First approximation stability theorems for differential inclusions. (Q556555)

A family of a multi-valued mapping \(A_{\rho}:R_{+}\times V\to V^{*}\) is introduced by \[ A_{\rho}(t,v) = A(t,v) + \rho\| v\| B_{V^{*}}, \] where \(\rho \geq 0, t\geq 0, v\in V, B_{V^{*}}=\{v^{*}\in V^{*}, \| v^{*}\| _{*} \leq 1\},\) \(V\) is a reflexive separable \(B\)-space with norm \(\| \cdot\|\) and \(V^{*}\) is its dual space with norm \(\| \cdot\| _{*}.\) Suppose that the multi-valued mapping \(A\) satisfies some conditions and that the zero solution of the inclusion \( 0\in y' + A(t,y) \) is exponentially stable in the small. Then there is a \(\rho_{0}>0\) such that, for \(\rho\in (0,\rho_{0}),\) the trivial solution of the inclusion \( 0\in y' + A_{\rho}(t,y)\) is also exponentially stable in the small.