A note on the viable solutions for a class of nonconvex differential inclusions
The existence of a viable solution to the following differential inclusion is established: \[ x^{\prime}(t) \in F(x(t)), \;x(0) = x_0, \;x(t) \in M, \] where \(F\) is upper semi-continuous with closed values, \(M\) is convex and locally compact. The author unifies and generalizes results of \textit{P. Rossi} [Differ. Integral Equ. 5, No. 2, 455--459 (1992; Zbl 0757.34019)] and his previous results [Math. Rep., Bucur. 6(56), No. 3, 217--224 (2004; Zbl 1077.34018)]. It is assumed that the intersection between \(F(x)\), the cotingent cone to \(M\) at \(x\), and the Fréchet subdifferential of \(V\) at \(x\) is nonempty for some \(V\), a proper lower semi-continuous \(\phi\)-convex function of order 2. This type of function was introduced by \textit{Degiovanni at al.} [Nonlinear Anal. 9, 1401--1443 (1995)]. The viable solution is obtained as the limit of a constructed sequence of piecewise linear maps.
scientific article; zbMATH DE number 5121511