Periodic solutions for functional differential equations with sign-changing nonlinearities. (English) Zbl 1200.34077

Graef, John R.; Kong, Lingju

Proc. R. Soc. Edinb., Sect. A, Math. 140, No. 3, 597-616 (2010).

The problem of existence of non-trivial \(T\)-periodic solutions is investigated for the functional differential equation
\[ u'(t) = -a(t)u(t) + b(t)f(u(t-\tau(t))) \]
and criteria are derived for the existence of non-trivial \(T\)-periodic solutions of the eigenvalue problem
\[ u'(t) = -a(t)u(t) +\lambda b(t)f(u(t-\tau(t))), \]
where \(\lambda\) is a positive parameter. Applications to related eigenvalue problems are also discussed. The analysis mainly relies on the topological degree theory.

Reviewer: Zhanyuan Hou (London)

MSC:

34K13	Periodic solutions to functional-differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

functional differential equations; periodic solutions