Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
VolumeΒ 2011, Article IDΒ 767186, 13 pages
http://dx.doi.org/10.1155/2011/767186
Research Article

Monotone Iterative Technique for Fractional Evolution Equations in Banach Spaces

Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730000, China

Received 7 June 2011; Accepted 3 July 2011

Academic Editor: Elsayed M. E.Β Zayed

Copyright Β© 2011 Jia Mu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 𝑋,𝐷𝛼𝑒(𝑑)+𝐴𝑒(𝑑)=𝑓(𝑑,𝑒(𝑑)),π‘‘βˆˆπΌ,𝑒(0)=π‘₯0βˆˆπ‘‹,(1.1) where 𝐷𝛼 is the Caputo fractional derivative of order 0<𝛼<1, 𝐼=[0,𝑇], 𝐴∢𝐷(𝐴)βŠ‚π‘‹β†’π‘‹ is a linear closed densely defined operator, βˆ’π΄ is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators 𝑇(𝑑) (𝑑β‰₯0), and π‘“βˆΆπΌΓ—π‘‹β†’π‘‹ 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 [1], neurons [2], viscoelastic materials [3], and electrical capacitors [4–6]. 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 [9–23]. 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 π›Όβˆˆ(0,1), namely, πœ•π›Όπ‘‘π‘’(𝑦,𝑑)=𝐴𝑒(𝑦,𝑑),𝑑β‰₯0,π‘¦βˆˆπ‘…,(1.2) where 𝐴 may be linear fractional partial differential operator. For fractional diffusion equations, we can see [24–26] 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 [27] 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 (𝛼=1), Li [28–32] and Yang [33] used this method, in which positive 𝐢0-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 [34–43].

However, when many partial differential equations involving time-variable 𝑑 turn to evolution equations in Banach spaces, they always generate an unbounded closed operator term 𝐴, such as (1.2). 𝐴 is corresponding to linear partial differential operator with certain boundary conditions. In this case, the results in [34–43] 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 βˆ’π΄ is the infinitesimal generator of an analytic semigroup in a Banach space, then βˆ’(𝐴+π‘žπΌ) generates a uniformly bounded analytic semigroup for π‘ž>0 large enough. This allows us to reduce the general case in which βˆ’π΄ 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 βˆ’π΄ is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators 𝑇(𝑑) (𝑑β‰₯0).

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 𝑓, which are analogous to those in Li and liu [44], Li [28–32], Chen and li [45], Chen [46], 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 [7]). The Riemann-Liouville fractional integral of order 𝛼>0 with the lower limit zero, of function π‘“βˆˆπΏ1(ℝ+), is defined as 𝐼𝛼1𝑓(𝑑)=ξ€œΞ“(𝛼)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓(𝑠)𝑑𝑠,(2.1) where Ξ“(β‹…) is the Euler gamma function.

Definition 2.2 (see [7]). The Caputo fractional derivative of order 𝛼>0 with the lower limit zero, π‘›βˆ’1<𝛼<𝑛, is defined as 𝐷𝛼1𝑓(𝑑)=ξ€œΞ“(π‘›βˆ’π›Ό)𝑑0(π‘‘βˆ’π‘ )π‘›βˆ’π›Όβˆ’1𝑓(𝑛)(𝑠)𝑑𝑠,(2.2) where the function 𝑓(𝑑) has absolutely continuous derivatives up to order π‘›βˆ’1. If 0<𝛼<1, then 𝐷𝛼1𝑓(𝑑)=ξ€œΞ“(1βˆ’π›Ό)𝑑0π‘“ξ…ž(𝑠)(π‘‘βˆ’π‘ )𝛼𝑑𝑠.(2.3) If 𝑓 is an abstract function with values in 𝑋, then the integrals and derivatives which appear in (2.1) and (2.2) are taken in Bochner,s sense.

Proposition 2.3. For 𝛼,𝛽>0 and 𝑓 as a suitable function (for instance, [7]), one has (i)𝐼𝛼𝐼𝛽𝑓(𝑑)=𝐼𝛼+𝛽𝑓(𝑑)(ii)𝐼𝛼𝐼𝛽𝑓(𝑑)=𝐼𝛽𝐼𝛼𝑓(𝑑)(iii)𝐼𝛼(𝑓(𝑑)+𝑔(𝑑))=𝐼𝛼𝑓(𝑑)+𝐼𝛼𝑔(𝑑)(iv)𝐼𝛼𝐷𝛼𝑓(𝑑)=𝑓(𝑑)βˆ’π‘“(0),0<𝛼<1(v)𝐷𝛼𝐼𝛼𝑓(𝑑)=𝑓(𝑑)(vi)𝐷𝛼𝐷𝛽𝑓(𝑑)≠𝐷𝛼+𝛽𝑓(𝑑)(vii)𝐷𝛼𝐷𝛽𝑓(𝑑)≠𝐷𝛽𝐷𝛼𝑓(𝑑),(viii)𝐷𝛼𝐢=0, 𝐢 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, 48–50].

