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

Diana Otrocol
(original), Approximation Theory, paper

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

    See the expanding block below.

    Paper coordinates

    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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    [1] Agratini, O., Rus, I.A., Iterates of a class of discrete linear operators via contraction principle, Comment. Math. Univ. Caroline 44 (2003), 555-563.
    [2] Agratini, O., Rus, I.A., Iterates of some bivariate approximation process via weakly Picard operators, Nonlinear Anal. Forum 8(2) (2003), 159-168.
    [3] Blaga, P., Catinas, T., Coman, G., Bernstein-type operators on triangle with one curved side, Mediterr. J. Math. 9 (2012), No. 4, 833-845.
    [4] Blaga, P., Catinas, T., Coman, G., Bernstein-type operators on a square with one and two curved sides, Studia Univ. Babe¸s–Bolyai Math. 55 (2010), No. 3, 51-67.
    [5] Blaga, P., C˘atina¸s, T., Coman, G., Bernstein-type operators on triangle with all curved sides, Appl. Math. Comput. 218 (2011), 3072-3082.
    [6] Catinas, T., Extension of some Cheney-Sharma type operators to a triangle with one curved side, 2017, submitted.
    [7] Catinas, 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] Catinas, T., Otrocol, D., Iterates of multivariate Cheney-Sharma operators, J. Comput. Anal. Appl. 15 (2013), No. 7, 1240-1246.
    [9] Coman, G., Catinas, T., Interpolation operators on a triangle with one curved side, BIT Numerical Mathematics 50 (2010), No. 2, 243-267.
    [10] Gavrea, I., Ivan, M., The iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl. 372 (2010), 366-368.
    [11] Gavrea, I., Ivan, M., The iterates of positive linear operators preserving the constants, Appl. Math. Lett. 24 (2011), No. 12, 2068-2071.
    [12] Gavrea, I., Ivan, M., On the iterates of positive linear operators, J. Approx. Theory 163 (2011), No. 9, 1076-1079.
    [13] Gonska, H., Kacso, D., Pitul, P., The degree of convergence of over-iterated positive linear operators, J. Appl. Funct. Anal. 1 (2006), 403-423.
    [14] Gonska, H., Pitul, P., Rasa, I., Over-iterates of Bernstein-Stancu operators, Calcolo 44 (2007), 117-125.
    [15] Gonska, H., Rasa, I., The limiting semigroup of the Bernstein iterates: degree of convergence, Acta Math. Hungar. 111 (2006), No. 1-2, 119-130.
    [16] Gonska, H., Ra¸sa, I., On infinite products of positive linear operators reproducing linear functions, Positivity 17 (2013), No. 1, 67-79.
    [17] Gwozdz- Lukawska, G., Jachymski, J., IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem, J. Math. Anal. Appl. 356(2) (2009), 453-463.
    [18] Karlin, S. , Ziegler, Z., Iteration of positive approximation operators, J. Approx. Theory 3 (1970), 310-339.
    [19] Kelisky, R.P., Rivlin, T.J., Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967), 511-520.
    [20] Rasa, I., C0-Semigroups and iterates of positive linear operators: asymptotic behaviour, Rend. Circ. Mat. Palermo, Ser. II, Suppl. 82 (2010), 123-142.
    [21] Rus, I.A., Picard operators and applications, Sci. Math. Jpn. 58 (2003), 191-219.
    [22] Rus, I.A., Iterates of Stancu operators, via contraction principle, Studia Univ. Babes–Bolyai Math. 47 (2002), No. 4, 101-104.
    [23] Rus, I.A., Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. 292 (2004), 259-261.
    [24] Rus, I.A., 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.

    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 T~h\tilde{T}_{h} with vertices V1=(0,h),V_{1}=(0,h), V2=(h,0)V_{2}=(h,0) and V3=(0,0),V_{3}=(0,0), with two straight sides Γ1,\Gamma_{1}, Γ2,\Gamma_{2}, along the coordinate axes, and with the third side Γ3\Gamma_{3} (opposite to the vertex V3V_{3}) defined by the one-to-one functions ff and g,g, where gg is the inverse of the function f,f, i.e., y=f​(x)y=f(x) and x=g​(y)x=g(y), with f​(0)=g​(0)=h,f(0)=g(0)=h,\ for h>0h>0. Also, we have f​(x)≤hf(x)\leq h and g​(y)≤h,g(y)\leq h, for x,y∈[0,h].x,y\in\left[0,h\right].

