Multiple bounded positive solutions to integral type BVPs for singular second order ODEs on the whole line (Q696048)

The Leggett-Williams three-fixed-points theorem on a cone is applied and gives sufficient conditions for the existence of three bounded positive solutions of the equation \[ [\rho(t)\Phi(x'(t))]'+f(t,x(t),x'(t))=0 \text{ for } t\in(-\infty, +\infty) \] satisfying the limiting conditions \[ \lim_{t\to-\infty}x(t)=\int_{-\infty}^{+\infty}g(s)x(s)\,ds, \;\lim_{t\to+\infty}x(t)=\int_{-\infty}^{+\infty}h(s)x(s)\,ds. \] Here, \(f\) is a nonnegative Carathéodory function, \(g, h:\mathbb{R}\to[0,+\infty)\) are \(L^1\) functions, \(\rho\) is a positive and continuous function and \(\Phi(x):=|x|^{p-2}x\) with \(p>1.\) An example closes the paper.