Posts by Diana Otrocol

Abstract

Given a function defined on a square with one curved side, we consider some Bernstein-type operators as well as their product and Boolean sum. Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of these operators.

Authors

T. Catinas
(Babes Bolyai Univ.)

D. Otrocol
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Keywords

Square with curved side, Bernstein operators, contraction principle, weakly Picard operators.

Cite this paper as:

T. Catinas, D. Otrocol, Iterates of Bernstein type operators on a square with one curved side via contraction principle, Fixed Point Theory, 14(2013), no. 1, pp. 97-106

PDF

https://ictp.acad.ro/otrocol/papers/2013-CatinasOtrocol-FPT-Iterates%20of%20Bernstein.pdf

http://www.math.ubbcluj.ro/∼nodeacj/sfptcj.html

http://www.math.ubbcluj.ro/~nodeacj/download.php?f=131Catinas-914-final.pdf

About this paper

Journal

Fixed Point Theory

Publisher Name

Casa Cartii de Stiinta, Cluj-Napoca, Romania

DOI

http://doi.org/10.24193/fpt-ro.2018.2.43

Print ISSN

1583-5022

Online ISSN

2066-9208

MR

MR3821782

ZBL

1397.34108

Google Scholar

https://scholar.google.ro/scholar?oi=bibs&hl=en&cites=13124375127534453666

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[2] O. Agratini, I.A. Rus, Iterates of some bivariate approximation process via weakly Picard operators, Nonlinear Analysis Forum, 8(2003), no. 2, 159-168.

[3] P. Blaga, T. Catinas, G. Coman, Bernstein-type operators on triangle with one curved side, Mediterr. J. Math., 10(2013), 10.1007/s00009-011-0156-2, in press.

[4] P. Blaga, T. Catinas, G. Coman, Bernstein-type operators on a square with one and two curved sides, Studia Univ. Babes–Bolyai Math., 55(2010), no. 3, 51-67.

[5] P. Blaga, T. Catinas, G. Coman, Bernstein-type operators on triangle with all curved sides, Appl. Math. Comput., 218(2011), 3072-3082.

[6] G. Coman, T. Catinas, Interpolation operators on a triangle with one curved side, BIT Numerical Mathematics, 50(2010), no. 2, 243-267.

[7] I. Gavrea, M. Ivan, The iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl., 372(2010), 366-368.

[8] I. Gavrea, M. Ivan, The iterates of positive linear operators preserving the constants, Appl. Math. Lett., 24(2011), no. 12, 2068-2071.

[9] I. Gavrea, M. Ivan, On the iterates of positive linear operators, J. Approximation Theory, 163(2011), no. 9, 1076-1079.

[10] H. Gonska, D. Kacso, P. Pitul, The degree of convergence of over-iterated positive linear operators, J. Appl. Funct. Anal., 1(2006), 403-423.

[11] H. Gonska, P. Pitul, I. Rasa, Over-iterates of Bernstein-Stancu operators, Calcolo, 44(2007), 117-125.

[12] H. Gonska, I. Rasa, The limiting semigroup of the Bernstein iterates: degree of convergence, Acta Math. Hungar., 111(2006), no. 1-2, 119-130.

[13] S. Karlin, Z. Ziegler, Iteration of positive approximation operators, J. Approximation Theory 3(1970), 310-339.

[14] R.P. Kelisky, T.J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math., 21(1967), 511-520.

[15] I.A. Rus, Generalized Contractions and Applications, Cluj Univ. Press, 2001.

[16] I.A. Rus, Iterates of Stancu operators, via contraction principle, Studia Univ. Babes-Bolyai Math., 47(2002), no. 4, 101-104.

[17] I.A. Rus, Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl., 292(2004), 259-261.

[18] I.A. Rus, Fixed point and interpolation point set of a positive linear operator on C(D), Studia Univ. Babes–Bolyai Math., 55(2010), no. 4, 243-248.

Fixed Point Theory, 14(2013), No.1, …-…

http://www.math.ubbcluj.ro/^∼nodeacj/sfptcj.html

Iterates of Bernstein type operators on a square with one curved side via contraction principle

Teodora Cătinaş^∗ and Diana Otrocol^∗∗

^∗Babeş-Bolyai University, Faculty of Mathematics and Computer Science, Str. M. Kogălniceanu Nr. 1, RO-400084 Cluj-Napoca, Romania
E-mail: tcatinas@math.ubbcluj.ro
^∗∗Tiberiu Popoviciu Institute of Numerical Analysis of Romanian Academy,
Cluj-Napoca, Romania
E-mail: dotrocol@ictp.acad.ro

Abstract. Given a function defined on a square with one curved side, we consider some Bernstein-type operators as well as their product and Boolean sum. Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of these operators.

Key Words and Phrases: Square with curved side, Bernstein operators, contraction principle, weakly Picard operators.

