## Asymptotic behavior of solutions of Volterra integro-differential equations

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- by M. Rama Mohana Rao and P. Srinivas
- Proc. Amer. Math. Soc.
**94**(1985), 55-60 - DOI: https://doi.org/10.1090/S0002-9939-1985-0781056-5
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## Abstract:

The asymptotic behavior of solutions of Volterra integrodifferential equations of the form \[ x’(t) = A(t)x(t) + \int _0^t {K(t,s)} x(s)ds + F(t)\] is discussed in which $A$ is not necessarily a stable matrix. An equivalent equation which involves an arbitrary function is derived and a proper choice of this function would pave a way for the new coefficient matrix $B$ (corresponding $A$) to be stable.## References

- T. A. Burton,
*Stability theory for Volterra equations*, J. Differential Equations**32**(1979), no. 1, 101–118. MR**532766**, DOI 10.1016/0022-0396(79)90054-8 - T. A. Burton,
*Uniform stabilities for Volterra equations*, J. Differential Equations**36**(1980), no. 1, 40–53. MR**571126**, DOI 10.1016/0022-0396(80)90074-1 - T. A. Burton,
*An integro-differential equation*, Proc. Amer. Math. Soc.**79**(1980), no. 3, 393–399. MR**567979**, DOI 10.1090/S0002-9939-1980-0567979-6 - W. A. Coppel,
*Stability and asymptotic behavior of differential equations*, D. C. Heath and Company, Boston, Mass., 1965. MR**0190463** - W. A. Coppel,
*Dichotomies in stability theory*, Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, Berlin-New York, 1978. MR**0481196** - Ronald Grimmer and George Seifert,
*Stability properties of Volterra integrodifferential equations*, J. Differential Equations**19**(1975), no. 1, 142–166. MR**388002**, DOI 10.1016/0022-0396(75)90025-X - S. I. Grossman and R. K. Miller,
*Nonlinear Volterra integrodifferential systems with $L^{1}$-kernels*, J. Differential Equations**13**(1973), 551–566. MR**348417**, DOI 10.1016/0022-0396(73)90011-9
—, - J. J. Levin,
*The asymptotic behavior of the solution of a Volterra equation*, Proc. Amer. Math. Soc.**14**(1963), 534–541. MR**152852**, DOI 10.1090/S0002-9939-1963-0152852-8 - V. Lakshmikantham and M. Rama Mohana Rao,
*Integro-differential equations and extension of Lyapunov’s method*, J. Math. Anal. Appl.**30**(1970), 435–447. MR**254549**, DOI 10.1016/0022-247X(70)90175-7 - 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 - M. Rama Mohana Rao,
*Ordinary differential equations*, Affiliated East-West Press Pvt. Ltd., New Delhi, 1980. Theory and applications; With a foreword by V. Lakshmikantham. MR**587850** - M. Rama Mohana Rao and Chris P. Tsokos,
*Integro-differential equations of Volterra type*, Bull. Austral. Math. Soc.**3**(1970), 9–22. MR**278027**, DOI 10.1017/S0004972700045603 - George Seifert,
*Liapunov-Razumikhin conditions for stability and boundedness of functional differential equations of Volterra type*, J. Differential Equations**14**(1973), 424–430. MR**492745**, DOI 10.1016/0022-0396(73)90058-2 - George Seifert,
*Liapunov-Razumikhin conditions for asymptotic stability in functional differential equations of Volterra type*, J. Differential Equations**16**(1974), 289–297. MR**460837**, DOI 10.1016/0022-0396(74)90016-3

*Perturbation theory for Volterra integro-differential systems*, J. Differential Equations

**8**(1970), 451-474.

## Bibliographic Information

- © Copyright 1985 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**94**(1985), 55-60 - MSC: Primary 45J05
- DOI: https://doi.org/10.1090/S0002-9939-1985-0781056-5
- MathSciNet review: 781056