Mixed problem for quasilinear hyperbolic system with coefficients functionally dependent on solution

The mixed problem for quasilinear hyperbolic system with coefficients functionally dependent on the solution is studied. We assume that the coefficients are continuous nonlinear operators in the Banach space C¹(ℝ) satisfying some additional assumptions. Under these assumptions we prove the uniqueness and existence of local in time C¹ solution, provided that the initial data are also of class C¹.