Approximate weak invariance for differential inclusions in Banach spaces (Q1935309)

The authors consider the differential inclusion \[ x^{\prime }(t)\in F(x(t)) \] in a Banach space and study the relationship between approximate weak invariance and a tangency condition. The definition of approximate weak invariance of a set \(K\) requires that an approximate solution starting in \(K\) remains near \(K\) for sufficiently small time. In Theorem 1, sufficient conditions for a locally closed set to be approximately weakly invariant are given, namely that \(F\) is locally bounded and that \(F(x)\) is quasi-tangent to \(K\) at every point \(x\in K\). Theorem 2 specifies conditions sufficient for approximate weak invariance of a locally closed set \(K\) with the restriction that one considers only exact solutions in the definition. \(F\) is specified to be bounded-valued, satisfy a global Lipschitz condition and \(F(x)\) is quasi-tangent to \(K\) at every point \(x\in K\). As an application, the authors prove a result on Lipschitz dependence of approximate solutions on the initial state. Theorem 3 specifies that the quasi-tangency assumption is necessary for a set \(K\) to be approximately weakly invariant in a separable Banach space under the assumptions that \(F\) satisfy an upper semicontinuity condition and a growth condition. The paper is motivated by the desire to extend to Banach spaces the results in Hilbert spaces found by \textit{F. H. Clarke} et al. [J. Dyn. Control Syst. 3, No. 4, 493--518 (1997; Zbl 0951.49007)].