Optimal Estimates On Rotation Number Of Almost Periodic Systems

Rotation number appears frequently in nonlinear ordinary differential equations,because rotation number has very important value in theory and application and hasnot been completely understood. Considering the nonlinear and asymmetric equations:(x|¨) + p(t)x₊ + q(t)x = 0, where p(t) and q(t) are almost periodic functions, this dis-sertation studies systematically the existence of the rotation number, gives some optimalestimates on the rotation number of the linear equation x¨ + p(t)x = 0 and that of theasymmetric equation (x|¨) +p(t)x₊ + q(t)x₌ 0 by introducing some kind of new normsin the spaces of almost periodic functions.The main novelties and results include: 1. We introduce rotation number to theasymmetric equations:(x|¨) + p(t)x₊ + q(t)x = 0, where p(t) and q(t) are almost periodicfunctions. 2. For any given exponent α ∈ [1, ∞], using the L^α norm ‖·‖α in the scalarfunction space AP of almost periodic functions (see (2.9) and (2.10)), we will obtain thefollowing optimal estimates on the rotation number ρ(p) of (x|¨) + p(t)x = 0:ρ(p) ≤ C(α)(‖p+‖α)^1/2 , (1)where p+(t) denotes the non-negative part of p(t) and C(α) is some constant related withthe Sobolev constant found in [30, p. 357]. Moreover, the constant C(α) in (1) is optimal.Note that when α = ∞, (1) coincides with the well-known resultp(t) ≤ M (M ≥ 0) →ρ(p) ≤ M^1/2.3. For any given exponent α ∈ [1, ∞], any given natural number n ∈ N and any givenreal number l > 0, we define some new norm L_α,n,l(·, ·) on the two dimensional almostperiodic vector function space AP². Using these norms, we will obtain the followingresult to the asymmetric case, which is a partial generalization of (1):Again, the constant (C|⁾(α) in (2) is optimal and is related with the Sobolev constant in [30].