A note on existence of viable solutions for a third order differential inclusion

Abstract

We prove the existence of viable solutions for the problem
$x'''\in F(t,x,x',x'')+G(x,x',x'')$, $x(0)=x_0$, $x'(0)=x^1_0$,
$x''(0)=x^2_0$, $x''(t)\in K$, where $K\subset {\mathbf{R}}^n$ is
a locally compact set, $F$ is a continuous set-valued map and $G$
is an upper semicontinuous set-valued map contained in the
Fr\'{e}chet subdifferential of a $\phi $- convex function of order two.