Periodic Boundary Value Problems for Semilinear Fractional Differential Equations

We study the periodic boundary value problem for semilinear fractional differential equations in an ordered Banach space. The method of upper and lower solutions is then extended. The results on the existence of minimal and maximal mild solutions are obtained by using the characteristics of positive operators semigroup and the monotone iterative scheme. The results are illustrated by means of a fractional parabolic partial differential equations.


Introduction
In this paper, we consider the periodic boundary value problem PBVP for semilinear fractional differential equation in an ordered Banach space X, D α u t Au t f t, u t , t ∈ I, where D α is the Caputo fractional derivative of order 0 < α < 1, I 0, ω , −A : D A ⊂ X → X is the infinitesimal generator of a C 0 -semigroup i.e., strongly continuous semigroup {T t } t≥0 of uniformly bounded linear operators on X, and f : I × X → X is a continuous function.
Fractional calculus is an old mathematical concept dating back to the 17th century and involves integration and differentiation of arbitrary order.In a later dated 30th of September 1695, L'Hospital wrote to Leibniz asking him about the differentiation of order 1/2.Leibniz' response was "an apparent paradox from which one day useful consequences will be drawn."In the following centuries, fractional calculus developed significantly within pure mathematics.However, the applications of fractional calculus just emerged in the last few decades.The advantage of fractional calculus becomes apparent in science and engineering.In recent years, fractional calculus attracted engineers' attention, because it can describe the behavior of real dynamical systems in compact expressions, taking into account nonlocal characteristics like infinite memory 1-3 .Some instances are thermal diffusion phenomenon 4 , botanical electrical impedances 5 , model of love between humans 6 , the relaxation of water on a porous dyke whose damping ratio is independent of the mass of moving water 7 , and so forth.On the other hand, directing the behavior of a process with fractional-order controllers would be an advantage, because the responses are not restricted to a sum of exponential functions; therefore, a wide range of responses neglected by integerorder calculus would be approached 8 .For other advantages of fractional calculus, we can see real materials 9-13 , control engineering 14, 15 , electromagnetism 16 , biosciences 17 , fluid mechanics 18 , electrochemistry 19 , diffusion processes 20 , dynamic of viscoelastic materials 21 , viscoelastic systems 22 , continuum and statistical mechanics 23 , propagation of spherical flames 24 , robotic manipulators 25 , gear transmissions 26 , and vibration systems 27 .It is well known that the fractional-order differential and integral operators are nonlocal operators.This is one reason why fractional differential operators provide an excellent instrument for description of memory and hereditary properties of various physical processes.
In recent years, there have been some works on the existence of solutions or mild solutions for semilinear fractional differential equations, see 28-36 .They use mainly Krasnoselskii's fixed-point theorem, Leray-Schauder fixed-point theorem, or contraction mapping principle.They established various criteria on the existence and uniqueness of solutions or mild solutions for the semilinear fractional differential equations by considering an integral equation which is given in terms of probability density functions and operator semigroups.Many partial differential equations involving time-variable t can turn to semilinear fractional differential equations in Banach spaces; they always generate an unbounded closed operator term A, such as the time fractional diffusion equation of order α ∈ 0, 1 , namely, where A may be linear fractional partial differential operator.So, 1.1 has the extensive application value.However, to the authors' knowledge, no studies considered the periodic boundary value problems for the abstract semilinear fractional differential equations involving the operator semigroup generator −A.Our results can be considered as a contribution to this emerging field.We use the method of upper and lower solutions coupled with monotone iterative technique and the characteristics of positive operators semigroup.
The method of upper and lower solutions has been effectively used for proving the existence results for a wide variety of nonlinear problems.When coupled with monotone iterative technique, one obtains the solutions of the nonlinear problems besides enabling the study of the qualitative properties of the solutions.The basic idea of this method is that using the upper and lower solutions as an initial iteration, one can construct monotone sequences, and these sequences converge monotonically to the maximal and minimal solutions.In some papers, some existence results for minimal and maximal solutions are obtained by establishing comparison principles and using the method of upper and lower solutions and the monotone iterative technique.The method requires establishing comparison theorems which play an important role in the proof of existence of minimal and maximal solutions.In abstract semilinear fractional differential equations, positive operators semigroup can play this role, see  In Section 2, we introduce some useful preliminaries.In Section 3, in two cases: T t is compact or noncompact, we establish various criteria on existence of the minimal and maximal mild solutions of PBVP 1.1 .The method of upper and lower solutions coupled with monotone iterative technique, and the characteristics of positive operators semigroup are applied effectively.In Section 4, we give also an example to illustrate the applications of the abstract results.

