The iterates of positive linear operators with the set of constant functions as the fixed point set

Diana Otrocol
(original), Approximation Theory, ISI/JCR, paper

Abstract

Let Omega subset of R-p, p is an element of N* be a nonempty subset and B(Omega) be the Banach lattice of all bounded real functions on Omega, equipped with sup norm.

Let X subset of B(Omega) be a linear sublattice of B(Omega) and A: X -> X be a positive linear operator with constant functions as the fixed point set.

In this paper, using the weakly Picard operators techniques, we study the iterates of the operator A.

Some relevant examples are also given.

Authors

T. Catinas
(Babes Bolyai Univ.)

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

I.A. Rus
(Babes Bolyai Univ.)

Keywords

Banach lattice; iterates of an operator; linear operator; linear and positive operator; linear fixed point partition; fixed point; weakly Picard operator

Cite this paper as:

T. Catinas, D. Otrocol, I.A. Rus, The iterates of positive linear operators with the set of constant functions as the fixed point set, Carpathian J. Math., Vol. 32(2016) no. 2, pp. 165-172.

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About this paper

Journal

Carpathian Journal of Mathematics

Publisher Name

North Univ. Baia Mare, Romania

DOI
Print ISSN

1584-2851

Online ISSN

1843-4401

MR

MR3587884

ZBL

Google Scholar

https://scholar.google.ro/scholar?oi=bibs&hl=en&cites=635617757058991429

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The iterates of positive linear operators with the set of constant functions as the fixed point set

Teodora Cătinaş∗, Diana Otrocol∗∗ and Ioan A. Rus∗ ∗ Babeş-Bolyai University
Department of Mathematics
M. Kogălniceanu St., No. 1, RO-400084, Cluj-Napoca, Romania tcatinas@math.ubbcluj.ro, iarus@math.ubbcluj.ro ∗∗ Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy
Fântânele St., No. 57, 400110, Cluj-Napoca, Romania dotrocol@ictp.acad.ro
(Received: date)
Key words and phrases:
Banach lattice, iterates of an operator, linear operator, linear and positive operator, linear fixed point partition, fixed point, weakly Picard operator.
2010 Mathematics Subject Classification:
47H10, 46B42, 47B65.

1. Introduction

Let XX be a real Banach space and A:X→XA\colon X\rightarrow X be a linear operator. Let us denote(AD)A(x∗):={x∈X|An(x)→x∗\ (AD)_{A}(x^{\ast}):=\{x\in X|\ A^{n}(x)\rightarrow x^{\ast} as n→∞}n\rightarrow\infty\} the attraction domain of a fixed point x∗x^{\ast} of the operator AA. Let FA:={x∈X|A​(x)=x}F_{A}:=\{x\in X|\ A(x)=x\} be the fixed point set of AA. By definition, the operator AA is weakly Picard operator (WPO) if, X=∪x∗∈FA​(A​D)A​(x∗).X=\underset{x^{\ast}\in F_{A}}{\cup}(AD)_{A}(x^{\ast}).

Let us denote by θ\theta the zero element of XX. It is clear that (A​D)A​(θ)(AD)_{A}(\theta) is a linear subspace of XX and (A​D)A​(x∗)={x∗}+(A​D)A​(θ),(AD)_{A}(x^{\ast})=\{x^{\ast}\}+(AD)_{A}(\theta), i.e., is an affine subspace of XX. This remark gives rise to the following notion (see [15]).

A partition of XX, X=∪x∗∈FA​Xx∗,X=\underset{x^{\ast}\in F_{A}}{\cup}X_{x^{\ast}}, is a linear fixed point partition (LFPP) of XX with respect to a linear operator AA iff:

  • (i)

    Xx∗∩FA={x∗},∀x∗∈FA;X_{x^{\ast}}\cap F_{A}=\{x^{\ast}\},\ \forall x^{\ast}\in F_{A};

  • (ii)

    A​(Xx∗)⊂Xx∗,∀x∗∈FA;A(X_{x^{\ast}})\subset X_{x^{\ast}},\ \forall x^{\ast}\in F_{A};

  • (iii)

    XθX_{\theta} is a linear subspace of X;X;

  • (iv)

    Xx∗={x∗}+Xθ.X_{x^{\ast}}=\{x^{\ast}\}+X_{\theta}.

The aim of this paper is to study the iterates of a linear operator, in the case of function spaces, using the technique of LFPP of the space.

2. Preliminaries

Let Ω⊂ℝp,\Omega\subset\mathbb{R}^{p}, p∈ℕ∗p\in\mathbb{N}^{\ast} be a nonempty subset, B​(Ω)B(\Omega) be the Banach lattice of all bounded real valued functions on Ω\Omega, equipped with s​u​p​n​o​r​msup\ norm. Let X⊂B​(Ω)X\subset B(\Omega) be a linear sublattice of XX and A:X→XA\colon X\rightarrow X be a linear operator with constant functions as the fixed point set.

Following [15], we consider some notions that will be used in the sequel.

Definition 2.1.

The operator A:X→XA\colon X\rightarrow X is a weakly Picard operator (WPO) if the sequence (An​(f))n∈ℕ(A^{n}(f))_{n\in\mathbb{N}} converges, for all f∈B​(Ω)f\in B(\Omega), and the limit (which may depend on ff) is a fixed point of AA.

