Existence of the Vejvoda-Shtedry spaces

Abstract

We investigate the linear periodic problem u tt - Uxx = F(x, t ), u (x + 2~, t) = u (x, t + T) = u (x, t), (x, t)a IR 2, and establish conditions for the existence of its classical solution in spaces that are subspaces of the Vejvoda-Shtedry spaces.We investigate the solvability of the linear problem 2re-periodic in a variable x and T-periodic in a variable t for a linear hyperbolic equation of the second order. Existence of a SolutionIn many works (see, e.g., [1,2]), a boundary-value periodic problemis considered and special spaces of functions are indicated in which this problem can be solved. In [2], it is shown that problem (1)-(3) can have a unique solution only in three spaces A l, A 2, and A 3 of functions which correspond to theperiods TI =(2p-1)rc/q, T:=4rcp/(2s-1), and T3 =2(2p-1)Tz/(2s-l), p ~ 7~, q,s~ ~. Let us prove that, in subspaces A ~ of the spaces A k, there exists a classical solution of the periodic problemu (x + 2rC, t) = u(x,t), u(x,t+T) = u(x,t),and establish analytic conditions of the appearance of the subspaces A ~ To study the existence of a solution of problem (4)- (6), we consider the operator (5) ix i (RF)(x,t,b) =-~ d~ {F(~,t+~-rl)+F(~,t-~+rl)}dr 1 oTemopol' Pedagogical Institute, Ternopol'.