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Positive solution of singular Dirichlet boundary value problems for second order differential equation system. (English) Zbl 1115.34025

The paper is concerned with the existence of a positive solution to the singular Dirichlet boundary value problem for the second-order ordinary differential system for $0

$\left\{\begin{array}{c}{x}_{1}^{\text{'}\text{'}}+{f}_{1}\left(t,{x}_{1},{x}_{2}\right)=0,\\ {x}_{2}^{\text{'}\text{'}}+{f}_{2}\left(t,{x}_{1},{x}_{2}\right)=0\end{array}\right\$

subject to

$\left\{\begin{array}{c}{x}_{1}\left(0\right)={x}_{1}\left(1\right)=0,\\ {x}_{2}\left(0\right)={x}_{2}\left(1\right)=0,\end{array}\right\$

where the nonlinearities ${f}_{1}$ and ${f}_{2}$ satisfy certain sublinear conditions and may be singular at ${x}_{1}=0,\phantom{\rule{0.166667em}{0ex}}{x}_{2}=0,\phantom{\rule{0.166667em}{0ex}}t=0$ and/or $t=1$. Using the method of lower and upper solutions both with the Schauder’s fixed point theorem, the author gives a necessary and sufficient condition for the existence of $C\left[0,1\right]×C\left[0,1\right]$ positive solutions as well as ${C}^{1}\left[0,1\right]×{C}^{1}\left[0,1\right]$ positive solutions. The paper ends with an example.

MSC:
 34B16 Singular nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 34B18 Positive solutions of nonlinear boundary value problems for ODE