The Abstract Cauchy Problem

Abstract

Consider the following Cauchy problem in an arbitrary Banach space E:

$$ v'\left( t \right) + Av\left( t \right) = f\left( t \right),{\text{ }}0 \leqslant t \leqslant 1,{\text{ }}v\left( 0 \right) = v_0 . $$

(1.1)

Here v(t) and f(t) are the unknown and the given function, respectively, defined on [0,1] with values in E. The derivative v′(t) is understood as the limit in the norm of E of the corresponding ratio of differences. A is a linear operator acting in E, with domain D(A). Finally, v₀ is a given element of E.