Positive solutions of asymptotically linear singular boundary value problems (Q5890222)

The authors consider the singular boundary value problem NEWLINE\[NEWLINE(p(t) x'(t))'+\lambda q(t) f(t,x(t))= 0,\quad t\in (0,1),NEWLINE\]NEWLINE NEWLINE\[NEWLINE\alpha x(0)- \beta\lim_{t\to 0+} p(t) x'(t)= 0,\quad \gamma x(1)+ \delta\lim_{t\to 1-} p(t) x'(t)= 0,NEWLINE\]NEWLINE where \(p\in C[0, 1]\cap C^1(0, 1)\), \(p(t)> 0\) for \(t\in (0,1)\), \(q\) is allowed to be singular at \(t= 0\) or \(t= 1\), \(f: [0,1]\times [0,\infty)\to [0,\infty)\) is continuous, and \(\alpha\), \(\beta\), \(\gamma\), \(\delta\geq 0\), \(\alpha\delta+ \beta\gamma+ \alpha\gamma> 0\). Assuming that \(f_\infty(t)= \lim_{x\to+\infty} {f(t,x)\over x}\) uniformly for \(t\in [0,1]\), \(f_\infty(t)\in C[0,1]\), \(f_\infty(t)> 0\) for \(t\in [0,1]\) and some other reasonable assumptions on \(p\) and \(q\), they show, for each \(\lambda\) in some bounded interval, the existence of at least one positive solution. They make use of fixed-point theorems of cone expansion and compression.NEWLINENEWLINENEWLINEActually, the paper contains other similar results which in the whole generalize those by \textit{R. Ma} [Acta Math. Sin. 41, No. 6, 1225-1230 (1998; Zbl 1027.34025)], \textit{L. H. Erbe} and \textit{H. Wang} [Proc. Am. Math. Soc. 120, No. 3, 743-748 (1994; Zbl 0802.34018)] and \textit{R. Dalmasso} [Nonlinear Anal., Theory Methods Appl. 27, No. 6, 645-652 (1996; Zbl 0860.34008)].