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.

**[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**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