Let 𝑋 be an ordered Banach space with norm β€–β‹…β€– and partial order ≀, whose positive cone 𝑃={π‘¦βˆˆπ‘‹βˆ£π‘¦β‰₯πœƒ} (πœƒ is the zero element of 𝑋) is normal with normal constant 𝑁. Let 𝐢(𝐼,𝑋) be the Banach space of all continuous 𝑋-value functions on interval 𝐼 with norm ‖𝑒‖𝐢=maxπ‘‘βˆˆπΌβ€–π‘’(𝑑)β€–. For 𝑒,π‘£βˆˆπΆ(𝐼,𝑋), 𝑒≀𝑣⇔𝑒(𝑑)≀𝑣(𝑑), for all π‘‘βˆˆπΌ. For 𝑣,π‘€βˆˆπΆ(𝐼,𝑋), denote the ordered interval [𝑣,𝑀]={π‘’βˆˆπΆ(𝐼,𝑋)βˆ£π‘£β‰€π‘’β‰€π‘€} and [𝑣(𝑑),𝑀(𝑑)]={π‘¦βˆˆπ‘‹βˆ£π‘£(𝑑)≀𝑦≀𝑀(𝑑)}, π‘‘βˆˆπΌ. Set 𝐢𝛼,0(𝐼,𝑋)={π‘’βˆˆπΆ(𝐼,𝑋)βˆ£π·π›Όπ‘’ exists and π·π›Όπ‘’βˆˆπΆ(𝐼,𝑋)}. By 𝑋1, we denote the Banach space 𝐷(𝐴) with the graph norm β€–β‹…β€–1=β€–β‹…β€–+‖𝐴⋅‖. We note that βˆ’π΄ is the infinitesimal generator of a uniformly bounded analytic semigroup 𝑇(𝑑) (𝑑β‰₯0). This means that there exists 𝑀β‰₯1 such that ‖𝑇(𝑑)‖≀𝑀,𝑑β‰₯0.(2.4)

Definition 2.4. If 𝑣0βˆˆπΆπ›Ό,0(𝐼,𝑋)∩𝐢(𝐼,𝑋1) and satisfies 𝐷𝛼𝑣0(𝑑)+𝐴𝑣0ξ€·(𝑑)≀𝑓𝑑,𝑣0ξ€Έ(𝑑),π‘‘βˆˆπΌ,𝑣0(0)≀π‘₯0,(2.5) then 𝑣0 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 β„Ž satisfies a uniform HΜˆπ‘œlder condition, with exponent π›½βˆˆ(0,1], then the unique solution of the Cauchy problem 𝐷𝛼𝑒(𝑑)+𝐴𝑒(𝑑)=β„Ž(𝑑),π‘‘βˆˆπΌ,𝑒(0)=π‘₯0βˆˆπ‘‹(2.6) is given by 𝑒(𝑑)=π‘ˆ(𝑑)π‘₯0+ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑉(π‘‘βˆ’π‘ )β„Ž(𝑠)𝑑𝑠,(2.7) where ξ€œπ‘ˆ(𝑑)=∞0πœπ›Ό(πœƒ)𝑇(π‘‘π›Όξ€œπœƒ)π‘‘πœƒ,𝑉(𝑑)=π›Όβˆž0πœƒπœπ›Ό(πœƒ)𝑇(π‘‘π›Όπœπœƒ)π‘‘πœƒ,(2.8)𝛼(1πœƒ)=π›Όπœƒβˆ’1βˆ’(1/𝛼)πœŒπ›Όξ€·πœƒβˆ’1/π›Όξ€ΈπœŒ,(2.9)𝛼1(πœƒ)=πœ‹βˆžξ“π‘›=0(βˆ’1)π‘›βˆ’1πœƒβˆ’π›Όπ‘›βˆ’1Ξ“(𝑛𝛼+1)𝑛!sin(π‘›πœ‹π›Ό),πœƒβˆˆ(0,∞),(2.10)πœπ›Ό(πœƒ) is a probability density function defined on (0,∞).

Remark 2.6 (see [19, 20, 22]). πœπ›Ό(πœƒ)β‰₯0,πœƒβˆˆ(0,∞), ∫∞0πœπ›Ό(πœƒ)π‘‘πœƒ=1, ∫∞0πœƒπœπ›Ό(πœƒ)π‘‘πœƒ=1/Ξ“(1+𝛼).

Definition 2.7. By the mild solution of the Cauchy problem (2.6), we mean the function π‘’βˆˆπΆ(𝐼,𝑋) satisfying the integral equation 𝑒(𝑑)=π‘ˆ(𝑑)π‘₯0+ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑉(π‘‘βˆ’π‘ )β„Ž(𝑠)𝑑𝑠,(2.11) where π‘ˆ(𝑑) and 𝑉(𝑑) are given by (2.8) and (2.9), respectively.

Definition 2.8. An operator family 𝑆(𝑑)βˆΆπ‘‹β†’π‘‹(𝑑β‰₯0) in 𝑋 is called to be positive if for any π‘’βˆˆπ‘ƒ and 𝑑β‰₯0 such that 𝑆(𝑑)𝑒β‰₯πœƒ.

From Definition 2.8, if 𝑇(𝑑)(𝑑β‰₯0) is a positive semigroup generated by βˆ’π΄, β„Žβ‰₯πœƒ, π‘₯0β‰₯πœƒ, then the mild solution π‘’βˆˆπΆ(𝐼,𝑋) of (2.6) satisfies 𝑒β‰₯πœƒ. For positive semigroups, one can refer to [28–32].

Now, we recall some properties of the measure of noncompactness will be used later. Let πœ‡(β‹…) denote the Kuratowski measure of noncompactness of the bounded set. For the details of the definition and properties of the measure of noncompactness, see [51]. For any π΅βŠ‚πΆ(𝐼,𝑋) and π‘‘βˆˆπΌ, set 𝐡(𝑑)={𝑒(𝑑)βˆ£π‘’βˆˆπ΅}. If 𝐡 is bounded in 𝐢(𝐼,𝑋), then 𝐡(𝑑) is bounded in 𝑋, and πœ‡(𝐡(𝑑))β‰€πœ‡(𝐡).

