Local analyticity in weighted spaces and applications to stability problems for Volterra equations
Authors:
G. S. Jordan, Olof J. Staffans and Robert L. Wheeler
Journal:
Trans. Amer. Math. Soc. 274 (1982), 749782
MSC:
Primary 45M05; Secondary 46J99
MathSciNet review:
675078
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Abstract: We study the qualitative properties of the solutions of linear convolution equations such as and . We are especially concerned with finding conditions which ensure that these equations have resolvents which belong to, or are determined up to a term belonging to, certain weighted spaces. Our results are obtained as consequences of more general Banach algebra results on functions that are locally analytic with respect to the elements of a weighted space. In particular, we derive a proposition of WienerLévy type for weighted spaces which underlies all subsequent results. Our method applies equally well to equations more general than those mentioned above. We unify and sharpen the results of several recent papers on the asymptotic behavior of Volterra convolution equations of the types mentioned above, and indicate how many of them can be extended to the Fredholm case. In addition, we give necessary and sufficient conditions on the perturbation term for the existence of bounded or integrable solutions in some critical cases when the corresponding limit equations have nontrivial solutions.
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 W. F. Donoghue, Jr., Distributions and Fourier transforms, Academic Press, New York, 1969.
 [2]
 I. M. Gelfand, Über absolut konvergente trigonometrische Reihen und Integrale, Mat. Sb. 9 (1941), 5166. MR 0004727 (3:51g)
 [3]
 I. M. Gelfand, D. A. Raikov and G. E. Shilov, Commutative normed rings, Chelsea, New York, 1964.
 [4]
 G. Gripenberg, On the asymptotic behavior of resolvents of Volterra equations, SIAM J. Math. Anal. 11 (1980), 654662. MR 579557 (81h:45003)
 [5]
 , Integrability of resolvents of systems of Volterra equations, SIAM J. Math. Anal. 12 (1981), 585594. MR 617717 (84a:45005)
 [6]
 , Decay estimates for resolvents of Volterra equations, J. Math. Anal. Appl. 85 (1982), 473487. MR 649187 (83d:45004)
 [7]
 K. B. Hannsgen, A Volterra equation with completely monotonic convolution kernel, J. Math. Anal. Appl. 31 (1970), 459471. MR 0265897 (42:806)
 [8]
 , A WienerLévy Theorem for quotients, with applications to Volterra equations, Indiana Univ. Math. J. 29 (1980), 103120. MR 554820 (81a:45001)
 [9]
 E. Hille and R. S. Phillips, Functional analysis and semigroups, rev. ed., Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R. I., 1957. MR 0089373 (19:664d)
 [10]
 G. S. Jordan and R. L. Wheeler, Asymptotic behavior of unbounded solutions of linear Volterra integral equations, J. Math. Anal. Appl. 55 (1976), 596615. MR 0425557 (54:13511)
 [11]
 , A generalization of the WienerLévy Theorem applicable to some Volterra equations, Proc. Amer. Math. Soc. 57 (1976), 109114. MR 0405023 (53:8819)
 [12]
 , Rates of decay of resolvents of Volterra equations with certain nonintegrable kernels, J. Integral Equations 2 (1980), 103110. MR 572481 (81d:45003)
 [13]
 , Weighted remainder theorems for resolvents of Volterra equations, SIAM J. Math. Anal. 11 (1980), 885900. MR 586916 (81j:45003)
 [14]
 R. K. Miller, Structure of solutions of unstable linear Volterra integrodifferential equations, J. Differential Equations 15 (1974), 129157. MR 0350351 (50:2844)
 [15]
 R. E. A. C. Paley and N. Wiener, Fourier transforms in the complex domain, Amer. Math. Soc. Colloq. Publ., vol. 19, Amer. Math. Soc., Providence, R. I., 1934. MR 1451142 (98a:01023)
 [16]
 D. F. Shea and S. Wainger, Variants of the WienerLévy Theorem, with applications to stability problems for some Volterra integral equations, Amer. J. Math. 97 (1975), 312343. MR 0372521 (51:8728)
 [17]
 O. J. Staffans, On asymptotically almost periodic solutions of a convolution equation, Trans. Amer. Math. Soc. 266 (1981), 603616. MR 617554 (83b:46056)
 [18]
 J. S. W. Wong and R. Wong, Asymptotic solutions of linear Volterra integral equations with singular kernels, Trans. Amer. Math. Soc. 189 (1974), 185200. MR 0338718 (49:3482)
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DOI:
http://dx.doi.org/10.1090/S00029947198206750784
PII:
S 00029947(1982)06750784
Article copyright:
© Copyright 1982 American Mathematical Society
