Iterates of Cheney-Sharma type operators on a triangle with curved side

5 years ago

Abstract

We consider some Cheney-Sharma type operators as well as their product and Boolean sum for a function defined on a triangle with one curved side.

Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of these operators.

Authors

Teodora Cătinaș
Babes-Bolyai University, Cluj Napoca, Romania

Diana Otrocol
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Technical University of Cluj Napoca, Romania

Keywords

Triangle with curved side; Cheney-Sharma operators; contraction principle; weakly Picard operators

References

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T. Cătinaș, D. Otrocol, Iterates of Cheney-Sharma type operators on a triangle with curved side, Journal Computational Analysis and Applications, 28 (2020) no. 4, pp. 737-744.

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Iterates of Cheney-Sharma type operators on a triangle with curved side

Teodora Cătinaş^∗, Diana Otrocol^∗∗

Abstract.

We consider some Cheney–Sharma type operators as well as their product and Boolean sum for a function defined on a triangle with one curved side. Using the weakly Picard operators technique and the contraction principle, we study the convergence of the iterates of these operators.

Keywords: Triangle with curved side, Cheney-Sharma operators, contraction principle, weakly Picard operators.

MSC 2010 Subject Classification: 41A36, 41A25, 39B12, 47H10.

^∗Babeş-Bolyai University, Faculty of Mathematics and Computer Science, M. Kogălniceanu St. 1, RO-400084 Cluj Napoca, Romania, E-mail: tcatinas@math.ubbcluj.ro

^∗∗T. Popoviciu Institute of Numerical Analysis, Romanian Academy, Cluj Napoca, Romania; Technical University of Cluj Napoca, Memorandumului St. 28, RO-400114, Cluj Napoca, Romania, E-mail: Diana.Otrocol@math.utcluj.ro

1. Cheney-Sharma type operators

We recall some results regarding Cheney-Sharma type operators on a triangle with one curved side, introduced in [6]. Similar operators were introduced and studied in [3], [4], [5] and [9].

We consider the standard triangle $\tilde{T}_{h}$ with vertices $V_{1}=(0,h),$ $V_{2}=(h,0)$ and $V_{3}=(0,0),$ with two straight sides $\Gamma_{1},$ $\Gamma_{2},$ along the coordinate axes, and with the third side $\Gamma_{3}$ (opposite to the vertex $V_{3}$ ) defined by the one-to-one functions $f$ and $g,$ where $g$ is the inverse of the function $f,$ i.e., $y=f(x)$ and $x=g(y)$ , with $f(0)=g(0)=h,\$ for $h>0$ . Also, we have $f(x)\leq h$ and $g(y)\leq h,$ for $x,y\in\left[0,h\right].$

Let $F$ be a real-valued function defined on $\widetilde{T}_{h}$ and $(0,y),$ $(g(y),y),$ respectively, $(x,$ $0),$ $(x,f(x))$ be the points in which the parallel lines to the coordinate axes, passing through the point $(x,y)\in\widetilde{T}_{h},$ intersect the sides $\Gamma_{i},$ $i=1,2,3.$ (See Figure 1.)

Refer to caption — Figure 1. Triangle $\tilde{T}_{h}.$

In [6], we have obtained the following extensions of Cheney-Sharma operator of second kind, to the case of functions defined on $\widetilde{T}_{h}$ :

(1.1)		$\displaystyle(Q_{m}^{x}F)(x,y)$	$\displaystyle={\textstyle\sum\limits_{i=0}^{m}}{q}_{m,i}(x,y)F\left(i\tfrac{g(y)}{m},y\right),$
	$\displaystyle(Q_{n}^{y}F)(x,y)$	$\displaystyle={\textstyle\sum\limits_{j=0}^{n}}{q}_{n,j}(x,y)F\left(x,j\tfrac{f(x)}{n}\right),$

with

	$\displaystyle q_{m,i}\left(x,y\right)$	$\displaystyle={\tbinom{m}{i}}\tfrac{1}{(1+m\beta)^{m-1}}\tfrac{x}{g(y)}(\tfrac{x}{g(y)}+i\beta)^{i-1}(1-\tfrac{x}{g(y)})[1-\tfrac{x}{g(y)}+(m-i)\beta]^{m-i-1},$
	$\displaystyle q_{n,j}\left(x,y\right)$	$\displaystyle={\tbinom{n}{j}}\tfrac{1}{(1+nb)^{n-1}}\tfrac{y}{f(x)}(\tfrac{y}{f(x)}+jb)^{j-1}(1-\tfrac{y}{f(x)})[1-\tfrac{y}{f(x)}+(n-j)b]^{n-j-1},$

where

\Delta_{m}^{x}=\left\{\left.i\tfrac{g(y)}{m}\right|\ i=\overline{0,m}\right\}\ \text{and }\Delta_{n}^{y}=\left\{\left.j\tfrac{f(x)}{n}\right|\ j=\overline{0,n}\right\}

are uniform partitions of the intervals $[0,g(y)]$ and $[0,f(x)]$ and $m,n\in\mathbb{N},$ $\beta,b\in\mathbb{R}_{+}.$

Remark 1.1.

