Almost automorphic solutions of semilinear evolution equations
We are concerned with the semilinear differential equation in a Banach space double-struck X sign, x′(t) = Ax(t) + F(t,x(t)), t ∈ ℝ, where A generates an exponentially stable C 0-semigroup and F(t, x) : ℝ × double-struck X sign → double-struck X sign is a function of the form F(t, x) = P(t)Q(x). Under appropriate conditions on P and Q, and using the Schauder fixed point theorem, we prove the existence of an almost automorphic mild solution to the above equation. © 2005 American Mathematical Society.
Proceedings of the American Mathematical Society
Goldstein, J., & N'Guérékata, G. (2005). Almost automorphic solutions of semilinear evolution equations. Proceedings of the American Mathematical Society, 133 (8), 2401-2408. https://doi.org/10.1090/S0002-9939-05-07790-7