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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 $f:[0,1]\times\bbfR^+\times \bbfR^-\to\bbfR^+$ 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:
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
74G25Global existence of solutions for equilibrium problems in solid mechanics
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References:
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