New solvability conditions for the Neumann problem for ordinary singular differential equations (Q5951171)

The solvability of the boundary value problem \[ x''+(k/t)x'=f(t,x,x'),\;x'(0)=0, \;x'(\tau)=a-\sigma x(\tau), \] is studied with \(f\in C(I\times \mathbb{R}^2)\), \(\tau>0\), \(k,\sigma\geq 0\), \(a\in\mathbb{R}\) and \(I=[0,\tau]\). The existence of a \(C^2(I)\) solution to this problem is established under the assumption that there exist \(\nu\in[0,2)\) and \(Q>0\) such that \[ f(t_2,x_2,P)-f(t_1,x_1,P)\geq -QP^\nu,\;(t_1,x_1),(t_2,x_2)\in E,\;t_1<t_2,\;x_1<x_2,\;\text{for all }P>m^*\;\text{and} \] \[ f(t_2,x_2,P)-f(t_1,x_1,P)\leq -Q|P|^\nu,\;(t_1,x_1),(t_2,x_2)\in E,\;t_1<t_2,\;x_1>x_2,\;\text{for all }P<-m^*, \] with \(E=\{(t,x):t\in I, x\in [\alpha(t),\beta(t)]\}\), and \(m^*\) a suitable constant depending on the upper solution \(\beta(t)\) and the lower solution \(\alpha(t)\) to the problem considered.