Research Article | Open Access
Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations
We investigate mild solutions of the fractional order nonhomogeneous Cauchy problem , where When is the generator of a -semigroup on a Banach space , we obtain an explicit representation of mild solutions of the above problem in terms of the semigroup. We then prove that this problem under the boundary condition admits a unique mild solution for each if and only if the operator is invertible. Here, we use the representation in which is a Wright type function. For the first order case, that is, , the corresponding result was proved by Prüss in 1984. In case is a Banach lattice and the semigroup is positive, we obtain existence of solutions of the semilinear problem
Let be a complex Banach space and the generator of a -semigroup in . We consider the following linear differential equation: where is the Caputo fractional derivative.
In the integer case , it is well known that there exists a strong connection between the spectrum of and solutions of the inhomogeneous differential equation (1) satisfying the condition , where is a forcing term. A complete characterization of the class of generators such that for any given , (1) with the condition has a unique solution which was obtained by Prüss  in 1984, extending earlier results by Haraux (see ).
Denoting the resolvent set of an operator by , the result of Prüss reads as follows: if and only if for any , the equation admits exactly one mild solution satisfying .
After Prüss theorem, many interesting consequences and related results have appeared. For example, the corresponding connection with the spectrum of strongly continuous sine functions , cosine functions , and connections with maximal -regularity are discussed in [5–7].
More recently, Nieto  studied periodic boundary valued solutions of (1) considering the scalar case. Nieto considers the Riemann-Liouville fractional derivative, and the meaning he gives to a “periodic” boundary condition is the following: Further results along these lines are given in . One disadvantage of this condition is that continuity of for forces the condition . It thus appears that Riemann-Liouville is not the most appropriate choice when one considers periodic boundary valued problems. In contrast, the Caputo derivative needs higher regularity conditions of than the Riemann-Liouville derivative.
Our objective in this paper is twofold: first, we reformulate Nieto's results for the vector-valued case of (1) considering Caputo's fractional derivative and the natural periodic boundary condition: Even more, we are successful in extending to the range the above mentioned characterization given by Prüss, in terms of the following strongly continuous resolvent family associated to (1): where is a Wright type function defined by for every (see Section 2 for more properties of the function ). We observe that is a probability density function on , whose Laplace transform is the Mittag-Leffler function in the whole complex plane.
A remarkable consequence of our extension result given in Theorem 9 is the following: if generates a uniformly stable semigroup, then for each (1) admits exactly one mild solution fulfilling the boundary conditions . In order to do this, we study mild solutions of (1) and show that any mild solution has the following representation: where is given by (4) and is a second operator family associated with (see Section 2).
Secondly, we study positivity of mild solutions and obtain a simple spectral condition that ensures positivity thereof in the periodic boundary value case. More precisely, let and be the generator of a positive -semigroup . Suppose for all and assume that is a mild solution of (1) and that Then, for all .
Finally, we study in Banach lattices existence of mild solutions for the semilinear problem: under the hypothesis that generates a positive -semigroup. This is an extension of recent results given by Zhang  in the integer case (cf. Theorem 16).
Typical operators to which the results apply are elliptic operators in divergence form: namely, let be an open subset of . We consider on the operator formally given by in which are bounded real valued functions. Under various boundary conditions (including Dirichlet, Neumann, Robin, and Wentzell), the results apply (see Section 4).
While in the present paper, we concentrate on periodic boundary conditions, we mention the recent papers [11–13] dealing with fractional differential equations. The first two deal with nonlocal Cauchy problems, while the third considers the fractional evolution problem governed by an almost sectional operator and proceeds to construct the corresponding evolution operators by mean of a certain functional calculus.
The paper is organized as follows. In Section 2, we present some preliminaries on the resolvent families needed in the sequel. In Section 3, assuming that generates a -semigroup, we represent the resolvent families of Section 2 using the subordination principle. In Section 4, we study mild solutions in general and in the periodic boundary valued case in particular. Positivity of mild solutions as well as the semilinear equation are considered in Section 5.
The algebra of bounded linear operators on a Banach space will be denoted by , the resolvent set of a linear operator by , and the spectral radius of a bounded operator will be denoted by . Let be a real number. The space of continuous functions is denoted by and its norm by . We denote , , where is the usual gamma function. It will be convenient to write , the Dirac measure concentrated at . Note the semigroup property: for all .
The Riemann-Liouville fractional integral of order , , of a function is given by for example, when is locally integrable on .
The Caputo fractional derivative of order of a function is defined by where is the distributional derivative of , under appropriate assumptions. The definition can be extended in a natural way to . Then, when is a natural number, we get .
The Laplace transform of a locally integrable function is defined by provided that the integral converges for some . If, for example, is exponentially bounded, that is, , for some , and , then the integral converges absolutely for and defines an analytic function there.