Definition 2.2.

If AA is WPO, then we define the operator A∞,A∞:X→XA^{\infty},\;A^{\infty}\colon X\rightarrow X, by

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

We remark that A∞​(X)=FA.A^{\infty}(X)=F_{A}.

Definition 2.3.

Let A:X→XA\colon X\rightarrow X be a weakly Picard operator and ψ:ℝ+→ℝ+\psi\colon\mathbb{R}_{+}\mathbb{\rightarrow R}_{+} an increasing function, continuous in 0 and ψ​(0)=0.\psi(0)=0. The operator AA is said to be a ψ\psi-weakly Picard operator (ψ\psi-WPO) iff

d​(f,A∞​(f))≤ψ​d​(f,A​(f)),∀f∈X.d(f,A^{\infty}(f))\leq\psi d(f,A(f)),\ \forall f\in X.
Definition 2.4.

The operator A:X→XA\colon X\rightarrow X is a Picard operator (PO) if AA is WPO and FAF_{A} is a unit set.

Remark 2.1.

We observe that Ak​(X)A^{k}(X) is an invariant subset of AA for each k∈ℕ∗k\in\mathbb{N}^{\ast} and FA⊂Ak​(X)F_{A}\subset A^{k}(X). We suppose that

Ak:=A|Ak​(X):Ak​(X)→Ak​(X)​ is WPO.A_{k}:=A|_{A^{k}(X)}\colon A^{k}(X)\rightarrow A^{k}(X)\text{ is WPO.}

Then we have that

Akn​(u)→Ak∞​(u)​ as ​n→∞,∀u∈Ak​(X),A_{k}^{n}(u)\rightarrow A_{k}^{\infty}(u)\text{ as }n\rightarrow\infty,\ \forall u\in A^{k}(X),

i.e., An​(Ak​(f))→Ak∞​(Ak​(f))=A∞​(f).A^{n}(A^{k}(f))\rightarrow A_{k}^{\infty}(A^{k}(f))=A^{\infty}(f).\ So, if for some k∈ℕ∗,A|Ak​(X)k\in\mathbb{N}^{\ast},\ A|_{A^{k}(X)} is WPO then, A:X→XA\colon X\rightarrow X is WPO and A∞​(f)=Ak∞​(Ak​(f)).A^{\infty}(f)=A_{k}^{\infty}(A^{k}(f)).

Remark 2.2.

Let ϕ:X→ℝ\phi\colon X\rightarrow\mathbb{R} be a linear functional and A:X→XA\colon X\rightarrow X a linear operator. We suppose that ϕ\phi is an invariant functional of AA, i.e., ϕ​(A​(f))=ϕ​(f)\phi(A(f))=\phi(f), ∀f∈X\forall f\in X. Let us denote, for λ∈ℝ,\lambda\in\mathbb{R},

Xλ:={f∈X|ϕ​(f)=λ}.X_{\lambda}:=\{f\in X|\ \phi(f)=\lambda\}.

Then X=∪λ∈ℝ​XλX=\underset{\lambda\in\mathbb{R}}{\cup}X_{\lambda} is a partition of X.X. If Xλ∩FA={fλ∗},∀λ∈ℝX_{\lambda}\cap F_{A}=\{f_{\lambda}^{\ast}\},\ \forall\lambda\in\mathbb{R}, then, X=∪λ∈ℝ​XλX=\underset{\lambda\in\mathbb{R}}{\cup}X_{\lambda} is a LFPP of XX with respect to A.A.

We also need the following result.

Lemma 2.1.

([15]) Let A:X→XA\colon X\rightarrow X be a linear operator. We suppose that X=∪f∗∈FA​Xf∗X=\underset{f^{\ast}\in F_{A}}{\cup}X_{f^{\ast}} is a LFPP of XX with respect to AA. Then:

  • (i)

    If A|Xθ:Xθ→XθA|_{X_{\theta}}\colon X_{\theta}\rightarrow X_{\theta} is PO, then AA is WPO.

  • (ii)

    If AA is WPO, then A∞​(f)=f∗,∀f∈Xf∗,A^{\infty}(f)=f^{\ast},\ \forall f\in X_{f^{\ast}}, f∗∈FAf^{\ast}\in F_{A}.

As a suggestion for finding a LFPP of the space, the following result is useful.

Theorem 2.1.

(Characterization theorem, [11]) 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)

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

  • (b)

    A|Xλ:Xλ→Xλ\left.A\right|_{X_{\lambda}}\colon X_{\lambda}\rightarrow X_{\lambda} is a Picard operator, ∀λ∈Λ.\forall\lambda\in\Lambda.

The limit behavior for the iterates of some classes of positive linear operators were also studied, for example, in [2], [3], [7], [8], [9], [12], [13], [14].

3. Basic results

Let Ω⊂ℝp,\Omega\subset\mathbb{R}^{p}, p∈ℕ∗p\in\mathbb{N}^{\mathbb{\ast}} be a nonempty subset, B​(Ω)B(\Omega) be the Banach lattice of all bounded real valued functions on Ω,\Omega, equipped with s​u​p​n​o​r​msup\ norm. Let X⊂B​(Ω)X\subset B(\Omega) a linear sublattice of B​(Ω)B(\Omega) and A:X→XA\colon X\rightarrow X be a linear operator. We have

Theorem 3.2.

