#### Abstract

We mainly study the fractional evolution equation in an ordered Banach space , , , . Using the monotone iterative technique based on lower and upper solutions, the existence and uniqueness results are obtained. The necessary perturbation results for accomplishing this approach are also developed.

#### 1. Introduction

In this paper, we use the perturbation theory and the monotone iterative technique based on lower and upper solutions to investigate the existence and uniqueness of mild solutions for the fractional evolution equation in an ordered Banach space : where is the Caputo fractional derivative, , is a linear closed densely defined operator, is continuous, is the zero element of , and is a Volterra integral operator with integral kernel , .

In particular, when , we study the existence and uniqueness of mild solutions for the fractional evolution equation in an ordered Banach space : where is the Caputo fractional derivative, , is a linear closed densely defined operator, is continuous, and is the zero element of .

The fractional calculus (i.e., calculus of integrals and derivatives of any arbitrary real or complex order) goes back to Newton and Leibnitz in the seventieth century. It has gained considerable popularity and importance during the past three decades or so, due mainly to its demonstrated applications in numerous seemingly diverse and widespread fields such as physics, chemistry, aerodynamics, viscoelasticity, porous media, electrodynamics of complex medium, and electrochemistry, control, electromagnetic. For instance, fractional calculus concepts have been used in the modeling of transmission lines [1], neurons [2], viscoelastic materials [3], and electrical capacitors [4]. Other examples from fractional order dynamics can be found in [5, 6] and the references therein.

One of the branches of fractional calculus is the theory of fractional evolution equations, that is evolution equations where the integer derivative with respect to time is replaced by a derivative of any order. Also, in recent years, fractional evolution equations have attracted increasing attention; see [7β21].

The monotone iterative technique based on lower and upper solutions is an effective and a flexible mechanism that offers theoretical as well as constructive existence results in a closed set. It yields monotone sequences of lower and upper approximate solutions that converge to the minimal and maximal solutions between the lower and upper solutions. Since under suitable conditions each member of the sequences happens to be the unique solution of a certain nonlinear problem, the advantage and importance of the technique is remarkable. For differential equations of integer order, many papers used the monotone iterative technique based on lower and upper solutions; see [22β24] and the references therein. Recently, there have been some papers which deal with the existence of the solutions of initial value problems or boundary value problems for fractional ordinary differential equations by using this method; see [25β31]. They mainly involve Riemann-Liouville fractional derivatives.

However, to the best of the authors' knowledge, no results yet exist for the fractional evolution equations by using the monotone iterative technique based on lower and upper solutions. Our results can be considered as a contribution to this emerging field.

In comparison with fractional ordinary differential equations, we have great difficulty in using the monotone iterative technique for the fractional evolution equations. Firstly, how to introduce a suitable concept of a mild solution for fractional evolution equations based on the corresponding solution operator? A pioneering work has been reported by El-Borai [10, 11]. Later on, some authors introduced the definitions of mild solutions for fractional evolution equations. Wang and Zhou [17], Wang et al. [16, 18], and Zhou and Jiao [20, 21] also introduced a suitable definition of the mild solutions based on the well-known theory of Laplace transform and probability density functions. Moreover, HernΓ‘ndez et al. [12] used an approach to treat abstract equations with fractional derivatives based on the well-developed theory of resolvent operators for integral equations. Shu et al. [15] give the definition of a mild solution by investigating the classical solutions of the corresponding system. Secondly, do the solution operators for fractional evolution equations have the perturbation properties analogous to those for the -semigroup? For evolution equations of integer order, perturbation properties play a significant role in monotone iterative technique; see [24].

Our paper copes with the above difficulties, and the new features of this paper mainly include the following aspects. We firstly introduce a new concept of a mild solution based on the well-known theory of Laplace transform, and the form is very easy. Secondly, we discuss the perturbation properties for the corresponding solution operators. Thirdly, by the monotone iterative technique based on lower and upper solutions, we obtain results on the existence and uniqueness of mild solutions for problem (1.1) and (1.3).

#### 2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts which are used throughout this paper.

*Definition 2.1 (see [5]). *The Riemann-Liouville fractional integral operator of order of function is defined as
where is the Euler gamma function.

*Definition 2.2 (see [5]). *The Caputo fractional derivative of order , , is defined as
where the function has absolutely continuous derivatives up to order . If is an abstract function with values in , then the integrals and derivatives which appear in (2.1) and (2.2) are taken in Bochner sense.

Proposition 2.3. *For and as a suitable function (e.g., [5]) one has the following: *(i)*; *(ii)*; *(iii)*; *(iv)*; *(v)*; *(vi)*. *

We observe from the above that the Caputo fractional differential operators do not possess neither semigroup nor commutative properties, which are inherent to the derivatives on integer order. For basic facts about fractional integrals and fractional derivatives one can refer to the books [5, 32β34].

