Nonlocal Problem for Fractional Evolution Equations of Mixed Type with the Measure of Noncompactness

Abstract

A general class of semilinear fractional evolution equations of mixed type with nonlocal conditions on infinite dimensional Banach spaces is concerned. Under more general conditions, the existence of mild solutions and positive mild solutions is obtained by utilizing a new estimation technique of the measure of noncompactness and a new fixed point theorem with respect to convex-power condensing operator.

1. Introduction

In this paper, we use a new estimation technique of the measure of noncompactness and fixed point theorem with respect to convex-power condensing operator to discuss the existence of mild solutions and positive mild solutions for nonlocal problem of fractional evolution equations (NPFEE) of mixed type with noncompact semigroup in Banach space 𝐸: ^𝐶𝐷^𝑞_𝑡𝑢(𝑡)(𝑡)((𝑡)(𝑡)(𝑡𝑢(0)(𝑢),+𝐴𝑢=𝑓𝑡,𝑢,𝐺𝑢,𝑆𝑢)),𝑡∈𝐽,=𝑔 (1) where ^𝐶𝐷^𝑞_𝑡 is the Caputo fractional derivative of order 𝑞; 0<𝑞<1, 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 is a closed linear operator and −𝐴 generates a uniformly bounded 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) in 𝐸, 𝑓∶𝐽×𝐸×𝐸×𝐸→𝐸 is a Carathéodory type function, 𝐽=[0,𝑎], 𝑎>0 is a constant, 𝑔 mapping from some space of functions to be specified later, and 𝐺𝑢(𝑡)=∫^𝑡₀𝐾()𝑢(𝑠)(𝑡)𝑡,𝑠𝑑𝑠,𝑡∈𝐽𝑆𝑢=∫^𝑎₀𝐻(𝑡,𝑠)𝑢(𝑠)𝑑𝑠,𝑡∈𝐽 (2) are integral operators with integral kernels 𝐾∈𝐶(Δ,ℝ), Δ={(𝑡,𝑠)∣0≤𝑠≤𝑡≤𝑎}, and 𝐻∈𝐶(Δ₀,ℝ), Δ₀={(𝑡,𝑠)∣0≤𝑡,𝑠≤𝑎}.

In recent years, fractional calculus has attracted many physicists, mathematicians, and engineers, and notable contributions have been made to both theory and applications of fractional differential equations. It has been found that the differential equations involving fractional derivatives in time are more realistic to describe many phenomena in practical cases than those of integer order in time. For more details about fractional calculus and fractional differential equations we refer to the books by Miller and Ross [1], Podlubny [2], and Kilbas et al. [3] and the papers by Eidelman and Kochubei [4], Lakshmikantham and Vatsala [5], Agarwal et al. [6], Darwish and Ntouyas [7–10], and Darwish et al. [11]. One of the branches of fractional calculus is the theory of fractional evolution equations. Since fractional order semilinear evolution equations are abstract formulations for many problems arising in engineering and physics, fractional evolution equations have attracted increasing attention in recent years; see [12–26] and the references therein.

The study of abstract nonlocal Cauchy problem was initiated by Byszewski and Lakshmikantham [27]. Since it is demonstrated that the nonlocal problems have better effects in applications than the traditional Cauchy problems, differential equations with nonlocal conditions were studied by many authors and some basic results on nonlocal problems have been obtained; see [17–32] and the references therein for more comments and citations. In the past few years, the existence, uniqueness, and some other properties of mild solutions to nonlocal problem of fractional evolution equations (1) with 𝑓(𝑡,𝑢(𝑡),𝐺𝑢(𝑡),𝑆𝑢(𝑡))=𝑓(𝑡,𝑢(𝑡)) have been extensively studied by using Banach contraction mapping principal, Schauder’s fixed point theorem and Krasnoselskii’s fixed point theorem, when 𝑇(𝑡)(𝑡≥0) is a compact semigroup. For more details on the basic theory of nonlocal problem for fractional evolution equations, one can see the papers of Diagana et al. [17], Wang et al. [18], Li et al. [19], Zhou and Jiao [20], Wang et al. [21], Wang et al. [22], Wang et al. [23], Chang et al. [24], Balachandran and Park [25], and Balachandran and Trujillo [26]. However, for the case that the semigroup 𝑇(𝑡)(𝑡≥0) is noncompact, there are very few papers studied nonlocal problem of fractional evolution equations; that only Wang et al. [14] discussed the existence of mild solutions for nonlocal problem of fractional evolution equations under the situation that 𝑇(𝑡)(𝑡≥0) is an analytic semigroup of uniformly bounded linear operators.

