Geodesic
Qualitative properties of solutions to fourth-order differential equations on graphs
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 1, Tome 208 (2022), pp. 37-48

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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.
Mots-clés : oscillation
Keywords: graph equation, fourth-order equation. 
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R. Ch. Kulaev; A. A. Urtaeva. Qualitative properties of solutions to fourth-order differential equations on graphs. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh International Spring Mathematical School "Modern Methods of the Theory of Boundary-Value Problems. Pontryagin Readings – XXXII”, Voronezh, May 3–9, 2021, Part 1, Tome 208 (2022), pp. 37-48. http://geodesic.mathdoc.fr/item/INTO_2022_208_a5/