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

Diana Otrocol
(original), ISI/JCR, paper

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

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

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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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[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.

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[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.

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

FA\displaystyle F_{A} :={x∈X|A​(x)=x}​-the fixed point set of ​A​;\displaystyle:=\{x\in X~|~A(x)=x\}\text{-the fixed point set of }A\text{;}
I​(A)\displaystyle I(A) :={Y⊂X|A​(Y)⊂Y,Y≠∅}​-the family of the nonempty invariant\displaystyle:=\{Y\subset X~|~A(Y)\subset Y,\ Y\neq\emptyset\}\text{-the family of the nonempty invariant }
subset of ​A\displaystyle\text{subset of }A
A0\displaystyle A^{0} :=1X,A1:=A,…,An+1:=A∘An,n∈ℕ​.\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→XA:X\rightarrow X is a Picard operator if there exists x∗∈Xx^{\ast}\in X such that:

(i) FA={x∗};F_{A}=\{x^{*}\};

(ii) the sequence (An​(x0))n∈ℕ(A^{n}(x_{0}))_{n\in\mathbb{N}} converges to x∗x^{*} for all x0∈Xx_{0}\in X.

Definition 1.2.

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

Definition 1.3.

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

A∞​(x):=limn→∞​An​(x).A^{\infty}(x):=\underset{n\rightarrow\infty}A^{n}(x).
Theorem 1.4.

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

  • (a)

    Xλ∈I​(A),X_{\lambda}\in I(A), ∀λ∈Λ;\forall\lambda\in\Lambda;

  • (b)

    A|Xλ:Xλ→Xλ\left.A\right|_{X_{\lambda}}:X_{\lambda}\rightarrow X_{\lambda} is a Picard operator, ∀λ∈Λ.\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,h>0, let DhD_{h} be the square with one curved side having the vertices V1=(0,0),V_{1}=(0,0), V2=(h,0),V_{2}=(h,0), V3=(h,h)V_{3}=(h,h) and V4=(0,h),V_{4}=(0,h), three straight sides Γ1,\Gamma_{1}, Γ2,\Gamma_{2}, along the coordinate axes and Γ3\Gamma_{3} parallel to axis O​x,Ox, and the curved side Γ4\Gamma_{4} which is defined by the function gg, such that g​(h)=g​(0)=hg(h)=g(0)=h (see Figure 1).

[Uncaptioned image]

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

Let FF be a real-valued function defined on DhD_{h} and (0,y)(0,y), (g​(y),y),(g(y),y), respectively, (x,0),(x,0), (x,h)(x,h) be the points in which the parallel lines to the coordinate axes, passing through the point (x,y)∈Dh,(x,y)\in D_{h}, intersect the sides Γ2,\Gamma_{2}, Γ4,\Gamma_{4}, respectively Γ1\Gamma_{1} and Γ3.\Gamma_{3}. We consider the uniform partitions of the intervals [0,g​(y)][0,g(y)] and [0,h][0,h], y∈[0,h],y\in[0,h], Δmx={im​g​(y)|i=0,m¯}\Delta_{m}^{x}=\left\{\left.\tfrac{i}{m}g(y)\right|\ i=\overline{0,m}\right\} and Δny={jn​h|j=0,n¯}\Delta_{n}^{y}=\left\{\left.\tfrac{j}{n}h\right|\ j=\overline{0,n}\right\} and the Bernstein-type operators BmxB_{m}^{x} and BnyB_{n}^{y} defined by

(Bmx​F)​(x,y)=∑i=0mpm,i​(x,y)​F​(im​g​(y),y),\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

pm,i​(x,y)=(mi)​[xg​(y)]i​[1−xg​(y)]m−i,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,

(Bny​F)​(x,y)=∑j=0nqn,j​(x,y)​F​(x,jn​h)\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

qn,j​(x,y)=(nj)​(yh)j​(1−yh)n−j.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 FF is a real-valued function defined on DhD_{h}\ then we have

  1. (1)

    Bmx​F=FB_{m}^{x}F=F\ on Γ2∪Γ4;\Gamma_{2}\cup\Gamma_{4};

    Bny​F=FB_{n}^{y}F=F\ on Γ1∪Γ3,\Gamma_{1}\cup\Gamma_{3},

  2. (2)

    (Bmx​ei​j)​(x,y)=xi​yj,i=0,1;\left(B_{m}^{x}e_{ij}\right)\left(x,y\right)=x^{i}y^{j},\ \ i=0,1; j∈ℕ;j\in\mathbb{N};

    (Bny​ei​j)​(x,y)=xi​yj,\left(B_{n}^{y}e_{ij}\right)\left(x,y\right)=x^{i}y^{j},\ i∈ℕ;i\in\mathbb{N}; j=0,1.j=0,1.

