Abstract
The main aim of this paper is to investigate generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions to a class of abstract (semilinear) multiterm fractional differential inclusions with Caputo derivatives. We illustrate our abstract results with several examples and possible applications.
1. Introduction and Preliminaries
Almost periodic and asymptotically almost periodic solutions of differential equations in Banach spaces have been considered by many authors so far (for the basic information on the subject, we refer the reader to the monographs [1–10]). Concerning almost automorphic and asymptotically almost automorphic solutions of abstract differential equations, one may refer, for example, to the monographs by Diagana [4], N’Guérékata [5], and references cited therein.
Of concern is the following abstract multiterm fractional differential inclusion:where , are bounded linear operators on a Banach space , is a closed multivalued linear operator on , , , is an -valued function, and denotes the Caputo fractional derivative of order ([11, 12]). In this paper, we provide the notions of -regularized -existence and uniqueness propagation families for (1) and -regularized -propagation families for (1). In Section 4, we profile these solution operator families in terms of vector-valued Laplace transform, while in Section 5 we consider asymptotical behaviour of analytic integrated solution operator families for (1). The main result of paper, Theorem 18, enables one to consider asymptotically periodic solutions, asymptotically almost periodic solutions, and asymptotically almost automorphic solutions of certain classes of abstract integrodifferential equations in Banach spaces. In a similar way, we can give the basic information about the following abstract semilinear multiterm fractional differential inclusion:where , are bounded linear operators on a Banach space , is a closed multivalued linear operator on , , , and is an -valued function satisfying certain assumptions.
Since we essentially follow the method proposed by Kostić et al. [13] (see also [12, Subsection 2.10.1]), the boundedness of linear operator is crucial for applications of vector-valued Laplace transform and therefore will be the starting point in our work.
The organization and main ideas of this paper can be briefly described as follows. In Section 2, we present the basic information about Stepanov and Weyl generalizations of asymptotically almost periodic functions and asymptotically almost automorphic functions (Proposition 4 is the only new contribution in this section). The main aim of third section is to give a brief recollection of results and definitions about multivalued linear operators in Banach spaces; in a separate Section 3.1, we analyze degenerate -regularized -resolvent families subgenerated by multivalued linear operators. Section 4, which is written almost in an expository manner, is devoted to the study of -regularized -propagation families for (1). The main result of fifth section is Theorem 18, where we investigate the asymptotic behaviour of -regularized -propagation families for (1). In the proof of this theorem, we use the well-known results on analytical properties of vector-valued Laplace transform established by Sova in [14] (see, e.g., [2, Theorem 2.6.1]) in place of Cuesta’s method established in the proof of [15, Theorem 2.1]. The proof of Theorem 18 is much simpler and transparent than that of [15, Theorem ] because of the simplicity of contour in our approach. We will essentially use this fact for improvement of some known results on the asymptotic behaviour of solution operator families governing solutions of abstract two-term fractional differential equations, established recently by Keyantuo et al. [16] and Luong [17]. Contrary to a great number of papers from the existing literature, Theorem 18 is applicable to the almost sectorial operators, generators of integrated or -regularized semigroups, and multivalued linear operators employing in the analysis of (fractional) Poisson heat equations in -spaces ([18, 19]). For more details, see Section 6.
We use the standard notation throughout the paper. By we denote a complex Banach space. If is also such a space, then by we denote the space of all continuous linear mappings from into ; If is a linear operator acting on , then the domain, kernel space, and range of will be denoted by , , and , respectively. The symbol denotes the identity operator on . By we denote the space consisted of all bounded continuous functions from into ; the symbol denotes the closed subspace of consisting of functions vanishing at infinity. By we denote the space consisted of all bounded uniformly continuous functions from to This space becomes one of Banach’s spaces when equipped with the sup-norm. Let us recall that a subset of is said to be total in iff its linear span is dense in .
Let . Consider the Laplace integralfor . If exists for some , then we define the abscissa of convergence of byotherwise, . It is said that is Laplace transformable or equivalently that belongs to the class (P1)-, iff ; in scalar-valued case, we write - and .
If , then we define , ; the Dirac delta distribution. Here, denotes the Gamma function. Set (), , and ().
During the past few decades, considerable interest in fractional calculus and fractional differential equations has been stimulated due to their numerous applications in many areas of physics and engineering. A great number of important phenomena in electromagnetics, acoustics, viscoelasticity, aerodynamics, electrochemistry, and cosmology are well described and modelled by fractional differential equations. For basic information about fractional calculus and nondegenerate fractional differential equations, one may refer, for example, to [11, 12, 20–25] and the references cited therein.