Let be an ordered Banach space with norm and partial order β€, whose positive cone ( is the zero element of ) is normal with normal constant . Let be the Banach space of all continuous -value functions on interval with norm . For , for all . For , denote the ordered interval , and , . By we denote the space of all bounded linear operators from to .

*Definition 2.4. *If , and satisfies
then is called a lower solution of problem (1.1); if all inequalities of (2.3) are inverse, we call it an upper solution of problem (1.1).

Similarly, we give the definitions of lower and upper solutions of problem (1.3).

*Definition 2.5. *If , and satisfy
then is called a lower solution of problem (1.3); if all inequalities of (2.4) are inverse, we call it an upper solution of problem (1.3).

Consider the following problem:

*Definition 2.6 (see [9]). *A family is called a solution operator for (2.5) if the following conditions are satisfied: (1) is strongly continuous for and ; (2) and for all , ; (3) is a solution of
for all , .

In this case, is called the generator of the solution operator and is called the solution operator generated by .

*Definition 2.7 (see [9]). *The solution operator is called exponentially bounded if there are constants and such that

An operator is said to belong to , or for short, if problem (2.5) has a solution operator satisfying (2.7). Denote , . In these notations and are the sets of all infinitesimal generators of -emigroups and cosine operator families (COF), respectively. Next, we give a characterization of .

Lemma 2.8 (see [9]). *Let , and let be the corresponding solution operator. Then for , one has and
*

Lemma 2.9 (see [9]). *Let and . Then the corresponding solution operator is given by
**
where are given by the recurrence relations: **, **, , , **, , . ** The convergence is uniform on bounded subsets of for any fixed .*

Lemma 2.10 (see [9]). *Let . Then if and only if and
*

Lemma 2.11. *Assume . For the linear Cauchy problem
** has the form
**
where is the solution operator generated by , and .*

*Proof. *For , applying the Laplace transform to (2.11), we have that
By Lemma 2.8, , from the above equation, we obtain
Since , by Lemma 2.8 and the inverse Laplace transform, we have that
where

*Remark 2.12. *If ( is a constant), we know that is the solution of (2.11) by [5, Exampleββ4.10], where and are the Mittag-Leffler functions. We also find that is the solution of the problem (2.5), and ; see [5].

*Definition 2.13. *A function is called a mild solution of (2.11) if and satisfies the following equation:
where is the solution operator generated by , and .

*Remark 2.14. *It is easy to verify that a classical solution of (2.11) is a mild solution of the same system.

Lemma 2.15. *If , , is the solution operator generated by , and , then one has that
*

*Proof. *By (2.7), for , we have that . Thus,

Now, we discuss the perturbation properties of the solution operators.

*Definition 2.16. *An operator is called a positive operator in if and such that .

From Definition 2.1, we can easily obtain the following result.

Lemma 2.17. *Assume that is the solution operator generated by and . Then is a positive operator if and only if is a positive operator.*

By Lemmas 2.8 and 2.9 and the closedness of the positive cone, we can obtain the following result.

Lemma 2.18. *Assume that and is the solution operator generated by . The following results are true. *(i)*If is a positive solution operator, then for any and , we have .*(ii)*If there is a , for any and such that , then is a positive solution operator. *

Lemma 2.19. *Assume , , ; then the following results hold. *(i)*, where is the Mittag-Leffler function. *(ii)*If and such that , then for and , one has . *

*Proof. *(i) If is the solution operator generated by , in view of [9, Theoremβ2.26], we have that
That is, .

(ii) If , by (i) and Lemma 2.8, we have , . Then by Lemma 2.10,
When ,
Therefore, for such the operator is invertible and
For any , in view of , and (2.22), then

*Remark 2.20. *If , Lemma 2.19 (i) is not true; see [9, Exampleββ2.24]. However, a classical perturbation result for or (see [35, 36]) is as follows: if is the generator of -semigroup (or COF) and , then is again a generator of a -semigroup (or COF).

By Definition 2.13, we can obtain the following result.

Lemma 2.21. *The linear Cauchy problem
**
where , , has the unique mild solution given by
**
where is the solution operator generated by , and .*

By Lemmas 2.17, 2.18, and 2.19, the following result holds.

Lemma 2.22. *Assume that and are the solution operators generated by and , respectively, and . Then is a positive operator and are positive operators.*

Now, we recall some properties of the measure of noncompactness that will be used later. Let denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [37]. For any and , set . If is bounded in , then is bounded in , and .

