Spectral Criteria for Solvability of Boundary Value Problems and Positivity of Solutions of Time-Fractional Differential Equations

and Applied Analysis 3 t ≥ 0 for someM > 0, and ω ∈ R, then the integral converges absolutely for Re(λ) > ω and defines an analytic function there. Regarding the fractional derivative, we have the following important property: forf ∈ C([0,∞);X) such thatg 1−α ∗f ∈ W 1,1 ((0,∞);X), D α t f (λ) = λ α f (λ) − λ α−1 f (0) . (12) The power function λα is uniquely defined as λα = |λ| α e i arg(λ), with −π < arg(λ) < π. The Mittag-Leffler function (see, e.g., [14, 15]) is defined as follows:


Introduction
Let  be a complex Banach space and  the generator of a  0semigroup (()) ≥0 in .We consider the following linear differential equation: where    is the Caputo fractional derivative.In the integer case  = 1, it is well known that there exists a strong connection between the spectrum of (()) ≥0 and solutions of the inhomogeneous differential equation (1) satisfying the condition (0) =  (1), where  is a forcing term.A complete characterization of the class of generators  such that for any given  ∈ ([0, 1]; ), (1) with the condition (0) = (1) has a unique solution which was obtained by Prüss [1] in 1984, extending earlier results by Haraux (see [2]).
After Prüss theorem, many interesting consequences and related results have appeared.For example, the corresponding connection with the spectrum of strongly continuous sine functions [3], cosine functions [4], and connections with maximal   -regularity are discussed in [5][6][7].
More recently, Nieto [8] 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 [9].One disadvantage of this condition is that continuity of () for  ≥ 0 forces the condition (1) = 0.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:  (0) =  (1) . ( Even more, we are successful in extending to the range 0 <  < 1 the above mentioned characterization given by Prüss, in terms of the following strongly continuous resolvent family associated to (1): where Φ  is a Wright type function defined by Φ  () = ∑ ∞ =0 ((−)  /!Γ(−+1−)) for every  ∈ C (see Section 2 for more properties of the function Φ  ).We observe that Φ  () is a probability density function on [0, ∞), 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  ∈ ([0, 1]; ) (1) admits exactly one mild solution fulfilling the boundary conditions (0) =  (1).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 (()) ≥0 (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  ∈ (0, 1) and  be the generator of a positive  0 -semigroup (()) ≥0 .Suppose ( −   (1)) −1  ≥ 0 for all  ∈  + and assume that  is a mild solution of (1) and that Then, () ≥ 0 for all  ∈ [0, 1].
Finally, we study in Banach lattices existence of mild solutions for the semilinear problem: under the hypothesis that  generates a positive  0semigroup.This is an extension of recent results given by Zhang [10] in the integer case  = 1 (cf.Theorem 16).Typical operators to which the results apply are elliptic operators in divergence form: namely, let Ω be an open subset of R  .We consider on   (Ω) the operator formally given by in which ( , ) 1≤,≤ 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][12][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  0semigroup, 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 Riemann-Liouville fractional integral of order , 0 <  < 1, of a function  : [0, 1] →  is given by for example, when  is locally integrable on (0, 1).The Caputo fractional derivative of order 0 <  < 1 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  > 0.Then, when  =  is a natural number, we get    :=   /  .The Laplace transform of a locally integrable function  : [0, ∞) →  is defined by (11) provided that the integral converges for some  ∈ C. If, for example,  is exponentially bounded, that is, ‖()‖ ≤   , Abstract and Applied Analysis 3  ≥ 0 for some  > 0, and  ∈ R, then the integral converges absolutely for Re() >  and defines an analytic function there.
The Mittag-Leffler function (see, e.g., [14,15]) is defined as follows: where Ha is a Hankel path, that is, a contour which starts and ends at −∞ and encircles the disc || ≤ || Using this formula, we obtain for 0 <  < 1 and the identity To see this, it is sufficient to write and invert the Laplace transform.
The following two definitions are taken from [16,17], respectively.Definition 1.Let  be a closed and linear operator with domain () defined on a Banach space  and  > 0. We call  the generator of an (, )-resolvent family if there exists  ≥ 0 and a strongly continuous function 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  > 0. We call  the generator of an (, 1)-resolvent family if there exists  ≥ 0 and a strongly continuous function such that {  : Re() > } ⊂ () and In this case,   () is called the (, 1)-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 (1, 1)-resolvent family is the same as a  0 -semigroup; a (2, 2)-resolvent family corresponds to the concept of sine family, while a (2, 1)-resolvent family corresponds to a cosine family.See, for example, [18] and the references therein for an overview on these concepts.A systematic study in the fractional case is carried out in [17].
Some properties of (  ()) and (  ()) are included in the following lemma.Their proof uses techniques from the general theory of (, )-regularized resolvent families [19] (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.
Proof.Let  be as in Definitions 1 or 2. Let ,  >  and  ∈ ().Then,  = ( −  − ) −1  for some  ∈ .Since (− − ) −1 and (− − ) −1 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  > 0. From these two equalities and the continuity of   on [0, ∞), 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).
The proof of the lemma is finished.
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 [10].
Theorem 4. Let  be a Banach lattice and  :  →  a nonlinear operator.Suppose that there exists a positive linear bounded operator  :  →  with   () < 1 and for all ,  ∈ ,  ≥ .Then, the equation  = () has a unique solution in .