Remark 2.2.

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

[Uncaptioned image]

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

[Uncaptioned image]

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

Let Pm​n=Bmx​Bny,P_{mn}=B_{m}^{x}B_{n}^{y}, respectively, Qn​m=Bny​BmxQ_{nm}=B_{n}^{y}B_{m}^{x} be the products of the operators BmxB_{m}^{x} and Bny.B_{n}^{y}. We have

(Pm​n​F)​(x,y)=∑i=0m∑j=0npm,i​(x,y)​qn,j​(i​g​(y)m,y)​F​(i​g​(y)m,j​hn),\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,

(Qn​m​F)​(x,y)=∑i=0m∑j=0npm,i​(x,j​hn)​qn,j​(x,y)​F​(im​g​(j​hn),j​hn).\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 FF is a real-valued function defined on DhD_{h} then:

  1. (1)

    (Pm​n​F)​(Vi)=F​(Vi),i=1,…,4;(P_{mn}F)(V_{i})=F(V_{i}),\ \ \ \ i=1,...,4;

    (Qn​m​F)​(Vi)=F​(Vi),i=1,…,4.(Q_{nm}F)(V_{i})=F(V_{i}),\ \ \ \ i=1,...,4.

  2. (2)

    (Pm​n​ei​j)​(x,y)=xi​yj,i=0,1;\left(P_{mn}e_{ij}\right)\left(x,y\right)=x^{i}y^{j},\ \ i=0,1; j=0,1;j=0,1;

    (Qn​m​ei​j)​(x,y)=xi​yj,\left(Q_{nm}e_{ij}\right)\left(x,y\right)=x^{i}y^{j},\ i=0,1;i=0,1; j=0,1.j=0,1.

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

Sm​n:=Bmx⊕Bny=Bmx+Bny−Bmx​Bny,S_{mn}:=B_{m}^{x}\oplus B_{n}^{y}=B_{m}^{x}+B_{n}^{y}-B_{m}^{x}B_{n}^{y}, (5)

respectively,

Tn​m:=Bny⊕Bmx=Bny+Bmx−Bny​Bmx.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 FF be a real-valued function defined on DhD_{h}, h∈ℝ+.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 BmxB_{m}^{x} and BnyB_{n}^{y} are weakly Picard operators and

(Bmx,∞​F)​(x,y)\displaystyle\left(B_{m}^{x,\infty}F\right)\left(x,y\right) =F​(0,y)+F​(g​(y),y)−F​(0,y)g​(y)​x,\displaystyle=F\left(0,y\right)+\frac{F(g(y),y)-F(0,y)}{g(y)}x, (7)
(Bny,∞​F)​(x,y)\displaystyle\left(B_{n}^{y,\infty}F\right)\left(x,y\right) =F​(x,0)+F​(x,h)−F​(x,0)h​y.\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 BmxB_{m}^{x} and Bny,B_{n}^{y}, let

Xφ|Γ2,φ|Γ4(1)\displaystyle X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)} ={F∈C​(Dh)|F​(0,y)=φ|Γ2,F​(g​(y),y)=φ|Γ4},for ​y∈[0,h],\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],
Xψ|Γ1,ψ|Γ3(2)\displaystyle X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)} ={F∈C​(Dh)|F​(x,0)=ψ|Γ1,F​(x,h)=ψ|Γ3},for ​x∈[0,h],\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

Fφ|Γ2,φ|Γ4(1)​(x,y)\displaystyle F_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}(x,y) :=φ|Γ2+φ|Γ4−φ|Γ2g​(y)​x,\displaystyle:=\left.\varphi\right|_{\Gamma_{2}}+\frac{\left.\varphi\right|_{\Gamma_{4}}-\left.\varphi\right|_{\Gamma_{2}}}{g(y)}x,
Fψ|Γ1,ψ|Γ3(2)​(x,y)\displaystyle F_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}(x,y) :=ψ|Γ1+ψ|Γ3−ψ|Γ1h​y,\displaystyle:=\left.\psi\right|_{\Gamma_{1}}+\frac{\left.\psi\right|_{\Gamma_{3}}-\left.\psi\right|_{\Gamma_{1}}}{h}y,

with φ,ψ∈C​(Dh).\varphi,\psi\in C\mathbb{(}D_{h}).

