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Zbl 1181.34035
Jiang, Jiqiang; Liu, Lishan; Wu, Yonghong
Second-order nonlinear singular Sturm-Liouville problems with integral boundary conditions. (English)
[J] Appl. Math. Comput. 215, No. 4, 1573-1582 (2009). ISSN 0096-3003

Summary: This paper is concerned with the second-order singular Sturm-Liouville integral boundary value problems $$\cases -u''(t)=\lambda h(t)f(t,u(t)),\quad 0<t<1,\\ \alpha u(0)=\beta u'(0)=\int^1_0 a(s)u(s)\,ds\\ \gamma u(0)=\delta u'(1)=\int^1_0 b(s)u(s)\,ds,\endcases$$ where $\lambda>0$, $h$ is allowed to be singular at $t=0,1$ and $f(t,x)$ may be singular at $x=0$. By using the fixed point theory in cones, an explicit interval for $\lambda$ is derived such that for any $\lambda$ in this interval, the existence of at least one positive solution to the boundary value problem is guaranteed. Our results extend and improve many known results including singular and non-singular cases.

MSC 2000:: *34B24 Sturm-Liouville theory
34L15 Estimation of eigenvalues for OD operators

Keywords: second-order singular differential equation; positive solutions; fixed point theory

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