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Integrated semigroups amd integrodifferential equations. (English) Zbl 0788.45011
The authors prove some existence, uniqueness and continuous dependence results for two classes of integro-differential equations in a Banach space by using the theory of nondegenerate, locally Lipschitz continuous integrated semigroups. Furthermore, they apply the abstract results to the study of some specific problems in one-dimensional viscoelasticity.
Reviewer: I.Vrabie (Iaşi)

MSC:
45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
74Hxx Dynamical problems in solid mechanics
47D06 One-parameter semigroups and linear evolution equations
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References:
[1] Arendt, W.,Vector valued Laplace transforms and Cauchy problems, Israel J. Math.59 (1987), 327–352. · Zbl 0637.44001 · doi:10.1007/BF02774144
[2] DaPrato, G., and Sinestrari, E.,Differential operatros with non-dense domain, Ann. Scuola Norm. Sup. Pisa14 (2) (1987), 285–344.
[3] Desch, W., Grimmer, R., and Schappacher, W.,Some considerations for linear integrodifferential equations, J. Math. Anal. Appl.104 (1984), 219–234. · Zbl 0595.45027 · doi:10.1016/0022-247X(84)90044-1
[4] Desch, W., Grimmer, R., and Schappacher, W.,Wellposedness and Wave Propagation for a Class of Integrodifferential Equations in Banach Space, J. Diff. Eq.,74 (1988), 391–411. · Zbl 0663.45008 · doi:10.1016/0022-0396(88)90011-3
[5] Goldstein, J.,Semigroups of Linear Operators and Applications, Oxford University Press, New York, 1985, p. 83. · Zbl 0592.47034
[6] Grimmer, R., and Liu, J.,Integrodifferential equations with non-densely defined operators, Differential Equations with Applications in Biology, Physics and Engineering, J. Goldstein, F. Kappel and W. Schappacher (eds.), Marcel Dekker Inc., 1991, 185–199. · Zbl 0745.45005
[7] Grimmer, R., and Sinestrari, E.,Maximum Norm in One-dimensional Hyperbolic Problems, Diff. & Integ. Eq.,5 (1992), 421–432. · Zbl 0782.47037
[8] Kellerman, H., and Hieber, M.,Integrated semigroups, J. Funct. Anal.,84, (1989), 160–180. · Zbl 0689.47014 · doi:10.1016/0022-1236(89)90116-X
[9] Miller, R.,Volterra integral equations in a Banach space, Funkcial. Ekvac.,18 (1975), 163–193. · Zbl 0326.45007
[10] Thieme, H.,”Integrated semigroups” and Integrated Solutions to Abstract Cauchy Problems, J. Math. anal. Appl.,152 (1990), 416–447. · Zbl 0738.47037 · doi:10.1016/0022-247X(90)90074-P
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