It is well known that the famous Sadovskii's fixed point theorem is an important tool to study various differential equations and integral equations on infinite dimensional Banach spaces. Early on, Lakshmikantham and Leela [33] studied the following initial value problem (IVP) of ordinary differential equation in Banach space 𝐸: 𝑢^′(𝑡)((𝑡𝑢(0)=𝑓𝑡,𝑢)),𝑡∈𝐽,=𝑢₀, (3) and they proved that if for any 𝑅>0, 𝑓 is uniformly continuous on 𝐽×𝐵_𝑅 and satisfies the noncompactness measure condition 𝛼𝑓(𝑡,𝐷)≤𝐿𝛼(𝐷),∀𝑡∈𝐽,𝐷⊂𝐵_𝑅, (4) where 𝐵_𝑅={𝑢∈𝐸∶‖𝑢‖≤𝑅}, 𝐿 is a positive constant, 𝛼(⋅) denotes the Kuratowski measure of noncompactness in 𝐸, then IVP (3) has a global solution provided that 𝐿 satisfies the condition 1𝐿<𝑎. (5) In fact, there are a large amount of authors who studied ordinary differential equations in Banach spaces similar to (3) by using Sadovskii's fixed point theorem and hypothesis analogous to (4); they also required that the constants satisfy a strong inequality similar to (5). For more details on this fact, we refer to Guo [34], Liu et al. [35] and Liu et al. [36].

It is easy to see that the inequality (5) is a strong restrictive condition, and it is difficult to be satisfied in applications. In order to remove the strong restriction on the constant 𝐿, Sun and Zhang [37] generalized the definition of condensing operator to convex-power condensing operator. And based on the definition of this new kind of operator, they established a new fixed point theorem with respect to convex-power condensing operator which generalizes the famous Schauder's fixed point theorem and Sadovskii's fixed point theorem. As an application, they investigated the existence of global mild solutions and positive mild solutions for the initial value problem of evolution equations in 𝐸: 𝑢^′(𝑡)(𝑡)((𝑡𝑢(0)+𝐴𝑢=𝑓𝑡,𝑢)),𝑡∈𝐽,=𝑢₀; (6) they assume that −𝐴 generate a equicontinuous 𝐶₀-semigroup; the nonlinear term 𝑓 is uniformly continuous on 𝐽×𝐵_𝑅 and satisfies a suitable noncompactness measure condition similar to (4). Recently, Shi et al. [38] developed the IVP (6) to the case that the nonlinear term is 𝑓(𝑡,𝑢(𝑡),𝐺𝑢(𝑡),𝑆𝑢(𝑡)) and obtained the existence of global mild solutions and positive mild solutions by using the new fixed point theorem with respect to convex-power condensing operator established in [37], but they also require that the nonlinear term 𝑓 is uniformly continuous on 𝐽×𝐵_𝑅×𝐵_𝑅×𝐵_𝑅.

We observed that, in [33–38], the authors all demand that the nonlinear term 𝑓 is uniformly continuous; this is a very strong assumption. As a matter of fact, if 𝑓(𝑡,𝑢) is Lipschitz continuous on 𝐽×𝐵_𝑅 with respect to 𝑢, then the condition (4) is satisfied, but 𝑓 may not be necessarily uniformly continuous on 𝐽×𝐵_𝑅.

Motivated by the above mentioned aspects, in this paper we studied the existence of mild solutions and positive mild solutions for the NPFEE (1) by utilizing a new fixed point theorem with respect to convex-power condensing operator due to Sun and Zhang [37] (see Lemma 8). Furthermore, we deleted the assumption that 𝑓 is uniformly continuous by using a new estimation technique of the measure of noncompactness (see Lemma 7).

2. Preliminaries

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

Let 𝐸 be a real Banach space with the norm ‖⋅‖. We denote by 𝐶(𝐽,𝐸) the Banach space of all continuous 𝐸-value functions on interval 𝐽 with the supnorm ‖𝑢‖_𝐶=sup_𝑡∈𝐽‖𝑢(𝑡)‖, and by 𝐿¹(𝐽,𝐸) the Banach space of all 𝐸-value Bochner integrable functions defined on 𝐽 with the norm ‖𝑢‖₁=∫¹₀‖𝑢(𝑡)‖𝑑𝑡.

Definition 1 (see [3]). The fractional integral of order 𝑞>0 with the lower limit zero for a function 𝑢∈𝐿¹(𝐽,𝐸) is defined as 𝐼^𝑞_𝑡𝑢(𝑡)=1Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1𝑢(𝑠)𝑑𝑠,𝑡>0, (7) where Γ(⋅) is the Euler gamma function.

