In this paper, we deepen the study of a sequence $(C_n)_{n \geq 1}$ of positive linear operators, first introduced in \cite{altomarecappellettileonessa1}, that generalize the classical Sz\'{a}sz-Mirakjan-Kantorovich operators. In particular, we present some qualitative properties and an asymptotic formula for such a sequence. Moreover, we prove that, under suitable assumptions, the Feller semigroups generated by the second order differential operator $V_c(u)(x)=xu''(x)+c u'(x)$ ($x \geq 0, c \in [0, 1])$ on suitable domains of continuous or integrable functions may be approximated by means of iterates of the $C_n$'s.

In this paper we study a class of elliptic second-order differential operators on finite dimensional convex compact sets whose principal part degenerates on a subset of the boundary of the domain. We show that the closures of these operators generate Feller semigroups. Moreover, we approximate these semigroups by iterates of suitable positive linear operators which we also introduce and study in this paper for the first time, and which we refer to as modified Bernstein-Schnabl operators. As a consequence of this approximation we investigate some regularity properties preserved by the semigroup. Finally, we consider the special case of the finite dimensional simplex and the well-known Wright-Fisher diffusion model of gene frequency used in population genetics.

In this paper, by using the Lyapunov method, we establish sufficient conditions for the global asymptotic stability of the positive periodic solution to diffusive Holling–Tanner predator–prey models with periodic coefficients and no-flux conditions.

Deepening the study of a new approximation sequence of positive linear operators we introduced and studied in [10], in this paper we disclose its relationship with the Markov semigroup (pre)generation problem for a class of degenerate second-order elliptic dierential operators which naturally arise through an asymptotic formula, as well as with the approximation of the relevant Markov semigroups in terms of the approximating operators themselves. The analysis is carried out in the context of the space C (K) of all continuous functions dened on an arbitrary compact convex subset K of Rd, d 1, having non-empty interior and a not necessarily smooth boundary, as well as, in some particular cases, in Lp(K) spaces, 1 <p < +\infty. The approximation formula also allows to infer some preservation properties of the semigroup such as the preservation of the Lipschitz-continuity as well as of the convexity. We nally apply the main results to some noteworthy particular settings such as balls and ellipsoids, the unit interval and multidimensonal hypercubes and simplices. In these settings the relevant dierential operators fall into the class of Fleming-Viot operators.

In this paper we deepen the study of a sequence of positive linear operators acting on $L^1([0,1]^N)$, $N \geq 1$, that have been introduced in \cite{AltomareCappellettiLeonessa} and that generalize the multidimensional Kantorovich operators (see \cite{zhou}). We show that particular iterates of these operators converge on $\mathscr{C}([0,1]^N)$ to a Markov semigroup and on $L^p([0,1]^N)$, $1 \leq p &lt;+\infty$, to a positive $C_0$-semigroup (that is an extension of the previous one). The generators of these $C_0$-semigroups are determined in a core of their domains, where they coincide with an elliptic second-order differential operator whose principal part degenerates on the vertices of the hypercube $[0,1]^N$.

