Stability criteria for Volterra equations

Burton, T. A.; Mahfoud, W. E.

doi:10.1090/S0002-9947-1983-0704607-8

Stability criteria for Volterra equations
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by T. A. Burton and W. E. Mahfoud

Trans. Amer. Math. Soc. 279 (1983), 143-174

DOI: https://doi.org/10.1090/S0002-9947-1983-0704607-8

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Abstract:

We consider a system of integro-differential equations of the form (1.1) \[ x’ = A(t)x + \int _0^t {C(t,s)x(s)\;ds} \] with $A$ and $C$ being $n \times n$ matrices. Various types of stability are defined and results are obtained showing when one type of stability is equivalent to another type. We also construct a number of Lyapunov functional from which we obtain necessary and sufficient conditions for stability of (1.1). Finally, we prove several results concerning qualitative behavior of solutions of (1.1).

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