The unique solvability of the doubly periodic problem for a system of nonlinear telegraph partial differential equations (Q5951220)

The author considers the solvability of the doubly periodic problem \[ u_{tt}- u_{xx}=Au+ Bu_t+f(t,x,u, u_t,u_x),\quad u\in\mathbb{R}^m, \] \[ u(t+ \omega,x)= u(t,x)= u(t,x+\omega), \quad u(t,-x)=-u(t,x), \] where \(A\) and \(B\) are square matrices, \(B\) is nondegenerate, and the vector-valued function \(f(t,x,u, p,q)\) is assumed to be continuous in \(t\) and \(x\) and sufficiently smooth in the remaining variables. By using an equivalent system of integral equations for the above problem, he studies the existence of the generalized solutions for the problem, extending the result by \textit{D. K. Lika} and \textit{Yu. A. Ryabov} [Differ. Equations 12 (1976), 463-471 (1977; Zbl 0365.34016)]. It is a very interesting and beautiful result.