For a linear inhomogeneous differential equation in a Banach space, we find a criterion for the existence of solutions that are bounded on the entire real axis under the assumption that the homogeneous equation admits an exponential dichotomy on the semiaxes. This result is a generalization of the Palmer lemma to the case of infinite-dimensional spaces. We consider examples of countable systems of ordinary differential equations that have bounded solutions.
Statement of the ProblemIn a Banach space B, we consider the differential equationwhere the operator functionthe Banach space of functions continuous and bounded on R, the operator function A(t) is strongly continuous [1, p. 141], |||A||| = sup t∈R A(t) < ∞, and a solution x(t) of the equation x(t) = x 0 + t t 0 (A(s)x(s) + f (s)) ds is continuously differentiable at every point t ∈ R and satisfies Eq. (1) everywhere in R. We seek a bounded solution x(t) of Eq. (1) in the Banach space BC 1 (R, B) of functions continuously differentiable on R and bounded together with their derivatives. Let us find conditions for the existence of solutions x(t) ∈ BC 1 (R, B) of Eq.(1) bounded on the entire axis R under the assumption that the corresponding homogeneous equationadmits an exponential dichotomy on the semiaxes. In the case of finite-dimensional spaces, where B = R n , this problem is solved by the well-known Palmer lemma [2, p. 245].
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.