On a multivalued version of the Sharkovskii theorem and its application to differential inclusions. II
This work is dedicated to the extension of the Sharkovskii theorem for \(M\)-maps and its applications to differential equations and inclusions. One of the main results of this paper is the following theorem: Let an \(M\)-map \(\varphi:\mathbb R\to \mathbb R\) have an \(n\)-orbit. Then \(\varphi\) has also a \(k\)-orbit, for every \(k < n\), with the exception of at most three orbits. A multi-valued map \(\varphi:\mathbb R\to \mathbb R\) is called an \(M\)-map if it is upper-semicontinuous and its nonempty sets of values are either single points or compact intervals. An \(n\)-orbit associated to an \(M\)-map \(\varphi\) is a sequence \(\{ x_1,\dots,x_n\}\) of points \(x_j\in {\mathbb{R}}\) such that \(x_{j+1}\in \varphi(x_j)\), \(j=1,2,\dots ,n\), where \(x_{n+1}=x_1\) and the orbit cannot be formed by going \(p\)-times arround a shorter \(m\)-orbit, where \(mp=n\). The order relation \(k < n\) is in the sense of Sharkovskii. These results complete results given in the paper of the first author, J. Fišer and the second author [Set-Valued Anal. 10, 1--14 (2002; Zbl 1082.37048)], and it is proved that they are sharp by constructing counter-examples.
application to differential equations and inclusions
scientific article; zbMATH DE number 2176207
scientific article