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Zhan-Dong Mei, Ji-Gen Peng, Jing-Huai Gao, "Convoluted Fractional -Semigroups and Fractional Abstract Cauchy Problems", Abstract and Applied Analysis, vol. 2014, Article ID 357821, 9 pages, 2014. https://doi.org/10.1155/2014/357821
Convoluted Fractional -Semigroups and Fractional Abstract Cauchy Problems
We present the notion of convoluted fractional -semigroup, which is the generalization of convoluted -semigroup in the Banach space setting. We present two equivalent functional equations associated with convoluted fractional -semigroup. Moreover, the well-posedness of the corresponding fractional abstract Cauchy problems is studied.
Let be a Banach space and let ( denotes the space of bounded linear operators from to itself) be injective. Let , where is locally integrable on . A strongly continuous operator family is called a -convoluted -semigroup if , , , and there holds If, in addition, , , implies , is called a nondegenerate -convoluted -semigroup. For more details, we refer to Kostić . Obviously, -convoluted -semigroups are a generalization of classical -semigroups. The extension of Widder’s representation theorem by Arendt  stimulated the development of the theory of -times integrated semigroups, which is the special case that , (see [3–6] for being an integer and [7–9] for being noninteger). Li and Shaw [10–12] introduced exponentially bounded -times integrated -semigroups which are the special case, where , and they studied their connection with the associated abstract Cauchy problem. Kuo and Shaw  were concerned with the case with being noninteger and they called it -times integrated -semigroups. Convoluted semigroups, which are the special case that , were introduced by Cioranescu and Lumer [14–16]. It is a generalization of integrated semigroups. Moreover, Kunstmann  showed that there exists a global convoluted semigroup whose generator is not stationary dense and therefore it cannot be the generator of a local integrated semigroup. After that, Melnikova and Filinkov , Keyantuo et al. , and Kostić and Pilipović  systematically studied the properties of convoluted semigroups and related them to associated abstract Cauchy problems.
In , Kostić presented the notion -convoluted -semigroup and found a sufficient condition for a nondegenerate strongly continuous linear operator family to be a -Convoluted -semigroup; that is, , , , , , and . Moreover, Kostić showed that if the sufficient condition holds, the abstract Cauchy problem is well-posed. In Proposition 2.3 of , it was proved that if the following abstract Cauchy problem is well-posed, then the above sufficient conditions hold. Motivated by such facts, Kostić gave in  the definition of -convoluted semigroup by the above sufficient conditions and proved the equivalence of the two definitions. Later, Kostić and Pilipović  gave the definition of -convoluted -semigroups by using the sufficient condition.
Observe that the abstract Cauchy problems related to the above “semigroups” are of integral order. Recently, fractional differential equations have received increasing attention because the behavior of many physical systems, such as fluid flows, electrical networks, viscoelasticity, chemical physics, electron-analytical chemistry, biology, and control theory, can be properly described by using the fractional order system theory and so forth (see [22–26]). Fractional derivatives appear in the theory of fractional differential equations; they describe the property of memory and heredity of materials, and it is the major advantage of fractional derivatives compared with integer order derivatives.
In , Bajlekova developed the notion of solution operator to study the well-posedness of fractional differential equations. da Prato and Iannelli  introduced the concept of resolvent families, which can be regarded as an extension of -semigroups, to study a class of integrodifferential equations. After that, the theory of resolvent families was developed rapidly to investigate the abstract Volterra equation: where (see [29–31]). Recently, Chen and Li  gave the family of functional equations, using resolvent operator function to equivalently describe resolvent families. Motivated by this, Kexue and Jigen [33, 34], Li et al. , and Mei et al.  studied different types of fractional resolvents describing the functional equation and applied them to fractional differential equations. Lizama and Poblete  studied the functional equation associated with -regularized resolvents. However, the above functional equations are not expressed in terms of the sum of time variables: . This is very important in concrete applications of the functional equation, just like -semigroups, integrated semigroups, integrated -semigroups, and -convoluted -semigroups (see (1)). In , Peng and Li derived a new characterization of -order semigroup () as follows: which is proved to be equivalent to fractional resolvents introduced in . Motivated by this, Mei et al.  presented a characterization as follows: which is proved to be equivalent to fractional resolvent presented by Kexue and Jigen . One of the tasks of the present paper is to find a characterization of -convoluted -order -semigroup (see Section 2 for the definition) in terms of .
In this paper, we consider the following fractional abstract Cauchy problem (FACP): where , , and is the modified Caputo fractional derivative defined by operator Observe that fractional integral of is defined by Then it is easy to show that, for , can be calculated by which is just the original definition of Caputo fractional derivative (see, e.g., ).
We plan to present in Section 2 the notion of -convoluted fractional -semigroups, which is a generalization of the notions of -convoluted semigroups  and -convoluted -semigroups . Moreover, some of their properties are studied. Section 3 is to give two equivalent descriptions of -convoluted -semigroup. The final section concerns the study of the well-posedness of the fractional abstract Cauchy problem.