We will use only the Caputo fractional derivatives. Let Then the Caputo fractional derivative ([11, 12]) is defined for those functions for which , byAssuming that the Caputo fractional derivative exists, then for each number the Caputo fractional derivative exists, as well.
The Mittag-Leffler function (, ) is defined bySet ,
The asymptotic behaviour of the Mittag-Leffler function is given in the following lemma (see, e.g., [12]):
Lemma 1. Let Then, for every and ,where is defined by and the first summation is taken over all those integers satisfying
If , , and , then the following special cases of Lemma 1 hold good:where
For further information about the Mittag-Leffler functions, compare [11, 12] and the references cited there.
2. Stepanov and Weyl Generalizations of (Asymptotically) Almost Periodic and Almost Automorphic Functions
The class of almost periodic functions was introduced by H. Bohr in 1925 and later generalized by many other mathematicians. Let or , and let be continuous, where is a Banach space with the norm . For any number given in advance, we say that a number is an -period for iff , The set consisting of all -periods for is denoted by We say that is almost periodic, a.p. for short, iff for each the set is relatively dense in , which means that there exists such that any subinterval of of length meets . For basic information about various classes of almost periodic functions and their generalizations, we refer the reader to [4–8, 10, 12, 13, 16, 19, 21, 26–34]. The space consisting of all almost periodic functions from the interval into will be denoted by
It is well known that the vector space consisting of all bounded continuous -periodic functions, denoted by , , is a vector subspace of Set .
Suppose that , , and , where or Define the Stepanov “metric” byThen, in scalar-valued case, there existsin The distance appearing in (11) is called the Weyl distance of and The Stepanov and Weyl “norm” of are introduced byrespectively. We say that a function is Stepanov -bounded, -bounded shortly, iffThe space consisting of all -bounded functions becomes a Banach space when equipped with the above norm. A function is called Stepanov -almost periodic, -almost periodic shortly, iff the function , defined by , , is almost periodic. It is said that is asymptotically Stepanov -almost periodic, asymptotically -almost periodic for short, iff is asymptotically almost periodic.
It is a well-known fact that if is an almost periodic (resp., a.a.p.) function then is also -almost periodic (resp., asymptotically -a.a.p.) for The converse statement is not true, in general.
By we denote the space consisted of all -almost periodic functions A function is said to be asymptotically Stepanov -almost periodic, asymptotically -almost periodic for short, iff is asymptotically almost periodic. By and we denote the vector spaces consisting of all Stepanov -almost periodic functions and asymptotically Stepanov -almost periodic functions, respectively.
Let us recall that any asymptotically almost periodic function is also asymptotically Stepanov -almost periodic (). The converse statement is clearly not true because an asymptotically Stepanov -almost periodic function need not be continuous.
We are continuing by explaining the basic definitions and results about the (asymptotically) Weyl-almost periodic functions.
Definition 2 (see [35]). Assume that or Let and .(i)It is said that the function is equi-Weyl--almost periodic, for short, iff for each we can find two real numbers and such that any interval of length contains a point such that(ii)It is said that the function is Weyl--almost periodic, for short, iff for each we can find a real number such that any interval of length contains a point such that
We know that in the set theoretical sense and that any of these two inclusions can be strict ([26]).
We refer the reader to [35] for basic definitions and results about asymptotically Weyl-almost periodic functions.
Definition 3. We say that is Weyl--vanishing iff
It is clear that for any function we can replace the limits in (16). It is said that is equi-Weyl--vanishing iff
If and is equi-Weyl--vanishing, then is Weyl--vanishing. The converse statement does not hold, in general ([35]). By and we denote the vector spaces consisting of all Weyl--vanishing functions and equi-Weyl--vanishing functions, respectively.
It can be simply proved that the limit of any uniformly convergent sequence of bounded continuous functions that are (asymptotically) almost periodic or automorphic, respectively (asymptotically), Stepanov almost periodic or automorphic, has again this property. The following result holds for the Weyl class.
Proposition 4. Let be a uniformly convergent sequence of functions from , respectively, , where If is the corresponding limit function, then , respectively, .
Proof. We will prove the part (i) only for the equi-Weyl--almost periodic functions. It is clear that Let be given in advance. Then there exists an integer such that for each we have thatBy definition, we know that there exist two real numbers and such that any interval of length contains a point such thatThen, for the proof of equi-Weyl--almost periodicity of function , we can choose the same and , and the same from any subinterval ; speaking-matter-of-factly, we havefor all , so that a simple calculation involving (18) gives the existence of a finite constant such thatThen the final result simply follows from (19).
