Positive Solutions for the Initial Value Problems of Fractional Evolution Equation

Abstract

This paper discusses the existence of positive solutions for the initial value problem of fractional evolution equation with noncompact semigroup 𝐷^𝑞𝑢(𝑡)+𝐴𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)), 𝑡≥0; 𝑢(0)=𝑢₀ in a Banach space 𝑋, where 𝐷^𝑞 denotes the Caputo fractional derivative of order 𝑞∈(0,1), 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a closed linear operator, −𝐴 generates an equicontinuous 𝐶₀ semigroup, and 𝑓∶[0,∞)×𝑋→𝑋 is continuous. In the case where 𝑓 satisfies a weaker measure of noncompactness condition and a weaker boundedness condition, the existence results of positive and saturated mild solutions are obtained. Particularly, an existence result without using measure of noncompactness condition is presented in ordered and weakly sequentially complete Banach spaces. These results are very convenient for application. As an example, we study the partial differential equation of parabolic type of fractional order.

1. Introduction

The theory of fractional differential equations is a new and important branch of differential equation theory, which has an extensive physical background and realistic mathematical model; see [1–6]. Correspondingly, the existence of solutions to fractional evolution equations in Banach space has also been studied by several authors; see [7–17]. In [7, 8], El-Borai first constructed the type of mild solutions to fractional evolution equations in terms of a probability density. And then they investigated the existence, uniqueness, and regularity of solutions to fractional integrodifferential equations in [9, 10]. Recently, this theory was developed by Zhou et al. [11–14]. In [15–17], the authors studied the existence of mild solutions to fractional impulsive evolutions equations. But as far as we know, there are seldom results on the existence of positive solutions to the fractional evolution equations; see [18–20].

In this paper, we use the Sadovskii’s fixed point theorem and monotone iterative technique to discuss the existence of positive and saturated mild solutions for the initial value problem (IVP) of fractional evolution equations: 𝐷^𝑞𝑢(𝑡)(𝑡)((𝑡𝑢(0)+𝐴𝑢=𝑓𝑡,𝑢)),𝑡≥0,=𝑢₀ (1) in Banach space 𝑋, where 𝐷^𝑞 denotes the Caputo fractional derivative of order 𝑞∈(0,1), 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 is a closed linear operator, −𝐴 generates a 𝐶₀-semigroup 𝑆(𝑡) (𝑡≥0) in 𝑋, and 𝑓∶ℝ⁺×𝑋→𝑋 is continuous and will be specified later, ℝ⁺=[0,∞).

In some existing articles, the fractional evolution equations were treated under the hypothesis that (I) −𝐴 generates a compact semigroup or (II) the nonlinearity 𝑓(𝑡,𝑢) is Lipschitz continuous in 𝑢 on a bounded set. For the case (I), the continuity of nonlinearity 𝑓 can guarantee the local existence of solutions. Hence it is convenient to apply to partial differential equations with compact resolvent. But for the case of noncompact semigroup, the condition (II) is not easy to verify sometimes. To make the things more applicable, in this work, we will prove the existence of mild solutions of the IVP(1) under the measure of noncompactness conditions. We will see that our conditions are weaker than the condition (II). In addition, we obtain the existence of positive mild solutions of the IVP(1) in this work, which is studied seldom before.

The rest of this paper is organized as follows. In Section 2, some preliminaries are given on the fractional calculus and the measure of noncompactness. In Section 3, we study the existence of positive and saturated mild solutions of the IVP(1). An example is given in Section 4 to illustrate the applicability of the abstract results obtained in Section 3.

2. Preliminaries

In this section, we introduce some basic facts about the fractional calculus and the measure of noncompactness that are used throughout this paper.

Let 𝑋 be a Banach space with norm ‖⋅‖, let 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be a closed linear operator, and −𝐴 generates a 𝐶₀-semigroup 𝑆(𝑡) (𝑡≥0) in 𝑋. It is well known that there exist 𝑀>0 and 𝛿∈ℝ such that ‖𝑆(𝑡)‖≤𝑀𝑒^𝛿𝑡,𝑡≥0. (2) Let 𝑇>0 be a constant. If 𝑡∈[0,𝑇], it follows from (2) that there exists a constant 𝑀>0 such that ‖𝑆(𝑡)‖≤𝑀.

Let us recall the following known definitions in fractional calculus. For more details, see [7, 8, 11–14, 16, 17] and the reference therein.

Definition 1. The fractional integral of order 𝜎>0 with the lower limits zero for a function 𝑓 is defined by 𝐼^𝜎𝑓(𝑡)=1Γ(𝜎)^𝑡₀(𝑡−𝑠)^𝜎−1𝑓(𝑠)𝑑𝑠,𝑡>0, (3) where Γ is the gamma function.
The Riemann-Liouville fractional derivative of order 𝑛−1<𝜎<𝑛 with the lower limits zero for a function 𝑓 can be written as ^𝐿𝐷^𝜎𝑓(𝑡)=1Γ()𝑑𝑛−𝜎𝑑𝑡^𝑛𝑛𝑡₀(𝑡−𝑠)^{𝑛−𝜎−1}𝑓(𝑠)𝑑𝑠,𝑡>0,𝑛∈ℕ. (4) Also the Caputo fractional derivative of order 𝑛−1<𝜎<𝑛 with the lower limits zero for a function 𝑓∈𝐶^𝑛[0,∞) can be written as 𝐷^𝜎𝑓(𝑡)=1Γ(𝑛−𝜎)^𝑡₀(𝑡−𝑠)^{𝑛−𝜎−1}𝑓^(𝑛)(𝑠)𝑑𝑠,𝑡>0,𝑛∈ℕ. (5)

Remark 2. (1) The Caputo derivative of a constant is equal to zero.
(2) If 𝑓 is an abstract function with values in 𝑋, then integrals which appear in Definition 1 are taken in Bochner’s sense.

