#### Abstract

We investigate the initial value problem for a class of fractional evolution equations in a Banach space. Under some monotone conditions and noncompactness measure conditions of the nonlinearity, the well-known monotone iterative technique is then extended for fractional evolution equations which provides computable monotone sequences that converge to the extremal solutions in a sector generated by upper and lower solutions. An example to illustrate the applications of the main results is given.

#### 1. Introduction

In this paper, we use the monotone iterative technique to investigate the existence and uniqueness of mild solutions of the fractional evolution equation in an ordered Banach space , where is the Caputo fractional derivative of order , , is a linear closed densely defined operator, is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators (), and is continuous.

The origin of 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. We observe that the fractional order can be complex in viewpoint of pure mathematics, and there is much interest in developing the theoretical analysis and numerical methods to fractional equations, because they have recently proved to be valuable in various fields such as physics, chemistry, aerodynamics, viscoelasticity, porous media, electrodynamics of complex medium, electrochemistry, control, and electromagnetic. For instance, fractional calculus concepts have been used in the modeling of transmission lines , neurons , viscoelastic materials , and electrical capacitors . References [5, 6] used modified Riemann-Liouville fractional derivatives (Jumarie's fractional derivatives) and proposed the method of fractional complex transform to find exact solutions which are much needed in engineering applications. Other examples from fractional-order dynamics can be found in [7, 8] and the references therein.

Fractional evolution equations are evolution equations where the integer derivative with respect to time is replaced by a derivative of any order. In recent years, fractional evolution equations have attracted increasing attention, see . A strong motivation for investigating the Cauchy problem (1.1) comes from physics. For example, fractional diffusion equations are abstract partial differential equations that involve fractional derivatives in space and time. The main physical purpose for investigating these type of equations is to describe phenomena of anomalous diffusion appearing in transport processes and disordered systems. The time fractional diffusion equation is obtained from the standard diffusion equation by replacing the first-order time derivative with a fractional derivative of order , namely, where may be linear fractional partial differential operator. For fractional diffusion equations, we can see  and the references therein.

It is well known that the method of monotone iterative technique has been proved to be an effective and a flexible mechanism. It yields monotone sequences of lower and upper approximate solutions that converge to the minimal and maximal solutions between the lower and upper solutions. Early on, Du and Lakshmikantham  established a monotone iterative method for an initial value problem for ordinary differential equation. Later on, many papers used the monotone iterative technique to establish existence and comparison results for nonlinear problems. For evolution equations of integer order (), Li  and Yang  used this method, in which positive -semigroup plays an important role. 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 .

However, when many partial differential equations involving time-variable turn to evolution equations in Banach spaces, they always generate an unbounded closed operator term , such as (1.2). is corresponding to linear partial differential operator with certain boundary conditions. In this case, the results in  are not suitable. To the best of the authors' knowledge, no results yet exist for the fractional evolution equations involving a closed operator term by using the monotone iterative technique. The approach via fractional differential inequalities is clearly better suited as in the case of classical results of differential equations, and therefore this paper choose to proceed in that setup.

If is the infinitesimal generator of an analytic semigroup in a Banach space, then generates a uniformly bounded analytic semigroup for large enough. This allows us to reduce the general case in which is the infinitesimal generator of an analytic semigroup to the case in which the semigroup is uniformly bounded. Hence, for convenience, throughout this paper, we suppose that is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators ().

Our contribution in this work is to establish the monotone iterative technique for the fractional evolution (1.1). Under some monotone conditions and noncompactness measure conditions of nonlinearity , which are analogous to those in Li and liu , Li , Chen and li , Chen , and Yang [33, 47], we obtain results on the existence and uniqueness of mild solutions of the problem (1.1). In this paper, positive semigroup also plays an important role. At last, to illustrate our main results, we examine sufficient conditions for the main results to a fractional partial differential diffusion equation.

#### 2. Preliminaries

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

Definition 2.1 (see ). The Riemann-Liouville fractional integral of order with the lower limit zero, of function , is defined as where is the Euler gamma function.

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

Proposition 2.3. For and as a suitable function (for instance, ), one has (i)(ii)(iii)(iv)(v)(vi)(vii),(viii), is a constant.

We observe from the above that the Caputo fractional differential operators 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 [7, 4850].

Let be an ordered Banach space with norm and partial order ≤, 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 , . Set exists and . By , we denote the Banach space with the graph norm . We note that is the infinitesimal generator of a uniformly bounded analytic semigroup (). This means that there exists such that

Definition 2.4. If and satisfies then is called a lower solution of the problem (1.1); if all inequalities of (2.5) are inverse, we call it an upper solution of problem (1.1).

Lemma 2.5 (see [12, 19, 20]). If satisfies a uniform Hlder condition, with exponent , then the unique solution of the Cauchy problem is given by where is a probability density function defined on .

