Pairs of positive periodic solutions of nonlinear ODEs with indefinite weight: a topological degree approach for the super-sublinear case (Q2832271)

In this interesting paper sufficient conditions are provided which guarantee the existence of a pair of positive periodic solutions of the second order ordinary differential equation NEWLINE\[NEWLINE(*)\enskip u''+cu'+\lambda a(t)g(u)=0.NEWLINE\]NEWLINE More exactly the authors, by using Mawhin's coincidence degree theory, adopted to the case of locally compact operators, prove the followingNEWLINENEWLINE Theorem. Let \(g:\mathbb{R}^+\to\mathbb{R}^+\) be a continuous function satisfying the conditions NEWLINE\[NEWLINE\begin{aligned} \lim_{s\to 0^+}{g(s)}/{s}=0&,\;\lim_{{s\to +\infty}}{g(s)}/{s}=0,\\ \lim_{{s\to 0^+},{\omega\to 1}}{g(\omega s)}/{g(s)}=1&, \;\lim_{{s\to +\infty},{\omega\to 1}}{g(\omega s)}/{g(s)}=1.\end{aligned}NEWLINE\]NEWLINE Let \(a:\mathbb{R}\to\mathbb{R}\) be locally integrable \(T\)-periodic function satisfying the condition \(\int_0^Ta(t)dt<0\), while for some interval \(I\subseteq [0,T]\) it holds \(a(t)\geq 0\) for \(t\in I\) a.e. and moreover \(\int_Ia(t)dt>0.\) Then there exists a \(\lambda^*>0\) such that, for each \(\lambda>\lambda^*\), equation \((*)\) has at least two positive \(T\)-periodic solutions. The advantage of using the degree theory for the proof is that pairs of positive \(T\)-periodic solutions of small state perturbations of the differential equation \((*)\) are, also, guaranteed. A corollary for nonexistence of \(T\)-periodic solutions is, finally, given.