Self-adjoint fourth order differential operators with eigenvalue parameter dependent and periodic boundary conditions

Résumé

Fourth order eigenvalue problems with periodic and separated boundary conditions are considered. One of the separated boundary conditions depends linearly on the eigenvalue parameter λ. These problems can be represented by an operator polynomial $L(\lambda)=\lambda ^{2}M-i\alpha\lambda K-A$ , where $\alpha>0$ , M and K are self-adjoint operators. Necessary and sufficient conditions are given such that A is self-adjoint.

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