X:S(\xi )=\xi \}\). If an operator B is a WPO and \(F_{S}\) has exactly one element, then S is called a Picard operator (PO).
First, we give a characterization for WPOs.
Theorem 1.2
([33])
The operator S on a metric space \((X,d)\) is a weakly Picard operator ⇔ there exists a partition \(\{X_{\lambda }:\lambda \in \Lambda \}\) such that for every \(\lambda \in \Lambda \) one has \(X_{\lambda }\in I(S)\) and \(S|_{X_{\lambda }}: X_{\lambda }\rightarrow X_{\lambda }\) is a PO, where \(I(S):= \{\emptyset \neq Y\subset X:S(Y)\subset Y\}\) denotes the collection of all non-empty subsets invariant under S.
Moreover, for a WPO S, we take \(S^{\infty }\in X\) defined as
Clearly, \(S^{\infty }(x) = F_{S}\). Also, if S is WPO, then we have \(F_{S}^{m}=F_{S}\neq \emptyset \), \(m\in \mathbb{N}\).
2 Iterates of Lupaş q-Bernstein operators
In the last decades the iterates of positive linear operators in various classes were intensively investigated. The study of convergence of iterates of Bernstein operators has connections to probability theory, matrix theory, spectral theory, and so on. We emphasize here the importance of the works of some mathematicians such as [1, 7, 9,10,11, 13, 14, 16, 20, 25], and [35].
We want to extend the study of the iterates of Bernstein operators using \((p,q)\)-calculus. Our aim is to study the convergence for iterates of Lupaşş \((p,q)\)-Bernstein operators using the contraction principle (theory of weakly Picard operators).
Theorem 2.1
([15])
For \(f\in C[0,1]\) and fixed \(n\in \mathbb{N}^{\ast }\), we have
Rus [35] proved this result by using the contraction principle.
In [29] the authors defined
and
In the same paper, the authors proved the convergence of iterates of q-Bernstein operators for \(q>0\) using q-differences, Stirling polynomials, and matrix techniques. Ostrovska [30] used eigenvalues and Radu [33] used the contraction principle to prove the convergence of these iterates. The following result gives the convergence of Lupaş q-Bernstein operators by using also the contraction principle. The following result gives the convergence of Lupaş q-Bernstein operators by using the contraction principle.
Theorem 2.2
Let \(L_{n}(f,q;x)\) be the Lupaş q-Bernstein operators defined in (1.2). Then for all \(q>0\),
for all \(f\in C[0,1]\) and \(x\in [ 0,1]\).
Proof
First, we define \(X_{\alpha ,\beta }= \{f\in C[0,1]:f(0)=\alpha ,f(1)= \beta \}\), \(\alpha ,\beta \in {\mathbb{R}}\). Clearly, every \(X_{\alpha ,\beta } \) is a closed subset of \(C[0,1]\), and \(\{X_{ \alpha ,\beta },(\alpha ,\beta )\in {\mathbb{R}\times \mathbb{R}}\}\) is a partition of the space \(C[0,1]\).
It follows directly from the definition that Lupaş q-Bernstein polynomials possess the end-point interpolation property.
So we have that \(X_{\alpha ,\beta }\) is an invariant subset of \(f\mapsto L_{n}(f,q;\cdot )\) for all \((\alpha ,\beta )\in {\mathbb{R} \times \mathbb{R}}\) and \(n\in \mathbb{N}\).
We show that the restriction of \(f\mapsto L_{n}(f,q;\cdot )\) to \(X_{\alpha ,\beta }\) is a contraction for any \(\alpha ,\beta \in \mathbb{R}\). Put \(t_{n}=\min_{x\in [ 0,1]} ( b_{n,0}(x;q)+b_{n,n}(x;q) ) \), that is,
Then \(0< t_{n}\leq 1\).
Let \(f,g\in X_{\alpha ,\beta }\). Then
and, finally,
The restriction of \(f\mapsto L_{n}(f,q;x)\) to \(X_{\alpha ,\beta }\) is a contraction.
