We consider fractional abstract Cauchy problems on infinite intervals. A fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces is considered. This form of degeneration may be strong and some convenient assumptions about the involved operators are required to handle the direct problem. Required conditions on spaces are also given, guaranteeing the existence and uniqueness of solutions. The fractional powers of the involved operator have been investigated in the space which consists of continuous functions u on without assuming . This enables us to refine some previous results and obtain the required abstract results when the operator is not necessarily densely defined.
In recent years, many studies were devoted to the problem of recovering the solution u to
where B, M, and L are closed linear operators on the complex Banach space E with , , and u is unknown. The first approach to handle existence and uniqueness of the solution u to Equation (1) was given by Favini-Yagi  (see in particular the monograph ). By using the real interpolation space , (see [3,4]), suitable assumptions on the operators B, M, L guarantee that Equation (1) has a unique solution. This result was improved by Favini, Lorenzi, and Tanabe  (see also [6,7,8]).
In all cases, the basic assumptions read as follows:
Operator B has a resolvent for any , Re, satisfying
Operators L, M satisfy
for any .
Let A be the possibly multivalued linear operator , . Then, A and B commute in the resolvent sense:
Very recently, Al Horani et al. , see also , generalized the previous results to the interpolation space , , i.e.,
Let B, M, L be three closed linear operators on the complex Banach space E satisfying (H)–(H), , . Then, for all , , , Equation (1) admits a unique solution u such that , .
There are many choices of the operator B verifying Assumption (H). In , the authors handled the abstract equation of the form
in the Banach space X with initial condition , where
and is the Gamma function.
For Riemann–Liouville derivative of order , we address the monograph  (see also [12,13]). Very recent applications concerning Caputo fractional derivative operator are also discussed in  by the same authors using a completely different method than Sviridyuk’s group (see [15,16]). Some related topics can be found in [17,18,19].
In , the authors extended the results of direct and inverse problems, given in , to degenerate differential equations on the half line . Precisely, let X be a complex Banach space and
endowed with the sup norm. If is the operator defined by
and are two closed linear operators in the complex Banach space E satisfying
, , then for all , equation
admits a unique solution u. Moreover, .
In this paper, we refine our results in  by investigating the fractional power of the operator in the space of continuous functions u defined on without assuming , i.e.,
where X is a complex Banach space. In this case, is not densely defined. In such a case, it is not known whether is true or not, since in the proof of Lemma A2 of T. Kato  it seems it is essentially used that A is densely defined. To obtain our results on such a new space E, we should investigate the previous fractional power problem in case .
The interpolation space , , could be characterized by using the famous results of P. Grisvard. Since the operator is of type and , is the infinitesimal generator of an analytic semigroup (see , Proposition 0.9, p. 19), the interpolation space could be characterized by
where denotes the space of all strongly measurable valued functions f on such that
The following lemma is also needed:
Section 2 is devoted to our main results. In Section 3, we present our conclusions and remarks.
2. Main Results
Let X be a complex Banach space and
Let be an operator defined by
Let , . Consider the problem
The solution is
Hence, u is bounded in , and so is . This implies that u is uniformly continuous in . Furthermore, is uniformly continuous. Therefore, and . Since is arbitrary, one concludes that and
Here, we make some preparations. Suppose that A is a not necessarily densely defined closed linear operator in a Banach space X satisfying
is uniformly bounded in each smaller sector ; and
for with some .
The first Assumption (i) is equivalent to .
Case. Set for
where C runs in the resolvent set of A from to , , avoiding the negative real axis and 0, where . Let . Let be another contour which has the same property as C and is located to the right of C without intersecting C. Then,
Hence, is a pseudo resolvent:
The first term of the last side vanishes, and the second term is equal to
Therefore, the following formula is obtained:
By virtue of Cauchy’s representation formula of holomorphic functions, one has
Let . Then,
Therefore, if , then . This implies . Hence, has an inverse. Since
implies , and hence , has an inverse . Since
one observes , and
Let and . Then, , and
Set for some . Then, , and . This implies
Letting , one gets . Hence, . Thus, writing
It is not difficult to show that the following relation holds if :
Let A be a not necessarily densely defined closed linear operator in a Banach space X satisfying (i)–(iii) and . Then, the fractional power of A is defined for , and the followings hold:
Consequently, the following proposition is established:
Let be the operator defined by Equations (7) and (8). Then, satisfies and Equations (11) and (12) hold. The fractional power of is defined implicitly by Equation (38) or Equation (39). has an inverse and for Equation (41) holds.
Especially if , then the function belongs to E. The converse is given in the next proposition.
Suppose that both functions f and belong to E. Then, and Equation (41) holds.
As a preparation, we first consider the case of a finite interval. Let . Let
Therefore, satisfies the assumptions of Proposition 1 with , and hence its fractional power is defined for , and we have:
Analogously to Equation (40), the following statement is established:
It follows from Equations (42) and (43) that for ,
Especially if ,
For and ,
From Equation (45) with f replaced by and Equation (46), it follows that
By the changes of the order of integration,
Substituting Equation (49) (with s and interchanged) into Equation (48), one deduces
From Equations (46), (47), and (50), it follows that the following equality holds :
This yields that, for :
Since is arbitrary, one concludes that Equation (51) holds for .
We return to the case of the infinite interval . Suppose that both functions f and belong to E. One has by virtue of Equation (39) with
Adding Equations (52) and (53), and using Equation (51) with , one observes
This yields that and . Consequently, and . □
In view of Propositions 3 and 4, the following statement is obtained:
Let . Then, if and only if . For , Equation (41) holds.
Suppose . Then, in view of Corollary 1 . Hence,
Therefore, under the assumption if and only if , and in this case holds. In particular, it is obtained that
Let , . Let L and M be densely defined closed linear operators in X such that , and
Consequently, it has been proved that Equation (81) holds with
Next, we show that v satisfies Equation (70). Since
Clearly, . By assumption is holomorphc in . Since
is also holomorphic in . If , then
Hence, lies in the region where is holomorphc. If
Hence, one observes
Let R be a large positive number. The set consists of two points . Let be the closed curve which consists of the part of in the disk and the part of the circle in . Since is holomorphic in the region , which contains the closed set surrounded by ,
Since and , , one has
This implies as , and hence as , . Therefore, by virtue of Equation (69) for
i.e., . In view of it follows that . Therefore, . Thus, the following result is established:
Let be two closed linear operators in the complex Banach space E satisfying , and let . Let be the operator defined by Equations (7) and (8) and . Then, for all equation admits a unique solution u. Moreover, .
The fractional powers of the involved operator are investigated in the space of continuous functions which do not necessarily vanish at the origin. This enables us to prove some previous results in the case where the involved operator is not necessarily densely defined. Precisely, a fractional abstract Cauchy problem for possibly degenerate equations in Banach spaces is considered and refined.
All authors have equally contributed to this work. All authors wrote, read, and approved the final manuscript.
This research received no external funding.
Conflicts of Interest
The authors declare no conflict of interest.
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