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Rev. Anal. Num´er. Th´eor. Approx., vol. 36 (2007) no. 1, pp. 9–23 ictp.acad.ro/jnaat

CONTINUOUS SELECTIONS OF BOREL MEASURES, POSITIVE OPERATORS AND DEGENERATE EVOLUTION PROBLEMS

FRANCESCO ALTOMAREand VITA LEONESSA∗∗

Abstract. In this paper we continue the study of a sequence of positive lin- ear operators which we have introduced in [9] and which are associated with a continuous selection of Borel measures on the unit interval. We show that the iterates of these operators converge to a Markov semigroup whose generator is a degenerate second-order elliptic differential operator on the unit interval. Some qualitative properties of the semigroup, or equivalently, of the solutions of the corresponding degenerate evolution problems, are also investigated.

MSC 2000. 47D06, 35A35, 41A36.

Keywords. Degenerate differential operator, diffusion equation, Markov semi- group, Borel measure, positive approximation process, asymptotic formula.

INTRODUCTION

In the previous paper [9] we have undertaken the study of a new sequence (Cn)n≥1 of positive linear operators acting on the space of Lebesgue functions on the unit interval. We have investigated their approximation and shape preserving properties, presenting some estimates of the rate of convergence by means of suitable moduli of smoothness.

In this paper we continue the study of these operators by establishing an asymptotic formula. This formula leads to a one-dimensional second-order differential operator of the form

(1) Au(x) :=α(x)u00(x) +d2xu0(x) (0< x <1)

defined on a suitable domain of C([0,1])∩ C2(]0,1[). Here 0< d≤2 and α is a continuous function on [0,1] such that α(0) = α(1) = 0 and α(x) > 0 for 0< x <1.

Under additional hypotheses on α, we show that the operatorA defined on the subspace

DM(A) :=u∈ C([0,1])∩ C2(]0,1[)| lim

x→0+Au(x)∈R, lim

x→1Au(x)∈R

Dipartimento di Matematica, Universit`a degli Studi di Bari, Via E. Orabona, 4 70125 Bari-Italy, e-mail: {altomare}@dm.uniba.it.

∗∗Dipartimento di Matematica e Informatica, Universit`a degli Studi della Basilicata, Via dell’Ateneo Lucano, 10, 85100 Potenza, Italy, e-mail: {vita.leonessa}@unibas.it.

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is the generator of a Markov semigroup (T(t))t≥0 onC([0,1]). Furthermore we prove that for every t ≥ 0, for every sequence (k(n))n≥1 of positive integers such thatk(n)/nt (n→ ∞) and for every f ∈ C([0,1]),

(2) lim

n→∞Cnk(n)(f) =T(t)f uniformly on [0,1].

Thanks to formula (2), we obtain some qualitative properties of the semi- group (T(t))t≥0and hence of the solutions of the diffusion equations associated with the operator (A, DM(A)).

In the last part of the paper we discuss a converse problem and we show that, given a differential operator of the form (1) generating a Markov semigroup there exists a continuous selection of Borel measures whose corresponding operators Cn represent the semigroup by means of their iterates.

1. THE OPERATORSCN

In this section we recall the definition and the main properties of the se- quence of the operators Cn, introduced in [9], whose iterates will be studied in the subsequent sections as we quoted in the Introduction.

As usual, we shall denote byC([0,1]) the space of all real valued continuous functions on [0,1] endowed with the sup-norm k · k.

Let B([0,1]) be the σ-algebra of all Borel subsets of [0,1] and denote by M+([0,1]) the cone of all (regular) Borel measures on [0,1] endowed with the vague topology. For every x ∈ [0,1] we shall denote by εx the point-mass measure concentrated at x, i.e.,

εx(B) :=

1 ifxB,

0 ifx /B, for everyB ∈ B([0,1]).

The symbol 1 stands for the constant function 1 and, for every n ≥ 1, en∈ C([0,1]) denotes the functions en(t) :=tn (0≤t≤1).

A continuous selection of probability Borel measures on [0,1] is a family (µx)0≤x≤1of probability Borel measures on [0,1] such that for everyf ∈ C([0,1]) the function x 7−→

Z 1 0

fx is continuous on [0,1]. Such a function will be denoted by T(f), i.e.,

(1.1) T(f)(x) :=

Z 1 0

fx (0≤x≤1).

