<p>In this paper, we study a fourth-order differential operator <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$L_\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation> on a graph depending on a real parameter <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>. The main question studied in the paper is determining the set of positive values of the spectral parameter <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> for which the operator <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L_\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation> is positively invertible. We prove that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$L_\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\lambda >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is positively invertible if and only if there is a fundamental system of solutions of the corresponding homogeneous equation consisting of functions that are positive on the graph. We formulate a necessary and sufficient condition for the differential operator <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L_\lambda $</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>λ</mi> </msub> </math></EquationSource> </InlineEquation> to be positively invertible for all positive values of the spectral parameter less than the smallest eigenvalue of the differential operator <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$L_0$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mn>0</mn> </msub> </math></EquationSource> </InlineEquation> corresponding to the value <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\lambda =0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. 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.</p>

MAXIMUM PRINCIPLE FOR A FOURTH-ORDER DIFFERENTIAL OPERATOR ON A GRAPH

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发表日期：

2026年2月1日

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V. A. Eloeva

摘要

In this paper, we study a fourth-order differential operator $L_\lambda $ $L_{λ}$ on a graph depending on a real parameter $\lambda $ $λ$ . The main question studied in the paper is determining the set of positive values of the spectral parameter $\lambda $ $λ$ for which the operator $L_\lambda $ $L_{λ}$ is positively invertible. We prove that $L_\lambda $ $L_{λ}$ for $\lambda >0$ $λ > 0$ 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 $ $L_{λ}$ to be positively invertible for all positive values of the spectral parameter less than the smallest eigenvalue of the differential operator $L_0$ $L_{0}$ corresponding to the value $\lambda =0$ $λ = 0$ . 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.