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Eigenvalues of fourth-order singular Sturm-Liouville boundary value problems. (English) Zbl 1131.34022

Summary: In this paper, by establishing a new comparison theorem and constructing upper and lower solutions, some sufficient conditions of existence of positive solutions for the following nonlinear fourth-order singular Sturm-Liouville eigenvalue problem: \[ \begin{cases}\frac{1}{p(t)}\,(p(t)u'''(t))'=\lambda f(t,u),\quad & t\in(0,1),\\ u(0)=u(1)=0,\\ \alpha u''(0)-\beta\lim_{t\to 0+}p(t)u'''(t)=0,\\ \gamma u''(1)+\delta\lim_{t\to 1-}p(t)u'''(t)=0,\end{cases} \] are established due to the Schauder’s fixed point theorem for \(\lambda\) large enough, where \(\alpha,\beta,\gamma,\delta\geq 0\), \(\beta\gamma+\alpha\gamma+\alpha\delta>0\), \(f\) and \(p\) can be singular at \(t=0\) and/or 1; moreover \(f\) can also be singular at \(u=0\). In addition, some peculiar cases are discussed and some further results are obtained.

MSC:

34B24 Sturm-Liouville theory
34B15 Nonlinear boundary value problems for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34B16 Singular nonlinear boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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References:

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