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Positive solutions for eigenvalue problems of fourth-order elastic beam equations. (English) Zbl 1072.34022

The following fourth-order eigenvalue problem

w (4) (t)=λf(t,w(t)),0<t<1,λ>0,w(0)=w(1)=w ' (0)=w ' (1)=0,(1)

is studied. The author uses the Krasnoselskii-Guo fixed-point theorem on cone expansion and compression and the properties of the Green function of the corresponding homogeneous BVP to obtain several results on the existence of positive solutions of (1). Some multiplicity results are established, too.

MSC:
34B18Positive solutions of nonlinear boundary value problems for ODE
74K10Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
34B15Nonlinear boundary value problems for ODE