Regarding the fractional derivative, we have the following important property: for such that , The power function is uniquely defined as , with .
The Mittag-Leffler function (see, e.g., [14, 15]) is defined as follows: where is a Hankel path, that is, a contour which starts and ends at and encircles the disc counterclockwise. It is an entire function which provides a generalization of several usual functions, for example,(i)exponential function: ;(ii)cosine functions: and ;(iii)sine functions: and .Let . The Laplace transform of the th-order derivative of the Mittag-Leffler function is given by  Using this formula, we obtain for and the identity To see this, it is sufficient to write and invert the Laplace transform.
Definition 1. Let be a closed and linear operator with domain defined on a Banach space and . We call the generator of an -resolvent family if there exists and a strongly continuous function (resp., in case ) such that and In this case, is called the -resolvent family generated by .
Definition 2. Let be a closed and linear operator with domain defined on a Banach space and . We call the generator of an -resolvent family if there exists and a strongly continuous function such that and In this case, is called the -resolvent family generated by .
In the above definitions, the integrals involved are taken in the sense of Riemann, more precisely as improper Riemann integrals.
By the uniqueness theorem for the Laplace transform, a -resolvent family is the same as a -semigroup; a -resolvent family corresponds to the concept of sine family, while a -resolvent family corresponds to a cosine family. See, for example,  and the references therein for an overview on these concepts. A systematic study in the fractional case is carried out in .
Some properties of and are included in the following lemma. Their proof uses techniques from the general theory of -regularized resolvent families  (see also [16, 17]). It will be of crucial use in the investigation of mild solutions in Section 4.
Lemma 3. The following properties hold.(i). (ii) and for all , .(iii)For all , .(iv)For all and (v) and for all , .(vi)For all , .(vii)For all and (viii)For , , and and for , .
Proof. Let be as in Definitions 1 or 2. Let , and . Then, for some . Since and are bounded and commute and since the operator is closed, we obtain from the definition of
and, analogously, from the definition of
Hence, by uniqueness of the Laplace transform,
for all . From these two equalities and the continuity of on , we immediately get (ii) and (v).
On the other hand, from the convolution theorems we obtain, for , The uniqueness theorem for the Laplace transform yields (iii) and (vi).
We now prove (iv) and (vii). Let and define , where is fixed. Let , . We have to show that and . Indeed, using (ii) and (iii), we obtain that proving the claim. Analogously, we prove that and From the continuity of on and from (iv), we obtain (i). That and for follow from (vi) by using the fact that , for , and that the operator is closed. We notice that (iv) implies that the domain of the operator is necessarily dense in . Now, if , the first assertion in (viii), that is, , for , follows from (vi), and we obtain that , for , by using this and the fact that is dense in . The proof of the lemma is finished.
Note that it follows from (vii) and (viii) that exhibits a singular behavior at the origin if . However, is in since by (viii), we have that if is near zero, then Recall the definition of the Wright type function [20, page 10, Formula (28)]: where is a contour which starts and ends at and encircles the origin once counterclockwise. By [17, page 14], is a probability density function, that is, We also have , and as , has the following asymptotic expansion: for any , with , where are real numbers.
The following identity holds: for every and , See [20, Formulas (40) and (41)]. We note that the above Laplace transform was formerly first given by Pollard and Mikusinski (see  and references therein). For more details on the Wright type functions, we refer to the monographs [17, 20–22] and the references therein.
Let be a Banach lattice with positive cone . We recall that a semigroup on is positive if for any and , . Similarly, an operator is resolvent positive if there is such that and for all and any .
It is a well-known fact that a strongly continuous semigroup is positive if and only if its generator is resolvent positive. We finally will need the following result due to Zhang .
Theorem 4. Let be a Banach lattice and a nonlinear operator. Suppose that there exists a positive linear bounded operator with and for all , . Then, the equation has a unique solution in .
Let be a linear closed densely defined operator in a complex Banach space . If generates a -semigroup then, generates an -resolvent family for all and they are related by the following formula : A change of variables shows that the above is equivalent to In particular, it follows from the above representation formulas that is analytic and .
Concerning -resolvent families, we prove the following important theorem, which is the main result of this section.
Theorem 5. Let . If generates a -semigroup , then generates an -resolvent family given for every and by Moreover, for all , , and
Proof. Since generates a -semigroup , there exists such that and
In particular, . It is clear that is strongly continuous (and in fact analytic) for .
We next show that for large enough. In fact, by (32) and Fubini's theorem, we obtain for every , for all sufficiently large, proving the claim. We conclude that is an resolvent family with generator .
On the other hand, from (35) and the fact that is closed, we obtain for all that and the identity proving (37). Integrating the above identity, we obtain Finally, from (vi) in Lemma 3, we get proving the theorem.
