A local evolution system
{U(t,s)} is defined and constructed from a family of nonlinear, multi-valued
operators {A(t)} with common domain D, in a real Banach space X. In particular, it
is shown that there is a family of operators {U(t,s)} with domains {D(t,s)}
satisfying:
U(t,s) : D(t,s) →D,
D ⊂⋃s<tD(t,s) for each s,
D(t,r) ⊂ D(s,r) for r ≦ s ≦ t,
U(t,t)x = x for x ∈ D(t,t) ⊃ D, and
U(s,r)D(t,r) ⊂ D(t,s) and U(t,s)U(s,r) ⊃ U(t,r).
The existence of {U(t,s)} is established by showing that lim∏st(I − ΔtiA(ti))−1x
exists for x ∈ D, where “lim” denotes the refinement limit. When this limit exists it
is called the product integral, and U(t,s)x is defined to be this product
integral.
The time dependent evolution equation
is also studied, and it is shown that when X∗ is uniformly convex, a strong solution
exists on [s,T]. Finally, the notion of a solution of
with respect to {Dn} is defined, where {Dn} is a non-decreasing sequence of sets
whose union is D. Such solutions are shown to be unique, and an existence theorem
is proved in the case when X∗ is uniformly convex.
