Internet Mathematical Olympiads

Modern Internet technologies open new possibilities in a wide spectrum of traditional methods, used in mathematical education. One of the areas, where these technologies can be efficiently used, is an organization of mathematical competitions. Contestants can stay in their schools or universities in different cities and even different countries and try to solve as many mathematical problems as possible and then submit their solutions to organizers through the Internet. Simple Internet technologies supply audio and video connection between participants and organizers in a time of the competitions

About differential inequalities for nonlocal boundary value problems with impulsive delay equations

summary:We propose results about sign-constancy of Green's functions to impulsive nonlocal boundary value problems in a form of theorems about differential inequalities. One of the ideas of our approach is to construct Green's functions of boundary value problems for simple auxiliary differential equations with impulses. Careful analysis of these Green's functions allows us to get conclusions about the sign-constancy of Green's functions to given functional differential boundary value problems, using the technique of theorems about differential and integral inequalities and estimates of spectral radii of the corresponding compact operators in the space of essential bounded functions

Semicycles and correlated asymptotics of oscillatory solutions to second-order delay differential equations

We obtain several new comparison results on the distance between zeros and local extrema of solutions for the second order delay differential equation \begin{equation*} x^{\prime \prime }(t)+p(t)x(t-\tau (t))=0,~~t\geq s\text{ }\ \end{equation*} where $\tau :\mathbb{R}\rightarrow \lbrack 0,+\infty )$, $p:\mathbb{R}$ are Lebesgue measurable and uniformly essentially bounded, including the case of a sign-changing coefficient. We are thus able to calculate upper bounds on the semicycle length, which guarantee that an oscillatory solution is bounded or even tends to zero. Using the estimates of the distance between zeros and extrema, we investigate the classification of solutions in the case $p(t)\leq 0,t\in \mathbb{R}.$Comment: 23 page

On the Kneser-Type Solutions for Two-Dimensional Linear Differential Systems with Deviating Arguments

For the differential system u1'(t)=p(t)u2(Ä(t)), u2'(t)=q(t)u1(Ã(t)), t∈[0,+∞), where p,q∈Lloc(â„Â+;â„Â+), Ä,Ã∈C(â„Â+;â„Â+), limt→+∞Ä(t)=limt→+∞Ã(t)=+∞, we get necessary and sufficient conditions that this system does not have solutions satisfying the condition u1(t)u2(t)0). The inequality (δ+Δ)pq>2/e is necessary and sufficient for nonexistence of solutions satisfying this condition

Hopf bifurcation of integro-differential equations

A method reducing integro-differential equations (IDEs) to system of ordinary ones is proposed. On this base stability and bifurcation phenomena in critical cases are studied. Analog of Hopf bifurcation for scalar IDEs of first order is obtained. Conditions of periodic solution existence are proposed. One of the conclusions is the following: phenomena characterized by two dimension systems of ODEs appear for scalar IDEs