Definition 2 (see [3]). The Caputo fractional derivative of order 𝑞>0 with the lower limit zero for a function 𝑢 is defined as ^𝐶𝐷^𝑞_𝑡𝑢(𝑡)=1Γ𝑛−𝑞^𝑡₀(𝑡−𝑠)^{𝑛−𝑞−1}𝑢^(𝑛)(𝑠)𝑑𝑠,𝑡>0,0≤𝑛−1<𝑞<𝑛, (8) where the function 𝑢(𝑡) has absolutely continuous derivatives up to order 𝑛−1.

If 𝑢 is an abstract function with values in 𝐸, then the integrals which appear in Definitions 1 and 2 are taken in Bochner's sense.

For 𝑢∈𝐸, define two operators 𝒯_𝑞(𝑡)(𝑡≥0) and 𝒮_𝑞(𝑡)(𝑡≥0) by 𝒯_𝑞(𝑡)𝑢=^∞₀ℎ_𝑞(𝑠)𝑇𝑡^𝑞𝑠𝒮𝑢𝑑𝑠,_𝑞(𝑡)𝑢=𝑞^∞₀𝑠ℎ_𝑞(𝑠)𝑇𝑡^𝑞𝑠𝑢𝑑𝑠, (9) where ℎ_𝑞(𝑠)=1𝜋𝑞^∞_𝑛=1(−𝑠)^𝑛−1Γ𝑛𝑞+1()𝑛!sin𝑛𝜋𝑞,𝑠∈0,∞ (10) is the function of Wright type defined on (0,∞) which satisfies ℎ_𝑞(𝑠)≥0,𝑠∈(0,∞),^∞₀ℎ_𝑞(𝑠)𝑑𝑠=1,_∞0𝑠^𝑣ℎ_𝑞(𝑠)Γ()𝑑𝑠=1+𝑣Γ[].1+𝑞𝑣,𝑣∈0,1 (11)

Let 𝑀=sup_{𝑡∈[0,+∞)}‖𝑇(𝑡)‖_ℒ(𝐸), where ℒ(𝐸) stands for the Banach space of all linear and bounded operators in 𝐸. The following lemma follows from the results in [12, 13, 20].

Lemma 3. The operators 𝒯_𝑞(𝑡)(𝑡≥0) and 𝒮_𝑞(𝑡)(𝑡≥0) have the following properties.(1)For any fixed 𝑡≥0, 𝒯_𝑞(𝑡) and 𝒮_𝑞(𝑡) are linear and bounded operators; that is, for any 𝑢∈𝐸, 𝒯_𝑞(𝑡)𝑢‖𝑢‖,𝒮≤𝑀_𝑞(𝑡)𝑢≤𝑞𝑀Γ‖𝑢‖=𝑀1+𝑞Γ𝑞‖𝑢‖. (12)(2)For every 𝑢∈𝐸, 𝑡→𝒯_𝑞(𝑡)𝑢 and 𝑡→𝒮_𝑞(𝑡)𝑢 are continuous functions from [0,∞) into 𝐸.(3)The operators 𝒯_𝑞(𝑡)(𝑡≥0) and 𝒮_𝑞(𝑡)(𝑡≥0) are strongly continuous, which means that, for all 𝑢∈𝐸 and 0≤𝑡^′<𝑡^′′≤𝑎, one has 𝒯_𝑞𝑡^′′𝑢−𝒯_𝑞𝑡^′𝑢𝒮⟶0,_𝑞𝑡^′′𝑢−𝒮_𝑞𝑡^′𝑢⟶0𝑎𝑠𝑡^′′−𝑡^′⟶0. (13)

Definition 4. A function 𝑢∈𝐶(𝐽,𝐸) is said to be a mild solution of the NPFEE (1) if it satisfies 𝑢(𝑡)=𝒯_𝑞(𝑡)𝑔(𝑢)+^𝑡₀(𝑡−𝑠)^𝑞−1×𝒮_𝑞(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠),𝑆𝑢(𝑠))𝑑𝑠. (14)

Next, we recall some properties of the measure of noncompactness that will be used in the proof of our main results. Since no confusion may occur, we denote by 𝛼(⋅) the Kuratowski measure of noncompactness on both the bounded sets of 𝐸 and 𝐶(𝐽,𝐸). For the details of the definition and properties of the measure of noncompactness, we refer to the monographs [39, 40]. For any 𝐷⊂𝐶(𝐽,𝐸) and 𝑡∈𝐽, set 𝐷(𝑡)={𝑢(𝑡)∣𝑢∈𝐷}⊂𝐸. If 𝐷⊂𝐶(𝐽,𝐸) is bounded, then 𝐷(𝑡) is bounded in 𝐸 and 𝛼(𝐷(𝑡))≤𝛼(𝐷).

