Contractive multifunctions, fixed point inclusions and iterated multifunction systems

We study the properties of multifunction operators that are contractive in the Covier-Nadler sense. In this situation, such operators $T$ possess fixed points satisying the relation $x \in Tx$. We introduce an iterative method involving projections that guarantees convergence from any starting point $x_0 \in X$ to a point $x \in X_T$, the set of all fixed points of a multifunction operator $T$. We also prove a continuity result for fixed point sets $X_T$ as well as a ``generalized collage theorem'' for contractive multifunctions. These results can then be used to solve inverse problems involving contractive multifunctions. Two applications of contractive multifunctions are introduced: (i) integral inclusions and (ii) iterated multifunction systems.