Lemma 2.9 (see [52]). Let 𝐡={𝑒𝑛}βŠ‚πΆ(𝐼,𝑋)(𝑛=1,2,…) be a bounded and countable set, then πœ‡(𝐡(𝑑)) is Lebesgue integral on 𝐼, πœ‡ξ€œξ‚΅ξ‚»πΌπ‘’π‘›ξ€œ(𝑑)π‘‘π‘‘βˆ£π‘›=1,2,…≀2πΌπœ‡(𝐡(𝑑))𝑑𝑑.(2.12)
In order to prove our results, one also needs a generalized Gronwall inequality for fractional differential equation.

Lemma 2.10 (see [53]). Suppose that 𝑏β‰₯0, 𝛽>0, and π‘Ž(𝑑) is a nonnegative function locally integrable on 0≀𝑑<𝑇 (some 𝑇≀+∞), and suppose that 𝑒(𝑑) is nonnegative and locally integrable on 0≀𝑑<𝑇 with ξ€œπ‘’(𝑑)β‰€π‘Ž(𝑑)+𝑏𝑑0(π‘‘βˆ’π‘ )π›½βˆ’1𝑒(𝑠)𝑑𝑠(2.13) on this interval, then ξ€œπ‘’(𝑑)β‰€π‘Ž(𝑑)+𝑑0ξƒ¬βˆžξ“π‘›=1(𝑏Γ(𝛽))𝑛Γ(𝑛𝛽)(π‘‘βˆ’π‘ )π‘›π›½βˆ’1ξƒ­π‘Ž(𝑠)𝑑𝑠,0≀𝑑<𝑇.(2.14)

3. Main Results

Theorem 3.1. Let 𝑋 be an ordered Banach space, whose positive cone 𝑃 is normal with normal constant 𝑁. Assume that 𝑇(𝑑)(𝑑β‰₯0) is positive, the Cauchy problem (1.1) has a lower solution 𝑣0∈𝐢(𝐼,𝑋) and an upper solution 𝑀0∈𝐢(𝐼,𝑋) with 𝑣0≀𝑀0, and the following conditions are satisfied. (𝐻1) There exists a constant 𝐢β‰₯0 such that 𝑓𝑑,π‘₯2ξ€Έξ€·βˆ’π‘“π‘‘,π‘₯1ξ€Έξ€·π‘₯β‰₯βˆ’πΆ2βˆ’π‘₯1ξ€Έ,(3.1) for any π‘‘βˆˆπΌ, and 𝑣0(𝑑)≀π‘₯1≀π‘₯2≀𝑀0(𝑑), that is, 𝑓(𝑑,π‘₯)+𝐢π‘₯ is increasing in π‘₯ for π‘₯∈[𝑣0(𝑑),𝑀0(𝑑)]. (𝐻2) There exists a constant 𝐿β‰₯0 such that πœ‡π‘“ξ€·ξ€·ξ€½π‘‘,π‘₯𝑛π‘₯ξ€Έξ€Ύξ€Έβ‰€πΏπœ‡ξ€·ξ€½π‘›ξ€Ύξ€Έ,(3.2) for any π‘‘βˆˆπΌ, and increasing or decreasing monotonic sequences {π‘₯𝑛}βŠ‚[𝑣0(𝑑),𝑀0(𝑑)], then the Cauchy problem (1.1) has the minimal and maximal mild solutions between 𝑣0 and 𝑀0, which can be obtained by a monotone iterative procedure starting from 𝑣0 and 𝑀0, respectively.

