Research Article | Open Access
Periodic Solutions and -Asymptotically Periodic Solutions to Fractional Evolution Equations
This paper deals with the existence and uniqueness of periodic solutions, -asymptotically periodic solutions, and other types of bounded solutions for some fractional evolution equations with the Weyl-Liouville fractional derivative defined for periodic functions. Applying Fourier transform we give reasonable definitions of mild solutions. Then we accurately estimate the spectral radius of resolvent operator and obtain some existence and uniqueness results.
It is well known that fractional order differential equations provide an excellent setting for capturing in a model framework real-world problems in many disciplines, such as chemistry, physics, engineering, and biology [1–4]. In recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Podlubny , Kilbas et al. , Zhou , and the recent papers [5–8] and the references therein.
As a matter of fact, periodic motion is a very important and special phenomenon not only in natural science but also in social science such as climate, food supplement, insecticide population, and sustainable development.
In our previous paper , we discussed a class periodic boundary value problem to fractional evolution equations and obtained the existence and uniqueness results for positive mild solutions. However, since the fractional derivatives provide the description of memory property, the solution of periodic boundary value problems cannot be periodically extended to the time .
On the other hand, several authors have showed that for some fractional order systems the solutions do not show any periodic behavior if the lower terminal of the derivative is finite; see [10–14]. Let and . If is a nonconstant -periodic function of class , [10, 11] tell us that cannot be -periodic function, where is understood as one of the fractional derivatives (Caputo, Riemann-Liouville or Grunwald-Letnikov) with the lower terminal finite. Nevertheless, the authors also point out in  that fractional order derivative of other form, such as Weyl-Liouville fractional derivatives defined for periodic function , perhaps preserves periodicity. As indicated in [2, 15], the Weyl-Liouville derivative coincides with the Caputo, Riemann-Liouville or Grunwald-Letnikov derivative with lower limit , which is denoted by . There is essential difference between finity and infinity. As in, for instance, [2, 11], for , while for , where is the Caputo fractional derivative with order and the lower terminal , and denotes the two parameters Mittag-Leffler functions. It is obvious that the Weyl-Liouville fractional derivative is suitable for the study of periodic solutions to differential equations.
In real life, many phenomena are not strictly periodic; therefore many other generalized periodic cases need to be studied, such as almost periodic, asymptotically almost periodic, -asymptotically periodic, asymptotically periodic, pseudoperiodic, and pseudo-almost periodic. As the advantages of fractional derivatives, such as the memorability and heredity, many papers concern these types of solutions for fractional differential equations. Since -asymptotically periodic functions in Banach space were first studied by Henríquez et al. , there are some papers about -asymptotically periodic solutions for fractional equations; one can refer to [17–19]. For almost periodic solutions, asymptotically almost periodic and other types of bounded solutions to fractional differential equations, one can refer to [13, 17, 20–22]. Ponce  studied the existence and uniqueness of bounded solutions for semilinear fractional integrodifferential equationwhere is a closed linear operator defined on a Banach space , is Weyl fractional derivative of order with the lower limit , is a scalar-valued kernel, and satisfies some Lipschitz type conditions. Assume that is the generator of an -resolvent family which is uniformly integrable. The mild solutions of (1) was given byBy Banach contraction principle, existence and uniqueness results of almost periodic, asymptotically almost periodic and other types of bounded solutions are established. In addition, Lizama and Poblete  gave some sufficient conditions ensuring the existence and uniqueness of bounded solutions to a fractional semilinear equation of order .
In this paper, we study the fractional evolution equations in an ordered Banach space where is the Weyl-Liouville fractional derivative of order which is defined in Section 2 and is the infinitesimal generator of a -semigroup . Applying Fourier transform, we give reasonable definitions of mild solutions of (3). Then the existence and uniqueness results for the corresponding linear fractional evolution equations are established, and the spectral radius of resolvent operator is accurately estimated. Finally, some sufficient conditions are established for the existence and uniqueness of periodic solutions, -asymptotically periodic solutions, and other types of bounded solutions when satisfies some ordered or Lipschitz conditions.
