A Volterra equation with nonintegrable resolvent

Author:
Gustaf Gripenberg

Journal:
Proc. Amer. Math. Soc. **73** (1979), 57-60

MSC:
Primary 45D05; Secondary 45E10

MathSciNet review:
512058

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Abstract: A counterexample, which shows that the resolvent associated with a nonnegative, nonincreasing function need not be integrable, is constructed.

**[1]**G. Gripenberg,*Integrability properties of resolvents of Volterra equations*, Report-HTKK-MAT-A117, Helsinki Univ. of Technology, 1978.**[2]**Stanley I. Grossman,*Integrability of resolvents of certain Volterra integral equations*, J. Math. Anal. Appl.**48**(1974), 785–793. MR**0352909****[3]**G. S. Jordan and Robert L. Wheeler,*A generalization of the Wiener-Lévy theorem applicable to some Volterra equations*, Proc. Amer. Math. Soc.**57**(1976), no. 1, 109–114. MR**0405023**, 10.1090/S0002-9939-1976-0405023-0**[4]**J. J. Levin,*Resolvents and bounds for linear and nonlinear Volterra equations*, Trans. Amer. Math. Soc.**228**(1977), 207–222. MR**0433162**, 10.1090/S0002-9947-1977-0433162-2**[5]**R. K. Miller,*On Volterra integral equations with nonnegative integrable resolvents*, J. Math. Anal. Appl.**22**(1968), 319–340. MR**0227707****[6]**Raymond E. A. C. Paley and Norbert Wiener,*Fourier transforms in the complex domain*, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR**1451142****[7]**Daniel F. Shea and Stephen Wainger,*Variants of the Wiener-Lévy theorem, with applications to stability problems for some Volterra integral equations*, Amer. J. Math.**97**(1975), 312–343. MR**0372521****[8]**O. J. Staffans,*On the integrability of the resolvent of a Volterra equation*, Report-HTKK-MAT-A103, Helsinki Univ. of Technology, 1977.

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DOI:
https://doi.org/10.1090/S0002-9939-1979-0512058-9

Article copyright:
© Copyright 1979
American Mathematical Society