On a nonlocal problem for fractional differential equations via resolvent operators
© Chen et al.; licensee Springer. 2014
Received: 27 June 2014
Accepted: 8 September 2014
Published: 24 September 2014
Using the techniques of approximate solutions, the analytic resolvent method, and the uniform continuity of the resolvent, we discuss the existence of mild solutions for nonlocal fractional differential equations governed by a linear closed operator which generates a resolvent. An example is also given to illustrate the application of our theory.
where is the Caputo fractional derivative of order α with , is the infinitesimal generator of a resolvent , , X is a real Banach space, denotes the order fractional integral of . f and g are appropriate continuous functions to be specified later.
Byszewski and Lakshmikantham  introduced nonlocal initial conditions into the initial-value problems and argued that the corresponding models more accurately describe the phenomena since more information was taken into account at the onset of the experiment, thereby reducing the ill effects incurred by a single initial measurement. Concerning the motivations, relevant developments, and current status of the theory we refer the reader to [2–4]. We remark that the main difficulty in dealing with the nonlocal problem is how to get the compactness of the solution operator at zero. Many methods and techniques have been developed to avoid this difficulty in nonlocal problems, we refer the reader to the papers [5–9] and the references therein.
On the other hand, the theory of fractional differential equations has received much attention over the past 20 years, since they are important in describing the natural models such as diffusion processes, stochastic processes, finance and hydrology. Concerning the literature of fractional equations we refer to [10–18]. However, few authors have considered the fractional nonlocal problem (1.1) governed by a linear closed operator which generates a resolvent. The main difficulty is the resolvent does not have the property of semigroups, even the continuity in the uniform operator topology. Fortunately, we have proven the continuity of the resolvent in the uniform operator topology and given the characterization of compactness for resolvents in the case of an analytic resolvent. For more details, we refer the reader to the recent paper  by Fan.
In this paper, we study the existence of the nonlocal fractional differential equation (1.1) governed by operator A generating an analytic resolvent. A standard approach in deriving the mild solution of (1.1) is define the solution operator Q. Then conditions are given such that some fixed point theorems such as Schauder’s and Browder’s can be applied to get a fixed point for solution operator Q, which gives rise to a mild solution of (1.1). As, the compactness or Lipschitz condition is required in this standard method. In this paper, to get rid of these restrictive conditions, based on the works of Fan and Li  and Zhu and Li , we mainly apply the techniques of approximate solutions, the analytic resolvent method and the uniform continuity of the resolvent to get the mild solution of the nonlocal fractional differential equation (1.1) without the compactness or Lipschitz continuity assumption on the nonlocal item g. Therefore, our results essentially generalize and improve many previous ones in this field [20–23].
The outline of this paper is as follows. In Section 2, we recall some definitions on Caputo fractional derivatives, analytic resolvent and the mild solutions to (1.1). In Section 3, we establish the existence of mild solutions of (1.1) when the nonlocal item is only continuous. An example is also given to illustrate our abstract results in the last section.
Let be a real Banach space. We denote the space of all X-valued continuous functions on with the norm , the space of X-valued Bochner integrable functions on with the norm , where . Also, we denote by the space of bounded linear operators from X into X endowed with the norm of operators.
Now let us recall some basic definitions and results on fractional derivatives and resolvents.
Definition 2.1 ()
where Γ is the Gamma function.
Definition 2.2 ()
In the remainder of this paper, we always suppose that , , , and A is a closed and densely defined linear operator on X.
Definition 2.3 ()
A family of bounded linear operators in X is called a resolvent (or solution operator) generated by A if the following conditions are satisfied.
(S1) is strong continuous on and ;
(S2) and for all and ;
Since A is a closed and densely defined operator on X, it is easy to show that the resolvent equation holds for all (see ).
Definition 2.4 ()
A resolvent is called analytic, if the function admits analytic extension to a sector for some . An analytic resolvent is said to be of analyticity type if for each and there is such that for , where Rez denotes the real part of z.
