On the convergence of Lupaş $(p,q)$-Bernstein operators via contraction principle

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

$$ B_{n}(f,x)=\sum_{k=0}^{n} \binom{n}{k}x^{k}(1-x)^{n-k}f \biggl( \frac{k}{n} \biggr), \quad x\in [ 0,1], n\in \mathbb{N}, $$

(1.1)

is known as Bernstein polynomials [6].

Lupaş [18] defined the first q-analogue of Bernstein operators (rational) for $q>0$ as follows:

$$ L_{n}(f,q;x)=\sum_{k=0}^{n}f \biggl( \frac{[k]_{q}}{[n]_{q}} \biggr) b _{n,k}(x;q), $$

(1.2)

where

$$b_{n,k}(x;q)=\frac{{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{q}^{\frac{n(n-1)}{2}}{x}^k{(1-x)}^{n-k}}{{\prod }_{j=0}^{n-1}\{(1-x)+q^{j}x\}}.$$

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

$$ [ k]_{q}= \textstyle\begin{cases} \frac{1-q^{k}}{1-q} & \text{if }q\neq 1, \\ k & \text{if }q=1. \end{cases} $$

Set $[0]_{q}={0}$. The q-factorial coefficients are defined by

$$ [ k]_{q}!= \textstyle\begin{cases} [ k]_{q}[k-1]_{q}\cdots [ 1]_{q} & \text{if }k\in \mathbb{N}, \\ 1 & \text{if }k=0, \end{cases} $$

and the q-binomial coefficients by

$${\left[\begin{array}{c}n\\ k\end{array}\right]}_q=\frac{{[n]}_q!}{{[k]}_q!{[n-k]}_q!},0\le k\le n,$$

with ${\left[\begin{array}{c}n\\ 0\end{array}\right]}_q=1$ and ${\left[\begin{array}{c}n\\ k\end{array}\right]}_q=0$ for $k>n$.

Also,

$$ (x-a)_{q}^{n}= \textstyle\begin{cases} 1 & \text{if }n=0, \\ (x-a)(x-qa)\cdots (x-q^{n-1}a) & \text{if }n\geq {1}. \end{cases} $$

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

$${\mathrm{\Delta }}_q^k{f}_i=\sum _{r=0}^k{(-1)}^r{q}^{\frac{r(r-1)}{2}}\left[\begin{array}{c}k\\ r\end{array}\right]f_{i+k-r}$$

(see [36] and [17]).

Note that (1.2) may be written in the q-difference form

$$L_n(f,q;x)=\sum _{k=0}^n{\left[\begin{array}{c}n\\ k\end{array}\right]}_q{\mathrm{\Delta }}_q^k{f}_0{x}^k.$$

(1.3)

We may deduce that

$$ L_{n}(ax+b,q;x)=ax+b, \quad a,b\in \mathbb{R}. $$

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

$$ S^{\infty }(x)=\lim_{m\rightarrow \infty }S^{m}(x),\quad x\in X. $$

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

$$ \lim_{M\rightarrow \infty }L_{n}^{M}(f;x)=f(0)+ \bigl(f(1)-f(0)\bigr)x,\quad x \in C[0,1]. $$

Rus [35] proved this result by using the contraction principle.

In [29] the authors defined

$$ L_{n}^{M+1}(f,q;x)=L_{n}\bigl(L_{n}^{M}(f,q;x) \bigr),\quad M=1,2,\ldots, $$

and

$$ L_{n}^{1}(f,q;x)=L_{n}(f,q;x). $$

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 $q>0$,

$$ \lim_{M\rightarrow \infty }L_{n}^{M}(f,q;x)=f(0)+ \bigl(f(1)-f(0)\bigr)x $$

(2.1)

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,

$$\begin{aligned} t_{n} =&\min_{x\in [ 0,1]} \biggl( \frac{(1-x)^{n}}{\prod_{j=0}^{n-1}\{(1-x)+q^{j}x\}}+ \frac{q^{\frac{n(n-1)}{2}}x^{n}}{ \prod_{j=0}^{n-1}\{(1-x)+q^{j}x\}} \biggr) \\ =&\min_{x\in [ 0,1]} \biggl( \frac{(1-x)^{n}+q^{ \frac{n(n-1)}{2}}x^{n}}{\prod_{j=0}^{n-1}\{(1-x)+q^{j}x\}} \biggr) . \end{aligned}$$

Then $0< t_{n}\leq 1$.

