On the Structure of the Set of Periods for Periodic Solutions of Some Linear Integro-differential Equations on the Multidimensional Sphere

The problem of periodic solutions for the family of linear differential equations (L− λ)u ≡ (1 i ∂ ∂t − a∆− λ ) u(x, t) = νG(u− f) is considered on the multidimensional sphere x ∈ Sn under the periodicity condition u|t=0 = u|t=b. Here a and λ are given reals, ν is a fixed complex number, Gu(x, t) is a linear integral operator, and ∆ is the Laplace operator on Sn. It is shown that the set of parameters (ν, b) for which the above problem admits a unique solution is a measurable set of full measure in C× R+. §1 In [4, 5] it was discovered that, for some partial differential equations, the set of periods for which a periodic solution is unique may have an unexpectedly complicated structure. In this paper, we study this issue for a class of linear equations on the multidimensional sphere. We consider the problem of periodic solutions for the nonlocal Schrödinger type equation (1) (1 i ∂ ∂t − a∆− λ ) u(x, t) = νG(u− f) with the t-periodicity condition (2) u|t=0 = u|t=b. Here u(x, t) is a complex function on S× [0, b], where S is the multidimensional sphere, n ≥ 2; a = 0, λ, and ν are given complex numbers; f(x, t) is a given function. The change of variables t = bτ reduces our problem to a problem with a fixed period, but with a new equation in which the coefficient of the τ -derivative is equal to 1b , (1 i ∂ b ∂τ − a∆− λ ) u(x, bτ) = νG(u(x, bτ)− f(x, bτ)). §2 Thus, problem (1), (2) turns into a problem on periodic solutions of the equation (3) (L− λ)u ≡ (1 i ∂ b ∂t − a∆− λ ) u(x, t) = νG(u− f) with the fixed periodicity condition (4) u|t=0 = u|t=1. 2000 Mathematics Subject Classification. Primary 35K20.