- Open Access
Existence of mild solutions for fractional impulsive neutral evolution equations with nonlocal conditions
© Liang and Mei; licensee Springer. 2014
- Received: 15 November 2013
- Accepted: 17 March 2014
- Published: 3 April 2014
In this paper, by using the fractional power of an operator and some fixed point theorems, we study the existence of mild solutions for the nonlocal problem of Caputo fractional impulsive neutral evolution equations in Banach spaces. In the end, an example is given to illustrate the applications of the abstract results.
- fractional impulsive neutral evolution equation
- compact and analytic semigroup
- mild solutions
- fixed point theorem
in a Banach space X, where is a constant, denotes the Caputo fractional derivative of order , is a closed linear operator and −A generates a -semigroup () in X, is continuous, , are the elements of X, , and represent the right and left limits of at , respectively. By using some fixed point theorems of compact operator, they derive many existence and uniqueness results concerning the mild solutions for problem (1) under the different assumptions on the nonlinear term f. For more articles about the fractional impulsive evolution equations, we refer to [12–14] and the references therein.
On the other hand, the fractional neutral differential equations have also been studied by many authors. Many methods of nonlinear analysis have been employed to research this problem; see [6, 15–18]. But, as far as we know, papers considering the fractional impulsive neutral evolution equations are seldom.
in a Banach space X, where , denotes the Caputo fractional derivative of order , −A is the infinitesimal generator of an analytic semigroup () in X, () are the impulsive functions, f, h, g are given functions and will be specified later. By utilizing the fixed point theorems, we derive many existence results concerning the mild solutions for problem (2) under different assumptions on the nonlinear term and nonlocal term.
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 the fractional calculus. In Section 3, we study the existence of mild solutions of the problem (2). An example is given in Section 4 to illustrate the applications of the abstract results.
In this section, we introduce some basic facts as regards the fractional power of the generator of an analytic semigroup and the fractional calculus.
It follows that each is an injective continuous endomorphism of X. Hence we can define by , which is a closed bijective linear operator in X. It can be shown that each has dense domain and that for . Moreover, for every and , where . , I is the identity in X. (For proofs of these facts we refer to [19, 20].)
We denote by the Banach space of equipped with norm for , which is equivalent to the graph norm of . Then we have for (with ), and the embedding is continuous. Moreover, has the following basic properties.
Lemma 1 
for each and .
for each and .
- (iii)For every , is bounded in X and there exists such that
is a bounded linear operator for in X.
From Lemma 1(iv), there exists a constant such that for .
For any , denote by the restriction of to . From Lemma 1(i) and (ii), () is a strongly continuous semigroup in , and for all . To prove our main results, the following lemma is needed.
Lemma 2 
() is an immediately compact semigroup in , and hence it is immediately norm-continuous.
where Γ is the gamma function.
Remark 1 If f is an abstract function with values in X, then integrals which appear in Definition 1 are taken in Bochner’s sense.
A measurable function is Bochner integrable if is Lebesgue integrable.
- (i)For fixed and any , we haveFor fixed and any , we have
The operators and are strongly continuous for all .
If () is a compact semigroup, then and are compact operators in X for .
If () is a compact semigroup, then the restriction of to and the restriction of to are compact operators in for every .
Lemma 5 (Krasnoselskii’s fixed point theorem)
Let E be a Banach space, B be a bounded closed and convex subset of E and , be maps of B into E such that for every pair . If is a contraction and is completely continuous, then the equation has a solution on B.
This means that is uniformly convergent to in for all . Hence we get and in as . That is is a Banach space endowed with norm for .
Hence, by using a completely similar technique as in [, Section 3], we obtain the following definition.
In this section, we introduce the existence theorems of mild solutions of the problem (2). The discussions are based on fixed point theorems. Our main results are as follows.
Theorem 1 Assume that the following conditions are satisfied.
(H1) () is a compact analytic semigroup;
for all and ;
For a.e. , the function is continuous, and for every , the function is strongly measurable.
- (ii)For each and , there exists a constant and a function such that
for all .
for all .
Proof Let , . Direct calculation shows that for . In view of Lemma 4, a similar argument as in the proof of [, Theorem 3.1] shows that is Bochner integrable with respect to for all .
Thus, is Lebesgue integrable with respect to for all . From Lemma 3, it follows that is Bochner integrable with respect to for all .
which contradicts (4). Hence there exists a positive constant such that for any .
Next, we will show that is a completely continuous operator and is a contraction on . Our proof will be divided into three steps.
Step I. is continuous on .
as , which implies that is continuous.
Step II. is relatively compact. It suffices to show that the family of functions is uniformly bounded and equicontinuous, and for any , is relatively compact in .
For any , we see from above that , which means that is uniformly bounded. In the following, we will show that is a family of equicontinuous functions.
for , where . Since Lemma 2 implies the continuity of in in the uniformly operator topology, it is easy to see that as independently of . Thus, as independently of , which means that the set is equicontinuous.
It remains to prove that for any , the set is relatively compact in .
where . Therefore, there are relatively compact sets arbitrarily close to the set , . Hence the set , is also relatively compact in .
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.
Step III. is a contraction on .
Since , we know that is a contraction on . Hence, Krasnoselskii’s fixed point theorem guarantees that the operator equation has a solution on , which is the mild solution of the problem (2) on . □
Theorem 2 Assume that (H1)-(H3) hold. Further, the following conditions are also satisfied.
A similar proof as in [, Theorem 3.1] shows that the set is relatively compact. Hence is a completely continuous operator. By Krasnoselskii’s fixed point theorem, the equation has a solution on , which is the mild solution of the problem (2) on . □
Theorem 3 Assume that (H2), (H4), (H5) hold. Further, the following condition is also satisfied.
for any and .
Since , it follows that Q is a contraction on . By the Banach contraction principle, Q has a unique fixed point in , which is the unique mild solution of the problem (2). □
where is a Caputo fractional partial derivative of order , and .
Then −A generates a compact and analytic semigroup (), and . It is well known that , and so the fractional powers of A are well defined. Moreover, the eigenvalues of A are and the corresponding normalized eigenvectors are , . We define by for each . From  we know that if , then z is absolutely continuous with and .
We define the Banach space by , where for any . It is well known that .
For solving the problem (7), we need the following assumptions.
- (i)is well defined and measurable with
For each , the function is differentiable and .
- (ii)There exists a constant such that
- (i)belongs to and
Moreover, if , we defined by . Thus, the system (7) can be reformed as the nonlocal problem (2).
for each . Hence h satisfies the hypothesis (H2).
for each . This implies that the assumptions (H4) and (H6) hold.
for each . By [, Theorem 4.3(b)], g is a compact operator. Thus, the assumptions (H5) and (H7) hold.
We can take and . Since for any , we have . So, we choose , then the assumptions (H3) and (H8) hold. Hence, if , according to Theorem 1 or Theorem 2, the system (7) has at least one mild solution provided that (4) or (5) holds. From Theorem 3, the system (7) has a unique mild solution provided that (6) holds.
The authors are grateful to the referees for their helpful comments and suggestions. The second author is supported by Zhangjiakou Science and Technology Bureau (No. 13110039I-4) and youth fund of natural science of Hebei North University (No. Q2013007).
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