Proof. It is easy to see that βˆ’(𝐴+𝐢𝐼) generates an analytic semigroup 𝑆(𝑑)=π‘’βˆ’πΆπ‘‘π‘‡(𝑑), and 𝑆(𝑑)(𝑑β‰₯0) is positive. Let ∫Φ(𝑑)=∞0πœπ›Ό(πœƒ)𝑆(π‘‘π›Όβˆ«πœƒ)π‘‘πœƒ,Ξ¨(𝑑)=π›Όβˆž0πœƒπœπ›Ό(πœƒ)𝑆(π‘‘π›Όπœƒ)π‘‘πœƒ. By Remark 2.6, Ξ¦(𝑑)(𝑑β‰₯0) and Ξ¨(𝑑)(𝑑β‰₯0) are positive. By (2.4) and Remark 2.6, we have that 𝛼‖Φ(𝑑)‖≀𝑀,β€–Ξ¨(𝑑)‖≀Γ(𝛼+1)π‘€β‰œπ‘€1,𝑑β‰₯0.(3.3)
Let 𝐷=[𝑣0,𝑀0], we define a mapping π‘„βˆΆπ·β†’πΆ(𝐼,𝑋) by 𝑄𝑒(𝑑)=Ξ¦(𝑑)π‘₯0+ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1[]Ξ¨(π‘‘βˆ’π‘ )𝑓(𝑠,𝑒(𝑠))+𝐢𝑒(𝑠)𝑑𝑠,π‘‘βˆˆπΌ.(3.4) By Lemma 2.5 and Definition 2.7, π‘’βˆˆπ· is a mild solution of the problem (1.1) if and only if 𝑒=𝑄𝑒.(3.5) For 𝑒1,𝑒2∈𝐷 and 𝑒1≀𝑒2, from the positivity of operators Ξ¦(𝑑) and Ξ¨(𝑑), and (𝐻1), we have that 𝑄𝑒1≀𝑄𝑒2.(3.6) Now, we show that 𝑣0≀𝑄𝑣0, 𝑄𝑀0≀𝑀0. Let 𝐷𝛼𝑣0(𝑑)+𝐴𝑣0(𝑑)+𝐢𝑣0(𝑑)β‰œπœŽ(𝑑), by Definition 2.4, Lemma 2.5, and the positivity of operators Ξ¦(𝑑) and Ξ¨(𝑑), we have that 𝑣0(𝑑)=Ξ¦(𝑑)𝑣0(ξ€œ0)+𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1Ξ¨(π‘‘βˆ’π‘ )𝜎(𝑠)𝑑𝑠≀Φ(𝑑)π‘₯0+ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓Ψ(π‘‘βˆ’π‘ )𝑠,𝑣0ξ€Έ(𝑠)+𝐢𝑣0ξ€»(𝑠)𝑑𝑠=𝑄𝑣0(𝑑),π‘‘βˆˆπΌ,(3.7) namely, 𝑣0≀𝑄𝑣0. Similarly, we can show that 𝑄𝑀0≀𝑀0. For π‘’βˆˆπ·, in view of (3.6), then 𝑣0≀𝑄𝑣0≀𝑄𝑒≀𝑄𝑀0≀𝑀0. Thus, π‘„βˆΆπ·β†’π· is an increasing monotonic operator. We can now define the sequences 𝑣𝑛=π‘„π‘£π‘›βˆ’1,𝑀𝑛=π‘„π‘€π‘›βˆ’1,𝑛=1,2,…,(3.8) and it follows from (3.6) that 𝑣0≀𝑣1≀⋯𝑣𝑛≀⋯≀𝑀𝑛≀⋯≀𝑀1≀𝑀0.(3.9) Let 𝐡={𝑣𝑛}(𝑛=1,2,…) and 𝐡0={π‘£π‘›βˆ’1}(𝑛=1,2,…). It follows from 𝐡0=𝐡βˆͺ{𝑣0} that πœ‡(𝐡(𝑑))=πœ‡(𝐡0(𝑑)) for π‘‘βˆˆπΌ. Let πœ‘ξ€·π΅(𝑑)=πœ‡(𝐡(𝑑))=πœ‡0ξ€Έ(𝑑),π‘‘βˆˆπΌ.(3.10) For π‘‘βˆˆπΌ, from (𝐻2), (3.3), (3.4), (3.8), (3.10), Lemma 2.9, and the positivity of operator Ξ¨(𝑑), we have that πœ‘ξ€·(𝑑)=πœ‡(𝐡(𝑑))=πœ‡π‘„π΅0ξ€Έξ€œ(𝑑)=πœ‡ξ‚΅ξ‚»π‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1Ψ𝑓(π‘‘βˆ’π‘ )𝑠,π‘£π‘›βˆ’1ξ€Έ(𝑠)+πΆπ‘£π‘›βˆ’1ξ€»ξ€œ(𝑠)π‘‘π‘ βˆ£π‘›=1,2,…≀2𝑑0πœ‡ξ€·ξ€½(π‘‘βˆ’π‘ )π›Όβˆ’1ξ€ΊΞ¨(π‘‘βˆ’π‘ )𝑓(𝑠,π‘£π‘›βˆ’1(𝑠)+πΆπ‘£π‘›βˆ’1ξ€»(𝑠)βˆ£π‘›=1,2,…𝑑𝑠≀2𝑀1ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1𝐡(𝐿+𝐢)πœ‡0ξ€Έ(𝑠)𝑑𝑠=2𝑀1ξ€œ(𝐿+𝐢)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1πœ‘(𝑠)𝑑𝑠.(3.11) By (3.11) and Lemma 2.10, we obtain that πœ‘(𝑑)≑0 on 𝐼. This means that 𝑣𝑛(𝑑)(𝑛=1,2,…) is precompact in 𝑋 for every π‘‘βˆˆπΌ. So, 𝑣𝑛(𝑑) has a convergent subsequence in 𝑋. In view of (3.9), we can easily prove that 𝑣𝑛(𝑑) itself is convergent in 𝑋. That is, there exists 𝑒(𝑑)βˆˆπ‘‹ such that 𝑣𝑛(𝑑)→𝑒(𝑑) as π‘›β†’βˆž for every π‘‘βˆˆπΌ. By (3.4) and (3.8), for any π‘‘βˆˆπΌ, we have that 𝑣𝑛(𝑑)=Ξ¦(𝑑)π‘₯0+ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓Ψ(π‘‘βˆ’π‘ )𝑠,π‘£π‘›βˆ’1(𝑠)+πΆπ‘£π‘›βˆ’1(𝑠)𝑑𝑠.(3.12) Let π‘›β†’βˆž, then by Lebesgue-dominated convergence theorem, for any π‘‘βˆˆπΌ, we have that 𝑒(𝑑)=Ξ¦(𝑑)π‘₯0+ξ€œπ‘‘0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑓Ψ(π‘‘βˆ’π‘ )𝑠,𝑒(𝑠)+𝐢𝑒(𝑠)𝑑𝑠,(3.13) and π‘’βˆˆπΆ(𝐼,𝑋), then 𝑒=𝑄𝑒. Similarly, we can prove that there exists π‘’βˆˆπΆ(𝐼,𝑋) such that 𝑒=𝑄𝑒. By (3.6), if π‘’βˆˆπ·, and 𝑒 is a fixed point of 𝑄, then 𝑣1=𝑄𝑣0≀𝑄𝑒=𝑒≀𝑄𝑀0=𝑀1. By induction, 𝑣𝑛≀𝑒≀𝑀𝑛. By (3.9) and taking the limit as π‘›β†’βˆž, we conclude that 𝑣0≀𝑒≀𝑒≀𝑒≀𝑀0. That means that 𝑒,𝑒 are the minimal and maximal fixed points of 𝑄 on [𝑣0,𝑀0], respectively. By (3.5), they are the minimal and maximal mild solutions of the Cauchy problem (1.1) on [𝑣0,𝑀0], respectively.

Remark 3.2. Theorem 3.1 extends [37, Theorem  2.1]. Even if 𝐴=0 and 𝑋=ℝ, our results are also new.

