On a problem with time-nonlocal conditions for a system of equations of hyperbolic type

In the domain Ω={(x,t):0<x<l, 0<t<T} we consider a mixed problem with time-nonlocal conditions for a system of equations of the form $$\frac{{\partial u_i }}{{\partial _t }} - \lambda _i (x,t)\frac{{\partial u_i }}{{\partial x}} = fi(x,t,u),{\text{ }}i = \overline {1,n.} $$ On the bases of the method of characteristics and the method of contraction mappings we prove an existence and uniqueness theorem for the solution of this problem.