And, just a few words about (generalized) automorphic extensions of introduced classes, where our results clearly apply. Let be continuous. As it is well known, is called almost automorphic, a.a. for short, iff for every real sequence there exist a subsequence of and a map such thatpointwise for If this is the case, then it is well known that and that the limit function must be bounded on but not necessarily continuous on Furthermore, it is clear that the uniform convergence of one of the limits appearing in (22) implies the convergence of the second one in this equation and that, in this case, the function has to be almost periodic and the function has to be continuous. If the convergence of limits appearing in (22) is uniform on compact subsets of , then we say that is compactly almost automorphic, c.a.a. for short. The vector space consisting of all almost automorphic, respectively, compactly almost automorphic functions, is denoted by , respectively, By Bochner’s criterion [4], any almost periodic function has to be compactly almost automorphic. The converse statement is not true, however [36]. It is also worth noting that P. Bender proved in doctoral dissertation that that a.a. function is c.a.a. iff it is uniformly continuous (1966, Iowa State University).
It is well-known that the reflexion at zero keeps the spaces and unchanged and that the function from (22) satisfies and , later needed to be a compact subset of An interesting example of an almost automorphic function that is not almost periodic has been constructed by W. A. Veech
A continuous function is called asymptotically (compact) almost automorphic, a.(c.)a.a. for short, iff there exist a function and a (compact) almost automorphic function such that , Using Bochner’s criterion again, it readily follows that any asymptotically almost periodic function is asymptotically (compact) almost automorphic. It is well known that the spaces of almost periodic, almost automorphic, compactly almost automorphic functions and asymptotically (compact) almost automorphic functions are closed subspaces of when equipped with the sup-norm.
We refer the reader to [28] for the notion of Stepanov-like almost automorphic functions. The concepts of Weyl-almost automorphy and Weyl pseudo almost automorphy, more general than those of Stepanov almost automorphy and Stepanov pseudo almost automorphy, were introduced by Abbas [37] in 2012. Besides the concepts of Stepanov-like almost automorphic functions, our results apply also to the classes of Weyl-almost automorphic functions and Besicovitch almost automorphic functions, introduced in [38] (cf. [7, 39] for more details).
3. Multivalued Linear Operators in Banach Spaces
In this section, we will present some necessary definitions and auxiliary results from the theory of multivalued linear operators in Banach spaces. For further information in this direction, the reader may consult the monographs by Cross [40] and Favini and Yagi [18].
Let and be two Banach spaces over the field of complex numbers. A multivalued mapping is said to be a multivalued linear operator (MLO) iff the following two conditions hold:(i) is a linear subspace of ;(ii), , and , , .
In the case that , then we say that is an MLO in It is well-known that the equality holds for every and for every with If is an MLO, then is always a linear subspace of and for any and Put Then the set is called the kernel of The inverse of an MLO is generally defined by and . It is checked at once that is an MLO in , and that and If , that is, if is single-valued, then is called injective. If are two MLOs, then we define its sum by and , It is evident that is likewise an MLO. We write iff and for all
Let and be two MLOs, where is a complex Banach space. The product of and is defined by and A simple proof shows that is an MLO and The scalar multiplication of an MLO with the number , for short, is defined by and , Then is an MLO and , .
It is said that an MLO is closed iff for any two sequences in and in such that ; for all we have that and imply and .
We need the following lemma from [19].
Lemma 5. Let be a locally compact, separable metric space, and let be a locally finite Borel measure defined on Suppose that is a closed MLO. Let and be -integrable, and let , Then and
Henceforward, will always be an appropriate subspace of and will always be the Lebesgue measure defined on
Denote by (P1)- the vector space consisting of all Laplace transformable functions ; by we denote the Laplace transform of , defined as in [2]. We need also the following lemma from [19].
Lemma 6. Assume that is a closed MLO and that (P1), (P1) and , for Then for any which is a point of continuity of both functions and .
Suppose that is an MLO in and that is possibly noninjective operator satisfying Then the -resolvent set of , for short, is defined as the union of those complex numbers for which(i);(ii) is a single-valued linear continuous operator on .
The operator is called the -resolvent of ; the resolvent set of is defined by , .
We will use the following extension of [19, Theorem 1.2.4(i)], whose proof can be left to the reader as an easy exercise (see also the proof of [18, Theorem 1.7, p. 24]).