2000 Mathematics Subject Classification: 41A36, 41A25, 39B12, 47H10.

1. Weakly Picard operators

We recall some results regarding weakly Picard operators that will be used in the sequel (see, e.g., [15]).

Let $(X,d)$ be a metric space and $A:X\rightarrow X$ an operator. We denote by

	$\displaystyle F_{A}$	$\displaystyle:=\{x\in X~\|~A(x)=x\}\text{-the fixed point set of }A\text{;}$
	$\displaystyle I(A)$	$\displaystyle:=\{Y\subset X~\|~A(Y)\subset Y,\ Y\neq\emptyset\}\text{-the family of the nonempty invariant }$
		$\displaystyle\text{subset of }A$
	$\displaystyle A^{0}$	$\displaystyle:=1_{X},\ A^{1}:=A,\ ...,\ A^{n+1}:=A\circ A^{n},\ \ n\in\mathbb{N}\text{.}$

Definition 1.1.

The operator $A:X\rightarrow X$ is a Picard operator if there exists $x^{\ast}\in X$ such that:

(i) $F_{A}=\{x^{*}\};$

(ii) the sequence $(A^{n}(x_{0}))_{n\in\mathbb{N}}$ converges to $x^{*}$ for all $x_{0}\in X$ .

Definition 1.2.

The operator $A$ is a weakly Picard operator if the sequence $(A^{n}(x))_{n\in\mathbb{N}}$ converges, for all $x\in X$ , and the limit (which may depend on $x$ ) is a fixed point of $A$ .

Definition 1.3.

If $A$ is weakly Picard operator then we consider the operator $A^{\infty},\;A^{\infty}:X\rightarrow X$ , defined by

A^{\infty}(x):=\underset{n\rightarrow\infty}A^{n}(x).

Theorem 1.4.

[15] An operator $A$ is a weakly Picard operator if and only if there exists a partition of $X,$ $X={\textstyle\bigcup\limits_{\lambda\in\Lambda}}X_{\lambda},$ such that

(a)

$X_{\lambda}\in I(A),$ $\forall\lambda\in\Lambda;$
(b)

$\left.A\right|_{X_{\lambda}}:X_{\lambda}\rightarrow X_{\lambda}$ is a Picard operator, $\forall\lambda\in\Lambda.$

2. Bernstein type operators on a square with one curved side

In [4] there are introduced some Bernstein-type operators on a square with one curved side. In [3], [5] and [6] there have been introduced interpolation and Berstein-type operators on triangles with some curved sides.

Given $h>0,$ let $D_{h}$ be the square with one curved side having the vertices $V_{1}=(0,0),$ $V_{2}=(h,0),$ $V_{3}=(h,h)$ and $V_{4}=(0,h),$ three straight sides $\Gamma_{1},$ $\Gamma_{2},$ along the coordinate axes and $\Gamma_{3}$ parallel to axis $Ox,$ and the curved side $\Gamma_{4}$ which is defined by the function $g$ , such that $g(h)=g(0)=h$ (see Figure 1).

Figure 1. The square $D_{h}.$

Let $F$ be a real-valued function defined on $D_{h}$ and $(0,y)$ , $(g(y),y),$ respectively, $(x,0),$ $(x,h)$ be the points in which the parallel lines to the coordinate axes, passing through the point $(x,y)\in D_{h},$ intersect the sides $\Gamma_{2},$ $\Gamma_{4},$ respectively $\Gamma_{1}$ and $\Gamma_{3}.$ We consider the uniform partitions of the intervals $[0,g(y)]$ and $[0,h]$ , $y\in[0,h],$ $\Delta_{m}^{x}=\left\{\left.\tfrac{i}{m}g(y)\right|\ i=\overline{0,m}\right\}$ and $\Delta_{n}^{y}=\left\{\left.\tfrac{j}{n}h\right|\ j=\overline{0,n}\right\}$ and the Bernstein-type operators $B_{m}^{x}$ and $B_{n}^{y}$ defined by

\left(B_{m}^{x}F\right)\left(x,y\right)=\sum_{i=0}^{m}p_{m,i}\left(x,y\right)F\left(\tfrac{i}{m}g(y),y\right),

(1)

with

p_{m,i}\left(x,y\right)=\binom{m}{i}\left[\tfrac{x}{g(y)}\right]^{i}\left[1-\tfrac{x}{g(y)}\right]^{m-i},

respectively,

\left(B_{n}^{y}F\right)\left(x,y\right)=\sum_{j=0}^{n}q_{n,j}\left(x,y\right)F\left(x,\tfrac{j}{n}h\right)

(2)

with

q_{n,j}\left(x,y\right)=\binom{n}{j}\left(\tfrac{y}{h}\right)^{j}\left(1-\tfrac{y}{h}\right)^{n-j}.