Lemma 3 (see [12]). A measurable function ℎ∶[0,𝑇]→𝑋 is Bochner integrable if ‖ℎ‖ is Lebesgue integrable.

For 𝑥∈𝑋, we define two families {𝑈(𝑡)}_𝑡≥0 and {𝑉(𝑡)}_𝑡≥0 of operators by 𝑈(𝑡)𝑥=^∞₀𝜂_𝑞(𝜃)𝑆𝑡^𝑞𝜃𝑉(𝑡)𝑥𝑑𝜃,𝑥=𝑞^∞₀𝜃𝜂_𝑞(𝜃)𝑆𝑡^𝑞𝜃𝑥𝑑𝜃,0<𝑞<1, (6) where 𝜂_𝑞(𝜃)=1𝑞𝜃^{−1−(1/𝑞)}𝜌_𝑞𝜃^−1/𝑞,𝜌_𝑞(𝜃)=1𝜋^∞_𝑛=1(−1)^𝑛−1𝜃^{−𝑞𝑛−1}Γ𝑛𝑞+1,(),𝑛!sin𝑛𝜋𝑞𝜃∈0,∞ (7) where 𝜂_𝑞 is a probability density function defined on (0,∞), which has properties 𝜂_𝑞(𝜃)≥0 for all 𝜃∈(0,∞) and ∫^∞₀𝜂_𝑞(𝜃)𝑑𝜃=1. Clearly, if the semigroup 𝑆(𝑡) (𝑡≥0) is positive, then the operators 𝑈(𝑡) and 𝑉(𝑡) are also positive for all 𝑡≥0.

The following lemma is needed in the proof of the main results.

Lemma 4. The operators 𝑈(𝑡) and 𝑉(𝑡) have the following properties.(i)For any fixed 𝑡≥0 and any 𝑥∈𝑋, one has ‖𝑈(𝑡)𝑥‖‖𝑥‖,‖𝑉(𝑡)𝑥‖≤≤𝑀𝑞𝑀Γ‖𝑥‖=𝑀1+𝑞Γ𝑞‖𝑥‖. (8)(ii)The operators 𝑈(𝑡) and 𝑉(𝑡) are strongly continuous for all 𝑡≥0. (iii)If 𝑆(𝑡) (𝑡≥0) is a equicontinuous semigroup, then 𝑈(𝑡) and 𝑉(𝑡) are equicontinuous in 𝑋 for 𝑡>0.

Proof. (i) and (ii) can be found in [12, 13], and we only need to prove (iii). For any 0≤𝑡₁<𝑡₂≤𝑇, we have 𝑈𝑡₂−𝑈𝑡₁=^∞₀𝜂_𝑞(𝜃)𝑆𝑡^𝑞₂𝜃−𝑆𝑡^𝑞₁𝜃𝑉𝑡𝑑𝜃,₂−𝑉𝑡₁=𝑞^∞₀𝜃𝜂_𝑞(𝜃)𝑆𝑡^𝑞₂𝜃−𝑆𝑡^𝑞₁𝜃𝑑𝜃. (9) According to the equicontinuity of 𝑆(𝑡) for 𝑡>0, we see that ‖𝑈(𝑡₂)−𝑈(𝑡₁)‖ and ‖𝑉(𝑡₂)−𝑉(𝑡₁)‖ tend to zero as 𝑡₂−𝑡₁→0, which means that the operators 𝑈(𝑡) and 𝑉(𝑡) are equicontinuous in 𝑋 for 𝑡>0.

Let 𝑡₀≥0 be a constant and 𝐼=[𝑡₀,𝑇]. We denote by 𝐶(𝐼,𝑋) the Banach space of all continuous 𝑋 value functions on interval 𝐼 with the norm ‖𝑢‖_𝐶=max_𝑡∈𝐼‖𝑢(𝑡)‖. Let 𝛼(𝐵) denote the Kuratowski measure of noncompactness of the bounded set 𝐵 in 𝑋 and 𝐶(𝐼,𝑋). It is clear that 0≤𝛼(𝐵)<∞. If 𝛼(𝐵)=0, then the set 𝐵 is relatively compact. For more details of the definition and properties of the measure of noncompactness; see [21]. For any 𝐵⊂𝐶(𝐼,𝑋) and 𝑡∈𝐼, set 𝐵(𝑡)={𝑢(𝑡)∶𝑢∈𝐵}⊂𝑋. If 𝐵 is bounded in 𝐶(𝐼,𝑋), then 𝐵(𝑡) is bounded in 𝑋, and 𝛼(𝐵(𝑡))≤𝛼(𝐵). A mapping 𝑄∶𝐵→𝐵 is said to be condensing if 𝛼(𝑄(𝐵))<𝛼(𝐵). The following Lemmas will be used in the proof of the main results.

Lemma 5 (see [22]). Let 𝐵∈𝐶(𝐼,𝑋) be bounded and equicontinuous. Then 𝛼(𝐵(𝑡)) is continuous on 𝐼 and 𝛼(𝐵)=max_𝑡∈𝐼𝛼(𝐵(𝑡))=𝛼(𝐵(𝐼)), (10) where 𝐵(𝐼)={𝑥(𝑡)∶𝑥∈𝐵,𝑡∈𝐼}.

Lemma 6 (see [23]). Let 𝐵={𝑢_𝑛}⊂𝐶(𝐼,𝑋) be countable. If there exists 𝜓∈𝐿¹(𝐼) such that ‖𝑢_𝑛(𝑡)‖≤𝜓(𝑡) a.e. 𝑡∈𝐼, 𝑛=1,2,…, then 𝛼(𝐵(𝑡)) is Lebesgue integral on 𝐼 and 𝛼_𝐼𝑢_𝑛(𝑡)𝑑𝑡∶𝑛∈ℕ≤2_𝐼𝛼(𝐵(𝑡))𝑑𝑡. (11)

Lemma 7 (see [24]). Let 𝐵⊂𝐶(𝐼,𝑋) be bounded. Then there exists a countable subset 𝐵₀ of 𝐵 such that 𝛼(𝐵)≤2𝛼(𝐵₀).

