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 š‘‹,š·š›¼š‘¢(š‘”)+š“š‘¢(š‘”)=š‘“(š‘”,š‘¢(š‘”)),š‘”āˆˆš¼,š‘¢(0)=š‘„0āˆˆš‘‹,(1.1) where š·š›¼ is the Caputo fractional derivative of order 0<š›¼<1, š¼=[0,š‘‡], š“āˆ¶š·(š“)āŠ‚š‘‹ā†’š‘‹ is a linear closed densely defined operator, āˆ’š“ is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators š‘‡(š‘”) (š‘”ā‰„0), and š‘“āˆ¶š¼Ć—š‘‹ā†’š‘‹ 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 [4ā€“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 [9ā€“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 š›¼āˆˆ(0,1), namely, šœ•š›¼š‘”š‘¢(š‘¦,š‘”)=š“š‘¢(š‘¦,š‘”),š‘”ā‰„0,š‘¦āˆˆš‘…,(1.2) where š“ may be linear fractional partial differential operator. For fractional diffusion equations, we can see [24ā€“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 (š›¼=1), Li [28ā€“32] and Yang [33] used this method, in which positive š¶0-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 [34ā€“43].

However, when many partial differential equations involving time-variable š‘” turn to evolution equations in Banach spaces, they always generate an unbounded closed operator term š“, such as (1.2). š“ is corresponding to linear partial differential operator with certain boundary conditions. In this case, the results in [34ā€“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 āˆ’š“ is the infinitesimal generator of an analytic semigroup in a Banach space, then āˆ’(š“+š‘žš¼) generates a uniformly bounded analytic semigroup for š‘ž>0 large enough. This allows us to reduce the general case in which āˆ’š“ 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 āˆ’š“ is the infinitesimal generator of an analytic semigroup of uniformly bounded linear operators š‘‡(š‘”) (š‘”ā‰„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 š‘“, which are analogous to those in Li and liu [44], Li [28ā€“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 š›¼>0 with the lower limit zero, of function š‘“āˆˆšæ1(ā„+), is defined as š¼š›¼1š‘“(š‘”)=ī€œĪ“(š›¼)š‘”0(š‘”āˆ’š‘ )š›¼āˆ’1š‘“(š‘ )š‘‘š‘ ,(2.1) where Ī“(ā‹…) is the Euler gamma function.

Definition 2.2 (see [7]). The Caputo fractional derivative of order š›¼>0 with the lower limit zero, š‘›āˆ’1<š›¼<š‘›, is defined as š·š›¼1š‘“(š‘”)=ī€œĪ“(š‘›āˆ’š›¼)š‘”0(š‘”āˆ’š‘ )š‘›āˆ’š›¼āˆ’1š‘“(š‘›)(š‘ )š‘‘š‘ ,(2.2) where the function š‘“(š‘”) has absolutely continuous derivatives up to order š‘›āˆ’1. If 0<š›¼<1, then š·š›¼1š‘“(š‘”)=ī€œĪ“(1āˆ’š›¼)š‘”0š‘“ī…ž(š‘ )(š‘”āˆ’š‘ )š›¼š‘‘š‘ .(2.3) If š‘“ is an abstract function with values in š‘‹, then the integrals and derivatives which appear in (2.1) and (2.2) are taken in Bochner,s sense.

Proposition 2.3. For š›¼,š›½>0 and š‘“ as a suitable function (for instance, [7]), one has (i)š¼š›¼š¼š›½š‘“(š‘”)=š¼š›¼+š›½š‘“(š‘”)(ii)š¼š›¼š¼š›½š‘“(š‘”)=š¼š›½š¼š›¼š‘“(š‘”)(iii)š¼š›¼(š‘“(š‘”)+š‘”(š‘”))=š¼š›¼š‘“(š‘”)+š¼š›¼š‘”(š‘”)(iv)š¼š›¼š·š›¼š‘“(š‘”)=š‘“(š‘”)āˆ’š‘“(0),0<š›¼<1(v)š·š›¼š¼š›¼š‘“(š‘”)=š‘“(š‘”)(vi)š·š›¼š·š›½š‘“(š‘”)ā‰ š·š›¼+š›½š‘“(š‘”)(vii)š·š›¼š·š›½š‘“(š‘”)ā‰ š·š›½š·š›¼š‘“(š‘”),(viii)š·š›¼š¶=0, š¶ 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, 48ā€“50].

