Abstract
At the end of the nineteenth century A. Poincare began to study equations which were unsolved with respect to high derivative equations. The systematical study of such equations began in S.L. Sobolev`s works in the second part of the last century. Therefore, such equations are called Sobolev type equations. The increased interest to Sobolev type equations led to the necessity to consider them in quasi-Banach spaces. This article presents the results of the existence of exponential dichotomies of solutions of dynamical Sobolev type equations studied in quasi-Banach spaces. The relatively spectral theorem and the problem of the existence of invariant solution spaces were considered. The interest to such solution is explained by the fact that it is the most popular and reflects experimental data while solving practical tasks. Besides the introduction and the references the article contains two parts. The first part provides necessary notions and a relatively spectral theorem in qiasi-Banach spaces. The second one represents the existence of invariant spaces and exponential dichotomies of solutions of the dynamical Sobolev type equation in quasi-Banach spaces
Highlights
Рассматривается вопрос существования экспоненциальных дихотомий решений динамических уравнений соболевского типа, рассматриваемых в квазибанаховых пространствах
The increased interest to Sobolev type equations led to the necessity to consider them in quasi-Banach spaces
This article presents the results of the existence of exponential dichotomies of solutions of dynamical Sobolev type equations studied in quasi-Banach spaces
Summary
(i) U u ≥ 0 при всех u ∈U , причем U u = 0 точно тогда, когда u = 0, где 0 – нуль линеала U ; (ii) U α u =| α | U u при всех u ∈U , ∀α ∈ R ;. ( ) (iii) U u + v ≤ C U u + U v при всех u,v ∈U , где константа C ≥ 1. Широко известным примером квазибанахова пространства служит пространство последовательностей, l q при q ∈ (0,1) , константа. В [5] построены так называемые квазисоболевы пространства l m q при всех m∈R , q ∈ R+ , причем l0q = lq. Что линейный оператор L :U → F с областью определения dom L = U непрерывен точно тогда, когда он ограничен Линеал L (U ; F ) линейных ограниченных операторов – квазибанахово пространство с квазинормой.
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