Implicit second order differential inclusions in reflexive smooth Banach spaces (Q1938519)

The authors study the existence of solutions to the second-order differential inclusion \[ 0\in H\left( t,u\left( t\right) ,\dot{u}\left( t\right),\ddot{u}\left( t\right) \right) \text{ a.e. on }\left[0,T\right], \] subject to the initial conditions \(u\left( 0\right) =u_{0},\) \(\dot{u}\left( 0\right) =x_{0}\), where \(H\) has the form \(H\left( t,x,y,z\right) =N\left( C\left( t,x,y\right) ,z\right) +F\left(t,x,y\right) +l\mathbb{B}\). Here, \(C:\left[ 0,T\right] \times X\times X\rightrightarrows X\) is a set-valued mapping with convex values such that \(C\left( t,x,y\right) \subset l\mathbb{B},\) \(F:\left[ 0,T\right]\times X\times X\rightrightarrows X\) is a continuous set-valued mapping, and \(X\) is a separable reflexive smooth Banach space.