On the other hand, \(K_{\alpha ,\beta }^{\ast }=\alpha e_{0}+(\beta - \alpha )e_{1}\in X_{\alpha ,\beta }\). Since \(L_{n}(e_{0},q;x)=e_{0}\) and \(L_{n}(e_{1},q;x)=e_{1}\), it follows that \(K_{\alpha ,\beta }^{\ast }\) is a fixed point of \(L_{n}(f,q;\cdot )\). For any \(f\in C[0,1]\), we have \(f\in X_{f(0),f(1)}\), and by using the contraction principle we obtain the desired result (2.1). □
In terms of WPOs, using (1.3), we can formulate the above theorem as follows.
Theorem 2.3
The Lupaş q-Bernstein operator \(f \mapsto L_{n}(f,q;\cdot )\) is WPO, and
Proof
The operator \(S:X\rightarrow X\) is WPO if the sequence \((S^{M}(x))_{M \geq 1}\) converges to a fixed point of S for all \(x\in X\).
For a WPO, we consider the operator \(S^{\infty }:X\rightarrow X\) defined as
Now, using Theorem 2.2 and (2.3), we get the result (2.2). □
3 Lupaş \((p,q)\)-Bernstein operator
Mursaleen et al. [23] defined the \((p,q)\)-analogs of Bernstein operators and studied their approximation properties (see [2, 3, 21, 22, 24,25,26,27,28]). For further reading, we refer to [4, 21, 22, 26, 27].
For any \(p>0\) and \(q>0\), we have
\(n=0,1,2,3,4,\ldots \) . Also,
and
Khan et al. [16] introduced the following Lupaş-type \((p,q)\)-analog of Bernstein operators (rational):
For any \(p>0\) and \(q>0\), we get
where
and \(b_{n,0}(x;p,q),b_{n,1}(x;p,q),\ldots,b_{n,n}(x;p,q)\) are the Lupaş \((p,q)\)-Bernstein basis functions [12]. We recall the following auxiliary results:
4 Iterates of Lupaş \((p,q)\)-Bernstein operator
Now we extend the study of the iterates of Bernstein operators in the framework of \((p,q)\)-calculus. The iterates of the Lupaş \((p,q)\)-Bernstein polynomial are defined as
and
We study the convergence of the iterates of Lupaş \((p,q)\)-Bernstein operators.
Theorem 4.1
Let \(L_{n}(f,p,q;x)\) be the Lupaş \((p,q)\)-Bernstein operators defined in (3.1), where \(p>0\), \(q>0\).
Then the Lupaş \((p,q)\)-Bernstein operator is a weakly Picard operator, and its sequence \(( L_{n}^{M} ) _{M\geq 1}\) of iterates satisfies
Proof
The proof follows the same steps as in Theorem 2.2. Let
The Lupaş \((p,q)\)-Bernstein polynomial possess the end-point interpolation property
Then \(X_{\alpha ,\beta }\) is an invariant subset of \(f\mapsto L_{n}(f,p,q; \cdot )\) for all \((\alpha ,\beta )\in \mathbb{R}\times \mathbb{R}\) and \(n\in \mathbb{N}\).
We prove that the restriction of \(f\mapsto L_{n}(f,p,q;\cdot )\) to \(X_{\alpha ,\beta }\) is a contraction for any \(\alpha \in \mathbb{R}\) and \(\beta \in \mathbb{R}\). Let \((\alpha ,\beta )\in \mathbb{R}\) and \(f,g\in X_{\alpha ,\beta }\). From the definition of the \((p,q)\)-Bernstein operator,
Let \(w_{n}=\min_{x\in [ 0,1]} ( b_{n,0}(x;p,q)+b_{n,n}(x;p,q) ) \), that is,
Then \(0< w_{n}\leq 1\). Therefore
for any \(f,g\in X_{\alpha ,\beta }\), that is, the restriction of \(f\mapsto L_{n}(f,p,q;\cdot )\) to \(X_{\alpha ,\beta }\) is a contraction.
On the other hand, \(K_{\alpha ,\beta }^{\ast }=\alpha e_{0}+(\beta - \alpha )e_{1}\in X_{\alpha ,\beta }\), where \(e_{0}(x)=1\) and \(e_{1}(x)=x\) for all \(x\in [ 0,1]\). Since \(L_{n}(e_{0},p,q;x)=e _{0}\), and \(L_{n}(e_{1},p,q;x)=e_{1}\), it follows that \(K_{\alpha , \beta }^{\ast }\) is a fixed point of \(L_{n}(f,p,q;\cdot )\).
