Russian Academy of Sciences. Sbornik. Mathematics, 75 (1993) 1, 125-135. Reviews: MR 93g:34055, ZM 751.34034.
A classical method of studying dissipativity (ultimate boundedness of solutions) of systems of ordinary differential equations on the plane is considered. The method consists in the construction of a system of compact sets covering the plane, whose boundaries are given by trajectories of a certain auxiliary system, and the trajectories of the given system intersect them from outside in. In this connection the problem of coincidence and intersection of trajectories of two differential inclusions (multivalued differential equations) is solved. In addition, this method is used to prove a theorem that generalizes certain well-known results on the existence of a periodic solution and the dissipativity of the Lienard x''+f(x,x')x'+g(x)=e(t,x,x') and Rayleigh x"+f(x')+g(x)=e(t,x,x') equations. These results follow from the dissipativity and the Brouwer Fixed Point Theorem, but for differential inclusions one needs to apply (in a nontrivial fashion) the Eilenberg-Montgomery Fixed Point Theorem.