Positive solutions for singular Sturm-Liouville boundary value problems with integral boundary conditions

Xiping Liu, University of Shanghai for Science and Technology, Shanghai, P. R. China
Yu Xiao, University of Shanghai for Science and Technology, Shanghai, P. R. China
Jianming Chen, University of Shanghai for Science and Technology, Shanghai, P. R. China

E. J. Qualitative Theory of Diff. Equ., No. 77. (2010), pp. 1-15.

Communicated by J. R. L. Webb.Received on 2010-07-22
Appeared on 2010-12-27

Abstract: In this paper, we study the second-order nonlinear singular Sturm-Liouville boundary value problems with Riemann-Stieltjes integral boundary conditions
\begin{equation*}\begin{cases}
-(p(t)u'(t))'+q(t)u(t)=f(t,u(t)),\; 0<t<1,\\
\alpha_{1}u(0)-\beta_{1}u'(0)=\int_{0}^{1}u(\tau)\mathrm{d}\alpha(\tau),\\
\alpha_{2}u(1)+\beta_{2}u'(1)=\int_{0}^{1}u(\tau)\mathrm{d}\beta(\tau),
\end{cases}\end{equation*}
where $f(t,u)$ is allowed to be singular at $t=0,1$ and $u=0$. Some new results for the existence of positive solutions of the boundary value problems are obtained. Our results extend some known results from the nonsingular case to the singular case, and we also improve and extend some results of the singular cases.


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