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), 749-782

MSC:
Primary 45M05; Secondary 46J99

MathSciNet review:
675078

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 Wiener-Lévy 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.

**[1]**W. F. Donoghue, Jr.,*Distributions and Fourier transforms*, Academic Press, New York, 1969.**[2]**I. Gelfand,*Über absolut konvergente trigonometrische Reihen und Integrale*, Rec. Math. [Mat. Sbornik] N. S.**9 (51)**(1941), 51–66 (German, with Russian summary). MR**0004727****[3]**I. M. Gelfand, D. A. Raikov and G. E. Shilov,*Commutative normed rings*, Chelsea, New York, 1964.**[4]**Gustaf Gripenberg,*On the asymptotic behavior of resolvents of Volterra equations*, SIAM J. Math. Anal.**11**(1980), no. 4, 654–662. MR**579557**, 10.1137/0511060**[5]**Gustaf Gripenberg,*Integrability of resolvents of systems of Volterra equations*, SIAM J. Math. Anal.**12**(1981), no. 4, 585–594. MR**617717**, 10.1137/0512051**[6]**Gustaf Gripenberg,*Decay estimates for resolvents of Volterra equations*, J. Math. Anal. Appl.**85**(1982), no. 2, 473–487. MR**649187**, 10.1016/0022-247X(82)90013-0**[7]**Kenneth B. Hannsgen,*A Volterra equation with completely monotonic convolution kernel*, J. Math. Anal. Appl.**31**(1970), 459–471. MR**0265897****[8]**Kenneth B. Hannsgen,*A Wiener-Lévy theorem for quotients, with applications to Volterra equations*, Indiana Univ. Math. J.**29**(1980), no. 1, 103–120. MR**554820**, 10.1512/iumj.1980.29.29008**[9]**Einar Hille and Ralph S. Phillips,*Functional analysis and semi-groups*, American Mathematical Society Colloquium Publications, vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR**0089373****[10]**G. S. Jordan and Robert L. Wheeler,*Asymptotic behavior of unbounded solutions of linear Volterra integral equations*, J. Math. Anal. Appl.**55**(1976), no. 3, 596–915. MR**0425557****[11]**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**[12]**G. S. Jordan and Robert L. Wheeler,*Rates of decay of resolvents of Volterra equations with certain nonintegrable kernels*, J. Integral Equations**2**(1980), no. 2, 103–110. MR**572481****[13]**G. S. Jordan and Robert L. Wheeler,*Weighted 𝐿¹-remainder theorems for resolvents of Volterra equations*, SIAM J. Math. Anal.**11**(1980), no. 5, 885–900. MR**586916**, 10.1137/0511079**[14]**R. K. Miller,*Structure of solutions of unstable linear Volterra integrodifferential equations*, J. Differential Equations**15**(1974), 129–157. MR**0350351****[15]**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****[16]**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****[17]**Olof J. Staffans,*On asymptotically almost periodic solutions of a convolution equation*, Trans. Amer. Math. Soc.**266**(1981), no. 2, 603–616. MR**617554**, 10.1090/S0002-9947-1981-0617554-5**[18]**J. S. W. Wong and R. Wong,*Asymptotic solutions of linear Volterra integral equations with singular kernels*, Trans. Amer. Math. Soc.**189**(1973), 185–200. MR**0338718**, 10.1090/S0002-9947-1974-0338718-0

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
45M05,
46J99

Retrieve articles in all journals with MSC: 45M05, 46J99

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1982-0675078-4

Article copyright:
© Copyright 1982
American Mathematical Society