Let š‘‹ be an ordered Banach space with norm ā€–ā‹…ā€– and partial order ā‰¤, whose positive cone š‘ƒ={š‘¦āˆˆš‘‹āˆ£š‘¦ā‰„šœƒ} (šœƒ is the zero element of š‘‹) is normal with normal constant š‘. Let š¶(š¼,š‘‹) be the Banach space of all continuous š‘‹-value functions on interval š¼ with norm ā€–š‘¢ā€–š¶=maxš‘”āˆˆš¼ā€–š‘¢(š‘”)ā€–. For š‘¢,š‘£āˆˆš¶(š¼,š‘‹), š‘¢ā‰¤š‘£ā‡”š‘¢(š‘”)ā‰¤š‘£(š‘”), for all š‘”āˆˆš¼. For š‘£,š‘¤āˆˆš¶(š¼,š‘‹), denote the ordered interval [š‘£,š‘¤]={š‘¢āˆˆš¶(š¼,š‘‹)āˆ£š‘£ā‰¤š‘¢ā‰¤š‘¤} and [š‘£(š‘”),š‘¤(š‘”)]={š‘¦āˆˆš‘‹āˆ£š‘£(š‘”)ā‰¤š‘¦ā‰¤š‘¤(š‘”)}, š‘”āˆˆš¼. Set š¶š›¼,0(š¼,š‘‹)={š‘¢āˆˆš¶(š¼,š‘‹)āˆ£š·š›¼š‘¢ exists and š·š›¼š‘¢āˆˆš¶(š¼,š‘‹)}. By š‘‹1, we denote the Banach space š·(š“) with the graph norm ā€–ā‹…ā€–1=ā€–ā‹…ā€–+ā€–š“ā‹…ā€–. We note that āˆ’š“ is the infinitesimal generator of a uniformly bounded analytic semigroup š‘‡(š‘”) (š‘”ā‰„0). This means that there exists š‘€ā‰„1 such that ā€–š‘‡(š‘”)ā€–ā‰¤š‘€,š‘”ā‰„0.(2.4)

Definition 2.4. If š‘£0āˆˆš¶š›¼,0(š¼,š‘‹)āˆ©š¶(š¼,š‘‹1) and satisfies š·š›¼š‘£0(š‘”)+š“š‘£0ī€·(š‘”)ā‰¤š‘“š‘”,š‘£0ī€ø(š‘”),š‘”āˆˆš¼,š‘£0(0)ā‰¤š‘„0,(2.5) then š‘£0 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 ā„Ž satisfies a uniform HĢˆš‘œlder condition, with exponent š›½āˆˆ(0,1], then the unique solution of the Cauchy problem š·š›¼š‘¢(š‘”)+š“š‘¢(š‘”)=ā„Ž(š‘”),š‘”āˆˆš¼,š‘¢(0)=š‘„0āˆˆš‘‹(2.6) is given by š‘¢(š‘”)=š‘ˆ(š‘”)š‘„0+ī€œš‘”0(š‘”āˆ’š‘ )š›¼āˆ’1š‘‰(š‘”āˆ’š‘ )ā„Ž(š‘ )š‘‘š‘ ,(2.7) where ī€œš‘ˆ(š‘”)=āˆž0šœš›¼(šœƒ)š‘‡(š‘”š›¼ī€œšœƒ)š‘‘šœƒ,š‘‰(š‘”)=š›¼āˆž0šœƒšœš›¼(šœƒ)š‘‡(š‘”š›¼šœšœƒ)š‘‘šœƒ,(2.8)š›¼(1šœƒ)=š›¼šœƒāˆ’1āˆ’(1/š›¼)šœŒš›¼ī€·šœƒāˆ’1/š›¼ī€øšœŒ,(2.9)š›¼1(šœƒ)=šœ‹āˆžī“š‘›=0(āˆ’1)š‘›āˆ’1šœƒāˆ’š›¼š‘›āˆ’1Ī“(š‘›š›¼+1)š‘›!sin(š‘›šœ‹š›¼),šœƒāˆˆ(0,āˆž),(2.10)šœš›¼(šœƒ) is a probability density function defined on (0,āˆž).

Remark 2.6 (see [19, 20, 22]). šœš›¼(šœƒ)ā‰„0,šœƒāˆˆ(0,āˆž), āˆ«āˆž0šœš›¼(šœƒ)š‘‘šœƒ=1, āˆ«āˆž0šœƒšœš›¼(šœƒ)š‘‘šœƒ=1/Ī“(1+š›¼).

