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Zbl 1081.34025
Yang, Bo
Positive solutions for a fourth order boundary value problem.
(English)
[J] Electron. J. Qual. Theory Differ. Equ. 2005, Paper No. 3, 17 p., electronic only (2005). ISSN 1417-3875/e

The following boundary value problem is considered: $$u^{(4)}(t)=g(t)f(u(t)),\quad t\in [0,1],\qquad u(0)=u'(0)=u''(1)=u'''(1)=0. \tag1$$ The author establishes several results on the existence of at least one positive solution to (1) applying the Krasnoselskii-Guo fixed-point theorem on cone expansion and compression in the space $C[0,1]$ with the cones $$P=\{v\in C[0,1]:v(1)\ge 0,a(t)v(1)\le v(t)\le tv(1), t\in[0,1]\}$$ and $$P_1=\{v\in C[0,1]:v(1)\ge 0, v \text{ nondecreasing on } [0,1],\ a(t)v(1)\le v(t)\le b_1(t)v(1), t\in[0,1]\},$$ with $a(t)=\frac 32 t^2-\frac12 t^3$ and $b_1(t)=2t^2-\frac43 t^3+\frac13 t^4$, $t\in [0,1]$.
[Mirosława Zima (Rzeszow)]
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems
34B15 Nonlinear boundary value problems of ODE

Keywords: beam equation; cone; positive solution; Krasnoselskii's fixed-point theorem

Cited in: Zbl 1242.34037

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