Fixed points and stability of a nonconvolution equation

Burton, T.

doi:10.1090/S0002-9939-04-07497-0

Fixed points and stability of a nonconvolution equation
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by T. A. Burton PDF

Proc. Amer. Math. Soc. 132 (2004), 3679-3687 Request permission

Abstract:

In this note we consider an equation of the form \[ x’(t)=-\int ^{t}_{t-r} a(t,s)g(x(s))ds\] and give conditions on $a$ and $g$ to ensure that the zero solution is asymptotically stable. When applied to the classical case of $a(t,s)=a(t-s)$, these conditions do not require that $a(r)=0$, nor do they involve the sign of $a(t)$ or the sign of any derivative of $a(t)$.

References

T. A. Burton and W. E. Mahfoud, Stability criteria for Volterra equations, Trans. Amer. Math. Soc. 279 (1983), no. 1, 143–174. MR 704607, DOI 10.1090/S0002-9947-1983-0704607-8
Tadasi Nakayama, On Frobeniusean algebras. I, Ann. of Math. (2) 40 (1939), 611–633. MR 16, DOI 10.2307/1968946
Jack K. Hale, Sufficient conditions for stability and instability of autonomous functional-differential equations, J. Differential Equations 1 (1965), 452–482. MR 183938, DOI 10.1016/0022-0396(65)90005-7
Hale, Jack, Dynamical systems and stability, J. Math. Anal. Appl. 26(1969), 39-59.
Jack Hale, Theory of functional differential equations, 2nd ed., Applied Mathematical Sciences, Vol. 3, Springer-Verlag, New York-Heidelberg, 1977. MR 0508721
N. N. Krasovskiĭ, Stability of motion. Applications of Lyapunov’s second method to differential systems and equations with delay, Stanford University Press, Stanford, Calif., 1963. Translated by J. L. Brenner. MR 0147744
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
J. J. Levin, A nonlinear Volterra equation not of convolution type, J. Differential Equations 4 (1968), 176–186. MR 225117, DOI 10.1016/0022-0396(68)90034-X
J. J. Levin and J. A. Nohel, On a nonlinear delay equation, J. Math. Anal. Appl. 8 (1964), 31–44. MR 163142, DOI 10.1016/0022-247X(64)90080-0
John A. Nohel, A class of nonlinear delay differential equations, J. Math. and Phys. 38 (1959/60), 295–311. MR 114104
Volterra, V., Sur la théorie mathématique des phénomènes héréditaires, J. Math. Pures Appl. 7(1928), 249-298.
Taro Yoshizawa, Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan, vol. 9, Mathematical Society of Japan, Tokyo, 1966. MR 0208086