Weak solvability of a hyperbolic integro-differential equation with integral condition

The study of boundary value problems with non-local conditions has known a great development in the recent years. This is due to the importance of nonlocal conditions appearing in the mathematical modeling of various phenomena of physics, ecology, biology, etc. It is the case when the values of function on the boundary are related to values inside the domains or when the direct measurements on the boundary are not possible. Several methods are used to solve such problems as functional methods, approximation methods, a priori estimates... The importance of approximation methods is that they don’t only prove the existence and uniqueness of the solution but they also allow the construction of algorithms for numerical solutions. These methods as the Galerkin method and the method of discretization in time also called Rothe’s method, are very effective tools in the study of the approximate solution and its convergence to the solution of problems [1-13]. In general it is difficult to find the exact solution in such cases, so the approximation methods provide other ways to find approximate solutions.