We suppose that:

  • (i)

    XX contains all constant functions on Ω;\Omega;

  • (ii)

    FAF_{A} consists of all constant functions on Ω;\Omega;

  • (iii)

    X=∪λ∈ℝ​XλX=\underset{\lambda\in\mathbb{R}}{\cup}X_{\lambda} is a LFPP of XX with respect to AA such that, for some k∈ℕk\in\mathbb{N}, we have that:

    ‖A​(u)‖≤l​‖u‖,∀u∈Ak​(X0), with some ​0<l<1.\left\|A(u)\right\|\leq l\left\|u\right\|,\ \forall u\in A^{k}(X_{0}),\text{ with some }0<l<1.

Then:

  • (a)

    AA is WPO and A∞​(f)=λ,∀f∈Xλ,λ∈ℝ;A^{\infty}(f)=\lambda,\forall f\in X_{\lambda},\lambda\in\mathbb{R};

  • (b)

    ‖Ak​(f)−A∞​(f)‖≤11−l​‖Ak​(f)−A​(Ak​(f))‖,∀f∈X;\left\|A^{k}(f)-A^{\infty}(f)\right\|\leq\frac{1}{1-l}\left\|A^{k}(f)-A(A^{k}(f))\right\|,\forall f\in X;

  • (c)

    ‖An​(Ak​(f))−A∞​(f)‖≤ln1−l​‖Ak​(f)−A​(Ak​(f))‖,∀f∈X,∀n∈ℕ;\left\|A^{n}(A^{k}(f))-A^{\infty}(f)\right\|\leq\frac{l^{n}}{1-l}\left\|A^{k}(f)-A(A^{k}(f))\right\|,\forall f\in X,\forall n\in\mathbb{N};

  • (d)

    if M>0:‖Ak​(f)−A​(Ak​(f))‖≤M,M>0:\left\|A^{k}(f)-A(A^{k}(f))\right\|\leq M, ∀f∈X,\forall f\in X, then An​(Ak​(f))​→u​n​i​f​A∞​(f)A^{n}(A^{k}(f))\overset{unif}{\rightarrow}A^{\infty}(f) as n→∞,n\rightarrow\infty, on X.X.

Proof.
  • (a)

    Let f,g∈Xλf,g\in X_{\lambda}. By linearity of AA and by the fact that f−g∈X0,f-g\in X_{0}, it follows that

    ‖A​(f)−A​(g)‖=‖A​(f−g)‖≤l​‖f−g‖, with​ 0<l<1, ​∀f,g∈Xλ,λ∈ℝ.\left\|A(f)-A(g)\right\|=\left\|A(f-g)\right\|\leq l\left\|f-g\right\|,\text{ with}\ \ 0<l<1,\text{ }\forall f,g\in X_{\lambda},\ \lambda\in\mathbb{R}.

    The constant function λ∈Xλ\lambda\in X_{\lambda} is a fixed point of A.A. Cosequently, we have that AA is a Picard operator, and taking into account (iii), by Theorem 2.1, it follows that the operator AA is a weakly Picard operator, with A∞​(f)=λ,∀f∈Xλ,λ∈ℝA^{\infty}(f)=\lambda,\forall f\in X_{\lambda},\lambda\in\mathbb{R}.

  • (b)

    We have

    (3.1) ‖Ak​(f)−A∞​(f)‖=‖Ak​(f)−λ‖≤‖Ak​(f)−A​(Ak​(f))‖+‖A​(Ak​(f))−λ‖,∀f∈X.\left\|A^{k}(f)-A^{\infty}(f)\right\|=\left\|A^{k}(f)-\lambda\right\|\leq\left\|A^{k}(f)-A(A^{k}(f))\right\|+\left\|A(A^{k}(f))-\lambda\right\|,\forall f\in X.

    By (iii), it follows

    (3.2) ‖A​(Ak​(f))−λ‖≤l​‖Ak​(f)−λ‖, with ​f−λ∈A​(X0).\left\|A(A^{k}(f))-\lambda\right\|\leq l\left\|A^{k}(f)-\lambda\right\|,\text{ with }f-\lambda\in A(X_{0}).

    From (3.1) and (3.2) we get

    ‖Ak​(f)−λ‖≤‖Ak​(f)−A​(Ak​(f))‖+l​‖Ak​(f)−λ‖.\left\|A^{k}(f)-\lambda\right\|\leq\left\|A^{k}(f)-A(A^{k}(f))\right\|+l\left\|A^{k}(f)-\lambda\right\|.

