Abstract and Applied Analysis
Volume 7 (2002), Issue 12, Pages 637-661
doi:10.1155/S1085337502208015

Bounded solutions of nonlinear Cauchy problems

Abstract

For a given closed and translation invariant subspace Y of the bounded and uniformly continuous functions, we will give criteria for the existence of solutions u∈Y to the equation u′(t)+A(u(t))+ωu(t)∍f(t),t∈ℝ, or of solutions u asymptotically close to Y for the inhomogeneous differential equation u′(t)+A(u(t))+ωu(t)∍f(t),t>0,u(0)=u0, in general Banach spaces, where A denotes a possibly nonlinear accretive generator of a semigroup. Particular examples for the space Y are spaces of functions with various almost periodicity properties and more general types of asymptotic behavior.