Existence of mild solutions for fractional evolution equations with nonlocal conditions

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.

Let X be a Banach space with norm , and let be a constant. Consider the existence and uniqueness of mild solutions of fractional evolution equation with nonlocal condition in the form

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 [6] 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 [6] 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.

Let X be a Banach space with norm , and let be the infinitesimal generator of an analytic semigroup () of a uniformly bounded linear operator in X, that is, there exists such that for all . Without loss of generality, let . Then for any , we can define by

is injective, and can be defined by with the domain . For , let .

Lemma 1 ([14])

has the following properties:

(i) is a Banach space with the normfor;

(ii) for each;

(iii) for eachand;

(iv) is a bounded linear operator onXwith;

(v) If, then.

Let be the Banach space of endowed with the norm . Denote by the Banach space of all continuous functions from J into with the supnorm given by for . From Lemma 1(iv), since is a bounded linear operator for , we denote by the operator norm of in X, that is, . For any , denote by the restriction of to . From Lemma 1(ii) and (iii), for any , we have

and

as . Therefore, () is a strongly continuous semigroup in , and for all . To prove our main results, the following lemma is needed.

Lemma 2 ([15])

If () is a compact semigroup inX, then () is an immediately compact semigroup in, and hence it is immediately norm-continuous.

For , define two families and of operators by

where

where is the probability density function defined on , which has properties for all and

The following lemma follows from the results in [6-8,10].

Lemma 3The following properties are valid:

(i) For fixedand any, we have

(ii) The operatorsandare strongly continuous for all.

(iii) If () is a compact semigroup inX, thenandare norm-continuous inXfor.

(iv) If () is a compact semigroup inX, thenandare compact operators inXfor.

In this paper, we adopt the following definition of a mild solution of Eq. (1).

Definition 1 By a mild solution of Eq. (1), we mean a function satisfying

for all .

To prove our main results, we also need the following two lemmas.

Lemma 4A measurable functionis Bochner integrable ifis Lebesgue integrable.

Lemma 5 (Krasnoselskii’s fixed point theorem)

LetXbe a Banach space, letBbe a bounded closed and convex subset ofXand letandbe mappings fromBintoXsuch thatfor every pair. Ifis a contraction andis completely continuous, then the operator equationhas a solution onB.

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.

() There exists a constant such that the functions satisfy the following conditions:

(i) For each , the functions , are measurable.

(ii) For each , the functions , are continuous.

() For and , there exist positive functions satisfying and such that

and there are positive constants and such that

() and for , there exists a positive constant L such that

for all .

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.

Theorem 1Suppose that the assumptions ()-() hold. Ifand the following inequality holds:

then Eq. (1) has at least one mild solution onJ.

Proof Define two operators and as follows:

Obviously, u is a mild solution of Eq. (1) if and only if u is a solution of the operator equation on J. To prove the operator equation has solutions, we first show that there is a positive number r such that for every pair . If this were not the case, then for each , there would exist and such that . Thus, from Lemma 3, () and (), we have

Dividing on both sides by r and taking the lower limit as , we have

which contradicts (3). Hence, for some , for every pair .

The next proof will be given in two steps.

Step 1. is a contraction on .

For any and , according to Lemma 3 and assumption (), we have

which implies that . It follows from (3) that , hence is a contraction on .

Step 2. is a completely continuous operator on .

We first prove that is continuous on . Let with as . Then for any , , by assumption (), we have

as , and from assumption (), we have

This together with the Lebesgue dominated convergence theorem gives that

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.

For any , we have for some , which means that is uniformly bounded. In what follows, we show that is a family of equicontinuous functions. For , we have

Hence, it is only necessary to consider . For , from Lemma 3 and assumption (), we have

where

For any , we have

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.

Obviously, is relatively compact in . Let be fixed. For each , and , we define an operator by

Then the sets are relatively compact in since by Lemma 2, the operator is compact for in . Moreover, for every , from Lemma 3 and assumption (), we have

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 , there exist functions such that

for any and .

Theorem 2Let the assumptions ()-() be satisfied. Ifand the inequalities (3) and

hold, then Eq. (1) has a unique mild solution.

Proof From Lemma 4 and assumption (), it is easy to see that is Bochner integrable with respect to for all . For any , we define an operator Q by

According to the proof of Theorem 1, we know that for some . For any and , from Lemma 3, assumptions () and (), we have

Thus,

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. □

Let equip with its natural norm and inner product . Consider the following system:

where is a constant, , and will be specified later.

Let the operator be defined by

Then −A generates a compact analytic semigroup of uniformly bounded linear operators and for all . Hence, we take . Moreover, the eigenvalues of A are , and the corresponding normalized eigenvectors are ,  . The operator is given by

for each and .

Lemma 6 ([16])

If, thenξis absolutely continuous, and.

Let , where for all . Assume that

(P₁) The function , , and the partial derivative belongs to .

Define

Let , it follows from

and

that f and h are functions from into and they are continuous. Moreover, a similar computation of [17] together with Lemma 6 and assumption (P₁) shows that whenever .

Then for any , we see that the assumptions ()-() are satisfied with

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 .

The author declares that they have no competing interests.

Research was supported by the Fundamental Research Funds for the Gansu Universities.

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