Corollary 3.3. Let 𝑋 be an ordered Banach space, whose positive cone 𝑃 is regular. Assume that 𝑇(𝑑)(𝑑β‰₯0) is positive, the Cauchy problem (1.1) has a lower solution 𝑣0∈𝐢(𝐼,𝑋) and an upper solution 𝑀0∈𝐢(𝐼,𝑋) with 𝑣0≀𝑀0, and (𝐻1) holds, then the Cauchy problem (1.1) has the minimal and maximal mild solutions between 𝑣0 and 𝑀0, which can be obtained by a monotone iterative procedure starting from 𝑣0 and 𝑀0, respectively.

Proof. Since (𝐻1) is satisfied, then (3.9) holds. In regular positive cone 𝑃, any monotonic and ordered-bounded sequence is convergent, then there exist π‘’βˆˆπΆ(𝐼,𝐸), π‘’βˆˆπΆ(𝐼,𝐸), and limπ‘›β†’βˆžπ‘£π‘›=𝑒, limπ‘›β†’βˆžπ‘€π‘›=𝑒. Then by the proof of Theorem 3.1, the proof is then complete.

Corollary 3.4. Let 𝑋 be an ordered and weakly sequentially complete Banach space, whose positive cone 𝑃 is normal with normal constant 𝑁. Assume that 𝑇(𝑑)(𝑑β‰₯0) is positive, the Cauchy problem (1.1) has a lower solution 𝑣0∈𝐢(𝐼,𝑋) and an upper solution 𝑀0∈𝐢(𝐼,𝑋) with 𝑣0≀𝑀0, and (𝐻1) holds, then the Cauchy problem (1.1) has the minimal and maximal mild solutions between 𝑣0 and 𝑀0, which can be obtained by a monotone iterative procedure starting from 𝑣0 and 𝑀0, respectively.

Proof. Since 𝑋 is an ordered and weakly sequentially complete Banach space, then the assumption (𝐻2) holds. In fact, by [54, Theorem  2.2], any monotonic and ordered bounded sequence is precompact. Let π‘₯𝑛 be an increasing or decreasing sequence. By (𝐻1), {𝑓(𝑑,π‘₯𝑛)+𝐢π‘₯𝑛} is a monotonic and ordered bounded sequence. Then, by the properties of the measure of noncompactness, we have πœ‡π‘“ξ€·ξ€·ξ€½π‘‘,π‘₯π‘›π‘“ξ€·ξ€Έξ€Ύξ€Έβ‰€πœ‡ξ€·ξ€½π‘‘,π‘₯𝑛+𝐢π‘₯𝑛+πœ‡ξ€·ξ€½πΆπ‘₯𝑛=0.(3.14) So, (𝐻2) holds. By Theorem 3.1, the proof is then complete.

Theorem 3.5. Let 𝑋 be an ordered Banach space, whose positive cone 𝑃 is normal with normal constant 𝑁. Assume that 𝑇(𝑑)(𝑑β‰₯0) is positive, the Cauchy problem (1.1) has a lower solution 𝑣0∈𝐢(𝐼,𝑋) and an upper solution 𝑀0∈𝐢(𝐼,𝑋) with 𝑣0≀𝑀0, (𝐻1) holds, and the following condition is satisfied: (𝐻3) there is constant 𝑆β‰₯0 such that 𝑓𝑑,π‘₯2ξ€Έξ€·βˆ’π‘“π‘‘,π‘₯1ξ€Έξ€·π‘₯≀𝑆2βˆ’π‘₯1ξ€Έ,(3.15) for any π‘‘βˆˆπΌ, 𝑣0(𝑑)≀π‘₯1≀π‘₯2≀𝑀0(𝑑).Then the Cauchy problem (1.1) has the unique mild solution between 𝑣0 and 𝑀0, which can be obtained by a monotone iterative procedure starting from 𝑣0 or 𝑀0.

Proof. We can find that (𝐻1) and (𝐻3) imply (𝐻2). In fact, for π‘‘βˆˆπΌ, let {π‘₯𝑛}βŠ‚[𝑣0(𝑑),𝑀0(𝑑)] be an increasing sequence. For π‘š,𝑛=1,2,… with π‘š>𝑛, by (𝐻1) and (𝐻3), we have that ξ€·πœƒβ‰€π‘“π‘‘,π‘₯π‘šξ€Έξ€·βˆ’π‘“π‘‘,π‘₯𝑛π‘₯+πΆπ‘šβˆ’π‘₯𝑛≀π‘₯(𝑆+𝐢)π‘šβˆ’π‘₯𝑛.(3.16) By (3.16) and the normality of positive cone 𝑃, we have ‖‖𝑓𝑑,π‘₯π‘šξ€Έξ€·βˆ’π‘“π‘‘,π‘₯𝑛‖‖≀‖‖π‘₯(𝑁𝑆+𝑁𝐢+𝐢)π‘šβˆ’π‘₯𝑛‖‖.(3.17) From (3.17) and the definition of the measure of noncompactness, we have that πœ‡π‘“ξ€·ξ€·ξ€½π‘‘,π‘₯𝑛π‘₯ξ€Έξ€Ύξ€Έβ‰€πΏπœ‡ξ€·ξ€½π‘›ξ€Ύξ€Έ,(3.18) where 𝐿=𝑁𝑆+𝑁𝐢+𝐢. Hence, (𝐻2) holds.
Therefore, by Theorem 3.1, the Cauchy problem (1.1) has the minimal mild solution 𝑒 and the maximal mild solution 𝑒 on 𝐷=[𝑣0,𝑀0]. In view of the proof of Theorem 3.1, we show that 𝑒=𝑒. For π‘‘βˆˆπΌ, by (3.3), (3.4), (3.5), (𝐻3), and the positivity of operator Ξ¨(𝑑), we have that πœƒβ‰€π‘’(𝑑)βˆ’π‘’(𝑑)=𝑄𝑒(𝑑)βˆ’π‘„π‘’=ξ€œ(𝑑)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1Ψ𝑓(π‘‘βˆ’π‘ )𝑠,𝑒(𝑠)βˆ’π‘“π‘ ,𝑒(𝑠)+𝐢𝑒(𝑠)βˆ’π‘’β‰€ξ€œ(𝑠)𝑑𝑠𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1ξ€·Ξ¨(π‘‘βˆ’π‘ )(𝑆+𝐢)𝑒(𝑠)βˆ’π‘’ξ€Έ(𝑠)𝑑𝑠≀𝑀1ξ€œ(𝑆+𝐢)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1𝑒(𝑠)βˆ’π‘’ξ€»(𝑠)𝑑𝑠.(3.19) By (3.19) and the normality of the positive cone 𝑃, for π‘‘βˆˆπΌ, we obtain that ‖‖𝑒(𝑠)βˆ’π‘’(‖‖𝑠)≀𝑁𝑀1(ξ€œπ‘†+𝐢)𝑑0(π‘‘βˆ’π‘ )π›Όβˆ’1‖‖𝑒(𝑠)βˆ’π‘’(‖‖𝑠)𝑑𝑠.(3.20) By Lemma 2.10, then 𝑒(𝑑)≑𝑒(𝑑) on 𝐼. Hence, 𝑒=𝑒 is the the unique mild solution of the Cauchy problem (1.1) on [𝑣0,𝑀0]. By the proof of Theorem 3.1, we know it can be obtained by a monotone iterative procedure starting from 𝑣0 or 𝑀0.

