Contractive multifunctions, fixed point inclusions and iterated multifunction systems

We study the properties of multifunction operators that are contractive in the Covitz–Nadler sense. In this situation, such operators T possess fixed points satisfying the relation x ∈ T x.We introduce an iterative method involving projections that guarantees convergence from any starting point x0 ∈ X to a point x ∈ XT , the set of all fixed points of a multifunction operator T . We also prove a continuity result for fixed point sets XT 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.