Remark 6. We observe that the paper  uses a different approach for the evolution operators and . More precisely, the authors consider an almost sectorial operator in a Banach space and give a direct construction using the Mittag-Leffler functions.
4. Boundary Conditions
In this section, we give a spectral characterization of existence of mild solutions of (1) with the boundary condition . The approach is based on the representation of solutions using the solution families and of the previous section. Assume that generates a semigroup . Let and be given by (34) and (36), respectively. The linear fractional equation with initial condition has the unique (classical) solution given by whenever and . Indeed, note that by (37), Hence, using (37), (38), and Lemma 3, we obtain
Note that for , the representation (46) is nothing else but the well-known variation of constant formula for the abstract Cauchy problem of first order, and corresponds exactly to , the -semigroup generated by .
Definition 7. Let with domain be a closed linear operator on a Banach space . Let and . Let be the operator defined in (9). A function is called a mild solution of the equation if and Equivalently,
We have the following representation of mild solutions.
Lemma 8. Suppose that the operator generates a -semigroup , and let such that the mapping is exponentially bounded. Let , and . Then, the following assertions are equivalent:(i), and , , that is, is a mild solution of (49),(ii) for all .
Note that the mapping is exponentially bounded if, for example, the function or itself is exponentially bounded.
Proof of Lemma 8. (i) (ii): Assume that assertion (i) holds. Then, . Taking the Laplace transform of this equality, we get that, , that is, . Therefore, , and . Hence,
Taking the inverse Laplace transform of this equality, we get the assertion (ii).
(ii) (i): As a consequence of (iv) and (vii) in Lemma 3, we have proving the lemma.
Uniqueness of the classical solution follows from the lemma upon observing that any classical solution is necessarily a mild solution.
The following problem was considered by Prüss  when and generates a strongly continuous semigroup. If one starts with and solves the problem with the boundary condition , then the resulting solution can be extended to a periodic continuous function on . We observe that Haraux  had considered similar problems earlier.
For the fractional differential Equation (44), we obtain a mild solution on . In the next result (Theorem 9), we obtain a necessary and sufficient condition that the mild solution will satisfy the boundary condition .
We remark that the concept of periodic boundary valued solutions for fractional differential equations has been introduced in the literature by Belmekki et al. in the paper  as described in the introduction. In this line of research, we note that the paper  by Kaslik and Sivasundaram considers existence and nonexistence of periodic solutions of fractional differential equations for various definitions of the fractional derivative.
We consider the following problem
Theorem 9. Let be a Banach space, and assume that generates a semigroup . Let be the subordinated -resolvent family. Then, if and only if for any ; (54) admits precisely one mild solution.
Proof. Suppose that . Note that if the solution of the differential equation in (54) satisfies the condition , then Lemma 8(ii) implies
We notice that the existence of solutions follows, if one defines by (56).
Conversely, define by means of , where denotes the unique mild solution of (54). It is clear that is linear and everywhere defined. Moreover, it is not difficult to show, using the closed graph theorem, that is bounded. Now, for , consider and define . Clearly, is linear and bounded. We claim that Indeed, the Laplace transform of is given by Let be the left-hand side of (58). Then, taking the Laplace transform and using (59), we get that From the uniqueness of the Laplace transform, we obtain (58) and the claim is proved. Now, using (55) and (58), we get that This shows that is surjective. Now, let be such that . Then, using (58), we get that Using (62), (55) (which follows from Lemma 8(ii)), and (58) again, we get that We have shown that proving that is injective. Hence, is invertible and the proof is finished.
Remark 10. An alternative proof of the injectivity of in the preceding proof runs as follows: let be such that and set . Then, is a mild solution of (54) with the forcing term . Since the function is also of mild solution of (54) (with the forcing term ), the uniqueness of the solution yields , proving that is injective.
We remark that the condition is trivially satisfied if .
Corollary 11. Suppose that the operator generates a -semigroup satisfying . Then, for , (54) admits exactly one mild solution.
Proof. First, observe that , and recall that , , and . Since is a nonzero analytic function, it follows that for each , we have . We first assume that is contractive, that is, , . Then, for , Let . We can choose such that for all . It follows that Therefore, and hence, is invertible. If is not contractive, we renorm the space with . This norm is equivalent to the original one and . The proof is complete.
Note that by [24, Prop. V.1.2 & V.1.7], the assumption on is equivalent to the fact that is exponentially stable.
In the following examples, the semigroups are exponentially stable.
Example 12. (1) Let be a bounded open set. Define the operator on by Then, is a realization of the Laplace operator on with Dirichlet boundary conditions and it generates a -semigroup on which is exponentially stable. Moreover, the semigroup interpolates on all