As the Cheney-Sharma operator of second kind interpolates a given function at the endpoints of the interval, we may use the operators $Q_{m}^{x}$ and $Q_{n}^{y}$ as interpolation operators on $\widetilde{T}_{h}.$

Theorem 1.2.

[6] If $F$ is a real-valued function defined on $\widetilde{T}_{h}\$ then the following properties hold:

(i)

$Q_{m}^{x}F=F\$ on $\Gamma_{1}\cup\Gamma_{3};$
(ii)

$Q_{n}^{y}F=F\$ on $\Gamma_{2}\cup\Gamma_{3};$
(iii)

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

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

Let $P_{mn}^{1}=Q_{m}^{x}Q_{n}^{y},$ respectively, $P_{nm}^{2}=Q_{n}^{y}Q_{m}^{x}$ be the products of the operators $Q_{m}^{x}$ and $Q_{n}^{y}.$ We have

(1.2)

\left(P_{mn}^{1}F\right)\left(x,y\right)\!=\!\sum_{i=0}^{m}\sum_{j=0}^{n}q_{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{f(i\tfrac{g(y)}{m})}{n}\Big),

respectively,

\left(P_{nm}^{2}F\right)\left(x,y\right)\!=\!\sum_{i=0}^{m}\sum_{j=0}^{n}q_{m,i}\left(x,j\tfrac{f(x)}{n}\right)q_{n,j}\left(x,y\right)F\Big(i\tfrac{g(j\tfrac{f(x)}{n})}{m},j\tfrac{f(x)}{n}\Big).

Theorem 1.3.

If $F$ is a real-valued function defined on $\widetilde{T}_{h}$ then

(i)

$(P_{mn}^{1}F)(V_{i})=F(V_{i}),\ \ \ \ i=1,...,3;$

$(P_{mn}^{1}F)(\Gamma_{3})=F(\Gamma_{3}),\$
(ii)

$(P_{nm}^{2}F)(V_{i})=F(V_{i}),\ \ \ \ i=1,...,3;$

$(P_{nm}^{2}F)(\Gamma_{3})=F(\Gamma_{3}).$

We consider the Boolean sums of the operators $Q_{m}^{x}$ and $Q_{n}^{y}$ ,

(1.3)		$\displaystyle S_{mn}^{1}$	$\displaystyle:=Q_{m}^{x}\oplus Q_{n}^{y}=Q_{m}^{x}+Q_{n}^{y}-Q_{m}^{x}Q_{n}^{y},$
	$\displaystyle S_{nm}^{2}$	$\displaystyle:=Q_{n}^{y}\oplus Q_{m}^{x}=Q_{n}^{y}+Q_{m}^{x}-Q_{n}^{y}Q_{m}^{x}.$

Theorem 1.4.

If $F$ is a real-valued function defined on $\widetilde{T}_{h},$ then

	$\displaystyle S_{mn}^{1}F\left\|{}_{\partial\widetilde{T}_{h}}=F\right\|_{\partial\widetilde{T}_{h}},$
	$\displaystyle S_{mn}^{2}F\left\|{}_{\partial\widetilde{T}_{h}}=F\right\|_{\partial\widetilde{T}_{h}}.$

2. Weakly Picard operators

We recall some results regarding weakly Picard operators that will be used in the sequel (see, e.g., [21]).

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

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

Definition 2.1.

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

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

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

Definition 2.2.

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 2.3.

If $A$ is a 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 2.4.

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, $\forall\lambda\in\Lambda.$

3. Iterates of Cheney-Sharma type operators

We study the convergence of the iterates of the Cheney-Sharma type operators (1.1) and of their product and Boolean sum operators, using the weakly Picard operators technique and the contraction principle. The same approach for some other linear and positive operators lead to similar results in [1], [2], [7], [8], [22]-[24].

