## Admissibility and nonlinear Volterra integral equations

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- by R. K. Miller
- Proc. Amer. Math. Soc.
**25**(1970), 65-71 - DOI: https://doi.org/10.1090/S0002-9939-1970-0257674-9
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## Abstract:

Nonlinear perturbations of linear Volterra integral equations are studied in an abstract setting which contains and generalizes some earlier results on the same problem. The perturbed problem is first written as a variation of constants equation on a Fréchet space. It is then shown that standard fixed point theorems may be applied if the linear equation is admissible w.r.t. a Banach subspace of the Fréchet space. This theory is applied to an example where ${L^2}$-stability is proved.## References

- José Luis Massera and Juan Jorge Schäffer,
*Linear differential equations and function spaces*, Pure and Applied Mathematics, Vol. 21, Academic Press, New York-London, 1966. MR**0212324** - C. Corduneanu,
*Problèmes globaux dans la théorie des équations intégrales de Volterra*, Ann. Mat. Pura Appl. (4)**67**(1965), 349–363 (French). MR**182849**, DOI 10.1007/BF02410815 - C. Corduneanu,
*Some perturbation problems in the theory of integral equations*, Math. Systems Theory**1**(1967), 143–155. MR**213919**, DOI 10.1007/BF01705524 - H. A. Antosiewicz,
*Un analogue du principe du point fixe de Banach*, Ann. Mat. Pura Appl. (4)**74**(1966), 61–64 (French). MR**205127**, DOI 10.1007/BF02416448 - R. K. Miller,
*On the linearization of Volterra integral equations*, J. Math. Anal. Appl.**23**(1968), 198–208. MR**230070**, DOI 10.1016/0022-247X(68)90127-3 - R. K. Miller, J. A. Nohel, and J. S. W. Wong,
*Perturbations of Volterra integral equations*, J. Math. Anal. Appl.**25**(1969), 676–691. MR**240573**, DOI 10.1016/0022-247X(69)90265-0 - I. W. Sandberg,
*On the ${\cal L}_{2}$-boundedness of solutions of nonlinear functional equations*, Bell System Tech. J.**43**(1964), 1581–1599. MR**171185**, DOI 10.1002/j.1538-7305.1964.tb04098.x
G. Zames, - 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**, DOI 10.1090/coll/019

*On the input-output stability of time-varying nonlinear feedback systems*. I, II, IEEE Trans. Automatic Control

**AC-11**(1966), 228-238; 465-476.

## Bibliographic Information

- © Copyright 1970 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**25**(1970), 65-71 - MSC: Primary 45.13
- DOI: https://doi.org/10.1090/S0002-9939-1970-0257674-9
- MathSciNet review: 0257674