Theorem 2.1.

[4] If $F$ is a real-valued function defined on $D_{h}\$ then we have

(1)

$B_{m}^{x}F=F\$ on $\Gamma_{2}\cup\Gamma_{4};$

$B_{n}^{y}F=F\$ on $\Gamma_{1}\cup\Gamma_{3},$
(2)

$\left(B_{m}^{x}e_{ij}\right)\left(x,y\right)=x^{i}y^{j},\ \ i=0,1;$ $j\in\mathbb{N};$

$\left(B_{n}^{y}e_{ij}\right)\left(x,y\right)=x^{i}y^{j},\$ $i\in\mathbb{N};$ $j=0,1.$

Remark 2.2.

The interpolation properties of $B_{m}^{x}F$ and $B_{n}^{y}F$ are illustrated in Figures 2 and 3. The bold sides indicate the interpolation sets.

Figure 2. Interpolation domain for $B_{m}^{x}F.$

Figure 3. Interpolation domain for $B_{y}^{n}F.$

Let $P_{mn}=B_{m}^{x}B_{n}^{y},$ respectively, $Q_{nm}=B_{n}^{y}B_{m}^{x}$ be the products of the operators $B_{m}^{x}$ and $B_{n}^{y}.$ We have

\left(P_{mn}F\right)\left(x,y\right)\!=\!\sum_{i=0}^{m}\sum_{j=0}^{n}p_{m,i}\left(x,y\right)q_{n,j}\left(i\tfrac{g(y)}{m},y\right)F\Big(i\tfrac{g(y)}{m},j\tfrac{h}{n}\Big),

(3)

respectively,

\left(Q_{nm}F\right)\left(x,y\right)\!=\!\sum_{i=0}^{m}\sum_{j=0}^{n}p_{m,i}\left(x,j\tfrac{h}{n}\right)q_{n,j}\left(x,y\right)F\Big(\tfrac{i}{m}g(j\tfrac{h}{n}),j\tfrac{h}{n}\Big).

(4)

Theorem 2.3.

[4] If $F$ is a real-valued function defined on $D_{h}$ then:

(1)

$(P_{mn}F)(V_{i})=F(V_{i}),\ \ \ \ i=1,...,4;$

$(Q_{nm}F)(V_{i})=F(V_{i}),\ \ \ \ i=1,...,4.$
(2)

$\left(P_{mn}e_{ij}\right)\left(x,y\right)=x^{i}y^{j},\ \ i=0,1;$ $j=0,1;$

$\left(Q_{nm}e_{ij}\right)\left(x,y\right)=x^{i}y^{j},\$ $i=0,1;$ $j=0,1.$

We consider the Boolean sums of the operators $B_{m}^{x}$ and $B_{n}^{y},$ i.e.,

S_{mn}:=B_{m}^{x}\oplus B_{n}^{y}=B_{m}^{x}+B_{n}^{y}-B_{m}^{x}B_{n}^{y},

(5)

respectively,

T_{nm}:=B_{n}^{y}\oplus B_{m}^{x}=B_{n}^{y}+B_{m}^{x}-B_{n}^{y}B_{m}^{x}.

(6)

3. Iterates of Bernstein type operators

Let $F$ be a real-valued function defined on $D_{h}$ , $h\in\mathbb{R}_{+}.$

Using the weakly Picard operators technique and the contraction principle, we obtain the following results regarding the convergence of the iterates of the Bernstein-type operators (1) and (2) and of their product and Boolean sum operators (3), (4), (5) and (6). The same approach for some other linear and positive operators lead to similar results in [1], [2], [16], [17] and [18].

The limit behavior for the iterates of some classes of positive linear operators were also studied, for example, in [14], [13], [7], [8], [9], [10], [11], [12].

Theorem 3.1.

The operators $B_{m}^{x}$ and $B_{n}^{y}$ are weakly Picard operators and

	$\displaystyle\left(B_{m}^{x,\infty}F\right)\left(x,y\right)$	$\displaystyle=F\left(0,y\right)+\frac{F(g(y),y)-F(0,y)}{g(y)}x,$		(7)
	$\displaystyle\left(B_{n}^{y,\infty}F\right)\left(x,y\right)$	$\displaystyle=F\left(x,0\right)+\frac{F(x,h)-F(x,0)}{h}y.$		(8)

Proof.