By Theorem 1.2 the Lupaş \((p,q)\)-Bernstein operator is a WPO, and using the contraction principle, we obtain the claim (4.1). □
5 Conclusion
In this paper, we have studied the convergence for Lupaş q-Bernstein operators by using the contraction principle. We further extended the study of the iterates of Bernstein operators using \((p,q)\)-calculus.
References
Abel, U., Ivan, M.: Over-iterates of Bernstein operators: a short and elementary proof. Am. Math. Mon. 116, 535–538 (2009)
Acar, T.: \((p,q)\)-Generalization of Szász–Mirakyan operators. Math. Methods Appl. Sci. 39(10), 2685–2695 (2016)
Acar, T., Mohiuddine, S.A., Mursaleen, M.: Approximation by \((p,q)\)-Baskakov–Durrmeyer–Stancu operators. Complex Anal. Oper. Theory (2018). https://doi.org/10.1007/s11785-016-0633-5
Acar, T., Mursaleen, M., Mohiuddine, S.A.: Stancu type \((p,q)\)-Szász–Mirakyan–Baskakov operators. Commun. Fac. Sci. Univ. Ank. Sér. A1 67(1), 116–128 (2018)
Acu, A.M., Radu, V.A.: About the iterates of some operators depending on a parameter and preserving the affine functions. Iran. J. Sci. Technol., Trans. A, Sci. https://doi.org/10.1007/s40995-017-0461-0
Bernstein, S.N.: Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun. Soc. Math. Kharkow 2(13), 1–2 (1912)
Bica, A., Galea, L.F.: On Picard iterates properties of the Bernstein operators and an application to fuzzy numbers. Commun. Math. Anal. 5(1), 8–19 (2008)
Brinde, V.: Approximating fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 9(1), 43–53 (2004)
Cătinaş, T., Otrocol, D.: Iterates of Bernstein type operators on a square with one curved side via contraction principle. Int. J. Fixed Point Theory Comput. Appl. 14(1), 97–106 (2013)
Devdhara, A.R., Mishra, V.N.: Stancu variant of \((p,q)\)-Szász–Mirakyan operators. J. Inequal. Spec. Funct. 8(5), 1–7 (2017)
Gonska, H.H., Raşa, I.: The limiting semigroup of the Bernstein iterates: degree of convergence. Acta Math. Hung. 111, 119–130 (2006)
Han, L.-W., Chu, Y., Qiu, Z.-Y.: Generalized Bèzier curves and surfaces based on Lupaş q-analogue of Bernstein operator. J. Comput. Appl. Math. 261, 352–363 (2014)
Kadak, U.: On weighted statistical convergence based on \((p,q)\)-integers and related approximation theorems for functions of two variables. J. Math. Anal. Appl. 443, 752–764 (2016)
Kadak, U.: Weighted statistical convergence based on generalized difference operator involving \((p,q)\)-gamma function and its applications to approximation theorems. J. Math. Anal. Appl. 448, 1633–1650 (2017)
Kelisky, R.P., Rivlin, T.J.: Iterates of Bernstein polynomials. Pac. J. Math. 21, 511–520 (1967)
Khan, K., Lobiyal, D.K.: Bèzier curve based on Lupaş \((p,q)\)-analogue of Bernstein functions in CAGD. J. Comput. Appl. Math. 317, 458–477 (2017)
Lee, S.L., Phillips, G.M.: Polynomial interpolation at points of a geometric mesh on a triangle. Proc. R. Soc. Edinb. 108A, 75–87 (1988)
Lupaş, A.: A q-analogue of the Bernstein operator. Univ. Cluj-Napoca Semin. Numer. Stat. Calc. 9, 85–92 (1987)
Mahmudov, N.I., Sabancgil, P.: Some approximation properties of Lupaş q-analogue of Bernstein operators. arXiv:1012.4245v1 [math.FA]. Accessed 20 Dec 2010
Mishra, V.N., Khatri, K., Mishra Deepmala, L.N.: Inverse result in simultaneous approximation by Baskakov–Durrmeyer–Stancu operators. J. Inequal. Appl. 2013, 586 (2013)
