On solvability of boundary value problems for higher order nonlinear hyperbolic equations

In the rectangle Omega = [0, a] x [0, b] for the nonlinear hyperbolic equation u(m, n) = Sigma(m=1)(i=0) h(]i)(x)u((i, n)) + Sigma(n=1)(k=0) h(2k) (y)u((m, k)) + f(x, y, u,..., u((m-1, n-1))) the boundary value problems of the type l(]i) (u(., y)) = 0 (i = 1,..., m), l(2k) (u(x,.)) = 0 (k=1,..., n) are considered, where l(1i) : Cm-1([0, a]) -> R (i = 1, m) and l(2k) Cn-1 ([0, b]) -> R (k = 1,..., n) are linear bounded functionals. Sufficient conditions of solvability and unique solvability of the general problem and its particular cases (Nicoletti type, Dirichlet, Lidstone and Periodic problems) are established. (C) 2007 Elsevier Ltd. All rights reserved.