Abstract and Applied Analysis

Volume 2014 (2014), Article ID 430418, 9 pages

http://dx.doi.org/10.1155/2014/430418

Research Article

Integrated Fractional Resolvent Operator Function and Fractional Abstract Cauchy Problem

Ya-Ning Li and Hong-Rui Sun

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received 31 October 2013; Accepted 11 December 2013; Published 8 January 2014

Academic Editor: Yong Zhou

Copyright © 2014 Ya-Ning Li and Hong-Rui Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We firstly prove that -times integrated -resolvent operator function (-ROF) satisfies a functional equation which extends that of -times integrated semigroup and -resolvent operator function. Secondly, for the inhomogeneous -Cauchy problem , , , if is the generator of an -ROF, we give the relation between the function and mild solution and classical solution of it. Finally, for the problem , , , where is a linear closed operator. We show that generates an exponentially bounded -ROF on a Banach space if and only if the problem has a unique exponentially bounded classical solution and Our results extend and generalize some related results in the literature.

1. Introduction

This paper is concerned with the properties of -integrated -resolvent operator function (-ROF) and two inhomogeneous fractional Cauchy problems.

Throughout this paper, , denotes the set of natural numbers. . Let be Banach spaces, denote the space of all bounded linear operators from to , . If is a closed linear operator, denotes the resolvent set of and denotes the resolvent operator of . denotes the space of -valued Bochner integrable functions: with the norm , it is a Banach space. By we denote the convolution of functions

denotes the function

and , the Dirac delta function.

In 1997, Mijatović et al. [1] introduced the concept of -times integrated semigroup which extends -times integrated semigroup [2], they showed an to be the pseudoresolvent of a -times integrated semigroup if and only if satisfies the following functional equation:

In the special case of , the corresponding result is summarized in [2].

For the inhomogeneous Cauchy problem

where , , , and is the generator of a -times integrated semigroup on a Banach space for some . Let . Lemmas and of [2] show that if there is a mild(classical) solution of (4), then and . On the other hand, if , then is also a mild (classical) solution of it.

Furthermore, if generates an exponential bounded -times integrated semigroup on a Banach space , then, for any , is the unique exponential bounded classical solution of the following problem:

In recent years, a considerable interest has been paid to fractional evolution equation due to its applications in different areas such as stochastic, finance, and physics; see [3–8]. One of the most important tools in the theory of fractional evolution equation is the solution operator (fractional resolvent family) [9–15]. The notion of solution operator was developed to study some abstract Volterra integral equations [16] and was first used by Bajlekova [17] to study a class of fractional order abstract Cauchy problem. In [9], Chen and Li introduced -resolvent operator functions (-ROF for short) defined by purely algebraic equation. They showed that a family is an -ROF if and only if is a solution of abstract fractional Cauchy problem

When , Peng and Li [18] proved that the solution operator for (6) satisfies the following equality:

We refer to [5, 15, 16, 19] for further information concerning general resolvent operator functions. In addition, Chen and Li [9] also introduced the concept of integrated fractional resolvent operator function in an algebraic notion as follows.

Definition 1 (see [9, Definition 3.7]). Let , . A function is called a -times integrated -resolvent operator function or an -resolvent operator function (()-ROF for short) if the following conditions hold:

(a) is strongly continuous on and ;(b) for all ;(c)the functional equation

 holds for , where is the Riemann-Liouville fractional integral of order .

The generator of is defined by

Note that an -ROF is just an -ROF.

In this paper, we firstly show that -ROF satisfies an equality which extends (3) and (7) for -integrated semigroup and -ROF, respectively. Then, we consider the inhomogeneous fractional order abstract Cauchy problem

where , , , and is assumed to be the generator of an -ROF on . We give the relation between the function and solution of (10). We also study the problem

where , , is the smallest integer greater than or equal to . We prove that if generates an exponentially bounded -ROF on if and only if the problem (11) has a unique exponentially bounded classical solution and . If , , our Theorem 13 reduces to Lemma in [2]. When ,  , it is easy to see that our Theorem 15 extends and generalizes Theorem in [2].

