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On a nonlocal problem for fractional differential equations via resolvent operators
Advances in Difference Equations volume 2014, Article number: 251 (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.
In this paper, we are concerned with the existence of mild solutions for the following fractional differential equation:
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 ()
Suppose . The fractional order integral of the function of order α is defined by
where Γ is the Gamma function.
Definition 2.2 ()
Suppose . The Caputo fractional order derivative of order α of a function given on the interval is defined by
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 ;
(S3) the resolvent equation holds:
Since A is a closed and densely defined operator on X, it is easy to show that the resolvent equation holds for all (see ).
For , let
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])
Suppose is a compact analytic resolvent of analyticity type . Then
Now, we consider the following fractional differential equation:
Suppose that , , and x is a solution of (2.1). Then we can give the following variation of the constant formula:
In fact, if x satisfies (2.1), then for ,
Thus, it follows from the definition of resolvent that
which implies that , .
So, we can give the following definition of mild solutions for (1.1).
Definition 2.7 A function is called a mild solution of fractional differential equation (1.1) if it satisfies
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 , .
We list the following hypotheses:
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 , .
Theorem 3.1 Assume that (HA), (Hf), and (Hg) are satisfied. Then the nonlocal problem (1.1) has at least one mild solution on provided that
In the proof of the above theorem, we will need the following auxiliary result.
For fixed , we consider the following approximate problem:
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 .
Proof For fixed , set defined by
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 .
Firstly, for any , we will show that is equicontinuous on . To this end, let . If , then for any ,
By Lemma 2.6, we find that is equicontinuous at .
If , for any , with and , it follows from (Hf) that
Note that, by Lemma 2.6(i), we may infer that
uniformly for all . Therefore, from the arbitrariness of ϵ, we can conclude that is equicontinuous on .
Secondly, we shall demonstrate that is relatively compact in X for all , where , . If , then . Since , is compact and the condition (Hg) holds, it is obvious that is relatively compact in X. If , for any , set
Further, is compact at , then is relatively compact at . In the following, we shall verify that
uniformly for all . In fact, for arbitrary , we have
From Lemma 2.6(ii), we know
Then, from the Lebesgue dominated convergence theorem and the arbitrariness of δ that
Hence, for fixed , we can derive by applying (3.2) that
Therefore, is relatively compact in X for all . By Schauder’s fixed point theorem, the operator has a fixed point in . □
Now, define the solution set D and by
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 .
Proof For , , we have
Let , , , . From hypothesis (Hf), there exists such that
Note that, from Lemma 2.6(iii), we have
uniformly as . Thus, for any , we can conclude that
Then , , is precompact since , , is compact.
Finally, we will show that D is equicontinuous for . Similar to the proof Lemma 3.2, for any , with and , we have
From Lemma 2.6(i), we know
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.
For , set
where δ comes from the condition (Hg). Then, by condition (Hg), .
At the same time, by Lemma 3.3, without loss of generality, we may suppose that , as . Since
i.e., is relatively compact in X.
On the other hand, for ,
uniformly as since is relatively compact. Thus, we obtain the result that the set is equicontinuous at . Therefore, D is precompact in . Without loss of generality, we may suppose that as . By the definition of a mild solution for (3.1), we have
for . Taking the limit in both sides, we obtain
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.
Corollary 3.5 Assume the hypotheses (HA), (Hg), and (Hf) are true for each . Moreover,
then the nonlocal problem (1.1) has at least one mild solution in .
Remark 3.6 The condition (3.3) is satisfied if there exist constant and such that
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.
Example 4.1 Consider the following fractional partial differential equation:
where , , is the Caputo fractional derivative, , , and , , are given real numbers.
Take , with the norm , and consider the operator to be , where . It is well known that A is the infinitesimal generator of an analytic semigroup for on X. Furthermore, A has discrete spectrum with eigenvalues of the form , , and corresponding normalized eigenfunctions given by . In addition, is an orthonormal basis for X, for all and every . From this expression it follows that , , is a uniformly bounded compact semigroup. Moreover,
i.e. , .
By the subordination principle [, Theorem 3.11], we know that A also generates a compact α-order fractional analytic resolvent of analyticity type for some , and
is the Wright function. First, we prove that there exists a constant such that , . In fact, by (4.2), we have
Therefore, , .
We now assume that:
is a continuous function defined by
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 .
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The work was supported by the National Science Foundation of China (11001034, 11271316) and the Qing Lan Project of Jiangsu Province of China.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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Chen, L., Fan, Z. & Li, G. On a nonlocal problem for fractional differential equations via resolvent operators. Adv Differ Equ 2014, 251 (2014). https://doi.org/10.1186/1687-1847-2014-251
- nonlocal conditions
- analytic resolvent
- Caputo fractional order derivative
- mild solution