We have the following properties:

  • (i)

    Xφ|Γ2,φ|Γ4(1)X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)} and Xψ|Γ1,ψ|Γ3(2)X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)} are closed subsets of C​(Dh)C(D_{h});

  • (ii)

    Xφ|Γ2,φ|Γ4(1)X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)} is an invariant subset of BmxB_{m}^{x} and Xψ|Γ1,ψ|Γ3(2)X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)} is an invariant subset of BnyB_{n}^{y}, for φ,ψ∈C​(Dh)\varphi,\psi\in C\mathbb{(}D_{h}) and n,m∈ℕ∗;n,m\in\mathbb{N}^{\ast};

  • (iii)

    C​(Dh)=∪φ∈C​(Dh)​Xφ|Γ2,φ|Γ4(1)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​(Dh)=∪ψ∈C​(Dh)​Xψ|Γ1,ψ|Γ3(2)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​(Dh)C(D_{h});

  • (iv)

    Fφ|Γ2,φ|Γ4(1)∈Xφ|Γ2,φ|Γ4(1)∩FBmxF_{\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 Fψ|Γ1,ψ|Γ3(2)∈Xψ|Γ1,ψ|Γ3(2)∩FBny,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 FBmxF_{B_{m}^{x}} and FBnyF_{B_{n}^{y}} denote the fixed points sets of BmxB_{m}^{x} and Bny.B_{n}^{y}.

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

(i​i)(ii) By linearity of Bernstein operators and Theorem 2.1, it follows that ∀Fφ|Γ2,φ|Γ4(1)∈Xφ|Γ2,φ|Γ4(1)\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 ∀Fψ|Γ1,ψ|Γ3(2)∈Xψ|Γ1,ψ|Γ3(2)\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

Bmx​Fφ|Γ2,φ|Γ4(1)​(x,y)\displaystyle B_{m}^{x}F_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}(x,y) =Fφ|Γ2,φ|Γ4(1)​(x,y),\displaystyle=F_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}(x,y),
Bny​Fψ|Γ1,ψ|Γ3(2)​(x,y)\displaystyle B_{n}^{y}F_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}(x,y) =Fψ|Γ1,ψ|Γ3(2)​(x,y).\displaystyle=F_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}(x,y).

So, Xφ|Γ2,φ|Γ4(1)X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)} and Xψ|Γ1,ψ|Γ3(2)X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)} are invariant subsets of BmxB_{m}^{x} and, respectively, of Bny,B_{n}^{y},\ for φ,ψ∈C​(Dh)\varphi,\psi\in C\mathbb{(}D_{h}) and n,m∈ℕ∗;n,m\in\mathbb{N}^{\ast};

(i​v)(iv) We prove that

Bmx|Xφ|Γ2,φ|Γ4(1):Xφ|Γ2,φ|Γ4(1)→Xφ|Γ2,φ|Γ4(1)​ and ​Bny|Xψ|Γ1,ψ|Γ3(2):Xψ|Γ1,ψ|Γ3(2)→Xψ|Γ1,ψ|Γ3(2)\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 φ,ψ∈C​(Dh)\varphi,\psi\in C\mathbb{(}D_{h}) and n,m∈ℕ∗.n,m\in\mathbb{N}^{\ast}.

Let F,G∈Xφ|Γ2,φ|Γ4(1)F,G\in X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}. From (1) we have

|Bmx​(F)​(x,y)−Bmx​(G)​(x,y)|=|Bmx​(F−G)​(x,y)|≤\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
≤|1−(1−xg​(y))m−(xg​(y))m|⋅‖F−G‖∞≤\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
≤(1−12m−1)​‖F−G‖∞,\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,

‖Bmx​(F)​(x,y)−Bmx​(G)​(x,y)‖∞≤(1−12m−1)​‖F−G‖∞,∀F,G∈Xφ|Γ2,φ|Γ4(1),\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., Bmx|Xφ|Γ2,φ|Γ4(1)\left.B_{m}^{x}\right|_{X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}} is a contraction for φ∈C​(Dh)\varphi\in C\mathbb{(}D_{h}).