Definition 2.7. By the mild solution of the Cauchy problem (2.6), we mean the function š‘¢āˆˆš¶(š¼,š‘‹) satisfying the integral equation š‘¢(š‘”)=š‘ˆ(š‘”)š‘„0+ī€œš‘”0(š‘”āˆ’š‘ )š›¼āˆ’1š‘‰(š‘”āˆ’š‘ )ā„Ž(š‘ )š‘‘š‘ ,(2.11) where š‘ˆ(š‘”) and š‘‰(š‘”) are given by (2.8) and (2.9), respectively.

Definition 2.8. An operator family š‘†(š‘”)āˆ¶š‘‹ā†’š‘‹(š‘”ā‰„0) in š‘‹ is called to be positive if for any š‘¢āˆˆš‘ƒ and š‘”ā‰„0 such that š‘†(š‘”)š‘¢ā‰„šœƒ.

From Definition 2.8, if š‘‡(š‘”)(š‘”ā‰„0) is a positive semigroup generated by āˆ’š“, ā„Žā‰„šœƒ, š‘„0ā‰„šœƒ, then the mild solution š‘¢āˆˆš¶(š¼,š‘‹) of (2.6) satisfies š‘¢ā‰„šœƒ. For positive semigroups, one can refer to [28ā€“32].

Now, we recall some properties of the measure of noncompactness will be used later. Let šœ‡(ā‹…) 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 šµāŠ‚š¶(š¼,š‘‹) and š‘”āˆˆš¼, set šµ(š‘”)={š‘¢(š‘”)āˆ£š‘¢āˆˆšµ}. If šµ is bounded in š¶(š¼,š‘‹), then šµ(š‘”) is bounded in š‘‹, and šœ‡(šµ(š‘”))ā‰¤šœ‡(šµ).

Lemma 2.9 (see [52]). Let šµ={š‘¢š‘›}āŠ‚š¶(š¼,š‘‹)(š‘›=1,2,ā€¦) be a bounded and countable set, then šœ‡(šµ(š‘”)) is Lebesgue integral on š¼, šœ‡ī€œī‚µī‚»š¼š‘¢š‘›ī€œ(š‘”)š‘‘š‘”āˆ£š‘›=1,2,ā€¦ī‚¼ī‚¶ā‰¤2š¼šœ‡(šµ(š‘”))š‘‘š‘”.(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 š‘ā‰„0, š›½>0, and š‘Ž(š‘”) is a nonnegative function locally integrable on 0ā‰¤š‘”<š‘‡ (some š‘‡ā‰¤+āˆž), and suppose that š‘¢(š‘”) is nonnegative and locally integrable on 0ā‰¤š‘”<š‘‡ with ī€œš‘¢(š‘”)ā‰¤š‘Ž(š‘”)+š‘š‘”0(š‘”āˆ’š‘ )š›½āˆ’1š‘¢(š‘ )š‘‘š‘ (2.13) on this interval, then ī€œš‘¢(š‘”)ā‰¤š‘Ž(š‘”)+š‘”0īƒ¬āˆžī“š‘›=1(š‘Ī“(š›½))š‘›Ī“(š‘›š›½)(š‘”āˆ’š‘ )š‘›š›½āˆ’1īƒ­š‘Ž(š‘ )š‘‘š‘ ,0ā‰¤š‘”<š‘‡.(2.14)

3. Main Results

Theorem 3.1. Let š‘‹ be an ordered Banach space, whose positive cone š‘ƒ is normal with normal constant š‘. Assume that š‘‡(š‘”)(š‘”ā‰„0) is positive, the Cauchy problem (1.1) has a lower solution š‘£0āˆˆš¶(š¼,š‘‹) and an upper solution š‘¤0āˆˆš¶(š¼,š‘‹) with š‘£0ā‰¤š‘¤0, and the following conditions are satisfied. (š»1) There exists a constant š¶ā‰„0 such that š‘“ī€·š‘”,š‘„2ī€øī€·āˆ’š‘“š‘”,š‘„1ī€øī€·š‘„ā‰„āˆ’š¶2āˆ’š‘„1ī€ø,(3.1) for any š‘”āˆˆš¼, and š‘£0(š‘”)ā‰¤š‘„1ā‰¤š‘„2ā‰¤š‘¤0(š‘”), that is, š‘“(š‘”,š‘„)+š¶š‘„ is increasing in š‘„ for š‘„āˆˆ[š‘£0(š‘”),š‘¤0(š‘”)]. (š»2) There exists a constant šæā‰„0 such that šœ‡š‘“ī€·ī€·ī€½š‘”,š‘„š‘›š‘„ī€øī€¾ī€øā‰¤šæšœ‡ī€·ī€½š‘›ī€¾ī€ø,(3.2) for any š‘”āˆˆš¼, and increasing or decreasing monotonic sequences {š‘„š‘›}āŠ‚[š‘£0(š‘”),š‘¤0(š‘”)], then the Cauchy problem (1.1) has the minimal and maximal mild solutions between š‘£0 and š‘¤0, which can be obtained by a monotone iterative procedure starting from š‘£0 and š‘¤0, respectively.

