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Existence of positive solutions for singular second-order $m$-point boundary value problems. (English) Zbl 1080.34517

The singular second-order $m$-point boundary value problem

$\begin{array}{c}{\left(p\left(t\right){x}^{\text{'}}\left(t\right)\right)}^{\text{'}}+q\left(t\right)x\left(t\right)+h\left(t\right)f\left(x\left(t\right)\right)=0,\phantom{\rule{1.em}{0ex}}0

is considered, with $p\in {C}^{1}\left[0,1\right]$, $p>0$, $q\in {C}^{0}\left[0,1\right]$, $q\le 0$, $h\in {C}^{0}\left(0,1\right)\cap {L}_{1}\left[0,1\right]$, $h\ge 0$, $f\in {C}^{0}\left(0,\infty \right)$, $f\ge 0$, $0<{\xi }_{1}<\cdots <{\xi }_{m-2}<1$ and ${a}_{j}>0$. The existence of positive solutions of problem (1) is obtained by means of fixed-point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operators. The existence theorems generalize the results by R. Ma and H. Wang [J. Math. Anal. Appl. 279, 216–227 (2003; Zbl 1028.34014)] and R. Ma [Acta Math. Sin. 46, No. 4, 785–794 (2003; Zbl 1056.34015)].

##### MSC:
 34B16 Singular nonlinear boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B10 Nonlocal and multipoint boundary value problems for ODE
##### References:
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