Properties of eigenvalues of positive operators depending nonlinearly on a parameter

It is proved that in a Banach space with a cone, the nonlinear (with respect to the spectral parameter ) eigenvalue problem (here is a scalar function and M is an operator-function, positive with respect to the cone for 0) admits a simple positive eigenvalue ₁. The number ₁ is characterized by variational principles. The obtained results are illustrated on the example of the quadratic pencil y+y+²B²x²y=0, which arises in the theory of structural analysis (see Ref. Zh. 1974, 9B1134).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 56, pp. 177–181, 1976.The author expresses his deep gratitude to D. F. Kharazov and Yu. Sh. Abramov for the interest shown and the advice given during the preparation of this paper.