Proof. It is easy to see that āˆ’(š“+š¶š¼) generates an analytic semigroup š‘†(š‘”)=š‘’āˆ’š¶š‘”š‘‡(š‘”), and š‘†(š‘”)(š‘”ā‰„0) is positive. Let āˆ«Ī¦(š‘”)=āˆž0šœš›¼(šœƒ)š‘†(š‘”š›¼āˆ«šœƒ)š‘‘šœƒ,ĪØ(š‘”)=š›¼āˆž0šœƒšœš›¼(šœƒ)š‘†(š‘”š›¼šœƒ)š‘‘šœƒ. By Remark 2.6, Ī¦(š‘”)(š‘”ā‰„0) and ĪØ(š‘”)(š‘”ā‰„0) are positive. By (2.4) and Remark 2.6, we have that š›¼ā€–Ī¦(š‘”)ā€–ā‰¤š‘€,ā€–ĪØ(š‘”)ā€–ā‰¤Ī“(š›¼+1)š‘€ā‰œš‘€1,š‘”ā‰„0.(3.3)
Let š·=[š‘£0,š‘¤0], we define a mapping š‘„āˆ¶š·ā†’š¶(š¼,š‘‹) by š‘„š‘¢(š‘”)=Ī¦(š‘”)š‘„0+ī€œš‘”0(š‘”āˆ’š‘ )š›¼āˆ’1[]ĪØ(š‘”āˆ’š‘ )š‘“(š‘ ,š‘¢(š‘ ))+š¶š‘¢(š‘ )š‘‘š‘ ,š‘”āˆˆš¼.(3.4) By Lemma 2.5 and Definition 2.7, š‘¢āˆˆš· is a mild solution of the problem (1.1) if and only if š‘¢=š‘„š‘¢.(3.5) For š‘¢1,š‘¢2āˆˆš· and š‘¢1ā‰¤š‘¢2, from the positivity of operators Ī¦(š‘”) and ĪØ(š‘”), and (š»1), we have that š‘„š‘¢1ā‰¤š‘„š‘¢2.(3.6) Now, we show that š‘£0ā‰¤š‘„š‘£0, š‘„š‘¤0ā‰¤š‘¤0. Let š·š›¼š‘£0(š‘”)+š“š‘£0(š‘”)+š¶š‘£0(š‘”)ā‰œšœŽ(š‘”), by Definition 2.4, Lemma 2.5, and the positivity of operators Ī¦(š‘”) and ĪØ(š‘”), we have that š‘£0(š‘”)=Ī¦(š‘”)š‘£0(ī€œ0)+š‘”0(š‘”āˆ’š‘ )š›¼āˆ’1ĪØ(š‘”āˆ’š‘ )šœŽ(š‘ )š‘‘š‘ ā‰¤Ī¦(š‘”)š‘„0+ī€œš‘”0(š‘”āˆ’š‘ )š›¼āˆ’1ī€ŗš‘“ī€·ĪØ(š‘”āˆ’š‘ )š‘ ,š‘£0ī€ø(š‘ )+š¶š‘£0ī€»(š‘ )š‘‘š‘ =š‘„š‘£0(š‘”),š‘”āˆˆš¼,(3.7) namely, š‘£0ā‰¤š‘„š‘£0. Similarly, we can show that š‘„š‘¤0ā‰¤š‘¤0. For š‘¢āˆˆš·, in view of (3.6), then š‘£0ā‰¤š‘„š‘£0ā‰¤š‘„š‘¢ā‰¤š‘„š‘¤0ā‰¤š‘¤0. Thus, š‘„āˆ¶š·ā†’š· is an increasing monotonic operator. We can now define the sequences š‘£š‘›=š‘„š‘£š‘›āˆ’1,š‘¤š‘›=š‘„š‘¤š‘›āˆ’1,š‘›=1,2,ā€¦,(3.8) and it follows from (3.6) that š‘£0ā‰¤š‘£1ā‰¤ā‹Æš‘£š‘›ā‰¤ā‹Æā‰¤š‘¤š‘›ā‰¤ā‹Æā‰¤š‘¤1ā‰¤š‘¤0.(3.9) Let šµ={š‘£š‘›}(š‘›=1,2,ā€¦) and šµ0={š‘£š‘›āˆ’1}(š‘›=1,2,ā€¦). It follows from šµ0=šµāˆŖ{š‘£0} that šœ‡(šµ(š‘”))=šœ‡(šµ0(š‘”)) for š‘”āˆˆš¼. Let šœ‘ī€·šµ(š‘”)=šœ‡(šµ(š‘”))=šœ‡0ī€ø(š‘”),š‘”āˆˆš¼.