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Zbl 1208.34016
Ma, Ruyun; Chen, Tianlan
Existence of positive solutions of fourth-order problems with integral boundary conditions.
(English)
[J] Bound. Value Probl. 2011, Article ID 297578, 17 p. (2011). ISSN 1687-2770/e

Summary: We study the existence of positive solutions of the following fourth-order boundary value problem with integral boundary conditions $$u^{(4)}(t)=f(t,u(t),u''(t)),\quad t\in (0,1),$$ $$u(0)=\int_0^1g(s)u(s)\,ds,\quad u(1)=0,\quad u''(0)=\int^1_0 h(s)u''(s)\,ds,\ u''(1)=0,$$ where $f:[0,1]\times [0,+\infty)\times (-\infty,0]\to [0,+\infty)$ is continuous, $g,h\in L^1[0,1]$ are nonnegative. The proof of our main result is based upon the Krein-Rutman theorem and global bifurcation techniques.
MSC 2000:
*34B10 Multipoint boundary value problems
34B18 Positive solutions of nonlinear boundary value problems
47N20 Appl. of operator theory to differential and integral equations
34C23 Bifurcation (periodic solutions)

Keywords: Krein-Rutman theorem; bifurcation techniques

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