In this paper, we study an inverse problem consisting of determining the unknown local changes of a population densities independent of the missing initial population age distribution from the knowledge of some measurement in the state. The main idea is to transform the study of the inverse problem into an optimal control problem with incomplete data. To solve this kind of problem, we use the method of no-regret control approximated by a sequence of low-regret control. We show the existence and uniqueness of the sequence of low-regret control, which converges weakly to the unique no-regret control. The local changes are characterized by a coupled optimality system.

Let D be a hexagon with two straight and four right angles. We consider a sevenelement difference equation with holomorphic coefficients generated by this hexagon. A method for regularizing this equation is proposed. The solution is sought in the class of functions holomorphic outside the "half"of the boundary ᲛD and vanishing at infinity. It is represented as a Cauchy-type integral over the "half"of the boundary with an unknown density. Applications to the problem of moments for entire functions of exponential type (e.f.e.t.) are indicated.

Let $K_n(x)$ denote the Fej\'er kernel given by
$$K_n(x)=\sum_{j=-n}^n\left(1-\frac{|j|}{n+1}\right)e^{-ijx}$$
and  let $\sigma_nf(x)=(K_n\ast f)(x)$,  where as usual $f\ast g$ denotes the convolution of $f$ and $g$. Let the sequence $\{n_k\}$ be lacunary. Then the series 
$$\mathcal{G}f(x)=\sum_{k=1}^\infty \left(\sigma_{n_{k+1}}f(x)-\sigma_{n_k}f(x)\right)$$ 
converges unconditionally for all $f\in L^2(\mathbb{R})$. Let $(n_k)$ be a lacunary sequence, and $\{c_k\}_{k=1}^\infty \in \ell^\infty$. Define
$$\mathcal{R}f(x)=\sum_{k=1}^\infty c_k\left(\sigma_{n_{k+1}}f(x)-\sigma_{n_k}f(x)\right).$$ 
Then  there exists a constant $C>0$ such that
$$\|\mathcal{R}f\|_2\leq C\|f\|_2$$
for all $f\in L^2(\mathbb{R})$, i.e., $\mathcal{R}f$ is of strong type $(2,2)$. As a special case it follows that $\mathcal{G}f$ also is of strong type $(2,2)$.

The problem of solving systems of linear algebraic equations (SLAE) with an ill-conditioned or degenerate exact matrix and an approximate right-hand side is considered. A scheme for solving such a problem is proposed and justified, which makes it possible to improve the conditionality of the SLAE matrix. As a result, an approximate solution that is stable to perturbations of the right-hand side is obtained with a higher accuracy than when using some other methods. The scheme is implemented by an algorithm that uses minimal pseudoinverse matrices. The results of numerical experiments are presented, confirming the theoretical provisions of the article.

The question of a priori estimation of periodic solutions for a system of non-linear ordinary differential equations of the second order with a distinguished main positively homogeneous part is investigated. In this question, the well-known methods for deriving an a priori estimation of periodic solutions for similar systems of first-order ordinary differential equations are not directly applicable.
Combining these methods with the idea of qualitative research of singularly perturbed ordinary differential equations, conditions are found that provide an a priori estimation of periodic solutions for the  system of second-order equations.
The conditions for the a priori estimation are formulated in terms of the properties of the main positively homogeneous part of the system of equations. The existence of periodic solutions is proved to be invariant under a continuous change of the main positively homogeneous part and the conditions of a priori estimation  are preserved. Based on the results obtained, in the future, it is possible to investigate the existence of periodic solutions.

We study two-dimensional nonlinear partial differential equations of the second order with variable coefficients. The left-hand side of these equations is a homogeneous polynomial of the second degree in unknown function and its derivatives. We consider a set of linear multiplicative transformations of the unknown function which keep the form of the initial equation. By analogy with linear equations, the Laplace invariants are determined as the invariants of this transformation. The expressions for the Laplace invariants in terms of the coefficients of the equation and their first derivatives are obtained. For the considered equations, we found the equivalent systems of the first order equations containing the Laplace invariants. It is shown that if one of the Laplace invariants equals zero, the corresponding system is reduced to one equation of the first order. Also in this case, the solution of the initial equation can be obtained in quadratures if some additional conditions on the coefficients are met. The investigations are executed for a hyperbolic equation with a mixed derivative and for a nonlinear second order equation of the general form with a homogeneous polynomial of the second degree in unknown function and its derivatives. We obtained for these cases the Laplace invariants and equivalent systems of the first order equations.

In this paper, on the base of the theory of attractors for non-invariant trajectory spaces, the limiting behavior of solutions to a periodic in spatial variables problem for the Bingham model is investigated. For the considered problem, necessary exponential estimates are established, the trajectory space is determined, and the existence of a minimal uniform trajectory and uniform global attractors is proved.

The proposed method of optimization of calculations is of a combined nature. First, using a piecewise uniform grid, a difference scheme of the second order of accuracy of the McCormack type is constructed, and then — to weaken the stability condition — an implicit version of the scheme is used, which even in the nonlinear case allows an iteration-free implementation. When constructing a grid, a preliminary study is required, which is based on the analytical properties of the solution of the differential problem. Based on the calculations carried out, it can be concluded that the potential reduction in the amount of calculations is dozens of times without a noticeable loss of accuracy.

A symmetric partial differential eigenvalue problem with nonlinear dependence on the spectral parameter arising in plasma physics is studied. We propose and justify new conditions for the existence of a positive eigenvalue and the corresponding positive eigenfunction. A finite element approximation of the problem preserving the property of positivity of solutions is constructed. The existence and convergence of approximate solutions are established.