    Then

    ‖Ak​(f)−A∞​(f)‖≤11−l​‖Ak​(f)−A​(Ak​(f))‖,∀f∈X.\left\|A^{k}(f)-A^{\infty}(f)\right\|\leq\frac{1}{1-l}\left\|A^{k}(f)-A(A^{k}(f))\right\|,\forall f\in X.
  • (c)

    We have

    ‖An​(Ak​(f))−An+p​(Ak​(f))‖\displaystyle\left\|A^{n}(A^{k}(f))-A^{n+p}(A^{k}(f))\right\|
    ≤‖An​(Ak​(f))−An+1​(Ak​(f))‖+‖An+1​(Ak​(f))−An+2​(Ak​(f))‖\displaystyle\leq\left\|A^{n}(A^{k}(f))-A^{n+1}(A^{k}(f))\right\|+\left\|A^{n+1}(A^{k}(f))-A^{n+2}(A^{k}(f))\right\|
    +⋯+‖An+p−1​(Ak​(f))−An+p​(Ak​(f))‖\displaystyle\quad+\cdots+\left\|A^{n+p-1}(A^{k}(f))-A^{n+p}(A^{k}(f))\right\|
    ≤ln​‖Ak​(f)−A​(Ak​(f))‖+ln+1​‖Ak​(f)−A​(Ak​(f))‖\displaystyle\leq l^{n}\left\|A^{k}(f)-A(A^{k}(f))\right\|+l^{n+1}\left\|A^{k}(f)-A(A^{k}(f))\right\|
    +⋯+ln+p−1​‖Ak​(f)−A​(Ak​(f))‖\displaystyle\quad+\cdots+l^{n+p-1}\left\|A^{k}(f)-A(A^{k}(f))\right\|
    =(ln+⋯+ln+p−1)​‖Ak​(f)−A​(Ak​(f))‖\displaystyle=(l^{n}+\cdots+l^{n+p-1})\left\|A^{k}(f)-A(A^{k}(f))\right\|
    ≤ln1−l​‖Ak​(f)−A​(Ak​(f))‖,for ​n∈ℕ,p∈ℕ∗.\displaystyle\leq\frac{l^{n}}{1-l}\left\|A^{k}(f)-A(A^{k}(f))\right\|,\text{for }n\in\mathbb{N},p\in\mathbb{N}^{\ast}.

    So, ‖An​(Ak​(f))−A∞​(f)‖≤ln1−l​‖Ak​(f)−A​(Ak​(f))‖,∀f∈X,∀n∈ℕ​.\left\|A^{n}(A^{k}(f))-A^{\infty}(f)\right\|\leq\frac{l^{n}}{1-l}\left\|A^{k}(f)-A(A^{k}(f))\right\|,\forall f\in X,\forall n\in\mathbb{N}\text{.}

  • (d)

    By (c) we obtain ‖An​(Ak​(f))−A∞​(f)‖≤ln1−l​M​→u​n​i​f​0\left\|A^{n}(A^{k}(f))-A^{\infty}(f)\right\|\leq\frac{l^{n}}{1-l}M\overset{unif}{\rightarrow}0.

∎

Let us give a class of operators for which the condition (iii) in Theorem 3.2 is satisfied.

Let X:=C​([0,1]p),p∈ℕ∗,X:=C([0,1]^{p}),\ p\in\mathbb{N}^{\ast}, ak∈[0,1]p,k=0,m¯,a_{k}\in[0,1]^{p},\ k=\overline{0,m}, p,m∈ℕ∗,p,m\in\mathbb{N}^{\ast}, are distinct points such that I:=c​o​{ak|k=0,m¯}I:=co\{a_{k}|\ k=\overline{0,m}\} has a nonempty interior, and ψk∈C​([0,1]p,ℝ+),k=0,m¯\psi_{k}\in C([0,1]^{p},\mathbb{R}_{+}),k=\overline{0,m}. We suppose that:

  1. (1)

    {ψk|k=0,m¯}\{\psi_{k}|\ k=\overline{0,m}\} is linearly independent;

  2. (2)

    ∑k=0mψk​(x)=1,∀x∈[0,1]p.\sum\limits_{k=0}^{m}\psi_{k}(x)=1,\ \forall x\in[0,1]^{p}.

Now we consider the following positive linear operator, A:C​([0,1]p)→C​([0,1]p)A\colon C([0,1]^{p})\rightarrow C([0,1]^{p}), A​(f):=∑k=0mf​(ak)​ψk.A(f):=\sum\limits_{k=0}^{m}f(a_{k})\psi_{k}.

Condition (2)(2) implies that the constant functions are the fixed points of AA. On the other hand, the condition

  1. (3)

    r​a​n​k​[ψi​(ak)−Im+1]=mrank[\psi_{i}(a_{k})-I_{m+1}]=m

implies that, FA=F_{A}= the set of constant functions.

To study the iterates of A,A, we shall give conditions in which the operator AA has an invariant functional of the following form

ϕ​(f):=∑i=0mci​f​(ai),\phi(f):=\sum\limits_{i=0}^{m}c_{i}f(a_{i}),

with some ci≥0,i=0,m¯.c_{i}\geq 0,\ i=\overline{0,m}.

We have the following result.

Theorem 3.3.