The limit behavior for the iterates of some classes of positive linear operators were also studied, for example, in [10]-[20]. In the papers [10]-[12] were introduced new methods for the study of the asymptotic behavior of the iterates of positive linear operators. These techniques enlarge the class of operators for which the limit of the iterates can be calculated.

Let $F$ be a real-valued function defined on $\widetilde{T}_{h}$ , $h\in\mathbb{R}_{+}.$ First we study the convergence of the iterates of the Cheney–Sharma type operators given in (1.1).

Theorem 3.1.

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

(3.1)		$\displaystyle\left(Q_{m}^{x,\infty}F\right)\left(x,y\right)$	$\displaystyle=\frac{F\left(g(y),y\right)-F\left(0,y\right)}{g(y)}x+F\left(0,y\right),$
(3.2)		$\displaystyle\left(Q_{n}^{y,\infty}F\right)\left(x,y\right)$	$\displaystyle=\frac{F\left(x,f(x)\right)-F\left(x,0\right)}{f(x)}y+F\left(x,0\right).$

Proof.

Taking into account the interpolation properties of $Q_{m}^{x}$ and $Q_{n}^{y}$ (from Theorem 1.2), let us consider the following sets:

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

and for $\varphi,\psi\in C\mathbb{(}\widetilde{T}_{h})$ we denote by

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

We have the following properties:

(i)

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

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

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

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

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

$(ii)$ By linearity of Cheney-Sharma operators and Theorem 1.2, it follows that $\forall F_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}\in X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}$ and $\forall F_{\left.\psi\right|_{\Gamma_{2}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}\in X_{\left.\psi\right|_{\Gamma_{2}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}$ we have

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

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

$(iv)$ We prove that

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

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

Let $F,G\in X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}$ . From (1.1) and (3.3) we get

	$\displaystyle\left\|Q_{m}^{x}(F)(x,y)-Q_{m}^{x}(G)(x,y)\right\|=$
	$\displaystyle=\left\|Q_{m}^{x}(F-G)(x,y)\right\|\leq$
	$\displaystyle\leq\left\|q_{m,0}\left(x;y\right)\left[F\left(0,0\right)-G(0,0)\right]\right\|$
	$\displaystyle\quad+\left\|\sum_{i=1}^{m}q_{m,i}\left(x;y\right)\left[F\left(\tfrac{ig(y)}{m},y\right)-G\left(x,\tfrac{jf(x)}{n}\right)\right]\right\|$
	$\displaystyle=\left\|\sum_{i=1}^{m}q_{m,i}\left(x;y\right)\left[F\left(\tfrac{ig(y)}{m},y\right)-G\left(x,\tfrac{jf(x)}{n}\right)\right]\right\|$
	$\displaystyle\leq\sum_{i=1}^{m}q_{m,i}\left(x;y\right)\left\\|F-G\right\\|_{\infty}$
	$\displaystyle=\left[\sum_{i=0}^{m}q_{m,i}\left(x;y\right)-q_{m,0}\left(x;y\right)\right]\left\\|F-G\right\\|_{\infty}$
	$\displaystyle=\left\{1-\left(1-\tfrac{x}{g(y)}\right)\left[1-\tfrac{x}{g(y)(1+m\beta)}\right]^{m-1}\right\}\left\\|F-G\right\\|_{\infty}$
	$\displaystyle\leq\left[1-\left(1-\tfrac{1}{1+m\beta}\right)^{m-1}\right]\left\\|F-G\right\\|_{\infty},$

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

Hence,

(3.4)		$\displaystyle\left\\|Q_{m}^{x}(F)(x,y)-Q_{m}^{x}(G)(x,y)\right\\|_{\infty}\leq$
	$\displaystyle\leq\left[1-\left(1-\tfrac{1}{1+m\beta}\right)^{m-1}\right]\left\\|F-G\right\\|_{\infty},\ \forall F,G\in X_{\left.\varphi\right\|_{\Gamma_{1}},\left.\varphi\right\|_{\Gamma_{3}}}^{(1)},$

i.e., $\left.Q_{m}^{x}\right|_{X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}}$ is a contraction for $\varphi\in C\mathbb{(}\widetilde{T}_{h})$ .