Preliminaries
In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.
If −A is the infinitesimal generator of a C 0 -semigroup in a Banach space, then − A qI generates a uniformly bounded C 0 -semigroup for q > 0 large enough.This allows us to reduce the general case in which −A is the infinitesimal generator of a C 0 -semigroup to the case in which the semigroup is uniformly bounded.Hence, for convenience, throughout this paper, we suppose that −A is the infinitesimal generator of a uniformly bounded C 0semigroup {T t } t≥0 .This means that there exists M ≥ 1 such that We need some basic definitions and properties of the fractional calculus theory which are used further in this paper.Definition 2.1 see 9, 32 .The fractional integral of order α with the lower limit zero for a function f ∈ AC 0, ∞ is defined as provided the right side is pointwise defined on 0, ∞ , where Γ • is the gamma function.
Definition 2.2 see 9, 32 .The Riemann-Liouville derivative of order α with the lower limit zero for a function f ∈ AC 0, ∞ can be written as

2.3
Definition 2.3 see 9, 32 .The Caputo fractional derivative of order α for a function f ∈ AC 0, ∞ can be written as 2.5 ii The Caputo derivative of a constant is equal to zero.
iii If f is an abstract function with values in X, then the integrals and derivatives which appear in Definitions 2.1-2.3 are taken in Bochner's sense.
For more fractional theories, one can refer to the books 9, 42-44 .Throughout this paper, let X be an ordered Banach space with norm • and partial order ≤, whose positive cone P {y ∈ X | y ≥ θ} θ is the zero element of X is normal with normal constant N. X 1 denotes the Banach space D A with the graph norm where where U t and V t are given by 2.9

2.18
Therefore, I − U ω has bounded inverse operator and Set is the unique mild solution of LIVP 2.7 and satisfies u 0 u ω .So set Proof.For the proof of i -iii , one can refer to 29, 31 .We only check iv , v , and vi as follows.
iv For 0 < t 1 ≤ t 2 , we have

2.24
Since T t is continuous in the uniform operator topology for t > 0, by Lebesquedominated convergence theorem and Remark 2.7 i , U t and V t are continuous in the uniform operator topology for t > 0.
v By Remark 2.7 i , the proof is then complete.
vi By v , 2.18 , and 2.19 , the proof is then complete.

Main Results
Case 1. {T t } t≥0 is compact.
Theorem 3.1.Assume that {T t } t≥0 is a compact and positive semigroup in X, PBVP 1.1 has a lower solution v 0 and an upper solution w 0 with v 0 ≤ w 0 and satisfies the following.

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H There exists a constant C > 0 such that for any t ∈ I, and Then PBVP 1.1 has the minimal and maximal mild solutions between v 0 and w 0 , which can be obtained by a monotone iterative procedure starting from v 0 and w 0 , respectively. Proof.It From Lemma 2.8, I − Φ ω has bounded inverse operator and By Lemma 2.16 v and vi , Φ t and Ψ t are positive for t ≥ 0, and I − Φ ω −1 is positive.Let D v 0 , w 0 , then we define a mapping Q : D → C I, X by where

3.8
In particular, By Definition 2.5, v 0 0 ≤ v ω , and by the positivity of operator I − Φ ω −1 , we have that Then by 3.8 and the positivity of operator Φ t ,

3.11
namely, v 0 ≤ Qv 0 .Similarly, we can show that Qw 0 ≤ w 0 .For u ∈ D, in view of 3.7 , then v 0 ≤ Qv 0 ≤ Qu ≤ Qw 0 ≤ w 0 .Thus, Q : D → D is a continuous increasing monotonic operator.We can now define the sequences 12 and it follows from 3.7 that In the following, we prove that {v n } and {w n } are convergent in C I, X .First, we show that Cu s ds, t ∈ I, 3.14 then we prove that for all 0 < t ≤ ω, WD t

3.15
For u ∈ D, by H , f t, v 0 t Cv 0 t ≤ f t, u t Cu t ≤ f t, w 0 t Cw 0 t for 0 ≤ t ≤ ω.By the normality of the cone P , there is Thus, by 3.16 and Remark 2.7 i , we have

3.17
Then by 3.15 , 3.17 and the compactness of S ε , for 0 Furthermore, by 3.16 and Lemma 2.16 i , we have

3.18
Therefore, for 0 < t ≤ ω, WD t is precompact in X.In particular, WD ω is precompact in X, and then Furthermore, for 0 ≤ t 1 < t 2 ≤ ω, by 3.16 and Lemma 2.16 i we have that

