Perturbation Results and Monotone Iterative Technique for Fractional Evolution Equations

Abstract

We mainly study the fractional evolution equation in an ordered Banach space 𝑋^𝐶𝐷^𝑎₀+𝑢(𝑡)+𝐴𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡),𝐺𝑢(𝑡)) , 1<𝛼<2, 𝑢(0)=𝑥∈𝑋, 𝑢^′(0)=𝜃. 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 𝑋:^𝐶𝐷^𝛼₀₊𝑢(𝑡)(𝑡)((𝑡)(𝑡[],𝑢(0)+𝐴𝑢=𝑓𝑡,𝑢,𝐺𝑢)),𝑡∈𝐼=0,𝑇=𝑥∈𝑋,𝑢^′(0)=𝜃, (1.1) where ^𝐶𝐷^𝛼₀₊ is the Caputo fractional derivative, 1<𝛼<2, 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a linear closed densely defined operator, 𝑓∶𝐼×𝑋×𝑋→𝑋 is continuous, 𝜃 is the zero element of 𝑋, and𝐺𝑢(𝑡)=^𝑡₀𝐾(𝑡,𝑠)𝑢(𝑠)𝑑𝑠 (1.2) is a Volterra integral operator with integral kernel 𝐾∈𝐶(Δ,ℝ⁺), Δ={(𝑡,𝑠)∣0≤𝑠≤𝑡≤𝑇}.

In particular, when 𝑓(𝑡,𝑢(𝑡),𝐺𝑢(𝑡))=𝑓(𝑡,𝑢(𝑡)), we study the existence and uniqueness of mild solutions for the fractional evolution equation in an ordered Banach space 𝑋:^𝐶𝐷^𝛼₀₊𝑢(𝑡)(𝑡)((𝑡𝑢(0)+𝐴𝑢=𝑓𝑡,𝑢)),𝑡∈𝐼,=𝑥∈𝑋,𝑢^′(0)=𝜃, (1.3) where ^𝐶𝐷^𝛼₀₊ is the Caputo fractional derivative, 1<𝛼<2, 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a linear closed densely defined operator, 𝑓∶𝐼×𝑋→𝑋 is continuous, and 𝜃 is the zero element of 𝑋.

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 [7–21].

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 [22–24] 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 [25–31]. 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, Hernández 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 𝐶₀-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 𝛼>0 of function 𝑓∈𝐿₁(ℝ⁺) is defined as 𝐼^𝛼₀₊𝑓(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)𝑑𝑠, (2.1) where Γ(⋅) is the Euler gamma function.

Definition 2.2 (see [5]). The Caputo fractional derivative of order 𝛼>0, 𝑛−1<𝛼<𝑛, is defined as ^𝐶𝐷^𝛼₀₊𝑓(𝑡)=1Γ(𝑛−𝛼)^𝑡₀(𝑡−𝑠)^{𝑛−𝛼−1}𝑓(𝑠)𝑑𝑠, (2.2) where the function 𝑓(𝑡) has absolutely continuous derivatives up to order 𝑛−1. 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 sense.

Proposition 2.3. For 𝛼,𝛽>0 and 𝑓 as a suitable function (e.g., [5]) one has the following: (i)𝐼^𝛼₀₊𝐼^𝛽₀₊𝑓(𝑡)=𝐼^𝛼+𝛽₀₊𝑓(𝑡); (ii)𝐼^𝛼₀₊𝐼^𝛽₀₊𝑓(𝑡)=𝐼^𝛽₀₊𝐼^𝛼₀₊𝑓(𝑡); (iii)𝐼^𝛼₀₊(𝑓(𝑡)+𝑔(𝑡))=𝐼^𝛼₀₊𝑓(𝑡)+𝐼^𝛼₀₊𝑔(𝑡); (iv)^𝐶𝐷^𝛼₀₊𝐼^𝛼₀₊𝑓(𝑡)=𝑓(𝑡); (v)^𝐶𝐷^𝛼𝐶₀₊𝐷^𝛽₀₊𝑓(𝑡)≠^𝐶𝐷^𝛼+𝛽₀₊𝑓(𝑡); (vi)^𝐶𝐷^𝛼𝐶₀₊𝐷^𝛽₀₊𝑓(𝑡)≠^𝐶𝐷^𝛽𝐶₀₊𝐷^𝛼₀₊𝑓(𝑡).

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, 32–34].

Let 𝑋 be an ordered Banach space with norm ‖⋅‖ and partial order ≤, whose positive cone 𝑃={𝑦∈𝑋∣𝑦≥𝜃} (𝜃 is the zero element of 𝑋) is normal with normal constant 𝑁. Let 𝐶(𝐼,𝑋) be the Banach space of all continuous 𝑋-value functions on interval 𝐼 with norm ‖𝑢‖_c=max_𝑡∈𝐼‖(𝑡)‖. For 𝑢,𝑣∈𝐶(𝐼,𝑋), 𝑢≤𝑣⇔𝑢(𝑡)≤𝑣(𝑡) for all 𝑡∈𝐼. For 𝑣,𝑤∈𝐶(𝐼,𝑋), denote the ordered interval [𝑣,𝑤]={𝑢∈𝐶(𝐼,𝑋)∣𝑣≤𝑢≤𝑤}, and [𝑣(𝑡),𝑤(𝑡)]={𝑦∈𝑋∣𝑣(𝑡)≤𝑦≤𝑤(𝑡)}, 𝑡∈𝐼. By ℬ(𝑋) we denote the space of all bounded linear operators from 𝑋 to 𝑋.

