Existence of periodic solutions of a system of damped wave equations in thin domains

Effect of resonance on the existence of periodic solutions for strongly damped wave equation

“…where β is the Hausdorff measure of noncompactness associated with | · | (1) and δ > 0 is a constant from the inequality (2.11).…”

“…and hence (−∞, 0] ⊂ ̺(A ε ). Therefore we can define H : 1], (x, y) ∈ U R . We claim that H has no zeros on ∂U R .…”

“…In particular a large part of these studies concerns the weakly damped wave equation u tt − cu t = ∆u + λu + f (t, x, u), t ≥ 0, x ∈ Ω u(t, x) = 0, t ≥ 0, x ∈ ∂Ω (1.2) where the Laplacian in the damping term appears in the zero fractional power. For instance the results obtained in the series of papers [1], [18], [21], [22], [23] provides the existence of T -periodic solutions for (1.2), in the case when Ω ⊂ R n is a thin domain, that is, a cartesian product of an open bounded subset of R n−1 and a small open interval. In these papers the periodic solutions are obtained as fixed points of the Poincaré operator by topological degree methods.…”