Qualitative properties of solutions to fourth-order differential equations on graphs

In this paper, we examine properties of solutions to fourth-order differential equations on geometric graphs (positivity, oscillatory behavior, distribution of zeros, etc.). We prove theorems on alternation of zeros of solutions and develop the theory of nonoscillation. The definition of nonoscillation for fourth-order equations on graphs is based on the concept of a double constancy zone introduced in the paper. The new approach allows one to generalize the basic principles of the theory of nonoscillation of second-order equations on a graph to fourth-order equations.