Nonnegative solutions of singular boundary value problems with sign changing nonlinearities

AbstractThe paper presents sufficient conditions for the existence of positive solutions of the equation x″(t) + q(t)f(t,x(t),x′(t)) = 0 with the Dirichlet conditions x(0) = 0, x(1) = 0 and of the equation (p(t)x′(t))′ + p(t)q(t)f(t,x(t),p(t)x′(t)) = 0 with the boundary conditions limt→o+ p(t)x′(t) = 0, x(1) = 0. Our nonlinearity f is allowed to change sign and f may be singular at x = 0. The proofs are based on a combination of the regularity and sequential techniques and the method of lower and upper functions

Histone Deacetylase Activity Modulates Alternative Splicing

There is increasing evidence to suggest that splicing decisions are largely made when the nascent RNA is still associated with chromatin. Here we demonstrate that activity of histone deacetylases (HDACs) influences splice site selection. Using splicing-sensitive microarrays, we identified ∼700 genes whose splicing was altered after HDAC inhibition. We provided evidence that HDAC inhibition induced histone H4 acetylation and increased RNA Polymerase II (Pol II) processivity along an alternatively spliced element. In addition, HDAC inhibition reduced co-transcriptional association of the splicing regulator SRp40 with the target fibronectin exon. We further showed that the depletion of HDAC1 had similar effect on fibronectin alternative splicing as global HDAC inhibition. Importantly, this effect was reversed upon expression of mouse HDAC1 but not a catalytically inactive mutant. These results provide a molecular insight into a complex modulation of splicing by HDACs and chromatin modifications

Two functional boundary-value problems with singularities in phase variables

Розглядається диференцiальне рiвняння x`` = f(t, x, x`) з двома функцiональними граничними умовами. Тут f(t, x, y) локально є функцiєю Каратеодорi, що може мати особливiсть вiдносно фазових змiнних x та y в точках x = 0 та y = 0. Основною спiльною властивiстю цих двох задач з особливостями є те, що будь-який розв’язок або похiдна будь-якого розв’язку „проходить” через особливостi f всерединi [0, T]. Результати про iснування доведено за допомогою регуляризацiї та послiдовностей, а також з використанням антимодальної теореми Барсука, степеня Лере – Шаудера та теореми Вiталi про збiжнiсть.The differential equation x`` = f(t, x, x`) together with two functional boundary conditions is considered. Here f(t, x, y) is local Caratheodory function which may be singular at the points x = 0 and y = 0 of the phase variables x and y. The main common feature for these two singular problems is the fact that any solution or the derivative of any solution “pass through” the singularities of f somewhere inside of [0, T]. Existence results are proved by the regularization and sequential techniques and using the Borsuk antipodal theorem, the Leray – Schauder degree and the Vitali’s convergence theorem

Nonnegative solutions of a class of second order nonlinear differential equations

A differential equation of the form (q(t)k(u)u')' = λf(t)h(u)u' depending on the positive parameter λ is considered and nonnegative solutions u such that u(0) = 0, u(t) > 0 for t > 0 are studied. Some theorems about the existence, uniqueness and boundedness of solutions are given

On solvability of singular periodic boundary-value problems

We present conditions ensuring the existence of a solution in the class C¹ ([0, T]) for the singular periodic boundary-value problem (r(x' ))' = H(p(t) + q(x))k(x')f(t, x, x'(0) = x(T), x'(0) = x'(T)