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Zbl 1221.34061
Cabada, Alberto; Enguiça, Ricardo Roque
Positive solutions of fourth order problems with clamped beam boundary conditions. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74, No. 10, 3112-3122 (2011). ISSN 0362-546X

The authors study the fourth order linear operator $u^{(4)} + M u$ 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 $u^{(4)} + M u = 0$. By using the method of lower and upper solutions, they also prove the existence of solutions of nonlinear problems coupled with these boundary conditions.
[Ruyun Ma (Lanzhou)]

MSC 2000:: *34B18 Positive solutions of nonlinear boundary value problems
34B27 Green functions
34B15 Nonlinear boundary value problems of ODE
34B05 Linear boundary value problems of ODE
34A40 Differential inequalities (ODE)

Keywords: clamped beam; fourth order boundary value problem; maximum principles; lower and upper solutions

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