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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 $0<t<1$ $$ \left\{ \matrix x_1''+f_1(t,x_1,x_2)=0,\\ x_2''+f_2(t,x_1,x_2)=0\\ \endmatrix \right.$$ subject to $$ \left\{ \matrix x_1(0)=x_1(1)=0,\\ x_2(0)=x_2(1)=0,\\ \endmatrix \right.$$ where the nonlinearities $f_1$ and $f_2$ satisfy certain sublinear conditions and may be singular at $x_1=0,\, x_2=0,\, t=0$ 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 $C[0,1]\times C[0,1]$ positive solutions as well as $C^1[0,1]\times C^1[0,1]$ positive solutions. The paper ends with an example.

34B16Singular nonlinear boundary value problems for ODE
34B15Nonlinear boundary value problems for ODE
34B18Positive solutions of nonlinear boundary value problems for ODE
Full Text: DOI
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