Inequalities of solutions of Volterra integral and differential equations

Tingxiu Wang, Missouri Western State University, Saint Joseph, MO, U.S.A.

E. J. Qualitative Theory of Diff. Equ., Spec. Ed. I, 2009 No. 28., pp. 1-10.

Abstract: In this paper, we study solutions of Volterra integral and differential equations,
$$
x'(t) = -a(t)x(t) + \int_{t-h}^tb(s)x(s)ds + f(t, x_t), \ \ x \in {\bf R},
$$
or
$$
X(t)=a(t)+\int_{t-\alpha}^tg(t,s)X(s)ds, \ \ X\in\bf R^n.
$$
With Lyapunov functionals, we obtain inequalities for the solutions of these equations. As a corollary, we also obtain a result on asymptotic stability which is simpler and better than some existing results.

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