Nonoscillation of ordinary differential equations and inequalities on spatial networks (Q5951328)

The author's introduction: The nonoscillation of an ordinary differential equation \[ Lu\equiv -(pu')' +qu= 0\tag{1} \] on an interval \([a, b]\subset\mathbb{R}\) means that every nontrivial solution to (1) has at most one zero on \([a, b]\). The nonoscillation property plays a key role in the theory of differential inequalities of the form \(Lu\geq 0\), where, by virtue of the Vallée-Poussin theorem, it is equivalent to the existence of a strictly positive solution to such an inequality on \([a, b]\). The latter property has various applications in the theory of boundary value problems for equation (1). It is natural that similar properties play an important role for boundary value problems on spatial networks, where an equation of form (1) is posed on each edge of a network and the solutions to the equations on adjacent edges must satisfy matching conditions at the common vertices. Boundary value problems on spatial networks arise in various fields of science. Elastic deformations of a plane network of strings like a tennis racket are one of the most illustrative prototypes of such problems. The pioneering papers in this area studied the equation in the product of function spaces (on the edges). From the qualitative viewpoint, this approach has soon been exhausted in view of the difficulty in analyzing the sign of the Green function even for the simplest operator \(L_0u\equiv -u''\). This has served as an impetus for the Voronezh school to develop a new approach to ``ordinary differential equations on graphs'' and work out qualitative methods similar to those in nonoscillation theory on an interval. These methods have been intensively used by the author and his students in the form of effective approaches and tools but have not been separately exposed. The present paper fills the gap.