Solution Branching in Linear Differential Equations

✍ Scribed by G. Gustafson; J. Ridenhour

• Publisher: Elsevier Science
• Year: 1993
• Tongue: English
• Weight: 555 KB
• Volume: 101
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1993.1016

✦ Synopsis

Given a nontrivial (y), solution branches of the form (z \equiv y+\sum_{i=1}^{m} c_{i} w_{i}) are constructed with (\left|w_{i}\right|=1,1 \leqslant i \leqslant m). Multiple zeros of (y) "unwind" to nearby simple zeros of the branch (z). Key assumptions are 1-dimensionality of the solution space of the boundary value problem, an exactness condition on the zeros of (y), and a restriction on multiple zero counts. The branching lemmas and theorems are applied to the theory of conjugate points and extremal solutions, and also to branching of solutions of (y^{(n)}+p(x) y=0). If 1 -dimensionality is dropped, then solution branches (z) exist but the constants (c_{\text {, }}) may not be small. 1993 Academic Press, Inc.