Toward a Sturm-Liouville theory for an equation with generalized coefficients
Consider the generalized equation \[ -(p u')' + Q' u = \lambda R' u + F' \] over \([0, 1]\), with \(p, Q, R, F\) having bounded variation in \([0, 1]\). Let \(\mu (x)\) be an increasing function such that \(Q, R, F\) are \(\mu\)-absolutely continuous (an example of such a \(\mu\) is given). Let \(E_\mu\) be the set of functions \(u(x)\) such that \(u(x)\) is absolutely continuous on \([0, 1]\), \(u'(x)\) has bounded variation in \([0, 1]\) as well as \(p(x) u'(x)\) is \(\mu\)-absolutely continuous. A ``solution'' of the above equation is defined to be a function in \(E_\mu\) that satisfies the equality almost everywhere (with respect to the \(\mu\)-measure).\ The authors establish some well-known classical results to the above mentioned generalized settings which include properties concerning the existence, uniqueness, continuity, Sturm-type oscillation of the solution of the initial value problems as well as the properties concerning Green's function and the eigenpairs of the Dirichlet boundary value problems.
scientific article; zbMATH DE number 2174270