(3.10) For š‘”āˆˆš¼, from (š»2), (3.3), (3.4), (3.8), (3.10), Lemma 2.9, and the positivity of operator ĪØ(š‘”), we have that šœ‘ī€·(š‘”)=šœ‡(šµ(š‘”))=šœ‡š‘„šµ0ī€øī€œ(š‘”)=šœ‡ī‚µī‚»š‘”0(š‘”āˆ’š‘ )š›¼āˆ’1ĪØī€ŗš‘“ī€·(š‘”āˆ’š‘ )š‘ ,š‘£š‘›āˆ’1ī€ø(š‘ )+š¶š‘£š‘›āˆ’1ī€»ī€œ(š‘ )š‘‘š‘ āˆ£š‘›=1,2,ā€¦ī‚¼ī‚¶ā‰¤2š‘”0šœ‡ī€·ī€½(š‘”āˆ’š‘ )š›¼āˆ’1ī€ŗĪØ(š‘”āˆ’š‘ )š‘“(š‘ ,š‘£š‘›āˆ’1(š‘ )+š¶š‘£š‘›āˆ’1ī€»(š‘ )āˆ£š‘›=1,2,ā€¦ī€¾ī€øš‘‘š‘ ā‰¤2š‘€1ī€œš‘”0(š‘”āˆ’š‘ )š›¼āˆ’1ī€·šµ(šæ+š¶)šœ‡0ī€ø(š‘ )š‘‘š‘ =2š‘€1ī€œ(šæ+š¶)š‘”0(š‘”āˆ’š‘ )š›¼āˆ’1šœ‘(š‘ )š‘‘š‘ .(3.11) By (3.11) and Lemma 2.10, we obtain that šœ‘(š‘”)ā‰”0 on š¼. This means that š‘£š‘›(š‘”)(š‘›=1,2,ā€¦) is precompact in š‘‹ for every š‘”āˆˆš¼. So, š‘£š‘›(š‘”) has a convergent subsequence in š‘‹. In view of (3.9), we can easily prove that š‘£š‘›(š‘”) itself is convergent in š‘‹. That is, there exists š‘¢(š‘”)āˆˆš‘‹ such that š‘£š‘›(š‘”)ā†’š‘¢(š‘”) as š‘›ā†’āˆž for every š‘”āˆˆš¼. By (3.4) and (3.8), for any š‘”āˆˆš¼, we have that š‘£š‘›(š‘”)=Ī¦(š‘”)š‘„0+ī€œš‘”0(š‘”āˆ’š‘ )š›¼āˆ’1ī€ŗš‘“ī€·ĪØ(š‘”āˆ’š‘ )š‘ ,š‘£š‘›āˆ’1(ī€øš‘ )+š¶š‘£š‘›āˆ’1(ī€»š‘ )š‘‘š‘ .(3.12) Let š‘›ā†’āˆž, then by Lebesgue-dominated convergence theorem, for any š‘”āˆˆš¼, we have that š‘¢(š‘”)=Ī¦(š‘”)š‘„0+ī€œš‘”0(š‘”āˆ’š‘ )š›¼āˆ’1ī€ŗš‘“ī€·ĪØ(š‘”āˆ’š‘ )š‘ ,š‘¢(ī€øš‘ )+š¶š‘¢(ī€»š‘ )š‘‘š‘ ,(3.13) and š‘¢āˆˆš¶(š¼,š‘‹), then š‘¢=š‘„š‘¢. Similarly, we can prove that there exists š‘¢āˆˆš¶(š¼,š‘‹) such that š‘¢=š‘„š‘¢. By (3.6), if š‘¢āˆˆš·, and š‘¢ is a fixed point of š‘„, then š‘£1=š‘„š‘£0ā‰¤š‘„š‘¢=š‘¢ā‰¤š‘„š‘¤0=š‘¤1. By induction, š‘£š‘›ā‰¤š‘¢ā‰¤š‘¤š‘›. By (3.9) and taking the limit as š‘›ā†’āˆž, we conclude that š‘£0ā‰¤š‘¢ā‰¤š‘¢ā‰¤š‘¢ā‰¤š‘¤0. That means that š‘¢,š‘¢ are the minimal and maximal fixed points of š‘„ on [š‘£0,š‘¤0], respectively. By (3.5), they are the minimal and maximal mild solutions of the Cauchy problem (1.1) on [š‘£0,š‘¤0], respectively.

