Massera-Type Results For CN -Almost Automorphic Solutions Of Some Differential Equations

Abstract

We are concerned with the existence of C(n)-almost automorphic solutions of the equation (1) x'(t) = A(t)x(t) + f(t) where A(t) is a τ-periodic operator and f is C(n)-almost automorphic. In 1950, Massera proved that if the equation (2) x'(t)=Ax(t) + f(t) where f(t) is periodic and A is a constant n x n matrix, has a bounded solution on R+, then it has a periodic solution on all of R. This result has since been generalized for many different cases, including when f(t) is almost periodic in an infinite dimensional space. In this thesis, we apply Floquet's Theory and some recent results on C(n)-almost automorphic functions to prove some Massera-type results for the nonautonomous differential equation (3) x'(t)=A(t)x(t)+f(t) in Ck. We show that if A(t) is periodic, f(t) is C(n)-almost automorphic, and Equation (3) has a bounded solution on R+, then it has a C(n+1)-almost automorphic solution on the whole real line. Moreover, every bounded solution of Equation (3) is C(n+1)-almost automorphic. We published these results in CUBO A Mathematical Journal in 2008. Secondly, we study the convolution operator of f with h, denoted (f*h) defined by (f*h)(t) := integral of f(σ)h(t-σ)dσ for all t in R where h is in L1(R) and explore the existence of C(n)-almost automorphic solutions of the convolution equation (4) u(t)=f(t) + integral of a(t-s)u(s)ds, for all t in R, where f: R /&rarr ; R is C(n)-almost automorphic and a is Lebesgue measurable. These generalize some recent results.