ON FORMALISM OF THE LEBESGUE-STIELTJES INTEGRAL AND RELATED DERIVATIVES IN THE THEORY OF A GENERALIZED STURM-LIOUVILLE PROBLEM

摘要

For solutions of the differential equationrn- (pu')' + (Q' - λM')u = F', 0 ≤ x ≤ l, (1)rnthat are continuous on [0, l] (where the primes denote the generalized differentiation and the functions p(x), Q(x), M(x), and F(x) are of bounded variation), we study the possibility of an adequate description in the form of the following equation that is pointwise defined (i.e., of the ordinary differential equation):rn- d/(dσ)(pd/(dx)u) + (q-λm)u=f, (2)rnwhere we have denoted q =d/(dσ)Q, m = d/(dσ)M, and f = d/(dσ), σ(x) is a function strictly increasing onrn[0, l] and defined only by the "exterior" parameters Q, M, and F of problem (1), and the symbol d/(dσ)rnmeans the pointwise σ-differentiation. At each of the endpoints of [0,l], Eq. (1) transforms into therntraditional Sturm-Liouville boundary condition. The passage to the pointwise form (2) allows us to study the qualitative properties of solutions of Eq. (1) (for example, the distribution and the alternation of zeros, the oscillation, etc.), which are characteristic for the classical Sturm-Liouville theory.