Antonín Slavík

Explicit solutions to dynamic diffusion-type equations and their time integrals

This paper deals with solutions of diffusion-type partial dynamic equations on discrete-space domains. We provide two methods for finding explicit solutions, examine their asymptotic behavior and time integrability. These properties depend significantly not only on the underlying time structure but also on the dimension and symmetry of the problem. Throughout the paper, the results are interpreted in the context of random walks and related stochastic processes.

Discrete-space partial dynamic equations on time scales and applications to stochastic processes

Abstract We consider a general class of discrete-space linear partial dynamic equations. The basic properties of solutions are provided (existence and uniqueness, sign preservation, maximum principle). Above all, we derive the following main results: first, we prove that the solutions depend continuously on the choice of the time scale. Second, we show that, under certain conditions, the solutions describe probability distributions of nonhomogeneous Markov processes, and that their time integrals remain the same for all underlying regular time scales.

On summability, multipliability, product integrability, and parallel translation

Abstract In this paper we provide necessary and sufficient conditions for the existence of the Kurzweil, McShane and Riemann product integrals of step mappings with well-ordered steps, and for right regulated mappings with values in Banach algebras. Our basic tools are the concepts of summability and multipliability of families in normed algebras indexed by well-ordered subsets of the real line. These concepts also lead to the generalization of some results from the usual theory of infinite series and products. Finally, we present two applications of product integrals: First, we describe the relation between Stieltjes-type product integrals, Haahti products, and parallel translation operators. Second, we provide a link between the theory of strong Kurzweil product integrals and strong solutions of linear generalized differential equations.

Well-posedness and maximum principles for lattice reaction-diffusion equations

Abstract Existence, uniqueness and continuous dependence results together with maximum principles represent key tools in the analysis of lattice reaction-diffusion equations. In this paper, we study these questions in full generality by considering nonautonomous reaction functions, possibly nonsymmetric diffusion and continuous, discrete or mixed time. First, we prove the local existence and global uniqueness of bounded solutions, as well as the continuous dependence of solutions on the underlying time structure and on initial conditions. Next, we obtain the weak maximum principle which enables us to get the global existence of solutions. Finally, we provide the strong maximum principle which exhibits an interesting dependence on the time structure. Our results are illustrated by the autonomous Fisher and Nagumo lattice equations and a nonautonomous logistic population model with a variable carrying capacity.

Mixing Problems with Many Tanks

Abstract We revisit the classical calculus problem of describing the flow of brine in a system of tanks connected by pipes. For various configurations involving an arbitrary number of tanks, we show that the corresponding linear system of differential equations can be solved analytically. Finally, we analyze the asymptotic behavior of solutions for a general closed system of tanks. It turns out that the problem is closely related to the study of Laplacian matrices for directed graphs.

ON A CHARACTERISTIC PROPERTY OF FINITE-DIMENSIONAL BANACH SPACES

This paper is inspired by a counter example of J. Kurzweil published in [5], whose intention was to demonstrate that a certain property of linear operators on finite-dimensional spaces need not be preserved in infinite dimension. We obtain a stronger result, which says that no infinite-dimensional Banach space can have the given property. Along the way, we will also derive an interesting proposition related to Dvoretzky’s theorem. 2010 Mathematics Subject Classification. 47A30, 47A63, 46B07, 15A45.

Discrete Bessel functions and partial difference equations

Abstract We introduce a new class of discrete Bessel functions and discrete modified Bessel functions of integer order. After obtaining some of their basic properties, we show that these functions lead to fundamental solutions of the discrete wave equation and discrete diffusion equation.

On Summability, Multipliability, and Integrability

We define and study summability and multipliability of families indexed by well-ordered sets of real numbers. These concepts generalize the classical notions of convergence of infinite series and products. The members of the families are assumed to be elements of general Banach spaces or Banach algebras, but most of our results are new even in the real-valued case. Our studies are also motivated by problems in integration theory of functions of one variable. In particular, we describe the relation between integrability and product integrability on one side, and summability and multipliability on the other side. Applications in the theory of differential equations with impulses and distributional differential equations are presented, and concrete examples are introduced to illustrate the derived theoretical results.

Intrinsically Defined Curves and Special Functions

Summary We study two classes of plane curves with prescribed curvature. First, we investigate spirals whose curvature is a power function, and express coordinates of the spirals’ centers in terms of the gamma function. For curves in the second family, the curvature is a multiple of the sine function. We show that this family contains infinitely many closed curves and provide their characterization in terms of the zeroth-order Bessel function.