Compared with the earlier related existence results which appeared in [21–23], there are at least four essence differences: (i) the order of the fractional derivative is ; (ii) in many papers concerning bounded solutions for differential equations, such as [21, 23], the solution is defined by the limit of the solution for the limit equation, without strict proofs of the solution for the given equation possessing that form, but, in this paper, we apply the Fourier transform to (3) itself and obtain the expression of the solution; naturally, we may define expression as a mild solution under suitable conditions; besides, the mild solution in this paper is expressed by the semigroups generated by , which are more convenient than some resolvent families applied in ; (iii) the spectral radius of resolvent operator is accurately estimated; (iv) the conditions on contain some ordered relations, and the associated method is monotone iterative technique.
The paper is organized as follows. Section 2 provides the definitions and preliminary results to be used in the article. In Section 3, the existence and uniqueness results for the linear equations are obtained. In Section 4, we consider the existence and uniqueness of the mild solutions for (3). In Section 5, we also give two examples to apply the abstract results.
This section is devoted to some preliminary facts needed in the sequel. Let be an ordered Banach space with norm and partial order , whose positive cone ( is the zero element of ) is normal with normal constant . Denote by the space of all linear bounded operator on Banach space , with the norm . The notation stands for the Banach space of all bounded continuous functions from into equipped with the sup norm ; that is,For , if for all . stands for the subspace of consisting of all -valued continuous -periodic functions. Set These functions in are called -asymptotically -periodic (see ). We note that and are Banach spaces (see ), and .
A function , if for any there is a real number and in arbitrary interval of length such that for all is said to be almost periodic (in the sense of Bohr). We denote by the set of all these functions. The space of almost automorphic functions (resp., compact almost automorphic functions) will be written as (resp., ). The bounded continuous function (resp., ) for every sequence there is a subsequence such that and for each (resp., uniformly on compact subsets of ). Clearly the compact almost automorphic function above is continuous on .
For convenience, we set . We regard the direct sum of and as the space of asymptotically periodic functions , the direct sum of and as the space of asymptotically almost periodic functions , the direct sum of and as the space of asymptotically compact almost automorphic functions, and the direct sum of and as the space of asymptotically almost automorphic functions.
Then we setand regard the direct sum of and as the space of pseudoperiodic functions, the direct sum of and as the space of pseudo-almost periodic functions, the direct sum of and as the space of pseudocompact almost automorphic functions, and the direct sum of and as the space of pseudo-almost automorphic functions.
The relationship of the above different classes of subspaces is presented in [21, p. 805]:
Denote by or simply the following function spaces: and the space of all functions satisfying uniformly for each in any bounded subset of . For fixed , then for and , the following conditions could ensure that : () is uniformly continuous with respect to on for each bounded subset of . More precisely, given and , there exists such that and imply that ;()for all bounded subset , is bounded;()if , where and , is uniformly continuous with respect to on for each bounded subset of .Then the results shown in Table 1 hold; see [21, Remark3.4 and p. 810].
Corollary 1 (see [21, Corollary 3.9]). Let and be given and fixed. Assume that there exists a constant such thatfor all and . Let ; then .
Now we recall the definitions of some fractional derivatives and integrals which are used in this paper (see ).
Definition 2. Let be periodic with period and such that its integral over a period vanishes. The Weyl fractional integral of order is defined aswherefor .
The above Weyl definition is accordant with the Riemann-Liouville definition for periodic functions whose integral over a periodic vanishes.
The Weyl-Liouville fractional derivative is defined asfor .
It is shown that the Weyl-Liouville derivative coincides with the Caputo, Riemann-Liouville, or Grunwald-Letnikov derivative with lower limit . It is known that for any , where denotes the identity operator and holds with ; see . For example, for the function,
Denote by the Fourier transform of ; that is,for and . Thus ; see [1, Remark2.11].
