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Positive solutions of fourth-order singular three point eigenvalue problems. (English) Zbl 1126.34321

Summary: By establishing a new comparison theorem and constructing upper and lower solutions, some sufficient conditions of existence of positive solutions for the following singular fourth order three point eigenvalue problem \[ \begin{cases} u^{(4)} (t)=\lambda f(t,u),\;t\in(0,1),\\ u(0)=\alpha u(\eta),\;u(1)=0,\\ u''(0)=\beta u''(\eta),\;u''(1)=0,\end{cases} \] are established due to Schauder’s fixed point theorem for \(\lambda\) large enough, where \(\alpha,\beta,\eta\in(0,1)\) are constants, \(f\) can be singular at \(t=0\) and/or 1, \(u=0\). Moreover, some peculiar cases are discussed and some further results are obtained.

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
34B24 Sturm-Liouville theory
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