Existence of mild solutions for fractional evolution equations with nonlocal conditions
© Yang; licensee Springer 2012
Received: 8 August 2012
Accepted: 28 September 2012
Published: 17 October 2012
This paper deals with the existence and uniqueness of mild solutions for a class of fractional evolution equations with nonlocal initial conditions. We present some local growth conditions on a nonlinear part and a nonlocal term to guarantee the existence theorems. An example is given to illustrate the applicability of our results.
MSC: 34A12, 35F25.
The differential equations involving fractional derivatives in time have recently been proved to be valuable tools in the modeling of many phenomena in various fields of engineering and science. Indeed, we can find numerous applications in electrochemistry, control, porous media, electromagnetic processing etc. (see [1–5]). Hence, the research on fractional differential equations has become an object of extensive study during recent years; see [6–11] and references therein.
On the other hand, the nonlocal initial condition, in many cases, has much better effect in applications than the traditional initial condition. As remarked by Byszewski and Lakshmikantham (see [12, 13]), the nonlocal initial value problems can be more useful than the standard initial value problems to describe many physical phenomena.
where is the Caputo fractional derivative of order , the linear operator −A is the infinitesimal generator of an analytic semigroup in X, the functions f, h and g will be specified later. , where , . Throughout this paper, we always assume that .
In some existing articles, the fractional differential equations with nonlocal initial conditions were treated under the hypothesis that the nonlocal term is completely continuous or global Lipschitz continuous. It is obvious that these conditions are not easy to verify in many cases. To make the things more applicable, in  the authors studied the existence and uniqueness of mild solutions of Eq. (1) under the case . In their main results, they did not assume the complete continuity of the nonlocal term, but they needed the following assumptions:
(F1) there exist a constant and such that for all and almost all ;
(F2) there exists a constant such that for ;
and some other conditions.
In this paper, we will improve the conditions (F1) and (F2). We only assume that f and h satisfy local growth conditions (see assumption ()) and g is local Lipschitz continuous (see assumption ()). We will carry out our investigation in the Banach space , , where is the domain of the fractional power of A. Finally, an example is given to illustrate the applicability of our main results. We can see that the main results in  cannot be applied to our example.
The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional power of the generator of an analytic semigroup and on the mild solutions of Eq. (1). In Section 3, we study the existence and uniqueness of mild solutions of Eq. (1). In Section 4, we give an example to illustrate the applicability of our results.
is injective, and can be defined by with the domain . For , let .
Lemma 1 ()
is a Banach space with the norm for ;
for each ;
for each and ;
is a bounded linear operator on X with ;
If , then .
as . Therefore, () is a strongly continuous semigroup in , and for all . To prove our main results, the following lemma is needed.
Lemma 2 ()
If () is a compact semigroup in X, then () is an immediately compact semigroup in , and hence it is immediately norm-continuous.
- (i)For fixed and any , we have
The operators and are strongly continuous for all .
If () is a compact semigroup in X, then and are norm-continuous in X for .
If () is a compact semigroup in X, then and are compact operators in X for .
In this paper, we adopt the following definition of a mild solution of Eq. (1).
for all .
To prove our main results, we also need the following two lemmas.
Lemma 4 A measurable function is Bochner integrable if is Lebesgue integrable.
Lemma 5 (Krasnoselskii’s fixed point theorem)
Let X be a Banach space, let B be a bounded closed and convex subset of X and let and be mappings from B into X such that for every pair . If is a contraction and is completely continuous, then the operator equation has a solution on B.
Lemmas 4 and 5, which can be found in many books, are classical.
The following are the basic assumptions of this paper.
() () is a compact operator semigroup in X.
For each , the functions , are measurable.
For each , the functions , are continuous.
for all .
3 Main results
In this section, we introduce the existence and uniqueness theorems of mild solutions of Eq. (1). The discussions are based on fractional calculus and Krasnoselskii’s fixed point theorem. Our main results are as follows.
then Eq. (1) has at least one mild solution on J.
which contradicts (3). Hence, for some , for every pair .
The next proof will be given in two steps.
Step 1. is a contraction on .
which implies that . It follows from (3) that , hence is a contraction on .
Step 2. is a completely continuous operator on .
as . Hence, . This means that is continuous on .
Next, we will show that the set is relatively compact. It suffices to show that the family of functions is uniformly bounded and equicontinuous, and for any , the set is relatively compact.
It follows from Lemma 3 that as and independently of . From the expressions of and , it is clear that and as independently of . Therefore, we prove that is a family of equicontinuous functions.
It remains to prove that for any , the set is relatively compact.
Therefore, there are relatively compact sets arbitrarily close to the set for and since it is compact at , we have the relative compactness of in for all .
Therefore, the set is relatively compact by the Ascoli-Arzela theorem. Thus, the continuity of and relative compactness of the set imply that is a completely continuous operator. Hence, Krasnoselskii’s fixed point theorem shows that the operator equation has a solution on . Therefore, Eq. (1) has at least one mild solution. The proof is completed. □
The following existence and uniqueness theorem for Eq. (1) is based on the Banach contraction principle. We will also need the following assumptions.
() There exists a constant such that the functions are strongly measurable.
for any and .
hold, then Eq. (1) has a unique mild solution.
which means that Q is a contraction according to (4). By applying the Banach contraction principle, we know that Q has a unique fixed point on , which is the unique mild solution of Eq. (1). This completes the proof. □
4 An example
where is a constant, , and will be specified later.
for each and .
Lemma 6 ()
If , then ξ is absolutely continuous, and .
Let , where for all . Assume that
(P1) The function , , and the partial derivative belongs to .
that f and h are functions from into and they are continuous. Moreover, a similar computation of  together with Lemma 6 and assumption (P1) shows that whenever .
Thus, the system (5) has at least one mild solution due to Theorem 1 provided that . And by Theorem 2, this mild solution of the system (5) is unique on .
Research was supported by the Fundamental Research Funds for the Gansu Universities.
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