Definition 2.4. If ^𝐶𝐷^𝛼₀₊𝑣₀,𝐴𝑣₀,𝑣^′₀∈𝐶(𝐼,𝑋), and 𝑣₀ satisfies ^𝐶𝐷^𝛼₀₊𝑣₀(𝑡)+𝐴𝑣₀(𝑡)≤𝑓𝑡,𝑣₀(𝑡),𝐺𝑣₀(𝑡)𝑣,𝑡∈𝐼,₀≤𝑥∈𝑋,𝑣^′₀(0)≤𝜃, (2.3) then 𝑣₀ 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 ^𝐶𝐷^𝛼₀₊𝑣₀,𝐴𝑣₀,𝑣^′₀∈𝐶(𝐼,𝑋), and 𝑣₀ satisfy ^𝐶𝐷^𝛼₀₊𝑣₀(𝑡)+𝐴𝑣₀(𝑡)≤𝑓𝑡,𝑣₀(𝑡)𝑣,𝑡∈𝐼,₀(0)≤𝑥∈𝑋,𝑣^′₀(0)≤𝜃, (2.4) then 𝑣₀ 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: ^𝐶𝐷^𝛼₀₊𝑢(𝑡)(𝑡)𝑢(0)+𝐴𝑢=𝜃,𝑡∈𝐼,=𝑥,𝑢^′(0)=𝜃. (2.5)

Definition 2.6 (see [9]). A family {𝑆_𝛼(𝑡)}_𝑡≥0⊂ℬ(𝑋) is called a solution operator for (2.5) if the following conditions are satisfied: (1)𝑆_𝛼(𝑡) is strongly continuous for 𝑡≥0 and 𝑆_𝛼(0)=𝐼; (2)𝑆_𝛼(𝑡)𝐷(𝐴)⊂𝐷(𝐴) and 𝐴𝑆_𝛼(𝑡)𝑥=𝑆_𝛼(𝑡)𝐴𝑥 for all 𝑥∈𝐷(𝐴), 𝑡≥0; (3)𝑆_𝛼(𝑡) is a solution of 𝑢(𝑡)1=𝑥−Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐴𝑢(𝑠)𝑑𝑠, (2.6) for all 𝑥∈𝐷(𝐴), 𝑡≥0.

In this case, −𝐴 is called the generator of the solution operator 𝑆_𝛼(𝑡) and 𝑆_𝛼(𝑡) is called the solution operator generated by −𝐴.

Definition 2.7 (see [9]). The solution operator 𝑆_𝛼(𝑡) is called exponentially bounded if there are constants 𝑀≥1 and 𝜔≥0 such that 𝑆_𝛼(𝑡)≤𝑀𝑒^𝜔𝑡,𝑡≥0. (2.7)
An operator −𝐴 is said to belong to 𝒞(𝑋;𝑀,𝜔), or 𝒞^𝛼(𝑀,𝜔) for short, if problem (2.5) has a solution operator 𝑆_𝛼(𝑡) satisfying (2.7). Denote 𝒞^𝛼(𝜔)=∪{𝒞^𝛼(𝑀,𝜔)∣𝑀≥1}, 𝒞^𝛼=∪{𝒞^𝛼(𝜔)∣𝜔≥0}. In these notations 𝒞¹ and 𝒞² are the sets of all infinitesimal generators of 𝐶₀-emigroups and cosine operator families (COF), respectively. Next, we give a characterization of 𝒞^𝛼(𝑀,𝜔).

Lemma 2.8 (see [9]). Let 1<𝛼<2, −𝐴∈C^𝛼(𝑀,𝜔) and let 𝑆_𝛼(𝑡) be the corresponding solution operator. Then for 𝜆>𝜔, one has 𝜆^𝛼∈𝜌(−𝐴) and 𝜆^𝛼−1𝑅𝜆^𝛼,−𝐴𝑥=₀^+∞𝑒^−𝜆𝑡𝑆_𝛼(𝑡)𝑥𝑑𝑡,𝑥∈𝑋. (2.8)

Lemma 2.9 (see [9]). Let 1<𝛼<2 and −𝐴∈C^𝛼. Then the corresponding solution operator is given by 𝑆_𝛼(𝑡)𝑥=lim_𝑛→∞1𝑛!^𝑛+1𝑘=1𝑏^𝛼_𝑘,𝑛+1𝑡𝐼+𝑛^𝛼𝐴^−𝑘𝑥=lim_𝑛→∞1𝑛!^𝑛+1𝑘=1𝑏^𝛼_𝑘,𝑛+1𝑛𝑡^𝛼𝑅𝑛𝑡^𝛼,−𝐴^𝑘𝑥, (2.9) where 𝑏^𝛼_𝑘,𝑛 are given by the recurrence relations: 𝑏^𝛼_1,1=1, 𝑏^𝛼_𝑘,𝑛=(𝑛−1−𝑘𝑎)𝑏^𝛼_{𝑘,𝑛−1}+𝛼(𝑘−1)𝑏^𝛼_{𝑘−1,𝑛−1}, 1≤𝑘≤𝑛, 𝑛=2,3,…, 𝑏^𝛼_𝑘,𝑛=0, 𝑘>𝑛, 𝑛=1,2,…. The convergence is uniform on bounded subsets of [0,+∞) for any fixed 𝑥∈𝑋.

Lemma 2.10 (see [9]). Let 1<𝛼<2. Then −𝐴∈C^𝛼(𝑀,𝜔) if and only if (𝜔^𝛼,∞)⊂𝜌(−𝐴) and 𝜕^𝑛𝜕𝜆^𝑛𝜆^𝛼−1𝑅𝜆^𝛼,−𝐴≤𝑀𝑛!(𝜆−𝜔)^𝑛+1𝜆>𝜔,𝑛=0,1,…. (2.10)

Lemma 2.11. Assume ℎ∈𝐶(𝐼,𝑋). For the linear Cauchy problem ^𝐶𝐷^𝛼₀₊𝑢(𝑡)(𝑡)(𝑡)𝑢(0)+𝐴𝑢=ℎ,𝑡∈𝐼,=𝑥∈𝑋,𝑢^′(0)=𝜃, (2.11)𝑢(𝑡) has the form 𝑢(𝑡)=𝑆_𝛼(𝑡)𝑥+^𝑡₀𝑇_𝛼(𝑡−𝑠)ℎ(𝑠)𝑑𝑠, (2.12) where 𝑆_𝛼(𝑡) is the solution operator generated by −𝐴, and 𝑇_𝛼(𝑡)=𝐼^𝛼−1₀₊𝑆_𝛼(𝑡).

