THE PERTURBATION OF ALMOST PERIODIC SOLUTION OF ALMOST PERIODIC SYSTEM

By using the exponential dichotomy and the averaging method,a perturbation theoryis established for the almost periodic solutions of an almost differential system.Suppose that the almost periodic differential system(dx)/(dt)=f(x,t) ε~2g(x,t,ε)(1)has an almost periodic solution x=x_0(t,M)for ε=0,where M=(m_1,…,m_k)is theparameter vector.The author discusses the conditions under which(1)has an almostperiodic solution x=x(t,ε)such that x(t,ε)=x_0(t,M)holds uniformly.The results obtained are quite complete.

THE PERTURBATION OF ALMOST PERIODIC SOLUTION OF ALMOST PERIODIC SYSTEM

By using the exponential dichotomy and the averaging methods a perturbation theory is established for the almost periodic solutions of an almost differential system. Suppose that the almost periodic differential system $$\\frac{{dx}}{{dt}} = f(x,t) + {s^2}g(x,t,s)\begin{array}{*{20}{c}} {}&{(1)} \end{array}\]$$ has an almost periodic solution \x = {x_0}(t,M)\] for s=0, where $\M = ({m_1}, \cdots ,{m_k})\]$ is the parameter vector. The author discusses the conditions under which (1) has an almost periodic solution $\x = x(t,s)\]$ such that $$\\mathop {\lim }\limits_{s \to 0} x(t,s) = {x_0}(t,M)\]$$ holds uniformly. The results obtained are quite complete.