Vol. 62, No. 1, 1976

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Local evolution systems in general Banach spaces

Alban J. Roques

Vol. 62 (1976), No. 1, 197–217

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

u′(t) ∈ A (t)u(t), u(s) = x,

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

u′(t) ∈ A(t)u(t), u(0) = x,

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.

Mathematical Subject Classification
Primary: 47H15
Received: 28 April 1975
Published: 1 January 1976
Alban J. Roques