DIFFERENTIAL GAME WITH A LIFELINE FOR THE INERTIAL MOVEMENTS OF PLAYERS

In this paper, we study the well-known problem of Isaacs called the "Life line" game when movements of players occur by acceleration vectors, that is, by inertia in Euclidean space. To solve this problem, we investigate the dynamics of the attainability domain of an evader through finding solvability conditions of the pursuit-evasion problems in favor of a pursuer or an evader. Here a pursuit problem is solved by a parallel pursuit strategy. To solve an evasion problem, we propose a strategy for the evader and show that the evasion is possible from given initial positions of players. Note that this work develops and continues studies of Isaacs, Petrosjan, Pshenichnii, Azamov, and others performed for the case of players' movements without inertia.

Factorization method for solving nonlocal boundary value problems in Banach space

This article deals with the factorization and solution of nonlocal boundary value problems in a Banach space of the abstract form B1u = Au − SΦ(u) − GΨ(A0u) = f, u ∈ D(B1),where A, A0 are linear abstract operators, S, G are vectors of functions, Φ, Ψ are vectors of linear bounded functionals, and u, f are functions. It is shown that the operator B1 under certain conditions can be factorized into a product of two simpler lower order operators as B1 = BB0. Then the solvability and the unique solution of the equation B1u = f easily follow from the solvability conditions and the unique solutions of the equations Bv = f and B0u = v. The universal technique proposed here is essentially different from other factorization methods in the respect that it involves decomposition of both the equation and boundary conditions and delivers the solution in closed form. The method is implemented to solve ordinary and partial Fredholm integro-differential equations.

Solvability Conditions and General Solution of a System of Matrix Equations Involving η-Skew-Hermitian Quaternion Matrices

In this article, we study the solvability conditions and the general solution of a system of matrix equations involving η-skew-Hermitian quaternion matrices. Several special cases of this system are discussed, and we recover some well-known results in the literature. An algorithm and a numerical example for the validation of our main result are also provided.

On Solvability Conditions for a Certain Conjugation Problem

We study a certain conjugation problem for a pair of elliptic pseudo-differential equations with homogeneous symbols inside and outside of a plane sector. The solution is sought in corresponding Sobolev–Slobodetskii spaces. Using the wave factorization concept for elliptic symbols, we derive a general solution of the conjugation problem. Adding some complementary conditions, we obtain a system of linear integral equations. If the symbols are homogeneous, then we can apply the Mellin transform to such a system to reduce it to a system of linear algebraic equations with respect to unknown functions.

Dirichlet-Neumann problem for the partial differential equations with deviation over the space argument

Dirichlet-Neumann problem for the typeless high order partial differential equation with deviating over the space argument is studied in the domain, which is the Cartesian product of the segment $(0,T)$ and the unit circle $\Omega=\mathbb R/(2\pi \mathbb Z)$. Dirichlet-Neumann problem for hyperbolic equations and their systems in case with absent argument deviation $h$ has been studied by the authors before. Correct solvability conditions have been established for these problems for almost all (with respect to Lebesgue measure) numbers $T>0$ and for almost all (with respect to Lebesgue measure) vectors, constructed by coefficients of the equation. In this paper, the solvability conditions of the problem for $h\ne0$ are described and the influence of the deviation $h$ on the solvability of the problem is studied. The solution of the problem is constructed in the form of the series with respect to the systems of orthogonal functions. Metric estimations (of exponential type) are proved for small denominators appearing during construction of the problem solution. These estimations guarantee the correctness of the problem in Sobolev spaces for almost all (with respect to Lebesgue measure) values $ T> 0 $ and for almost all (with respect to Lebesgue measure) values $ h \in [0,2\pi) $. The obtained results are based on the fact that the corresponding characteristic determinant permits factorization in the form of the product of hyperbolic functions with integer parameters.

The solvability conditions for the second order nonlinear differential equation with unbounded coefficients in L2(R)

The article deals with the existence of a generalized solution for the second order nonlinear differential equation in an unbounded domain. Intermediate and lower coefficients of the equation depends on the required function and considered smooth. The novelty of the work is that we prove the solvability of a nonlinear singular equation with the leading coefficient not separated from zero. In contrast to the works considered earlier, the leading coefficient of the equation can tend to zero, while the intermediate coefficient tends to infinity and does not depend on the growth of the lower coefficient. The result obtained formulated in terms of the coefficients of the equation themselves; there are no conditions on any derivatives of these coefficients.

Solvability conditions in the analytical form of a descriptor system of partial differential equations

A system of first-order partial differential-algebraic equations in a Banach space with constant degenerate operators in the case of a regular operator pencil is considered. In this case, under some additional condition, the original system splits into two subsystems in disjoint subspaces in order to search for the projections of the original unknown function in the subspaces. The matching conditions for the parameters of the systems are identified. A solution of the considered system of differential-algebraic equations is constructed.

New solvability condition of 2-d nonlocal boundary value problem for Poisson’s operator on rectangle

Differential and difference interpretations of a nonlocal boundary value problem for Poisson’s equation in open rectangular domain are studied. New solvability conditions are obtained in respect of existence, uniqueness and a priori estimate of the classical solution. Second order of accuracy difference scheme is presented.