Stability of solutions for nonlinear nonautonomous differential-delay equations in Hilbert spaces. (English) Zbl 1027.34083

Gil’, Michael I.

Electron. J. Differ. Equ. 2002, Paper No. 94, 15 p. (2002).

In a real separable Hilbert space the functional-differential equation \[ \dot{u}+Au+B\int_{0}^{h}u(t-\tau) d\mu(\tau)=F(t,u(\cdot)),\quad t>0, \] where \(F(t,u(\cdot))\) is a nonlinear Volterra-type operator, is considered. Estimates on the solutions are obtained. Explicit conditions for the absolute stability, and input-output stability of the zero solution are established.

Reviewer: Tamaz Tadumadze (Tbilisi)

MSC:

34K20	Stability theory of functional-differential equations
34K12	Growth, boundedness, comparison of solutions to functional-differential equations
34K30	Functional-differential equations in abstract spaces

Keywords:

functional-differential equations; absolute stability; input-output stability