Semigroup Theory

The objective of this chapter is to establish that, in an appropriate functional setting, there is a unique solution of the Cauchy problem \(\displaystyle {} \dot {y}(t) = Ay(t)+f(t) , \qquad y(0) = y_0 , \) where A is a linear operator on a Banach space X, and where \(y(t)\) and \(f(t)\) evolve in X, which is given by \(\displaystyle {} y(t) = S(t) y_0 + \int _0^t S(t-s)f(s)\, ds \) where \((S(t))_{t\geqslant 0}\) is the semigroup generated by the operator A.