Descripción
This talk is divided in two parts. First we analyse two fourth order problems - one with periodic conditions and another with simply supported conditions - allowing the nonlinearity to depend on x, u(x) and u″(x). In both cases the fourth order operator can be decomposed into two second order operators, and using maximum principles it is possible to prove existence of a solution between lower and upper solutions (eventually in reversed order). The second part deals with the fourth order operator u(4)+Mu coupled with the clamped beam conditions, for which the approach used previously is not possible. We obtain the exact values on the real parameter M for which this operator satisfies a maximum principle. When M<0 we obtain the best estimate by means of the spectral theory and for M>0 we 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 we deduce the existence of solutions for nonlinear problems (nonlinearity not depending on the derivatives) with this boundary conditions.
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