Diffusive Holling–Tanner predator–prey models in periodic environments

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

A Modification of Bernstein-Durrmeyer Operators with Jacobi Weights on the Unit Interval

The present paper is devoted to the study of a sequence of positive linear operators, acting on the space of all continuous functions on [0, 1] as well as on some weighted spaces of integrable functions on [0, 1]. These operators are, as a matter of fact, a generalization of the Bernstein-Durrmeyer operators with Jacobi weights. In particular, we present qualitative and approximation properties of these operators, also providing estimates of the rate of convergence. Moreover, by means of their asymptotic formula, we compare our operators with the Bernstein-Durrmeyer ones and a suitable modification of theirs, showing that, in suitable intervals, they provide a lower approximating error estimate

A generalization of Kantorovich operators for convex compact subsets

In this paper we introduce and study a new sequence of positive linear operators acting on function spaces defined on a convex compact subset. Their construction depends on a given Markov operator, a positive real number and a sequence of probability Borel measures. By considering special cases of these parameters for particular convex compact subsets we obtain the classical Kantorovich operators defined in the one-dimensional and multidimensional setting together with several of their wide-ranging generalizations scattered in the literature. We investigate the approximation properties of these operators by also providing several estimates of the rate of convergence. Finally, the preservation of Lipschitz-continuity as well as of convexity are discussedComment: Research articl

Kantorovich-type modifications of certain discrete-type operators on the positive real axis

The paper is concerned with the approximation properties of a modification of Kantorovich-type of a general class of operators of discrete-type. Such a modification was introduced by Agratini in 2015; in particular, we focus on extending its approximation properties in several function spaces, including polynomial weighted spaces of any degree as well as $L^p$-spaces. Some estimates of the rate of convergence are also obtained

On some density theorems in regular vector lattices of continuous functions

In this paper, we establish some density theorems in the setting of particular locally convex vector lattices of ontinuous functions defined on a locally compact Hausdorff space, which we introduced and studied in $[3, 4]$ and which we named regular vector lattices. In this framework, by using properties of the subspace of the so-called generalized affine functions, we give a simple description of the closed vector sublattice, the closed Stone vector sublattice and the closed subalgebra generated by a subset of a regular vector lattice. As a consequence, we obtain some density results. Finally, a connection with the Korovkin type approximation theory is also shown

Francesco Altomare - the remarkable mathematician and human being

We laconically describe the great contributions of Professor Francesco Altomare to mathematical research and PhD education, and his unique status in the mathematical community. In particular, we present and give examples of his innovative and great achievements related to the following areas of mathematics: Functional Analysis, Operator Theory, Potential Theory, Approximation Theory, Probability Theory, Function Spaces, Choquet's Theory, Dirichlet's Problem and Semigroup Theory. Moreover, we report on and give concrete examples of his unique way to work together with PhD students, both before and sometimes also after their dissertation. Finally, we shortly describe his remarkable “class travel” from “simple” conditions with no academic traditions in his family in the small hometown Giovinazzo to finally become the broad, ingenious, and powerful mathematician he is regarded to be today

On Sequences of J. P. King-Type Operators

This survey is devoted to a series of investigations developed in the last fifteen years, starting from the introduction of a sequence of positive linear operators which modify the classical Bernstein operators in order to reproduce constant functions and x2 on [0,1]. Nowadays, these operators are known as King operators, in honor of J. P. King who defined them, and they have been a source of inspiration for many scholars. In this paper we try to take stock of the situation and highlight the state of the art, hoping that this will be a useful tool for all people who intend to extend King's approach to some new contents within Approximation Theory. In particular, we recall the main results concerning certain King-type modifications of two well known sequences of positive linear operators, the Bernstein operators and the Szász-Mirakyan operators

Operator Methods in Approximation Theory

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