Lemma 7. Let and let be an MLO. If , , , and , then one has
Suppose that is an MLO in Then is said to be an eigenvalue of iff there exists an element such that ; we call an eigenvector of operator corresponding to the eigenvalue Let us recall that, in purely multivalued case, an element can be an eigenvector of operator corresponding to different values of scalars The point spectrum of , for short, is defined as the union of all eigenvalues of
3.1. Degenerate -Regularized -Resolvent Operator Families
If it is not stated otherwise, we assume that , , , , , is an MLO, , is injective, is injective, and .
We need the following notions from [19].
Definition 8. Suppose , , , , , is an MLO, , and is injective.(i)Then it is said that is a subgenerator of a (local, if ) mild -regularized -existence and uniqueness family iff the mappings , , and , , are continuous for every fixed and , and the following conditions hold: (ii)Let be strongly continuous. Then it is said that is a subgenerator of a (local, if ) mild -regularized -existence family iff (25) holds.(iii)Let be strongly continuous. Then it is said that is a subgenerator of a (local, if ) mild -regularized -uniqueness family iff (26) holds.
Definition 9. Suppose that , , , , , is an MLO, is injective, and Then it is said that a strongly continuous operator family is an -regularized -resolvent family with a subgenerator iff is a mild -regularized -uniqueness family having as subgenerator, , and .
If , is said to be exponentially bounded (bounded) iff there exists () such that the family is bounded. If , where , then it is also said that is an -times integrated -resolvent family; -times integrated -resolvent family is further abbreviated to -resolvent family. We accept a similar terminology for the classes of mild -regularized -existence families and mild -regularized -uniqueness families.
The integral generator of a mild -regularized -uniqueness family (mild -regularized -existence and uniqueness family ) is defined throughthe integral generator of an -regularized -regularized family is defined in a similar fashion. The integral generator is a closed MLO in which is, in fact, the maximal subgenerator of () with respect to the set inclusion. We refer the reader to [19] for the notion of an exponentially bounded, analytic -regularized -resolvent operator family.
Unless stated otherwise, we will always assume henceforth that the function is a scalar-valued kernel on and that the operator is injective. For more details about abstract degenerate differential equations, the reader may consult the monographs [18, 41–43].
4. -Regularized -Propagation Families for (1)
Recall that , are bounded linear operators on a Banach space , is a closed multivalued linear operator on , , , and is an -valued function. Henceforth, we always assume that are scalar-valued kernels and in Set , , , , and .
We will use the following definition.
Definition 10. A function is called a (strong) solution of (1) iff for , , and (1) holds.
Integrating both sides of (1) -times and employing the closedness of , Lemma 5, and the equality [11, ], it readily follows that any strong solution , of (1) satisfies the following:
If , then we define Plugging , , , in (28), we getwhere appears in the th place () starting from Proceeding as in nondegenerate case [12], this inclusion motivates us to introduce the following extension of [12, Definition 2.10.2] (cf. also [34, Definition ] and [32, Definitions and ] for similar notions).
Definition 11. Suppose that , , , and and are injective. A sequence of strongly continuous operator families in is called a (local, if ):(i)-regularized -existence propagation family for (1) iff the following holds: for any (ii)-regularized -uniqueness propagation family for (1) iff the following holds: provided and .(iii)-regularized -resolvent propagation family for (1), in short -regularized -propagation family for (1), iff is a -regularized -uniqueness propagation family for (1), and if for every , , and , one has , , and .
In the case that , where , then we also say that is a -times integrated -resolvent propagation family for (1); -times integrated -resolvent propagation family for (1) is simply called -resolvent propagation family for (1). For a -regularized -existence and uniqueness family , it is said that is exponentially bounded iff each single operator family is. The above terminological agreement is accepted for all other classes of -regularized -propagation families introduced so far.
If , where for , then it is also said that is a subgenerator of The notion of integral generator of is introduced as in nondegenerate case [12].
Hereafter, the following equality will play an important role in our analysis:for any The basic properties of subgenerators and integral generators continue to hold, with appropriate changes, in degenerate case; compare [12] and [19, Section ] for more details. We leave to the interested reader the problem of transferring the assertions of [12, Propositions 2.10.3–2.10.5, Theorem 2.10.7] to degenerate case.
The following is a degenerate version of [12, Definition 2.10.6].
Definition 12. Let Consider the following inhomogeneous Cauchy inclusion:A function is said to be(i)a strong solution of (33) iff there exists a continuous function such that for all and (ii)a mild solution of (33) iff
Clearly, every strong solution of (33) is also a mild solution of the same problem while the converse statement is not true, in general. We similarly define the notion of a strong (mild) solution of problem (28).
We have the following:(a)If is a -existence propagation family for (1), then the function , , is a mild solution of (28) with for (b)If is a -uniqueness propagation family for (1), and , , , , and , then the function , , is a strong solution of (28), provided for .