In the conditions (1),(2)(1),(2) and (3)(3), there exists c∗∈ℝm+1c^{\ast}\in\mathbb{R}^{m+1} such that:

  • (a)

    c∗≥0,∑i=0mci∗=1;c^{\ast}\geq 0,\sum\limits_{i=0}^{m}c_{i}^{\ast}=1;

  • (b)

    the functional, ϕ:C​([0,1]p)→ℝ,ϕ​(f):=∑i=0mci∗​f​(ai),\phi\colon C([0,1]^{p})\rightarrow\mathbb{R},\ \phi(f):=\sum\limits_{i=0}^{m}c_{i}^{\ast}f(a_{i}), is an invariant functional of the operator A;A;

  • (c)

    if C​([0,1]p)=∪λ∈ℝ​XλC([0,1]^{p})=\underset{\lambda\in\mathbb{R}}{\cup}X_{\lambda} is a LFPP corresponding to ϕ\phi, and ψi​(x)>0,∀x∈I,i=0,m¯,\psi_{i}(x)>0,\forall x\in I,i=\overline{0,m}, then ∥A|X0∩C​(I)∥=l<1\left\|A|_{X_{0}\cap C(I)}\right\|=l<1, and so, A|C​(I)A|_{C(I)} is WPO and A∞​(f)=∑i=1mci∗​f​(ai),f∈C​(I).A^{\infty}(f)=\sum\limits_{i=1}^{m}c_{i}^{\ast}f(a_{i}),f\in C(I).

Proof.

First, we remark that a functional, ϕ​(f)=∑i=0mci​f​(ai),\phi(f)=\sum\limits_{i=0}^{m}c_{i}f(a_{i}), ci≠0c_{i}\neq 0, is invariant for AA iff:

∑i=0mci​ψk​(ai)=ck,k=0,m¯.\sum\limits_{i=0}^{m}c_{i}\psi_{k}(a_{i})=c_{k},\ k=\overline{0,m}.

Let us consider the subset K⊂ℝm+1,K:={c∈ℝm+1|ci≥0,∑i=0mci=1}.K\subset\mathbb{R}^{m+1},\ K:=\{c\in\mathbb{R}^{m+1}|\ c_{i}\geq 0,\ \sum\limits_{i=0}^{m}c_{i}=1\}.\ We take in KK the following function:

T:K→ℝm+1,T​(c):=(∑i=0mci​ψ0​(ai),…,∑i=0mci​ψm​(ai)).T\colon K\rightarrow\mathbb{R}^{m+1},\ T(c):=\left(\sum\limits_{i=0}^{m}c_{i}\psi_{0}(a_{i}),\ldots,\sum\limits_{i=0}^{m}c_{i}\psi_{m}(a_{i})\right).

Since the matrix [φi​(ak)][\varphi_{i}(a_{k})] is a stochastic matrix, it follows that, T​(K)⊂K.T(K)\subset K. From the Brouwer fixed point theorem, there exists c∗∈Kc^{\ast}\in K such that T​(c∗)=c∗T(c^{\ast})=c^{\ast}. From the condition (3)(3), it follows that there exists such a unique fixed point. So, we have (a)(a) and (b)(b). Let f∈X0,f\in X_{0}, i.e., ∑i=0mci∗​f​(ai)=0,\sum\limits_{i=0}^{m}c_{i}^{\ast}f(a_{i})=0, and ci∗>0,i=0,m¯c_{i}^{\ast}>0,\ i=\overline{0,m}. For f∈C​(I)∩X0f\in C(I)\cap X_{0} we have

|A​(f)​(x)|\displaystyle\left|A(f)(x)\right| =|∑k=0mf​(ak)​ψk​(x)|≤max0≤k≤m​(1−ψk​(x))​‖f‖=\displaystyle=\left|\sum\limits_{k=0}^{m}f(a_{k})\psi_{k}(x)\right|\leq\underset{0\leq k\leq m}{\max}(1-\psi_{k}(x))\left\|f\right\|=
=l​‖f‖, with ​l<1.\displaystyle=l\left\|f\right\|,\text{ with }l<1.

So, ‖A​(f)‖≤l​‖f‖,∀f∈C​(I),\left\|A(f)\right\|\leq l\left\|f\right\|,\forall f\in C(I), with ϕ​(f)=0,\phi(f)=0, and we have (c)(c), from Theorem 3.2. ∎

Remark 3.3.

(see [13]) If ψi,i=0,m¯\psi_{i},\ i=\overline{0,m} are polynomial functions, then the operatorA:C​([0,1]p)→C​([0,1]p)A\colon C([0,1]^{p})\rightarrow C([0,1]^{p}) is WPO and, A∞​(f)=∑i=0mci∗​ψi​(ai).A^{\infty}(f)=\sum\limits_{i=0}^{m}c_{i}^{\ast}\psi_{i}(a_{i}).

For presenting the next result we need the following definition.

Definition 3.5.

Let Y⊂XY\subset X be a linear subspace of X.X. By definition, an element e∈Ye\in Y is an order unit element if, for any f∈Yf\in Y there exists Mf>0M_{f}>0 such that |f|≤Mf​e.|f|\leq M_{f}e. (See, e.g., [5], [6], [10].)

In this case we have on YY the Minkowski norm, ∥⋅∥e,\left\|\cdot\right\|_{e}, and it follows:

  • •

    |f|≤‖f‖e​e;\left|f\right|\leq\left\|f\right\|_{e}e;

  • •

    ‖f‖≤‖f‖e​‖e‖.\left\|f\right\|\leq\left\|f\right\|_{e}\left\|e\right\|.

