Singular perturbations of integrodifferential equations in Banach space

Author:
James H. Liu

Journal:
Proc. Amer. Math. Soc. **122** (1994), 791-799

MSC:
Primary 45M05; Secondary 34E15, 34K30

MathSciNet review:
1287101

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let and consider

*X*when . Here

*A*is the generator of a strongly continuous cosine family and a strongly continuous semigroup, and is a bounded linear operator for . With some convergence conditions on initial data and and smoothness conditions on , we prove that if , then in

*X*uniformly for for any fixed . We will apply this to an equation in viscoelasticity.

**[1]**G. W. Desch and R. Grimmer,*Propagation of singularities for integro-differential equations*, J. Differential Equations**65**(1986), no. 3, 411–426. MR**865070**, 10.1016/0022-0396(86)90027-6**[2]**W. Desch, R. Grimmer, and W. Schappacher,*Propagation of singularities by solutions of second order integrodifferential equations*, Volterra integrodifferential equations in Banach spaces and applications (Trento, 1987) Pitman Res. Notes Math. Ser., vol. 190, Longman Sci. Tech., Harlow, 1989, pp. 101–110. MR**1018875****[3]**Wolfgang Desch, Ronald Grimmer, and Wilhelm Schappacher,*Some considerations for linear integro-differential equations*, J. Math. Anal. Appl.**104**(1984), no. 1, 219–234. MR**765053**, 10.1016/0022-247X(84)90044-1**[4]**W. Desch and W. Schappacher,*A semigroup approach to integro-differential equations in Banach spaces*, J. Integral Equations**10**(1985), no. 1-3, suppl., 99–110. Integro-differential evolution equations and applications (Trento, 1984). MR**831237****[5]**H. O. Fattorini,*Second order linear differential equations in Banach spaces*, North-Holland Mathematics Studies, vol. 108, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 99. MR**797071****[6]**Jerome A. Goldstein,*Semigroups of linear operators and applications*, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR**790497****[7]**Ronald Grimmer and He Tao Liu,*Integrodifferential equations with non-densely defined operators*, Differential equations with applications in biology, physics, and engineering (Leibnitz, 1989) Lecture Notes in Pure and Appl. Math., vol. 133, Dekker, New York, 1991, pp. 185–199. MR**1171469****[8]**Ronald Grimmer and James H. Liu,*Singular perturbations in viscoelasticity*, Rocky Mountain J. Math.**24**(1994), no. 1, 61–75. 20th Midwest ODE Meeting (Iowa City, IA, 1991). MR**1270027**, 10.1216/rmjm/1181072452**[9]**Jack K. Hale,*Ordinary differential equations*, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1969. Pure and Applied Mathematics, Vol. XXI. MR**0419901****[10]**Jack K. Hale and Geneviève Raugel,*Upper semicontinuity of the attractor for a singularly perturbed hyperbolic equation*, J. Differential Equations**73**(1988), no. 2, 197–214. MR**943939**, 10.1016/0022-0396(88)90104-0**[11]**R. C. MacCamy,*An integro-differential equation with application in heat flow*, Quart. Appl. Math.**35**(1977/78), no. 1, 1–19. MR**0452184****[12]**R. C. MacCamy,*A model for one-dimensional, nonlinear viscoelasticity*, Quart. Appl. Math.**35**(1977/78), no. 1, 21–33. MR**0478939****[13]**Donald R. Smith,*Singular-perturbation theory*, Cambridge University Press, Cambridge, 1985. An introduction with applications. MR**812466****[14]**Kunio Tsuruta,*Bounded linear operators satisfying second-order integro-differential equations in a Banach space*, J. Integral Equations**6**(1984), no. 3, 231–268. MR**738871****[15]**C. C. Travis and G. F. Webb,*An abstract second-order semilinear Volterra integro-differential equation*, SIAM J. Math. Anal.**10**(1979), no. 2, 412–424. MR**523855**, 10.1137/0510038

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
45M05,
34E15,
34K30

Retrieve articles in all journals with MSC: 45M05, 34E15, 34K30

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1994-1287101-0

Keywords:
Singular perturbation,
convergence in solutions

Article copyright:
© Copyright 1994
American Mathematical Society