For our later purposes, it will be sufficient to characterize the introduced classes of -regularized propagation families by vector-valued Laplace transform; keeping in mind Lemmas 5–7, the proofs are almost the same as in nondegenerate case and we will only notify some details of the proof of Theorem 14 below because the formulation of [12, Theorem ] is slightly misleading since the injectivity of operator for with has not been clarified in a proper way and property (ii) in the formulation of this theorem is required to hold for all .
Theorem 13. Suppose satisfies (P1), is strongly continuous, and the family is bounded, provided Let be a closed MLO on , let , and let be injective. Set(i)Suppose , Then is a global -regularized -existence propagation family for (1) iff the following conditions hold:(a)The inclusion holds provided , , , and .(b)The inclusion holds provided , , , and .(ii)Suppose , Then is a global -regularized -uniqueness propagation family for (1) iff, for every with , , and , the following equality holds:
Theorem 14. Suppose satisfies (P1), is strongly continuous, and the family is bounded, provided .(I)Let the following two conditions hold:(i), , , , , , and , .(ii)There exists an index satisfying exactly one of the following two conditions:(a) and the operator is injective for every with and ,(b), , and the operator is injective for every with and If is a global -regularized -resolvent propagation family for (1) and (30) holds, then is injective for every with and and equalities (37)-(38) are fulfilled.(II)Suppose that is injective for every with and and equalities (37)-(38) are fulfilled. If condition (I)(i) holds, then is a global -regularized -resolvent propagation family for (1).
Proof. Concerning assertion (I), we will only sketch the main details of the proof of the injectivity of operator for every with and (we know that (37)-(38) hold on account of Theorem 13). Observe that we do not need the condition (I)(ii) for the proof of (II), where we only use an elementary argumentation as well as Lemmas 5–7 (the composition property (31) follows by applying the Laplace transform and Lemma 7, while the commutation of operator families with the operators and for is much simpler to show). The consideration is quite similar in the case that the condition (II) holds and, because of that, we will consider only the first case. Let with and be fixed, and let for some Using the fact that is a global -regularized -uniqueness propagation family for (1), we can simply prove thatby performing the Laplace transform at the both sides of the composition property (31). By the injectivity of the operator for , we obtain that and the claimed assertion follows.
These results enable one to simply clarify the Hille-Yosida type theorems for exponentially bounded -regularized -resolvent propagation families. The analytical properties of -regularized -resolvent propagation families can be analyzed similarly as in nondegenerate case [12]. We will use the following definition.
Definition 15. (i) Let , and let be a -regularized -resolvent propagation family for (1). Then it is said that is an analytic -regularized -resolvent propagation family of angle , iff for each there exists a function which satisfies that, for every , the mapping , is analytic and that(a), , and(b) for all and .(ii) Suppose that is an analytic -regularized -resolvent propagation family of angle Then it is said that is an exponentially bounded, analytic -regularized -resolvent propagation family of angle , respectively, bounded -regularized -resolvent propagation family, iff for every , there exists , respectively, , such that the family is bounded for all . Since there is no risk for confusion, we will identify in the sequel and .
For our purposes, the following result will be sufficiently enough (cf. Theorem 14 and [2, Theorem 2.6.1, Proposition 2.6.3 b]); we feel duty bound to say that the small inconsistencies in the formulation of [12, Theorem 2.10.11] have been made; see also [34].
Theorem 16. Assume satisfies (P1), , , and, for every , the function can be analytically extended to a function satisfying that, for every , the set is bounded.
Let the following three conditions hold:(i), , , , , , and , .(ii)The operator is injective for all (iii)Let satisfy that, for every , the mapping , is analytic and that for each there exists an operator such that provided , provided , and, in the case ,Then there exists an exponentially bounded, analytic -regularized -resolvent propagation family for (1), of angle . Furthermore, (30) holds, the family is bounded for all and , and , , and .
Remark 17. For the sequel, it will be very important to note that the notion introduced in Definition 11(iii) and Definition 15 can be introduced for any single operator family of the tuple The assertions of Theorems 14 and 16 can be simply reformulated for ; for example, if the index is given in advance, then in the formulation of Theorem 16 it suffices to assume that the function can be analytically extended to a function satisfying that, for every , the set is bounded, and that (i)-(ii) hold and (iii) holds only for this specified index It will be said that is an (exponentially bounded, analytic/analytic) -regularized -resolvent propagation family. All terminological agreements explained before will be accepted for -regularized -resolvent propagation families; the classes of -regularized -existence propagation families and -regularized -uniqueness propagation families are introduced similarly.
