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On the abstract Cauchy problem for operators in locally convex spaces. (English) Zbl 1264.34005

The principal obstacle to extending strongly continuous semigroup theory (and the associated theory of abstract Cauchy problems) from normed to linear topological spaces is the loss of exponential growth at infinity and, therefore, of the basic Laplace transform relation \[ R(\lambda; A)u = \int_0^\infty e^{- \lambda t}S(t)u\, dt \eqno(1) \] connecting the semigroup \(S(t)\) and the resolvent of the infinitesimal generator \(A\). This can be seen in very simple examples such as the translation semigroup \(S(t)u(x) = u(x + t)\) in the locally convex space \(E\) of all \(u\) continuous in \((-\infty, \infty)\) with the topology generated by the family of semi-norms \(\|u\|_n = \max \{|u(x)|: |x| \leq n\}\). Not only the right side of (1) fails to make sense in general, but the left side does not exist for any \(\lambda;\) in an obvious sense, the infinitesimal generator of the semigroup is \(Au(x) = u'(x),\) but the equation \(\lambda u(x) - u'(x) = v(x)\) has multiple solutions for every \(\lambda\) and \(v(x).\) If one assumes exponential growth, which in the present context means equicontinuity of \(\{e^{- \omega t}S(t); t \geq 0 \}\) for some \(\omega,\) the Banach space results generalize to locally convex spaces as suggested by L. Schwartz and [Lectures on mixed problems in partial differential equations and the representation of semi-groups. Bombay: Tata Institute for Fundamental Research (1958)]. The details were put in place by Miyadera, Komatsu and others (see the references of this paper).
The general case (possible hyperexponential growth) was handled by T. Komura in 1968 as follows. The integral on the right of (1) is given sense as in the theory of Fourier-Laplace transforms of distributions of arbitrary growth due to Gelfand and Silov, and the left side is replaced by the generalized resolvent \(\operatorname{Re} (\lambda; A)\), a vector-valued analytic functional rather than a function. The generalized resolvent can then be approached by “approximate resolvents” \(\operatorname{Re}_n(\lambda; A)\) obtained cutting off the domain of integration in (1). In other approaches to this problem, the sequence of approximate resolvents rather than the resolvent itself is emphasized. Lack of exponential growth also occurs in Banach spaces for distribution semigroups; here, the true resolvent can be actually constructed from the approximate resolvent as shown by Chazarain in 1971 (see the references of this paper). This approach has been extended by I. Cioranescu [Math. Ann. 211, 37–46 (1974; Zbl 0276.35077)] to vector-valued convolution equations \({\mathcal F}*S = \delta \otimes I\) (in the semigroup case \({\mathcal F} = \delta' \otimes I - \delta \otimes A).\) Still in Banach spaces, hyperexponential growth can also occur in the strong solution setting for higher-order equations like \(u''(t) + Bu'(t) + Au(t) = 0\) [the reviewer, Pac. J. Math. 33, 583–615 (1970; Zbl 0181.42601)].
In this paper, the authors present a treatment of the abstract Cauchy problem that applies to Laplace hyperfunction solutions of the Cauchy problem using their previously developed theory of Laplace transforms of Laplace hyperfunctions. They apply their results to equations of population dynamics.

MSC:

34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
47A10 Spectrum, resolvent
34G10 Linear differential equations in abstract spaces
44A10 Laplace transform
47D06 One-parameter semigroups and linear evolution equations
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
46F15 Hyperfunctions, analytic functionals
32A45 Hyperfunctions
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References:

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