Theorem 3.4.

We suppose that:

  • (i)

    AA is a linear positive operator;

  • (ii)

    XX contains all constant functions on Ω;\Omega;

  • (iii)

    FAF_{A} consists of all constant functions on Ω;\Omega;

  • (iv)

    X=∪λ∈ℝ​XλX=\underset{\lambda\in\mathbb{R}}{\cup}X_{\lambda} is a LFPP of XX with respect to AA such that, for some k∈ℕ∗k\in\mathbb{N}^{\ast}, Ak​(X0)A^{k}(X_{0}) has an order unit element ee.

Then:

  • (a)

    An​(e)​(x)→0A^{n}(e)(x)\rightarrow 0 as n→∞,∀x∈Ωn\rightarrow\infty,\forall x\in\Omega implies that An​(f)​(x)→λ,∀x∈Ω,∀f∈Xλ,λ∈ℝA^{n}(f)(x)\rightarrow\lambda,\ \forall x\in\Omega,\forall f\in X_{\lambda},\lambda\in\mathbb{R}, i.e., AA is WPO with respect to pointwise convergence on XX and A∞​(f)=λ,∀f∈Xλ;A^{\infty}(f)=\lambda,\forall f\in X_{\lambda};

  • (b)

    An​(e)​→∥⋅∥​0⇒AA^{n}(e)\overset{\left\|\cdot\right\|}{\rightarrow}0\Rightarrow A is WPO with respect to →∥⋅∥\overset{\left\|\cdot\right\|}{\rightarrow} and A∞​(f)=λ,∀f∈Xλ,λ∈ℝ;A^{\infty}(f)=\lambda,\forall f\in X_{\lambda},\lambda\in\mathbb{R};

  • (c)

    if there exists l∈]0,1[l\in]0,1[ such that A​(e)≤l​eA(e)\leq le then:

    • (1)

      ‖A​(f)‖e≤l​‖f‖e,∀f∈Ak​(X0);\left\|A(f)\right\|_{e}\leq l\left\|f\right\|_{e},\forall f\in A^{k}(X_{0});

    • (2)

      ‖A​(f)−A​(g)‖e≤l​‖f−g‖e,∀f,g∈Ak​(Xλ),λ∈ℝ;\left\|A(f)-A(g)\right\|_{e}\leq l\left\|f-g\right\|_{e},\forall f,g\in A^{k}(X_{\lambda}),\ \lambda\in\mathbb{R};

    • (3)

      ‖A​(f)−A2​(f)‖e≤l​‖f−A​(f)‖e,∀f∈Ak​(X);\left\|A(f)-A^{2}(f)\right\|_{e}\leq l\left\|f-A(f)\right\|_{e},\forall f\in A^{k}(X);

    • (4)

      ‖f−A∞​(f)‖e≤11−l​‖f−A​(f)‖e,∀f∈Ak​(X);\left\|f-A^{\infty}(f)\right\|_{e}\leq\frac{1}{1-l}\left\|f-A(f)\right\|_{e},\forall f\in A^{k}(X);

    • (5)

      AA is ψ−\psi-WPO on (X,∥⋅∥e)(X,\left\|\cdot\right\|_{e}) with ψ​(t)=t1−l,t∈ℝ+.\psi(t)=\frac{t}{1-l},\ t\in\mathbb{R}_{+}.

Proof.
  • (a),(b)

    By Lemma 2.1 it is sufficient to prove that A|Ak​(X0)A|_{A^{k}(X_{0})} is Picard operator. Let f∈Ak​(X0)f\in A^{k}(X_{0}). Since ee is an order unit for Ak​(X0)A^{k}(X_{0}) we have that:

    |f|≤λ​(f)​e​.\left|f\right|\leq\lambda(f)e\text{.}

    But AA is a non-decreasing linear operator and we have that

    0≤|An​(f)|≤An​(|f|)≤λ​(f)​An​e→0​ as n→∞,0\leq\left|A^{n}(f)\right|\leq A^{n}(\left|f\right|)\leq\lambda(f)A^{n}e\rightarrow 0\text{ as n}\rightarrow\infty,

    i.e., A|Ak​(X0)A|_{A^{k}(X_{0})} is a Picard operator.

  • (c)

    (1) We have

    |A​(f)|\displaystyle\left|A(f)\right| ≤A​(|f|)≤‖f‖e​A​(e)≤‖f‖e​l​e\displaystyle\leq A(\left|f\right|)\leq\left\|f\right\|_{e}A(e)\leq\left\|f\right\|_{e}le
    =l​‖f‖e​e\displaystyle=l\left\|f\right\|_{e}e
    ‖A​(f)‖≤‖A‖​‖f‖≤‖f‖e​A​(e)≤‖f‖e​l​e,\left\|A(f)\right\|\leq\left\|A\right\|\left\|f\right\|\leq\left\|f\right\|_{e}A(e)\leq\left\|f\right\|_{e}le,

    whence it follows

    ‖A​(f)‖e≤l​‖f‖e,∀f∈Ak​(X0);\left\|A(f)\right\|_{e}\leq l\left\|f\right\|_{e},\ \forall f\in A^{k}(X_{0});