Lemma 8 (see [25] (Sadovskii’s fixed point theorem)). Let 𝑋 be a Banach space and let Ω be a nonempty bounded convex closed set in 𝑋. If 𝑄∶Ω→Ω is a condensing mapping, then 𝑄 has a fixed point in Ω.

In the proof of the main results, we also need the following generalized Gronwall-Bellman inequality, which can be found in [26, Page 188].

Lemma 9. Suppose 𝑏≥0, 𝛽>0, and 𝑎(𝑡) is a nonnegative function locally integrable on 0≤𝑡<𝑇 (some 𝑇≤∞), and suppose 𝑢(𝑡) is nonnegative and locally integrable on 0≤𝑡<𝑇 with 𝑢(𝑡)≤𝑎(𝑡)+𝑏^𝑡₀(𝑡−𝑠)^𝛽−1𝑢(𝑠)𝑑𝑠 (12) on this interval, and then 𝑢(𝑡)≤𝑎(𝑡)+^𝑡₀^∞_𝑛=1𝑏Γ𝛽^𝑛Γ()𝑛𝛽𝑡−𝑠^𝑛𝛽−1𝑎(𝑠)𝑑𝑠,0≤𝑡<𝑇. (13)

Remark 10. In Lemma 9, if 𝑎(𝑡)≡0 for all 0≤𝑡<𝑇, we easily see that 𝑢(𝑡)=0.

For any 𝑥₀∈𝑋 and ℎ∈𝐶(𝐼,𝑋), a function 𝑢 is called the mild solution of the initial value problem 𝐷^𝑞𝑢(𝑡)(𝑡)(𝑡)𝑢𝑡+𝐴𝑢=ℎ,𝑡∈𝐼,₀=𝑥₀, (14) if 𝑢∈𝐶(𝐼,𝑋) satisfies the integral equation: 𝑢(𝑡)=𝑈𝑡−𝑡₀𝑥₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)ℎ(𝑠)𝑑𝑠,𝑡∈𝐼. (15) Hence, for the IVP(1), we have the following definition.

Definition 11. By a mild solution of the IVP(1), we mean a function 𝑢∈𝐶(𝐼,𝑋) satisfying 𝑢(𝑡)=𝑈(𝑡)𝑢₀+^𝑡₀(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠 (16) for all 𝑡≥0.

3. Existence of Positive Mild Solutions

In this section, we introduce the existence theorems of positive mild solutions of the IVP(1). The discussions are based on fractional calculus and fixed point theorems.

Let 𝜆₁ be the smallest positive real eigenvalue of the linear operator 𝐴, and let 𝑒₁∈𝐷(𝐴) be the positive eigenvector corresponding to 𝜆₁. Our main results are as follows.

Theorem 12. Let 𝑋 be a Banach space, let 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be a closed linear operator, and −𝐴 generate a positive and equicontinuous 𝐶₀-semigroup 𝑆(𝑡) (𝑡≥0) in 𝑋. Assume that 𝑓∈𝐶(ℝ⁺×𝑋,𝑋) and, for any 𝑇>0, 𝑓 satisfies the following conditions.(𝐻₁) There exist 𝑎,𝑏∈𝐿^1/𝑞1([0,𝑇],ℝ⁺), 𝑞₁∈(0,𝑞) such that 𝑓(𝑡,𝑥)≤𝑎(𝑡)‖𝑥‖+𝑏(𝑡),𝑡∈[0,𝑇],𝑥∈𝑋. (17) (𝐻₂) For any 𝑢∈𝐶([0,𝑇],𝑋) with 𝑢(𝑡)≥𝜎𝑒₁, 𝑡∈[0,𝑇], we have 𝑓(𝑡,𝑢(𝑡))≥𝑓𝑡,𝜎𝑒₁,𝑡∈[0,𝑇], (18) where 𝜎>0 is a constant.(𝐻₃) For any bounded set 𝐷⊂𝑋, there exists a constant 𝐿>0 such that 𝛼𝑓(𝑡,𝐷)≤𝐿𝛼(𝐷),𝑡∈[0,𝑇]. (19)
If 𝑓(𝑡,𝜎𝑒₁)≥𝜆₁𝜎𝑒₁ and 𝑢₀∈𝑋 with 𝑢₀≥𝜎𝑒₁, then the IVP(1) has at least one positive and saturated mild solution 𝑢∈𝐶([0,𝑇),𝑋). And if 𝑇<∞, one has lim_𝑡∈𝑇⁻‖𝑢(𝑡)‖=∞.