Proof. For 𝜆>𝜔, applying the Laplace transform to (2.11), we have that 𝜆^𝛼ℒ𝑢(𝜆)−𝜆^𝛼−1𝑢(0)−𝜆^𝛼−2𝑢^′(0)+𝐴ℒ𝑢(𝜆)=𝜆^𝛼ℒ𝑢(𝜆)−𝜆^𝛼−1𝑥+𝐴ℒ𝑢(𝜆)=ℒℎ(𝜆). (2.13) By Lemma 2.8, 𝜆^𝛼∈𝜌(−𝐴), from the above equation, we obtain ℒ𝑢(𝜆)=𝜆^𝛼−1𝜆^𝛼𝐼+𝐴⁻¹𝑥+𝜆^1−𝛼𝜆^𝛼−1𝜆^𝛼𝐼+𝐴⁻¹ℒℎ(𝜆). (2.14) Since ℒ[𝑡^𝛼−2/Γ(𝛼−1)](𝜆)=𝜆^1−𝛼, by Lemma 2.8 and the inverse Laplace transform, we have that 𝑢(𝑡)=𝑆_𝛼(𝑡)𝑥+^𝑡₀𝑇_𝛼(𝑡−𝑠)ℎ(𝑠)𝑑𝑠, (2.15) where 𝑇_𝛼(𝑡)=1Γ(𝛼−1)^𝑡₀(𝑡−𝑠)^𝛼−2𝑆_𝛼(𝑠)𝑑𝑠=𝐼^𝛼−1₀₊𝑆_𝛼(𝑡). (2.16)

Remark 2.12. If 𝐴=𝑎 (𝑎 is a constant), we know that 𝑢(𝑡)=𝐸_𝛼(−𝑎𝑡^𝛼)𝑥+∫^𝑡₀(𝑡−𝑠)^𝛼−1𝐸_𝛼,𝛼(−𝑎(𝑡−𝑠)^𝛼)ℎ(𝑠)𝑑𝑠 is the solution of (2.11) by [5, Example  4.10], where 𝐸_𝛼(−𝑎𝑡^𝛼) and 𝐸_𝛼,𝛼(−𝑎(𝑡−𝑠)^𝛼) are the Mittag-Leffler functions. We also find that 𝐸_𝛼(−𝑎𝑡^𝛼)𝑥 is the solution of the problem (2.5), and 𝑡^𝛼−1𝐸_𝛼,𝛼(−𝑎𝑡^𝛼)=𝐼^𝛼−1₀₊𝐸_𝛼(−𝑎𝑡^𝛼); see [5].

Definition 2.13. A function 𝑢∶𝐼→𝑋 is called a mild solution of (2.11) if 𝑢∈𝐶(𝐼,𝑋) and satisfies the following equation: 𝑢(𝑡)=𝑆_𝛼(𝑡)𝑥+^𝑡₀𝑇_𝛼(𝑡−𝑠)ℎ(𝑠)𝑑𝑠, (2.17) where 𝑆_𝛼(𝑡) is the solution operator generated by −𝐴, and 𝑇_𝛼(𝑡)=𝐼^𝛼−1₀₊𝑆_𝛼(𝑡).

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 1<𝛼<2, −𝐴∈C^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the solution operator generated by −𝐴, and 𝑇_𝛼(𝑡)=𝐼^𝛼−1₀₊𝑆_𝛼(𝑡), then one has that 𝑇_𝛼(𝑡≤𝑀)Γ(𝛼)𝑒^𝜔𝑡𝑡^𝛼−1,𝑡≥0. (2.18)

Proof. By (2.7), for 𝑠≥0, we have that ‖𝑆_𝛼(𝑠)‖≤𝑀𝑒^𝜔𝑠. Thus, 𝑇_𝛼(𝑡=1)Γ(𝛼−1)^𝑡₀(𝑡−𝑠)^𝛼−2𝑆_𝛼(𝑠)1𝑑𝑠≤𝑀Γ(𝛼−1)^𝑡₀(𝑡−𝑠)^𝛼−2𝑒^𝜔𝑠𝑑𝑠=𝑀𝑒^𝜔𝑡1Γ(𝛼−1)^𝑡₀(𝑡−𝑠)^𝛼−2𝑑𝑠=𝑀𝑒^𝜔𝑡𝑡^𝛼−1Γ(𝛼). (2.19)

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

Definition 2.16. An operator 𝑆(𝑡)∶𝑋→𝑋(𝑡≥0) is called a positive operator in 𝑋 if 𝑢∈𝑃 and 𝑡≥0 such that 𝑆(𝑡)𝑢≥𝜃.

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

Lemma 2.17. Assume that 𝑆_𝛼(𝑡) is the solution operator generated by −𝐴 and 𝑇_𝛼(𝑡)=𝐼^𝛼−1₀₊𝑆_𝛼(𝑡). Then 𝑆_𝛼(𝑡)(𝑡≥0) is a positive operator if and only if 𝑇_𝛼(𝑡)(𝑡≥0) 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 −𝐴∈C^𝛼(𝑀,𝜔) and 𝑆_𝛼(𝑡) is the solution operator generated by −𝐴. The following results are true. (i)If 𝑆_𝛼(𝑡)(𝑡≥0) is a positive solution operator, then for any 𝜆>𝜔 and 𝑢∈𝑃, we have 𝑅(𝜆^𝛼,−𝐴)𝑢≥𝜃.(ii)If there is a 𝜆₀>𝜔, for any 𝜆>𝜆₀ and 𝑢∈𝑃 such that 𝑅(𝜆^𝛼,−𝐴)𝑢≥𝜃, then 𝑆_𝛼(𝑡)(𝑡≥0) is a positive solution operator.

Lemma 2.19. Assume 𝐶>0, 1<𝛼<2, −𝐴∈C^𝛼(𝑀,𝜔); then the following results hold. (i)−(𝐴+𝐶𝐼)∈C^𝛼(𝑀𝐸_𝛼(𝑀𝐶𝑇^𝛼),𝜔), where 𝐸_𝛼(𝑀𝐶𝑇^𝛼) is the Mittag-Leffler function. (ii)If 𝜆>𝜔 and 𝑢∈𝑃 such that 𝑅(𝜆^𝛼,−𝐴)𝑢≥𝜃, then for 𝜆>𝜔+𝐶𝑀𝜔^1−𝛼 and 𝑢∈𝑃, one has 𝑅(𝜆^𝛼,−(𝐴+𝐶𝐼))𝑢≥𝜃.

