Nonlocal Problems for Fractional Differential Equations via Resolvent Operators

We discuss the continuity of analytic resolvent in the uniform operator topology and then obtain the compactness of Cauchy operator by means of the analytic resolvent method. Based on this result, we derive the existence of mild solutions for nonlocal fractional differential equations when the nonlocal item is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively. An example is also given to illustrate our theory.

On the other hand, abstract differential equations with nonlocal conditions have also been studied extensively in the literature, since it is demonstrated that the nonlocal problems have better effects in applications than the classical ones.It was Byszewski and Lakshmikantham [21] who first studied the existence and uniqueness of mild solutions for nonlocal differential equations.And the main difficulty in dealing with the nonlocal problem is how to get the compactness of solution operator at zero, especially when the nonlocal item is only assumed to be Lipschitz continuous or continuous.Many authors developed different techniques and methods to solve this problem.For more details on this topic, we refer to [10,11,[22][23][24][25][26][27][28][29][30][31][32][33] and references therein.
In this paper, we combine the above two directions and study the nonlocal fractional differential equation (1) governed by operator  generating an analytic resolvent.A standard approach in deriving the mild solution of ( 1) is to define the solution operator .Then, conditions are given such that some fixed point theorems such as Browder's and Schauder's fixed point theorems can be applied to get a fixed point for solution operator , which gives rise to a mild solution of (1).The key step of using this approach is International Journal of Differential Equations to prove the compactness of Cauchy operator  associated with solution operator .When the operator  generates a compact semigroup, it is well known that the Cauchy operator  is also compact.However, to the best of our knowledge, it is unknown when  generates a compact resolvent.The main difficulty of this problem lies in the fact that there is no property of semigroups for resolvent.
To this end, we will first discuss the continuity of resolvent in the uniform operator topology in this paper.In fact, we prove that the compact analytic resolvent is continuous in the uniform operator topology.Based on this result, we can prove the compactness of Cauchy operator.As a consequence, we obtain the existence of mild solutions for (1) when the nonlocal item is Lipschitz continuous.At the same time, we also derive the existence of mild solutions for (1) without the Lipschitz or compact assumption on the nonlocal item  by using the techniques developed in [24,30].Actually, we only assume that  is continuous on ([0, ], ) and  is completely determined on [, ] for some small  > 0 or  is continuous on ([0, ], ) with  1 ([0, ], ) topology (see . This paper has four sections.In Section 2, we recall some definitions on Caputo fractional derivatives, analytic resolvent, and mild solutions to (1).In Section 3, we prove the compactness of Cauchy operator.Finally, in Section 4 we establish the existence of mild solutions of (1) when the nonlocal item satisfies different conditions.An example is also given in this section.

Preliminaries
Throughout this paper, let  > 0 be fixed, and let N, R, and R + be the set of positive integers, real numbers, and nonnegative real numbers, respectively.We denote by  the Banach space with the norm ‖ ⋅ ‖, ([0, ], ) the space of all -valued continuous functions on [0, ] with the norm ‖‖ = sup{‖()‖,  ∈ [0, ]}, and   ([0, ], ) the space of valued Bochner integrable functions on [0, ] with the norm ‖‖   = (∫  0 ‖()‖  d) 1/ , where 1 ≤  < ∞.Also, we denote by L() the space of bounded linear operators from  into  endowed with the norm of operators.Now, let us recall some basic definitions and results on fractional derivative and fractional differential equations.
In the remainder of this paper, we always suppose that 0 <  < 1 and  is a closed and densely defined linear operator on .
Next, we introduce the Hausdorff measure of noncompactness (⋅) defined on each bounded subset Ω of Banach space  by Some basic properties of (⋅) are given in the following lemma.
The map  :  ⊆  →  is said to be a -contraction if there exists a positive constant  < 1 such that () ≤ () for any bounded closed subset  ⊆ , where  is a Banach space.

Lemma 9 ([34], Darbo-Sadovskii).
If  ⊆  is bounded closed and convex, the continuous map  :  →  is a -contraction, then the map  has at least one fixed point in .

Compactness of Cauchy Operators
Let Cauchy operator  : ([0, ], ) → ([0, ], ) be defined by If   () is a compact  0 -semigroup, it is well known that  is compact.However, it is unknown in case of compact resolvent.The main difficulty is that the resolvent does not have the property of semigroups.Thus, it seems to be more complicated to prove the compactness of Cauchy operator.
Here, we will first discuss the continuity of resolvent in the uniform operator topology.Then, we can give the positive answer to the above problem.
First, we claim that the set   is equicontinuous on ([0, ], ).In fact, let 0 ≤  1 ≤  2 ≤  and  ∈   ; then we have If  1 = 0, it is easy to see that lim If 0 <  1 < , for 0 <  <  1 , we have Note that from Lemma 10(i), we know   () is operator norm continuous uniformly for  ∈ [,].Combining this and the arbitrariness of  with the above estimation on , we can conclude that lim Thus,   is equicontinuous on ([0, ], ).

Nonlocal Problems
In this section, we always assume that  0 ∈  and that the operator  generates a compact analytic resolvent   () of analyticity type ( 0 ,  0 ), and we will prove the existence of mild solutions of (1) when the nonlocal item  is assumed to be Lipschitz continuous and neither Lipschitz nor compact, respectively.
Let  be a fixed positive real number and Clearly,   is a bounded closed and convex set.We make the following assumptions.
Under these assumptions, we can prove the first main result in this paper.
It is easy to see that the fixed point of  is the mild solution of nonlocal Cauchy problem (1).Subsequently, we will prove that  has a fixed point by using Lemma 9 (Darbo-Sadovskii's fixed point theorem).Firstly, we prove that the mapping  is continuous on ([0, ], ).For this purpose, let {  } ≥1 be a sequence in ([0, ], ) with lim  → ∞   =  in ([0, ], ).Then By the continuity of  and , we deduce that  is continuous on ([0, ], ).
Remark 13.It is easy to see that condition (H) is weaker than the compactness and convexity of .The same hypothesis can be seen from [24,30], where the authors considered the existence of mild solutions for semilinear nonlocal problems of integer order when  is a linear, densely defined operator on  which generates a  0 -semigroup.After the proof of our main results, we will give some special types of nonlocal item  which is neither Lipschitz nor compact, but satisfies the condition (H) in the next Corollaries.
Theorem 14. Assume that conditions (H1) and (H) are satisfied.Then the nonlocal problem (1) has at least one mild solution provided that (31) holds.
Proof.We will prove that  has a fixed point by using Schauder's fixed point theorem.According to the proof of Theorem 12, we have proven that  :   →   is continuous.Next, we will prove that there exists a set  ⊆   such that  :  →  is compact.For this purpose, let 0 <  ≤  and 0 <  < .It is easy to see that the set Proof.According to Theorem 14, it is sufficient to prove that the hypothesis (H) is satisfied.For arbitrary  > 0, there exists 0 <  <  such that ∫  0 ‖()‖d <  for all  ∈   .Let (46) From the proof of Theorem Finally, we give a simple example to illustrate our theory.
Under these assumptions, the fractional partial differential heat equation (47) can be reformulated as the abstract problem (1)
. If the inequality  > 0, there exists at least one mild solution for fractional equation (47) in view of Corollary 18.