Taking into account the interpolation properties, from Theorem 2.1, of $B_{m}^{x}$ and $B_{n}^{y},$ let

	$\displaystyle X_{\left.\varphi\right\|_{\Gamma_{2}},\left.\varphi\right\|_{\Gamma_{4}}}^{(1)}$	$\displaystyle=\{F\in C(D_{h})\ \|\ F(0,y)=\left.\varphi\right\|_{\Gamma_{2}},\ F(g(y),y)=\left.\varphi\right\|_{\Gamma_{4}}\},\ \ \text{for }y\in[0,h],$
	$\displaystyle X_{\left.\psi\right\|_{\Gamma_{1}},\left.\psi\right\|_{\Gamma_{3}}}^{(2)}$	$\displaystyle=\{F\in C(D_{h})\ \|\ F(x,0)=\left.\psi\right\|_{\Gamma_{1}},\ F(x,h)=\left.\psi\right\|_{\Gamma_{3}}\},\ \ \text{for }x\in[0,h],$

and denote by

	$\displaystyle F_{\left.\varphi\right\|_{\Gamma_{2}},\left.\varphi\right\|_{\Gamma_{4}}}^{(1)}(x,y)$	$\displaystyle:=\left.\varphi\right\|_{\Gamma_{2}}+\frac{\left.\varphi\right\|_{\Gamma_{4}}-\left.\varphi\right\|_{\Gamma_{2}}}{g(y)}x,$
	$\displaystyle F_{\left.\psi\right\|_{\Gamma_{1}},\left.\psi\right\|_{\Gamma_{3}}}^{(2)}(x,y)$	$\displaystyle:=\left.\psi\right\|_{\Gamma_{1}}+\frac{\left.\psi\right\|_{\Gamma_{3}}-\left.\psi\right\|_{\Gamma_{1}}}{h}y,$

with $\varphi,\psi\in C\mathbb{(}D_{h}).$

We have the following properties:

(i)

$X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}$ and $X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}$ are closed subsets of $C(D_{h})$ ;
(ii)

$X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}$ is an invariant subset of $B_{m}^{x}$ and $X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}$ is an invariant subset of $B_{n}^{y}$ , for $\varphi,\psi\in C\mathbb{(}D_{h})$ and $n,m\in\mathbb{N}^{\ast};$
(iii)

$C(D_{h})=\underset{\varphi\in C\mathbb{(}D_{h})}{\cup}X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}$ and $C(D_{h})=\underset{\psi\in C\mathbb{(}D_{h})}{\cup}X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}$ are partitions of $C(D_{h})$ ;
(iv)

$F_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}\in X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}\cap F_{B_{m}^{x}}$ and $F_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}\in X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}\cap F_{B_{n}^{y}},$ where $F_{B_{m}^{x}}$ and $F_{B_{n}^{y}}$ denote the fixed points sets of $B_{m}^{x}$ and $B_{n}^{y}.$

The statements $(i)$ and $(iii)$ are obvious.

$(ii)$ By linearity of Bernstein operators and Theorem 2.1, it follows that $\forall F_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}\in X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}$ and $\forall F_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}\in X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}$ we have

	$\displaystyle B_{m}^{x}F_{\left.\varphi\right\|_{\Gamma_{2}},\left.\varphi\right\|_{\Gamma_{4}}}^{(1)}(x,y)$	$\displaystyle=F_{\left.\varphi\right\|_{\Gamma_{2}},\left.\varphi\right\|_{\Gamma_{4}}}^{(1)}(x,y),$
	$\displaystyle B_{n}^{y}F_{\left.\psi\right\|_{\Gamma_{1}},\left.\psi\right\|_{\Gamma_{3}}}^{(2)}(x,y)$	$\displaystyle=F_{\left.\psi\right\|_{\Gamma_{1}},\left.\psi\right\|_{\Gamma_{3}}}^{(2)}(x,y).$

So, $X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}$ and $X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}$ are invariant subsets of $B_{m}^{x}$ and, respectively, of $B_{n}^{y},\$ for $\varphi,\psi\in C\mathbb{(}D_{h})$ and $n,m\in\mathbb{N}^{\ast};$

$(iv)$ We prove that

\left.B_{m}^{x}\right|_{X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}}\!:\!X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}\!\rightarrow X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}\text{ and }\left.B_{n}^{y}\right|_{X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}}\!:\!X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}\!\rightarrow X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}

are contractions for $\varphi,\psi\in C\mathbb{(}D_{h})$ and $n,m\in\mathbb{N}^{\ast}.$

Let $F,G\in X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}$ . From (1) we have

	$\displaystyle\left\|B_{m}^{x}(F)(x,y)-B_{m}^{x}(G)(x,y)\right\|=\left\|B_{m}^{x}(F-G)(x,y)\right\|\leq$
	$\displaystyle\leq\left\|1-\left(1-\frac{x}{g(y)}\right)^{m}-\left(\frac{x}{g(y)}\right)^{m}\right\|\cdot\left\\|F-G\right\\|_{\infty}\leq$
	$\displaystyle\leq\left(1-\frac{1}{2^{m-1}}\right)\left\\|F-G\right\\|_{\infty},$

where $\left\|\cdot\right\|_{\infty}$ denotes the Chebyshev norm. So,

\left\|B_{m}^{x}(F)(x,y)-B_{m}^{x}(G)(x,y)\right\|_{\infty}\leq\left(1-\frac{1}{2^{m-1}}\right)\left\|F-G\right\|_{\infty},\forall F,G\in X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)},

i.e., $\left.B_{m}^{x}\right|_{X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}}$ is a contraction for $\varphi\in C\mathbb{(}D_{h})$ .