During the last twenty years important progresses have been made, from the point of view of constructive approximation theory, in the study of initial-boundary value differential problems of parabolic type governed by positive 0-semigroups of operators. The main aim of this approach is to construct suitable positive approximation processes whose iterates strongly converge to the semigroups which, as it is well-known, in principle furnish the solutions to the relevant initial-boundary value differential problems. By means of such kind of approximation it is then possible to investigate, among other things, preservation properties and the asymptotic behavior of the semigroups, i.e., spatial regularity properties and asymptotic behavior of the solutions to the differential problems. This series of researches, for a survey on which we refer to [25] and the references therein, has its roots in several studies developed between 1989 and 1994 which are documented in Chapter 6 of the monograph [18]. These studies were concerned with special classes of second-order elliptic differential operators acting on spaces of smooth functions on finite dimensional compact convex subsets, which are generated by a positive projection. The projections themselves offer the tools to construct an approximation process whose iterates converge to the relevant semigroup, making possible the development of a qualitative analysis as above. This theory disclosed several interesting applications by stressing the relationship among positive semigroups, initial-boundary value problems, Markov processes and approximation theory, and by offering, among other things, a unifying approach to the study of diverse differential problems. Nevertheless, over the subsequent years, it has naturally arisen the need to extend the theory by developing a parallel one for positive operators rather than for positive projections and by trying to include in the same project of investigations more general differential operators having a first-order term. The aim of this research monograph is to accomplish such an attempt by considering complete second-order (degenerate) elliptic differential operators whose leading coefficients are generated by a positive operator, by means of which it is possible to construct suitable approximation processes which approximate the relevant semigroups. Some aspects of the theory are treated also in infinite dimensional settings. viii Preface The above described more general framework guarantees to notably enlarge the class of differential operators by including, in particular, those which can be obtained by usual operations with positive operators such as convex combinations, compositions, tensor products and so on. Moreover, this non-trivial generalization discloses new challenging problems as well. However, the approximation processes which we construct in terms of the given positive operator seem to have an interest in their own also for the approximation of continuous functions. For these reasons, a special emphasis is placed upon various aspects of this theme as well.

In this paper we introduce and study two new sequences of positive linear operators acting on the space of all Lebesgue integrable functions defined, respectively, on the $N$-dimensional hypercube and on the $N$-dimensional simplex ($N \geq 1$). These operators represent a natural generalization to the multidimensional setting of the ones introduced in \cite{AltomareLeonessaCn1} and, in a particular case, they turn into the multidimensional Kantorovich operators on these frameworks. We study the approximation properties of such operators with respect both to the sup-norm and to the $L^p$-norm and we give some estimates of their rate of convergence by means of certain moduli of smoothness.

In this paper we introduce and study a sequence of positive linear operators acting on suitable spaces of measurable functions on [0, +\infty[, including L^p([0, +\infty[) spaces, 1 \leq p <+\infty, and continuous function spaces with polynomial weights. These operators generalize the Sz\'{a}sz-Mirakjan-Kantorovich operators and they allow to approximate (or to reconstruct) suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,+\infty[ that do not constitute a subdivision of it. We also give some estimates of the rates of convergence by means of suitable moduli of smoothness.

In this paper we introduce and study a new class of elliptic second-order differential operators on a convex compact subset K of R^d, which are associated with a Markov operator T on C(K) and which degenerate on a suitable subset of K containing its extreme points. Among other things, we show that the closures of these operators generate Markov semigroups. Moreover, we prove that these semigroups can be approximated by means of iterates of suitable positive linear operators, which are referred to as the Bernstein-Schnabl operators associted with T. As a consequence we show that the semigroups preserve polynomials of a given degree as well as Holder continuity which gives rise some spatial regularity properties of the solutions of the relevant evolution equations.

The paper is concerned with a special class of positive linear operators acting on the space C(K) of all continuous functions defined on a convex compact subset K of R^d, having non-empty interior. Actually, this class consists of all positive linear operators T on C(K) which leave invariant the polynomials of degree at most $1$ and which, in addition, map polynomials into polynomials of the same degree. Among other things, we discuss the existence of such operators in the special case where K is strictly convex by also characterizing them within the class of positive projections. In particular we show that such operators exist if and only if the boundary of K is an ellipsoid. Furthermore, a characterization of balls of R^d in terms of a special class of them is furnished. Additional results and illustrative examples are presented as well.

In this paper we study a class of degenerate second-order elliptic differential operators, often referred to as Fleming-Viot type operators, in the framework of function spaces defined on the d-dimensional hypercube Qd of Rd, d ≥ 1. By making mainly use of techniques arising from approximation theory, we show that their closures generate positive semigroups both in the space of all continuous functions and in weighted Lp-spaces. In addition, we show that the semigroups are approximated by iterates of certain polynomial type positive linear operators, which we introduce and study in this paper and which generalize the Bernstein-Durrmeyer operators with Jacobi weights on [0, 1]. As a consequence, after determining the unique invariant measure for the approximating operators and for the semigroups, we establish some of their regularity properties along with their asymptotic behaviours.