Inverse Positivity of a Forth‐Order Differential Operator on a Network

We consider a boundary value problem for a network beam equation with a forcing term. The main aim of the paper is to obtain upper and lower bounds for the restoring force, ensuring the property of inverse positivity of the corresponding differential operator. These bounds are based on spectral theory, in fact the bounds of the obtained interval are given by suitable eigenvalues of the differential operator with different boundary conditions. The proof of this result is based on abstract methods of functional analysis of operators in ordered spaces. Thus, the proposed method allows to avoid the explicit calculation of the related Green's function.