The operatorT:C([0,1])−→ C([0,1]) is positive (hence continuous) andkTk= 1.

As in [9] we shall fix a continuous selection (µx)0≤x≤1 of probability Borel measures on [0,1] satisfying the following additional assumption:

(1.2)

Z 1 0

e1x =x (0≤x≤1) (i.e.,T(e1) =e1).

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Let (an)n≥1 and (bn)n≥1 be two real sequences such that, for every n≥1, 0 ≤ an < bn ≤ 1. For every n ≥ 1 consider the positive linear operator Cn:L1([0,1])−→ C([0,1]) defined, for every f ∈ L1([0,1]) andx∈[0,1], as

Cn(f)(x) : = Z

[0,1]n

bn+1

n−an

Z x1+···+xn+1n+bn

x1+···+xn+an

n+1

f(t)dt

nx(x1, . . . , xn)

= Z 1

0

· · · Z 1

0

bn+1

n−an

Z x1+···+xn+bn

n+1 x1+···+xn+an

n+1

f(t)dt

x(x1). . .x(xn), (1.3)

whereµnx denotes the tensor product of ncopies of µx.

The operatorCnis well-defined and maps the spaceL1([0,1]) into the space C([0,1]). Moreover each Cn is continuous from C([0,1]) into C([0,1]) and its norm is equal to 1.

It is worth pointing out that to a given continuous selection (µx)0≤x≤1 of probability Borel measures on [0,1] it is possible to associate another sequence of positive linear operators, namely the Bernstein-Schnabl operators, which are defined as

Bn(f)(x) : = Z

[0,1]n

f x1+···+xn nnx(x1, . . . , xn)

= Z 1

0

· · · Z 1

0

f x1+···+xn nx(x1)· · ·dµx(xn),

for every n ≥ 1, f ∈ C([0,1]) and 0 ≤ x ≤ 1. These operators have been extensively studied (see, e.g., [1], [7] and [10]).

There is a close relationship between the operators Cn and Bn. In [9, Remark 1.3] we showed that, for a givenf ∈ L1([0,1]), considering the function F ∈ C([0,1]) defined by F(x) =

Z x 0

f(t)dt (0≤x≤1) then, for every n≥1, the operatorCn can be written as

(1.4) Cn(f) = bn+1

n−anBnn(F)), where the mapping σn:C([0,1])−→ C([0,1]) is defined by

σn(F)(x) :=Fn+1n x+n+1bn Fn+1n x+ n+1an , for everyF ∈ C([0,1]) and x∈[0,1].

Another formula which relates the operatorsCnto the operatorsBnis given in [9, Remark 1.4]. It has been useful both for investigating the behaviour of the operatorsCnon convex functions and for suggesting a possible generaliza- tion of our results replacing the interval [0,1] with an arbitrary interval (not necessarily bounded) or with a convex subset of some locally convex space.

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We recall it here:

Cn(f)(x) = Z 1

0

· · · Z 1

0

f(bnan)s+n+1an+x1+···+xndsdµx(x1). . .x(xn)

= Z 1

0

· · · Z 1

0

fx1+···+xn+1n+1n(x1)dµx(x2). . .x(xn+1), where µn denote the image measure of the Borel-Lebesgue measure λ1 under the mapping Tn(x) = (bnan)x+an (0≤x≤1).

Some examples of operators Cn can be found in [9, Examples 1.5]. In particular we point out that, ifµx:=1+ (1−x)ε0,x∈[0,1], then they can be rewritten as

(1.5) Cn(f)(x) =

n

X

k=0 n k

xk(1−x)n−k

bn+1

n−an

Z k+bn+1n

k+an

n+1

f(t)dt

and, by taking an= 0, bn = 1 for eachn≥1, they turn into the well-known Kantorovich operators ([21]; [7, pp. 333–335]).

Another example of operators Cn can be obtained by considering the con- tinuous selection (νxλ)0≤x≤1 of the probability Borel measuresνx defined by

νxλ:=λ(x)µx+ (1−λ(x))εx (0≤x≤1),

where the measure µx is given by µx := 1 + (1−x)ε0 and λC([0,1]) is a function satisfying 0 ≤ λ ≤ 1. The operators Cn associated with the continuous selection (νxλ)0≤x≤1 are given by

Cn,λ(f)(x) =

=

n

X

h=0 n−h

X

k=0 n h

n−h k

bn+1

nan

Z k+hx+bn+1 n

k+hx+an

n+1

f(t)dt

xk(1−x)n−h−kλ(x)n−h(1−λ(x))h, for everyf ∈ L1([0,1]),x∈[0,1] andn≥1.

