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

In this paper, we study the regularity of mild solution for the following fractional abstract Cauchy problem ( ) ( ) ( ), (0, ] t D u t Au t f t t T α = + ∈ (0) u = 0 x on a Banach space X with order (0,1) α ∈ , 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 ([0, ]; ) p f L T X ∈ for some 1/ p α > , then the mild solution to the above equation is in 1/ [ , ] p C T α−  for every 0 >  . Moreover, if f is Hölder continuous, then so are the ( ) t D u t α and ( ) Au t .

, 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 ([0, ]; ) , then the mild solution to the above equation is in

Introduction
Recently there are increasing interests on fractional differential equations due to their wide applications in viscoelasticity, dynamics of particles, economic and science et al.For more details we refer to [1] [2].
Many evolution equations can be rewritten as an abstract Cauchy problem, and then they can be studied in an unified way.For example, a heat equation with different initial data or boundary conditions can be written as a first order Cauchy problem, in which the governing operator generates a C 0 -semigroup, and then the solution is given by the operation of this semigroup on the initial data.See for instance [3] [4].Prüss [5] developed the theory of solution operators to research some abstract Volterra integral equations and it was Bajlekova [6]  nerates a C 0 -semigroup, we can invoke an operator described by the C 0 -semigroup and a probability density function to solve this problem, for more details we refer to [7] [8] [9].The vector-valued Laplace transform developed in [3] is an important tool in the theory of fractional differential equations.
There are some papers devoted to the fractional differential equations in many different respects: the connection between solutions of fractional Cauchy problems and Cauchy problems of first order [10]; the existence of solution of several kinds of fractional equations [11] [12]; the Hölder regularity for a class of fractional equations [13] [14]; the maximal p L regularity for fractional order equations [6]; the boundary regularity for the fractional heat equation [15]; the relation of continuous regularity for fractional order equations with semi-variations [12].In this paper we are mainly interested in the Hölder regularity for abstract Cauchy problems of fractional order.
Pazy [4] considered the regularity for the abstract Cauchy problem of first order: where A is the infinitesimal generator of an analytic C 0 -semigroup.He showed T .If in addition f is Hölder continuous, then Pazy showed that there are some further regularity of ( ) Au t and d d u t .Li [16] gave similar results for fractional differential equations with order (1, 2) α ∈ . In this paper we will extend their results to fractional Cauchy problems with order in (0,1) .
Our paper is organized as follows.In Section 2 there are some preliminaries on fractional derivatives, fractional Cauchy problems and fractional resolvent families.In Section 3 we give the regularity of the mild solution under the condition that ([0, ], ) . And some further continuity results are given in Section 4.

Preliminaries
Let A be a closed densely defined linear operator on a Banach space X.In this paper we consider the following equation: where u and f are X-valued functions, 0 1 α < < , and t D α is the Caputo fractional derivative defined by and 0 ( ) g t is understood as the Dirac measure δ at 0. The convolution of two functions f and g is defined by when the above integrals exist.
The classical (or strong) solution to (2.1) is defined as: 2) 3) If there is an α-times resolvent family ( ) S t α for the operator A, then the mild solution of (2.1) is given by the following lemma.Lemma 2.4.[10] Let A generate an α-times resolvent family S α and let For the strong solution of (2.1), we have Lemma 2.5.[10] Let A generate an α-times resolvent family S α and let 0 ( ) α ∈ , then the following statements are equivalent: If in addition, the α-times resolvent family ( ) S t α admits an analytic extension to some sector If A α ∈  , then there exists constants C, ω and 0 θ such that ( ) . The α-times resolvent family generated by A can be given by Proof.(1) By the definition of ( ) e e e e d r , we can obtain that the above integral is bounded by .
By the closedness of the operator A, the assertion of (2) follows.
(3) By the proof of ( 2) and the closedness of A, .
And the second part of (3) can be proved similarly.□ Remark 2.7.Similar results for (1, 2) were given in [16].It is obvious that

Regularity of the Mild Solutions
In this section we consider the mild solution of (2.1) with 0 1 α < < .Suppose that the operator A generates an analytic α-resolvent family, then by Lemma 2.4 and Remark 2.7 the mild solution of (2.1) is given by Then for every 0 , where ( ) u t is given by (3.1).If moreover 0 ( ) Proof.Since ( ) S t α is analytic, we only need to show that 1 ( )( ) .    If we put more conditions on ( ) f t , the regularity of ( ) u t can be raised.

t h t t h t t P f t h P f t P t h s f s s P t s f s s P t h s f s s P t h s P t s f s s I I
Proof.If ( ) f t satisfies the assumption, by [[17], Theorem

Regularity of the Classical Solutions
Motivated by the results in [18] for the C 0 -semigroups, we first give the following proposition.
Proposition 4.1.Let 0 1 α < < and 0 1 Then the mild solution of (2.1) is the classical solution.Proof.By Lemma 2.5 we only need to show that ( )( ) ( ) By our assumption and Lemma 2.6 there exists a constant 0 C > such that .

AI t h AI t AP t h s AP t s f s f t s AP t h s f t f t h s AP t h s f s f t h s h h h
We can estimate 2 h as follows:

t f t h g AP t h g AP h f t f t h g AS t h g AS h f t f t h S t h f t f t h S h f t f t h
And it is easy to show that

Conclusion
In this paper, we proved the Hölder regularity of the mild and strong solutions to the α-order abstract Cauchy problem (2.1) with (0,1) α ∈ . Our results are complemental to the existing results of Pazy [18] for the case 1 α = and Li [16] who first use solution operators to discuss the fractional abstract Cauchy problems.If the coefficient operator of a fractional abstract Cauchy problem ge-C.Y. Li, M. Li DOI: 10.4236/jamp.2018.61030311 Journal of Applied Mathematics and Physics By Hölder's inequality and Lemma 2.6, C. Y. Li, M. Li 0 ,