Existence and Uniqueness Theorems on Periodic Solutions to Multidimensional Linear Hyperbolic Equations

We make use of following notations and definitions. Z+ is the set of all nonnegative integers; Z+ is the set of all multiindices α = (α1, . . . , αn); ‖α‖ = α1 + · · ·+ αn; 0 = (0, . . . , 0) ∈ Z+. The inequalities between the multiindices α = (α1 , . . . , αn) and β = (β1, . . . , βn) are understood componentwise. It will be assumed that m > 0. If for some multiindex α = (α1, . . . , αn) we have αi1 = · · · = αik = 0 (i1 < · · · < ik), and αj1 , . . . , αjn−k > 0 (j1 < · · · < jn−k), {j1, . . . , jn−k} = {1, . . . , n} \ {i1, . . . , ik}, then by xα (by x) denote the vector (xi1 , . . . , xik ) ∈ R k (the vector (xj1 , . . . , xjn−k ) ∈ R n−k). If α > 0, then in equation (1) by pα(xα) we understand a constant function. A multiindex α ∈ Z+ will be called even, if all its components are even. A multiindex α ∈ Z+ will be called odd, if ‖α‖ is odd. By E and O, respectively, denote the sets of all even and odd multiindices not exceeding m and different from m, i.e., E = {