Regularity of solutions to an inhomogeneous differential equation in Banach space
Let T(t), t > or = 0, be a strongly continuous semigroup of bounded linear operators in the Banach space X with infinitesimal generator, and let f be a continuous X-valued function on (0, infinity). It is well known that, without some restrictions on the semigroup T(t), t > or = 0, or the continuous function f, the weak solution v(t) = T(t)x + ..integral../sub 0//sup t/ T(t-s)f(s)ds (eq. A), need not be a strong solution of the inhomogeneous linear differential equation du(t)/dt = Au(t) + f(t), u(0) = x (eq. B). The purpose of this work is to characterize the class of strongly continuous semigroups for which a weak solution of eq. B is a strong solution when f is an element of C((o,r);X). It is shown that eq. A is a strong solution of eq. B for every continuous function f iff the semigroup T(t), t > or = 0, is of bounded semivariation. (RWR)
- Research Organization:
- Oak Ridge National Lab., TN (USA)
- Sponsoring Organization:
- USDOE
- DOE Contract Number:
- W-7405-ENG-26
- OSTI ID:
- 6130617
- Report Number(s):
- CONF-7906102-1
- Resource Relation:
- Conference: Functional differential and integral equations conference, Morgantown, WV, USA, 18 Jun 1979
- Country of Publication:
- United States
- Language:
- English
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