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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

x 1 '' +f 1 (t,x 1 ,x 2 )=0,x 2 '' +f 2 (t,x 1 ,x 2 )=0

subject to

x 1 (0)=x 1 (1)=0,x 2 (0)=x 2 (1)=0,

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]×C[0,1] positive solutions as well as C 1 [0,1]×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