Existence and uniqueness of positive solutions to a class of singular \(m\)-point boundary value problems. (English) Zbl 1158.34315

Du, Xinsheng; Zhao, Zengqin

Appl. Math. Comput. 198, No. 2, 487-493 (2008).

The paper deals with the problem of existence and uniqueness of positive solutions of the \(m\)-point boundary value problem
\[ -u''=f(t,u), \quad t\in (0,1) \]
with conditions
\[ u(0)= \sum^{m-2}_{i=1}\alpha_{i} u(t_{i}), \quad u(1)=0, \]
where \(\alpha_{i}> 0,\quad \sum^{m-2}_{i=1}\alpha_{i}<1\), \(0<t_{1} <t_{2}< \dots <t_{m-2}<1,\) and where \(f\) is a positive decreasing function with respect to \( u \) ( \(f \) may be singular at \(u=0\), \(t=0 \) and/or \(t=1\)). A sufficient condition for the existence and uniqueness of \(C[0,1]\) and \(C^{1}[0,1] \) positive solutions is given by constructing lower and upper solutions.

Reviewer: Carmen Simerská (Praha)

MSC:

34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B16	Singular nonlinear boundary value problems for ordinary differential equations
34B10	Nonlocal and multipoint boundary value problems for ordinary differential equations

Keywords:

m-point boundary value problem; existence and uniqueness; positive solutions

References:

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