Analogously we have

‖Bny​(F)​(x,y)−Bny​(G)​(x,y)‖∞≤(1−12n−1)​‖F−G‖∞, ​∀F,G∈Xψ|Γ1,ψ|Γ3(2),\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., Bny|Xψ|Γ1,ψ|Γ3(2)\left.B_{n}^{y}\right|_{X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}} is a contraction for ψ∈C​(Dh).\psi\in C\mathbb{(}D_{h}).

On the other hand, φ|Γ2+φ|Γ4−φ|Γ2g​(y)​(⋅)∈Xφ|Γ2,φ|Γ4(1),\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)}, ψ|Γ1+ψ|Γ3−ψ|Γ1h​(⋅)∈Xψ|Γ1,ψ|Γ3(2)\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 BmxB_{m}^{x} and BnyB_{n}^{y}, i.e.,

Bmx​(φ|Γ2+φ|Γ4−φ|Γ2g​(y)​(⋅))\displaystyle B_{m}^{x}(\left.\varphi\right|_{\Gamma_{2}}+\frac{\left.\varphi\right|_{\Gamma_{4}}-\left.\varphi\right|_{\Gamma_{2}}}{g(y)}(\cdot)) =φ|Γ2+φ|Γ4−φ|Γ2g​(y)​(⋅),\displaystyle=\left.\varphi\right|_{\Gamma_{2}}+\frac{\left.\varphi\right|_{\Gamma_{4}}-\left.\varphi\right|_{\Gamma_{2}}}{g(y)}(\cdot),
Bny​(ψ|Γ1+ψ|Γ3−ψ|Γ1h​(⋅))\displaystyle B_{n}^{y}(\left.\psi\right|_{\Gamma_{1}}+\frac{\left.\psi\right|_{\Gamma_{3}}-\left.\psi\right|_{\Gamma_{1}}}{h}(\cdot)) =ψ|Γ1+ψ|Γ3−ψ|Γ1h​(⋅).\displaystyle=\left.\psi\right|_{\Gamma_{1}}+\frac{\left.\psi\right|_{\Gamma_{3}}-\left.\psi\right|_{\Gamma_{1}}}{h}(\cdot).

From the contraction principle, Fφ|Γ2,φ|Γ4(1)​(x,y):=φ|Γ2+φ|Γ4−φ|Γ2g​(y)​xF_{\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 BmxB_{m}^{x} in Xφ|Γ2,φ|Γ4(1)X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)} and Bmx|Xφ|Γ2,φ|Γ4(1)\left.B_{m}^{x}\right|_{X_{\left.\varphi\right|_{\Gamma_{2}},\left.\varphi\right|_{\Gamma_{4}}}^{(1)}} is a Picard operator, with

(Bmx,∞​F)​(x,y)=F​(0,y)+F​(g​(y),y)−F​(0,y)g​(y)​x,\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, Fψ|Γ1,ψ|Γ3(2)​(x,y):=ψ|Γ1+ψ|Γ3−ψ|Γ1h​yF_{\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 BnyB_{n}^{y} in Xψ|Γ1,ψ|Γ3(2)X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)} and Bny|Xψ|Γ1,ψ|Γ3(2)\left.B_{n}^{y}\right|_{X_{\left.\psi\right|_{\Gamma_{1}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}} is a Picard operator, with

(Bny,∞​F)​(x,y)=F​(x,0)+F​(x,h)−F​(x,0)h​y,\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 (i​i)(ii), by Theorem 1.4 it follows that the operators BmxB_{m}^{x} and BnyB_{n}^{y} are weakly Picard operators. ∎

Theorem 3.2.