    (2) We have

    |f−g|≤‖f−g‖e​e\left|f-g\right|\leq\left\|f-g\right\|_{e}e
    |A​(f−g)|\displaystyle\left|A(f-g)\right| ≤A​(|f−g|)≤‖f−g‖e​A​(e)≤‖f−g‖e​l​e\displaystyle\leq A(\left|f-g\right|)\leq\left\|f-g\right\|_{e}A(e)\leq\left\|f-g\right\|_{e}le
    =l​‖f−g‖e​e,\displaystyle=l\left\|f-g\right\|_{e}e,

    so it follows

    ‖A​(f−g)‖e≤l​‖f−g‖e,∀f,g∈Ak​(Xλ),λ∈ℝ;\left\|A(f-g)\right\|_{e}\leq l\left\|f-g\right\|_{e},\ \forall f,g\in A^{k}(X_{\lambda}),\ \lambda\in\mathbb{R};

    (3) We have

    |f−A​(f)|≤‖f−A​(f)‖e​e\left|f-A(f)\right|\leq\left\|f-A(f)\right\|_{e}e
    |A​(f−A​(f))|\displaystyle\left|A(f-A(f))\right| ≤A​(|f−A​(f)|)≤‖f−A​(f)‖e​A​(e)≤‖f−A​(f)‖e​l​e\displaystyle\leq A(\left|f-A(f)\right|)\leq\left\|f-A(f)\right\|_{e}A(e)\leq\left\|f-A(f)\right\|_{e}le
    =l​‖f−A​(f)‖e​e,\displaystyle=l\left\|f-A(f)\right\|_{e}e,

    whence we obtain

    ‖A​(f)−A2​(f)‖e≤l​‖f−A​(f)‖e,∀f∈Ak​(Xλ),λ∈ℝ;\left\|A(f)-A^{2}(f)\right\|_{e}\leq l\left\|f-A(f)\right\|_{e},\ \forall f\in A^{k}(X_{\lambda}),\ \lambda\in\mathbb{R};

    (4) By Theorem 3.2, (b) we get the result.

    (5) By (4) and Definition 2.3 we get the result.

∎

4. Applications

Example 4.1.

Let Ω=([0,∞[)p,p≥1,X:=CB(Ω),ϕ:X→ℝ,ϕ(f):=f(0)\Omega=([0,\infty[)^{p},\ p\geq 1,\ X:=C_{B}(\Omega),\ \phi\colon X\rightarrow\mathbb{R},\ \phi(f):=f(0) and Ak:X→X,k=1,m¯A_{k}\colon X\rightarrow X,k=\overline{1,m} a linear operator such that

  • •

    Ak​(1~)=1~,k=1,m¯;A_{k}(\widetilde{1})=\widetilde{1},\ k=\overline{1,m};

  • •

    (Ak​(f))​(0)=f​(0),∀f∈X,k=1,m¯.(A_{k}(f))(0)=f(0),\ \forall f\in X,\ k=\overline{1,m}.

Let X=∪λ∈ℝ​XλX=\underset{\lambda\in\mathbb{R}}{\cup}X_{\lambda} be the linear partition corresponding to ϕ\phi.

Let ck∈ℝ∖{0},k=1,m¯c_{k}\in\mathbb{R}\setminus\{0\},k=\overline{1,m} be such that

  • •

    c1+⋯+cm=0c_{1}+\cdots+c_{m}=0

  • •

    |c1|+⋯+|cm|=l<1.\left|c_{1}\right|+\cdots+\left|c_{m}\right|=l<1.

Let A:X→XA\colon X\rightarrow X be defined by A​(f):=f​(0)~+∑k=1mck​Ak​(f)A(f):=\widetilde{f(0)}+\sum\limits_{k=1}^{m}c_{k}A_{k}(f). We remark that ϕ\phi is an invariant functional for A.A.

Now we suppose that: ∥Ak|X0∥≤1\left\|A_{k}|_{X_{0}}\right\|\leq 1. Then, from Theorem 3.2, we have:

  • (i)

    AA is WPO;

  • (ii)

    FA={λ~|λ∈ℝ};F_{A}=\{\widetilde{\lambda}|\ \lambda\in\mathbb{R}\};

  • (iii)

    FA∩Xλ={λ~};F_{A}\cap X_{\lambda}=\{\widetilde{\lambda}\};

  • (iv)

    A∞​(f)=f​(0),∀f∈XA^{\infty}(f)=f(0),\forall f\in X;

  • (v)

    ‖f−A∞​(f)‖≤11−l​‖f−A​(f)‖,∀f∈X\left\|f-A^{\infty}(f)\right\|\leq\frac{1}{1-l}\left\|f-A(f)\right\|,\forall f\in X;

Example 4.2.

Let α,β∈ℝ\alpha,\beta\in\mathbb{R}, 0≤α<β.0\leq\alpha<\beta. We consider the Stancu operator Sm,α,β:C​([0,1]×[0,1])→C​([0,1]×[0,1])S_{m,\alpha,\beta}\colon C([0,1]\times[0,1])\rightarrow C([0,1]\times[0,1]) defined by (see, e.g., [1], [4], [16])

(Sm,α,β​f)​(x,y)=∑i=0m∑j=0m(mi)​(mj)​xi​yj​(1−x)m−i​(1−y)m−j​f​(i+αm+β,j+αm+β).(S_{m,\alpha,\beta}f)(x,y)=\sum_{i=0}^{m}\sum_{j=0}^{m}\binom{m}{i}\binom{m}{j}x^{i}y^{j}(1-x)^{m-i}(1-y)^{m-j}f\left(\frac{i+\alpha}{m+\beta},\frac{j+\alpha}{m+\beta}\right).

