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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.

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
Full Text:
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