Almost periodic solutions of the equation \(\dot x=x^ 3+\lambda g(t)x+\mu f(t)\) and their stability (Q1063168)

By using the Lyapunov function and the contraction mapping principle, the author investigates the existence and stability of almost periodic solutions to the first order nonlinear equations \(dx/dt=-h_ 1(x)+h_ 2(x)g(t)+f(t)\) and \(dx/dt=r(t)x^ n+\lambda g(t)x+\mu f(t),\) where r(t), g(t), f(t) are given almost periodic functions, \(n(\geq 2)\) integer, and \(\lambda\), \(\mu\) real parameters. As a special case, for the equation \(dx/dt=-x^ 3+\lambda g(t)x+\mu f(t),\) under the conditions \(1\leq | g(t)| <3\), \(| f(t)| \leq 1\), the author constructs regions in the (\(\lambda\),\(\mu)\)-plane such that for \((\lambda,\mu)\) in these regions there are either one or three almost periodic solutions. Similar conditions and regions are also obtained such that the equation \(dx/dt=- x^ 2+\lambda g(t)x+\mu f(t)\) has either two or no almost periodic solutions. Moreover, by using the successive approximation method, a sufficient condition is obtained for the existence of an almost periodic solution of a quasilinear system.