Some well-posed functional equations which generate semigroups

Abstract

Consider the abstract linear functional equation (FE) ( Dx)( t) = f( t) ( t ⩾ 0), x( t) = ϑ( t) ( t ⩽ 0) in a Banach space B. A theorem is proven which contains the following result as a special case. Let Y(R; B; η) be a L p-space or C 0-space on R = (−t8, ∞), with a suitable weight function η, and with values in B. Let D be a closed (unbounded) causal linear operator in Y(R; B; η), which commutes with translations. Suppose that D + λI has a continuous causal inverse for some complex λ, and that D restricted to those functions in Y(R; B; η) which vanish on R − = (−∞, 0] has a continuous causal inverse. Then (FE) generates a strongly continuous semigroup of translation type on a Banach space, which is essentially the cross product of the restriction of the domain of D to R − and Y(R +; B; η). Examples with B = C n on how the theory applies to a neutral functional differential equation, a difference equation, a Volterra integrodifferential equation (with nonintegrable kernel but integrable resolvent), and a fractional order functional differential equation are given. Also, an abstract neutral functional differential equation in a Hilbert space is studied and applications to an abstract Volterra integrodifferential equation in a Banach space are indicated.