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

Corollary 3.6. Assume that 𝑇(𝑑)(𝑑β‰₯0) is positive, the Cauchy problem (1.1) has a lower solution 𝑣0∈𝐢(𝐼,𝑋) and an upper solution 𝑀0∈𝐢(𝐼,𝑋) with 𝑣0≀𝑀0, (𝐻1) and (𝐻3) hold, and one of the following conditions is satisfied: (i)X is an ordered Banach space, whose positive cone 𝑃 is regular, (ii)X is an ordered and weakly sequentially complete Banach space, whose positive cone 𝑃 is normal with normal constant 𝑁, then the Cauchy problem (1.1) has the unique mild solution between 𝑣0 and 𝑀0, which can be obtained by a monotone iterative procedure starting from 𝑣0 or 𝑀0.

4. Examples

Example 4.1. In order to illustrate our main results, we consider the fractional partial differential diffusion equation in 𝑋, πœ•π›Όπ‘‘π‘’βˆ’Ξ”π‘’=𝑔(𝑦,𝑑,𝑒),(𝑦,𝑑)βˆˆΞ©Γ—πΌ,π‘’βˆ£πœ•Ξ©π‘’=0,(𝑦,0)=πœ“(𝑦),π‘¦βˆˆΞ©,(4.1) where πœ•π›Όπ‘‘ is the Caputo fractional partial derivative with order 0<𝛼<1, Ξ” is the Laplace operator, 𝐼=[0,𝑇], Ξ©βŠ‚β„π‘ is a bounded domain with a sufficiently smooth boundary πœ•Ξ©, and π‘”βˆΆΞ©Γ—πΌΓ—β„β†’β„ is continuous.
Let 𝑋=𝐿2(Ξ©), 𝑃={π‘£βˆ£π‘£βˆˆπΏ2(Ξ©),𝑣(𝑦)β‰₯0π‘Ž.𝑒.π‘¦βˆˆΞ©}, then 𝑋 is a Banach space, and 𝑃 is a normal cone in 𝑋. Define the operator 𝐴 as follows: 𝐷(𝐴)=𝐻2(Ξ©)∩𝐻10(Ξ©),𝐴𝑒=βˆ’Ξ”π‘’.(4.2) Then βˆ’π΄ generates an analytic semigroup of uniformly bounded analytic semigroup 𝑇(𝑑)(𝑑β‰₯0) in 𝑋 (see [18]). 𝑇(𝑑)(𝑑β‰₯0) is positive (see [31, 32, 55, 56]). Let 𝑒(𝑑)=𝑒(β‹…,𝑑), 𝑓(𝑑,𝑒)=𝑔(β‹…,𝑑,𝑒(β‹…,𝑑)), then the problem (4.1) can be transformed into the following problem: 𝐷𝛼𝑒(𝑑)+𝐴𝑒(𝑑)=𝑓(𝑑,𝑒(𝑑)),π‘‘βˆˆπΌ,𝑒(0)=πœ“.(4.3) Let πœ†1 be the first eigenvalue of 𝐴, and πœ“1 is the corresponding eigenfunction, then πœ†1β‰₯0, πœ“1(𝑦)β‰₯0. In order to solve the problem (4.1), we also need the following assumptions: (𝑂1)πœ“(𝑦)∈𝐻2(Ξ©)∩𝐻10(Ξ©), 0β‰€πœ“(𝑦)β‰€πœ“1(𝑦), 𝑔(𝑦,𝑑,0)β‰₯0, 𝑔(𝑦,𝑑,πœ“1(𝑦))β‰€πœ†1πœ“1(𝑦), (𝑂2)the partial derivative 𝑔′𝑒(𝑦,𝑑,𝑒) is continuous on any bounded domain.

Theorem 4.2. If (𝑂1) and (𝑂2) are satisfied, then the problem (4.1) has the unique mild solution.

Proof. From Definition 2.4 and (𝑂1), we obtain that 0 is a lower solution of (4.3), and πœ“1(𝑦) is an upper solution of (4.3). Form (𝑂2), it is easy to verify that (𝐻1) and (𝐻3) 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.

