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Fixed points and stability of a nonconvolution equation
Author:
T. A. Burton
Journal:
Proc. Amer. Math. Soc. 132 (2004), 3679-3687
MSC (2000):
Primary 34K20, 47H10
Posted:
May 12, 2004
MathSciNet review:
2084091
Full-text PDF Free Access
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Additional Information
Abstract: In this note we consider an equation of the form
and give conditions on  and  to ensure that the zero solution is asymptotically stable. When applied to the classical case of  , these conditions do not require that  , nor do they involve the sign of  or the sign of any derivative of  .
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- Burton, T. A. and Mahfoud, W. E., Stability criteria for Volterra equations, Trans. Amer. Math. Soc. 279 (1983), 143-174. MR 84h:45004
- 2.
- Brownell, F. H. and Ergen, W. K., A theorem on rearrangements and its application to certain delay differential equations, J. Rational Mech. Anal. 3(1954), 565-579. MR 16:714c
- 3.
- Hale, Jack, Sufficient conditions for stability and instability of autonomous functional-differential equations, J. Differential Equations 1(1965), 452-482. MR 32:1414
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- Hale, Jack, Dynamical systems and stability, J. Math. Anal. Appl. 26(1969), 39-59.
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- Hale, Jack, Theory of Functional Differential Equations, Springer, New York, 1977. MR 58:22904
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- Krasovskii, N. N., Stability of Motion, Stanford Univ. Press, Stanford, CA, 1963. MR 26:5258
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- Levin, J. J., The asymptotic behavior of the solution of a Volterra equation, Proc. Amer. Math. Soc. 14(1963), 534-541. MR 27:2824
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- Levin, J. J., A nonlinear Volterra equation not of convolution type, J. Differential Equations 4(1968), 176-186. MR 37:712
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- Levin, J. J. and Nohel, J. A., On a nonlinear delay equation, J. Math. Anal. Appl. 8(1964), 31-44. MR 29:445
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- Nohel, J. A., A class of nonlinear delay differential equations, J. Math. Physics 38(1960), 295-311. MR 22:4931
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- Volterra, V., Sur la théorie mathématique des phénomènes héréditaires, J. Math. Pures Appl. 7(1928), 249-298.
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- Yoshizawa, T., Stability Theory by Liapunov's Second Method, Math. Soc. Japan, Tokyo, 1966.MR 34:7896
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Additional Information
T. A. Burton
Affiliation:
Northwest Research Institute, 732 Caroline St., Port Angeles, Washington 98362
Email:
taburton@olypen.com
DOI:
http://dx.doi.org/10.1090/S0002-9939-04-07497-0
PII:
S 0002-9939(04)07497-0
Keywords:
Delay equations,
fixed points,
stability
Received by editor(s):
July 8, 2003
Received by editor(s) in revised form:
September 3, 2003
Posted:
May 12, 2004
Communicated by:
Carmen C. Chicone
Article copyright:
© Copyright 2004 American Mathematical Society
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