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

    Refer to caption
    Figure 1. Triangle T~h.\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 T~h\widetilde{T}_{h}:

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

    with

    qm,i​(x,y)\displaystyle q_{m,i}\left(x,y\right) =(mi)​1(1+m​β)m−1​xg​(y)​(xg​(y)+i​β)i−1​(1−xg​(y))​[1−xg​(y)+(m−i)​β]m−i−1,\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},
    qn,j​(x,y)\displaystyle q_{n,j}\left(x,y\right) =(nj)​1(1+n​b)n−1​yf​(x)​(yf​(x)+j​b)j−1​(1−yf​(x))​[1−yf​(x)+(n−j)​b]n−j−1,\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

    Δmx={i​g​(y)m|i=0,m¯}​and ​Δny={j​f​(x)n|j=0,n¯}\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)][0,g(y)] and [0,f​(x)][0,f(x)] and m,n∈ℕ,m,n\in\mathbb{N}, β,b∈ℝ+.\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 QmxQ_{m}^{x} and QnyQ_{n}^{y} as interpolation operators on T~h.\widetilde{T}_{h}.

    Theorem 1.2.

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

    1. (i)

      Qmx​F=FQ_{m}^{x}F=F\ on Γ1∪Γ3;\Gamma_{1}\cup\Gamma_{3};

    2. (ii)

      Qny​F=FQ_{n}^{y}F=F\ on Γ2∪Γ3;\Gamma_{2}\cup\Gamma_{3};

    3. (iii)

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

    4. (iv)

      (Qny​ei​j)​(x,y)=xi​yj,\left(Q_{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, where ei​j​(x,y)=xi​yj,e_{ij}\left(x,y\right)=x^{i}y^{j},\ i,j∈ℕ.i,j\in\mathbb{N}.

    Let Pm​n1=Qmx​Qny,P_{mn}^{1}=Q_{m}^{x}Q_{n}^{y}, respectively, Pn​m2=Qny​QmxP_{nm}^{2}=Q_{n}^{y}Q_{m}^{x} be the products of the operators QmxQ_{m}^{x} and Qny.Q_{n}^{y}. We have

    (1.2) (Pm​n1​F)​(x,y)=∑i=0m∑j=0nqm,i​(x,y)​qn,j​(i​g​(y)m,y)​F​(i​g​(y)m,j​f​(i​g​(y)m)n),\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,

    (Pn​m2​F)​(x,y)=∑i=0m∑j=0nqm,i​(x,j​f​(x)n)​qn,j​(x,y)​F​(i​g​(j​f​(x)n)m,j​f​(x)n).\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 FF is a real-valued function defined on T~h\widetilde{T}_{h} then

    1. (i)

      (Pm​n1​F)​(Vi)=F​(Vi),i=1,…,3;(P_{mn}^{1}F)(V_{i})=F(V_{i}),\ \ \ \ i=1,...,3;

      (Pm​n1​F)​(Γ3)=F​(Γ3),(P_{mn}^{1}F)(\Gamma_{3})=F(\Gamma_{3}),\

    2. (ii)

      (Pn​m2​F)​(Vi)=F​(Vi),i=1,…,3;(P_{nm}^{2}F)(V_{i})=F(V_{i}),\ \ \ \ i=1,...,3;

      (Pn​m2​F)​(Γ3)=F​(Γ3).(P_{nm}^{2}F)(\Gamma_{3})=F(\Gamma_{3}).

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

    (1.3) Sm​n1\displaystyle S_{mn}^{1} :=Qmx⊕Qny=Qmx+Qny−Qmx​Qny,\displaystyle:=Q_{m}^{x}\oplus Q_{n}^{y}=Q_{m}^{x}+Q_{n}^{y}-Q_{m}^{x}Q_{n}^{y},
    Sn​m2\displaystyle S_{nm}^{2} :=Qny⊕Qmx=Qny+Qmx−Qny​Qmx.\displaystyle:=Q_{n}^{y}\oplus Q_{m}^{x}=Q_{n}^{y}+Q_{m}^{x}-Q_{n}^{y}Q_{m}^{x}.
    Theorem 1.4.

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

    Sm​n1F|=∂T~hF|∂T~h,\displaystyle S_{mn}^{1}F\left|{}_{\partial\widetilde{T}_{h}}=F\right|_{\partial\widetilde{T}_{h}},
    Sm​n2F|=∂T~hF|∂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)(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 points set of ​A​;\displaystyle:=\{x\in X~|~A(x)=x\}\text{-the fixed points 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 }
    subsets of ​A;\displaystyle\text{subsets 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 2.1.