5. Asymptotical Behaviour of -Regularized -Propagation Families for (1)
The main aim of this section is to investigate polynomial decaying of -regularized -propagation families for (1) as time goes to infinity. Applications of Theorem 16 (see also Remark 17) will be crucial in our work and we start by observing that it is not clear how one can prove the injectivity of operator , given by (36), in general case. Because of that, we will first focus our attention to the case that , where for , by exploring the generation of fractionally integrated -propagation families for (1) only. Moreover, we will assume that the numbers are nonnegative for and that (the case can be analyzed similarly) and the multivalued linear operator under our consideration is possibly not densely defined.
Theorem 18. Suppose that for , , is a closed MLO, is injective, , and the following condition holds:(H)There exist finite constants , , , and such thatAssume that the mapping , is strongly continuous. Assume also that satisfies , andSet if is densely defined, otherwise, and Then there exists an exponentially bounded, analytic -regularized -propagation family for (1), of angle Moreover, (30) holds and there exists a finite constant such that
Proof. Since we have assumed that the mapping , is strongly continuous, and its restriction to is strongly analytic on this region; see [19, Proposition 1.2.6(iii)]. Taking into account (47) and for , we getIt is clear that (46) impliesUsing an elementary argumentation, (46), (49), and (50), we can simply verify that the conditions of Theorem 16 hold with sufficiently large, , , and , in the case that the operator is not densely defined. Hence, is a subgenerator of an exponentially bounded, analytic -times integrated -propagation family for (1), of angle , as claimed. Estimate (48) remains to be proved. Fix the numbers and By the proof of [2, Theorem ] and Cauchy theorem, we have that, for every ,where is oriented counterclockwise and consists of and Keeping in mind (49) and the estimate from the condition (H), it readily follows that for each we have that , so thatBut, then estimate (48) follows from a simple integral computation that is very similar to that appearing in the proof of [2, Theorem ].
Remark 19. (i) It is worth noting that the value of exponent in (H) does not depend on the final estimate (48). In our proof, we only use the estimate ,
(ii) As mentioned in the introductory part, the proof of important result of Cuesta [15, Theorem ] for the classical fractional oscillation resolvent families generated by densely defined linear operators [11], satisfying the condition (H) with , , and , follows completely different lines. Furthermore, in our approach, the case in which or can occur, Theorem 18 is applicable in the qualitative analysis of fractional relaxation multiterm differential inclusions (but not in the analysis of generalized asymptotical almost periodicity and generalized asymptotical almost automorphy of solutions; see (53)).
Let , where the symbol denotes the space of scalar-valued asymptotically almost automorphic functions, and let be defined as above. If is the -regularized -propagation family for (1), constructed with the help of Theorem 18, and , then it can be easily checked that is a -regularized -propagation family for (1), satisfying additionally (30), where By Theorem 18, some known assertions concerning inheritance of asymptotical periodicity, almost asymptotical almost periodicity and asymptotical almost automorphy under the action of finite convolution products ([5, 16]), and the assertions (a)-(b) clarified above, this yields the following result (the uniqueness of solutions follows from the fact that [12, Theorem ] holds in degenerate case and that the condition stated in the formulation of [12, Proposition ] holds true, which can be verified by performing the Laplace transform).
Corollary 20. Let the requirements of Theorem 18 hold, let , and let Set , , Assume thatThen, for every , is a unique mild solution of the abstract Cauchy inclusionFurthermore, is a strong solution of (54) provided that .
Concerning the Stepanov asymptotical almost periodicity and (equi-) Weyl asymptotical almost periodicity, some extra conditions on the vanishing part of function must be imposed if we want that the solution defined above belongs to the same class of functions as For example, we have the following:(i)Stepanov class: suppose that and is asymptotically -almost periodic and that the locally -integrable functions , satisfy the conditions from [30, Lemma ], and Then the function is asymptotically -almost periodic (see [33, Proposition , Remark ]).(ii)Weyl classes: if is bounded and Weyl-almost periodic and satisfies the following conditions:(i)the mapping , , is bounded as ,(ii). then the function is in class , where ; see [35, Proposition 2.3(i), Example ]. Here, is chosen so that , .
Denote by the set consisting of all generalized (asymptotically) almost periodic function spaces and all generalized (asymptotically) almost automorphic function spaces considered so far. Let denote the collection of all spaces from that are not in any class of functions obtained as a sum of some spaces of (equi-) Weyl almost periodic (automorphic) functions and some space of (equi-) Weyl almost vanishing functions.
The second part of the following proposition is very similar to [7, Proposition ].
