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Existence of positive solutions to a system of singular boundary value problems. (English) Zbl 1215.34029

Summary: Existence results for positive solutions of a coupled system of nonlinear singular two point boundary value problems of the type
\[ \begin{aligned} -&x'' = p(t)f(t,y(t),x'(t)),\quad t\in (0,1),\\ -&y'' = q(t)g(t,x(t),y'(t)),\quad t\in (0,1),\\ & a_1x(0)-b_1x'(0)=x'(1)=0,\\ & a_2y(0)-b_2y'(0)=y'(1)=0,\end{aligned} \]
are established. The nonlinearities \(f,g:[0,1]\times (0,\infty)\to [t,\infty)\) are allowed to be singular at \(x'=0\) and \(y'=0\). The functions \(p,q\in C(0,1)\) are positive on \((0,1)\) and the constants \(a_i,b_i>0\) \((i=1,2)\). An example is included to show the applicability of our result.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
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