Volume 6, Issue 2

A function is called sup-measurable if F ƒ : ℝ → ℝ given by F ƒ ( x ) = F ( x , ƒ( x )), x ∈ ℝ, is measurable for each measurable function ƒ : ℝ → ℝ. It is known that under different set theoretical assumptions, including CH, there are sup-measurable non-measurable functions, as well as their category analogues. In this paper we will show that the existence of the category analogues of sup-measurable non-measurable functions is independent of ZFC. A similar result for the original measurable case is the subject of a work in prepartion by Ros lanowski and Shelah.

We present some properties of real valued functions of bounded generalized variation of Riesz–Orlicz type including weight and characterize Lipschitzian superposition Nemytskii operators which map between spaces (in fact, Banach algebras) of these functions.

In this paper we prove the existence of all moments of the solutions to the time-dependent spatially homogeneous transport equation describing elastic and inelastic scattering of particles. The proof uses the theory of resolvent positive operators and Desch's perturbation theorem. As an application we carry out the asymptotic analysis of the full transport equation with dominant elastic scattering and, using the results of the first part of the paper, we show that its solution can be approximated in the L 1 -norm by the solution of the limit equation obtained by formal asymptotic expansion.

The notion of even-outer-semicontinuity for set-valued maps is introduced and compared with related ones from [Bagh, Wets, Set-Valued Anal. 4: 333–360, 1996] and [Kowalczyk, Demonstratio Math. 27: 79–87, 1994]. The coincidence of these notions provides a new characterization of compactness and of local compactness. The following result is proved: Let X be a topological space, Y a uniform space, { F σ : σ ∈ Σ} be a net of set-valued maps from X to Y and F be a set valued map from X to Y . Then any two of the following conditions imply the third: (1) the net { F σ : σ ∈ Σ} is evenly-outer semicontinuous; (2) the net { F σ : σ ∈ Σ} is graph convergent to F ; (3) the net { F σ : σ ∈ Σ} is pointwise convergent to F . This theorem generalizes some results from [Bagh, Wets, Set-Valued Anal. 4: 333–360, 1996] and [Kowalczyk, Demonstratio Math. 27: 79–87, 1994].

In the paper the motion of a viscous compressible heat conducting capillary fluid in a domain bounded by a free surface is considered. We prove the local existence and uniqueness of a solution to a problem describing such a motion in anisotropic Sobolev–Slobodetskii spaces. This solution is such that the velocity and temperature belong to , and density to , , α ∈ [3/4, 1).

The main goal of this paper is to characterize both the quotients of quasi-continuous and the quotients of Darboux quasi-continuous functions. We prove also theorems concerning common divisor for the families of the quotients of quasi-continuous (Darboux quasi-continuous) functions with respect to quasi-continuity (Darboux property and quasi-continuity, respectively).

We show a connection between the polynomials whose inflection points coincide with their interior roots (let us write shorter PIPCIR), Legendre polynomials, and Jacobi polynomials, and study some properties of PIPCIRs (Part I). In addition, we give new formulas for some classical orthogonal polynomials. Then we use PIPCIRs to solve some partial differential equations (Part II).

In this paper, we give two new existence theorems for maximal elements in H -spaces. As their applications, we obtain a extended version of Fan–Yen minimax inequality and some new existence theorems of an equilibrium for a qualitative game and an abstract economy. Our main results not only generalize the corresponding results of [Fan, A minimax theorems and applications. Inequalities III, Academic Press, 1972, Tan and Yuan, J. Math. Econom. 23: 243–251, 1994, Yannelis and Prabhakar, J. Math. Econom. 12: 223–245, 1983, Yen, Pacific J. Math. 97: 477–481, 1981] to H -spaces, but also the open question raised by Yannelis and Prabhakar [J. Math. Econom. 12: 223–245, 1983] is answered in affirmative.

It is proved that every measurable, non-vanishing cocycle defined on the product of (0, ∞) and an arbitrary compact metric space is continuous. Some other sufficient conditions for continuity of a cocycle are also given.