Proposition 21. Suppose that satisfies (P1) and is a strongly Laplace transformable -regularized -propagation family for (1).(i)For every , there exists a function satisfying (P1)- and provided that , respectively, provided that (ii)Denote by the set consisting of all eigenvectors of operator which corresponds to eigenvalues of operator for which the mappingbelongs to the space . Then the mapping , , belongs to the space for all ; furthermore, the mapping , , belongs to the space for all provided additionally that is bounded.
Proof. We will examine the case only. The proof of (i) can be given following the lines of the proof of [9, Theorem ], with appropriate changes briefly described as follows. Since the operator is invertible in for all with sufficiently large, because the norm of bounded linear operator for such values of is strictly less than , we get that the termis well-defined for all with , for some SetandThen it is clear thatand that there exists a finite constant such thatwhere denotes the th convolution power of The function , , is well-defined since there exists a finite constant such thatand, due to Lemma 1,For the remaining part of proof of (i), it suffices to repeat literally the arguments from the proof of [9, Theorem ]. For the proof of (ii), observe first that, if for some , then performing the Laplace transform at the both sides of the composition property (31), as it has been done as in our previous examinations, immediately yields thatfor suff. large, and therefore , As a consequence, we have that the mapping , , belongs to the space for all The boundedness of implies the uniform convergence of to for any sequence converging to some element ; then the final result follows by combining the previously proved statement and the fact that the limit of a uniform convergent sequence of bounded continuous functions belonging to any space from belongs to this space again (see Proposition 4 for the class of (equi-) Weyl-almost periodic functions).
Remark 22. If and for some , then a simple calculation shows thatfor satisfying . To the best knowledge of the authors, in the handbooks containing tables of Laplace transforms, the explicit forms of functions like are not known, with the exception of some very special cases of the coefficients , (see, e.g., [12, Remark 3.3.10(vi)]).
The following theorem is motivated by some pioneering results of Ruess and Summers concerning integration of asymptotically almost periodic functions [24].
Theorem 23. Assume that is an exponentially bounded -regularized -propagation family for (1). Let , and let there exist a number such that , Assume, further, that the following conditions hold:(i)Let satisfy that there exists a function such that , (ii)Assume that , , where and .(iii)Let for all .(iv)Assume that for all and (i.e., does not contain an isomorphic copy of ), or is weakly relatively compact in for all (v)For every , we haveThen there exists a unique exponentially bounded mild solution of the abstract Cauchy inclusion (33). Furthermore, .
Proof. From our previous considerations of nondegenerate case, it is well known that any mild solution of the abstract Cauchy inclusion (33) has to satisfy the following equality:See, for example, [12, Theorem ]. Taking the Laplace transform, we getSince , , we getBy the proofs of Proposition 21 and [9, Theorem ], the right-hand side of above equality is really the Laplace transform of a continuous exponentially bounded function given byWith the help of Laplace transform and a simple calculation, it can be simply verified that the function , whose Laplace transform is given by (69), is a mild solution of abstract fractional inclusion (33). The growth order of implies that the function , , is in Since is closed in and converges uniformly to for , the asymptotical almost periodicity of immediately follows if we prove that the function , , is asymptotically almost periodic for all ; see (iii). But, this can be shown by making use of (iv)-(v) and applying successively [24, Theorem ]; here, we only want to observe that the vanishing term of function is given by , since The uniqueness of exponentially bounded mild solutions of (33) can be proved as follows. Let be such a solution. Taking the Laplace transform and multiplying after that with , we getHence, and for , By the uniqueness theorem for the Laplace transform, must be uniquely determined. The proof of the theorem is thereby complete.
Remark 24. Concerning Theorem 23, the case in which is a little bit complicated: it seems that the assumption for some complex numbers has to be imposed for establishing of any relevant result. Details can be left to the interested reader.
6. Examples and Applications
We have already noted that the method established in the proof of Theorem 18 will be further employed for reconsideration and improving some known results recently established by Keyantuo et al. [16] and Luong [17]. The main aim of the following example is to explain how we can do this.
Example 1. In [16], the authors have considered the abstract two-term fractional differential equationwhere , is a densely defined linear operator satisfying the condition (H) with , and , , and is a given -valued function; here, we have been forced to slightly change the notation used in [16]. For this, the notion of an -regularized family generated by , which is a special case of the notion introduced in Definition 9 with , , and , , has been introduced; compare [16, Lemma ]. Our main contributions will be given in the case that , which is used in the formulations and proofs of [16, Theorems and ], the main results of afore-mentioned paper (although possible applications can be given in the study of two-term fractional Poisson heat equations on the space , where is a bounded domain in with smooth boundary [18], we will pay our attention to the case that is a single-valued linear operator).