Let $f,g\in X_{\alpha ,\beta }$. Then

$$\begin{aligned}& \bigl\vert L_{n}(f,q;x)-L_{n}(g,q;x) \bigr\vert \\& \quad = \Biggl\vert \sum_{k=1}^{n-1}b_{n,k}(x;q) (f-g) \biggl( \frac{[k]_{q}}{[n]_{q}} \biggr) \Biggr\vert \\& \quad \leq \sum_{k=1}^{n-1}b_{n,k}(x;q) \Vert f-g \Vert _{[0,1]}= \bigl( 1-b_{n,0}(x;q)-b_{n,n}(x;q) \bigr) \Vert f-g \Vert _{[0,1]} \\& \quad = \biggl( 1-\frac{(1-x)^{n}}{\prod_{j=0}^{n-1}\{(1-x)+q^{j}x \}}-\frac{q^{\frac{n(n-1)}{2}}x^{n}}{\prod_{j=0}^{n-1}\{(1-x)+q ^{j}x\}} \biggr) \Vert f-g \Vert _{[0,1]} \\& \quad = \biggl( 1-\frac{((1-x)^{n}+q^{\frac{n(n-1)}{2}}x^{n})}{\prod_{j=0}^{n-1}\{(1-x)+q^{j}x\}} \biggr) \Vert f-g \Vert _{[0,1]} \\& \quad \leq ( 1-t_{n} ) \Vert f-g \Vert _{[0,1]}, \end{aligned}$$

and, finally,

$$ \bigl\Vert L_{n}(f,q;x)-L_{n}(g,q;x) \bigr\Vert _{[0,1]}\leq ( 1-t_{n} ) \Vert f-g \Vert _{[0,1]}. $$

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

$$ L^{\infty }_{n}(f,q;x) = f(0) + \bigl(f(1) - f(0) \bigr)x. $$

(2.2)

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

$$ S^{\infty }(x)=\lim_{M\rightarrow \infty }S^{M}(x). $$

(2.3)

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

$$ [ n]_{p,q}=p^{n-1}+p^{n-2}q+p^{n-3}q^{2}+ \cdots+pq^{n-2}+q^{n-1}= \textstyle\begin{cases} \frac{p^{n}-q^{n}}{p-q} & \text{when }p\neq q\neq 1, \\ {n}p^{n-1} & \text{when }p=q\neq 1, \\ [ n ] _{q} & \text{when }p=1, \\ {n} & \text{when }p=q=1, \end{cases} $$

$n=0,1,2,3,4,\ldots$ . Also,

$$\begin{array}{c}{[k]}_{p,q}!={[k]}_{p,q}{[k-1]}_{p,q}\dots {[1]}_{p,q},k=1,2,3,\dots ,\hfill \\ {\left[\begin{array}{c}n\\ k\end{array}\right]}_{p,q}=\frac{{[n]}_{p,q}!}{{[k]}_{p,q}!{[n-k]}_{p,q}!},k=1,2,3,\dots ,\hfill \end{array}$$

and

$$\begin{array}{c}{(ax+by)}_{p,q}^n=\sum _{k=0}^n{p}^{\frac{(n-k)(n-k-1)}{2}}{q}^{\frac{k(k-1)}{2}}{\left[\begin{array}{c}n\\ k\end{array}\right]}_{p,q}{a}^{n-k}{b}^k{x}^{n-k}{y}^k,\hfill \\ {(x+y)}_{p,q}^n=(x+y)(px+qy)(p^{2}x+q^{2}y)\dots (p^{n-1}x+q^{n-1}),\hfill \\ {(1-x)}_{p,q}^n=(1-x)(p-qx)(p^2-q^{2}x)\dots (p^{n-1}-q^{n-1}x).\hfill \end{array}$$