Lemma 2.23 (see [38]). *Let be a bounded and countable set. Then is Lebesgue integral on , and
*

#### 3. Main Results

Theorem 3.1. *Let be an ordered Banach space, whose positive cone β is normal with normal constant , is continuous, and is a linear closed densely defined operator. Assume that , is the positive solution operator generated by , the Cauchy problem (1.1) has a lower solution and an upper solution with , and the following conditions are satisfied. *()

*There exists a constant such that for any , , and .*()

*There exists a constant such that for any , and increasing or decreasing monotonic sequences and .*

*Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between and , which can be obtained by a monotone iterative procedure starting from and , respectively.*

*Proof. *Since , by Lemmas 2.15 and 2.19, we have that
Set
Since is the positive solution operator generated by , by Lemma 2.22, and are positive operators.

Let ; we define a mapping by
By Lemma 2.21, is a mild solution of problem (1.1) if and only if
By , for and , we have that
That is, is an increasing monotonic operator. Now, we show that , .

Let ; by Definition 2.1, Lemma 2.21, and the positivity of operators and , we have that
namely, . Similarly, we can show that . For , in view of (3.7), then . Thus, . We can now define the sequences
and it follows from (3.7) that
For convenience, by (1.2), we can denote
Let and . It follows from that for . Let
In view of (3.10), since the positive cone is normal, then and are bounded in . By Lemma 2.23 and (3.12), is Lebesgue integrable on . For , by (3.11) and Lemma 2.23, we have that
and therefore,
For , from Lemma 2.23, , (3.4), (3.5), (3.9), (3.12), (3.14), and the positivity of operator , we have that
By (3.15) and the Gronwall inequality, we obtain that on . This means that is precompact in for every . So, has a convergent subsequence in . In view of (3.7), we can easily prove that itself is convergent in . That is, there exist such that as for every . By (3.5) and (3.9), for any , we have that
Let ; then by Lebesgue-dominated convergence theorem, for any , we have that
and . Then . Similarly, we can prove that there exists such that . By (3.7), if , and is a fixed point of , then . By induction, . By (3.10) and taking the limit as , we conclude that . That means that are the minimal and maximal fixed points of on , respectively. By (3.6), they are the minimal and maximal mild solutions of the Cauchy problem (1.1) on , respectively.

*Remark 3.2. *Even if , our results are also new.

Corollary 3.3. *Let be an ordered Banach space, whose positive cone β is regular, is continuous, and is a linear closed densely defined operator. Assume that , is the positive solution operator generated by , the Cauchy problem (1.1) has a lower solution and an upper solution with , and holds. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between and , which can be obtained by a monotone iterative procedure starting from and , respectively.*

*Proof. *Since is satisfied, then (3.10) holds. In regular positive cone*β*, any monotonic and ordered-bounded sequence is convergent. Then there exist , and , . Then by the proof of Theorem 3.1, the proof is then complete.

Corollary 3.4. *Let be an ordered and weakly sequentially complete Banach space, whose positive cone β is normal with normal constant , is continuous, and is a linear closed densely defined operator. Assume that , is the positive solution operator generated by , the Cauchy problem (1.1) has a lower solution and an upper solution with , and holds. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between and , which can be obtained by a monotone iterative procedure starting from and , respectively.*

*Proof. *Since is an ordered and weakly sequentially complete Banach space, then the assumption holds. In fact, by [39, Theoremββ2.2], any monotonic and ordered-bounded sequence is precompact. Let and be two increasing or decreasing sequences. By , is monotonic and ordered-bounded sequence. Then, by the properties of the measure of noncompactness, we have
So, holds. By Theorem 3.1, the proof is then complete.

Theorem 3.5. *Let be an ordered Banach space, whose positive cone β is normal with normal constant , is continuous, is a linear closed densely defined operator. Assume , is the positive solution operator generated by , the Cauchy problem (1.1) has a lower solution and an upper solution with , holds, and the following condition is satisfied: *

*There are constants such that for any , , and .*

*Then the Cauchy problem (1.1) has the unique mild solution between and , which can be obtained by a monotone iterative procedure starting from or .*

*Proof. *We can find that and imply . In fact, for , let and be two increasing or decreasing monotonic sequence. For with , by and , we have that
By (3.20) and the normality of positive cone , we have
From (3.21) and the definition of the measure of noncompactness, we have that
where . Hence, holds.

Therefore, by Theorem 3.1, the Cauchy problem (1.1) has the minimal solution and the maximal solution on . In view of the proof of Theorem 3.1, we show that . For , by (3.4), (3.5), (3.6), (3.11), , and the positivity of operator , we have that
By (3.23) and the normality of the positive cone , for , we obtain that
By the Gronwall inequality, then on . Hence is the the unique mild solution of the Cauchy problem (1.1) on . By the proof of Theorem 3.1, we know that it can be obtained by a monotone iterative procedure starting from or .