Proof. For any 𝑡₀≥0 and 𝑥₀∈𝑋 with 𝑥₀≥𝜎𝑒₁, we first prove that there exists a constant ℎ_𝑡0=ℎ(𝑡₀,‖𝑥₀‖)>0 such that the initial value problem (IVP) 𝐷^𝑞𝑢(𝑡)+𝐴𝑢(𝑡)=𝑓(𝑡,𝑢(𝑡)),𝑡>𝑡₀,𝑢𝑡₀=𝑥₀ (20) has at least one positive mild solution on 𝐽=[𝑡₀,𝑡₀+ℎ_𝑡0]. For this purpose, we define an operator 𝑄 by (𝑄𝑢)(𝑡)=𝑈𝑡−𝑡₀𝑥₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠))𝑑𝑠,𝑡≥𝑡₀. (21) Then 𝑄∶𝐶(𝐽,𝑋)→𝐶(𝐽,𝑋) is continuous, and the mild solutions of the IVP(20) are equivalent to the fixed point of the operator 𝑄.
Let 𝑅_𝑡0∶=2𝑀(‖𝑥₀‖+1)+𝜎𝑒₁>0. Denote Ω_𝑅𝑡0∶=𝑢∈𝐶(𝐽,𝑋)∶‖𝑢(𝑡)‖≤𝑅_𝑡0,𝑢(𝑡)≥𝜎𝑒₁,𝑡∈𝐽. (22) Then Ω_𝑅𝑡0⊂𝐶(𝐽,𝑋) is a nonempty bounded convex closed set. Let ℎ_𝑡0=ℎ(𝑡₀,‖𝑥₀‖)=min{1,(Γ(𝑞)(1+𝑐)^1−𝑞1(‖𝑥₀‖+1)/(𝑅_𝑡0𝑎₀+𝑏₀))^{1/(𝑞−𝑞1)},(Γ(𝑞+1)/(4𝑀𝐿+1))^1/𝑞}, where 𝑐=((𝑞−1)/(1−𝑞₁))∈(−1,0), 𝑎₀=‖𝑎‖_𝐿1^1/𝑞_{([𝑡0,𝑡0+1],ℝ}⁺₎, 𝑏₀=‖𝑏‖_𝐿1^1/𝑞_{([𝑡0,𝑡0+1],ℝ}⁺₎. Then for any 𝑢∈Ω_𝑅𝑡0 and 𝑡∈𝐽, by Lemma 4(i), (𝐻₁), and (21), we have 𝑡≤𝑈‖(𝑄𝑢)()‖𝑡−𝑡₀𝑥₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉()𝑓((𝑠𝑥𝑡−𝑠𝑠,𝑢))𝑑𝑠≤𝑀₀+𝑀Γ𝑞^𝑡_𝑡0(𝑡−𝑠)^𝑞−1[𝑎(𝑠𝑢(𝑠(𝑠𝑥)‖)‖+𝑏)]𝑑𝑠=𝑀₀+𝑀Γ𝑞𝑅_𝑡0^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑎(𝑠)+𝑑𝑠_𝑡𝑡0(𝑡−𝑠)^𝑞−1𝑏(𝑠)𝑥𝑑𝑠≤𝑀₀+𝑀𝑅_𝑡0𝑎₀+𝑏₀Γ𝑞(1+𝑐)^1−𝑞1⋅ℎ^𝑞−𝑞1_𝑡0≤𝑅_𝑡0. (23)
Let 𝑣₀≡𝜎𝑒₁. Then 𝑣₀(𝑡)=𝜎𝑒₁ for any 𝑡∈𝐽 and 𝜙(𝑡)≜𝐷^𝑞𝑣₀(𝑡)+𝐴𝑣₀(𝑡)=𝜆₁𝜎𝑒₁≤𝑓𝑡,𝜎𝑒₁,𝑡∈𝐽. (24) By the positivity of semigroup 𝑆(𝑡) (𝑡≥0), the assumption (𝐻₂) and (21), for any 𝑢∈Ω_𝑅𝑡0 and 𝑡∈𝐽, we have 𝜎𝑒₁=𝑣₀(𝑡)=𝑈𝑡−𝑡₀𝑣₀𝑡₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝜙(𝑠)𝑑𝑠≤𝑈𝑡−𝑡₀𝜎𝑒₁+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)𝑓𝑠,𝜎𝑒₁𝑑𝑠≤𝑈𝑡−𝑡₀𝑥₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉()𝑓((𝑠=(𝑡).𝑡−𝑠𝑠,𝑢))𝑑𝑠𝑄𝑢)( (25) Thus, 𝑄∶Ω_𝑅𝑡0→Ω_𝑅𝑡0 is continuous and it implies that (𝑄𝜎𝑒₁)(𝑡)≤(𝑄𝑢)(𝑡) for any 𝑢∈Ω_𝑅𝑡0 and 𝑡∈𝐽.
Now, we prove that the set 𝑄(Ω_𝑅𝑡0)∶={𝑄𝑢∶𝑢∈Ω_𝑅𝑡0} is equicontinuous in 𝐶(𝐽,𝑋). For any 𝑢∈Ω_𝑅𝑡0 and 𝑡₀≤𝑡₁<𝑡₂≤𝑡₀+ℎ_𝑡0, it follows from assumption (𝐻₁) and (21) that (𝑄𝑢)𝑡₂−(𝑄𝑢)𝑡₁≤𝑈𝑡₂−𝑡₀𝑥₀−𝑈𝑡₁−𝑡₀𝑥₀+^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1×𝑉𝑡₂𝑓((𝑠+−𝑠𝑠,𝑢))𝑑𝑠_𝑡1𝑡0𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1⋅𝑉𝑡₂𝑓((𝑠+−𝑠𝑠,𝑢))𝑑𝑠_𝑡1𝑡0𝑡₁−𝑠^𝑞−1×𝑉𝑡₂−𝑠−𝑉𝑡₁((𝑠𝑈𝑡−𝑠×𝑓𝑠,𝑢))𝑑𝑠≤𝑀₂−𝑡₁𝑥₀−𝑥₀+𝑀Γ𝑞^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1×[𝑎(𝑠𝑢(𝑠(𝑠+𝑀)‖)‖+𝑏)]𝑑𝑠Γ𝑞^𝑡1_𝑡0𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1⋅[𝑎(𝑠𝑢(𝑠(𝑠+)‖)‖+𝑏)]𝑑𝑠_𝑡1𝑡0𝑡₁−𝑠^𝑞−1×𝑉𝑡₂−𝑠−𝑉𝑡₁((𝑠−𝑠×𝑓𝑠,𝑢))𝑑𝑠≜𝐼₁+𝐼₂+𝐼₃+𝐼₄. (26) By Lemma 4(ii), it is easy to see that 𝐼₁→0 independently of 𝑢∈Ω_𝑅𝑡0 as 𝑡₂−𝑡₁→0: 𝐼₂≤𝑀Γ𝑞𝑅_𝑡0^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝑎(𝑠)+𝑑𝑠_𝑡2𝑡1𝑡₂−𝑠^𝑞−1𝑏(𝑠)≤𝑀𝑅𝑑𝑠_𝑡0𝑎₀+𝑏₀Γ𝑞(1+𝑐)^1−𝑞1𝑡₂−𝑡₁^𝑞−𝑞1. (27)
