Systems of Equations
This chapter is devoted to the study of L p Lyapunov-type inequalities for linear systems of ordinary differential equations with different boundary conditions (which include the case of Neumann, Dirichlet, periodic, and antiperiodic boundary value problems) and for any constant p ≥ 1. Elliptic problems are also considered. As in the scalar case, the results obtained in the linear case are combined with Schauder fixed point theorem to provide several results about the existence and uniqueness of solutions for resonant nonlinear systems. In addition, we study the stable boundedness of linear periodic conservative systems. The proof uses in a fundamental way the nontrivial relation (proved in Chap. 2) between the best Lyapunov constants and the minimum value of some especial constrained or unconstrained minimization problems (depending on the considered problems are resonant or nonresonant, respectively).
- 9.Clark, S., Hinton, D.: Positive eigenvalues of second order boundary value problems and a theorem of M.G. Krein. Proc. Am. Math. Soc. 130, 3005–3015 (2002)Google Scholar
- 13.Krein, M.G.: On tests for stable boundedness of solutions of periodic canonical systems. Am. Math. Soc. Translat. Ser. (2) 120, 71–110 (1983)Google Scholar
- 16.Lang, S.: Analysis.II. Addison-Wesley, Reading, MA (1969)Google Scholar
- 20.Mawhin, J.: Contractive mappings and periodically perturbed conservative systems. Arch. Math. (Brno) 12, 67–74 (1976)Google Scholar