In [9, Section 2] we investigated the approximation properties of the op- erators Cn in the space C([0,1]) and, in some cases, in the space Lp([0,1]).

We also presented several estimates of the rate of convergence by means of suitable moduli of smoothness. Shape preserving properties of these operators were also discussed (see [9, Section 3]). In particular we proved that each operator Cn preserves both the class of H¨older continuous functions and the one of convex continuous functions.

We recall here some of these results which will be useful in Section 3. The first one shows that each operatorCn preserves the class of H¨older continuous functions. The next one gives information about the preservation of convex functions by the operators Cn. For more details see [9, Section 3].

For given M > 0 and 0≤ α ≤1, we shall denote by LipMα the subset of all f ∈ C([0,1]) such that

|f(x)−f(y)| ≤M|x−y|α for everyx, y∈[0,1].

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Moreover, for any f ∈ C([0,1]) the symbol ω(f,·) stands for the usual modulus of smoothness of the first order which is defined by

ω(f, δ) := sup{|f(x)−f(y)|:|x−y| ≤δ, x, y∈[0,1]} (δ >0).

Theorem 1.1. If T(Lip11) ⊂ T(Lipc1) for some c ≥ 1, then for every n≥1, f ∈ C([0,1]), δ >0, M >0 and 0< α≤1

ω(Cn(f), δ)≤(1 +c)ω(f, δ) and Cn(LipMα)⊂LipcαMα.

In particular, if T(Lip11)⊂Lip11, then

ω(Cn(f), δ)≤ 2ω(f, δ) and Cn(LipMα)⊂LipMα.

Theorem 1.2. Consider the operators Cn associated with the continuous selection of probability Borel measuresx)0≤x≤1 defined by (1.3). Suppose that:

(c1) The operatorT, given by(1.1), maps continuous convex functions into (continuous) convex functions;

(c2) For everyx, y∈[0,1]

Z

[0,1]2

ϕfd(µxµx+µyµy)≥2 Z

[0,1]2

ϕfd(µxµy),

where ϕf(s, t) :=f s+t2 , (s, t)∈[0,1]2 and the symboldenotes the tensor product among measures.

Then each operator Cn maps continuous convex functions into (continuous) convex functions.

Examples of selections of measures satisfying (c1) and (c2) can be found in [10, Examples 2.7].

2. AN ASYMPTOTIC FORMULA

In this section we establish an asymptotic formula for the sequence of the operators Cn defined by (1.3) with respect to the uniform norm.

The usefulness of asymptotic formulae in the representation of C0-semi- groups in terms of positive linear operators has been shown by many results in the last years, since the pioneer work of the first author ([1], [2], [3], [4]) (see also [6], [8], [11], [12], [13], [18], [19] and the references given there).

The first result about asymptotic formulae is due to Voronovskaja [24].

It states that, considering the sequence (Bn)n≥1 of the classical Bernstein operators defined on the unit interval [17] (see also, e.g., [7, pp. 218–220]), for any f ∈ C2([0,1])

n→∞lim n(Bn(f)(x)−f(x)) = x(1−x)2 f00(x),

uniformly with respect tox∈[0,1]. Such a result shows that for Bernstein op- erators the convergence cannot be too fast, even if the approximating function is smooth.

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In order to show an asymptotic formula for the sequence (Cn)n≥1, we shall use a generalization of Voronovskaja’s result due to Mamedov [22] (see also [5, Theorem 1]), which holds for an arbitrary sequence (Ln)n≥1 of positive linear operators acting onC([0,1]) and which is stated below.

As usual, for every x∈[0,1] the symbol ψx stands for the function (2.1) ψx(t) :=tx (0≤t≤1).

Theorem 2.1. Consider a sequence (Ln)n≥1 of positive linear operators from C([0,1]) into itself and let α, β and γ be functions defined on [0,1].

Assume that (i) lim

n→∞n(Ln(1)(x)−1) =γ(x) uniformly on [0,1], (ii) lim

n→∞nLnx)(x) =β(x) uniformly on [0,1], (iii) lim

n→∞nLnx2)(x) =α(x) uniformly on [0,1], (iv) lim

n→∞nLnxq)(x) = 0 uniformly on[0,1], for some even positive integer q≥4.

