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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 (4) +Mu coupled with the clamped beam conditions u(0)=u(1)=u ' (0)=u ' (1)=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 (4) +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:
34B18Positive solutions of nonlinear boundary value problems for ODE
34B27Green functions
34B15Nonlinear boundary value problems for ODE
34B05Linear boundary value problems for ODE
34A40Differential inequalities (ODE)
References:
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