Remark 2.6 (see [19, 20, 22]). , , .

Definition 2.7. By the mild solution of the Cauchy problem (2.6), we mean the function satisfying the integral equation where and are given by (2.8) and (2.9), respectively.

Definition 2.8. An operator family in is called to be positive if for any and such that .

From Definition 2.8, if is a positive semigroup generated by , , , then the mild solution of (2.6) satisfies . For positive semigroups, one can refer to .

Now, we recall some properties of the measure of noncompactness 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 . For any and , set . If is bounded in , then is bounded in , and .

Lemma 2.9 (see ). Let be a bounded and countable set, then is Lebesgue integral on ,
In order to prove our results, one also needs a generalized Gronwall inequality for fractional differential equation.

Lemma 2.10 (see ). Suppose that , , and is a nonnegative function locally integrable on (some ), and suppose that is nonnegative and locally integrable on with on this interval, then

#### 3. Main Results

Theorem 3.1. Let be an ordered Banach space, whose positive cone is normal with normal constant . Assume that is positive, 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 , that is, is increasing in for . () There exists a constant such that for any , and increasing or decreasing monotonic sequences , 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. It is easy to see that generates an analytic semigroup , and is positive. Let . By Remark 2.6, and are positive. By (2.4) and Remark 2.6, we have that
Let , we define a mapping by By Lemma 2.5 and Definition 2.7, is a mild solution of the problem (1.1) if and only if For and , from the positivity of operators and , and , we have that Now, we show that , . Let , by Definition 2.4, Lemma 2.5, and the positivity of operators and , we have that namely, . Similarly, we can show that . For , in view of (3.6), then . Thus, is an increasing monotonic operator. We can now define the sequences and it follows from (3.6) that Let and . It follows from that for . Let For , from , (3.3), (3.4), (3.8), (3.10), Lemma 2.9, and the positivity of operator , we have that By (3.11) and Lemma 2.10, we obtain that on . This means that is precompact in for every . So, has a convergent subsequence in . In view of (3.9), we can easily prove that itself is convergent in . That is, there exists such that as for every . By (3.4) and (3.8), 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 (3.6), if , and is a fixed point of , then . By induction, . By (3.9) and taking the limit as , we conclude that . That means that are the minimal and maximal fixed points of on , respectively. By (3.5), they are the minimal and maximal mild solutions of the Cauchy problem (1.1) on , respectively.

Remark 3.2. Theorem 3.1 extends [37, Theorem  2.1]. Even if and , our results are also new.

Corollary 3.3. Let be an ordered Banach space, whose positive cone is regular. Assume that is positive, 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 (3.9) holds. In regular positive cone , 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 is normal with normal constant . Assume that is positive, 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 [54, Theorem  2.2], any monotonic and ordered bounded sequence is precompact. Let be an increasing or decreasing sequence. By , is a 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 is normal with normal constant . Assume that is positive, the Cauchy problem (1.1) has a lower solution and an upper solution with , holds, and the following condition is satisfied: () there is constant such that for any , .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 be an increasing sequence. For with , by and , we have that By (3.16) and the normality of positive cone , we have From (3.17) 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 mild solution and the maximal mild solution on . In view of the proof of Theorem 3.1, we show that . For , by (3.3), (3.4), (3.5), , and the positivity of operator , we have that By (3.19) and the normality of the positive cone , for , we obtain that By Lemma 2.10, 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 it can be obtained by a monotone iterative procedure starting from or .

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

Corollary 3.6. Assume that is positive, 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)X is an ordered Banach space, whose positive cone is regular, (ii)X is an ordered and weakly sequentially complete Banach space, whose positive cone 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 .

#### 4. Examples

Example 4.1. In order to illustrate our main results, we consider the fractional partial differential diffusion equation in , where is the Caputo fractional partial derivative with order , is the Laplace operator, , is a bounded domain with a sufficiently smooth boundary , and is continuous.
Let , , then is a Banach space, and is a normal cone in . Define the operator as follows: Then generates an analytic semigroup of uniformly bounded analytic semigroup in (see ). is positive (see [31, 32, 55, 56]). Let , , then the problem (4.1) can be transformed into the following problem: Let be the first eigenvalue of , and is the corresponding eigenfunction, then , . In order to solve the problem (4.1), we also need the following assumptions: (), , , , ()the partial derivative is continuous on any bounded domain.

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

Proof. From Definition 2.4 and , we obtain that 0 is a lower solution of (4.3), and is an upper solution of (4.3). Form , it is easy to verify that and are satisfied. Therefore, by Theorem 3.5, the problem (4.1) has the unique mild solution.

#### Acknowledgments

This research was supported by NNSFs of China (10871160, 11061031) and project of NWNU-KJCXGC-3-47.