Then for every f ∈ C2([0,1])

n→∞limn(Ln(f)(x)−f(x)) = α(x)2 f00(x) +β(x)f0(x) +γ(x) uniformly on [0,1].

For a proof we refer the reader to [5, Theorem 1] where a more general result for not necessarily compact interval is presented.

Now we are in a position to state and prove the main result of this section.

Theorem 2.2. Consider the sequence (Cn)n≥1 of the operators Cn defined by (1.3) and assume that the sequence (an+bn)n≥1 is convergent. Then for every f∈ C2([0,1])

n→∞limn(Cn(f)(x)−f(x)) = T(e2)(x)−x2 2f00(x) +d2xf0(x), uniformly with respect tox∈[0,1], whereT(e2)(x) =

Z 1 0

e2x (see(1.1)) and d:= lim

n→∞(an+bn).

Proof. We shall apply Theorem 2.1 withq= 4. Observe that condition (i), (ii) and (iii) of the above result are satisfied with γ = 0, β(x) = d2x and α(x) =T(e2)(x)−x2 (0≤x≤1) because, for every 0≤x≤1,

Cn(1)(x) = 1, Cnx)(x) = n+11 an+b2 nx and

Cnx2)(x) = (n+1)1−n2 x2+(n+1)n 2 T(e2)(x)−(n+1)an+bn2x+ a2n+a3(n+1)nbn+b2 2n, (see [9, formulae (2.3), (2.4)]). In order to verify condition (iv), we shall explicitly determine the function Cn4x).

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Letx∈[0,1]. Sinceψ4x=e4−4xe3+ 6x2e2−4x3e1+x41, we have Cn4x) =Cn(e4)−4xCn(e3) + 6x2Cn(e2)−4x3Cn(e1) +x4Cn(1).

The expression ofCn(1), Cn(e1) andCn(e2) are the following Cn(1)(x) = bn+1

n−an

bn−an

n+1 = 1, Cn(e1)(x) = (n+1)n x+2(n+1)an+bn and

Cn(e2)(x) = n(n−1)(n+1)2 x2+ (n+1)n 2 T(e2)(x) +n(a(n+1)n+b2n)x+a2n+a3(n+1)nbn+b2 2n, (see (2), (3) and (4) of [9, Theorem 2.1]). Therefore we proceed to evaluate Cn on the functions e3 and e4, by using relation (1.4) between our operators Cn and the corresponding Bernstein-Schnabl operators Bn. A simply but laborious computations shows

Cn(e3)(x) =(n+1)n3 3 Bn(e3)(x)+32 n2(n+1)(an+b3n)Bn(e2)(x) +n(a2n+a(n+1)nbn3+b2n)x+(an+b4(n+1)n)(a2n3+b2n) and

Cn(e4)(x) =(n+1)n4 4 Bn(e4)(x)+2n(n+1)3(an+b4n)Bn(e3)(x)+2n2(a(n+1)2n+anb4n+b2n)

×Bn(e2)(x)+n(an+(n+1)bn)(a42n+b2n)x+b4n+b3nan+b5(n+1)2na2n+b5 na3n+a4n. Therefore

nCn4x)(x) =(n+1)n2 4 T(e3)(x)+4n(n+1)2(n1)4 x T(e3)(x)+3n(n+1)2(n1)4 T(e2)2(x)

+ 6n(n+1)2(5−n)4 x2T(e2)(x) + 3n3(n+1)−22n24+nx4+2n(n+1)2(an+b4n)T(e3)(x) + 6n2(n−1)(a(n+1)4n+bn)x T(e2)(x) +2n(5n−1)(a(n+1)4n+bn)x3+2n(n+1)2(n−1)4

×(a2n+anbn+b2n)x2+2n2(a(n+1)2n+anb4n+b2n)T(e2)(x)+n2(an(n+1)+bn)(a42n+b2n)x +n(b4n+b3nan+b2na2n+bna3n+a4n)

5(n+1)5(n+1)4n23x T(e3)(x)−(n+1)6n23x T(e2)(x)

4n2(a2n(n+1)+anb3n+b2n)x2n(an+b(n+1)n)(a32n+b2n)x+6n(n+1)2(n−1)2 x4 + (n+1)6n22 x2T(e2)(x) +2n(a2n+anbn+b2n)

(n+1)3 x2

and so condition (iv) follows.

