In the bounded cylinder $Q=\Omega\times[0,T]$ with arbitrary fixed $T>0$ the mixed problem with Dirichlet boundary conditions is considered for the quasilinear hyperbolic equation $$ u_{tt}+(-1)^m\cdot a\biggl(\int_\Omega|\nabla^mu|^2\,dx\biggr)\cdot\Delta^mu=f. $$ A particular class of functions is introduced in which there is an existence and uniqueness theorem for solutions of this problem. A theorem on the unique solvability of the Cauchy problem for a certain nonlinear differential equation in Hilbert space is first proved. This problem is a very simple abstract analogue of the indicated mixed problem for the quasilinear hyperbolic equation. Bibliography: 2 titles.