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
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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]
Goong
Chen, Control and stabilization for the wave equation in a bounded
domain, SIAM J. Control Optim. 17 (1979), no. 1,
66–81. MR
516857 (80b:93066), http://dx.doi.org/10.1137/0317007
 [2]
Goong
Chen and Ronald
Grimmer, Semigroups and integral equations, J. Integral
Equations 2 (1980), no. 2, 133–154. MR 572484
(81f:45026)
 [3]
, Integral equations as evolution equations, J. Differential Equations (to appear).
 [4]
Avner
Friedman and Marvin
Shinbrot, Volterra integral equations in Banach
space, Trans. Amer. Math. Soc. 126 (1967), 131–179. MR 0206754
(34 #6571), http://dx.doi.org/10.1090/S00029947196702067547
 [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), no. 2, 429–447. MR 0440311
(55 #13186)
 [6]
R.
C. Grimmer and R.
K. Miller, Wellposedness of Volterra integral equations in Hilbert
space, J. Integral Equations 1 (1979), no. 3,
201–216. MR
540827 (80i:45003)
 [7]
Ronald
Grimmer and George
Seifert, Stability properties of Volterra integrodifferential
equations, J. Differential Equations 19 (1975),
no. 1, 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]
Morton
E. Gurtin and A.
C. Pipkin, A general theory of heat conduction with finite wave
speeds, Arch. Rational Mech. Anal. 31 (1968),
no. 2, 113–126. MR
1553521, http://dx.doi.org/10.1007/BF00281373
 [10]
Kenneth
B. Hannsgen, The resolvent kernel of an integrodifferential
equation in Hilbert space, SIAM J. Math. Anal. 7
(1976), no. 4, 481–490. MR 0417861
(54 #5909)
 [11]
Kenneth
B. Hannsgen, Uniform 𝐿¹ behavior for an
integrodifferential equation with parameter, SIAM J. Math. Anal.
8 (1977), no. 4, 626–639. MR 0463848
(57 #3787)
 [12]
Tosio
Kato, Linear evolution equations of “hyperbolic”
type, J. Fac. Sci. Univ. Tokyo Sect. I 17 (1970),
241–258. MR 0279626
(43 #5347)
 [13]
Tosio
Kato, Linear evolution equations of “hyperbolic” type.
II, J. Math. Soc. Japan 25 (1973), 648–666. MR 0326483
(48 #4827)
 [14]
Richard
K. Miller, Volterra integral equations in a Banach space,
Funkcial. Ekvac. 18 (1975), no. 2, 163–193. MR 0410312
(53 #14062)
 [15]
R.
K. Miller, An integrodifferential equation for rigid heat
conductors with memory, J. Math. Anal. Appl. 66
(1978), no. 2, 313–332. MR 515894
(80g:45015), http://dx.doi.org/10.1016/0022247X(78)902342
 [16]
Richard
K. Miller, Nonlinear Volterra integral equations, W. A.
Benjamin, Inc., Menlo Park, Calif., 1971. Mathematics Lecture Note Series.
MR
0511193 (58 #23394)
 [17]
Richard
K. Miller and Robert
L. Wheeler, Asymptotic behavior for a linear Volterra integral
equation in Hilbert space, J. Differential Equations
23 (1977), no. 2, 270–284. MR 0440313
(55 #13188)
 [18]
Richard
K. Miller and Robert
L. Wheeler, Wellposedness and stability of linear Volterra
integrodifferential equations in abstract spaces, Funkcial. Ekvac.
21 (1978), no. 3, 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 infinitedimensional
systems, SIAM Rev. 23 (1981), no. 1,
25–52. MR
605439 (82f:93063), http://dx.doi.org/10.1137/1023003
 [21]
Hiroki
Tanabe, Equations of evolution, Monographs and Studies in
Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston,
Mass., 1979. Translated from the Japanese by N. Mugibayashi and H. Haneda.
MR 533824
(82g:47032)
 [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
