zbMATH — the first resource for mathematics

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)

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
Full Text: DOI EuDML
[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.