A Nonlinear Neumann-Type Problem of a System of High Order Hyperbolic Integro-Differential Equations

Neumann and mixed boundary value problems for second order hyperbolic equations and systems have been dealt with in many papers (cp. [6, 11, 16 19, 221 and the references therein). Papers devoted to higher order equations were not so numerous and, except paper [9] where the right-hand side of the equation may depend on the unknown function but not on its derivatives, concerned only linear problems (cp. [1, 2, 5, 7 10, 12 15, 21]). In most of these papers the domain considered is a half-space. In this paper we examine a nonlinear Neumann-type problem for a system of hyperbolic integro-differential equations of order 2p (where p is any positive integer) with two independent variables. The method of treating the problem is different from those in the quoted papers and similar to that in bur paper [4] we reduce the problem to a system of nonlinear integro-functional equations, via an auxiliary boundary value problem analogous to that in [201, and hence prove the existence of a local solution by using the Banach fixed point theorem. To the best of our knowledge, the problem in question has not been examined so far.