Analogously, we prove that $\left.Q_{n}^{y}\right|_{X_{\left.\psi\right|_{\Gamma_{2}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}}$ is a contraction for $\psi\in C\mathbb{(}\widetilde{T}_{h}).$

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

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

From the contraction principle, $F_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}(x,y):=\tfrac{\left.\varphi\right|_{\Gamma_{3}}-\left.\varphi\right|_{\Gamma_{1}}}{g(y)}x+\left.\varphi\right|_{\Gamma_{1}}$ is the unique fixed point of $Q_{m}^{x}$ in $X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}$ and $\left.Q_{m}^{x}\right|_{X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}}$ is a Picard operator, with

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

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

\left(Q_{n}^{y,\infty}F\right)\left(x,y\right)=\tfrac{F\left(x,f(x)\right)-F\left(x,0\right)}{f(x)}y+F\left(x,0\right).

Consequently, taking into account $(ii)$ , by Theorem 2.4, it follows that the operators $Q_{m}^{x}$ and $Q_{n}^{y}$ are weakly Picard operators. ∎

Further we study the convergence of the product and Boolean sum operators given in (1.2) and (1.3).

Theorem 3.2.

The operator $P_{mn}^{1}$ is a weakly Picard operator and

(3.5)

\left(P_{mn}^{1,\infty}F\right)\left(x,y\right)=\frac{F\left(g(y),y\right)}{g(y)}x.

Proof.

Let

X_{\alpha}=\{F\in C(\widetilde{T}_{h})\ |\ F(g(y),y)=\alpha\},\ \ \ \ \alpha\in\mathbb{R}

and denote by

F_{\alpha}(x,y):=\frac{\alpha}{g(y)}x.

We remark that:

(i)

$X_{\alpha}$ is a closed subset of $C(\widetilde{T}_{h})$ ;
(ii)

$X_{\alpha}$ is an invariant subset of $P_{mn}^{1}$ , for $\alpha\in\mathbb{R}$ and $n,m\in\mathbb{N}^{\ast};$
(iii)

$C(\widetilde{T}_{h})=\underset{\alpha}{\cup}X_{\alpha}$ is a partition of $C(\widetilde{T}_{h})$ ;
(iv)

$F_{\alpha}\in X_{\alpha}\cap F_{P_{mn}^{1}},$ where $F_{P_{mn}^{1}}$ denote the fixed points sets of $P_{mn}^{1}.$

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

$(ii)$ Similarly with the proof of Theorem 3.1, by linearity of Cheney-Sharma operators and Theorem 1.3, it follows that $X_{\alpha}$ is an invariant subset of $P_{mn}^{1}$ , for $\alpha\in\mathbb{R}$ and $n,m\in\mathbb{N}^{\ast};$

$(iv)$ We prove that

\left.P_{mn}^{1}\right|_{X_{\alpha}}:X_{\alpha}\rightarrow X_{\alpha}\text{ }

is a contraction for $\alpha\in\mathbb{R}$ and $n,m\in\mathbb{N}^{\ast}.$ Let $F,G\in X_{\alpha}$ . From [2, Lemma 8] and (3.4), it follows that

	$\displaystyle\left\|P_{mn}^{1}(F)(x,y)-P_{mn}^{1}(G)(x,y)\right\|=\left\|P_{mn}^{1}(F-G)(x,y)\right\|$
	$\displaystyle\leq\left[1-\left(\tfrac{m\beta}{1+m\beta}\right)^{m-1}\left(\tfrac{nb}{1+nb}\right)^{n-1}\right]\left\\|F-G\right\\|_{\infty},$

so, $\left.P_{mn}^{1}\right|_{X_{\alpha}}$ is a contraction for $\alpha\in\mathbb{R}.$

From the contraction principle we have that $F_{\alpha}$ is the unique fixed point of $P_{mn}^{1}$ in $X_{\alpha}$ and $\left.P_{mn}^{1}\right|_{X_{\alpha}}$ is a Picard operator, so (3.5) holds. Consequently, taking into account $(ii)$ , by Theorem 2.4, it follows that the operators $P_{mn}^{1}$ is a weakly Picard operator. ∎

Remark 3.3.

Similar results can be obtained for the operator $P_{mn}^{2}$ .

Theorem 3.4.

The operator $S_{mn}^{1}$ is a weakly Picard operator and

\left(S_{mn}^{1,\infty}F\right)\left(x,y\right)=\tfrac{-F\left(0,y\right)}{g(y)}x+\tfrac{F\left(x,f(x)\right)-F\left(x,0\right)}{f(x)}y+F\left(x,0\right)+F\left(0,y\right).

Proof.