3. MARKOV SEMIGROUPS ASSOCIATED WITH A CLASS OF ONE-DIMENSIONAL DIFFUSION EQUATIONS AND THEIR APPROXIMATION

The main aim of this section is to discuss some one-dimensional diffusion equations on the unit interval by means of the theory of C0-semigroups of operators and to represent the relevant solutions (or the corresponding semi- groups) by iterates of the operatorsCn. For more details about the theory of C0-semigroups we refer the reader to [16], [20], [23].

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Let (Cn)n≥1 be the sequence of the operators on C([0,1]) associated with the continuous selection (µx)0≤x≤1 of probability Borel measures on [0,1]

defined by (1.3). By Jensen’s inequality [15, Theorem 3.9] it follows that x2T(e2)(x) ≤ x (0 ≤ x ≤ 1), where the operator T is defined by (1.1).

Therefore

0≤T(e2)(x)−x2xx2 =x(1x) (0≤x≤1).

In particular, T(e2)(0) = 0 and so Suppµ0 = {0}, that is µ0 = ε0. Here Suppµ0 stands for the support of the measureµ0 (see, e.g., [7, Section 1.2]).

Moreover T(e2)(1) = 1, so Z 1

0

(e1e2)dµ1 = 0 and Suppµ1 ⊂ {0,1}.

Then there exist α, β ∈ [0,1], α+β = 1 such that µ1 = αε0 +βε1, so that 1 =

Z 1 0

e11=β. In conclusion we obtainα = 0 andµ1 =ε1. Finally we observe that, if 0≤x≤1, then

T(e2)(x) =x2 if and only if µx=εx.

Indeed, ifT(e2)(x) =x2, considering the functionψx defined by (2.1), we have Z 1

0

ψ2xx= 0, so Suppµx={x}and µx =εx. The converse is trivial.

From now on we suppose that the family (µx)0≤x≤1 satisfies the following further condition

(3.1) µx 6=εx for every 0< x <1.

Set

α(x) := 12T(e2)(x)−x2= 12 Z 1

0

e2xx2

(0≤x≤1).

Then α∈ C([0,1]),α(0) =α(1) = 0 and

0< α(x)x(1−x)2 for every 0< x <1.

Suppose in addition thatα is differentiable at 0 and 1 and

(3.2) α0(0)6= 06=α0(1).

Then

α(x) = x(1−x)2 λ(x) (0≤x≤1) where

(3.3) λ(x) =

0(0) ifx= 0,

2α(x)

x(1−x) if 0< x <1,

−2α0(1) ifx= 1, and λ∈ C([0,1]), 0< λ(x)≤1 for every x∈[0,1].

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Suppose that there existsd:= lim

n→∞(an+bn)>0 and consider the differential operatorA defined by setting for everyu∈ C2(]0,1[)

Au(x) :=α(x)u00(x) +d2xu0(x) (0< x <1).

Set (3.4)

DM(A) :=

u∈ C([0,1])∩ C2(]0,1[)| lim

x→0+A(u)(x)∈R, lim

x→1A(u)(x)∈R

and continue to denote withA:DM(A)−→ C([0,1]) the operator defined by setting for everyuDM(A) andx∈[0,1]

(3.5) A(u)(x) :=

lim

x→0+A(u)(x) if x= 0, α(x)u00(x) +d2xu0(x) if 0< x <1,

lim

x→1A(u)(x) if x= 1.

We recall that a core for a linear operator A : D(A) −→ E defined on a linear subspace D(A) of a Banach space E, is a subspace D0 of D(A) which is dense inD(A) for the graph normkukA:=kuk+kAuk (u∈D(A)).

A Feller semigroup onC([0,1]) is a strongly continuous semigroup of positive linear contractions onC([0,1]). A Markov semigroup on C([0,1]) is a strongly continuous positive semigroup (T(t))t≥0 on C([0,1]) satisfying T(t)1 = 1 for every t≥ 0. A strongly continuous positive semigroup on C([0,1]) with gen- erator (A, D(A)) is a Markov semigroup if and only if 1D(A) andA1= 0.

