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Existence and multiplicity of positive solutions for an elastic beam equation. (English) Zbl 1249.34082

Summary: This paper investigates the boundary value problem for an elastic beam equation of the form

${u}^{\text{'}\text{'}\text{'}\text{'}}\left(t\right)=q\left(t\right)f\left(t,u\left(t\right),{u}^{\text{'}}\left(t\right),{u}^{\text{'}\text{'}}\left(t\right),{u}^{\text{'}\text{'}\text{'}}\left(t\right)\right),\phantom{\rule{1.em}{0ex}}0

with the boundary conditions

$u\left(0\right)={u}^{\text{'}}\left(1\right)={u}^{\text{'}\text{'}}\left(0\right)={u}^{\text{'}\text{'}\text{'}}\left(1\right)=0·$

The boundary conditions describe the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. By using Leray-Schauder nonlinear alternative, Leray-Schauder fixed point theorem and a fixed point theorem due to R. I. Avery and A. C. Peterson [Comput. Math. Appl. 42, No. 3–5, 313–322 (2001; Zbl 1005.47051)], we establish some results on the existence and the multiplicity of positive solutions to the boundary value problem.

##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B15 Nonlinear boundary value problems for ODE 47N20 Applications of operator theory to differential and integral equations 74K10 Rods (beams, columns, shafts, arches, rings, etc.) in solid mechanics
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