Mishra, V.N., Mursaleen, M., Pandey, S., Alotaibi, A.: Approximation properties of Chlodowsky variant of \((p,q)\)-Bernstein–Stancu–Schurer operators. J. Inequal. Appl. 2017, 176 (2017)
Mishra, V.N., Pandey, S.: On \((p,q)\)-Baskakov–Durrmeyer–Stancu operators. Adv. Appl. Clifford Algebras 27(2), 1633–1646 (2017)
Mursaleen, M., Ansari, K.J., Khan, A.: On \((p,q)\)-analogue of Bernstein operators. Appl. Math. Comput. 266, 874–882 (2015). Erratum to “On \((p,q)\)-analogue of Bernstein operators”, Appl. Math. Comput. 278, 70–71 (2016)
Mursaleen, M., Ansari, K.J., Khan, A.: Some approximation results by \((p,q)\)-analogue of Bernstein–Stancu operators. Appl. Math. Comput. 264, 392–402 (2015)
Mursaleen, M., Khan, A.: Generalized q-Bernstein–Schurer operators and some approximation theorems. J. Funct. Spaces 2013, Article ID 719834 (2013)
Mursaleen, M., Khan, F., Khan, A.: Approximation by \((p,q)\)-Lorentz polynomials on a compact disk. Complex Anal. Oper. Theory 10(8), 1725–1740 (2016)
Mursaleen, M., Nasiruzzaman, Md., Khan, A., Ansari, K.J.: Some approximation results on Bleimann–Butzer–Hahn operators defined by \((p,q)\)-integers. Filomat 30(3), 639–648 (2016)
Mursaleen, M., Nasiruzzaman, Md., Nurgali, A.: Some approximation results on Bernstein–Schurer operators defined by \((p,q)\)-integers. J. Inequal. Appl. 2015, 249 (2015)
Oruç, H., Tuncer, N.: On the convergence and iterates of q-Bernstein polynomials. J. Approx. Theory 117, 301–313 (2002)
Ostrovska, S.: q-Bernstein polynomials and their iterates. J. Approx. Theory 123, 232–255 (2003)
Ostrovska, S.: On the Lupaş q-analogue of the Bernstein operator. Rocky Mt. J. Math. 36(5), 1615–1629 (2006)
Phillips, G.M.: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4, 511–518 (1997)
Radu, V.A.: Note on the iterates of q- and \((p,q)\)-Bernstein operators, Sci. Stud. Res. Ser. Math. Inform., 26(2), 83–94 (2016)
Rus, I.A.: Weakly Picard mappings. Comment. Math. Univ. Carol. 34(4), 769–773 (1993)
Rus, I.A.: Iterates of Bernstein operators, via contraction principle. J. Math. Anal. Appl. 292, 259–261 (2004)
Schoenberg, I.J.: On polynomial interpolation at the points of a geometric progression. Proc. R. Soc. Edinb. 90A, 195–207 (1981)
Srivastava, H.M., Mursaleen, M., Alotaibi, A.M., Nasiruzzaman, Md., Al-Abied, A.A.H.: Some approximation results involving the q-Szász–Mirakjan–Kantorovich type operators via Dunkl’s generalization. Math. Methods Appl. Sci. 40(15), 5437–5452 (2017)
Xiang, X., He, Q., Yang, W.: Convergence rate for iterates of q-Bernstein polynomials. Anal. Theory Appl. 23(3), 243–254 (2007)
Funding
This research project was supported by a grant from the “Research Center of the Female Scientific and Medical Colleges”, Deanship of Scientific Research, King Saud University.
Author information
Authors and Affiliations
Contributions
The first author has 50 percent contribution, the third author also has 50 percent contribution, whereas the second author checked and drafted the manuscript in its present form after making necessary corrections. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access 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.
About this article
Cite this article
Bin Jebreen, H., Mursaleen, M. & Ahasan, M. On the convergence of Lupaş \((p,q)\)-Bernstein operators via contraction principle. J Inequal Appl 2019, 34 (2019). https://doi.org/10.1186/s13660-019-1985-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13660-019-1985-y