Existence, uniqueness, and instability of large-period harmonics to third-order nonlinear ordinary differential equations
This paper studies \(T\)-periodic solutions for the three equations \[ \begin{aligned} x'''+ ax''+ bx'+ h(x) & = p(t),\tag{1}\\ x'''+ ax''+ g(x')+ cx & = p(t),\tag{2}\\ x'''+ f(x'')+ bx'+ cx & = p(t),\tag{3}\end{aligned} \] where \(a\), \(b\) and \(c\) are constants, \(f\), \(g\) and \(h\) are continuous and \(p\) is continuous and \(T\)-periodic. Explicit conditions are given on the equation to ensure existence of \(T\)-periodic solutions of (1). The author describes also assumptions to ensure uniqueness of \(T\)-periodic solutions for each of the equations (1), (2) and (3). In a third part, he investigates instability of each of these equations. A model example is the pendulum-like equation \(x'''+ ax''+ bx'+ c\sin x= p(t)\). The methods of proof are somewhat elementary. Existence makes use of Lyapunov-like functions, instability is based on the characteristic equation and Sturm's theorem.
scientific article; zbMATH DE number 902554