Maximum descent monotone solutions of an ordinary differential equation with a discontinuous right-hand side

Abstract

We present an existence theorem for absolutely continuous monotone solutions of the Cauchy problem

$$\dot x\left( t \right) = proj_{T_{P\left( {x\left( t \right)} \right)} x\left( t \right)} ( - \nabla w(x)(t))),x\left( 0 \right) = x_0 .$$

Moreover, we prove that the limit pointx* of any solution is a minimum forw(·) inP(x*). The results are applied to a decision problem for a firm which wants to satsify twoa priori incompatible criteria: (i) monotonicity with respect to a preorder; and (ii) minimization of costs.