Remark 3.2. Theorem 3.1 extends [37, Theoremā€‰ā€‰2.1]. Even if š“=0 and š‘‹=ā„, our results are also new.

Corollary 3.3. Let š‘‹ be an ordered Banach space, whose positive cone š‘ƒ is regular. Assume that š‘‡(š‘”)(š‘”ā‰„0) is positive, the Cauchy problem (1.1) has a lower solution š‘£0āˆˆš¶(š¼,š‘‹) and an upper solution š‘¤0āˆˆš¶(š¼,š‘‹) with š‘£0ā‰¤š‘¤0, and (š»1) holds, then the Cauchy problem (1.1) has the minimal and maximal mild solutions between š‘£0 and š‘¤0, which can be obtained by a monotone iterative procedure starting from š‘£0 and š‘¤0, respectively.

Proof. Since (š»1) is satisfied, then (3.9) holds. In regular positive cone š‘ƒ, any monotonic and ordered-bounded sequence is convergent, then there exist š‘¢āˆˆš¶(š¼,šø), š‘¢āˆˆš¶(š¼,šø), and limš‘›ā†’āˆžš‘£š‘›=š‘¢, limš‘›ā†’āˆžš‘¤š‘›=š‘¢. Then by the proof of Theorem 3.1, the proof is then complete.

Corollary 3.4. Let š‘‹ be an ordered and weakly sequentially complete Banach space, whose positive cone š‘ƒ is normal with normal constant š‘. Assume that š‘‡(š‘”)(š‘”ā‰„0) is positive, the Cauchy problem (1.1) has a lower solution š‘£0āˆˆš¶(š¼,š‘‹) and an upper solution š‘¤0āˆˆš¶(š¼,š‘‹) with š‘£0ā‰¤š‘¤0, and (š»1) holds, then the Cauchy problem (1.1) has the minimal and maximal mild solutions between š‘£0 and š‘¤0, which can be obtained by a monotone iterative procedure starting from š‘£0 and š‘¤0, respectively.

Proof. Since š‘‹ is an ordered and weakly sequentially complete Banach space, then the assumption (š»2) holds. In fact, by [54, Theoremā€‰ā€‰2.2], any monotonic and ordered bounded sequence is precompact. Let š‘„š‘› be an increasing or decreasing sequence. By (š»1), {š‘“(š‘”,š‘„š‘›)+š¶š‘„š‘›} is a monotonic and ordered bounded sequence. Then, by the properties of the measure of noncompactness, we have šœ‡š‘“ī€·ī€·ī€½š‘”,š‘„š‘›š‘“ī€·ī€øī€¾ī€øā‰¤šœ‡ī€·ī€½š‘”,š‘„š‘›ī€ø+š¶š‘„š‘›ī€¾ī€ø+šœ‡ī€·ī€½š¶š‘„š‘›ī€¾ī€ø=0.(3.14) So, (š»2) holds. By Theorem 3.1, the proof is then complete.