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

$$ L_{n}(f,p,q;x)=\sum_{k=0}^{n}f \biggl( \frac{p^{n-k}[k]_{p,q}}{[n]_{p,q}} \biggr) b_{n,k}(x;p,q),\quad x \in [ 0,1], $$

(3.1)

where

$$b_{n,k}(x;p,q)=\frac{{\left[\begin{array}{c}n\\ k\end{array}\right]}_{p,q}{p}^{\frac{(n-k)(n-k-1)}{2}}{q}^{\frac{k(k-1)}{2}}{x}^k{(1-x)}^{n-k}}{{\prod }_{j=0}^{n-1}\{p^j(1-x)+q^{j}x\}},$$

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:

$$\begin{aligned}& L_{n}(e_{0},p,q;x) = 1 , \\& L_{n}(e_{1},p,q;x) = x , \\& L_{n}(e_{2},p,q;x) = \frac{p^{n-1}x}{[n]_{p,q}} + \frac{q^{2}x^{2}}{p(1-x) + qx}\frac{[n-1]_{p,q}}{[n]_{p,q}}. \end{aligned}$$

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

$$ L_{n}^{M+1}(f,p,q;x)=L_{n} \bigl( L_{n}^{M}(f,p,q;x) \bigr), \quad M=1,2,\ldots, $$

and

$$ L_{n}^{1}(f,p,q;x)=L_{n}(f,p,q;x). $$

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

$$ \lim_{M\rightarrow \infty }L_{n}^{M}(f,p,q;x)=f(0)+ \bigl(f(1)-f(0)\bigr)x. $$

(4.1)

Proof

The proof follows the same steps as in Theorem 2.2. Let

$$ X_{\alpha ,\beta } = \bigl\{ f\in C[0,1]\mid f(0)=\alpha ,f(1)=\beta \bigr\} , \quad \alpha ,\beta \in {\mathbb{R}}. $$

The Lupaş $(p,q)$-Bernstein polynomial possess the end-point interpolation property

$$ L_{n}(f,p,q;0)=f(0),\quad\quad L_{n}(f,p,q;1)=f(1). $$

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,

$$\begin{aligned} \bigl\vert L_{n}(f,p,q;x)-L_{n}(g,p,q;x) \bigr\vert &= \Biggl\vert \sum_{k=1}^{n-1}b_{n,k}(x;p,q) (f-g) \biggl( \frac{p^{n-k}[k]_{p,q}}{[n]_{p,q}} \biggr) \Biggr\vert \\ &\leq \sum_{k=1}^{n-1}b_{n,k}(x;p,q) \Vert f-g \Vert _{[0,1]} \\ &= \bigl( 1-b_{n,0}(x;p,q)-b_{n,n}(x;p,q) \bigr) \Vert f-g \Vert _{[0,1]}. \end{aligned}$$

Let $w_{n}=\min_{x\in [ 0,1]} ( b_{n,0}(x;p,q)+b_{n,n}(x;p,q) )$, that is,

$$ w_{n}=\min_{x\in [ 0,1]} \biggl( \frac{p^{\frac{n(n-1)}{2}}(1-x)^{n}}{ \prod_{j=0}^{n-1}\{p^{j}(1-x)+q^{j}x\}}+ \frac{q^{ \frac{n(n-1)}{2}}x^{n}}{\prod_{j=0}^{n-1}\{p^{j}(1-x)+q^{j}x\}} \biggr). $$

Then $0< w_{n}\leq 1$. Therefore

$$\begin{aligned} \bigl\vert L_{n}(f,p,q;x)-L_{n}(g,p,q;x) \bigr\vert &= \biggl( 1-\frac{(p^{\frac{n(n-1)}{2}}(1-x)^{n}+q ^{\frac{n(n-1)}{2}}x^{n})}{\prod_{j=0}^{n-1}\{p^{j}(1-x)+q^{j}x \}} \biggr) \Vert f-g \Vert _{[0,1]} \\ &\leq ( 1-w_{n} ) \Vert f-g \Vert _{[0,1]} \end{aligned}$$

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.