Proof. (i) If 𝑆_𝛼(𝑡) is the solution operator generated by −(𝐴+𝐶𝐼), in view of [9, Theorem 2.26], we have that 𝑆_𝛼(𝑡)≤𝑀𝐸_𝛼𝑀𝐶𝑇^𝛼𝑒^𝜔𝑡,𝑡≥0. (2.20) That is, −(𝐴+𝐶𝐼)∈𝒞^𝛼(𝑀𝐸_𝛼(𝑀𝐶𝑇^𝛼),𝜔).
(ii) If 𝜆>𝜔, by (i) and Lemma 2.8, we have 𝜆^𝛼∈𝜌(−𝐴), 𝜆^𝛼∈𝜌(−(𝐴+𝐶𝐼)). Then by Lemma 2.10, 𝜆^𝛼−1𝑅𝜆^𝛼≤𝑀,−𝐴.𝜆−𝜔 (2.21) When 𝜆>𝜔+𝐶𝑀𝜔^1−𝛼, 𝐶𝑅𝜆^𝛼,−𝐴≤𝐶𝑀𝜆^1−𝛼≤𝜆−𝜔𝐶𝑀𝜔^1−𝛼𝜆−𝜔<1. (2.22) Therefore, for such 𝜆 the operator 𝐼+𝐶𝑅(𝜆^𝛼,−𝐴) is invertible and 𝑅𝜆^𝛼,−(𝐴+𝐶𝐼)=𝑅𝜆^𝛼,−𝐴𝐼+𝐶𝑅𝜆^𝛼,−𝐴⁻¹=𝑅𝜆^𝛼,−𝐴^∞_𝑛=0(−1)^𝑛𝐶𝑅𝜆^𝛼,−𝐴^𝑛. (2.23) For any 𝑢∈𝑃, in view of 𝑅(𝜆^𝛼,−𝐴)𝑢≥𝜃, 𝐶>0 and (2.22), then 𝑅𝜆^𝛼,−(𝐴+𝐶𝐼)𝑢≥𝜃. (2.24)

Remark 2.20. If 𝛼∈(0,1), Lemma 2.19 (i) is not true; see [9, Example  2.24]. However, a classical perturbation result for 𝒞¹ or 𝒞² (see [35, 36]) is as follows: if 𝐴 is the generator of 𝐶₀-semigroup (or COF) and 𝐵∈ℬ(𝑋), then 𝐴+𝐵 is again a generator of a 𝐶₀-semigroup (or COF).

By Definition 2.13, we can obtain the following result.

Lemma 2.21. The linear Cauchy problem ^𝐶𝐷^𝛼₀₊𝑢(𝑡)(𝑡)(𝑡)(𝑡)𝑢(0)+𝐴𝑢+𝐶𝑢=ℎ,𝑡∈𝐼,=𝑥∈𝑋,𝑢^′(0)=𝜃, (2.25) where 𝐶>0, ℎ∈𝐶(𝐼,𝑋), has the unique mild solution given by 𝑢(𝑡)=𝑆_𝛼(𝑡)𝑥+^𝑡₀𝑇_𝛼(𝑡−𝑠)ℎ(𝑠)𝑑𝑠, (2.26) where 𝑆_𝛼(𝑡) is the solution operator generated by −(𝐴+𝐶𝐼), and 𝑇_𝛼(𝑡)=𝐼^𝛼−1₀₊𝑆_𝛼(𝑡).

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

Lemma 2.22. Assume that 𝑆_𝛼(𝑡) and 𝑆_𝛼(𝑡) are the solution operators generated by −𝐴 and −(𝐴+𝐶𝐼), respectively, and 𝑇_𝛼(𝑡)=𝐼^𝛼−1₀₊𝑆_𝛼(𝑡). Then 𝑆_𝛼(𝑡) is a positive operator ⇒𝑆_𝛼(𝑡) and 𝑇_𝛼(𝑡) are positive operators.

Now, we recall some properties of the measure of noncompactness that will be used later. Let 𝜇(⋅) 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 𝐵⊂𝐶(𝐼,𝑋) and 𝑡∈𝐼, set 𝐵(𝑡)={𝑢(𝑡)∣𝑢∈𝐵}. If 𝐵 is bounded in 𝐶(𝐼,𝑋), then 𝐵(𝑡) is bounded in 𝑋, and 𝜇(𝐵(𝑡))≤𝜇(𝐵).

Lemma 2.23 (see [38]). Let 𝐵={𝑢_𝑛}⊂𝐶(𝐼,𝑋)(𝑛=1,2,…) be a bounded and countable set. Then 𝜇(𝐵(𝑡)) is Lebesgue integral on 𝐼, and 𝜇_𝐼𝑢_𝑛(𝑡)𝑑𝑡∣𝑛=1,2,…≤2_𝐼𝜇(𝐵(𝑡))𝑑𝑡. (2.27)

3. Main Results

Theorem 3.1. Let 𝑋 be an ordered Banach space, whose positive cone 𝑃 is normal with normal constant 𝑁, 𝑓∶𝐼×𝑋×𝑋→𝑋 is continuous, and 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a linear closed densely defined operator. Assume that −𝐴∈𝒞^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, the Cauchy problem (1.1) has a lower solution 𝑣₀∈𝐶(𝐼,𝑋) and an upper solution 𝑤₀∈𝐶(𝐼,𝑋) with 𝑣₀≤𝑤₀, and the following conditions are satisfied. (𝐻₁)There exists a constant 𝐶≥0 such that 𝑓𝑡,𝑥₂,𝑦₂−𝑓𝑡,𝑥₁,𝑦₁≥−𝐶𝑥₂−𝑥₁, (3.1) for any 𝑡∈𝐼, 𝑣₀(𝑡)≤𝑥₁≤𝑥₂≤𝑤₀(𝑡), and 𝐺𝑣₀(𝑡)≤𝑦₁≤𝑦₂≤𝐺𝑤₀(𝑡).(𝐻₂)There exists a constant 𝐿≥0 such that 𝜇𝑓𝑡,𝑥_𝑛,𝑦_𝑛≤𝐿𝜇𝑥_𝑛+𝜇𝑦_𝑛, (3.2) for any 𝑡∈𝐼, and increasing or decreasing monotonic sequences {𝑥_𝑛}⊂[𝑣₀(𝑡),𝑤₀(𝑡)] and {𝑦_𝑛}⊂[𝐺𝑣₀(𝑡),𝐺𝑤₀(𝑡)]. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ and 𝑤₀, respectively.

