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| Eigenvalues of Fourth-order SingularSturm-Liouville Boundary Value Problems |
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| KeyWord:Sturm-Liouville problems Eigenvalue Krasnoselskii’s fixed-point
theorem |
| Author Name | Affiliation | | Lina Zhou | School of Mathematical Science, Hebei Normal University, Shijiazhuang,
Hebei 050024, China | | Weihua Jiang | College of Science, Hebei University of Science and Technology, Shijiazhuang,
Hebei 050018, China | | Qiaoluan Li | School of Mathematical Science, Hebei Normal University, Shijiazhuang, Hebei 050024, China |
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| Abstract: |
| In this paper, by using Krasnoselskii's fixed-point theorem, some sufficient conditions of existence of positive solutions for the following fourth-order nonlinear Sturm-Liouville eigenvalue problem:\begin{equation*}\left\{\begin{array}{lll}
\frac{1}{p(t)}(p(t)u''')'(t)+ \lambda f(t,u)=0, t\in(0,1),
\\ u(0)=u(1)=0,
\\ \alpha u''(0)- \beta \lim_{t \rightarrow 0^{+}} p(t)u'''(t)=0,
\\ \gamma u''(1)+\delta\lim_{t \rightarrow 1^{-}} p(t)u'''(t)=0,
\end{array}\right.\end{equation*}
are established, where $\alpha,\beta,\gamma,\delta \geq 0,$ and $~\beta\gamma+\alpha\gamma+\alpha\delta >0$.
The function $p$ may be singular at $t=0$ or $1$, and $f$ satisfies Carath\'{e}odory condition. |
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