Time periodic problem for the compressible Navier-Stokes equation on the whole space is studied. The existence of a time periodic solution is proved for sufficiently small time periodic external force with some symmetry when the space dimension is greater than or equal to 3. The proof is based on the spectral properties of the time-T map associated with the linearized problem around the motionl...

Let T be a periodic time scale. We use a fixed point theorem due to Krasnoselskii to show that the nonlinear neutral dynamic equation with infinite delay x(t) = −a(t)x(t) + (Q(t, x(t− g(t))))) + ∫ t −∞ D (t, u) f (x(u)) ∆u, t ∈ T, has a periodic solution. Under a slightly more stringent inequality we show that the periodic solution is unique using the contraction mapping principle. Also, by the...

Let T be a periodic time scale. We use a fixed point theorem due to Krasnosel’skĭı to show that the nonlinear neutral dynamic equation with delay x(t) = −a(t)x(t) + (Q(t, x(t), x(t− g(t))))) +G ` t, x(t), x(t− g(t)) ́ , t ∈ T, has a periodic solution. Under a slightly more stringent inequality we show that the periodic solution is unique using the contraction mapping principle. Also, by the aid ...