Analogously we have

\left\|B_{n}^{y}(F)(x,y)-B_{n}^{y}(G)(x,y)\right\|_{\infty}\leq\left(1-\frac{1}{2^{n-1}}\right)\left\|F-G\right\|_{\infty},\text{\ }\forall F,G\in X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)},

i.e., $\left.B_{n}^{y}\right|_{X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}}$ is a contraction for $\psi\in C\mathbb{(}D_{h}).$

On the other hand, $\left.\varphi\right|_{\Gamma_{2}}+\frac{\left.\varphi\right|_{\Gamma_{4}}-\left.\varphi\right|_{\Gamma_{2}}}{g(y)}(\cdot)\in X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)},$ $\left.\psi\right|_{\Gamma_{1}}+\frac{\left.\psi\right|_{\Gamma_{3}}-\left.\psi\right|_{\Gamma_{1}}}{h}(\cdot)\in X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}$ are fixed points of $B_{m}^{x}$ and $B_{n}^{y}$ , i.e.,

	$\displaystyle B_{m}^{x}(\left.\varphi\right\|_{\Gamma_{2}}+\frac{\left.\varphi\right\|_{\Gamma_{4}}-\left.\varphi\right\|_{\Gamma_{2}}}{g(y)}(\cdot))$	$\displaystyle=\left.\varphi\right\|_{\Gamma_{2}}+\frac{\left.\varphi\right\|_{\Gamma_{4}}-\left.\varphi\right\|_{\Gamma_{2}}}{g(y)}(\cdot),$
	$\displaystyle B_{n}^{y}(\left.\psi\right\|_{\Gamma_{1}}+\frac{\left.\psi\right\|_{\Gamma_{3}}-\left.\psi\right\|_{\Gamma_{1}}}{h}(\cdot))$	$\displaystyle=\left.\psi\right\|_{\Gamma_{1}}+\frac{\left.\psi\right\|_{\Gamma_{3}}-\left.\psi\right\|_{\Gamma_{1}}}{h}(\cdot).$

From the contraction principle, $F_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}(x,y):=\left.\varphi\right|_{\Gamma_{2}}+\frac{\left.\varphi\right|_{\Gamma_{4}}-\left.\varphi\right|_{\Gamma_{2}}}{g(y)}x$ is the unique fixed point of $B_{m}^{x}$ in $X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}$ and $\left.B_{m}^{x}\right|_{X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}}$ is a Picard operator, with

\left(B_{m}^{x,\infty}F\right)\left(x,y\right)=F\left(0,y\right)+\frac{F(g(y),y)-F(0,y)}{g(y)}x,

and, similarly, $F_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}(x,y):=\left.\psi\right|_{\Gamma_{1}}+\frac{\left.\psi\right|_{\Gamma_{3}}-\left.\psi\right|_{\Gamma_{1}}}{h}y$ is the unique fixed point of $B_{n}^{y}$ in $X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}$ and $\left.B_{n}^{y}\right|_{X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}}$ is a Picard operator, with

\left(B_{n}^{y,\infty}F\right)\left(x,y\right)=F\left(x,0\right)+\frac{F(x,h)-F(x,0)}{h}y,

Consequently, taking into account $(ii)$ , by Theorem 1.4 it follows that the operators $B_{m}^{x}$ and $B_{n}^{y}$ are weakly Picard operators. ∎

Theorem 3.2.

The operators $P_{mn}$ and $Q_{nm}$ are weakly Picard operators and

$\displaystyle\left(P_{mn}^{\infty}F\right)\left(x,y\right)$	$\displaystyle=F\left(0,0\right)+\frac{F(h,0)-F(0,0)}{g(y)}x+\frac{F(0,h)-F(0,0)}{h}y$	(9)
	$\displaystyle\quad+\frac{F(0,0)-F(0,h)-F(h,0)+F(h,h)}{g(y)h}xy,$
$\displaystyle\left(Q_{nm}^{\infty}F\right)\left(x,y\right)$	$\displaystyle=F\left(0,0\right)+\frac{F(h,0)-F(0,0)}{g(y)}x+\frac{F(0,h)-F(0,0)}{h}y$	(10)
	$\displaystyle\quad+\frac{F(0,0)-F(0,h)-F(h,0)+F(h,h)}{g(y)h}xy.$

Proof.