Proof. Since −𝐴∈C^𝛼(𝑀,𝜔), by Lemmas 2.15 and 2.19, we have that 𝑆_𝛼(𝑡)≤𝑀𝐸_𝛼𝑀𝐶𝑇^𝛼𝑒^𝜔𝑡𝑇,𝑡≥0,_𝛼(𝑡≤𝑀)Γ(𝛼)𝐸_𝛼𝑀𝐶𝑇^𝛼𝑒^𝜔𝑡𝑡^𝛼−1,𝑡≥0. (3.3) Set 𝑀_𝑇=max_𝑡∈𝐼𝑇_𝛼(𝑡). (3.4) Since 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, by Lemma 2.22, 𝑆_𝛼(𝑡) and 𝑇_𝛼(𝑡) are positive operators.
Let 𝐷=[𝑣₀,𝑤₀]; we define a mapping 𝑄∶𝐷→𝐶(𝐼,𝑋) by 𝑄𝑢(𝑡)=𝑆_𝛼(𝑡)𝑥+^𝑡₀𝑇_𝛼(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))+𝐶𝑢(𝑠)𝑑𝑠,𝑡∈𝐼. (3.5) By Lemma 2.21, 𝑢∈𝐷 is a mild solution of problem (1.1) if and only if 𝑢=𝑄𝑢. (3.6) By (𝐻₁), for 𝑢₁,𝑢₂∈𝐷 and 𝑢₁≤𝑢₂, we have that 𝑄𝑢₁≤𝑄𝑢₂. (3.7) That is, 𝑄 is an increasing monotonic operator. Now, we show that 𝑣₀≤𝑄𝑣₀, 𝑄𝑤₀≤𝑤₀.
Let 𝜎(𝑡)≜^𝐶𝐷^𝛼₀₊𝑣₀(𝑡)+𝐴𝑣₀(𝑡)+𝐶𝑣₀(𝑡); by Definition 2.1, Lemma 2.21, and the positivity of operators 𝑆_𝛼(𝑡) and 𝑇_𝛼(𝑡), we have that 𝑣₀(𝑡)=𝑆_𝛼(𝑡)𝑣₀(0)+^𝑡₀𝑇_𝛼()𝜎(𝑠)≤𝑆𝑡−𝑠𝑑𝑠_𝛼(𝑡)𝑥+^𝑡₀𝑇_𝛼(𝑡−𝑠)𝑓𝑠,𝑣₀(𝑠),𝐺𝑣₀(𝑠)+𝐶𝑣₀(𝑠)𝑑𝑠=𝑄𝑣₀(𝑡),𝑡∈𝐼, (3.8) namely, 𝑣₀≤𝑄𝑣₀. Similarly, we can show that 𝑄𝑤₀≤𝑤₀. For 𝑢∈𝐷, in view of (3.7), then 𝑣₀≤𝑄𝑣₀≤𝑄𝑢≤𝑄𝑤₀≤𝑤₀. Thus, 𝑄∶𝐷→𝐷. We can now define the sequences 𝑣_𝑛=𝑄𝑣_𝑛−1,𝑤_𝑛=𝑄𝑤_𝑛−1,𝑛=1,2,…, (3.9) and it follows from (3.7) that 𝑣₀≤𝑣₁≤⋯𝑣_𝑛≤⋯≤𝑤_𝑛≤⋯≤𝑤₁≤𝑤₀. (3.10) For convenience, by (1.2), we can denote 𝐾₀=max_{(𝑡,𝑠)∈Δ}𝐾(𝑡,𝑠). (3.11) Let 𝐵={𝑣_𝑛}(𝑛=1,2,…) and 𝐵₀={𝑣_𝑛−1}(𝑛=1,2,…). It follows from 𝐵₀=𝐵∪{𝑣₀} that 𝜇(𝐵(𝑡))=𝜇(𝐵₀(𝑡)) for 𝑡∈𝐼. Let 𝜑(𝑡)=𝜇(𝐵(𝑡))=𝜇𝐵₀(𝑡),𝑡∈𝐼. (3.12) In view of (3.10), since the positive cone 𝑃 is normal, then 𝐵₀ and 𝐵 are bounded in 𝐶(𝐼,𝑋). By Lemma 2.23 and (3.12), 𝜑(𝑡) is Lebesgue integrable on 𝐼. For 𝑡∈𝐼, by (3.11) and Lemma 2.23, we have that 𝜇𝐺𝐵₀(𝑡)=𝜇^𝑡₀𝐾(𝑡,𝑠)𝑣_𝑛−1(𝑠)∣𝑛=1,2,…≤2𝐾₀^𝑡₀𝜑(𝑠)𝑑𝑠, (3.13) and therefore, ^𝑡₀𝜇𝐺𝐵₀(𝑠)𝑑𝑠≤2𝑇𝐾₀^𝑡₀𝜑(𝑠)𝑑𝑠. (3.14) For 𝑡∈𝐼, from Lemma 2.23, (𝐻₂), (3.4), (3.5), (3.9), (3.12), (3.14), and the positivity of operator 𝑇_𝛼(𝑡), we have that 𝜑(𝑡)=𝜇(𝐵(𝑡))=𝜇𝑄𝐵₀(𝑡)=𝜇^𝑡₀𝑇_𝛼(𝑡−𝑠)𝑓𝑠,𝑣_𝑛−1(𝑠),𝐺𝑣_𝑛−1(𝑠)+𝐶𝑣_𝑛−1(𝑠)𝑑𝑠∣𝑛=1,2,…≤2^𝑡₀𝜇𝑇_𝛼(𝑡−𝑠)𝑓𝑠,𝑣_𝑛−1(𝑠),𝐺𝑣_𝑛−1(𝑠)+𝐶𝑣_𝑛−1(𝑠)𝑀∣𝑛=1,2,…𝑑𝑠≤2_𝑇^𝑡₀𝐿𝜇𝐵₀(𝑠)+𝜇𝐺𝐵₀(𝑠)+𝐶𝜇𝐵₀(𝑠)𝑀𝑑𝑠=2_𝑇𝐿+2𝐿𝑇𝐾₀+𝐶^𝑡₀𝜑(𝑠)𝑑𝑠. (3.15) By (3.15) and the Gronwall inequality, we obtain that 𝜑(𝑡)≡0 on 𝐼. This means that 𝑣_𝑛(𝑡)(𝑛=1,2,…) is precompact in 𝑋 for every 𝑡∈𝐼. So, 𝑣_𝑛(𝑡) has a convergent subsequence in 𝑋. In view of (3.7), we can easily prove that 𝑣_𝑛(𝑡) itself is convergent in 𝑋. That is, there exist 𝑢(𝑡)∈𝑋 such that 𝑣_𝑛(𝑡)→𝑢(𝑡) as 𝑛→∞ for every 𝑡∈𝐼. By (3.5) and (3.9), for any 𝑡∈𝐼, we have that 𝑣_𝑛(𝑡)=𝑆_𝛼(𝑡)𝑥+^𝑡₀𝑇_𝛼(𝑡−𝑠)𝑓𝑠,𝑣_𝑛−1(𝑠),𝐺𝑣_𝑛−1(𝑠)+𝐶𝑣_𝑛−1(𝑠)𝑑𝑠. (3.16) Let 𝑛→∞; then by Lebesgue-dominated convergence theorem, for any 𝑡∈𝐼, we have that 𝑢(𝑡)=𝑆_𝛼(𝑡)𝑥+^𝑡₀𝑇_𝛼(𝑡−𝑠)𝑓𝑠,𝑢(𝑠),𝐺𝑢(𝑠)+𝐶𝑢(𝑠)𝑑𝑠, (3.17) and 𝑢∈𝐶(𝐼,𝑋). Then 𝑢=𝑄𝑢. Similarly, we can prove that there exists 𝑢∈𝐶(𝐼,𝑋) such that 𝑢=𝑄𝑢. By (3.7), if 𝑢∈𝐷, and 𝑢 is a fixed point of 𝑄, then 𝑣₁=𝑄𝑣₀≤𝑄𝑢=𝑢≤𝑄𝑤₀=𝑤₁. By induction, 𝑣_𝑛≤𝑢≤𝑤_𝑛. By (3.10) and taking the limit as 𝑛→∞, we conclude that 𝑣₀≤𝑢≤𝑢≤𝑢≤𝑤₀. That means that 𝑢,𝑢 are the minimal and maximal fixed points of 𝑄 on [𝑣₀,𝑤₀], respectively. By (3.6), they are the minimal and maximal mild solutions of the Cauchy problem (1.1) on [𝑣₀,𝑤₀], respectively.

