Hölder Regularity for Abstract Fractional Cauchy Problems with Order in (0,1)

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In this paper, we study the regularity of mild solution for the following fractional abstract Cauchy problem* D*_{t}^{ α}*u(t)=Au(t)+f(t), t ∈ (0,T] u(0)= x*_{0} on a Banach space *X* with order *α* ∈ (0,1), where the fractional derivative is understood in the sense of Caputo fractional derivatives. We show that if *A* generates an analytic α-times resolvent family on *X* and* f ∈ L*^{p}* ([0,T];X*) for some *p* > 1/α, then the mild solution to the above equation is in *C*^{α-1/p}*[ò,T]* for every ò > 0. Moreover, if *f* is Hölder continuous, then so are the *D*_{t}^{ α}*u(t)* and *Au(t)*.

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