    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^{\ast}\};

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

    Definition 2.2.

    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 2.3.

    If AA is a 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 2.4.

    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

    1. (a)

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

    2. (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.

    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 FF be a real-valued function defined on T~h\widetilde{T}_{h}, h∈ℝ+.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 QmxQ_{m}^{x} and QnyQ_{n}^{y} are weakly Picard operators and

    (3.1) (Qmx,∞​F)​(x,y)\displaystyle\left(Q_{m}^{x,\infty}F\right)\left(x,y\right) =F​(g​(y),y)−F​(0,y)g​(y)​x+F​(0,y),\displaystyle=\frac{F\left(g(y),y\right)-F\left(0,y\right)}{g(y)}x+F\left(0,y\right),
    (3.2) (Qny,∞​F)​(x,y)\displaystyle\left(Q_{n}^{y,\infty}F\right)\left(x,y\right) =F​(x,f​(x))−F​(x,0)f​(x)​y+F​(x,0).\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 QmxQ_{m}^{x} and QnyQ_{n}^{y} (from Theorem 1.2), let us consider the following sets:

    (3.3) Xφ|Γ1,φ|Γ3(1)\displaystyle X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)} ={F∈C​(T~h)|F​(0,y)=φ|Γ1,F​(g​(y),y)=φ|Γ3},for ​y∈[0,h],\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],
    Xψ|Γ2,ψ|Γ3(2)\displaystyle X_{\left.\psi\right|_{\Gamma_{2}},\left.\psi\right|_{\Gamma_{3}}}^{(2)} ={F∈C​(T~h)|F​(x,0)=ψ|Γ2,F​(x,f​(x))=ψ|Γ3},for ​x∈[0,h],\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 φ,ψ∈C​(T~h)\varphi,\psi\in C\mathbb{(}\widetilde{T}_{h}) we denote by

    Fφ|Γ1,φ|Γ3(1)​(x,y)\displaystyle F_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}(x,y) =φ|Γ3−φ|Γ1g​(y)​x+φ|Γ1,\displaystyle=\frac{\left.\varphi\right|_{\Gamma_{3}}-\left.\varphi\right|_{\Gamma_{1}}}{g(y)}x+\left.\varphi\right|_{\Gamma_{1}},
    Fψ|Γ2,ψ|Γ3(2)​(x,y)\displaystyle F_{\left.\psi\right|_{\Gamma_{2}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}(x,y) =ψ|Γ3−ψ|Γ2f​(x)​y+ψ|Γ2.\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:

    1. (i)

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

    2. (ii)

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

    3. (iii)

      C​(T~h)=∪φ∈C​(T~h)​Xφ|Γ1,φ|Γ3(1)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​(T~h)=∪ψ∈C​(T~h)​Xψ|Γ2,ψ|Γ3(2)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​(T~h)C(\widetilde{T}_{h});

    4. (iv)

      Fφ|Γ1,φ|Γ3(1)∈Xφ|Γ1,φ|Γ3(1)∩FQmxF_{\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 Fψ|Γ2,ψ|Γ3(2)∈Xψ|Γ2,ψ|Γ3(2)∩FQny,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 FQmxF_{Q_{m}^{x}} and FQnyF_{Q_{n}^{y}} denote the fixed points sets of QmxQ_{m}^{x} and Qny.Q_{n}^{y}.

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

    (i​i)(ii) By linearity of Cheney-Sharma operators and Theorem 1.2, it follows that ∀Fφ|Γ1,φ|Γ3(1)∈Xφ|Γ1,φ|Γ3(1)\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 ∀Fψ|Γ2,ψ|Γ3(2)∈Xψ|Γ2,ψ|Γ3(2)\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

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

    So, Xφ|Γ1,φ|Γ3(1)X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)} and Xψ|Γ2,ψ|Γ3(2)X_{\left.\psi\right|_{\Gamma_{2}},\left.\psi\right|_{\Gamma_{3}}}^{(2)} are invariant subsets of QmxQ_{m}^{x} and, respectively, of Qny,Q_{n}^{y},\ for φ,ψ∈C​(T~h)\varphi,\psi\in C\mathbb{(}\widetilde{T}_{h}) and n,m∈ℕ∗;n,m\in\mathbb{N}^{\ast};

    (i​v)(iv) We prove that

    Qmx|Xφ|Γ1,φ|Γ3(1):Xφ|Γ1,φ|Γ3(1)→Xφ|Γ1,φ|Γ3(1)​ and ​Qny|Xψ|Γ2,ψ|Γ3(2):Xψ|Γ2,ψ|Γ3(2)→Xψ|Γ2,ψ|Γ3(2)\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 φ,ψ∈C​(T~h)\varphi,\psi\in C\mathbb{(}\widetilde{T}_{h}) and n,m∈ℕ∗.n,m\in\mathbb{N}^{\ast}.

