Construction of an asymptotically periodic solution to one nonstationary system with delay

In this paper we consider an ensemble of two subsystems of linear differential equations with constant delay. One of the subsystems contains an exponential multiplier in its right-hand side. We construct an asymptotically periodic solution of this system.     

Stability of homogeneity almost everywhere

We consider approximately ϕ-homogeneous mappings almost everywhere, that is functions F such that the difference F(αx) − ϕ(α)F(x) is in some sense bounded almost everywhere in a product space. We will prove, under some assumptions, that then either we have some kind of boundedness of ϕ and F, or there exist a homomorphism and a -homogeneous function , which are almost everywhere equal to ϕ and F, respectively. From this result we derive the superstability effect for the multiplicativity almost everywhere.      

The averaging method and the asymptotic behavior of solutions to differential inclusions

We prove one version of the first Bogolyubov theorem for differential inclusions with multivalued mappings that satisfy certain one-sided constraints. We study the dependence of solutions to differential inclusions on the parameters.     

Some remarks on trigonometric sums

Let

Copositive approximation of periodic functions

Let f be a real continuous 2π-periodic function changing its sign in the fixed distinct points y _i ∈ Y:= {y _i } _i∈ℤ such that for x ∈ [y _i , y _i−1], f(x) ≧ 0 if i is odd and f(x) ≦ 0 if i is even. Then for each n ≧ N(Y) we construct a trigonometric polynomial P _n of order ≦ n, changing its sign at the same points y _i ∈ Y as f, and

Existence of solutions and periodic solutions for nonlinear evolution inclusions

In this paper we consider nonlinear-dependent systems with multivalued perturbations in the framework of an evolution triple of spaces. First we prove a surjectivity result for generalized pseudomonotone operators and then we establish two existence theorems: the first for a periodic problem and the second for a Cauchy problem. As applications we work out in detail a periodic nonlinear parabolic partial differential equation and an optimal control problem for a system driven by a nonlinear parabolic equation.     

A CRITERION OF THE NON-EXISTENCE OF PERIODIC SOLUTIONS FOR A GENERALIZED LI ENARD SYSTEM AND ITS APPLICATIONS

In this paper, a new criterion of the non-existence of periodic solutions for a generalized liénard system

Error estimation of approximate solutions to one inverse problem for a parabolic equation

This paper is devoted to solving the inverse boundary problem of the heat diagnostics by the projective regularization method. We obtain exact with respect to the order error estimates of the corresponding approximate solution.     

Asymptotically almost periodic solutions of abstract retarded functional differential equations of second order

We establish existence of asymptotically almost periodic mild solutions for a class of semi-linear second-order abstract retarded functional differential equations with infinite delay. Research supported in part by FONDECYT, grant 1050314.     

Localization of the method of guiding functions in the problem about periodic solutions of differential inclusions

In this paper we consider the problem on the existence of forced oscillations in nonlinear objects governed by differential inclusions. We propose certain modifications of the methods of generalized and integral guiding functions.     

Two Turán type inequalities

Turán’s book [2], in Section 19.4, refers to the following result of Gábor Halász. Let a ₀, a ₁, ..., a _n−1 be complex numbers such that the roots α ₁, ⋯, α _n of the polynomial x ⁿ + a _n−1 x ⁿ⁻¹ + ⋯ + a ₁ x + a ₀ satisfy min _j Re α _j ≧ 0 and let function Y(t) be a solution of the linear differential equation Y ⁽ⁿ⁾ + a _n−1 Y ⁽ⁿ⁻¹⁾ + ⋯ + a ₁ Y′ + a ₀ Y = 0. Then

Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources

The main result of this paper is an exponential sum bound in prime fields for multilinear expressions of the type under nearly optimal conditions on . It provides the expected generalization of the well-known inequality for r = 2. We also establish a new result on Gauss sums for multiplicative subgroups H of , obtaining a nontrivial estimate provided . This is a further improvement on [BGK]. Received: May 2007, Revision: October 2007, Accepted: October 2007     

On the stability of the quadratic functional equation on bounded domains

A result of Skof and Terracini will be generalized; More precisely, we will prove that if a functionf : [-t, t]ⁿ →E satisfies the inequality (1) for some δ > 0 and for allx, y ∈ [-t, t]ⁿ withx + y, x - y ∈ [-t, t]ⁿ, then there exists a quadratic functionq: ℝⁿ →E such that ∥f(x) -q(x)∥ < (2912n² + 1872n + 334)δ for anyx ∈ [-t, t] ⁿ .     

An exotic sphere with positive curvature almost everywhere

In this article we show that there is an exotic sphere with positive sectional curvature almost everywhere. In 1974 Gromoll and Meyer found a metric of nonnegative sectional on an exotic 7-sphere. They showed that the metric has positive curvature at a point and asserted, without proof, that the metric has positive sectional curvature almost everywhere [4]. We will show here that this assertion is wrong. In fact, the Gromoll-Meyer sphere has zero curvatures on an open set of points. Never the less, its metric can be perturbed to one that has positive curvature almost everywhere.     

On the maximum number of points at least one unit away from each other in the unit <Emphasis Type="Italic">n</Emphasis>-cube

We present a very short survey of known results and many new estimates and results on the maximum number of points that can be chosen in the n-dimensional unit cube so that every distance between them is at least 1. Research was supported by Slovak national grant VEGA 1/3839/06.     

Collapsing Construction with Nilpotent Structures

A fundamental result concerning collapsed manifolds with bounded sectional curvature is the existence of compatible local nilpotent symmetry structures whose orbits capture all collapsed directions of the local geometry [CFG]. The underlying topological structure is called an N-structure of positive rank. We show that if a manifold M admits such an N-structure , then M admits a one-parameter family of metrics g _∈ with curvature bounded in absolute value while injectivity radii and the diameters of -orbits away from the singular set of uniformly converge to zero as . Moreover, g _∈ is -invariant away from the singular set. This result extends collapsing results in [CG1], [Fu3] and [G]. Q.C. supported partially by a Jingshi research fund from Beijing Normal University. X.R. supported partially by NSF Grant DMS 0504534 and by a Jingshi research fund from Beijing Normal University.     

On continuation of a topology from a system of generators of a group to a topology on the group

In this paper we establish conditions under which a topology is continued from a subset of a group to a topology on the group coordinated with the group operation.     

Two solutions to a nonlinear Neumann problem without asymptotic conditions

We use critical point theory to establish the existence of at least two solutions to a nonlinear Neumann problem involving the one-dimensional p-Laplacian without assuming asymptotic conditions at infinity on the nonlinearity.     

The operad of finite labeled tournaments

We study the operad of finite labeled tournaments. We describe the structure of suboperads of this operad generated by simple tournaments. We prove that a suboperad generated by a tournament with two vertices (i.e., the operad of finite linearly ordered sets) is isomorphic to the operad of symmetric groups, and a suboperad generated by a simple tournament with more that two vertices is isomorphic to the quotient operad of the free operad with respect to a certain congruence. We obtain this congruence explicitly.     

Integrability of rough almost complex structures

We extend the Newlander-Nirenberg theorem to manifolds with almost complex structures that have somewhat less than Lipschitz regularity. We also discuss the regularity of local holomorphic coordinates in the integrable case, with particular attention to Lipschitz almost complex structures.