Abstract
This chapter presents some remarks on boundedness and asymptotic equivalence of ordinary differential equations. It presents linear and nonlinear equations where y, x, A, f are functions in a Banach space E and the independent variable. E1 is considered to be the subspace of E consisting of all the points of E which are values for t = 0 of bounded solutions. The chapter presents an assumption that E1 can be complemented in E by a closed subspace E2 and denoted by P1 P2 the corresponding projections of E onto E1, and E2. The necessary and sufficient conditions for the existence of bounded solutions are given. The theorem gives sufficient conditions for the existence of asymptotic equivalence.
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