Throughout this paper, , without any additional statement.
2. Convoluted Fractional -Semigroup
In this section, we will introduce the notion of -convoluted -order -semigroup and study some of its properties. The following definition is stimulated by the definitions of -convoluted semigroup  and -convoluted -semigroup .
Definition 1. Let be a closed linear operator on a Banach space . If there exists a strongly continuous operator family such that , , , , , and then is called a -convoluted -order -semigroup generated by , and is called the generator of . In the special case , is called -convoluted -order semigroup. One says that is an exponentially bounded -convoluted -order -semigroup generated by if, additionally, there exist and such that , .
Remark 2. We note that -convoluted -order -semigroup is an regularized -resolvent family ( and ), introduced by Kostić . We are concerned with this case because it is closely related to fractional differential equation. There are three reasons why we use such concept: (1) Kostić and Pilipović called the special case and to be -convoluted semigroup  and to be -convoluted -cosine function ; (2) Peng and Li called -regularized resolvent -order semigroup () ; (3) “-convoluted -order -semigroup” should be “-regularized -resolvent,” but the term “-regularized -resolvent” is too complex.
It can be checked directly that if generates a -convoluted -order -semigroup , then , , . If in , then it is clear that the -convoluted -order -semigroup is nondegenerate (let , ; (11) implies that ; thereby ).
Lemma 3. Assume that is a nondegenerate -convoluted -order -semigroup generated by on Banach space . Then .
Proof. Let . By Definition 1, it follows that , , and The combination of the closedness of and (11) implies that which gives that The two equalities (12) and (14) indicate that . Hence, Since is closed, it follows that Using the closedness of again, we obtain that , . Since is nondegenerate, it follows that , . The proof is complete.
Theorem 4. Assume that is a nondegenerate -convoluted -order -semigroup generated by on Banach space . Define the operator by Then(i)is single-valued;(ii) is closed;(iii).
Proof. We first prove that the nondegeneracy implies that is single-valued. Indeed, if, for any , satisfy, respectively,
By Titchmarsh’s theorem, it follows that , . The nondegeneracy implies that ; thereby is single valued. Hence we have .
Let the sequence satisfy and as . Then we have Letting , (43) gives that which implies that and . Hence is closed.
If , then , . The combination of (11) and (17) implies that Similar to the proof of Lemma 3, using the closedness of , we obtain that is, . Since and , we obtain that . Combining Lemma 3, Definition 1, and (23), it follows that . Since is nondegenerate, it follows that ; that is, . So . Conversely, let ; that is, and . By Definition 1, and which implies that Since is closed, (26) indicates that Hence and . Hence (iii) holds.
By the above theorem, for a nondegenerate -convoluted -order -semigroup generated by , we have . Moreover, in the special case , we have .
3. Two Characterizations of Convoluted Fractional -Semigroup
In this section we will introduce two functional equations characterizing -convoluted -order -semigroups, just like -convoluted -semigroups and integrated -semigroups.
Lemma 6. Assume that is a strongly continuous family of bounded linear operators on Banach space . Then is a -convoluted fractional -semigroup if and only if
Necessity. Assume that is a -convoluted -order -semigroup generated by . Let . Denote by the generator of . By (11), it follows that In (11), replacing with , it follows that Observe that and The combination of (28), (29), and (30) implies that
Sufficiency. Assume that (27) holds. Define operator as in (17). Then is closed. Moreover, (27) implies that and . Hence is a -convoluted -order -semigroup generated by . The proof is therefore complete.
Remark 7. We have to mention that the theory of -regularized -resolvent families is a generalization of the theory of -regularized resolvent families developed by Lizama and Poblete (see the survey paper ) and has been used by the same mentioned author in the development of many properties concerning the study of fractional differential equations . Therefore, Lemma 6 is a natural extension to the case of the results in  and the proof is also motivated by that of .
We note that (27) does not express the functional equation in terms of the sum of time variables: . As we describe semigroups and convoluted -semigroups, this is very important in concrete applications of the functional equation modeling evolution in time. Below we will find a novel functional equation in terms of the sum of time variables to describe -convoluted -order -semigroup.
Theorem 8. Assume that is a -convoluted -order -semigroup generated by on the Banach space . Then(i);(ii);(iii)for any , where the integrals are taken in the strong sense.
Proof. Denote by and the left and right hand sides of equality (32), respectively. We only have to prove that for all . For brevity, we introduce the following notations. Let
Moreover, for sufficiently large denote by the truncation of at and by , , , and the quantities obtained by replacing with in , , , and , respectively.