    Let F,G∈Xφ|Γ1,φ|Γ3(1)F,G\in X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}. From (1.1) and (3.3) we get

    |Qmx​(F)​(x,y)−Qmx​(G)​(x,y)|=\displaystyle\left|Q_{m}^{x}(F)(x,y)-Q_{m}^{x}(G)(x,y)\right|=
    =|Qmx​(F−G)​(x,y)|≤\displaystyle=\left|Q_{m}^{x}(F-G)(x,y)\right|\leq
    ≤|qm,0​(x;y)​[F​(0,0)−G​(0,0)]|\displaystyle\leq\left|q_{m,0}\left(x;y\right)\left[F\left(0,0\right)-G(0,0)\right]\right|
    +|∑i=1mqm,i​(x;y)​[F​(i​g​(y)m,y)−G​(x,j​f​(x)n)]|\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|
    =|∑i=1mqm,i​(x;y)​[F​(i​g​(y)m,y)−G​(x,j​f​(x)n)]|\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|
    ≤∑i=1mqm,i​(x;y)​‖F−G‖∞\displaystyle\leq\sum_{i=1}^{m}q_{m,i}\left(x;y\right)\left\|F-G\right\|_{\infty}
    =[∑i=0mqm,i​(x;y)−qm,0​(x;y)]​‖F−G‖∞\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}
    ={1−(1−xg​(y))​[1−xg​(y)​(1+m​β)]m−1}​‖F−G‖∞\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}
    ≤[1−(1−11+m​β)m−1]​‖F−G‖∞,\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) ‖Qmx​(F)​(x,y)−Qmx​(G)​(x,y)‖∞≤\displaystyle\left\|Q_{m}^{x}(F)(x,y)-Q_{m}^{x}(G)(x,y)\right\|_{\infty}\leq
    ≤[1−(1−11+m​β)m−1]​‖F−G‖∞,∀F,G∈Xφ|Γ1,φ|Γ3(1),\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., Qmx|Xφ|Γ1,φ|Γ3(1)\left.Q_{m}^{x}\right|_{X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}} is a contraction for φ∈C​(T~h)\varphi\in C\mathbb{(}\widetilde{T}_{h}).

    Analogously, we prove that Qny|Xψ|Γ2,ψ|Γ3(2)\left.Q_{n}^{y}\right|_{X_{\left.\psi\right|_{\Gamma_{2}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}} is a contraction for ψ∈C​(T~h).\psi\in C\mathbb{(}\widetilde{T}_{h}).

    On the other hand, φ|Γ3−φ|Γ1g​(y)​(⋅)+φ|Γ1∈Xφ|Γ1,φ|Γ3(1)\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 ψ|Γ3−ψ|Γ2f​(x)​(⋅)+ψ|Γ2∈Xψ|Γ2,ψ|Γ3(2)\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 QmxQ_{m}^{x} and QnyQ_{n}^{y}, i.e.,

    Qmx​(φ|Γ3−φ|Γ1g​(y)​(⋅)+φ|Γ1)=φ|Γ3−φ|Γ1g​(y)​(⋅)+φ|Γ1,\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}},
    Qny​(ψ|Γ3−ψ|Γ2f​(x)​(⋅)+ψ|Γ2)=ψ|Γ3−ψ|Γ2f​(x)​(⋅)+ψ|Γ2.\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, Fφ|Γ1,φ|Γ3(1)​(x,y):=φ|Γ3−φ|Γ1g​(y)​x+φ|Γ1F_{\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 QmxQ_{m}^{x} in Xφ|Γ1,φ|Γ3(1)X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)} and Qmx|Xφ|Γ1,φ|Γ3(1)\left.Q_{m}^{x}\right|_{X_{\left.\varphi\right|_{\Gamma_{1}},\left.\varphi\right|_{\Gamma_{3}}}^{(1)}} is a Picard operator, with

