Resolvent operators for integral equations in a Banach space
Author:
R. C. Grimmer
Journal:
Trans. Amer. Math. Soc. 273 (1982), 333349
MSC:
Primary 45N05; Secondary 34G10, 45D05
MathSciNet review:
664046
Fulltext PDF Free Access
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Abstract: Conditions are given which ensure the existence of a resolvent operator for an integrodifferential equation in a Banach space. The resolvent operator is similar to an evolution operator for nonautonomous differential equations in a Banach space. As in the finite dimensional case, this operator is used to obtain a variation of parameters formula which can be used to obtain results concerning the asymptotic behaviour of solutions and weak solutions.
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 [1]
 G. Chen, Control and stabilization for the wave equation in a bounded domain, SIAM J. Control 17 (1979), 6681. MR 516857 (80b:93066)
 [2]
 G. Chen and R. Grimmer, Semigroups and integral equations, J. Integral Equations 2 (1980), 133154. MR 572484 (81f:45026)
 [3]
 , Integral equations as evolution equations, J. Differential Equations (to appear).
 [4]
 A. Friedman and M. Shinbrot, Volterra integral equations in Banach space, Trans. Amer. Math. Soc. 126 (1967), 131179. MR 0206754 (34:6571)
 [5]
 R. C. Grimmer and R. K. Miller, Existence, uniqueness and continuity for integral equations in a Banach space, J. Math. Anal. Appl. 57 (1977), 429447. MR 0440311 (55:13186)
 [6]
 , Well posedness of Volterra integral equations in Hilbert space, J. Integral Equations 1 (1979), 201216. MR 540827 (80i:45003)
 [7]
 R. C. Grimmer and G. Seifert, Stability properties of Volterra integrodifferential equations, J. Differential Equations 19 (1975), 142166. MR 0388002 (52:8839)
 [8]
 S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems, J. Differential Equations 8 (1970), 457474. MR 0270095 (42:4988)
 [9]
 M. E. Gurtin and A. C. Pipkin, A general theory of heat conduction with finite wave speeds, Arch. Rational Mech. Anal. 31 (1968), 113126. MR 1553521
 [10]
 K. B. Hannsgen, The resolvent kernel of an integrodifferential equation in Hilbert space, SIAM J. Math. Anal. 7 (1976), 481490. MR 0417861 (54:5909)
 [11]
 , Uniform behavior for an integrodifferential equation with parameter, SIAM J. Math. Anal. 8 (1977), 626639. MR 0463848 (57:3787)
 [12]
 T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo Sec. I 17 (1970), 241258. MR 0279626 (43:5347)
 [13]
 , Linear evolution equations of "hyperbolic" type. II, J. Math. Soc. Japan 25 (1973), 648666. MR 0326483 (48:4827)
 [14]
 R. K. Miller, Volterra integral equations in a Banach space, Funkcial. Ekvac. 18 (1975), 163193. MR 0410312 (53:14062)
 [15]
 , An integrodifferential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), 313332. MR 515894 (80g:45015)
 [16]
 , Nonlinear Volterra integral equations, Benjamin, Menlo Park, Calif., 1971. MR 0511193 (58:23394)
 [17]
 R. K. Miller and R. L. Wheeler, Asymptotic behavior for a linear Volterra integral equation in Hilbert space, J. Differential Equations 23 (1977), 270284. MR 0440313 (55:13188)
 [18]
 , Wellposedness and stabiltiy of linear Volterra integrodifferential equations in abstract spaces, Funkcial. Ekvac. 21 (1978), 279305. MR 540397 (80j:45017)
 [19]
 A. Pazy, Semigroups of linear operators and applications to partial differential equations, Dept. Math. Lecture Note # 10, University of Maryland, 1974.
 [20]
 A. J. Pritchard and J. Zabczyk, Stability and stabilizability of infinte dimensional systems, SIAM Rev. 23 (1981), 2552. MR 605439 (82f:93063)
 [21]
 H. Tanabe, Equations of evolution, Pittman, London, 1979. MR 533824 (82g:47032)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198206640464
PII:
S 00029947(1982)06640464
Article copyright:
© Copyright 1982
American Mathematical Society
