J. Myjak

On the structure of the set of solutions of the Darboux problem for hyperbolic equations

Consider the Darboux problem where φ,ψ: I → R d ( I =[0,1]) are given absolutely continuous functions with φ(0)=ψ(0), and the mapping f : Q × R d → R d ( Q = I × I ) satisfies the following hypotheses: (A 1 ) f (.,., z ) is measurable for every z ∈ R d ; (A 2 ) f ( x, y ,.) is continuous for a.a. (almost all) ( x, y ) ∈ Q ; (A 3 ) there exists an integrable function α: Q →[0, + ∞) such that | f(x, y, z )|≦α( x, y ) for every ( x, y, z )∈ Q × R d .

On mutually nearest and mutually furthest points of sets in Banach spaces

Abstract Let A be a nonempty closed bounded subset of a uniformly convex Banach space E . Let b ( E ) denote the space of all nonempty closed convex and bounded subsets of E , endowed with the Hausdorff metric. We prove that the set of all X ϵ E ( E ) such that the maximization problem max( A , X ) is well posed is a G δ dense subset of b ( E ). A similar result is proved for the minimization problem min( A , X ), with X in an appropriate subspace of b ( E ).

Markov operators with a unique invariant measure

Let M be the set of all finite Borel measures on a Polish space X. Let P be a Markov operator on M and π the transition function corresponding to P. Set Γ(x)=suppπ(x,·), x∈X. It is proved that, if P admits a unique invariant measure μ∗, then μ∗(D)=0 or μ∗(⋂n=0∞Γn(D))=1 for every Borel set D such that Γ(D)⊂D. Moreover, if P is nonexpansive, then a trajectory of every Markov chain corresponding to P and starting from suppμ∗ is dense in suppμ∗. The last statement fails if we drop nonexpansivity condition.

FRACTALS, SEMIFRACTALS AND MARKOV OPERATORS

The paper contains a review of results concerning the theory of iterated function systems (IFS) acting on an arbitrary metric space (without any assumption of compactness). First we discuss IFS acting on sets and we define fractals and semifractals using topological limits. Then we study IFS with probabilities acting on measures and we show a relationship with the theory of Markov operators and Markov processes.

Some generic properties of functional differential equations in Banach spaces

x’ =f(t, x), x&J = x0 , (0.1) with f continuous, is well posed in the sense that it has the property of existence uniqueness and continuous dependence of solutions upon the data. Moreover, when a solution exists, it is very useful to approach it by means of some constructive method, for instance using the classical Peano-Picard method of the successive approximations. It is a well known fact that the continuity of the vector fieldf is not sufficient to ensure uniqueness or convergence of successive approximations and, in infinite dimension, even existence of solutions [16, 61. These pathologies are however ruled out if f is supposed to satisfy some additional hypotheses, for instance if f is locally Lipschitzian. It is then quite natural to raise the problem whether the set of all “nice” continuous f for which problem (0.1) is well posed and/or the corresponding successive approximations converge to a solution of (0.1) is larger, in a sense to be specified, than its complement. To make the question meaningful we say that a property is generic in a Baire space if the subset on which it is not true is meager or, equivalently, if the property is true on a residual subset. Here by meager set we mean a set of Baire first category. Thus a generic property is one which is possessed by almost all

Ambiguous loci of the metric projection onto compact starshaped sets in a Banach space

LetE be a real Banach space andL(E) the family of all nonempty compact starshaped subsets ofE. Under the Hausdorff distance,L(E) is a complete metric space. The elements of the complement of a first Baire category subset ofL(E) are called typical elements ofL(E). ForX∈L(E) we denote byπχ the metrical projection ontoX, i.e. the mapping which associates to eacha∈E the set of all points inX closest toa. In this note we prove that, ifE is strictly convex and separable with dimE≥2, then for a typicalX∈L(E) the mapπχ is not single valued at a dense set of points. Moreover, we show that a typical element ofL(E) has kernel consisting of one point and set of directions dense in the unit sphere ofE.

Generic properties of functional equations

A PROPERTY is said to be generic on a Baire space if the subset on which it is true is residual, that is its complement is a set of Baire first category. The study of generic properties of differential equations was started by W. Orlicz [l] who showed that the subset allf for which the Cauchy problem for the equation y’ = f(t, y) has not uniqueness is of first category in the space of all continuous and boundedf with values in R”, equipped with a natural metric. An analogous result for hyperbolic equations was proved by A. Alexiewicz and W. Orlicz [2]. More recently A. Lasota and J. Yorke [3] studied properties concerning existence, uniqueness and continuous dependence of solutions of the Cauchy problem for ordinary differential equations in a Banach space. Similar problems for functional differential equations y’ = f(t, y,) in a Banach space have been discussed in [4]. In [5] ([6]) it has been shown that the convergence of successive approximations for ordinary differential equations in a Banach space (for a class of hyperbolic partial differential equations) is a generic property in the corresponding spaces of continuous functions. (For the R” case see [7].) Generic properties of fured points for nonlinear nonexpansive mappings in a Banach space have been investigated in [8,9]. Other problems in the same spirit of the aforementioned ones have been treated in [l&12]. In this note we shall present some contributions to the theory of generic properties for functional equations and functional differential equations. Section 1 contains results concerning generic properties of existence uniqueness, continuous dependence of solutions and convergence of successive approximations for abstract functional equations. In Sections 2 and 3 we establish, by using similar arguments, generic properties of existence, uniqueness, continuous dependence of solutions and convergence of successive approximations for a class of functional integral equations. In Section 4 the results of Section 1 are used to obtain generic properties of nonexpansive mappings in a Banach space. The question how those maps, for which the generic properties under consideration fail to be true, are scattered is discussed in Section 5.

Capacity of invariant measures related to Poisson-driven stochastic differential equations

Sufficient conditions for the asymptotic stability of Poisson-driven stochastic differential equations (SDEs) on a separable Banach space are presented. It is also proved that the lower capacity of a unique invariant measure with respect to the semigroup generated by this equation is ≥1.

On the Hausdorff dimension of Cantor-like sets with overlaps

Abstract We give a lower estimate of the Hausdorff dimension for Cantor-like sets which are obtained by an overlapping Moran-like construction.

Stability versus chaos for a partial differential equation

Abstract We investigate a semiflow generated by a first-order partial differential equation. We show that stability and chaos for such a semiflow depend on the proper choice of the phase space. Chaos results from the existence of invariant measures having strong ergodic and analytic properties.