Theorem3.1. Under the assumptions(3.1) and(3.2), if moreoverα0(0)≤ d/2≤1 +α0(1) and the function r(x) := d/2−xλ(x) (0≤x≤1) is H¨older contin- uous at 0 and 1, then the operator (A, DM(A)) is the generator of a Markov semigroup (T(t))t≥0 on C([0,1]) andC2([0,1])is a core for (A, DM(A)).

Proof. We introduce the auxiliary operator

(1) Bu(x) = x(1−x)2 u00(x) +r(x)u0(x) defined on the domain D(B) :=DM(A). Thus,B =λA and

D(B) :=

u∈ C([0,1])∩ C2(]0,1[)| lim

x→0+B(u)(x)∈R, lim

x→1B(u)(x)∈R

. Then, (B, D(B)) is the generator of a Feller semigroup on C([0,1]) (see [13, pp. 120–121]). Therefore, since A = λB, the result follows by a well-known result about the generation of the multiplicative perturbation of generators (see [7, Theorem 1.6.11]). Since 1DM(A) and A1 = 0, the semigroup is a Markov semigroup. Finally in order to prove that C2([0,1]) is a core for (A, DM(A)), we use Theorem 2.3 of [13] applied to the operator B defined by (1), obtaining that C2([0,1])∩D(B) is a core for (B, D(B)). But, since

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C2([0,1]) ⊂ D(A) = D(B), C2([0,1]) is a core for (B, D(B)) and hence for

(A, D(A)).

At this point we are in a position to obtain a result about the approximation of the semigroup above.

Thep-th power (p≥1) of the operatorCn:C([0,1])−→ C([0,1]) is defined as

Cnp :=

(Cn if p= 1, CnCnp−1 if p≥2.

Clearly

(3.6) kCnpk ≤1 for everyn≥1 andp≥1, sincekCnk ≤1.

Note that, if u∈ C2([0,1]), then

Au(x) =α(x)u00(x) +d2xu0(x) for every x∈[0,1]

and hence, by Theorem 2.2,

(3.7) lim

n→∞n(Cn(u)−u) =Au uniforlmy on [0,1].

We finally need the following general result which can be obtained by Trotter’s theorem on the approximation of semigroups (see, e.g., [20, Corollary 5.8] or [7, Theorem 1.6.7]).

Theorem 3.2. Let (T(t))t≥0 be a strongly continuous semigroup on a Ba- nach space E with generator (A, D(A)). Consider a sequence (Ln)n≥1 of bounded linear operators on E and assume that

(i) There existsM ≥1 and ω∈R such that

kT(t)k ≤Mexp (ωt) and kLpnk ≤Mexp ωpn for everyt≥0, n≥1 p≥1.

(ii) There exists a coreD0 for (A, D(A)) such that

n→∞lim n(Ln(u)−u) =Au for every uD0.

Then for every t ≥ 0, for every sequence (k(n))n≥1 of positive integers such that k(n)/nt(n→ ∞) and for every fE

T(t)f = lim

n→∞Lk(n)n f.

From (3.6) and (3.7) and from Theorems 3.1 and 3.2 the next result imme- diately follows.

Theorem 3.3. Under the same assumptions of Theorem 3.1, considering the Markov semigroup (T(t))t≥0 generated by(A, DM(A)), for everyt≥0, for every sequence (k(n))n≥1 of positive integers such that k(n)/nt (n→ ∞) and for every f ∈ C([0,1]),

(3.8) T(t)f = lim

n→∞Cnk(n)(f) uniformly on[0,1].

(11)

By using the representation formula (3.8), it is possible to obtain some properties of the semigroup from the preservation properties of the operators Cn (see Theorems 1.1 and 1.2).

Proposition 3.4. Under the same assumptions of Theorem 3.3, consid- ering the Markov semigroup (T(t))t≥0 generated by the operator (A, DM(A)) defined by (3.5), the following statements hold true:

(1) If the operatorT, defined by (1.1), maps Lip11 into Lip11, then T(t)(LipMα)⊂LipMα

for everyM ≥1 and0< α≤1.

(2) If the operator T, given by (1.1), satisfies the hypotheses of Theorem 1.2, then for every t≥0, T(t) maps continuous convex functions into (continuous) convex functions.

It is possible to obtain a further property which holds for the semigroup (T(t))t≥0. We need the following lemma.

Lemma 3.5. Under the same assumptions of Theorem 3.1 consider the Markov semigroup (T(t))t≥0 generated by (A, DM(A)). Then for every t≥0

T(t)e1 = e−te1+d2(1−e−t).

