Differentiability of weak solutions to an abstract inhomogeneous differential equation

Travis, C. C.

doi:10.1090/S0002-9939-1981-0612734-2

Differentiability of weak solutions to an abstract inhomogeneous differential equation
HTML articles powered by AMS MathViewer

by C. C. Travis PDF

Proc. Amer. Math. Soc. 82 (1981), 425-430 Request permission

Abstract:

A characterization is given of those strongly continuous semigroups $T(t)$ for which the function $\int _0^tT(t - s)f(s)ds$ is continuously differentiable on $t > 0$ for each continuous $f:[0,\infty ) \to X$.

References

Caractère bone’ de certains générateurs de semigroupes linéaire dans les espaces de Banach

J. M. Ball, Strongly continuous semigroups, weak solutions, and the variation of constants formula, Proc. Amer. Math. Soc. 63 (1977), no. 2, 370–373. MR 442748, DOI 10.1090/S0002-9939-1977-0442748-6
M. G. Crandall and A. Pazy, On the differentiability of weak solutions of a differential equation in Banach space, J. Math. Mech. 18 (1968/1969), 1007–1016. MR 0242014
Chaim Samuel Hönig, The Green function of a linear differential equation with a lateral condition, Bull. Amer. Math. Soc. 79 (1973), 587–593. MR 312019, DOI 10.1090/S0002-9904-1973-13214-8
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Amnon Pazy, Semi-groups of linear operators and applications to partial differential equations, University of Maryland, Department of Mathematics, College Park, Md., 1974. Department of Mathematics, University of Maryland, Lecture Note, No. 10. MR 0512912
R. S. Phillips, Perturbation theory for semi-groups of linear operators, Trans. Amer. Math. Soc. 74 (1953), 199–221. MR 54167, DOI 10.1090/S0002-9947-1953-0054167-3
G. F. Webb, Regularity of solutions to an abstract inhomogeneous linear differential equation, Proc. Amer. Math. Soc. 62 (1977), no. 2, 271–277. MR 432996, DOI 10.1090/S0002-9939-1977-0432996-3