## Motions in semidynamical systems.(Russian)Zbl 0567.54023

Let (X,$$\rho)$$ be a metric space and C(X) the collection of all nonempty compact subsets of X. For A,B$$\in C(X)$$ the following distances are defined: $\rho (a,B)=\inf \{\rho (a,b);\quad b\in B\},$
$\beta (A,B)=\sup \{\rho (a,B);\quad a\in A\}$ and $\alpha (A,B)=\max \{\beta (A,B);\quad \beta (B,A)\}.$ A mapping $$f: X\times R^+\to C(X)$$ is said to define a quasi-semidynamical system on X if it satisfies the axioms: (1) $$f(x,0)=x$$ for all $$x\in X$$ and (2) $$f(f(x,t_ 1),t_ 2)=f(x,t_ 1+t_ 2)$$ for all $$x\in X$$; $$t_ 1,t_ 2\in R^+$$. It is called a semidynamical system if additionally $$(4)\quad \lim_{t\to t_ 0}\alpha (f(x,t),f(x,t_ 0))=0.$$ A (quasi-) semidynamical system is called semicontinuous with respect to x if $$(3)\quad \lim_{x\to x_ 0}\beta (f(x,t),f(x_ 0,t))=0.$$ For these types of systems the author studies problems related to the existence and the continuity of motions or the classification of maximal motions.