Linear boundary value problems for generalized differential equations - Tvrdý, Milan (Annales Mathematicae Silesianae, 2022)
The paper deals with boundary value problems of the form (0.1) x(t) - x(0) - ∫0td[A(s)]x(s) = f(t) - f(0), t∈[0,1], (0.2) Mx(0) + ∫01K(τ)d[x(τ)] = r. Their solutions are functions regulated on [0,1] and regular on (0,1) (i.e. 2x(t) = x(t-)+x(t+) for all t∈(0,1)). We assume that A and K have bounded variations on [0,1], f is regulated on [0,1] and all of them are regular on (0,1). We derive conditions for the existence and uniqueness of solutions to the given problem. Furthermore, the relationship between the dimensions of the spaces of solutions of the corresponding homogeneous problem and of its adjoint is established. Special attention is paid to the case when the additional condition (0.2) reduces to the periodic boundary condition x(0) = x(1). It is known (cf. [13]) that in the case that A and f are continuous from the right at t = 0 and from the left at t = 1, the equation (0.1) reduces to the distributional differential equation (0.3) x' - A'x = f'. Related results concerning the case of solutions left-continuous on (0,1) were obtained in [18] and similar questions for periodic problems and for linear differential equations with distributional coefficients of the form (0.3) were recently treated by Z. Wyderka [21], cf. also [2], [3] or [10].