Conditions of Existence and Uniqueness of Solutions of the Multipoint Boundary Value Problem for a System of Generalized Ordinary Differential Equations

Effective sufficient conditions are established for the solvability and unique solvability of the boundary value problem where is a matrix-function with bounded variation components, is a vector-function belonging to the Carathéodory class corresponding to are continuous functionals (in general nonlinear) defined on the set of all vector-functions of bounded variation.     

On the correctness of nonlinear boundary value problems for systems of generalized ordinary differential equations

The concept of a strongly isolated solution of the nonlinear boundary value problem $$dx(t) = dA(t) \cdot f(t,x(t)),h(x) = 0,$$ is introduced, whereA: [a, b]→R ^n×n is a matrix-function of bounded variation,f: [a, b]×R ⁿ →R ⁿ is a vector-function belonging to a Carathéodory class, andh a continuous operator from the space ofn-dimensional vector-functions of bounded variation intoR ⁿ . It is stated that the problems with strongly isolated solutions are correct. Sufficient conditions for the correctness of these problems are given.     

A multipoint boundary value problem for systems of linear generalized differential equations with singularities

We study a multipoint boundary value problem for systems of Kurzweil generalized linear differential equations with singularities on a finite closed interval of the real line. We assume that the offdiagonal entries of the matrix function corresponding to the system, as well as the elements of the right-hand side of the system, have bounded variation on the entire interval; however, the diagonal entries of the matrix function are not assumed to have bounded variation on the entire interval. This is what we mean by saying that the system is singular. We study the unique solvability of the problem. We prove a general theorem and use it to we obtain efficient optimal (in particular, spectral) solvability conditions for the problem.     

On some boundary value problems for linear generalized differential systems with singularities

For a linear Kurzweil generalized differential system with singularities, we consider two-point boundary value problems of two forms. The singularity is understood in the sense that the matrix and vector functions defining the system may have infinite total variation.     

On the stability of solutions of linear boundary value problems for a system of ordinary differential equations

Linear boundary value problems for a system of ordinary differential equations are considered. The stability of the solution with respect to small perturbations of coefficients and boundary values is investigated.     

On boundary value problems for systems of nonlinear generalized ordinary differential equations

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R ⁿ → R ⁿ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R ^n×n with bounded total variation components, and h: BV_s([a, b],R ⁿ ) → R ⁿ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t₁(x)) = B(x) · x(t ₂(x))+c ₀, where t _i: BV_s([a, b],R ⁿ ) → [a, b] (i = 1, 2) and B: BV_s([a, b], R ⁿ ) → R ⁿ are continuous operators, and c ₀ ∈ R ⁿ .