Theorem 3.5. Let š‘‹ be an ordered Banach space, whose positive cone š‘ƒ is normal with normal constant š‘. Assume that š‘‡(š‘”)(š‘”ā‰„0) is positive, the Cauchy problem (1.1) has a lower solution š‘£0āˆˆš¶(š¼,š‘‹) and an upper solution š‘¤0āˆˆš¶(š¼,š‘‹) with š‘£0ā‰¤š‘¤0, (š»1) holds, and the following condition is satisfied: (š»3) there is constant š‘†ā‰„0 such that š‘“ī€·š‘”,š‘„2ī€øī€·āˆ’š‘“š‘”,š‘„1ī€øī€·š‘„ā‰¤š‘†2āˆ’š‘„1ī€ø,(3.15) for any š‘”āˆˆš¼, š‘£0(š‘”)ā‰¤š‘„1ā‰¤š‘„2ā‰¤š‘¤0(š‘”).Then the Cauchy problem (1.1) has the unique mild solution between š‘£0 and š‘¤0, which can be obtained by a monotone iterative procedure starting from š‘£0 or š‘¤0.

Proof. We can find that (š»1) and (š»3) imply (š»2). In fact, for š‘”āˆˆš¼, let {š‘„š‘›}āŠ‚[š‘£0(š‘”),š‘¤0(š‘”)] be an increasing sequence. For š‘š,š‘›=1,2,ā€¦ with š‘š>š‘›, by (š»1) and (š»3), we have that ī€·šœƒā‰¤š‘“š‘”,š‘„š‘šī€øī€·āˆ’š‘“š‘”,š‘„š‘›ī€øī€·š‘„+š¶š‘šāˆ’š‘„š‘›ī€øā‰¤ī€·š‘„(š‘†+š¶)š‘šāˆ’š‘„š‘›ī€ø.(3.16) By (3.16) and the normality of positive cone š‘ƒ, we have ā€–ā€–š‘“ī€·š‘”,š‘„š‘šī€øī€·āˆ’š‘“š‘”,š‘„š‘›ī€øā€–ā€–ā‰¤ā€–ā€–š‘„(š‘š‘†+š‘š¶+š¶)š‘šāˆ’š‘„š‘›ā€–ā€–.(3.17) From (3.17) and the definition of the measure of noncompactness, we have that šœ‡š‘“ī€·ī€·ī€½š‘”,š‘„š‘›š‘„ī€øī€¾ī€øā‰¤šæšœ‡ī€·ī€½š‘›ī€¾ī€ø,(3.18) where šæ=š‘š‘†+š‘š¶+š¶. Hence, (š»2) holds.
Therefore, by Theorem 3.1, the Cauchy problem (1.1) has the minimal mild solution š‘¢ and the maximal mild solution š‘¢ on š·=[š‘£0,š‘¤0]. In view of the proof of Theorem 3.1, we show that š‘¢=š‘¢. For š‘”āˆˆš¼, by (3.3), (3.4), (3.5), (š»3), and the positivity of operator ĪØ(š‘”), we have that šœƒā‰¤š‘¢(š‘”)āˆ’š‘¢(š‘”)=š‘„š‘¢(š‘”)āˆ’š‘„š‘¢=ī€œ(š‘”)š‘”0(š‘”āˆ’š‘ )š›¼āˆ’1ĪØī€ŗš‘“ī€·(š‘”āˆ’š‘ )š‘ ,š‘¢ī€øī€·(š‘ )āˆ’š‘“š‘ ,š‘¢ī€øī€·(š‘ )+š¶š‘¢(š‘ )āˆ’š‘¢ā‰¤ī€œ(š‘ )ī€øī€»š‘‘š‘ š‘”0(š‘”āˆ’š‘ )š›¼āˆ’1ī€·ĪØ(š‘”āˆ’š‘ )(š‘†+š¶)š‘¢(š‘ )āˆ’š‘¢ī€ø(š‘ )š‘‘š‘ ā‰¤š‘€1ī€œ(š‘†+š¶)š‘”0(š‘”āˆ’š‘ )š›¼āˆ’1ī€ŗš‘¢(š‘ )āˆ’š‘¢ī€»(š‘ )š‘‘š‘ .(3.19) By (3.19) and the normality of the positive cone š‘ƒ, for š‘”āˆˆš¼, we obtain that ā€–ā€–š‘¢(š‘ )āˆ’š‘¢(ā€–ā€–š‘ )ā‰¤š‘š‘€1(ī€œš‘†+š¶)š‘”0(š‘”āˆ’š‘ )š›¼āˆ’1ā€–ā€–š‘¢(š‘ )āˆ’š‘¢(ā€–ā€–š‘ )š‘‘š‘ .(3.20) By Lemma 2.10, then š‘¢(š‘”)ā‰”š‘¢(š‘”) on š¼. Hence, š‘¢=š‘¢ is the the unique mild solution of the Cauchy problem (1.1) on [š‘£0,š‘¤0]. By the proof of Theorem 3.1, we know it can be obtained by a monotone iterative procedure starting from š‘£0 or š‘¤0.

