Neubrander, Frank Well-posedness of abstract Cauchy problems. (English) Zbl 0542.34053 Semigroup Forum 29, 75-85 (1984). In this paper a unified treatment of the Hille-Phillips-Fattorini results on well-posedness of the abstract Cauchy problem for \(u'(t)=Au(t)\) and \(u''(t)=Au(t)\) is presented. Namely, we give six equivalent definitions of well-posedness for the first and for the second order equation. The proofs of their equivalence to each other and to A being a semigroup or cosine family generator appear to be simpler than any of the other proofs in the literature. In the author’s dissertation (Tübingen, 1984) these results have been extended to the Cauchy problem for \[ u^{(n+1)}(t)- Au^{(n)}(t)-B_ 1u^{(n-1)}(t)-...-B_ nu(t)=0, \] where A is a semigroup generator and \(B_ 1,...,B_ n\) are closed operators whose domains contain the domain of A. Cited in 1 ReviewCited in 7 Documents MSC: 34G10 Linear differential equations in abstract spaces 47D03 Groups and semigroups of linear operators Keywords:semigroups of operators; cosine families of operators; abstract Cauchy problem PDFBibTeX XMLCite \textit{F. Neubrander}, Semigroup Forum 29, 75--85 (1984; Zbl 0542.34053) Full Text: DOI EuDML References: [1] Da Prato, G. and E. Giusti, Una catterizzatione dei generatori di funzioni coseno astratto, Boll. Unione Mat. Ital. 22 (1967), 357–362. · Zbl 0186.47702 [2] Fattorini, H. O., Ordinary differential equations in linear topological spaces, I, J. Diff. Equ. 5 (1968), 72–105. · Zbl 0175.15101 · doi:10.1016/0022-0396(69)90105-3 [3] Fattorini, H. O., The Abstract Cauchy Problem, Addison-Wesley, Reading, Mass., 1983. · Zbl 0493.34005 [4] Goldstein, J. A., Semigroups of Operators and Abstract Cauchy Problems. Tulane University Lecture Notes, New Orleans, 1970. · Zbl 0219.47037 [5] Goldstein, J. A., Semigroups of Operators and Applications, in preparation. [6] Hille, E., Une généralisation du problème de Cauchy. Ann. Inst. Fourier (Grenoble) 4 (1952), 31–48. · Zbl 0055.34503 [7] Hille, E., and R. S. Phillips, Functional Analysis and Semi-Groups, Amer. Math. Soc. Colloq. Publ. Vol. 31, Providence, R. I., 1957. · Zbl 0078.10004 [8] Kisynski, Y., On cosine operator functions and one-parameter groups of operators. Studia Math. 44 (1972) 93–105. · Zbl 0232.47045 [9] Lutz, D., Strongly continuous operator cosine functions, in: Functional Analysis, Proceedings, Dubrovnik 1981, Lecture Notes in Mathematics 948, Springer-Verlag Berlin-Heidelberg-New York, 1982. [10] Phillips, R. S., A note on the abstract Cauchy problem, Proc. Nat. Acad. Sci. U.S.A. 40 (1954), 244–248. · Zbl 0058.10603 · doi:10.1073/pnas.40.4.244 [11] Sandefur, J. T., Higher order abstract Cauchy problems, J. Math. Anal. Appl. 60 (1977), 728–742. · Zbl 0358.35068 · doi:10.1016/0022-247X(77)90012-9 [12] Sova, M., Cosine operator functions, Rozprawy Math. 49 (1966), 1–46. [13] Travis, C. C., and G. F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Sci. Hung. 32 (1978), 75–96. · Zbl 0388.34039 · doi:10.1007/BF01902205 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.