We remark that the operator Sm,α,βS_{m,\alpha,\beta} satisfies the conditions of Theorem 3.3. By this theorem we have the following properties:

  • (a)

    the operator Sm,α,βS_{m,\alpha,\beta}\ is W​P​O;WPO;

  • (b)

    Sm,α,β∞​(f)=∑i=0m∑j=0mci​j∗​f​(i+αm+β,j+αm+β),S_{m,\alpha,\beta}^{\infty}(f)=\sum\limits_{i=0}^{m}\sum\limits_{j=0}^{m}c_{ij}^{\ast}f\left(\frac{i+\alpha}{m+\beta},\frac{j+\alpha}{m+\beta}\right), where ci​j∗c_{ij}^{\ast} are the unique solutions in KK of the following system

    (4.3) ∑i=0m∑j=0m(mi)​(mj)​(i+αm+β)k​(1−i+αm+β)m−k​(j+αm+β)l​(1−j+αm+β)m−l​ci​j=ck,l,\sum_{i=0}^{m}\sum_{j=0}^{m}\tbinom{m}{i}\tbinom{m}{j}\left(\tfrac{i+\alpha}{m+\beta}\right)^{k}\left(1-\tfrac{i+\alpha}{m+\beta}\right)^{m-k}\left(\tfrac{j+\alpha}{m+\beta}\right)^{l}\left(1-\tfrac{j+\alpha}{m+\beta}\right)^{m-l}c_{ij}=c_{k,l},

       for k,l=0,m¯.k,l=\overline{0,m}.

For example, for m=1m=1 the system (4.3) implies:

{((1+β−α)2−(1+β)2)​c00+(1+β−α)​(β−α)​c01+(β−α)​(1+β−α)​c10+(β−α)2​c11=0(1+β−α)​α​c00+((1+β−α)​(1+α)−(1+β)2)​c01+(β−α)​α​c10+(β−α)​(1+α)​c11=0α​(1+β−α)​c00+α​(β−α)​c01+((1+α)​(1+β−α)−(1+β)2)​c10+(1+α)​(β−α)​c11=0α2​c00+α​(1+α)​c01+α​(1+α)​c10+((1+α)2−(1+β)2)​c11=0c00+c01+c10+c11=1\left\{\begin{array}[c]{l}((1+\beta-\alpha)^{2}-(1+\beta)^{2})c_{00}+(1+\beta-\alpha)(\beta-\alpha)c_{01}+(\beta-\alpha)(1+\beta-\alpha)c_{10}\\ +(\beta-\alpha)^{2}c_{11}=0\\ (1+\beta-\alpha)\alpha c_{00}+((1+\beta-\alpha)(1+\alpha)-(1+\beta)^{2})c_{01}+(\beta-\alpha)\alpha c_{10}\\ +(\beta-\alpha)(1+\alpha)c_{11}=0\\ \alpha(1+\beta-\alpha)c_{00}+\alpha(\beta-\alpha)c_{01}+((1+\alpha)(1+\beta-\alpha)-(1+\beta)^{2})c_{10}\\ +(1+\alpha)(\beta-\alpha)c_{11}=0\\ \alpha^{2}c_{00}+\alpha(1+\alpha)c_{01}+\alpha(1+\alpha)c_{10}+((1+\alpha)^{2}-(1+\beta)^{2})c_{11}=0\\ c_{00}+c_{01}+c_{10}+c_{11}=1\end{array}\right.

and we get

S1,α,β∞​(f)\displaystyle S_{1,\alpha,\beta}^{\infty}(f) =(α−β)2β2​f​(α1+β,α1+β)−α​(α−β)β2​f​(α1+β,1+α1+β)\displaystyle=\frac{(\alpha-\beta)^{2}}{\beta^{2}}f\left(\frac{\alpha}{1+\beta},\frac{\alpha}{1+\beta}\right)-\frac{\alpha(\alpha-\beta)}{\beta^{2}}f\left(\frac{\alpha}{1+\beta},\frac{1+\alpha}{1+\beta}\right)
−α​(α−β)β2​f​(1+α1+β,α1+β)+α2β2​f​(1+α1+β,1+α1+β).\displaystyle\quad-\frac{\alpha(\alpha-\beta)}{\beta^{2}}f\left(\frac{1+\alpha}{1+\beta},\frac{\alpha}{1+\beta}\right)+\frac{\alpha^{2}}{\beta^{2}}f\left(\frac{1+\alpha}{1+\beta},\frac{1+\alpha}{1+\beta}\right).

Particular cases. For α=0\alpha=0, β>0\beta>0 we have S1,α,β∞​(f)=f​(0,0)S_{1,\alpha,\beta}^{\infty}(f)=f\left(0,0\right).

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2016

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