Smoothness property on parameters of periodic systems (Q366036)

In this paper, the author studies the differentiability with parameters for a hyperbolic type periodic system. By employing \(C_0\) semigroup theory of bounded linear operators and fixed point theory, he proves that, when the nonhomogeneous term is a periodic function with respect to the parameter, then the solution is also periodic, and it is continuously (Fréchet) differentiable with respect to the parameter. Then he further proves that the nonlinear equation has a similar property. As an application, he gives an example for the 1-dimensional wave equation with periodic boundary conditions.