Hence 𝐼₂→0 independently of 𝑢∈Ω_𝑅𝑡0 as 𝑡₂−𝑡₁→0: 𝐼₃≤𝑀Γ𝑞𝑅_𝑡0^𝑡1_𝑡0𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1(𝑠)+×𝑎𝑑𝑠_𝑡1𝑡0𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1𝑏(𝑠)≤𝑀𝑑𝑠Γ𝑞𝑅_𝑡0^𝑡1_𝑡0𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^{𝑞−11/(1−𝑞1)}𝑑𝑠^1−𝑞1⋅‖𝑎‖_𝐿1^1/𝑞_{([𝑡0⋅𝑡1],ℝ}⁺₎+^𝑡1_𝑡0𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^{𝑞−11/(1−𝑞1)}𝑑𝑠^1−𝑞1⋅‖𝑏‖_𝐿1^1/𝑞_{([𝑡0⋅𝑡1],ℝ}⁺₎≤𝑀𝑅_𝑡0𝑎₀+𝑏₀Γ𝑞×^𝑡1_𝑡0𝑡₁−𝑠^𝑐−𝑡₂−𝑠^𝑐𝑑𝑠^1−𝑞1=𝑀𝑅_𝑡0𝑎₀+𝑏₀Γ𝑞(1+𝑐)^1−𝑞1×𝑡₁−𝑡₀^1+𝑐−𝑡₂−𝑡₀^1+𝑐+𝑡₂−𝑡₁^{1+𝑐1−𝑞1}≤𝑀𝑅_𝑡0𝑎₀+𝑏₀Γ𝑞(1+𝑐)^1−𝑞1𝑡₂−𝑡₁^𝑞−𝑞1. (28) It follows that 𝐼₃→0 independently of 𝑢∈Ω_𝑅𝑡0 as 𝑡₂−𝑡₁→0. For 𝑡₁=𝑡₀, 𝑡₀<𝑡₂≤𝑡₀+ℎ_𝑡0, it is easy to see that 𝐼₄=0. Let 𝑡₁>𝑡₀ and 𝜖∈(0,𝑡₁−𝑡₀) be small enough, and we have 𝐼₄≤^𝑡1_𝑡^−𝜖₀𝑡₁−𝑠^𝑞−1𝑉𝑡₂−𝑠−𝑉𝑡₁((𝑠+−𝑠×𝑓𝑠,𝑢))𝑑𝑠_{𝑡1𝑡1−𝜖}𝑡₁−𝑠^𝑞−1𝑉𝑡₂−𝑠−𝑉𝑡₁((𝑠≤−𝑠×𝑓𝑠,𝑢))𝑑𝑠_{𝑡1𝑡−𝜖0}𝑡₁−𝑠^𝑞−1𝑉𝑡₂−𝑠−𝑉𝑡₁⋅𝑓((𝑠+−𝑠𝑠,𝑢))𝑑𝑠2𝑀Γ𝑞^𝑡1_𝑡1−𝜖𝑡₁−𝑠^𝑞−1𝑓((𝑠≤𝑠,𝑢))𝑑𝑠_{𝑡1𝑡−𝜖0}𝑡₁−𝑠^𝑞−1[𝑎(𝑠)‖𝑢(𝑠)‖+𝑏(𝑠)]𝑑𝑠⋅sup_{𝑠∈[𝑡0,𝑡1−𝜖]}𝑉𝑡₂−𝑠−𝑉𝑡₁+−𝑠2𝑀Γ𝑞^𝑡1_𝑡1−𝜖𝑡₁−𝑠^𝑞−1×[𝑎(𝑠𝑢(𝑠(𝑠≤𝑅)‖)‖+𝑏)]𝑑𝑠_𝑡0𝑎₀+𝑏₀(1+𝑐)^1−𝑞1𝑡₁−𝑡₀^1+𝑐−𝜖^{1+𝑐1−𝑞1}⋅sup_{𝑠∈[𝑡0,𝑡1−𝜖]}𝑉𝑡₂−𝑠−𝑉𝑡₁+𝑅−𝑠2𝑀_𝑡0𝑎₀+𝑏₀Γ𝑞(1+𝑐)^1−𝑞1𝜖^𝑞−𝑞1. (29) Since Lemma 4(iii) implies the continuity of 𝑉(𝑡) for 𝑡>0 in the uniform operator topology, it is easy to see that 𝐼₄→0 independently of 𝑢∈Ω_𝑅𝑡0 as 𝑡₂−𝑡₁→0 and 𝜖→0. Thus, ‖(𝑄𝑢)(𝑡₂)−(𝑄𝑢)(𝑡₁)‖→0 independently of 𝑢∈Ω_𝑅𝑡0 as 𝑡₂−𝑡₁→0, which means that the set 𝑄(Ω_𝑅𝑡0) is equicontinuous.
It remains to prove that 𝑄∶Ω_𝑅𝑡0→Ω_𝑅𝑡0 is a condensing mapping. Let 𝐵⊂Ω_𝑅𝑡0 be a bounded set. By Lemma 7, there exists 𝐵₀={𝑢_𝑛}⊂𝐵 such that 𝛼(𝑄(𝐵))≤2𝛼(𝑄(𝐵₀)). Since 𝑄(𝐵₀)⊂𝑄(Ω_𝑅𝑡0)⊂Ω_𝑅𝑡0 is bounded and equicontinuous, by Lemma 5, it follows that 𝛼(𝑄(𝐵₀))=max_𝑡∈𝐽𝛼(𝑄(𝐵₀)(𝑡)). Thus, for any 𝑡∈𝐽, by (21), one has 𝛼𝑄𝐵₀(𝑡)=𝛼𝑈𝑡−𝑡₀𝑥₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)×𝑓𝑠,𝑢_𝑛(𝑠)𝑑𝑠∶𝑛∈ℕ=𝛼^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)×𝑓𝑠,𝑢_𝑛(𝑠)𝑑𝑠∶𝑛∈ℕ≤2^𝑡_𝑡0(𝑡−𝑠)^𝑞−1‖𝑉(𝑓𝑡−𝑠)‖⋅𝛼𝑠,𝐵₀(𝑠)≤𝑑𝑠2𝑞𝑀𝐿Γ𝑞+1^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝛼𝐵₀(𝑠)=𝑑𝑠2𝑀𝐿ℎ^𝑞_𝑡0Γ𝛼𝐵𝑞+1⁰≤2𝑀𝐿ℎ^𝑞_𝑡0Γ𝛼(𝐵).𝑞+1 (30) Thus, 𝛼(𝑄(𝐵))≤2𝛼(𝑄(𝐵₀))=2max_𝑡∈𝐽𝛼(𝑄(𝐵₀)(𝑡))≤(4𝑀𝐿ℎ^𝑞_𝑡0/Γ(𝑞+1))𝛼(𝐵), which means that 𝑄∶Ω_𝑅𝑡0→Ω_𝑅𝑡0 is a condensing mapping. By Lemma 8, the operator 𝑄 has at least one fixed point 𝑢^∗ in Ω_𝑅𝑡0, and 𝑢^∗(𝑡)≥𝜎𝑒₁>0 for all 𝑡∈𝐽. Hence 𝑢^∗∈𝐶(𝐽,𝑋) is a positive mild solution of the IVP(20).
Hence, for the IVP(1), there exists an interval [0,ℎ₀] such that the IVP(1) has a positive mild solution 𝑢 on [0,ℎ₀]. Now, by the extension theorem of initial value problem, 𝑢 can be extended to a saturated solution 𝑢∈𝐶([0,𝑇),𝑋) of the IVP(1), whose existence interval is [0,𝑇), and if 𝑇<∞, one has lim_𝑡→𝑇⁻‖𝑢(𝑡)‖=∞.