By [38, (20)], it follows that the Laplace transform of with respect to and is given by
We now compute the Laplace transform of with respect to and . It can be shown that, for any , with which we can further take Laplace transform of with respect to as follows:
We set By Lemma 6, it follows that for any . Thus, for all ,
The Laplace transform of is
Therefore, where stands for the Laplace transform of the function of . Thus, we obtain
By virtue of inverse Laplace transform, we obtain from (39) and (41) that The combination of (38) and (42) implies that Therefore, , for all . This completes the proof.
Theorem 9. Assume that is a strongly continuous nondegenerate operator family on Banach space . Suppose that the conditions (i), (ii), and (iii) in Theorem 8 hold. Then is a nondegenerate -convoluted -order -semigroup generated by , where is defined as in Theorem 4.
Proof. Obviously, the right hand side of (32) is symmetric with respect to and . Hence the left hand side is symmetric with respect to and . By the proof of Proposition 1 of , it follows that is commutative; that is,
For any , there holds Applying on both sides of (45) and using the commutativity (44) of , we obtain This implies by definition that and .
Denote by and the left and right hand sides of equality (32), respectively. For any , denote by the truncation of at ; that is, , and otherwise. Denote by and the quantities obtained by replacing with in , , respectively.
Let and be fixed. Define the function for by Obviously, for , Taking the Laplace transform with respect to and successively for both sides of (47), we derive The combination of (43) and (49) implies that
By virtue of inverse Laplace transform, it follows that Here the Laplace transform formula is used.
By the definition of , it follows that for all ; we have that In particular, we obtain Combining (48) and (54), we derive that This implies by definition that and Therefore, is a -convoluted -order -semigroup generated by .
Theorem 10. Assume that is a strongly continuous operator family on Banach space . Then is a -convoluted -order -semigroup if and only if (i), (ii), and (iii) in Theorem 8 hold.
Remark 11. Letting , (32) becomes the -convoluted -semigroup defined in . Indeed, if for each in some dense set the mapping is continuously differentiable in , then the limits of two sides of the equality (32) multiplied with equal and , respectively. The combination of the above fact and Theorem 10 implies that -convoluted -order -semigroup can be deemed as a natural extension of -convoluted -semigroup.
4. Fractional Abstract Cauchy Problems
In this section we will apply the theory of -convoluted -order -semigroups developed in Sections 2 and 3 to fractional abstract Cauchy problems (FACP). We begin with the definition of classical solution of FACP.
Definition 12. A function is called a classical solution of FACP, if , , and the mapping is continuously differentiable such that , holds.
Remark 13. In Definition 12, is just the modified Caputo derivative of because .
Definition 14. The fractional abstract Cauchy problem (FACP) is said to be -well-posed if for any , there exists a unique solution , and implies that as in , uniformly on any compact subinterval of .
Theorem 15. Assume that is closed, , and FACP is -well-posed. Then generates a -convoluted -order -semigroup.
Proof. For any , denote by , , the solution of FACP. Denote by , . Obviously, and is strongly continuous. Since , we have
By the uniqueness of the solution of ACP, we have . Then
By (1.21) of , it follows that
Below we will show that is continuous, a.e. , . By (59), it follows that is the solution of ACP.
Let . Define . Then . Since , , and is closed, it follows that and Then and . Hence , . Since is closed, we have . By Titchmarsh’s theorem, , . The proof is complete.
Theorem 16. Assume that generates a nondegenerate -convoluted -order -semigroup. Then, FACP is -well-posed.
Proof. Since is a nondegenerate strongly continuous family generated by , for any , we have , the mapping
is continuously differentiable, and there holds
Therefore, is a solution of FACP.
Now we prove the uniqueness of solution of FACP. Assume that is another solution of FACP. Then Taking -time integrals, we derive that which implies that Observe that is closed and communicative with , and is communicative with . By (67), it follows that Since is nondegenerate, it follows that , . Then , because is injective.
For any , let ; we have which implies that tends to uniformly on . This completes the proof.
Theorem 17. The operator generates a -convoluted -order -semigroup if and only if is closed, , and FACP is -well-posed.
Corollary 19. Assume that there exists a nondegnerate strongly continuous operator family on Banach space satisfying (i), (ii), and (iii) in Theorem 8, and defined in Theorem 4 is equal to . Then FACP is -well-posed.
Corollary 20. Assume that is a nondegnerate strongly continuous operator family on the Banach space . Suppose that . Then the following statements are equivalent:(1) is a -convoluted -order -semigroup generated by ;(2) is a closed linear operator on , and FACP is -well-posed;(3) satisfies (27), and , where is defined as in Theorem 4;(4) satisfies (i), (ii), and (iii) in Theorem 8, and , where is defined as in Theorem 4.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by NSF of China (Grant nos. 11301412, 11131006, 41390450, and 11271297), Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20130201120053), the Fundamental Research Funds for the Central Universities (Grant no. 2012jdhz52), and the National Basic Research Program of China (Grant no. 2013CB329404).
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