A multivalued version of Sharkovskii's theorem holds with at most two exceptions

A multivalued version of Sharkovskii's theorem is formulated for M-maps on linear continua and, more generally, for triangular M-maps on a Cartesian product of linear continua. This improves the main result of [AP1] in the sense that our multivalued analogue holds with at most two exceptions. A further specification requires some additional restrictions. For instance, 3-orbits of m-maps imply the existence of k-orbits for all k is an element of N, except possibly for k is an element of {4, 6}. It is also shown that, on every connected linearly ordered topological space, an M-map with orbits of all periods can always be constructed. This demonstrates that Baldwin's classification of linear continua in terms of admissible periods [Ba] is useless for multivalued maps.