The proof follows the same steps as the proof of Theorem 3.2, using the following inequality

	$\displaystyle\left\\|S_{mn}(F)(x,y)-S_{mn}(G)(x,y)\right\\|_{\infty}$
	$\displaystyle\leq\left\{1-\left[\left(\tfrac{m\beta}{1+m\beta}\right)^{m-1}+\left(\tfrac{nb}{1+nb}\right)^{n-1}-\left(\tfrac{m\beta}{1+m\beta}\right)^{m-1}\left(\tfrac{nb}{1+nb}\right)^{n-1}\right]\right\}\left\\|F-G\right\\|_{\infty},$

for proving that $S_{mn}^{1}$ is a contraction. ∎

Remark 3.5.

We have a similar result for the operator $S_{nm}^{2}$ .

References

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[6] Cătinaş, T., Extension of some Cheney-Sharma type operators to a triangle with one curved side, 2017, submitted.
[7] Cătinaş, T., Otrocol, D., Iterates of Bernstein type operators on a square with one curved side via contraction principle, Fixed Point Theory 14 (2013), No. 1, 97-106.
[8] Cătinaş, T., Otrocol, D., Iterates of multivariate Cheney-Sharma operators, J. Comput. Anal. Appl. 15 (2013), No. 7, 1240-1246.
[9] Coman, G., Cătinaş, T., Interpolation operators on a triangle with one curved side, BIT Numerical Mathematics 50 (2010), No. 2, 243-267.
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(3.3)		$\displaystyle X_{\left.\varphi\right\|_{\Gamma_{1}},\left.\varphi\right\|_{\Gamma_{3}}}^{(1)}$	$\displaystyle=\{F\in C(\widetilde{T}_{h})\ \|\ F\left(0,y\right)=\left.\varphi\right\|_{\Gamma_{1}},\ F\left(g(y),y\right)=\left.\varphi\right\|_{\Gamma_{3}}\},\ \ \text{for }y\in[0,h],$
	$\displaystyle X_{\left.\psi\right\|_{\Gamma_{2}},\left.\psi\right\|_{\Gamma_{3}}}^{(2)}$	$\displaystyle=\{F\in C(\widetilde{T}_{h})\ \|\ F\left(x,0\right)=\left.\psi\right\|_{\Gamma_{2}},\ F(x,f(x))=\left.\psi\right\|_{\Gamma_{3}}\},\ \ \text{for }x\in[0,h],$

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

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

	$\displaystyle\left\|Q_{m}^{x}(F)(x,y)-Q_{m}^{x}(G)(x,y)\right\|=$
	$\displaystyle=\left\|Q_{m}^{x}(F-G)(x,y)\right\|\leq$
	$\displaystyle\leq\left\|q_{m,0}\left(x;y\right)\left[F\left(0,0\right)-G(0,0)\right]\right\|$
	$\displaystyle\quad+\left\|\sum_{i=1}^{m}q_{m,i}\left(x;y\right)\left[F\left(\tfrac{ig(y)}{m},y\right)-G\left(x,\tfrac{jf(x)}{n}\right)\right]\right\|$
	$\displaystyle=\left\|\sum_{i=1}^{m}q_{m,i}\left(x;y\right)\left[F\left(\tfrac{ig(y)}{m},y\right)-G\left(x,\tfrac{jf(x)}{n}\right)\right]\right\|$
	$\displaystyle\leq\sum_{i=1}^{m}q_{m,i}\left(x;y\right)\left\\|F-G\right\\|_{\infty}$
	$\displaystyle=\left[\sum_{i=0}^{m}q_{m,i}\left(x;y\right)-q_{m,0}\left(x;y\right)\right]\left\\|F-G\right\\|_{\infty}$
	$\displaystyle=\left\{1-\left(1-\tfrac{x}{g(y)}\right)\left[1-\tfrac{x}{g(y)(1+m\beta)}\right]^{m-1}\right\}\left\\|F-G\right\\|_{\infty}$
	$\displaystyle\leq\left[1-\left(1-\tfrac{1}{1+m\beta}\right)^{m-1}\right]\left\\|F-G\right\\|_{\infty},$

Iterates of Cheney-Sharma type operators on a triangle with curved side

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Iterates of Cheney-Sharma type operators on a triangle with curved side

Abstract.

1. Cheney-Sharma type operators

Remark 1.1.

Theorem 1.2.

Theorem 1.3.

Theorem 1.4.

2. Weakly Picard operators

Definition 2.1.

Definition 2.2.

Definition 2.3.

Theorem 2.4.

3. Iterates of Cheney-Sharma type operators

Theorem 3.1.

Proof.

Theorem 3.2.

Proof.

Remark 3.3.

Theorem 3.4.

Proof.

Remark 3.5.

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