    (Qmx,∞​F)​(x,y)=F​(g​(y),y)−F​(0,y)g​(y)​x+F​(0,y),\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, Fψ|Γ2,ψ|Γ3(2)​(x,y):=ψ|Γ3−ψ|Γ2f​(x)​y+ψ|Γ2F_{\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 QnyQ_{n}^{y} in Xψ|Γ2,ψ|Γ3(2)X_{\left.\psi\right|_{\Gamma_{2}},\left.\psi\right|_{\Gamma_{3}}}^{(2)} and Qny|Xψ|Γ2,ψ|Γ3(2)\left.Q_{n}^{y}\right|_{X_{\left.\psi\right|_{\Gamma_{2}},\left.\psi\right|_{\Gamma_{3}}}^{(2)}} is a Picard operator, with

    (Qny,∞​F)​(x,y)=F​(x,f​(x))−F​(x,0)f​(x)​y+F​(x,0).\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 (i​i)(ii), by Theorem 2.4, it follows that the operators QmxQ_{m}^{x} and QnyQ_{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 Pm​n1P_{mn}^{1} is a weakly Picard operator and

    (3.5) (Pm​n1,∞​F)​(x,y)=F​(g​(y),y)g​(y)​x.\left(P_{mn}^{1,\infty}F\right)\left(x,y\right)=\frac{F\left(g(y),y\right)}{g(y)}x.
    Proof.

    Let

    Xα={F∈C​(T~h)|F​(g​(y),y)=α},α∈ℝX_{\alpha}=\{F\in C(\widetilde{T}_{h})\ |\ F(g(y),y)=\alpha\},\ \ \ \ \alpha\in\mathbb{R}

    and denote by

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

    We remark that:

    1. (i)

      XαX_{\alpha} is a closed subset of C​(T~h)C(\widetilde{T}_{h});

    2. (ii)

      XαX_{\alpha} is an invariant subset of Pm​n1P_{mn}^{1}, for α∈ℝ\alpha\in\mathbb{R} and n,m∈ℕ∗;n,m\in\mathbb{N}^{\ast};

    3. (iii)

      C​(T~h)=∪𝛼​XαC(\widetilde{T}_{h})=\underset{\alpha}{\cup}X_{\alpha} is a partition of C​(T~h)C(\widetilde{T}_{h});

    4. (iv)

      Fα∈Xα∩FPm​n1,F_{\alpha}\in X_{\alpha}\cap F_{P_{mn}^{1}}, where FPm​n1F_{P_{mn}^{1}} denote the fixed points sets of Pm​n1.P_{mn}^{1}.

    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 Cheney-Sharma operators and Theorem 1.3, it follows that XαX_{\alpha} is an invariant subset of Pm​n1P_{mn}^{1}, for α∈ℝ\alpha\in\mathbb{R} and n,m∈ℕ∗;n,m\in\mathbb{N}^{\ast};

    (i​v)(iv) We prove that

    Pm​n1|Xα:Xα→Xα​ \left.P_{mn}^{1}\right|_{X_{\alpha}}:X_{\alpha}\rightarrow X_{\alpha}\text{ }

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

    |Pm​n1​(F)​(x,y)−Pm​n1​(G)​(x,y)|=|Pm​n1​(F−G)​(x,y)|\displaystyle\left|P_{mn}^{1}(F)(x,y)-P_{mn}^{1}(G)(x,y)\right|=\left|P_{mn}^{1}(F-G)(x,y)\right|
    ≤[1−(m​β1+m​β)m−1​(n​b1+n​b)n−1]​‖F−G‖∞,\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, Pm​n1|Xα\left.P_{mn}^{1}\right|_{X_{\alpha}} is a contraction for α∈ℝ.\alpha\in\mathbb{R}.

    From the contraction principle we have that FαF_{\alpha} is the unique fixed point of Pm​n1P_{mn}^{1} in XαX_{\alpha} and Pm​n1|Xα\left.P_{mn}^{1}\right|_{X_{\alpha}} is a Picard operator, so (3.5) holds. Consequently, taking into account (i​i)(ii), by Theorem 2.4, it follows that the operators Pm​n1P_{mn}^{1} is a weakly Picard operator. ∎

    Remark 3.3.

    Similar results can be obtained for the operator Pm​n2P_{mn}^{2}.

    Theorem 3.4.