Therefore, for every t >0

e1T(t)e1 if and only if d= 2.

Proof. We use the representation formula (3.8) in order to show the first part of the claim. For every n≥1 we have indeed

Cn(e1) = n+1n e1+2(n+1)an+bn, Cn2(e1) = n+1n 2e1+

1−n+1n 2

an+bn

2

and, reasoning by induction, for every p≥3

Cnp(e1) =n+1n pe1+1−n+1n pan+b2 n.

Given t ≥ 0 and considering a sequence (k(n))n≥1 of positive integers such that k(n)/nt, we get

T(t)e1 = lim

n→∞Cnk(n)e1= e−te1+ d2(1−e−t),

sincen+1n k(n)→et.The second part of the statement follows from the first one, since d= lim

n→∞(an+bn)≤2.

Proposition3.6. Under the same hypotheses of Theorem 3.1the following propositions are equivalent:

(a) For every increasing convex functionf ∈ C([0,1]) and for everyt≥0, fT(t)f.

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(b) d= 2.

Proof. (a) ⇒ (b) For f = e1, we get e1T(t)e1 for every t ≥ 0 and so d= 2 by the previous lemma.

(b) ⇒ (a) Fix t > 0. Since d= 2, e1T(t)e1. Let f ∈ C([0,1]) convex and increasing. Let x ∈]0,1[ and let ϕ be an increasing affine function on [0,1]

such that f(x) =ϕ(x) and ϕf. Let a, b∈R,a≥0 such that ϕ=ae1+b.

Then T(t)ϕ=aT(t)e1+band so ϕT(t)ϕ. Accordingly f(x) =ϕ(x)T(t)ϕ(x)T(t)f(x).

By continuity the inequality fT(t)f can be extended on the whole interval

[0,1] and so the result follows.

Remark 3.7. From the general theory of C0-semigroups of operators and from Theorem 3.1 it follows that for every u0DM(A) the following initial- boundary differential problem of diffusion type

(3.9)

∂u

∂t(x, t) =α(x)∂x2u2(x, t) +d2x∂u∂x(x, t) 0< x <1, t≥0,

u(x,0) =u0(x) 0≤x≤1,

lim

x→0+ x→1

α(x)∂x2u2(x, t) +d2x∂u∂x(x, t)∈R t≥0, has a unique solution given by

u(x, t) =T(t)u0(x) (0≤x≤1, t≥0).

Moreover |u(x, t)| ≤ ku0k (0≤ x ≤1, t ≥0) and u(·, t) is positive for every t≥0 provided thatu0≥0.

Furthermore, by Theorem 3.3, the solutions can be approximate by means of iterates of the operators Cn. Finally, Propositions 3.4 and 3.6 give some

qualitative properties of them as well.

We end the paper by considering a kind of converse problem. Let d ∈R, 0 < d≤2 and α ∈ C([0,1]), α(0) = α(1) = 0, α(x)>0 for every 0< x < 1.

Moreover suppose that α is differentiable at 0 and 1, α0(0) 6= 0 6= α0(1) and the function

r(x) :=

d

0(0) if x= 0,

x(1−x)(d/2−x)

2α(x) if 0< x <1,

1−d/2

0(1) if x= 1, is H¨older continuous at 0 and 1.

Consider the differential operator (A, DM(A)), defined in (3.4) and (3.5), that is,

Au(x) :=α(x)u00(x) +d2xu0(x) (0< x <1)

(13)

for everyuDM(A) :=

(

u∈ C([0,1])∩ C2(]0,1[)| lim

x→0+ x→1

A(u)(x)∈R )

. Ifα0(0)≤d/2≤1 +α0(1), then (A, DM(A)) generates a Markov semigroup (T(t))t≥0 on the space C([0,1]), as the same proof of Theorem 3.1 shows.

The problem is then to find a continuous selection (νx)0≤x≤1 of probability Borel measures on [0,1], satisfying (1.2), and two sequences (an)n≥1, (bn)n≥1 in [0,1] such that the corresponding operators Cn represent the semigroup by means of their iterates, as in Theorem 3.3. To this respect we have the following result.