References

  1. G. Chen and G. Friedman, β€œAn RLC interconnect model based on Fourier analysis,” IEEE Transactions on Computer Aided Design of Integrated Circuits and Systems, vol. 24, no. 2, pp. 170–183, 2005. View at Publisher Β· View at Google Scholar
  2. B. Lundstrom, M. Higgs, W. Spain, and A. Fairhall, β€œFractional differentiation by neocortical pyramidal neurons,” Nature Neuroscience, vol. 11, no. 11, pp. 1335–1342, 2008. View at Publisher Β· View at Google Scholar
  3. Y. Rossikhin and M. Shitikova, β€œApplication of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems,” Acta Mechanica, vol. 120, no. 1–4, pp. 109–125, 1997. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  4. S. Westerlund and L. Ekstam, β€œCapacitor theory,” IEEE Transactions on Dielectrics and Electrical Insulation, vol. 1, no. 5, pp. 826–839, 1994. View at Publisher Β· View at Google Scholar
  5. Z. B. Li, β€œAn extended fractional complex transform,” International Journal of Nonlinear Sciences and Numerical Simulation, vol. 11, pp. 335–338, 2010. View at Publisher Β· View at Google Scholar
  6. Z. B. Li and J. H. He, β€œFractional complex transform for fractional differential equations,” Mathematical and Computational Applications, vol. 15, no. 5, pp. 970–973, 2010. View at Google Scholar Β· View at Zentralblatt MATH
  7. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, The Netherlands, 2006.
  8. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. View at Publisher Β· View at Google Scholar
  9. K. Balachandran, S. Kiruthika, and J. J. Trujillo, β€œExistence results for fractional impulsive integrodifferential equations in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 4, pp. 1970–1977, 2011. View at Publisher Β· View at Google Scholar
  10. K. Balachandran and J. Y. Park, β€œNonlocal Cauchy problem for abstract fractional semilinear evolution equations,” Nonlinear Analysis, vol. 71, no. 10, pp. 4471–4475, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. E. Bazhlekova, Fractional Evolution Equations in Banach Spaces, Ph.D. thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 2001.
  12. M. M. El-Borai, β€œSome probability densities and fundamental solutions of fractional evolution equations,” Chaos, Solitons and Fractals, vol. 14, no. 3, pp. 433–440, 2002. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  13. M. M. El-Borai, β€œThe fundamental solutions for fractional evolution equations of parabolic type,” Journal of Applied Mathematics and Stochastic Analysis, no. 3, pp. 197–211, 2004. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  14. E. Hernndez, D. Oregan, and K. Balachandran, β€œOn recent developments in the theory of abstract differential equations with fractional derivatives,” Nonlinear Analysis, vol. 73, no. 10, pp. 3462–3471, 2010. View at Publisher Β· View at Google Scholar
  15. E. I. Kaikina, β€œNonlinear evolution equations with a fractional derivative on a half-line,” Nonlinear Analysis, vol. 71, no. 3-4, pp. 766–781, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. Z. M. Odibat, β€œCompact and noncompact structures for nonlinear fractional evolution equations,” Physics Letters A, vol. 372, no. 8, pp. 1219–1227, 2008. View at Publisher Β· View at Google Scholar
  17. X.-B. Shu et al., β€œThe existence of mild solutions for impulsive fractional partial differential equations,” Nonlinear Analysis, vol. 74, no. 5, pp. 2003–2011, 2011. View at Publisher Β· View at Google Scholar
  18. J. Wang, Y. Zhou, and W. Wei, β€œA class of fractional delay nonlinear integrodifferential controlled systems in Banach spaces,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 4049–4059, 2011. View at Publisher Β· View at Google Scholar
  19. J. Wang and Y. Zhou, β€œA class of fractional evolution equations and optimal controls,” Nonlinear Analysis, vol. 12, no. 1, pp. 262–272, 2011. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  20. J. Wang, Y. Zhou, W. Wei, and H. Xu, β€œNonlocal problems for fractional integrodifferential equations via fractional operators and optimal controls,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1427–1441, 2011. View at Publisher Β· View at Google Scholar
  21. R.-N. Wang et al., β€œA note on the fractional Cauchy problems with nonlocal initial conditions,” Applied Mathematics Letters, vol. 24, no. 8, pp. 1435–1442, 2011. View at Publisher Β· View at Google Scholar
  22. Y. Zhou and F. Jiao, β€œExistence of mild solutions for fractional neutral evolution equations,” Computers and Mathematics with Applications, vol. 59, no. 3, pp. 1063–1077, 2010. View at Google Scholar Β· View at Zentralblatt MATH
  23. Y. Zhou and F. Jiao, β€œNonlocal Cauchy problem for fractional evolution equations,” Nonlinear Analysis, vol. 11, no. 5, pp. 4465–4475, 2010. View at Publisher Β· View at Google Scholar
  24. B. Baeumer, S. Kurita, and M. M. Meerschaert, β€œInhomogeneous fractional diffusion equations,” Fractional Calculus and Applied Analysis, vol. 8, no. 4, pp. 375–397, 2005. View at Google Scholar Β· View at Zentralblatt MATH
  25. J. Henderson and A. Ouahab, β€œFractional functional differential inclusions with finite delay,” Nonlinear Analysis, vol. 70, no. 5, pp. 2091–2105, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  26. M. M. Meerschaert, D. A. Benson, H. Scheffler, and B. Baeumer, β€œStochastic solution of space-time fractional diffusion equations,” Physical Review E, vol. 65, no. 4, pp. 1103–1106, 2002. View at Publisher Β· View at Google Scholar
  27. S. Du and V. Lakshmikantham, β€œMonotone iterative technique for differential equations in a Banach space,” Journal of Mathematical Analysis and Applications, vol. 87, no. 2, pp. 454–459, 1982. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  28. Y. Li, β€œExistence and uniqueness of positive periodic solutions for abstract semilinear evolution equations,” Journal of Systems Science and Mathematical Sciences, vol. 25, no. 6, pp. 720–728, 2005. View at Google Scholar Β· View at Zentralblatt MATH