The operators Pm​nP_{mn} and Qn​mQ_{nm} are weakly Picard operators and

(Pm​n∞​F)​(x,y)\displaystyle\left(P_{mn}^{\infty}F\right)\left(x,y\right) =F​(0,0)+F​(h,0)−F​(0,0)g​(y)​x+F​(0,h)−F​(0,0)h​y\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)
+F​(0,0)−F​(0,h)−F​(h,0)+F​(h,h)g​(y)​h​x​y,\displaystyle\quad+\frac{F(0,0)-F(0,h)-F(h,0)+F(h,h)}{g(y)h}xy,
(Qn​m∞​F)​(x,y)\displaystyle\left(Q_{nm}^{\infty}F\right)\left(x,y\right) =F​(0,0)+F​(h,0)−F​(0,0)g​(y)​x+F​(0,h)−F​(0,0)h​y\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)
+F​(0,0)−F​(0,h)−F​(h,0)+F​(h,h)g​(y)​h​x​y.\displaystyle\quad+\frac{F(0,0)-F(0,h)-F(h,0)+F(h,h)}{g(y)h}xy.
Proof.

Let

Xα,β,γ,δ={F∈C​(Dh)|F​(0,0)=α,F​(0,h)=β,F​(h,h)=γ,F​(h,0)=δ}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

Fα,β,γ,δ​(x,y):=α+δ−αg​(y)​x+β−αh​y+α−β−δ+γg​(y)​h​x​yF_{\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 α,β,γ,δ∈ℝ.\alpha,\beta,\gamma,\delta\in\mathbb{R}.

We remark that

  • (i)

    Xα,β,γ,δX_{\alpha,\beta,\gamma,\delta} is closed subset of C​(Dh)C(D_{h});

  • (ii)

    Xα,β,γ,δX_{\alpha,\beta,\gamma,\delta} is an invariant subset of Pm​nP_{mn} and Qn​mQ_{nm}, for α,β,γ,δ∈ℝ\alpha,\beta,\gamma,\delta\in\mathbb{R} and n,m∈ℕ∗;n,m\in\mathbb{N}^{\ast};

  • (iii)

    C​(Dh)=∪α,β,γ,δ​Xα,β,γ,δC(D_{h})=\underset{\alpha,\beta,\gamma,\delta}{\cup}X_{\alpha,\beta,\gamma,\delta} is a partition of C​(Dh)C(D_{h});

  • (iv)

    Fα,β,γ,δ∈Xα,β,γ,δ∩FPm​nF_{\alpha,\beta,\gamma,\delta}\in X_{\alpha,\beta,\gamma,\delta}\cap F_{P_{mn}} and Fα,β,γ,δ∈Xα,β,γ,δ∩FQn​m,F_{\alpha,\beta,\gamma,\delta}\in X_{\alpha,\beta,\gamma,\delta}\cap F_{Q_{nm}}, where FPm​nF_{P_{mn}} and FQn​mF_{Q_{nm}} denote the fixed points sets of Pm​nP_{mn} and Qn​m.Q_{nm}.

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

(i​i)(ii) Similarly with the proof of Theorem 3.1, by linearity of Bernstein operators and Theorem 2.3, it follows that Xα,β,γ,δX_{\alpha,\beta,\gamma,\delta} is an invariant subset of Pm​nP_{mn} and, respectively, of Qn​mQ_{nm}, for α,β,γ,δ∈ℝ\alpha,\beta,\gamma,\delta\in\mathbb{R} and n,m∈ℕ∗;n,m\in\mathbb{N}^{\ast};

(i​v)(iv) We prove that

Pm​n|Xα,β,γ,δ:Xα,β,γ,δ→Xα,β,γ,δ​ and ​Qn​m|Xα,β,γ,δ:Xα,β,γ,δ→Xα,β,γ,δ\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 α,β,γ,δ∈ℝ\alpha,\beta,\gamma,\delta\in\mathbb{R} and n,m∈ℕ∗.n,m\in\mathbb{N}^{\ast}. Let F,G∈Xα,β,γ,δF,G\in X_{\alpha,\beta,\gamma,\delta}. From [2, Lemma 8] it follows that

|Pm​n​(F)​(x,y)−Pm​n​(G)​(x,y)|=|Pm​n​(F−G)​(x,y)|≤\displaystyle\left|P_{mn}(F)(x,y)-P_{mn}(G)(x,y)\right|=\left|P_{mn}(F-G)(x,y)\right|\leq
≤(1−12m+n−2)​‖F−G‖∞.\displaystyle\leq\left(1-\frac{1}{2^{m+n-2}}\right)\left\|F-G\right\|_{\infty}.

