Solvability of the Boundary-Value Problem for a Variable-Order Differential Equation on a Geometric Graph

The solvability of the boundary-value problem for a string-beam model is studied. The model is described by an equation of orders 2 and 4 on different edges of an arbitrary graph. Criteria for the problem to be degenerate and nondegenerate are obtained; in particular, it is proved that the nondegeneracy of the problem is equivalent to the maximum principle.