Let

X_{\alpha,\beta,\gamma,\delta}=\{F\in C(D_{h})\ |\ F(0,0)=\alpha,\ F(0,h)=\beta,F(h,h)=\gamma,F(h,0)=\delta\}

and denote by

F_{\alpha,\beta,\gamma,\delta}(x,y):=\alpha+\frac{\delta-\alpha}{g(y)}x+\frac{\beta-\alpha}{h}y+\frac{\alpha-\beta-\delta+\gamma}{g(y)h}xy

with $\alpha,\beta,\gamma,\delta\in\mathbb{R}.$

We remark that

(i)

$X_{\alpha,\beta,\gamma,\delta}$ is closed subset of $C(D_{h})$ ;
(ii)

$X_{\alpha,\beta,\gamma,\delta}$ is an invariant subset of $P_{mn}$ and $Q_{nm}$ , for $\alpha,\beta,\gamma,\delta\in\mathbb{R}$ and $n,m\in\mathbb{N}^{\ast};$
(iii)

$C(D_{h})=\underset{\alpha,\beta,\gamma,\delta}{\cup}X_{\alpha,\beta,\gamma,\delta}$ is a partition of $C(D_{h})$ ;
(iv)

$F_{\alpha,\beta,\gamma,\delta}\in X_{\alpha,\beta,\gamma,\delta}\cap F_{P_{mn}}$ and $F_{\alpha,\beta,\gamma,\delta}\in X_{\alpha,\beta,\gamma,\delta}\cap F_{Q_{nm}},$ where $F_{P_{mn}}$ and $F_{Q_{nm}}$ denote the fixed points sets of $P_{mn}$ and $Q_{nm}.$

The statements $(i)$ and $(iii)$ are obvious.

$(ii)$ Similarly with the proof of Theorem 3.1, by linearity of Bernstein operators and Theorem 2.3, it follows that $X_{\alpha,\beta,\gamma,\delta}$ is an invariant subset of $P_{mn}$ and, respectively, of $Q_{nm}$ , for $\alpha,\beta,\gamma,\delta\in\mathbb{R}$ and $n,m\in\mathbb{N}^{\ast};$

$(iv)$ We prove that

\left.P_{mn}\right|_{X_{\alpha,\beta,\gamma,\delta}}:X_{\alpha,\beta,\gamma,\delta}\rightarrow X_{\alpha,\beta,\gamma,\delta}\text{ and }\left.Q_{nm}\right|_{X_{\alpha,\beta,\gamma,\delta}}:X_{\alpha,\beta,\gamma,\delta}\rightarrow X_{\alpha,\beta,\gamma,\delta}

are contractions for $\alpha,\beta,\gamma,\delta\in\mathbb{R}$ and $n,m\in\mathbb{N}^{\ast}.$ Let $F,G\in X_{\alpha,\beta,\gamma,\delta}$ . From [2, Lemma 8] it follows that

	$\displaystyle\left\|P_{mn}(F)(x,y)-P_{mn}(G)(x,y)\right\|=\left\|P_{mn}(F-G)(x,y)\right\|\leq$
	$\displaystyle\leq\left(1-\frac{1}{2^{m+n-2}}\right)\left\\|F-G\right\\|_{\infty}.$

So,

\left\|P_{mn}(F)(x,y)-P_{mn}(G)(x,y)\right\|_{\infty}\leq\left(1-\frac{1}{2^{m+n-2}}\right)\left\|F-G\right\|_{\infty},\forall F,G\in X_{\alpha,\beta,\gamma,\delta},

i.e., $\left.P_{mn}\right|_{X_{\alpha,\beta,\gamma,\delta}}$ is a contraction for $\alpha,\beta,\gamma,\delta\in\mathbb{R}.$ Analogously, we have

\left\|Q_{nm}(F)(x,y)-Q_{nm}(G)(x,y)\right\|_{\infty}\leq\left(1-\frac{1}{2^{m+n-2}}\right)\left\|F-G\right\|_{\infty},\text{\ }\forall F,G\in X_{\alpha,\beta,\gamma,\delta},

i.e., $\left.Q_{nm}\right|_{X_{\alpha,\beta,\gamma,\delta}}$ is a contraction for $\alpha,\beta,\gamma,\delta\in\mathbb{R}$ .