For any 𝑇>0 and 𝑟>0, define a set Ω_𝑟 by Ω_𝑟=])∶‖𝑢(𝑡(𝑡)𝑢∈𝐶([0,𝑇,𝑋)‖≤𝑟,𝑢≥𝜎𝑒₁,𝑡∈[0,𝑇]. (31) If 𝑓(𝑡,𝑢) is increasing in Ω_𝑟, that is, 𝑓(𝑡,𝑢) satisfies the condition:(𝐻₄) for any 𝑢₁,𝑢₂∈Ω_𝑟 with 𝑢₁(𝑡)≤𝑢₂(𝑡),𝑡∈[0,𝑇], we have 𝑓𝑡,𝑢₁(𝑡)≤𝑓𝑡,𝑢₂(𝑡),𝑡∈[0,𝑇], (32)then we have 𝑓(𝑡,𝑢(𝑡))≥𝑓(𝑡,𝜎𝑒₁) for any 𝑢∈Ω_𝑟 and 𝑡∈[0,𝑇]. Hence by Theorem 12, we have the following existence result.

Corollary 13. Let 𝑋 be a Banach space, let 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be a closed linear operator, and −𝐴 generates a positive and equicontinuous 𝐶₀-semigroup 𝑆(𝑡) (𝑡≥0) in 𝑋. Assume that 𝑓∈𝐶(ℝ⁺×𝑋,𝑋) and, for any 𝑇>0, 𝑓 satisfies the conditions (𝐻₁), (𝐻₃), and (𝐻₄). If 𝑓(𝑡,𝜎𝑒₁)≥𝜆₁𝜎𝑒₁ and 𝑢₀∈𝑋 with 𝑢₀≥𝜎𝑒₁, then the IVP(1) has at least one positive and saturated mild solution 𝑢∈𝐶([0,𝑇),𝑋). And if 𝑇<∞, one has lim_𝑡∈𝑇⁻‖𝑢(𝑡)‖=∞.

Noticing that the condition (𝐻₃) is not easy to verify in applications, we can weaken or delete the condition (𝐻₃) in ordered Banach space.

Theorem 14. Let 𝑋 be an ordered Banach space, whose positive cone 𝐾 is normal, let 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be a closed linear operator, and −𝐴 generates a positive and equicontinuous 𝐶₀-semigroup 𝑆(𝑡) (𝑡≥0) in 𝑋. Assume that 𝑓∈𝐶(ℝ⁺×𝐾,𝑋) and for any 𝑇>0, 𝑓 satisfies the conditions (𝐻₁), (𝐻₄), and(𝐻₅) there exists a constant 𝐿₁>0 such that 𝛼𝑓(𝑡,𝐷(𝑡))≤𝐿₁𝛼(𝐷(𝑡)),𝑡∈[0,𝑇] (33)for any increasing sequence 𝐷={𝑥_𝑛}⊂Ω_𝑟.
If 𝑓(𝑡,𝜎𝑒₁)≥𝜆₁𝜎𝑒₁ and 𝑢₀∈𝑋 with 𝑢₀≥𝜎𝑒₁, then the IVP(1) has at least one positive and saturated mild solution 𝑢∈𝐶([0,𝑇),𝐾). And if 𝑇<∞, one has lim_𝑡∈𝑇⁻‖𝑢(𝑡)‖=∞.

