Fixed points of iterates of multivalued mappings

Abstract

Multivalued mappings arise in a number of applications including games theory [7, 101, economics [ 1,6], and systems and control theory [2, 111, to give a few scant representative references from a vast literature. A functional analytic synthesis of many of these aspects has been accomplished by Browder [3] and Fan [S]. Underpinning much of this work are the fixed point theorems of Kakutani [7], Browder [3], and Fan [S], while Nadler [9] has proved a fixed point theorem for multivalued contraction mappings. In view of the recent interest in chaotic mappings (see [ 121 for references and [S] for a clear definition), it is not unreasonable to seek analogues for multivalued functions. One characteristic of many classes of dynamical systems is that periodic points of any period exist [13]. This is the same as saying that there exist fixed points (of any order) of a diffeomorphism. Herein, sufficient conditions are stated for the existence of fixed points of iterates (of all orders) of multivalued mappings. Of course, these are generally not periodic. These conditions are similar to those for chaotic single valued maps [4,8] but instead of using Brouwer and Schauder fixed point theorems, those of Nadler and Kakutani are involved. The sufficient conditions are stated in the next section and the theorems proved in Section 3. Two examples are discussed in Section 4.