We have that

F_{\alpha,\beta,\gamma,\delta}(x,y):=\alpha+\frac{\delta-\alpha}{g(y)}x+\frac{\beta-\alpha}{h}y+\frac{\alpha-\beta-\delta+\gamma}{g(y)h}xy

and

	$\displaystyle P_{mn}$	$\displaystyle\left(\alpha+\frac{\delta-\alpha}{g(y)}x+\frac{\beta-\alpha}{h}y+\frac{\alpha-\beta-\delta+\gamma}{g(y)h}xy\right)$
		$\displaystyle=\alpha+\frac{\delta-\alpha}{g(y)}x+\frac{\beta-\alpha}{h}y+\frac{\alpha-\beta-\delta+\gamma}{g(y)h}xy,$

	$\displaystyle Q_{nm}$	$\displaystyle\left(\alpha+\frac{\delta-\alpha}{g(y)}x+\frac{\beta-\alpha}{h}y+\frac{\alpha-\beta-\delta+\gamma}{g(y)h}xy\right)$
		$\displaystyle=\alpha+\frac{\delta-\alpha}{g(y)}x+\frac{\beta-\alpha}{h}y+\frac{\alpha-\beta-\delta+\gamma}{g(y)h}xy.$

From the contraction principle we have that $F_{\alpha,\beta,\gamma,\delta}$ is the unique fixed point of $P_{mn}$ in $X_{\alpha,\beta,\gamma,\delta}$ and $\left.P_{mn}\right|_{X_{\alpha,\beta,\gamma,\delta}}$ is a Picard operator and, respectively, $F_{\alpha,\beta,\gamma,\delta}$ is the unique fixed point of $Q_{nm}$ in $X_{\alpha,\beta,\gamma,\delta}$ and $\left.Q_{nm}\right|_{X_{\alpha,\beta,\gamma,\delta}}$ is a Picard operator, so (9) and (10) hold. Consequently, taking into account $(ii)$ , by Theorem 1.4 it follows that the operators $P_{mn}$ and $Q_{nm}$ are weakly Picard operators. ∎

Theorem 3.3.

The operator $S_{mn}$ is weakly Picard operator and

	$\displaystyle\left(S_{mn}^{\infty}F\right)\left(x,y\right)$	$\displaystyle=F\left(0,y\right)+F\left(x,0\right)-F\left(0,0\right)$
		$\displaystyle+\frac{F(g(y),y)-F(0,y)-F(h,0)+F(0,0)}{g(y)}x$
		$\displaystyle+\frac{F(x,h)-F(x,0)-F(0,h)+F(0,0)}{h}y$
		$\displaystyle-\frac{F(0,0)-F(0,h)-F(h,0)+F(h,h)}{g(y)h}xy.$

Proof.

The proof follows the same steps as in the previous theorems but using the following inequality

\left\|S_{mn}(F)(x,y)-S_{mn}(G)(x,y)\right\|_{\infty}\leq\left[1-\left(\frac{1}{2^{m-1}}+\frac{1}{2^{n-1}}-\frac{1}{2^{m+n-2}}\right)\right]\left\|F-G\right\|_{\infty},

in order to prove that $S_{mn}$ is a contraction. ∎

Remark 3.4.

We have an analogous result for the operator $T_{nm}$ .

Acknowledgement The authors are grateful to professor I. A. Rus for his helpful comments and suggestions.

References

[1] O. Agratini, I.A. Rus, Iterates of a class of discrete linear operators via contraction principle, Comment. Math. Univ. Caroline, 44(2003), 555-563.
[2] O. Agratini, I.A. Rus, Iterates of some bivariate approximation process via weakly Picard operators, Nonlinear Analysis Forum, 8(2)(2003), 159-168.
[3] P. Blaga, T. Cătinaş, G. Coman, Bernstein-type operators on triangle with one curved side, Mediterr. J. Math., 10(2013), 10.1007/s00009-011-0156-2, in press.
[4] P. Blaga, T. Cătinaş, G. Coman, Bernstein-type operators on a square with one and two curved sides, Studia Univ. Babeş–Bolyai Math., 55(2010), No. 3, 51-67.
[5] P. Blaga, T. Cătinaş, G. Coman, Bernstein-type operators on triangle with all curved sides, Appl. Math. Comput., 218(2011), 3072-3082.
[6] G. Coman, T. Cătinaş, Interpolation operators on a triangle with one curved side, BIT Numerical Mathematics, 50(2010), No. 2, 243-267.
[7] I. Gavrea, M. Ivan, The iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl., 372(2010), 366-368.
[8] I. Gavrea, M. Ivan, The iterates of positive linear operators preserving the constants, Appl. Math. Lett., 24(2011), No. 12, 2068-2071.
[9] I. Gavrea, M. Ivan, On the iterates of positive linear operators, J. Approximation Theory, 163(2011), No. 9, 1076-1079.
[10] H. Gonska, D. Kacsó, P. Piţul, The degree of convergence of over-iterated positive linear operators, J. Appl. Funct. Anal., 1(2006), 403-423.
[11] H. Gonska, P. Piţul, I. Raşa Over-iterates of Bernstein-Stancu operators, Calcolo, 44(2007), 117-125.
[12] H. Gonska, I. Raşa The limiting semigroup of the Bernstein iterates: degree of convergence, Acta Math. Hungar., 111(2006), No. 1-2, 119-130.
[13] S. Karlin, Z. Ziegler, Iteration of positive approximation operators, J. Approximation Theory 3(1970), 310-339.
[14] R.P. Kelisky, T.J. Rivlin, Iterates of Bernstein polynomials, Pacific J. Math., 21(1967), 511-520.
[15] I.A. Rus, Generalized contractions and applications, Cluj Univ. Press, 2001.
[16] I.A. Rus, Iterates of Stancu operators, via contraction principle, Studia Univ. Babeş–Bolyai Math., 47(2002), No. 4, 101-104.
[17] I.A. Rus, Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl., 292(2004), 259-261.
[18] I.A. Rus, Fixed point and interpolation point set of a positive linear operator on $C(\overline{D})$ , Studia Univ. Babeş–Bolyai Math., 55(2010), No. 4, 243-248.