Proof. For any 𝑡₀≥0 and 𝑥₀∈𝑋 with 𝑥₀≥𝜎𝑒₁, we first prove that the IVP(20) has at least one positive mild solution on 𝐽=[𝑡₀,𝑡₀+ℎ_𝑡0], where ℎ_𝑡0=min{1,(Γ(𝑞)(1+𝑐)^1−𝑞1(‖𝑥₀‖+1)/(𝑅_𝑡0𝑎₀+𝑏₀))^{1/(𝑞−𝑞1)}}. Define an operator 𝑄 as in (21). Let 𝑅_𝑡0=2𝑀(‖𝑋₀‖+1)+𝜎𝑒₁. Write Ω_𝑅𝑡0 as in (22). A similar argument as in the proof of Theorem 12 shows that 𝑄∶Ω_𝑅𝑡0→Ω_𝑅𝑡0 is continuous and the set 𝑄(Ω_𝑅𝑡0) is equicontinuous. From the assumption (𝐻₄), it is easy to see that 𝑄∶Ω_𝑅𝑡0→Ω_𝑅𝑡0 is an increasing operator.
Let 𝑣₀≡𝜎𝑒₁∈Ω_𝑅𝑡0. Define a sequence {𝑣_𝑛} by the iterative scheme 𝑣_𝑛=𝑄𝑣_𝑛−1,𝑛=1,2,…. (34) Since 𝑣₀=𝜎𝑒₁≤𝑄(𝜎𝑒₁)=𝑄(𝑣₀)=𝑣₁, by the increasing property of the operator 𝑄, we have 𝑣₀≤𝑣₁≤𝑣₂≤⋯≤𝑣_𝑛≤⋯. (35) By the equicontinuity property of the set 𝑄(Ω_𝑅𝑡0), the set {𝑣_𝑛}={𝑄𝑣_𝑛−1}⊂𝑄(Ω_𝑅𝑡0) is equicontinuous. Next, we prove that the set {𝑣_𝑛} is uniformly convergent on 𝐽.
For convenience, let 𝐵={𝑣_𝑛∶𝑛∈ℕ} and 𝐵₀={𝑣_𝑛−1∶𝑛∈ℕ}. From 𝐵₀=𝐵∪{𝑣₀}, it follows that 𝛼(𝐵₀(𝑡))=𝛼(𝐵(𝑡)) for any 𝑡∈𝐽. Let 𝜑(𝑡)∶=𝛼(𝐵(𝑡))=𝛼(𝐵₀(𝑡)). By Lemma 6, assumption (𝐻₅), and (21), we have 𝜑(𝑡)(𝐵(𝑡𝑈=𝛼))=𝛼𝑡−𝑡₀𝑥₀+^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)×𝑓𝑠,𝑣_𝑛−1(𝑠)𝑑𝑠∶𝑛∈ℕ=𝛼^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝑉(𝑡−𝑠)×𝑓𝑠,𝑣_𝑛−1(𝑠)𝑑𝑠∶𝑛∈ℕ≤2^𝑡_𝑡0(𝑡−𝑠)^𝑞−1‖𝑉(𝑓𝑡−𝑠)‖⋅𝛼𝑠,𝐵₀(𝑠)≤𝑑𝑠2𝑀𝐿₁Γ𝑞^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝛼𝐵₀(𝑠)=𝑑𝑠2𝑀𝐿₁Γ𝑞^𝑡_𝑡0(𝑡−𝑠)^𝑞−1𝜑(𝑠)𝑑𝑠. (36) Hence by Lemma 9, 𝛼(𝐵(𝑡))=𝜑(𝑡)≡0 for any 𝑡∈𝐽. By Lemma 5, 𝛼(𝐵)=max_𝑡∈𝐽𝛼(𝐵(𝑡))=0, from which we obtain that the set {𝑣_𝑛} is relatively compact. Thus, there is a subset {𝑣_𝑛𝑘}⊂{𝑣_𝑛} such that 𝑣_𝑛𝑘→𝑢^∗∈Ω_𝑅𝑡0. Combining this with the monotonicity (35), we easily prove that {𝑣_𝑛} itself is convergent in Ω_𝑅𝑡0, that is, 𝑣_𝑛→𝑢^∗∈Ω_𝑅𝑡0 as 𝑛→∞.
Letting 𝑛→∞ in (34), by the continuity of the operator 𝑄, we have 𝑢^∗=𝑄𝑢^∗ and 𝑢^∗(𝑡)≥𝜎𝑒₁>0 for all 𝑡∈𝐽. Hence 𝑢^∗∈𝐶(𝐽,𝑋) is a positive mild solution of the IVP(20).
Hence, for the IVP(1), there exists an interval [0,ℎ₀] such that the IVP(1) has a positive mild solution 𝑢 on [0,ℎ₀]. By the extension theorem of the initial value problem, 𝑢 can be extended to a saturated solution 𝑢∈𝐶([0,𝑇),𝑋) of the IVP(1), whose existence interval is [0,𝑇), and if 𝑡<∞, then lim_𝑡→𝑇⁻‖𝑢(𝑡)‖=∞.

In Theorem 14, if 𝑋 is weakly sequentially complete, the condition (𝐻₅) holds automatically. In fact, by [27, Theorem 2.2], any monotonic and order-bounded sequence is precompact. Let 𝐷={𝑥_𝑛}⊂Ω_𝑟 be an increasing sequence. Then by the conditions (𝐻₁) and (𝐻₄), {𝑓(𝑡,𝑥_𝑛)} is a monotonic increasing and order-bounded sequence. By the property of the measure of noncompactness, we have 𝛼𝑓𝑡,𝑥_𝑛=0. (37) Thus, the condition (𝐻₅) holds. From Theorem 14, we have the following.

