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A pencil of fourth-order differential equations on the entire axis with multiple characteristics is considered. Using special solutions of a fourth-order differential equation, the spectrum of a differential pencil is studied. Necessary and sufficient conditions are found for a non-real number to be an eigenvalue of this pencil. It is proved that the operator pencil has no eigenvalues on the real axis.
Gafarova et al. (Fri,) studied this question.