So,

‖Pm​n​(F)​(x,y)−Pm​n​(G)​(x,y)‖∞≤(1−12m+n−2)​‖F−G‖∞,∀F,G∈Xα,β,γ,δ,\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., Pm​n|Xα,β,γ,δ\left.P_{mn}\right|_{X_{\alpha,\beta,\gamma,\delta}} is a contraction for α,β,γ,δ∈ℝ.\alpha,\beta,\gamma,\delta\in\mathbb{R}. Analogously, we have

‖Qn​m​(F)​(x,y)−Qn​m​(G)​(x,y)‖∞≤(1−12m+n−2)​‖F−G‖∞, ​∀F,G∈Xα,β,γ,δ,\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., Qn​m|Xα,β,γ,δ\left.Q_{nm}\right|_{X_{\alpha,\beta,\gamma,\delta}} is a contraction for α,β,γ,δ∈ℝ\alpha,\beta,\gamma,\delta\in\mathbb{R}.

We have that

Fα,β,γ,δ​(x,y):=α+δ−αg​(y)​x+β−αh​y+α−β−δ+γg​(y)​h​x​yF_{\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

Pm​n\displaystyle P_{mn} (α+δ−αg​(y)​x+β−αh​y+α−β−δ+γg​(y)​h​x​y)\displaystyle\left(\alpha+\frac{\delta-\alpha}{g(y)}x+\frac{\beta-\alpha}{h}y+\frac{\alpha-\beta-\delta+\gamma}{g(y)h}xy\right)
=α+δ−αg​(y)​x+β−αh​y+α−β−δ+γg​(y)​h​x​y,\displaystyle=\alpha+\frac{\delta-\alpha}{g(y)}x+\frac{\beta-\alpha}{h}y+\frac{\alpha-\beta-\delta+\gamma}{g(y)h}xy,
Qn​m\displaystyle Q_{nm} (α+δ−αg​(y)​x+β−αh​y+α−β−δ+γg​(y)​h​x​y)\displaystyle\left(\alpha+\frac{\delta-\alpha}{g(y)}x+\frac{\beta-\alpha}{h}y+\frac{\alpha-\beta-\delta+\gamma}{g(y)h}xy\right)
=α+δ−αg​(y)​x+β−αh​y+α−β−δ+γg​(y)​h​x​y.\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 Fα,β,γ,δF_{\alpha,\beta,\gamma,\delta} is the unique fixed point of Pm​nP_{mn} in Xα,β,γ,δX_{\alpha,\beta,\gamma,\delta} and Pm​n|Xα,β,γ,δ\left.P_{mn}\right|_{X_{\alpha,\beta,\gamma,\delta}} is a Picard operator and, respectively, Fα,β,γ,δF_{\alpha,\beta,\gamma,\delta} is the unique fixed point of Qn​mQ_{nm} in Xα,β,γ,δX_{\alpha,\beta,\gamma,\delta} and Qn​m|Xα,β,γ,δ\left.Q_{nm}\right|_{X_{\alpha,\beta,\gamma,\delta}} is a Picard operator, so (9) and (10) hold. Consequently, taking into account (i​i)(ii), by Theorem 1.4 it follows that the operators Pm​nP_{mn} and Qn​mQ_{nm} are weakly Picard operators. ∎

Theorem 3.3.

The operator Sm​nS_{mn} is weakly Picard operator and

(Sm​n∞​F)​(x,y)\displaystyle\left(S_{mn}^{\infty}F\right)\left(x,y\right) =F​(0,y)+F​(x,0)−F​(0,0)\displaystyle=F\left(0,y\right)+F\left(x,0\right)-F\left(0,0\right)
+F​(g​(y),y)−F​(0,y)−F​(h,0)+F​(0,0)g​(y)​x\displaystyle+\frac{F(g(y),y)-F(0,y)-F(h,0)+F(0,0)}{g(y)}x
+F​(x,h)−F​(x,0)−F​(0,h)+F​(0,0)h​y\displaystyle+\frac{F(x,h)-F(x,0)-F(0,h)+F(0,0)}{h}y
−F​(0,0)−F​(0,h)−F​(h,0)+F​(h,h)g​(y)​h​x​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

‖Sm​n​(F)​(x,y)−Sm​n​(G)​(x,y)‖∞≤[1−(12m−1+12n−1−12m+n−2)]​‖F−G‖∞,\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 Sm​nS_{mn} is a contraction. ∎

Remark 3.4.

We have an analogous result for the operator Tn​mT_{nm}.

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

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