Syrotenko A. Bounded and intagrable with power solutions to difference and differential equations in Banach space

M.G. Krein's classical theorem asserts that a differential equation with a bounded operator coefficient has a unique bounded solution for an arbitrary "input" function, which is continuous and bounded on the real axis, if and only if the spectrum of the operator coefficient does not intersect the imaginary axis of the complex plane. A similar result for the difference-operator equation was obtained by V. Yu. Slyusarchuk, who stated that for such an equation, the spectrum of the operator coefficient should not intersect the unit circle centered at zero. In the thesis, the problem of the existence of bounded solutions to difference and differential equations with operator coefficient is studied in the case when the specified conditions on the spectrum of the operator coefficient are not fulfilled. Thus, the difference-operator and differential-operator equations, in which the "input" sequences or functions belong to some special classes, are considered. For an difference equation with a discrete argument in a Banach space the criterion for the existence and uniqueness of a bounded on or -power summable on solution is proved for any "input" sequence with elements belonging to some special set of the Banach space. In addition to boundedness, in the second section we study the existence of a unique p-power summable solution to the difference-operator equation. This result is a generalization of the result obtained by M.F. Horodnii and O.V. Vyatchaninov for a difference equation in the Banach space considered on N. In this case, the "input" sequences take the values in the set of all such elements u of the Banach space that the Cauchy problem for a homogeneous difference equation with the initial condition u has a bounded solution.