PERIODIC ORBITS FOR MULTIVALUED MAPS WITH CONTINUOUS MARGINS OF INTERVALS

Let I be a bounded connected subset of R containing more than one point, and L(I) be the family of all nonempty connected subsets of I. Each map from I to L(I) is called a multivalued map. A multivalued map F: I -> L(I) is called a multivalued map with continuous margins if both the left endpoint and the right endpoint functions of F are continuous. We show that the well-known Sharkovskii theorem for interval maps also holds for every multivalued map with continuous margins F: I -> L(I), that is, if F has an n-periodic orbit and n > m (in the Sharkovskii ordering), then F also has an m-periodic orbit.