One method for investigating the solvability of boundary value problems for an implicit differential equation

Abstract

The article concernes a boundary value problem with linear boundary conditions of general form for the scalar differential equation f(t,x(t),x ̇(t))=y ̂(t), not resolved with respect to the derivative x ̇ of the required function. It is assumed that the function f satisfies the Caratheodory conditions, and the function y ̂ is measurable. The method proposed for studying such a boundary value problem is based on the results about operator equation with a mapping acting from a metric space to a set with distance (this distance satisfies only one axiom of a metric: it is equal to zero if and only if the elements coincide). In terms of the covering set of the function f(t,x_1,•): R→R and the Lipschitz set of the function f(t,•,x_2): R →R, conditions for the existence of solutions and their stability to perturbations of the function f generating the differential equation, as well as to perturbations of the right-hand sides of the boundary value problem: the function y ̂ and the value of the boundary condition, are obtained.

Extension of the Kantorovich Theorem to Equations in Vector Metric Spaces: Applications to Functional Differential Equations

“…However, in the general case, "implicit" FDEs remain practically unexamined because many classical methods of analysis, fixedpoint theorems in particular, cannot be applied here. It is possible that statements about the existence of solutions and their estimates and dependence on parameters can be obtained with the use of contemporary results on covering maps (see [17][18][19][20]), which have recently been successfully applied to ordinary differential equations unsolved with respect to the derivative (see, e.g., articles [21][22][23][24][25] and other works by the same authors). But until now, such studies of "implicit" FDEs were fragmentary; we note only the works [26][27][28].…”

“…Given u 0 ∈ L n and e ∈ L + , let the functions φ, Φ satisfy the conditions M1[u 0 , e], M2[u 0 , e], let the function φ(t, •, ς) : R + → R be nonincreasing for a.e. t ∈ [0, T] and any ς ∈ R + , and let the inequality Φ t, (Ku 0 )(t), (S h u 0 )(t), u 0 (t) ≤ −φ(t, 0, 0, 0) (25) hold. Then, if the set of solutions to Equation (13) is not empty, and ς * ∈ E + is its smallest element, then there exists a solution u * ∈ B L n (u 0 , ς * ) to Equation (11).…”