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On the existence of positive solutions for the bending elastic beam equations. (English) Zbl 1118.74032
Summary: We discuss the existence of positive solutions of the fourth-order boundary value problem $$u^{(4)}=f(t,u,u'')$$, $$0<t<1$$, $$u(0)=u(1)= u''(0)=u''(1)=0$$, which models a statically bending elastic beam whose two ends are simply supported, where $$f:[0,1]\times\mathbb{R}^+\times \mathbb{R}^-\to\mathbb{R}^+$$ is continuous. We derive conditions on $$f$$ guaranteeing the existence of positive solution. The discussion is based on the fixed point index theory in cones.

##### MSC:
 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74G25 Global existence of solutions for equilibrium problems in solid mechanics (MSC2010)
##### Keywords:
cone; fixed point index
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##### References:
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