This paper is organized as follows. In Section 2, we provide some preliminaries of the fractional calculus and -ROF. Section 3 is devoted to present an equality characteristic of the -ROF. Finally, as an application of -ROF, we discuss the solutions of fractional abstract Cauchy problem in Section 4.

2. Preliminary

Recall that the Riemann-Liouville fractional integral of order of is defined by

and the Caputo fractional derivative of order of can be written as

where is the smallest integer greater than or equal to . For more details in fractional calculus, we refer to [5, 20, 21].

The Mittag-Leffler function is defined by

And if , , then

where

and the -term is uniform in if .

We now recall some properties of -ROF.

Lemma 2 (see [9, Proposition 3.10]). Let be an -ROF generated by . The following assertions hold:

(a) and for and ;(b)for all , and , ;(c) and if and only if , ;(d) is closed.

Lemma 3 (see [9, Proposition 3.5, Theorem 3.11]). Let , . generates an -ROF satisfying , , for some constants and , if and only if and there exists a strongly continuous function such that for all and , , for all . Furthermore, is .

Lemma 4 (see [2, Proposition B.6]). Let . If function satisfies , then there is an operator on such that for all if and only if .

3. An Novel Equality Characteristic for ()-ROF

The following theorem shows that an -ROF satisfies a functional equation and the treatment bases on the technique of Laplace transform. For convenience, we drop the subscript from in this theorem.

Theorem 5. Let satisfy . If is an -ROF, then it satisfies the following equality:

Proof. Denote by and the left and right sides of equality (17), respectively, and denote by the truncation of at , that is, for and otherwise.

We will show that the Laplace transform of and with respect to and is equivalent, and by the uniqueness of Laplace transform, we can get that .

Taking Laplace transform of with respect to as follows

then taking Laplace transform with respect to , we have

where the last equality follows from Lemma 3.

On the other hand, observing that

Then taking Laplace transform with respect to and , respectively, we deduce

where the last equality follows from the resolvent identity. In view of (19), (21), and the uniqueness of Laplace transform, we obtain . The arbitrariness of implies for .

Remark 6. (a) If , then -ROF is an -ROF and the equality (17) degenerates to be equality (7).

(b) If we assume that, for each , the map is continuously differentiable on and the limit of -ROF exists as , then multiplying both sides of (17) with and integrating by parts to the right side of (17) and letting , we can get that (3) is just the limit state of (17).

By Lemma 3, -ROF generated by operator is exactly operator valued functions whose Laplace transforms are . In the following theorem, we show that this property corresponds to the functional equation (17) for . The proof of this theorem is proved by Ardent [2, proposition 3.2.4] for . Our proof is different since we could not use the binomial formula as in [2].

Theorem 7. Let be a strongly continuous function satisfying for some . Let satisfy that , set

Then the following assertions are equivalent.

(i)There exists an operator such that and for .(ii)For , the equality

 holds and for all implies that .

Proof. Assume that (i) holds; then , for ; from Lemma 3, we know that is the -ROF generated by ; then Theorem 5 shows that equality (23) holds. It follows from and for that is injective. If for all , from , we have ; thus .

If (ii) is satisfied, similar as the calculations of (19) and (21), we can get that the Laplace transform of the left side and the right side of (17) are

respectively. So,

On the other hand, if , by and uniqueness of Laplace transform, we have for all , then from (ii) we know , so, , by (25) and Lemma 4, we get the conclusion.

4. Fractional Abstract Cauchy Problems

In this section, we study the following inhomogeneous fractional abstract Cauchy problem:

where , is a linear closed operator.

First, we give the definitions of solutions to (26).

Definition 8. A function is called a mild solution of (26), if and .

Definition 9. A function is called a classical solution of (26) if satisfies the following.

(a).(b).(c) satisfies (26).

From the above definitions, it is clear that a classical solution of (26) is a mild solution of it. The following assertion shows that a mild solution of the problem (26) with suitable regularity is also a classical solution.

Theorem 10. Let be a mild solution of (26) and , if , and for any , ; then is also a classical solution of (26).