Remark 3.2. Even if 𝐴=0, our results are also new.

Corollary 3.3. Let 𝑋 be an ordered Banach space, whose positive cone 𝑃 is regular, 𝑓∶𝐼×𝑋×𝑋→𝑋 is continuous, and 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a linear closed densely defined operator. Assume that −𝐴∈C^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, the Cauchy problem (1.1) has a lower solution 𝑣₀∈𝐶(𝐼,𝑋) and an upper solution 𝑤₀∈𝐶(𝐼,𝑋) with 𝑣₀≤𝑤₀, and (𝐻₁) holds. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ and 𝑤₀, respectively.

Proof. Since (𝐻₁) is satisfied, then (3.10) holds. In regular positive cone 𝑃, any monotonic and ordered-bounded sequence is convergent. Then there exist 𝑢∈𝐶(𝐼,𝐸), 𝑢∈𝐶(𝐼,𝐸) and lim_𝑛→∞𝑣_𝑛=𝑢, lim_𝑛→∞𝑤_𝑛=𝑢. 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, 𝑓∶𝐼×𝑋×𝑋→𝑋 is continuous, and 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a linear closed densely defined operator. Assume that −𝐴∈C^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, the Cauchy problem (1.1) has a lower solution 𝑣₀∈𝐶(𝐼,𝑋) and an upper solution 𝑤₀∈𝐶(𝐼,𝑋) with 𝑣₀≤𝑤₀, and (𝐻₁) holds. Then the Cauchy problem (1.1) has the minimal and maximal mild solutions between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ and 𝑤₀, respectively.

Proof. Since 𝑋 is an ordered and weakly sequentially complete Banach space, then the assumption (𝐻₂) holds. In fact, by [39, Theorem  2.2], any monotonic and ordered-bounded sequence is precompact. Let 𝑥_𝑛 and 𝑦_𝑛 be two increasing or decreasing sequences. By (𝐻₁), {𝑓(𝑡,𝑥_𝑛,𝑦_𝑛)+𝐶𝑥_𝑛} is monotonic and ordered-bounded sequence. Then, by the properties of the measure of noncompactness, we have 𝜇𝑓𝑡,𝑥_𝑛,𝑦_𝑛≤𝜇𝑓𝑡,𝑥_𝑛,𝑦_𝑛+𝐶𝑥_𝑛+𝐶𝜇𝑥_𝑛=0. (3.18) 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, 𝑓∶𝐼×𝑋×𝑋→𝑋 is continuous, 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a linear closed densely defined operator. Assume −𝐴∈𝒞^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, the Cauchy problem (1.1) has a lower solution 𝑣₀∈𝐶(𝐼,𝑋) and an upper solution 𝑤₀∈𝐶(𝐼,𝑋) with 𝑣₀≤𝑤₀, (𝐻₁) holds, and the following condition is satisfied: (𝐻₃) There are constants 𝑆₁,𝑆₂≥0 such that 𝑓𝑡,𝑥₂,𝑦₂−𝑓𝑡,𝑥₁,𝑦₁≤𝑆₁𝑥₂−𝑥₁+𝑆₂𝑦₂−𝑦₁, (3.19) for any 𝑡∈𝐼, 𝑣₀(𝑡)≤𝑥₁≤𝑥₂≤𝑤₀(𝑡), and 𝐺𝑣₀(𝑡)≤𝑦₁≤𝑦₂≤𝐺𝑤₀(𝑡).
Then the Cauchy problem (1.1) has the unique mild solution between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ or 𝑤₀.