    The operator Sm​n1S_{mn}^{1} is a weakly Picard operator and

    (Sm​n1,∞​F)​(x,y)=−F​(0,y)g​(y)​x+F​(x,f​(x))−F​(x,0)f​(x)​y+F​(x,0)+F​(0,y).\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

    ‖Sm​n​(F)​(x,y)−Sm​n​(G)​(x,y)‖∞\displaystyle\left\|S_{mn}(F)(x,y)-S_{mn}(G)(x,y)\right\|_{\infty}
    ≤{1−[(m​β1+m​β)m−1+(n​b1+n​b)n−1−(m​β1+m​β)m−1​(n​b1+n​b)n−1]}​‖F−G‖∞,\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 Sm​n1S_{mn}^{1} is a contraction. ∎

    Remark 3.5.

    We have a similar result for the operator Sn​m2S_{nm}^{2}.

    References

    • [1] Agratini, O., Rus, I.A., Iterates of a class of discrete linear operators via contraction principle, Comment. Math. Univ. Caroline 44 (2003), 555-563.
    • [2] Agratini, O., Rus, I.A., Iterates of some bivariate approximation process via weakly Picard operators, Nonlinear Anal. Forum 8(2) (2003), 159-168.
    • [3] Blaga, P., Cătinaş, T., Coman, G., Bernstein-type operators on triangle with one curved side, Mediterr. J. Math. 9 (2012), No. 4, 833-845.
    • [4] Blaga, P., Cătinaş, T., Coman, G., Bernstein-type operators on a square with one and two curved sides, Studia Univ. Babeş–Bolyai Math. 55 (2010), No. 3, 51-67.
    • [5] Blaga, P., Cătinaş, T., Coman, G., Bernstein-type operators on triangle with all curved sides, Appl. Math. Comput. 218 (2011), 3072-3082.
    • [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.
    • [10] Gavrea, I., Ivan, M., The iterates of positive linear operators preserving the affine functions, J. Math. Anal. Appl. 372 (2010), 366-368.
    • [11] Gavrea, I., Ivan, M., The iterates of positive linear operators preserving the constants, Appl. Math. Lett. 24 (2011), No. 12, 2068-2071.
    • [12] Gavrea, I., Ivan, M., On the iterates of positive linear operators, J. Approx. Theory 163 (2011), No. 9, 1076-1079.
    • [13] Gonska, H., Kacso, D., Piţul, P., The degree of convergence of over-iterated positive linear operators, J. Appl. Funct. Anal. 1 (2006), 403-423.
    • [14] Gonska, H., Piţul, P., Raşa, I., Over-iterates of Bernstein-Stancu operators, Calcolo 44 (2007), 117-125.
    • [15] Gonska, H., Raşa, I., The limiting semigroup of the Bernstein iterates: degree of convergence, Acta Math. Hungar. 111 (2006), No. 1-2, 119-130.
    • [16] Gonska, H., Raşa, I., On infinite products of positive linear operators reproducing linear functions, Positivity 17 (2013), No. 1, 67-79.
    • [17] Gwóźdź-Łukawska, G., Jachymski, J., IFS on a metric space with a graph structure and extensions of the Kelisky-Rivlin theorem, J. Math. Anal. Appl. 356(2) (2009), 453-463.
    • [18] Karlin, S. , Ziegler, Z., Iteration of positive approximation operators, J. Approx. Theory 3 (1970), 310-339.
    • [19] Kelisky, R.P., Rivlin, T.J., Iterates of Bernstein polynomials, Pacific J. Math. 21 (1967), 511-520.
    • [20] Raşa, I., C0C_{0}-Semigroups and iterates of positive linear operators: asymptotic behaviour, Rend. Circ. Mat. Palermo, Ser. II, Suppl. 82 (2010), 123-142.
    • [21] Rus, I.A., Picard operators and applications, Sci. Math. Jpn. 58 (2003), 191-219.
    • [22] Rus, I.A., Iterates of Stancu operators, via contraction principle, Studia Univ. Babeş–Bolyai Math. 47 (2002), No. 4, 101-104.
    • [23] Rus, I.A., Iterates of Bernstein operators, via contraction principle, J. Math. Anal. Appl. 292 (2004), 259-261.
    • [24] Rus, I.A., Fixed point and interpolation point set of a positive linear operator on C​(D¯)C(\overline{D}), Studia Univ. Babeş–Bolyai Math. 55 (2010), No. 4, 243-248.
    2020

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