By Corollaries 3.3 and 3.4, and Theorem 3.5, we have the following results.

Corollary 3.6. *Let be continuous, and let be a linear closed densely defined operator. Assume that , is the positive solution operator generated by , the Cauchy problem (1.1) has a lower solution and an upper solution with , and hold, and one of the following conditions is satisfied: *(i)* is an ordered Banach space, whose positive cone β is regular; *(ii)

*is an ordered and weakly sequentially complete Banach space, whose positive cone*

*β*is normal with normal constant .*Then the Cauchy problem (1.1) has the unique mild solution between and , which can be obtained by a monotone iterative procedure starting from or .*

Next, we consider the existence and uniqueness results of the Cauchy problem (1.3). Substituting for in Theorem 3.1, Corollaries 3.3 and 3.4, and Theorem 3.5, we can obtain the following results.

Corollary 3.7. *Let be an ordered Banach space, whose positive cone β is normal with normal constant , is continuous, is a linear closed densely defined operator. Assume , is the positive solution operator generated by , the Cauchy problem (1.3) has a lower solution and an upper solution with , and the following conditions are satisfied. *()

*There exists a constant such that*

*for any , and ;*()

*There exists a constant such that for any , and increasing or decreasing monotonic sequence .*

*Then the Cauchy problem (1.3) has the minimal and maximal mild solutions between and , which can be obtained by a monotone iterative procedure starting from and , respectively.*

Corollary 3.8. *Let be an ordered Banach space, whose positive cone β is regular, is continuous, and is a linear closed densely defined operator. Assume that , is the positive solution operator generated by , the Cauchy problem (1.3) has a lower solution and an upper solution with , and holds. Then the Cauchy problem (1.3) has the minimal and maximal mild solutions between and , which can be obtained by a monotone iterative procedure starting from and , respectively.*

Corollary 3.9. *Let be an ordered and weakly sequentially complete Banach space, whose positive cone β is normal with normal constant , is continuous, and is a linear closed densely defined operator. Assume that , is the positive solution operator generated by , the Cauchy problem (1.3) has a lower solution and an upper solution with , and holds. Then the Cauchy problem (1.3) has the minimal and maximal mild solutions between and , which can be obtained by a monotone iterative procedure starting from and , respectively.*

Corollary 3.10. *Let be an ordered Banach space, whose positive cone β is normal with normal constant , is continuous, and is a linear closed densely defined operator. Assume that , is the positive solution operator generated by , the Cauchy problem (1.3) has a lower solution and an upper solution with , holds, and the following condition is satisfied. *()

*There exists a constant such that for any , and .*

*Then the Cauchy problem (1.3) has the unique mild solution between and , which can be obtained by a monotone iterative procedure starting from or .*

By Corollaries 3.8, 3.9, and 3.10, we have the following results.

Corollary 3.11. *Let be continuous, and let be a linear closed densely defined operator. Assume that , is the positive solution operator generated by , the Cauchy problem (1.3) has a lower solution and an upper solution with , and hold, and one of the following conditions is satisfied: *(i)* is an ordered Banach space, whose positive cone β is regular; *(ii)

*is an ordered and weakly sequentially complete Banach space, whose positive cone*

*β*is normal with normal constant .*Then the Cauchy problem (1.3) has the unique mild solution between and , which can be obtained by a monotone iterative procedure starting from or .*

#### 4. Examples

*Example 4.1. *In order to illustrate our main results, we consider the Cauchy problem in (-dimensional Euclidean space and ):
where is the Caputo fractional derivative, , is a real matrix, is continuous, is the zero element of , and
is a Volterra integral operator with integral kernel , .

For , and , we define the partial order . Set ; then is a normal cone*β*in and normal constant . It is easy to verify that generates a uniformly continuous positive cosine operator family :
By [9, Theoremββ3.1], there exist and such that , and the corresponding solution operator is
where , is a probability density function, , , and . Thus, is the positive solution operator generated by . In order to solve the problem (4.1), we give the following assumptions. (), for . ()There exist such that for . ()The partial derivative is continuous on any bounded domain and has upper bound.

Theorem 4.2. *If (), (), and () are satisfied, then the problem (4.1) has the unique mild solution , and .*

*Proof. *From () and (), we obtain that is a lower solution of (4.1), and is an upper solution of (4.1). Form (), it is easy to verify that and are satisfied. Therefore, by Theorem 3.5, the problem (4.1) has the unique solution , and .

#### Acknowledgments

The author is grateful to the anonymous referee for his/her valuable suggestions and comments. This research supported by the Project of NWNU-KJCXGC-3-47.