On principal eigenvalues of measure differential equations and a patchy Neumann eigenvalue problem

Minimization of lowest positive periodic eigenvalue for the Camassa–Holm equation with indefinite potential

“…The continuous dependence of eigenvalues on potentials or weights for different kinds of problems has also been extensively studied in the literature; see [9,22,34,41,43,47]. Finally, we prove the optimal lower bound of the lowest positive eigenvalues λ + 0 (µ) of (1.5) when the total variation of potentials µ is given (see Theorem 3.6).…”

“…As a result, we have to choose the general setting of so-called measure differential equations to understand the eigenvalues, eigenfunctions and the minimization. Measure differential equations are a special class of generalized ordinary differential equations [38], and the related theory has been extensively developed during the last two decades; see [33,34,35,41,42,45] and the references therein. Therefore, after recalling some basic results on measures, the space of Radon measures, and the weak * topology for measures, we will prove the main results for the eigenvalues of the second order measure differential equation…”

Linearization and Hölder continuity of generalized ODEs with application to measure differential equations

“…The GODEs include various types of other differential equations as special cases, such as the classical ODEs, measure differential equations (MDEs), impulsive differential equations (IDEs), functional differential equations (FDEs) as well as dynamic equations on time scales. In particular, a pretty application of GODEs is MDEs, which have well studied (see e.g., Federson and Mesquita [2], Federson, Mesquita and Slav ĺk [3], Piccoli [4], Piccoli and Rossi [5], Meng [6], Meng and Zhang [7], Zhang [8], Wen [9], Wen and Zhang [10]), another application of GODEs is IDEs (see Federson and Schwabik [11], Afonso, Bonotto, Federson and Schwabik [12]).…”