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Resolvent operators for integral equations in a Banach space


Author: R. C. Grimmer
Journal: Trans. Amer. Math. Soc. 273 (1982), 333-349
MSC: Primary 45N05; Secondary 34G10, 45D05
DOI: https://doi.org/10.1090/S0002-9947-1982-0664046-4
MathSciNet review: 664046
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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), 66-81. MR 516857 (80b:93066)
  • [2] G. Chen and R. Grimmer, Semigroups and integral equations, J. Integral Equations 2 (1980), 133-154. 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), 131-179. 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), 429-447. MR 0440311 (55:13186)
  • [6] -, Well posedness of Volterra integral equations in Hilbert space, J. Integral Equations 1 (1979), 201-216. MR 540827 (80i:45003)
  • [7] R. C. Grimmer and G. Seifert, Stability properties of Volterra integrodifferential equations, J. Differential Equations 19 (1975), 142-166. MR 0388002 (52:8839)
  • [8] S. I. Grossman and R. K. Miller, Perturbation theory for Volterra integrodifferential systems, J. Differential Equations 8 (1970), 457-474. 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), 113-126. MR 1553521
  • [10] K. B. Hannsgen, The resolvent kernel of an integrodifferential equation in Hilbert space, SIAM J. Math. Anal. 7 (1976), 481-490. MR 0417861 (54:5909)
  • [11] -, Uniform $ {L^1}$ behavior for an integrodifferential equation with parameter, SIAM J. Math. Anal. 8 (1977), 626-639. MR 0463848 (57:3787)
  • [12] T. Kato, Linear evolution equations of "hyperbolic" type, J. Fac. Sci. Univ. Tokyo Sec. I 17 (1970), 241-258. MR 0279626 (43:5347)
  • [13] -, Linear evolution equations of "hyperbolic" type. II, J. Math. Soc. Japan 25 (1973), 648-666. MR 0326483 (48:4827)
  • [14] R. K. Miller, Volterra integral equations in a Banach space, Funkcial. Ekvac. 18 (1975), 163-193. MR 0410312 (53:14062)
  • [15] -, An integrodifferential equation for rigid heat conductors with memory, J. Math. Anal. Appl. 66 (1978), 313-332. 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), 270-284. MR 0440313 (55:13188)
  • [18] -, Well-posedness and stabiltiy of linear Volterra integrodifferential equations in abstract spaces, Funkcial. Ekvac. 21 (1978), 279-305. 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), 25-52. MR 605439 (82f:93063)
  • [21] H. Tanabe, Equations of evolution, Pittman, London, 1979. MR 533824 (82g:47032)

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DOI: https://doi.org/10.1090/S0002-9947-1982-0664046-4
Article copyright: © Copyright 1982 American Mathematical Society

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