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Wronskian of Neutral FDE and Sturm Separation Theorem
Published in Leonid Berezansky, Alexander Domoshnitsky, Roman Koplatadze, Oscillation, Nonoscillation, Stability and Asymptotic Properties for Second and Higher Order Functional Differential Equations, 2020
Leonid Berezansky, Alexander Domoshnitsky, Roman Koplatadze
For ordinary homogeneous differential equation (10.3) the classical Sturm separation theorem is valid. Validity of the Sturm theorem follows from non-vanishing of Wronskian W(t) of a fundamental system of ODE (10.3). Really, let us suppose existence of two zeros t1 and t2 of nontrivial solution x2 between adjoint zeros of x1. Consider the following function y(t)=x2(t)x1(t). From the form of y(t) it follows that y(t1) = y(t2) = 0, but this contradicts with the fact that the derivative y′(t)=W(t)[x1(t)]2 preserves sign for t∈[t1,t2].
Sign-changing points of solutions of homogeneous Sturm–Liouville equations with measure-valued coefficients
Published in Applicable Analysis, 2022
When 1/p and q are real-valued locally integrable functions on , , and when p>0 on , zeros of every nontrivial real-valued solution of the famous homogeneous Sturm–Liouville differential equation are isolated in . This fact is the key to establish two big results in Sturm–Liouville theory, namely the Sturm separation theorem and the Sturm comparison theorem. These two celebrated results are due to Sturm [1] and they date back to 1836. The Sturm separation theorem states that zeros of two linearly independent solutions of Equation (1) are interlaced. The Sturm comparison theorem states that if u and v are nontrivial solutions of (1) and , respectively, u vanishes at s and t, and but different on a set of positive Lebesgue measure, then v vanishes at some point between s and t.