Definition 2.5 A resolvent is called compact for if for every , is a compact operator.
Lemma 2.6 ([, Lemmas 3.4, 3.5, 3.8])
which implies that , .
So, we can give the following definition of mild solutions for (1.1).
3 Existence results
In this section, by using the techniques of approximate solutions, analytic resolvent and fixed point, we prove an existence theorem for the nonlocal problem (1.1) when the nonlocal item g is only continuous in .
Let r be a fixed positive real number. We consider the sets , .
is a compact analytic resolvent of analyticity type and .
is a Caratheodory function, i.e., for a.e. , is continuous and for all , the function is measurable. Moreover, for any , there exists a function such that for a.e. and all .
is a continuous mapping, which maps into a bounded set and there is a such that for any with , .
In the proof of the above theorem, we will need the following auxiliary result.
Lemma 3.2 Assume that all the conditions in Theorem 3.1 are satisfied. Then for any , the nonlocal problem (3.1) has at least one mild solution .
It is easy to see that the fixed point of is the mild solution of nonlocal problem (3.1). Subsequently, we will prove that has a fixed point by using Schauder’s fixed point theorem. From assumption (Hf), (Hg), it is easy to check that the mapping is continuous and maps into itself. According to Schauder’s fixed point theorem, it remains to prove that is compact in .
By Lemma 2.6, we find that is equicontinuous at .
uniformly for all . Therefore, from the arbitrariness of ϵ, we can conclude that is equicontinuous on .
Therefore, is relatively compact in X for all . By Schauder’s fixed point theorem, the operator has a fixed point in . □
Lemma 3.3 Assume that all the conditions in Theorem 3.1 are satisfied. Then for each , is relatively compact in X and D is equicontinuous on .
Then , , is precompact since , , is compact.
Then it follows from the arbitrariness of ϵ that , i.e., D is equicontinuous on . □
Proof of Theorem 3.1 To prove that the solution set D of nonlocal problem (3.1) is precompact in , we should only prove that is relatively compact in X and D is equicontinuous at due to Lemmas 3.2, 3.3.
where δ comes from the condition (Hg). Then, by condition (Hg), .
i.e., is relatively compact in X.
which implies that is a mild solution of nonlocal problem (1.1). □
Remark 3.4 The continuity of the resolvent in the uniform operator topology plays a key role in the proof of our main results. Moreover, the technique of approximate solutions is very important in the proof of Theorem 3.1. The application of it not only allows us to get rid of the compactness of the nonlocal item successfully, but it also lets us solve the problem of the compactness of the solution operator at zero. Finally, the method in this paper also could resolve the fractional differential equations via resolvent operators such as the differential inclusion, differential equation with delay and differential equations with impulsive conditions etc.
then the nonlocal problem (1.1) has at least one mild solution in .
Corollary 3.7 Let conditions (HA), (Hf) be satisfied. Suppose that , where , , are given constants, and . Then the nonlocal problem (1.1) has at least one mild solution in provided that .
Proof It is easy to see that if , condition (Hg) holds with . Thus all the conditions in Theorem 3.1 are satisfied. Then the nonlocal problem (1.1) has at least one mild solution on . □
Finally, we give a simple example to illustrate our theory.
where , , is the Caputo fractional derivative, , , and , , are given real numbers.
i.e. , .
Therefore, , .
- (1)is a continuous function defined by
- (2)is a continuous function defined by
where , , . Moreover, for each , we have for , and all .
Under the above conditions, the problem (4.1) also can be reformulated as the abstract problem (1.1), and conditions (Hg), (Hf) are satisfied with , . On the other hand, there must be such that the inequality holds, then, according to Theorem 3.1, the problem (4.1) has at least one mild solution on .
The work was supported by the National Science Foundation of China (11001034, 11271316) and the Qing Lan Project of Jiangsu Province of China.
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