Let us define an exponentially bounded, analytic -regularized family of angle , subgenerated by , as the exponentially bounded, analytic -regularized -resolvent family of angle , subgenerated by the same operator, with and being defined as above. Then we haveand the assertion of [16, Theorem ] holds with the initial values and the inhomogeneity replaced therein with the initial values and the inhomogeneity , respectively, with the meaning clear.
Assume that the condition (H) holds with , , and , where . Then we can argue as follows. Similarly as in the proof of Theorem 18, we have that the operator is a subgenerator of an exponentially bounded, analytic -regularized family of angle ; compare also [12, Theorem ]. Estimating the term uniformly, as in the proof of Theorem 18, and using the integral computation given in the final part of this theorem and [2, Theorem ], we gethere we would like to note that the arguments used in proof of [16, Theorem ] are much more complicated than ours and that our estimate is better even in the case that because then, with the notion introduced in [16] accepted, we only require that the operator is of sectorial angle , not of , as the stronger estimate in [16] requires. Using the estimate (75), we can repeat literally the proofs of [16, Theorems and ] in order to see that their validity hold with -sectorial operators of smaller angle , with the meaning clear. In the case that , we can make applications to generators of analytic -regularized semigroups generated by nonelliptic differential operators in -spaces [44].
In [17], Luong has investigated the following abstract two-term fractional differential equation with nonlocal conditions in a Banach space :where , the functions and satisfy certain conditions, and is of sectorial angle The improvements of main results of this paper, Theorem 13, to -sectorial operators of angle , with being clarified above, can be proved straightforwardly ().
As mentioned before, Corollary 20 can be applied to the almost sectorial operators and multivalued linear operators used in the analysis of Poisson heat equations.
Example 2. (i) ([45]) Assume that , , is a bounded domain in with boundary of class , and Consider the operator given bywith domain In this place, , , and Let satisfy the following:(i) for all and (ii) for all , and(iii)there exists a constant such thatThen there exists a sufficiently large number such that the operator satisfies the condition (H) with , , , and some Let us remind ourselves that is not densely defined and that the value of exponent is sharp. Applications of Corollary 20 are clear and here we would like to illustrate just one of them: for , , , , and .
(ii) ([18]) Let , where is a bounded domain in , , a.e. , , and Suppose that the operator acts on with the Dirichlet boundary conditions and that is the multiplication operator by the function By the analysis contained in [18, Example ], the condition (H) is satisfied for the multivalued linear operator with , , and some numbers and ; here it is worth noting that the validity of additional condition [18, ] on the function enables us to get the better exponent in (H), provided that Applications in the study of the existence and uniqueness of asymptotically almost periodic and asymptotically almost automorphic solutions of multiterm fractional integrodifferential Poisson heat equationwhere for all and , are immediate ().
Example 3. In [12, Subsection ], we have analyzed hypercyclic and topologically mixing properties of solutions of abstract multiterm fractional Cauchy problem (1) with being single-valued and for some . Let it be the case. With the help of Proposition 21, we can reconsider a great number of examples given in the above-mentioned part of [12] and provide several interesting applications in the investigation of problem about the existence of a dense linear subspace of such that the mapping , , is asymptotically almost periodic for all ; here, is a given -regularized -propagation family for (1) with a subgenerator (cf. [7] for further information in this direction). For the sake of illustration, we will consider only the situation of [12, Example 3.3.12(ii)]; see also Ji and Weber [31]. Suppose that is a symmetric space of noncompact type and rank one, , and the parabolic domain and the positive real number possess the same meaning as in [31]. Suppose, further, that , is a nonconstant complex polynomial with , , , , , , , , and Then we already know that the operator is the integral generator of an exponentially bounded, analytic resolvent propagation family of angle , and that is topologically mixing provided the conditioncompare [12] for the notion. Our new assumption will be, instead of (80), that there exist a number and a sufficiently small number such that and that for Let Since there exists an such that due to our assumption , we can employ [12, Lemma ] in order to see thatsee also [12, p. 418]. Due to our assumption , the asymptotic expansion formulae (8)-(9), and the fact that the first term in the above expression can be continuously extended to the nonnegative real axis, we get the mapping , , is asymptotically almost periodic. Since the set s.t. and is total in , Proposition 21 implies that there exists a dense linear subspace of such that the mapping , , is asymptotically almost periodic for all .
Conflicts of Interest
The authors have no conflicts of interest with this publication.
Acknowledgments
The second author is partially supported by Ministry of Science and Technological Development, Republic of Serbia, Grant 174024.