Positive solutions of boundary value problems for second-order singular nonlinear differential equations (Q5942742)

The authors consider the singular boundary value problems \[ u''+ g(t) f(t)=0,\quad t\in (0,1),\quad \alpha u(0)-\beta u'(0)= 0,\quad \gamma u(1)+\delta u'(1)=0, \] with \(\alpha\), \(\beta\), \(\gamma\), \(\delta\geq 0\), \(\alpha\delta+ \beta\gamma+ \alpha\gamma>0\), \(f\in C([0,\infty)\times [0,\infty))\), \(g\in C((0,1)\times [0,\infty))\) and \(0< \int^1_0 G(s,s) g(s) ds< \infty\) where \(G(t,s)\) is the Green function of the associated linear problem. One can note that these assumptions allow \(g\) to be singular at \(t= 0\) and/or \(t=1\). Under the conditions \(0\leq f^+_0< M_1\), \(m_1< f^-_\infty\leq\infty\) or \(0\leq f^+_\infty< M_1\), \(m_1< f^-_0\leq\infty\), where \(f^+_0= \varlimsup_{u\to 0}f(u)/u\), \(f^-_\infty= \varliminf_{u\to\infty} f(u)/u\), \(f^-_0= \varliminf_{u\to 0} f(u)/u\), \(f^+_\infty= \varlimsup_{u\to\infty} f(u)/u\), and \(M_1= (\max_{0\leq t\leq 1} \int^1_0 G(t,s) g(s) ds)^{-1}\), \(m_1= (\min_{0\leq t\leq 1}\int^1_0 G(t,s) g(s) ds)^{-1}\), they prove the existence of at least one positive solution \(u\in C[0,1]\), \(u(t)> 0\), \(0< t< 1\). The proof uses a fixed-point theorem in cone theory. Their main result improve and extend results due to \textit{L. H. Erbe} and \textit{H. Wang} [Proc. Am. Math. Soc. 120, No. 3, 743-748 (1994; Zbl 0802.34018)] and \textit{R. Ma} [Acta Math. Sin. 41, No. 6, 1225-1230 (1998; Zbl 1027.34025)].