Corollary 15. Let 𝑋 be an ordered and weakly sequentially complete Banach space, whose positive cone 𝐾 is normal, let 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 be a closed linear operator, and −𝐴 generates a positive and equicontinuous 𝐶₀-semigroup 𝑆(𝑡) (𝑡≥0) in 𝑋. Assume that 𝑓∈𝐶(ℝ⁺×𝐾,𝑋) and, for any 𝑇>0, 𝑓 satisfies the conditions (𝐻₁) and (𝐻₄). If 𝑓(𝑡,𝜎𝑒₁)≥𝜆₁𝜎𝑒₁ and 𝑢₀∈𝑋 with 𝑢₀≥𝜎𝑒₁, then the IVP(1) has at least one positive and saturated mild solution 𝑢∈𝐶([0,𝑇),𝐾). And if 𝑇<∞, one has lim_𝑡∈𝑇⁻‖𝑢(𝑡)‖=∞.

4. Positive Mild Solutions of Parabolic Equations

Let Ω⊂ℝ^𝑁 be a bounded domain with a sufficiently smooth boundary 𝜕Ω, 𝐹∶Ω×ℝ⁺×ℝ→ℝ. We consider the following problem of parabolic type: 𝜕^𝑞𝜕𝑡^𝑞𝑢(𝑥,𝑡)+Δ𝑢(𝑥,𝑡)=𝐹(𝑥,𝑡,𝑢(𝑥,𝑡))inΩ×ℝ⁺,𝑢|_𝜕Ω𝑢()(𝑥)=0,𝑥,0=𝜑inΩ, (38) where 0<𝑞<1 is a constant, and Δ is the Laplace operator. Let 𝑋∶=𝐿²(Ω). Then 𝑋 is an ordered Banach space with the norm ‖𝑓‖₂=(∫_Ω|𝑓(𝑥)|²𝑑𝑥)^1/2 for any 𝑓∈𝑋 and the partial order “≤”. 𝐾∶={𝑢∈𝑋∶𝑢(𝑥)≥0a.e.𝑥∈Ω} is the positive cone in 𝑋. Consider the operator 𝐴∶𝐷(𝐴)⊂𝑋→𝑋 defined by 𝐷(𝐴)=𝐻²(Ω)∩𝐻¹₀(Ω),𝐴𝑢=−Δ𝑢. (39) Then −𝐴 generates a positive and analytic semigroup 𝑆(𝑡) (𝑡≥0) in 𝑋 (see [28, 29]). Let 𝜆₁ be the smallest positive real eigenvalue of operator 𝐴 under the Dirichlet boundary condition 𝑢|_𝜕Ω=0 and let 𝑒₁(𝑥) be the positive eigenvector corresponding to 𝜆₁. Then 𝜆₁>0 and 𝑒₁(𝑥)>0 for 𝑥∈Ω. For any 𝑇>0 and 𝑟>0, denote by 𝑃_𝑟=𝑢∈𝐶[0,𝑇],𝐿²(Ω)∶‖𝑢(𝑥,𝑡)‖₂≤𝑟,𝑢(𝑥,𝑡)≥𝜎𝑒₁(𝑥),𝑡∈[0,𝑇], (40) where 𝜎>0 is a constant. Assume that 𝐹∶Ω×ℝ⁺×ℝ→ℝ is continuous with 𝐹(𝑥,𝑡,𝜎𝑒₁(𝑥))≥𝜆₁𝜎𝑒₁(𝑥), 𝑥∈Ω, 𝑡∈ℝ⁺ and satisfies the following conditions.(𝐹₁) There exist 𝑎,𝑏∈𝐿^1/𝑞1([0,𝑇],ℝ⁺), 𝑞₁∈(0,𝑞) such that |𝐹(𝑥,𝑡,𝑢(𝑥,𝑡))|≤𝑎(𝑡)|𝑢(𝑥,𝑡)|+𝑏(𝑡),𝑥∈Ω,𝑡∈[0,𝑇]. (41)(𝐹₂) For any 𝑢₁,𝑢₂∈𝑃_𝑟 with 𝑢₁≤𝑢₂, we have 𝐹𝑥,𝑡,𝑢₁(𝑥,𝑡)≤𝐹𝑥,𝑡,𝑢₂(𝑥,𝑡),𝑥∈Ω,𝑡∈[0,𝑇]. (42)

Let 𝑓∶ℝ⁺×𝑋→𝑋 be defined by 𝑓(𝑡,𝑢(𝑡))(⋅)=𝐹(⋅,𝑡,𝑢(⋅,𝑡)). Then 𝑓∈𝐶(ℝ⁺×𝑋,𝑋) with 𝑓(𝑡,𝜎𝑒₁)≥𝜆₁𝜎𝑒₁ for 𝑡∈ℝ⁺ and satisfies the assumptions (𝐻₁) and (𝐻₄). Therefore, by Corollary 15, we have the following existence result for the problem (38).

Theorem 16. Assume that 𝐹∈𝐶(Ω×ℝ⁺×ℝ,ℝ) with 𝐹(𝑥,𝑡,𝜎𝑒₁(𝑥))≥𝜆₁𝜎𝑒₁(𝑥) for 𝑥∈Ω,𝑡∈ℝ⁺ and satisfies the assumptions (𝐹₁) and (𝐹₂). If 𝜑∈𝐶(Ω) with 𝜑(𝑥)≥𝜎𝑒₁(𝑥) for any 𝑥∈Ω, then the problem (38) has at least one positive mild solution 𝑢, satisfies 𝑢(𝑥,𝑡)≥𝜎𝑒₁(𝑥) for any 𝑥∈Ω and 𝑡∈[0,𝑇]. And if 𝑇<+∞, one has lim_𝑡→𝑇⁻|𝑢(𝑡)|=+∞.

Remark 17. In Theorem 16, we do not use the property of compactness of the semigroup 𝑆(𝑡) (𝑡≥0), which is a key assumption in [7–9, 11, 12].