Subordination
Let  be a linear closed densely defined operator in a complex Banach space .If  generates a  0 -semigroup (()) ≥0 then,  generates an (, 1)-resolvent family (  ()) for all 0 <  < 1 and they are related by the following formula [17]: A change of variables shows that the above is equivalent to In particular, it follows from the above representation formulas that (  ()) is analytic and   (0) = .Concerning (, )-resolvent families, we prove the following important theorem, which is the main result of this section.
We next show that P () = (  − ) −1 for Re() large enough.In fact, by (32) and Fubini's theorem, we obtain for every  ∈ , for all Re() 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 Abstract and Applied Analysis 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 [13] 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.

Boundary Conditions
In this section, we give a spectral characterization of existence of mild solutions of (1) with the boundary condition (0) = (1).The approach is based on the representation of solutions using the solution families (  ()) and (  ()) of the previous section.Assume that  generates a  0 semigroup (()) ≥0 .
Proof of Lemma 8. (i) ⇒ (ii): Assume that assertion (i) holds.Then, () − (0) =   () +   ().Taking the Laplace transform of this equality, we get that, û() − 1/(0 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 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 [1] when  = 1 and  generates a strongly continuous semigroup.If one starts with  ∈ ([0, 1]; ) and solves the problem   () = ()+() with the boundary condition (0) = (1), then the resulting solution can be extended to a periodic continuous function on R. We observe that Haraux [2] had considered similar problems earlier.
For the fractional differential Equation (44), we obtain a mild solution on [0, ∞).In the next result (Theorem 9), we obtain a necessary and sufficient condition that the mild solution will satisfy the boundary condition (0) =  (1).
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 [9] as described in the introduction.In this line of research, we note that the paper [23] by Kaslik and Sivasundaram considers existence and nonexistence of periodic solutions of fractional differential equations for various definitions of the fractional derivative.
In the following examples, the semigroups are exponentially stable.
Then,   is a realization of the Laplace operator on  2 (Ω) with Dirichlet boundary conditions and it generates a  0semigroup on  2 (Ω) which is exponentially stable.Moreover, the semigroup interpolates on all   (Ω) and each semigroup on   (Ω) (1 ≤  < ∞) is also exponentially stable (for a complete description we refer, e.g., to [25]).
(2) Assume that Ω ⊂ R  is a bounded domain with a Lipschitz continuous boundary Ω and let  ∈  ∞ (Ω) satisfy () ≥  0 > 0 for some constant  0 .Define the bilinear form   on  2 (Ω) by Then, the operator   on  2 (Ω) associated with the form   in the sense that is a realization of the Laplace operator with Robin boundary conditions.As in part (1), this operator generates an exponentially stable  0 -semigroup in  2 (Ω) which interpolates on   (Ω) and each semigroup is exponentially stable in   (Ω) (1 ≤  < ∞).