Theorem3.8. Under the previous hypotheses, further suppose thatα(x)

x(1−x)

2 (0≤x≤1) and for every x∈[0,1] set

(3.10) νx :=λ(x)µx+ (1−λ(x))εx,

where µx :=1+ (1−x)ε0 and the functionλ∈ C([0,1]) is defined by (3.3).

For every n≥1 set

(3.11) bn:=d/2 and an:=

0 1/n≥d/2, d/2−1/n 1/n < d/2.

Consider the sequence (Cn)n≥1 associated with the selectionx)0≤x≤1 and the sequences (an)n≥1 and (bn)n≥1. Then, for every t≥0, for each sequence (k(n))n≥1 of positive integers such that k(n)/nt (n → ∞) and for every f ∈ C([0,1]),

T(t)f = lim

n→∞Cnk(n)(f) uniformly on[0,1].

Proof. We preliminarily observe that 0 ≤ an < bn ≤ 1 and an+bnd.

Moreover the mapping x 7→ νx is continuous, the conditions (1.2) and (3.1) are verified and finally

1 2

Z 1

0

e2xx2

= λ(x)x(1−x)2 =α(x) (0≤x≤1).

From Theorem 3.3, the result follows.

We point out that, if d ≤ 1, then one can consider the sequences an = 0 and bn=d(n≥1) instead of the ones given by (3.11).

We also remark that, under the assumptions of Theorem 3.8, the operator T corresponding to the selection (3.10) via formula (1.1) is given by

T(f) = (1−λ)f +f(1)λe1+f(0)λ(1−e1) (f ∈ C([0,1])).

Therefore, if such an operatorT maps Lip11 into Lip11 and/or if it satisfies conditions (c1) and (c2), then the semigroup generated by (A, DM(A)) maps LipMα into itself (M > 0, 0 < α ≤ 1) and/or continuous convex functions into convex functions.

Moreover, ifd= 2, property (a) of Proposition 3.6 holds true as well.

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For instance, ifλis constant, i.e.,α(x) =λx(1−x)2 (0≤x≤1), thenT maps Lip11 into Lip11 and it satisfies (c1) and (c2).

We shall finally restrict our attention to the particular case d= 1 and α(x) = x(1−x)2 (0≤x≤1).

In this case Theorem 3.8 and Proposition 3.4 can be fully applied and the operators Cn,n≥1, whose iterates converge to the semigroup (T(t))t≥0 gen- erated by (A, DM(A)), are the Kantorovich operators defined by (1.5) with an= 0 andbn= 1, n≥1.

In this particular case we wish to point out another property of the semi- group (and hence of the solutions of problems (3.9)) which is concerned with functions of bounded variation on [0,1]. If f : [0,1]−→R is such a function, we denote by V[0,1](f) its total variation, i.e.,

V[0,1](f) := sup ( n

X

i=1

|f(xi)−f(xi−1)|(xi)0≤i≤npartition of [0,1]

) . We recall that, if (fn)n≥1 is a sequence of functions of bounded variation on [0,1] pointwise convergent to a function f: [0,1]−→Rand if, in addition, sup

n≥1

V[0,1](f)<+∞, then f is a function of bounded variation on [0,1] and V[0,1](f)≤sup

n≥1

V[0,1](fn).

In [14, Proposition 3.3], it is shown that for every function f : [0,1]−→R of bounded variation on [0,1] and for any n≥1

V[0,1](Cn(f))≤V[0,1](f).

An analogue inequality holds true for the iterates of the operatorsCn. Therefore from (3.8) and from the previous remark it follows that, for every continuous function of bounded variationf : [0,1]−→Rand, for everyt≥0,

T(t)f is a (continuous) function of bounded variation and

V[0,1](T(t)(f))≤V[0,1](f).

We leave as an open problem the question whether the above inequality is still true for other classes of semigroups as stated in Theorem 3.1.

Acknowledgement. The authors acknowledge the support of the Univer- sity of Bari. The second author also acknowledges the support of the “Istituto per Ricerche ed Attivit`a Educative” of Napoli.

REFERENCES

[1] Altomare, F.,Limit semigroups of Bernstein-Schnabl operators associated with posi- tive projections, Ann. Sc. Norm. Pisa, Cl. Sci.,16(4), no. 2, pp. 259–279, 1989.

[2] Altomare, F.,Lototsky-Schnabl operators on the unit interval, C. R. Acad. Sci. Paris, 313, S´erie I, pp. 371–375, 1991.

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