Volume 1, Issue 2

We establish in the offered paper that a ‘displacement postulate’ for the light sphere allows to obtain a metric in four–dimensional space–time from which the well–known Schwarzschild's metric may be simply deduced (by passing to the locational time). Probably, displacement postulate is a description of a profound mechanism of an interaction between the particles and the gravitational field.

Let ( X, d ) be a Hausdorff semimetric ( d need not satisfy the triangle inequality) and d –Cauchy complete space. Let ƒ be a selfmap on X , for which d (ƒ x , ƒ y ) ≤ φ ( d ( x, y )), ( x, y ∈ X ), where φ is a non– decreasing function from R + , the nonnegative reals, into R + such that φ n ( t ) → 0, for all t ∈ R + . We prove that ƒ has a unique fixed point if there exists an r > 0, for which the diameters of all balls in X with radius r are equi-bounded. Such a class of semimetric spaces includes the Frechet spaces with a regular ecart, for which the Contraction Principle was established earlier by M. Cicchese [Boll. Un. Mat. Ital 13–A: 175-179, 1976], however, with some further restrictions on a space and a map involved. We also demonstrate that for maps ƒ satisfying the condition d (ƒ x , ƒ y ) ≤ φ (max{ d ( x , ƒ x ), d ( y , ƒ y )}), ( x, y ∈ X ) (the Bianchini [Boll. Un. Mat. Ital. 5: 103–108, 1972] type condition), a fixed point theorem holds under substantially weaker assumptions on a distance function d .

In this paper we investigate relationships between the families of weakly Darboux, quasi-Darboux and Darboux functions which are quasi-continuous, and analyse the problem connected with the Morrey monotonicity of the restriction of a weakly Darboux function with the property ( P ) to the closure of the union of levels which are continua.

We consider some properties of sets and functions in connection with their translations. We give an application of those properties to the theory of invariant measures (in particular, to the theory of invariant extensions of the classical Lebesgue measure).

The Dirichlet problem to nonlinear diagonal parabolic system with some special right–hand sides but still satisfying the maximum growth conditions is considered. First applying the idea of Stampacchia a L ∞ – estimate in terms of data for a priori bounded weak solutions is found. Next following the methods of Ladyzhenskaya and Uraltseva the Hölder continuity with some exponent is proved. Next applying the results of Amann global existence of solutions is shown.

The main result of this paper is the theorem stating that every convex set–valued function F : I ↦ c ( Y ), where I ⊂ R is an interval and Y is a locally convex space, possesses an affine selection. In the case if Y = R and values of F are closed real intervals we can replace the assumption of convexity of F by the more general condition.

Let Perf( X ) denote the family of all perfect subsets of a perfect Polish space X . We show several properties and characterizations of coinitial subfamilies of Perf( X ). Connections with the Vietoris topology and Morgan's category bases are observed. We characterize the least cardinality of a family of coinitial ideals with noncoinitial intersection. Some examples of special noncoinitial families in Perf( X ) are presented. Much place is devoted to perfect isomorphisms between ideals of subsets of X .

Let C be a non–empty subset of a linear topological space X , and T be a selfmap of C such that the range of I – T is convex, where I denotes the identity map on X . We give conditions under which a map T has a fixed point or a V –fixed point (i.e. a point x 0 ∈ C such that Tx 0 ∈ X 0 + V , where V is a neighborhood of the origin). Our theorems generalize the recent results of M. Edelstein and K.-K. Tan ([Math. Japon. 38: 325–332, 1993], [J. Math. Anal. Appl. 181: 181–187, 1994]). As an application we provide a simple proof of the Markov–Kakutani theorem. We also establish a common V –fixed point theorem for commuting affine maps (possibly discontinuous).

In this article we study the convergence of convolution powers of normalized measures ( θ C n ν )* n on a Sturm–Liouville hypergroup (ℝ + ,∗). It is shown that this sequence converges for a suitable choice of the normalizing constants c n > 0 if and only if the usual regular variation conditions of the tail of ν are valid. The possible limit distributions are described in terms of their Fourier transform; they form a two dimensional family of probability measures on ℝ + .