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Almost automorphic mild solutions to fractional differential equations. (English) Zbl 1166.34033

Authors’ abstract: We introduce the concept of α-resolvent families to prove the existence of almost automorphic mild solutions to the differential equation

D t α u(t)=Au(t)+t n f(t),1α2,n

considered in a Banach space X, where f:RX is almost automorphic. We also prove the existence and uniqueness of an almost automorphic mild solution of the semilinear equation

D t α u(t)=Au(t)+f(t,u(t)),1α2

assuming f(t,x) is almost automorphic in t for each xX, satisfies a global Lipschitz condition and takes values on X. Finally, we prove also the existence and uniqueness of an almost automorphic mild solution of the semilinear equation

D t α u(t)=Au(t)+f(t,u(t),u ' (t)),1α2

under analogous conditions as in the previous case.

34G20Nonlinear ODE in abstract spaces
26A33Fractional derivatives and integrals (real functions)
43A60Almost periodic functions on groups, etc.; almost automorphic functions