ASYMPTOTICALLY -ALMOST PERIODIC SOLUTIONS TO DIFFERENTIAL EQUATIONS IN BANACH SPACES

In this paper, we establish the existence and uniqueness result of asymptotically $(\omega, c)$-almost periodic mild solutions to semilinear differential equations in a Banach space. For this purpose, we first give some properties of $(\omega, c)$-almost periodic functions and asymptotically $(\omega, c)$-almost periodic functions, including the composition theorems. Then we obtain the existence and uniqueness result of $(\omega, c)$-almost periodic mild solution to the semilinear differential equation, and the existence and uniqueness theorems for $(\omega, c)$-almost periodic and asymptotically $(\omega, c)$-almost periodic mild solutions for the corresponding linear equation. Moreover, an example of the heat equation is given to illustrate the abstract results.