Proof. Since is a mild solution of (26), we have

If we denote , then it follows from (27) that

Since , then is well defined, and by (28), we have

Thus,

On the other hand, from the closeness of and for , by Proposition in [2], we have

Then from (30) and the closeness of , we obtain

It is clear that . Thus, is a classical solution of (26).

Lemma 11. Let , . Suppose is the generator of an -ROF on for some . Then, for every , exists, and .

Proof. For every , since , we get , hence, from

we obtain that exists.

For and , we have

From the dominated convergence theorem and absolute continuity of integral, we deduce

So, .

Let

From Lemma 11, we know that is well defined, and .

The following theorem is proved by Arendt [2, Lemma ] for . Our proof is different because we could not use the formula of integration by parts as [2, Lemma ].

Theorem 12. Suppose that is the generator of an -ROF on for some . Let be defined by (36). Then one has the following results.

(a)If (26) has a mild solution , then and .(b)If there is a classical solution of (26), then and .

Proof. If is a mild solution of (26), then and

Using Lemma 2(b) and the closeness of , we have

that is, . So

Thus, it follows from that and . Hence (a) holds. If is a classical solution of (26), then is a mild solution of (26). So, assertion (b) follows immediately from (a).

Theorem 13. Let be defined by (36). Assume that , for , and ; then is a mild solution of the problem (26). Moreover, if , and for any , , then is also a classical solution of (26).

Proof. Consider the following steps.

S tep 1. We first claim that and

In view of definition of , we have

for .

From Lemma 2(b), for , we have

combining with the closeness of , one has

Thus , and

So

Step 2. We prove and .

Since for  , we have

So

where . From (45), we know that , and the closeness of implies that , and , by Step 1, exists, then , and

Step 3. We show that for implies

In fact, if , we have , and

If , we have , and

From the above discussion and for , we conclude that (49) holds.

Finally, in view of (40), (48), and (49), we have

Therefore, is a mild solution of (26).

Moreover, if , and for any , , applying Theorem 10, we have that is a classical solution of (26).

Remark 14. If , then (26) becomes (4). Theorem 13 degenerated to Lemma in [2]. Note that the condition for  is not necessary in Lemma of [2], since from its proof, it is easy to see that implies that for .

Now, we turn our attention to the problem

where , is a linear closed operator on and is the smallest integer greater than or equal to .

Theorem 15. Let be a closed operator on and ; the following two assertions are equivalent.

(i) generates an exponentially bounded -ROF on .(ii)For every , there exists a unique classical solution of (53) which is exponentially bounded and .

Proof. If is satisfied, for every , define by , then for . By Lemma 2(b), we have , and

Thus, is a classical solution of (53); it is unique by Theorem 12. Since is exponentially bounded, we have that is exponentially bounded. From

we know that . So (ii) is true.

Assume that (ii) holds. From linearity of (53) and the uniqueness of its solution, we get that is linear in . So, for each , there exists a linear mapping such that for any .

Next, we show that, for each , .

We consider the mapping by . Then, is a linear operator defined on . Now we show that is closed, if in and in . For , by the dominated convergence theorem, we have that converges to , since , from the closeness of , it follows that as , , which implies that and is closed. Therefore, by the closed graph theorem, is bounded. So, for each , . Then, the exponentially boundedness of and Lemma in [2], imply that for some constants . So is well defined for .

Since , then the Laplace transform of is well defined, and from the closeness of , for , we have

Now, we show that is injective for . Assume that for some and . Then, by the method of Laplace transform, we have that the solution of (53) is . Since , for all , combine with (15), it follows that . Hence for and is an -ROF. Let

then exists and for all and all . So

and taking the Laplace transform, we have

that is,

From Lemma 3, we know that is the -ROF generated by .

Remark 16. Theorem 15 extends and generalizes Theorem in [2]. In fact, when and , (53) becomes (5), is a -times integrated semigroup. For problem (5), the condition in (ii) is not necessary. Since from the proof of Theorem in [2], it is easy to see that the assumption that exponentially boundedness of the unique classical solution to the problem (5) imply that .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the Program for New Century Excellent Talents in University (NECT-12-0246) and FRFCU (lzujbky-2013-k02).

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