	$\displaystyle X_{\left.\varphi\right\|_{\Gamma_{2}},\left.\varphi\right\|_{\Gamma_{4}}}^{(1)}$	$\displaystyle=\{F\in C(D_{h})\ \|\ F(0,y)=\left.\varphi\right\|_{\Gamma_{2}},\ F(g(y),y)=\left.\varphi\right\|_{\Gamma_{4}}\},\ \ \text{for }y\in[0,h],$
	$\displaystyle X_{\left.\psi\right\|_{\Gamma_{1}},\left.\psi\right\|_{\Gamma_{3}}}^{(2)}$	$\displaystyle=\{F\in C(D_{h})\ \|\ F(x,0)=\left.\psi\right\|_{\Gamma_{1}},\ F(x,h)=\left.\psi\right\|_{\Gamma_{3}}\},\ \ \text{for }x\in[0,h],$

	$\displaystyle F_{\left.\varphi\right\|_{\Gamma_{2}},\left.\varphi\right\|_{\Gamma_{4}}}^{(1)}(x,y)$	$\displaystyle:=\left.\varphi\right\|_{\Gamma_{2}}+\frac{\left.\varphi\right\|_{\Gamma_{4}}-\left.\varphi\right\|_{\Gamma_{2}}}{g(y)}x,$
	$\displaystyle F_{\left.\psi\right\|_{\Gamma_{1}},\left.\psi\right\|_{\Gamma_{3}}}^{(2)}(x,y)$	$\displaystyle:=\left.\psi\right\|_{\Gamma_{1}}+\frac{\left.\psi\right\|_{\Gamma_{3}}-\left.\psi\right\|_{\Gamma_{1}}}{h}y,$

	$\displaystyle B_{m}^{x}F_{\left.\varphi\right\|_{\Gamma_{2}},\left.\varphi\right\|_{\Gamma_{4}}}^{(1)}(x,y)$	$\displaystyle=F_{\left.\varphi\right\|_{\Gamma_{2}},\left.\varphi\right\|_{\Gamma_{4}}}^{(1)}(x,y),$
	$\displaystyle B_{n}^{y}F_{\left.\psi\right\|_{\Gamma_{1}},\left.\psi\right\|_{\Gamma_{3}}}^{(2)}(x,y)$	$\displaystyle=F_{\left.\psi\right\|_{\Gamma_{1}},\left.\psi\right\|_{\Gamma_{3}}}^{(2)}(x,y).$

	$\displaystyle B_{m}^{x}(\left.\varphi\right\|_{\Gamma_{2}}+\frac{\left.\varphi\right\|_{\Gamma_{4}}-\left.\varphi\right\|_{\Gamma_{2}}}{g(y)}(\cdot))$	$\displaystyle=\left.\varphi\right\|_{\Gamma_{2}}+\frac{\left.\varphi\right\|_{\Gamma_{4}}-\left.\varphi\right\|_{\Gamma_{2}}}{g(y)}(\cdot),$
	$\displaystyle B_{n}^{y}(\left.\psi\right\|_{\Gamma_{1}}+\frac{\left.\psi\right\|_{\Gamma_{3}}-\left.\psi\right\|_{\Gamma_{1}}}{h}(\cdot))$	$\displaystyle=\left.\psi\right\|_{\Gamma_{1}}+\frac{\left.\psi\right\|_{\Gamma_{3}}-\left.\psi\right\|_{\Gamma_{1}}}{h}(\cdot).$

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Iterates of Bernstein type operators on a square with one curved side via contraction principle

1. Weakly Picard operators

Definition 1.1.

Definition 1.2.

Definition 1.3.

Theorem 1.4.

2. Bernstein type operators on a square with one curved side

Theorem 2.1.

Remark 2.2.

Theorem 2.3.

3. Iterates of Bernstein type operators

Theorem 3.1.

Proof.

Theorem 3.2.

Proof.

Theorem 3.3.

Proof.

Remark 3.4.

References

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