Let us recall the definitions and properties of operator semigroups; for details see . Assume that is the infinitesimal generator of a -semigroup . If there are and such that , thenA -semigroup is called exponentially stable if there exist constants and such thatThe growth bound of the semigroup is defined asFurthermore, could also be expressed byFor -semigroup , if there exists a constant such thatthen it is called uniformly bounded. A -semigroup is called compact if is compact for . The positive -semigroup is -semigroup satisfying for all and . For the positive operators semigroup, one can refer to .
In the following part, we shall recall the definitions and properties of Mittag-Leffler functions (see ). NoteThese functions have the following properties for and .
Lemma 5. (1)Assume that is a uniformly bounded semigroup and satisfies (21). Then, for any fixed , is a linear and bounded operator; that is, for any , we get(2)If is a semigroup, then is strongly continuous.(3)If is a positive -semigroup, then is positive for .(4)If is exponentially stable and satisfies (18), then
Next, we consider the following linear abstract fractional evolution equation:where generates a -semigroup of operators on Banach space and is continuous.
For convenience, we assume the following condition:(), , for , and satisfies (28), where
Proof. Applying Fourier transform to (28), we get for . In view of (17) and Remark 4, we have where . By the uniqueness of Fourier transform, we deduce that the assertion of lemma holds. This completes the proof.
3. Results for Linear Equations
Proof. : If , then Therefore, .
: by the hypotheses, for any , we can find a real number for any interval of length and there exists a number in this interval such that for all . From Lemmas 3, 5(4), and , we have and and are all almost periodic.
: since , there is and such that and as , uniformly on compact subsets of .therefore by Lebesgue dominated convergence theorem, when , we get for all .
Furthermore, for a compact set and , by Lemmas 3 and 5, we choose and such that where . For , by Lemmas 3 and 5, we estimate which implies that that the convergence is irrelevant to . Similarly, we can prove that if uniformly for on compact subsets of . The case of the space is similar too.
: set . For all sequence , there is of such thatSinceby (36) and Lemmas 3 and 5, for any , we get as . For any Lebesgue dominated convergence theorem implies that as . It is similar that: assume that . For any , there exists such that for . Then, by Lemmas 3 and 5, we get for . It follows that as . Thus, .
We next consider the asymptotic property of the solutions. For and , we have for some and . Then Lemmas 3 and 5 imply that which implies that as . Naturally, we can also get the results for the spaces , , , and .
Set . For we get We can find that the set is translation-invariant and get by Lebesgue dominated convergence theorem. Then , , , and have the maximal regularity property under the convolution defined by (33).
Remark 10. Equation (46) is an optimal estimation. In fact, for , the periodic solution of equation is .
Corollary 11. Let . Assume that generates a uniformly bounded -semigroup and satisfies (21). If , then linear fractional evolution equationhas a unique mild solution , and
Proof. generates a -semigroup , and . Then , so is exponentially stable for . The conclusion follows by Theorem 9.
Theorem 12. Let . Assume that generates an exponentially stable -semigroup , that is, the growth bound Then linear fractional evolution equation (28) has a unique mild solution , is bounded linear, and spectral radius .
Proof. By Theorem 9 and Lemma 5, we obtain that (28) has a unique mild solution , and is bounded linear.
For all , there exists such thatDefine a new norm in asSince , then is equivalent to . The norm of in is denoted by . Then, for , we have This implies that . Remark 4 implies thatwhere is defined by (24). Since , we have . By (55) and Lemma 3, we have which implies that . Then , and spectral radius . Since is any number in , we obtain .
Remark 13. Similarly to , if is a compact and positive analytic semigroup and is a regeneration cone, then , where is the first eigenvalue of .
Corollary 14. Assume that and positive cone is a regeneration cone, compact and positive -semigroup is generated by , whose first eigenvalueThen (28) possesses a unique mild solution , is positive and bounded linear, and the corresponding spectral radius .
Proof. The proof is similar to that in .
4. Results for Nonlinear Equations
Theorem 15. Let be an ordered Banach space, whose positive cone is normal with normal constant . Assume that generates a positive -semigroup , , and satisfies the results shown in Table 1, for , and the following assumptions hold: ()for , there is such that where ,