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A principle of linearization in theory of stability of solutions of variational inequalities. (English) Zbl 0847.34057

The principle of linearization and its correctness represents an interesting problem in the theory of stability of solutions of differential equations. In this paper it is shown that, in some assumptions, the uniform exponential stability and the uniform stability at permanently acting disturbances of a sufficiently smooth but not necessarily steady-state solution of the general variational inequality \[ \begin{aligned} \biggl\langle {{dU} \over {dt}}+ F(t,U), W-U \biggr\rangle \geq 0 \quad &\text{ for all }W\in K\\ U(t)\in K \quad &\text{ for all }t\in I(U) \text{ on the time interval } [0,+ \infty) \end{aligned} \] is a consequence of the uniform exponential stability of a zero solution of another (so called linearized) variational inequality \[ \begin{aligned} \biggl\langle {{dv} \over {dt}}+ A(t)v+ f(t), w-v \biggr\rangle \geq 0 \quad &\text{ for all } w\in K_1 (t),\\ v(t)\in K_1 (t) \quad &\text{ for all }t\in I(v). \end{aligned} \] {}.

MSC:

34D05 Asymptotic properties of solutions to ordinary differential equations
34G20 Nonlinear differential equations in abstract spaces
58E35 Variational inequalities (global problems) in infinite-dimensional spaces
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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