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A relationship between the existence of well-posed boundary-value problems and exponential dichotomy for functional equations and linear differential-operator equations on a half-line is considered. It is shown that well-posed boundary-value problems can exist for equations without the exponential dichotomy property. [ABSTRACT FROM AUTHOR]

We introduce a number of notions related to the Lyapunov transformation of linear differential operators with unbounded operator coefficients generated by a family of evolution operators. We prove statements about similar operators related to the Lyapunov transformation and describe their spectral properties. One of the main results of the paper is a similarity theorem for a perturbed differential operator with constant operator coefficient, an operator which is the generator of a bounded group of operators. For the perturbation, we consider the operator ofmultiplication by a summable operator function. The almost periodicity (at infinity) of the solutions of the corresponding homogeneous differential equation is established. [ABSTRACT FROM AUTHOR]

The problem of stability in part of the variables of a 'partial' equilibrium (this means that a given part of the phase vector coordinates is zero) is considered for nonlinear nonstationary systems of functional differential equations with aftereffect. The notions of stability in part of the variables, which admit more general (compared with the known ones) assumptions about the values of the supremum-norm of the components of the initial vector function corresponding to the variables that do not determine the given equilibrium, are introduced. The stability and asymptotic stability conditions of the the type mentioned above are obtained in the context of the method of Lyapunov-Krasovskii functionals; this conditions allow generalization of several well-known results. [ABSTRACT FROM AUTHOR]

The spectral properties of operators constructed from a family of evolution operators are studied in Banach spaces of vector functions defined by a weight function of pre-exponential growth. Necessary and sufficient conditions for the invertibility of such operators are obtained in terms of exponential dichotomy. We prove a spectrum mapping theorem for semigroups of Howland difference operators generated by the operator under study. [ABSTRACT FROM AUTHOR]

We consider a class of semigroups of operators in Hilbert space whose generators are linear relations. An analog of the Gearhart-Prüss theorem for semigroups of operators from the class under consideration is obtained. [ABSTRACT FROM AUTHOR]

We obtain necessary and sufficient conditions for the hyperbolicity of a semigroup of operators. In so doing, we use Lyapunov's equation in operator form constructed from its generator. [ABSTRACT FROM AUTHOR]

The paper deals with averaging problems for parabolic equations. We prove that exponential dichotomy is preserved without any assumption concerning the almost-periodicity of the coefficients. [ABSTRACT FROM AUTHOR]

We obtain a criterion for invertibility in Hölder spaces of linear parabolic operators of arbitrary order with any number of spatial variables and almost-periodic coefficients. [ABSTRACT FROM AUTHOR]

The article investigates the exponential dichotomy of the solutions for the linear difference system with a matrix that is almost periodic within the environment of the direct Lyapunov technique. Mathematical proof is provided that the system has exponential dichotomy because an indefinite Hermitian bounded nondegenerate matrix exists and the phase space E is divided into the direct sum of the subspaces E1 and E2.