On some boundary value problems with integral conditions for functional differential equations

For the functional differential equationu ⁽ⁿ⁾ (t)=f(u)(t) we have established the sufficient conditions for solvability and unique solvability of the boundary value problems $$u^{(i)} (0) = c_i (i = 0,...,m - 1), \smallint _0^{ + \infty } |u^{(m)} (t)|^2 dt< + \infty $$ and $$\begin{gathered} u^{(i)} (0) = c_i (i = 0),...,m - 1, \hfill \\ \smallint _0^{ + \infty } t^{2j} |u^{(j)} (t)|^2 dt< + \infty (j = 0,...,m), \hfill \\ \end{gathered} $$ wheren≥2,m is the integer part of $\tfrac{n}{2}$ ,c _i ∈R, andf is the continuous operator acting from the space of (n?1)-times continuously differentiable functions given on an interval [0,+∞] into the space of locally Lebesgue integrable functions.     

On some properties of solutions of second-order linear functional differential equations

The properties of solutions of the equationu″(t) =p ₁(t)u(τ₁(t)) +p ₂(t)u′(τ₂(t)) are investigated wherep _i :a, + ∞[→R (i=1,2) are locally summable functions τ₁ :a, + ∞[→R is a measurable function, and τ₂ :a, + ∞[→R is a nondecreasing locally absolutely continuous function. Moreover, τ _i (t) ≥t (i = 1,2),p ₁(t)≥0,p ₂ ² (t) ≤ (4 - ɛ)τ ₂ ^′ (t)p ₁(t), ɛ =const > 0 and . In particular, it is proved that solutions whose derivatives are square integrable on [α,+∞] form a one-dimensional linear space and for any such solution to vanish at infinity it is necessary and sufficient that .     

Finite element approximations of a nonlinear diffusion model with memory

The convergence of a finite element scheme approximating a nonlinear system of integro-differential equations is proven. This system arises in mathematical modeling of the process of a magnetic field penetrating into a substance. Properties of existence, uniqueness and asymptotic behavior of the solutions are briefly described. The decay of the numerical solution is compared with both the theoretical and finite difference results.     

Regioselective Synthesis of Trichloromethyl‐Substituted Salicylates and Cyclohexenones by One‐Pot Cyclizations of 1,3‐Bis(trimethylsilyloxy)buta‐1,3‐dienes

A variety of 6‐(trichloromethyl)salicylates (=2‐hydroxy‐6‐(trichloromethyl)benzoates) were prepared by TiCl₄‐mediated cyclization of 1,3‐bis(trimethylsilyloxy)buta‐1,3‐dienes with 1,1,1‐trichloro‐4,4‐dimethoxybut‐3‐en‐2‐one. The employment of trimethylsilyl trifluoromethanesulfonate (Me₃SiOTf) as Lewis acid resulted in the formation of trichloromethyl‐substituted cyclohexenones. The cyclizations proceeded with good‐to‐very‐good regioselectivities.     

On the Dirichlet Problem in a Characteristic Rectangle for Fourth Order Linear Singular Hyperbolic Equations

In the rectangle D = (0,

On a resonance periodic problem for nonautonomous higher-order differential equations

For nonlinear nonautonomous higher-order ordinary differential equations, we prove in a sense optimal criteria for the solvability and unique solvability of a resonance periodic problem.