Volume 52, Nº 6 (2017)

This paper is mainly concerned with the existence of solutions for a certain class of discrete fractional difference inclusions with boundary conditions.Under certain suitable conditions, the existence results are established by using fixed point theory for multi-valued upper semicontinuous maps. Also, an example is presented to illustrate the possible applications of the obtained results.

The present paper is a continuation of the author’s paper [1], where by means of a special integral representation of functions we prove embedding theorems for multianisotropic functional spaces. In contrast to [1], here we consider the case where the corresponding completely regular polyhedron has many anisotropy vertices.

Let E = E(a, b) be some Banach space of measurable functions on (a, b), I be the identity operator, and let \(\hat K\) be a Fredholm-type regular integral operator acting on E and \({\hat K_ \pm }\) be its triangular parts. We consider the representation \(I - \hat K = \left( {I - {{\hat K}_ - }} \right)\left( {I - \hat U} \right)\left( {I - {{\hat K}_ + }} \right)\), for some known classes of integral operators. In particular,we show that under certain conditions the operator \(\hat U\) is positive and its spectral radius satisfies the condition \(r\left( {\hat U} \right) < 1\). Also, we give some possible applications of the representation.

The paper studies the Banach spaces h_∞(φ), h₀(φ), and h¹(η) of harmonic functions over the unit ball in Rⁿ. These spaces depend on a weight function φ and a weight measure η. For a given function φ from a sufficiently broad class of functions, we solve the duality problem. that is, we construct measures η such that h¹(η)* ∼ h_∞(φ) and h₀(φ)* ∼ h¹(η).

This work focuses on the variance properties of isotropic Boolean random sets containing randomly-oriented cylinders with circular cross-section. Emphasis is put on cylinders with large aspect ratios, of the oblate and prolate types. A link is established between the power law decay of the covariance function and the variance of the estimates of the volume fraction of cylinders. The covariance and integral range of the Boolean mixtures are expressed in terms of the orientation-averaged covariogram of cylinders, for which exact analytical formulas and approximate expressions are provided.