Positive solutions for singular boundary value problems involving integral conditions. (English) Zbl 1282.34030
Summary: We are interested in the singular boundary value problem:
\[
\begin{aligned} &u''(t)+ \mu w(t)f(u(t))=0, \quad t\in(0,1) ,\\ &u(0)=0, \quad u(1)=\int_{0}^{1}u(s)dA(s), \end{aligned}
\]
where \(\mu > 0\) is a parameter and \(\int_{0}^{1}u(s)dA(s)\) is a Stieltjes integral. The function \({w\in C((0, 1), (0, +\infty))}\) may be singular at \(t = 0\) and/or \(t = 1\), \(f\in C([0,+\infty),(0,\infty))\). Some a priori estimates and the existence,
multiplicity and nonexistence of positive solutions are obtained. Our proofs are based
on the method of the global continuation theorem, the lower-upper solutions method
and the fixed point index theory. Furthermore, we also discuss the interval of parameter
\(\mu\) such that the problem has a positive solution.
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 |
| 47N20 | Applications of operator theory to differential and integral equations |
| 34B09 | Boundary eigenvalue problems for ordinary differential equations |
Keywords:
singularity; global continuous theorem; solution of boundedness; fixed point index; positive solutionReferences:
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