Product integrals for an ordinary differential equation in a Banach space

David Lovelady
1973 Pacific Journal of Mathematics  
Let Y be a Banach space with norm | |, and let R + be the interval [0, oo). Let A be a function on R + having the properties that if t is in R+ then A(t) is a function from Y to Y and that the function from R + X Y to Y described by it, x) -» A(£)M is continuous. Suppose there is a continuous real-valued function a on R + such that if t is in R + then A(t) -cx(t)I is dissipative. Now it is known that if z is in Y, the differential equation u'(t) = A(t)[u(t)]; u(θ) = z has exactly one solution
more » ... R + . It is shown in this paper that if t is in R+ then u(t) = 0 IP exp [(ds)A(s)] [z] = 0 IP[I-(ώ) AC*)]-1^] , where the exponentials are defined by the solutions of the associated family of autonomous equations.
doi:10.2140/pjm.1973.48.163 fatcat:gbdsnbjcovfmbefhfh3lrfkowu