Positive solutions for second-order \(m\)-point boundary value problems. (English) Zbl 1026.34016

Summary: By using the Schauder fixed-point theorem and analysis method, we establish the existence of solutions to the \(m\)-point boundary value problem \[ u''(t)+ a(t)f(u)=0, \quad u(0)=0,\;u(1)-\sum^{m-2}_{i=1} k_iu(\xi_i)= b, \] where \(b,k_i>0\), \(i=1,2,\dots, m-2\), \(0<\xi_1 <\xi_2< \cdots <\xi_{m-2}<1\) and \(a(t)\) is allowed to be singular at \(t=0,1\). Under some conditions, we show that there exists a positive number \(b^*\) such that the problem has at least one positive solution for \(0<b<b^*\) and no solution for \(b>b^*\).


34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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