  29. Y. Li, β€œExistence of solutions to initial value problems for abstract semilinear evolution equations,” Acta Mathematica Sinica, vol. 48, no. 6, pp. 1089–1094, 2005. View at Google Scholar
  30. Y. Li, β€œPeriodic solutions of semilinear evolution equations in Banach spaces,” Acta Mathematica Sinica, vol. 41, no. 3, pp. 629–636, 1998. View at Google Scholar
  31. Y. Li, β€œThe global solutions of inition value problems for abstract semilinear evolution equations,” Acta Analysis Functionalis Applicata, vol. 3, no. 4, pp. 339–347, 2001. View at Google Scholar
  32. Y. Li, β€œThe positive solutions of abstract semilinear evolution equations and their applications,” Acta Mathematica Sinica, vol. 39, no. 5, pp. 666–672, 1996. View at Google Scholar Β· View at Zentralblatt MATH
  33. H. Yang, β€œMonotone iterative technique for the initial value problems of impulsive evolution equations in ordered Banach spaces,” Abstract and Applied Analysis, vol. 2010, 11 pages, 2010. View at Publisher Β· View at Google Scholar
  34. V. Lakshmikantham, β€œTheory of fractional functional differential equations,” Nonlinear Analysis, vol. 69, no. 10, pp. 3337–3343, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  35. V. Lakshmikantham and A. S. Vatsala, β€œGeneral uniqueness and monotone iterative technique for fractional differential equations,” Applied Mathematics Letters, vol. 21, no. 8, pp. 828–834, 2008. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  36. X. Liu and M. Jia, β€œMultiple solutions for fractional differential equations with nonlinear boundary conditions,” Computers and Mathematics with Applications, vol. 59, no. 8, pp. 2880–2886, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  37. Z. Lv, J. Liang, and T. Xiao, β€œSolutions to the Cauchy problem for differential equations in Banach spaces with fractional order,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1303–1311, 2011. View at Publisher Β· View at Google Scholar
  38. F. A. McRae, β€œMonotone iterative technique and existence results for fractional differential equations,” Nonlinear Analysis, vol. 71, no. 12, pp. 6093–6096, 2009. View at Publisher Β· View at Google Scholar
  39. Z. Wei, W. Dong, and J. Che, β€œPeriodic boundary value problems for fractional differential equations involving a Riemann-Liouville fractional derivative,” Nonlinear Analysis, vol. 73, no. 10, pp. 3232–3238, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  40. Z. Wei, Q. Li, and J. Che, β€œInitial value problems for fractional differential equations involving Riemann-Liouville sequential fractional derivative,” Journal of Mathematical Analysis and Applications, vol. 367, no. 1, pp. 260–272, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  41. S. Zhang, β€œMonotone iterative method for initial value problem involving Riemann-Liouville fractional derivatives,” Nonlinear Analysis, vol. 71, no. 5-6, pp. 2087–2093, 2009. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  42. S. Zhang, β€œExistence of a solution for the fractional differential equation with nonlinear boundary conditions,” Computers and Mathematics with Applications, vol. 61, no. 4, pp. 1202–1208, 2011. View at Publisher Β· View at Google Scholar
  43. X. Zhao, C. Chai, and W. Ge, β€œPositive solutions for fractional four-point boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3665–3672, 2011. View at Publisher Β· View at Google Scholar
  44. Y. Li and Z. Liu, β€œMonotone iterative technique for addressing impulsive integro-differential equations in Banach spaces,” Nonlinear Analysis, vol. 66, no. 1, pp. 83–92, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  45. P. Chen and Y. Li, β€œMixed monotone iterative technique for a class of semilinear impulsive evolution equations in Banach spaces,” Nonlinear Analysis, vol. 74, no. 11, pp. 3578–3588, 2011. View at Publisher Β· View at Google Scholar
  46. P. Chen, β€œMixed monotone iterative technique for impulsive periodic boundary value problems in Banach spaces,” Boundary Value Problems, vol. 2011, 13 pages, 2011. View at Publisher Β· View at Google Scholar
  47. H. Yang, β€œMixed monotone iterative technique for abstract impulsive evolution equations in Banach spaces,” Journal of Inequalities and Applications, no. 2010, Article ID 293410, 15 pages, 2010. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  48. K. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, John Wiley and Sons, New York, NY, USA, 1993.
  49. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  50. S. Samko, A. Kilbas, and O. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Switzerland, 1993.
  51. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
  52. H. R. Heinz, β€œOn the behaviour of measures of noncompactness with respect to differentiation and integration of vector-valued functions,” Nonlinear Analysis, vol. 7, no. 12, pp. 1351–1371, 1983. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  53. H. Ye, J. Gao, and Y. Ding, β€œA generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  54. Y. Du, β€œFixed points of increasing operators in ordered Banach spaces and applications,” Applicable Analysis, vol. 38, no. 1-2, pp. 1–20, 1990. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  55. S. Campanato, β€œGeneration of analytic semigroups by elliptic operators of second order in Hölder spaces,” Annali della Scuola Normale Superiore di Pisa, vol. 8, no. 3, pp. 495–512, 1981. View at Google Scholar Β· View at Zentralblatt MATH
  56. J. Liang, J. Liu, and T. Xiao, β€œNonlocal problems for integrodifferential equations,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 15, no. 6, pp. 815–824, 2008. View at Google Scholar Β· View at Zentralblatt MATH