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Almost automorphic solutions to semilinear evolution equations. (English) Zbl 1102.34044

The authors investigate the semilinear differential equations

x ' (x)+Ax(t)=f(t,x(t)),t,(1) x ' (t)+(A+B)x(t)=f(t,x(t)),t,(2)

where -A is the infinitesimal generator of an analytic semigroup {T(t);t} on the Banach space X, B is a densely defined closed operator and f:×XX is an almost automorphic function. Using the fractional powers of A and the uniform spectrum of f, they prove the existence and uniqueness of almost automorphic (classical) solutions of (1) and (2).

34G20Nonlinear ODE in abstract spaces
47D03(Semi)groups of linear operators
34C27Almost and pseudo-almost periodic solutions of ODE