Almost automorphic mild solutions to fractional differential equations

We introduce the concept of αα-resolvent families to prove the existence of almost automorphic mild solutions to the differential equation Dtαu(t)=Au(t)+tnf(t),1≤α≤2,n∈Z+ considered in a Banach space XX, where f:R→Xf:R→X is almost automorphic. We also prove the existence and uniqueness of an almost automorphic mild solution of the semilinear equation Dtαu(t)=Au(t)+f(t,u(t)),1≤α≤2 assuming f(t,x)f(t,x) is almost automorphic in tt for each x∈Xx∈X, satisfies a global Lipschitz condition and takes values on XX. Finally, we prove also the existence and uniqueness of an almost automorphic mild solution of the semilinear equation Dtαu(t)=Au(t)+f(t,u(t),u′(t)),1≤α≤2, under analogous conditions as in the previous case.