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

Corollary 3.6. Assume that š‘‡(š‘”)(š‘”ā‰„0) is positive, the Cauchy problem (1.1) has a lower solution š‘£0āˆˆš¶(š¼,š‘‹) and an upper solution š‘¤0āˆˆš¶(š¼,š‘‹) with š‘£0ā‰¤š‘¤0, (š»1) and (š»3) hold, and one of the following conditions is satisfied: (i)X is an ordered Banach space, whose positive cone š‘ƒ is regular, (ii)X is an ordered and weakly sequentially complete Banach space, whose positive cone š‘ƒ is normal with normal constant š‘, then the Cauchy problem (1.1) has the unique mild solution between š‘£0 and š‘¤0, which can be obtained by a monotone iterative procedure starting from š‘£0 or š‘¤0.

4. Examples

Example 4.1. In order to illustrate our main results, we consider the fractional partial differential diffusion equation in š‘‹, šœ•š›¼š‘”š‘¢āˆ’Ī”š‘¢=š‘”(š‘¦,š‘”,š‘¢),(š‘¦,š‘”)āˆˆĪ©Ć—š¼,š‘¢āˆ£šœ•Ī©š‘¢=0,(š‘¦,0)=šœ“(š‘¦),š‘¦āˆˆĪ©,(4.1) where šœ•š›¼š‘” is the Caputo fractional partial derivative with order 0<š›¼<1, Ī” is the Laplace operator, š¼=[0,š‘‡], Ī©āŠ‚ā„š‘ is a bounded domain with a sufficiently smooth boundary šœ•Ī©, and š‘”āˆ¶Ī©Ć—š¼Ć—ā„ā†’ā„ is continuous.
Let š‘‹=šæ2(Ī©), š‘ƒ={š‘£āˆ£š‘£āˆˆšæ2(Ī©),š‘£(š‘¦)ā‰„0š‘Ž.š‘’.š‘¦āˆˆĪ©}, then š‘‹ is a Banach space, and š‘ƒ is a normal cone in š‘‹. Define the operator š“ as follows: š·(š“)=š»2(Ī©)āˆ©š»10(Ī©),š“š‘¢=āˆ’Ī”š‘¢.(4.2) Then āˆ’š“ generates an analytic semigroup of uniformly bounded analytic semigroup š‘‡(š‘”)(š‘”ā‰„0) in š‘‹ (see [18]). š‘‡(š‘”)(š‘”ā‰„0) is positive (see [31, 32, 55, 56]). Let š‘¢(š‘”)=š‘¢(ā‹…,š‘”), š‘“(š‘”,š‘¢)=š‘”(ā‹…,š‘”,š‘¢(ā‹…,š‘”)), then the problem (4.1) can be transformed into the following problem: š·š›¼š‘¢(š‘”)+š“š‘¢(š‘”)=š‘“(š‘”,š‘¢(š‘”)),š‘”āˆˆš¼,š‘¢(0)=šœ“.(4.3) Let šœ†1 be the first eigenvalue of š“, and šœ“1 is the corresponding eigenfunction, then šœ†1ā‰„0, šœ“1(š‘¦)ā‰„0. In order to solve the problem (4.1), we also need the following assumptions: (š‘‚1)šœ“(š‘¦)āˆˆš»2(Ī©)āˆ©š»10(Ī©), 0ā‰¤šœ“(š‘¦)ā‰¤šœ“1(š‘¦), š‘”(š‘¦,š‘”,0)ā‰„0, š‘”(š‘¦,š‘”,šœ“1(š‘¦))ā‰¤šœ†1šœ“1(š‘¦), (š‘‚2)the partial derivative š‘”ā€²š‘¢(š‘¦,š‘”,š‘¢) is continuous on any bounded domain.

Theorem 4.2. If (š‘‚1) and (š‘‚2) are satisfied, then the problem (4.1) has the unique mild solution.

Proof. From Definition 2.4 and (š‘‚1), we obtain that 0 is a lower solution of (4.3), and šœ“1(š‘¦) is an upper solution of (4.3). Form (š‘‚2), it is easy to verify that (š»1) and (š»3) 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.