About properties of solutions of some mixed spectral problems

Previously, boundary value problems, spectral and initial boundary value problems were investigated based on the symmetric sesquilinear form $(\eta, u)_{H^{1}(\Omega)}=\Phi_{0}(\eta, u)$. For example, M. S. Agranovich, N. D. Kopachevsky, V. I. Voititsky, P. A. Starkov, K. A. Radomirskaya and others were engaged in such research. Kopachevsky N. D. proposed to investigate problems for one and two domains in the case, where the scalar product $(\eta, u)_{H^{1}(\Omega)}=\Phi_{0}(\eta, u)$ is replaced with a sesquilinear nonsymmetric form $\Phi _{\varepsilon}(\eta, u)$. The form $\Phi _{\varepsilon}(\eta, u)$ is defined on the space $H^{1}(\Omega)$, bounded and uniformly accretive. Thus, the main goal of this work is to study problems in the nonsymmetric case. The parameter $\varepsilon \in \mathbb{R}$ was introduced for convenience of consideration, and all the problems turn into the corresponding unperturbed problems as $\varepsilon \rightarrow 0$. Based on the already considered problems in the case of one domain (see [1]), we study mixed spectral conjugation problems generated by the sesquilinear form. In this case, the principle of superposition is used. The principle makes it possible to represent the solution of the original problem as a sum of solutions to auxiliary problems. These auxiliary problems contain inhomogeneity in only one place, that is, either in the equation or in one of the boundary conditions. Studying of the spectral problems leads to the same operator pencil $L(\lambda, \mu)$, which is investigated with the methods of the spectral theory of operator pencils (see [2]). The properties of solutions of the spectral problems in the case $\lambda$ is fixed and $\mu$ is spectral parameter or $\mu$ is fixed and $\lambda$ is spectral parameter are investigated. The theorems on the spectrum structure, on the basis property of root elements, on the spectrum localization under the first and second conjugation conditions are proved.