3.19
By Remark 2.12 and Lemma 2.16 iv , Ψ t is continuous in the uniform operator topology for t > 0. Then by Lebesque-dominated convergence theorem, WD is equicontinuous in C I, X .By Lemma 2.16 ii , {Ψ t } t≥0 is strongly continuous.So, QD is equicontinuous in C I, X .Then by Ascoli-Arzela's theorem, QD {Qu | u ∈ D} is precompact in C I, X .By 3.12 and 3.13 , {v n } has a convergent subsequence in C I, X .Combining this with the monotonicity of {v n }, it is itself convergent in C I, X .Using a similar argument to that for {v n }, we can prove that {w n } is also convergent in C I, X .Set Let n → ∞, by the continuity of Q and 3.12 , we have u Qu, u Qu.

3.21
By 3.7 , if u ∈ D is a fixed-point of Q, then v 1 Qv 0 ≤ Qu u ≤ Qw 0 w 1 .By induction, v n ≤ u ≤ w n .By 3.13 and taking the limit as n → ∞, we conclude that v 0 ≤ u ≤ u ≤ u ≤ w 0 .This means that u, u are the minimal and maximal fixed-points of Q on v 0 , w 0 , respectively.By 3.6 , they are the minimal and maximal mild solutions of PBVP 1.1 on v 0 , w 0 , respectively.For 0 ≤ t ≤ ω, by 3.7 and 3.13 , { Qv n t } is monotone in X.Since the cone P is regular, then { Qv n t } is convergent in X.
By Ascoli-Arzela's theorem, {Qv n } is precompact in C I, X and {Qv n } has a convergent subsequence in C I, X .Combining this with the monotonicity of {Qv n }, it is itself convergent in C I, X .Using a similar argument to that for {Qw n }, we can prove that {Qw n } is also convergent in C I, X .Let then it is similar to the proof of Theorem 3.1 that u and u are the minimal and maximal mild solutions of PBVP 1.1 on v 0 , w 0 , respectively.
Corollary 3.4.Let X be an ordered and weakly sequentially complete Banach space.Assume that {T t } t≥0 is an equicontinuous and positive semigroup in X, PBVP 1.1 has a lower solution v 0 and an upper solution w 0 with v 0 ≤ w 0 , and (H) holds, then PBVP 1.1 has the minimal and maximal mild solutions between v 0 and w 0 , which can be obtained by a monotone iterative procedure starting from v 0 and w 0 , respectively.
Proof.In an ordered and weakly sequentially complete Banach space, the normal cone P is regular.Then the proof is complete.
Corollary 3.5.Let X be an ordered and reflective Banach space.Assume that {T t } t≥0 is an equicontinuous and positive semigroup in X, PBVP 1.1 has a lower solution v 0 and an upper solution w 0 with v 0 ≤ w 0 , and (H) holds, then PBVP 1.1 has the minimal and maximal mild solutions between v 0 and w 0 , which can be obtained by a monotone iterative procedure starting from v 0 and w 0 , respectively.
Proof.In an ordered and reflective Banach space, the normal cone P is regular.Then the proof is complete.
By Theorem 3.3, Corollaries 3.4 and 3.5, we have the following.
Corollary 3.6.Assume that {T t } t≥0 is an equicontinuous and positive semigroup in X, f t, θ ≥ θ for t ∈ I.If there is y ∈ X such that y ≥ θ, Ay ≥ f t, y for t ∈ I, f satisfies (H 1 ) and one of the following conditions: i X is an ordered Banach space, whose positive cone P is regular, ii X is an ordered and weakly sequentially complete Banach space, iii X is an ordered and reflective Banach space.

