Some remarks on Sobolev type inequalities

We extend two Sobolev type inequalities for balls to arbitrary smooth bounded domains. In the case of balls, one inequality is due to Brezis and Lieb and another is due to Escobar. The extension has been achieved by analyzing the asymptotic behaviour of solutions of certain semilinear Neumann problems.     

Some remarks on a linear stochastic differential equation

In this note we study the properties of the solution of a two-parameter linear stochastic differential equation with constant coefficients. We prove, in particular, that this solution may be negative with positive probability.     

Some remarks on reverse Hilbert and Hardy-Hilbert type inequalities

In this paper we obtain an extension of discrete Hilbert’s inequality, by using some numerical methods. We shall obtain, in a similar way as Yang did in [10], that the parameter from the kernel can be taken from the interval [3/2, 3). We also compare our findings with existing results, known from the literature.     

Extremal maps in Sobolev type inequalities: Some remarks

In this work we make some observations on the existence of extremal maps for sharp L²-Riemannian Sobolev type inequalities as Nash and logarithmic Sobolev ones. Among other results, we prove also that there exist smooth compact Riemannian manifolds with scalar curvature changing signal on which there exist extremal maps.     

Some remarks on linear groups

Three problems concerning linear groups are solved.Translated from Matematicheskie Zametki, Vol. 4, No. 6, pp. 635–638, December, 1968.     

Some remarks on homoclinic groups of hyperbolic toral automorphisms

The homoclinic group (an invariant with respect to topological conjugacy) for hyperbolic toral automorphisms is determined. Certain conditions are given for conjugacy of a homeomorphism of a compact space to hyperbolic toral automorphism. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 223, 1994, pp. 140–147. This paper is partially supported by Russian Foundation for Basic Research, grant 94-01-00921. Translated by V. V. Sadovskaya.     

Some remarks on the Jensen-Hadamard inequalities and applications

We provide some inequalities and integral inequalities connected to the Jensen-Hadamard inequalities for convex functions. In particular, we give some refinements to these inequalities. Some natural applications and further extensions are given.

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Sunto Forniamo alcune diseguaglianze e diseguaglianze integrali connesse alle dise-gueglianze di Jensen-Hadamard per funzioni convesse. In particolare, diamo qualche miglioramento di queste diseguaglianze. Alcune applicazioni naturali ed ulteriori estensioni sono date.

     

Some functional inequalities

LetM, N, O be open subsets of ? ⁿ and letF:M×N→O,f:O→?,g: M→?,h: N→? be functions, satisfying the functional inequality $$\forall (x,y) \in M \times N:f[F(x,y)] \leqslant g(x) + h(y).$$ IfF belongs to a certain extensive class of functions, we prove in this note, thatf is bounded above on every compact subset of ? ⁿ , wheneverh is bounded above on a Lebesgue-measurable set of positive Lebesgue-measure, contained inN (no assumptions aboutg are needed). Moreover we give applications of this theorem to generalized convex and subadditive functions.     

Some considerations on functional differential equations of advanced type

We prove some comparison results for the periodic boundary value problem related to a first‐order functional differential equation of advanced type. These maximum principles provide uniqueness results for nonlinear differential equations with advanced arguments. By a change of variable, we deduce analogous results for fuctional differential equations with delayed arguments (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Some remarks on the Chebyshev functional

Identities are provided for the Chebyshev functional defined on a real linear space equipped with a bilinear form. In addition, some inequalities for the functional are discussed.     

Some remarks on the asymptotic behavior of a nonautonomous,nonlinear functional differential equation

The asymptotic behavior of the abstract nonautonomous, nonlinear functional differential equation $\text{x}\text{?}\text{(t) = f(t, x(t)) + g(t, x}{}_{\mathrm{t}}\text{), x}{}_{\mathrm{s}}\text{= ?}$ is considered. Estimates on the growth of solutions are given and these estimates are shown to be the best possible.     

Differential and difference inequalities generated by mixed problems for hyperbolic functional differential equations with impulses

The paper deals with initial-boundary value problems for impulsive functional differential equations. Theorems on impulsive functional differential and difference inequalities are obtained. Comparison results implying uniqueness criteria are proved. A result on the convergence of difference methods is given.     

A note on Jordan type inequalities for hyperbolic functions

In this work the authors present some new lower and upper bounds for the functions sinx/x and x/sinhx, thus improving some inequalities put forward by Klén et al. (2010) in the paper [2].     

Some remarks on reachability of infinite-dimensional linear systems

In this paper we shall be concerned with the question of reachability when allowing distribution inputs. We show that a certain class of systems accept distribution inputs, but, in general, they cannot be exactly reachable. We shall also consider the problem of the uniqueness of canonical realizations in relation to exact reachability, and show that Matsuo's result on uniqueness (Ph.D. dissertation, in preparation) does not apply to the example given in Baras, Brockett, and Fuhrmann [IEEE Trans. Automatic ControlAC-19 (1974), 693–700].