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Positive solutions for a class of coupled system of singular three-point boundary value problems. (English) Zbl 1181.34030

Summary: The existence of positive solutions for a boundary value problem
\[ -x''(t)=f(t,x(t),y(t)),\quad t\in (0,1), \]
\[ -y''(t)=g(t,x(t),y(t)),\quad t\in(0,1), \]
\[ x(0)=y(0)=0, \quad x(1)=\alpha x(\eta),\quad y(1)=\alpha y(\eta), \]
is established. The nonlinearities \(f,g:(0,1)\times(0,\infty)\times(0,\infty)\to[0,\infty)\) are continuous and may be singular at \(t=0\), \(t=1\), \(x=0\), and/or \(y=0\), while the parameters \(\eta,\alpha\) satisfy \(\eta\in(0,1)\), \(0<\alpha<1/\eta\). An example is also included to show the applicability of our result.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
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References:

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