New results for the Liebau phenomenon via fixed point index
The authors are concerned with the second-order nonlinear differential equation: \[ x''+ax'+m^2x=f_m(t,x),\quad 0<t<T, \] together with periodic conditions \(x(0)=x(T)\) and \(x'(0)=x'(T)\). Here, \(f_m=rx^\alpha-sx^\beta+m^2x\) with \(0<\alpha<\beta<1\), \(m\in\mathbb{R}\), and \(r, s\) are continuous functions on \([0,T]\). This problem arises in the circulation of a fluid in some mechanical system. The authors formulate the problem with a fixed point Green operator then use Krasonesl'skii's fixed point index in a positive cone of a Banach space to prove the existence of at least one solution. Some conditions on the nonlinearity \(f_m\) are assumed. Examples of applications with numerical results are provided at the end of the paper.
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