Proof. We can find that (𝐻₁) and (𝐻₃) imply (𝐻₂). In fact, for 𝑡∈𝐼, let {𝑥_𝑛}⊂[𝑣₀(𝑡),𝑤₀(𝑡)] and {𝑦_𝑛}⊂[𝐺𝑣₀(𝑡),𝐺𝑤₀(𝑡)] be two increasing or decreasing monotonic sequence. For 𝑚,𝑛=1,2,… with 𝑚>𝑛, by (𝐻₁) and (𝐻₃), we have that 𝜃≤𝑓𝑡,𝑥_𝑚,𝑦_𝑚−𝑓𝑡,𝑥_𝑛,𝑦_𝑛+𝐶𝑥_𝑚−𝑥_𝑛≤𝑆₁+𝐶𝑥_𝑚−𝑥_𝑛+𝑆₂𝑦_𝑚−𝑦_𝑛. (3.20) By (3.20) and the normality of positive cone 𝑃, we have 𝑓𝑡,𝑥_𝑚,𝑦_𝑚−𝑓𝑡,𝑥_𝑛,𝑦_𝑛≤𝑁𝑆₁+𝑁𝐶+𝐶𝑥_𝑚−𝑥_𝑛+𝑁𝑆₂𝑦_𝑚−𝑦_𝑛. (3.21) From (3.21) and the definition of the measure of noncompactness, we have that 𝜇𝑓𝑡,𝑥_𝑛,𝑦_𝑛≤𝑁𝑆₁+𝑁𝐶+𝐶𝜇𝑥_𝑛+𝑁𝑆₂𝜇𝑦_𝑛≤𝐿𝜇𝑥_𝑛+𝜇𝑦_𝑛, (3.22) where 𝐿=𝑁𝑆₁+𝑁𝐶+𝐶+𝑁𝑆₂. Hence, (𝐻₂) holds.
Therefore, by Theorem 3.1, the Cauchy problem (1.1) has the minimal solution 𝑢 and the maximal solution 𝑢 on 𝐷=[𝑣₀,𝑤₀]. In view of the proof of Theorem 3.1, we show that 𝑢=𝑢. For 𝑡∈𝐼, by (3.4), (3.5), (3.6), (3.11), (𝐻₃), and the positivity of operator 𝑇_𝛼(𝑡), we have that 𝜃≤𝑢(𝑡)−𝑢(𝑡)=𝑄𝑢(𝑡)−𝑄𝑢(𝑡)=^𝑡₀𝑇_𝛼(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠))−𝑓𝑠,𝑢(𝑠),𝐺𝑢(𝑠)+𝐶𝑢(𝑠)−𝑢(𝑠)≤𝑑𝑠_𝑡0𝑇_𝛼(𝑡−𝑠)𝑆₁+𝐶𝑢(𝑠)−𝑢(𝑠)+𝑆₂𝐺𝑢(𝑠)−𝐺𝑢(𝑠≤𝑀)𝑑𝑠_𝑇𝑆₁+𝐶+𝑆₂𝐾₀𝑇^𝑡₀𝑢(𝑠)−𝑢(𝑠)𝑑𝑠. (3.23) By (3.23) and the normality of the positive cone 𝑃, for 𝑡∈𝐼, we obtain that 𝑢(𝑠)−𝑢(𝑠)≤𝑁𝑀_𝑇𝑆₁+𝐶+𝑆₂𝐾₀𝑇^𝑡₀𝑢(𝑠)−𝑢(𝑠)𝑑𝑠. (3.24) By the Gronwall inequality, then 𝑢(𝑡)≡𝑢(𝑡) on 𝐼. Hence 𝑢=𝑢 is the the unique mild solution of the Cauchy problem (1.1) on [𝑣₀,𝑤₀]. By the proof of Theorem 3.1, we know that it can be obtained by a monotone iterative procedure starting from 𝑣₀ or 𝑤₀.

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

Corollary 3.6. Let 𝑓∶𝐼×𝑋×𝑋→𝑋 be continuous, and let 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be a linear closed densely defined operator. Assume that −𝐴∈𝒞^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, the Cauchy problem (1.1) has a lower solution 𝑣₀∈𝐶(𝐼,𝑋) and an upper solution 𝑤₀∈𝐶(𝐼,𝑋) with 𝑣₀≤𝑤₀, (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 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ or 𝑤₀.

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

Corollary 3.7. Let 𝑋 be an ordered Banach space, whose positive cone 𝑃 is normal with normal constant 𝑁, 𝑓∶𝐼×𝑋→𝑋 is continuous, 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a linear closed densely defined operator. Assume −𝐴∈𝒞^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, the Cauchy problem (1.3) has a lower solution 𝑣₀∈𝐶(𝐼,𝑋) and an upper solution 𝑤₀∈𝐶(𝐼,𝑋) with 𝑣₀≤𝑤₀, and the following conditions are satisfied. (𝐻₁)There exists a constant 𝐶≥0 such that𝑓𝑡,𝑥₂−𝑓𝑡,𝑥₁≥−𝐶𝑥₂−𝑥₁, (3.25) for any 𝑡∈𝐼, and 𝑣₀(𝑡)≤𝑥₁≤𝑥₂≤𝑤₀(𝑡);(𝐻₂)There exists a constant 𝐿≥0 such that 𝜇𝑓𝑡,𝑥_𝑛≤𝐿𝜇𝑥_𝑛, (3.26) for any 𝑡∈𝐼, and increasing or decreasing monotonic sequence {𝑥_𝑛}⊂[𝑣₀(𝑡),𝑤₀(𝑡)].
Then the Cauchy problem (1.3) has the minimal and maximal mild solutions between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ and 𝑤₀, respectively.

Corollary 3.8. Let 𝑋 be an ordered Banach space, whose positive cone 𝑃 is regular, 𝑓∶𝐼×𝑋→𝑋 is continuous, and 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a linear closed densely defined operator. Assume that −𝐴∈𝒞^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, the Cauchy problem (1.3) has a lower solution 𝑣₀∈𝐶(𝐼,𝑋) and an upper solution 𝑤₀∈𝐶(𝐼,𝑋) with 𝑣₀≤𝑤₀, and (𝐻₁) holds. Then the Cauchy problem (1.3) has the minimal and maximal mild solutions between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ and 𝑤₀, respectively.

