An abstract semilinear Volterra integrodifferential equation
Author:
G. F. Webb
Journal:
Proc. Amer. Math. Soc. 69 (1978), 255260
MSC:
Primary 45K05
MathSciNet review:
0467214
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Abstract: The abstract semilinear Volterra integrodifferential equation is investigated, where A is the infinitesimal generator of a semigroup of linear operators in a Banach space X and g is nonlinear and unbounded in its second place. Some results are proved concerning local existence, global existence, continuous dependence upon initial values, and asymptotic stability. The method used treats the equation in the domain of A with the graph norm employing results from linear semigroup theory concerning abstract inhomogeneous linear differential equations.
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 C. M. Dafermos, An abstract Volterra equation with applications to linear viscoelasticity, J. Differential Equations 7 (1970), 554569. MR 0259670 (41:4305)
 [7]
 A. Friedman, Monotonicity of solutions of Volterra integral equations in Banach space, Trans. Amer. Math. Soc. 138 (1969), 129148. MR 0242024 (39:3359)
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 A. Friedman and M. Shinbrot, Volterra integral equations in Banach space, Trans. Amer. Math. Soc. 126 (1967), 131179. MR 0206754 (34:6571)
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 T. Kato, Perturbation theory for linear operators, SpringerVerlag, New York, 1966. MR 0203473 (34:3324)
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 S.O. Londen, An existence result on a Volterra equation in a Banach space, Trans. Amer. Math. Soc. (to appear). MR 0473770 (57:13432)
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 , On an integral equation in a Hilbert space, SIAM J. Math. Anal. (to appear). MR 0511229 (58:23401)
 [12]
 R. C. MacCamy, Stability theorems for a class of functional differential equations, SIAM J. Math. Anal. (to appear). MR 0404818 (53:8618)
 [13]
 , An integrodifferential equation with applications in heat flow (to appear).
 [14]
 R. C. MacCamy and J. S. W. Wong, Stability theorems for some functional differential equations, Trans. Amer. Math. Soc. 164 (1972), 137. MR 0293355 (45:2432)
 [15]
 R. K. Miller, Volterra integral equations in a Banach space, Funkcial. Ekvac. 18 (1975), 163194. MR 0410312 (53:14062)
 [16]
 M. Slemrod, A hereditary partial differential equation with applications in the theory of simple fluids, Arch. Rational Mech. Anal. (to appear). MR 0416245 (54:4320)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197804672144
PII:
S 00029939(1978)04672144
Keywords:
Abstract Volterra integrodifferential equation,
semigroup of bounded linear operators,
infinitesimal generator,
existence,
uniqueness,
asymptotic behavior
Article copyright:
© Copyright 1978 American Mathematical Society
