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Positive solutions of fourth order problems with clamped beam boundary conditions. (English) Zbl 1221.34061
The authors study the fourth order linear operator ${u}^{\left(4\right)}+Mu$ coupled with the clamped beam conditions $u\left(0\right)=u\left(1\right)={u}^{\text{'}}\left(0\right)={u}^{\text{'}}\left(1\right)=0$. They obtain the exact values of the real parameter $M$ for which this operator satisfies an anti-maximum principle. When $M<0$, they obtain the best estimate by means of the spectral theory and, for $M>0$, they attain the optimal value by studying the oscillation properties of the solutions of the homogeneous equation ${u}^{\left(4\right)}+Mu=0$. By using the method of lower and upper solutions, they also prove the existence of solutions of nonlinear problems coupled with these boundary conditions.
##### MSC:
 34B18 Positive solutions of nonlinear boundary value problems for ODE 34B27 Green functions 34B15 Nonlinear boundary value problems for ODE 34B05 Linear boundary value problems for ODE 34A40 Differential inequalities (ODE)
##### References:
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