Corollary 3.9. Let 𝑋 be an ordered and weakly sequentially complete Banach space, whose positive cone 𝑃 is normal with normal constant 𝑁, 𝑓∶𝐼×𝑋→𝑋 is continuous, and 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a linear closed densely defined operator. Assume that −𝐴∈𝒞^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, the Cauchy problem (1.3) has a lower solution 𝑣₀∈𝐶(𝐼,𝑋) and an upper solution 𝑤₀∈𝐶(𝐼,𝑋) with 𝑣₀≤𝑤₀, and (𝐻₁) holds. Then the Cauchy problem (1.3) has the minimal and maximal mild solutions between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ and 𝑤₀, respectively.

Corollary 3.10. Let 𝑋 be an ordered Banach space, whose positive cone 𝑃 is normal with normal constant 𝑁, 𝑓∶𝐼×𝑋→𝑋 is continuous, and 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a linear closed densely defined operator. Assume that −𝐴∈𝒞^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, the Cauchy problem (1.3) has a lower solution 𝑣₀∈𝐶(𝐼,𝑋) and an upper solution 𝑤₀∈𝐶(𝐼,𝑋) with 𝑣₀≤𝑤₀, (𝐻₁) holds, and the following condition is satisfied. (𝐻₃) There exists a constant 𝑆₁≥0 such that 𝑓𝑡,𝑥₂−𝑓𝑡,𝑥₁≤𝑆₁𝑥₂−𝑥₁, (3.27) for any 𝑡∈𝐼, and 𝑣₀(𝑡)≤𝑥₁≤𝑥₂≤𝑤₀(𝑡).
Then the Cauchy problem (1.3) has the unique mild solution between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ or 𝑤₀.

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

Corollary 3.11. Let 𝑓∶𝐼×𝑋→𝑋 be continuous, and let 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be a linear closed densely defined operator. Assume that −𝐴∈𝒞^𝛼(𝑀,𝜔), 𝑆_𝛼(𝑡) is the positive solution operator generated by −𝐴, the Cauchy problem (1.3) has a lower solution 𝑣₀∈𝐶(𝐼,𝑋) and an upper solution 𝑤₀∈𝐶(𝐼,𝑋) with 𝑣₀≤𝑤₀, (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 (1.3) has the unique mild solution between 𝑣₀ and 𝑤₀, which can be obtained by a monotone iterative procedure starting from 𝑣₀ or 𝑤₀.

4. Examples

Example 4.1. In order to illustrate our main results, we consider the Cauchy problem in 𝑋=ℝ^𝑛 (𝑛-dimensional Euclidean space and ‖𝑦‖=(∑^𝑛_𝑖=1𝑦²_𝑖)^1/2): ^𝐶𝐷^𝛼₀₊𝑢(𝑡)(𝑡)((𝑡)(𝑡[],𝑢(0)+𝐴𝑢=𝑓𝑡,𝑢,𝐺𝑢)),𝑡∈𝐼=0,𝑇=𝑥∈𝑋,𝑢^′=𝜃, (4.1) where ^𝐶𝐷^𝛼₀₊ is the Caputo fractional derivative, 1<𝛼<2, 𝐴=(𝑎_𝑖𝑗)_𝑛×𝑛(𝑎_𝑖𝑗≤0) is a real matrix, 𝑓∶𝐼×𝑋×𝑋→𝑋 is continuous, 𝜃=(0,0,…,0) is the zero element of 𝑋, and 𝐺𝑢(𝑡)=^𝑡₀𝐾(𝑡,𝑠)𝑢(𝑠)𝑑𝑠 (4.2) is a Volterra integral operator with integral kernel 𝐾∈𝐶(Δ,ℝ⁺), Δ={(𝑡,𝑠)∣0≤𝑠≤𝑡≤𝑇}.
For 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛), and 𝑧=(𝑧₁,𝑧₂,…,𝑧_𝑛), we define the partial order 𝑦≤𝑧⇔𝑦_𝑖≤𝑧_𝑖(𝑖=1,2,⋯,𝑛). Set 𝑃={𝑦∈𝑋∣𝑦≥𝜃}; then 𝑃 is a normal cone in 𝑋 and normal constant 𝑁=1. It is easy to verify that −𝐴 generates a uniformly continuous positive cosine operator family 𝒮₂(𝑡): 𝒮₂(𝑡)=^∞_𝑛=0𝑡^2𝑛(−𝐴)^𝑛2𝑛!,𝑡≥0. (4.3) By [9, Theorem  3.1], there exist 𝑀≥1 and 𝜔≥0 such that −𝐴∈C^𝛼(𝑀,𝜔^2/𝛼), and the corresponding solution operator is 𝒮_𝛼(𝑡)=^∞₀𝜑_𝑡,𝛼/2(𝑠)𝒮₂(𝑠)𝑑𝑠,𝑡>0, (4.4) where 𝜑_𝑡,𝛼/2(𝑠)=𝑡^−𝛼/2Φ_𝛼/2(𝑠𝑡^−𝛼/2), Φ_𝛼/2(𝜏) is a probability density function, Φ_𝛼/2(𝜏)≥0, 𝜏>0, and ∫^∞₀Φ_𝛼/2(𝜏)𝑑𝜏=1. Thus, 𝒮_𝛼(𝑡) is the positive solution operator generated by −𝐴. In order to solve the problem (4.1), we give the following assumptions. (𝑂₁)𝑥≥𝜃, 𝑓(𝑡,𝜃,𝜃)≥𝜃 for 𝑡∈𝐼. (𝑂₂)There exist 𝑥≤𝑥∈𝑋 such that 𝐴𝑥≥𝑓(𝑡,𝑥,𝐺𝑥) for 𝑡∈𝐼. (𝑂₃)The partial derivative 𝑓^′_𝑢(𝑡,𝑢,𝑣) is continuous on any bounded domain and 𝑓^′_𝑣(𝑡,𝑢,𝑣) has upper bound.

Theorem 4.2. If (𝑂₁), (𝑂₂), and (𝑂₃) are satisfied, then the problem (4.1) has the unique mild solution 𝑢(𝑡), and 𝜃≤𝑢≤𝑥.

Proof. From (𝑂₁) and (𝑂₂), we obtain that 𝜃 is a lower solution of (4.1), and 𝑥 is an upper solution of (4.1). Form (𝑂₃), it is easy to verify that (𝐻₁) and (𝐻₃) are satisfied. Therefore, by Theorem 3.5, the problem (4.1) has the unique solution 𝑢(𝑡), and 𝜃≤𝑢≤𝑥.

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.