Existence and uniqueness of almost automorphic mild solutions to some semilinear abstract differential equations. (English) Zbl 1077.47058

N’Guérékata, Gaston M.

Semigroup Forum 69, No. 1, 80-86 (2004).

The paper studies the existence of unique almost automorphic solutions to the equations \[ x'(t)=Ax(t)+f(t),\;x'(t)=Ax(t)+g \bigl(t,x(t)\bigr),\quad t\in\mathbb{R},\tag{1} \] when \(A\) generates an exponentially \(C_0\)-semigroup on a Banach space \(X\), and the functions \(f(t)\) and \(g(t,x)\) are almost automorphic in \(t\in\mathbb{R}\) for each \(x\in X\).

Reviewer: Lahcen Maniar (Marrakech)

MSC:

47N20	Applications of operator theory to differential and integral equations
34G10	Linear differential equations in abstract spaces
47D06	One-parameter semigroups and linear evolution equations

Keywords:

semilinear differential equations; exponentially \(C_0\)-semigroup