A linear periodic boundary-value problem for a second-order hyperbolic equation

Abstract

We study the boundary-value problemu _tt -u _xx =g(x, t),u(0,t) =u (π,t) = 0,u(x, t +T) =u(x, t), 0 ≤x ≤ π,t ∈ ℝ. We findexact classical solutions of this problem in three Vejvoda-Shtedry spaces, namely, in the classes of$\frac{\pi }{q} - , \frac{{2\pi }}{{2s - 1}} -$, and$\frac{{4\pi }}{{2s - 1}}$-periodic functions (q and s are natural numbers). We obtain the results only for sets of periods$T_1 = (2p - 1)\frac{\pi }{q}, T_2 = (2p - 1)\frac{{2\pi }}{{2s - 1}}$, and$T_3 = (2p - 1)\frac{{4\pi }}{{2s - 1}}$ which characterize the classes of π-, 2π -, and 4π-periodic functions.