Abstract:
In this paper, we study a fourth-order differential operator $L_\lambda$ on a graph depending on a real parameter $\lambda$. The main question studied in the paper is determining the set of positive values of the spectral parameter $\lambda $ for which the operator $L_\lambda$ is positively invertible. We prove that $L_\lambda$ for $\lambda>0$ is positively invertible if and only if there is a fundamental system of solutions of the corresponding homogeneous equation consisting of functions that are positive on the graph. We formulate a necessary and sufficient condition for the differential operator $L_\lambda$ to be positively invertible for all positive values of the spectral parameter less than the smallest eigenvalue of the differential operator $L_0$ corresponding to the value $\lambda=0$. We establish the positivity of eigenvalues and prove a comparison theorem for eigenvalues of the spectral problem. We formulate maximum principles for fourth-order differential inequalities on the graph.
Keywords:maximum principle, equation on a graph, spectral problem on a graph
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