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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)
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