Examples
Proof.Set v 0 0, by Theorem 3.3, PBVP 4.1 has the minimal and maximal solutions between 0 and w 0 .
Definition 2.11.A C 0 -semigroup {T t } t≥0 is called an equicontinuous semigroup if T t is continuous in the uniform operator topology i.e., uniformly continuous for t > 0.Remark 2.14.From Definition 2.13, if h ≥ θ, x 0 ≥ θ, and T t t ≥ 0 is a positive C 0 -semigroup generated by −A, the mild solution u ∈ C I, X given by 2.8 satisfies u ≥ θ.For the applications of positive operators semigroup, we can see 37-41 .It is easy to see that positive operators semigroup can play the role as the comparison principles.Definition 2.15.A bounded linear operator K on X is called to be positive if Kx ≥ θ for all x ≥ θ.The operators U and V given by 2.9 have the following properties:i For any fixed t ≥ 0, U t and V t are linear and bounded operators, that is, for any x ∈ X, s ds, 2.22 then Ph is the unique mild solution of LPBVP 2.13 .Remark 2.9.For sufficient conditions of exponentially stable C 0 -semigroup, one can see 49 .Definition 2.10.A C 0 -semigroup {T t } t≥0 is called a compact semigroup if T t is compact for t > 0. Definition 2.13.A C 0 -semigroup {T t } t≥0 is called a positive semigroup if T t x ≥ θ for all x ≥ θ and t ≥ 0. ii {U t } t≥0 and {V t } t≥0 are strongly continuous, iii {U t } t≥0 and {V t } t≥0 are compact operators if {T t } t≥0 is a compact semigroup, iv U t and V t are continuous in the uniform operator topology (i.e., uniformly continuous) for t > 0 if {T t } t≥0 is an equicontinuous semigroup, v U t and V t are positive for t ≥ 0 if {T t } t≥0 is a positive semigroup, vi I − U ω −1 is a positive operator if {T t } t≥0 is an exponentially and positive semigroup.
is easy to see that − A CI generates an exponentially stable and positive compact 16 the continuity of f and Lemma 2.16ii , Q : D → C I, X is continuous.By Lemma 2.8, u ∈ D is a mild solution of PBVP 1.1 if and only ifFor u 1 , u 2 ∈ D and u 1 ≤ u 2 , from H , the positivity of operators I − Φ ω −1 , Φ t , and Ψ t , we have thatQu 1 ≤ Qu 2 .3.7Now, we show that v 0 ≤ Qv 0 , Qw 0 ≤ w 0 .Let D α v 0 t Av 0 t Cv 0 t σ t , by Definition 2.5, the positivity of operator Ψ t , we have that Theorem 3.2.Assume that {T t } t≥0 is a compact and positive semigroup in X, f t, θ ≥ θ for t ∈ I.If there is y ∈ X such that y ≥ θ, Ay ≥ f t, y for t ∈ I, and f satisfies the following:H 1 There exists a constant C 1 > 0 such that f t, x 2 − f t, x 1 ≥ −C 1 x 2 − x 1 , 3.22for any t ∈ I, and θ ≤x 1 ≤ x 2 ≤ y, that is, f t, x C 1 x is increasing in x for x ∈ θ, y .Then PBVP 1.1 has a positive mild solution u: θ ≤ u ≤ y.Proof.Let v 0 θ and w 0 y, by Theorem 3.1, PBVP 1.1 has mild solution on v 0 , w 0 .Case 2. {T t } t≥0 is noncompact.Assume that the positive cone P is regular, {T t } t≥0 is an equicontinuous and positive semigroup in X, PBVP 1.1 has a lower solution v 0 and an upper solution w 0 with v 0 ≤ w 0 , and (H) holds, then PBVP 1.1 has the minimal and maximal mild solutions between v 0 and w 0 , which can be obtained by a monotone iterative procedure starting from v 0 and w 0 , respectively.Proof.By the proof of Theorem 3.1, 3.2 -3.13 and 3.19 are valid.By Lemma 2.16 iv , Ψ t is continuous in the uniform operator topology for t > 0. Then by Lebesque-dominated convergence theorem, WD is equicontinuous in C I, X .From Lemma 2.16 ii , {Ψ t } t≥0 is strongly continuous.So, QD is equicontinuous in C I, X .Thus, {Qv n } is equicontinuous in C I, X .
39 continuous, Bu b 0 x u δ ∂u/∂n is a regular boundary operator on ∂Ω, and strong elliptic operator of second order, whose coefficient functions are H ölder continuous in Ω.Let X L p Ω p ≥ 2 , P {v | v ∈ L p Ω , v x ≥ 0 a.e.x ∈ Ω}, then X is a Banach space, and P is a regular cone in X. Define the operator A as follows:Then −A generates a uniformly bounded analytic semigroup T t t ≥ 0 in X see 39 .By the maximum principle, we can easily find that T t t ≥ 0 is positive see39.Let Mathematical Problems in Engineering u t u •, t , f t, u g •, t, u •, t , then the problem 4.1 can be transformed into the following problem: Let f x, t, 0 ≥ 0. If there exists w 0 x, t ∈ C 2,α Ω × I such that ∂ α t w 0 A x, D w 0 ≥ g x, t, w 0 , x, t ∈ Ω × I, H 4 there exists a constant C 2 ≥ 0 such that g x, t, ξ 2 − g x, t, ξ 1 ≥ −C 2 ξ 2 − ξ 1 , 4.6 for any t ∈ I, and 0 ≤ ξ 1 ≤ ξ 2 ≤ w 0 .