Nonlocal problem with time-dependent Steklov's boundary conditions for hyperbolic equation

In this article, the solvability of boundary-value problem for hyperbolic equation with nonlocal conditions
$$a_1(t)u_x(0,t)+a_2(t)u_x(1,t)+a_3(t)u(0,t)+a_4(t)u(1,t)=0,$$

$$b_1(t)u_x(0,t)+b_2(t)u_x(1,t)+b_3(t)u(0,t)+b_4(t)u(1,t)=0 $$
is proved. The proof is mainly based on a priori estimates and Galerkin procedure.