Sturm-Liouville oscillation theory for impulsive problems.

*(English. Russian original)*Zbl 1170.34313
Russ. Math. Surv. 63, No. 1, 109-153 (2008); translation from Usp. Mat. Nauk. 63, No. 1, 111-154 (2008).

This very interesting and well-written paper summarizes the studies of the authors during the last decade aimed at the extension of the Sturm-Liouville oscillation theory to classes of problems with coefficients involving \(\delta \)-function type singularities. To this end, the classical ordinary differential equation
\[
-\left( pu^{\prime}\right) ^{\prime}+qu=\lambda mu\tag{1}
\]
is replaced with a much more general equation
\[
-\left( pu^{\prime}\right) (x)+\left( pu^{\prime}\right) (0)+\int_{0} ^{x}udQ=\lambda\int_{0}^{x}udM,\tag{2}
\]
where the functions \(p,\) \(Q,\) \(M\) belong to the space \(\text{BV}[0,l]\) of functions of bounded variation, and the integrals are understood in the sense of Riemann-Stieltjes. By a solution of (2) one means an absolutely continuous function that satisfies the equation and has a derivative in \(\text{BV}[0,l]\).

The idea of using the equation (2) originates from the model of the Stieltjes string (an elastic string with bead masses distributed along it) studied by F . V. Atkinson and M. G. Kreĭn in the form \[ u_{+}^{\prime}(x)=u_{-}^{\prime}(0)-\lambda\int_{0}^{x+0}udM, \] where \(u_{+}^{\prime}(x)\) is the right derivative and \(u_{-}^{\prime}(0)\) is some “extended value” of the derivative. The authors start by providing a nice brief description of the origins of the oscillation theory and its development in the twentieth century and describe the goals of the study. Remarkably, they manage to prove fundamental Sturm oscillation theorems for the general equation (2) preserving their exact formulation for the classical equation (1). This requires development of a completely new theory for the equation (2), analogous to the one known for ordinary differential equations, which is far from being a trivial task.

The paper is divided into four main sections. In Section 1, the Riemann-Stieltjes integral and the space of the functions of bounded variation are introduced, and their properties are discussed. Some fundamental results regarding differential inequalities and factorization of a quasi-differential expression are also stated and proved here. A detailed analysis of the equation (2) starts with the study of the structure of the solution space, interpretation of the equation at singular points, an existence and uniqueness theorem, and results on the continuous dependence of the solution on parameters. In Section 2, the linear theory is developed with the attention concentrated on the properties of the Wronskian, the integral representation of the solutions of a nondegenerate boundary value problem, and the distribution of zeros of the homogeneous equation. The authors proceed with the study of the spectral Sturm-Liouville problem, addressing in Section 3 the structure of the spectrum and oscillatory properties of the eigenfunctions. This delightful paper concludes with bibliographic comments and an extensive list of references including 89 items.

The idea of using the equation (2) originates from the model of the Stieltjes string (an elastic string with bead masses distributed along it) studied by F . V. Atkinson and M. G. Kreĭn in the form \[ u_{+}^{\prime}(x)=u_{-}^{\prime}(0)-\lambda\int_{0}^{x+0}udM, \] where \(u_{+}^{\prime}(x)\) is the right derivative and \(u_{-}^{\prime}(0)\) is some “extended value” of the derivative. The authors start by providing a nice brief description of the origins of the oscillation theory and its development in the twentieth century and describe the goals of the study. Remarkably, they manage to prove fundamental Sturm oscillation theorems for the general equation (2) preserving their exact formulation for the classical equation (1). This requires development of a completely new theory for the equation (2), analogous to the one known for ordinary differential equations, which is far from being a trivial task.

The paper is divided into four main sections. In Section 1, the Riemann-Stieltjes integral and the space of the functions of bounded variation are introduced, and their properties are discussed. Some fundamental results regarding differential inequalities and factorization of a quasi-differential expression are also stated and proved here. A detailed analysis of the equation (2) starts with the study of the structure of the solution space, interpretation of the equation at singular points, an existence and uniqueness theorem, and results on the continuous dependence of the solution on parameters. In Section 2, the linear theory is developed with the attention concentrated on the properties of the Wronskian, the integral representation of the solutions of a nondegenerate boundary value problem, and the distribution of zeros of the homogeneous equation. The authors proceed with the study of the spectral Sturm-Liouville problem, addressing in Section 3 the structure of the spectrum and oscillatory properties of the eigenfunctions. This delightful paper concludes with bibliographic comments and an extensive list of references including 89 items.

Reviewer: Yuri V. Rogovchenko (Kalmar)