Lemma 5 (see [39]). Let 𝐸 be a Banach space; let 𝐷⊂𝐶(𝐽,𝐸) be bounded and equicontinuous. Then 𝛼(𝐷(𝑡)) is continuous on 𝐽, and 𝛼(𝐷)=max_𝑡∈𝐽𝛼(𝐷(𝑡))=𝛼(𝐷(𝐽)). (15)

Lemma 6 (see [41]). Let 𝐸 be a Banach space; let 𝐷={𝑢_𝑛}⊂𝐶(𝐽,𝐸) be a bounded and countable set. Then 𝛼(𝐷(𝑡)) is Lebesgue integral on 𝐽, and 𝛼_𝐽𝑢_𝑛(𝑡)𝑑𝑡∣𝑛∈ℕ≤2_𝐽𝛼(𝐷(𝑡))𝑑𝑡. (16)

Lemma 7 (see [32, 42]). Let 𝐸 be a Banach space; let 𝐷⊂𝐸 be bounded. Then there exists a countable set 𝐷₀⊂𝐷, such that 𝛼(𝐷)≤2𝛼(𝐷₀).

Lemma 8 (fixed point theorem with respect to convex-power condensing operator (see [37])). Let 𝐸 be a Banach space; let 𝐷⊂𝐸 be bounded, closed, and convex. Suppose 𝑄∶𝐷→𝐷 is a continuous operator and 𝑄(𝐷) is bounded. For any 𝑆⊂𝐷 and 𝑥₀∈𝐷, set 𝑄^(1,𝑥0)(𝑆)(𝑆),𝑄≡𝑄_(𝑛,𝑥0)(𝑆)=𝑄co𝑄^{(𝑛−1,𝑥0)}(𝑆),𝑥₀,𝑛=2,3,…. (17) If there exist 𝑥₀∈𝐷 and positive integer 𝑛₀ such that for any bounded and nonprecompact subset 𝑆⊂𝐷, 𝛼𝑄^{(𝑛0,𝑥0)}(𝑆)<𝛼(𝑆), (18) then 𝑄 has at least one fixed point in 𝐷.

Lemma 9. For 𝜎∈(0,1] and 0<𝑎≤𝑏, one has 𝑎^𝜎−𝑏^𝜎≤(𝑏−𝑎)^𝜎. (19)

3. Existence of Mild Solutions

In this section, we discuss the existence of mild solutions for the NPFEE (1). We first make the following hypotheses. (H1) The function 𝑓∶𝐽×𝐸×𝐸×𝐸→𝐸 satisfies the Carathéodory type conditions; that is, 𝑓(⋅,𝑢,𝐺𝑢,𝑆𝑢) is strongly measurable for all 𝑢∈𝐸, and 𝑓(𝑡,⋅,⋅,⋅) is continuous for a.e. 𝑡∈𝐽.(H2) For some 𝑟>0, there exist constants 𝑞₁∈[0,𝑞), 𝜌₁>0 and functions 𝜑_𝑟∈𝐿^1/𝑞1(𝐽,ℝ⁺) such that for a.e. 𝑡∈𝐽 and all 𝑢∈𝐸 satisfying ‖𝑢‖≤𝑟, 𝑓(𝑡,𝑢,𝐺𝑢,𝑆𝑢)≤𝜑_𝑟(𝑡),liminf_𝑟→+∞𝜑_{𝑟𝐿11/𝑞[0,𝑎]}𝑟=𝜌₁<+∞. (20)(H3) There exist positive constants 𝐿_𝑖(𝑖=1,2,3) such that for any bounded and countable sets 𝐷_𝑖⊂𝐸(𝑖=1,2,3) and a.e. 𝑡∈𝐽: 𝛼𝑓𝑡,𝐷₁,𝐷₂,𝐷₃≤³_𝑖=1𝐿_𝑖𝛼𝐷_𝑖. (21)(H4) The nonlocal function 𝑔∶𝐶(𝐽,𝐸)→𝐸 is continuous and compact, and there exist a constant 𝜌₂>0 and a nondecreasing continuous function Φ∶ℝ⁺→ℝ⁺ such that, for some 𝑟>0 and all 𝑢∈Ω_𝑟={𝑢∈𝐶(𝐽,𝐸)∶‖𝑢‖_𝐶≤𝑟}, 𝑔(𝑢)≤Φ(𝑟),liminf_𝑟→+∞Φ(𝑟)𝑟=𝜌₂<+∞. (22)

Theorem 10. Let 𝐸 be a real Banach space; let 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 be a closed linear operator, and −𝐴 generate an equicontinuous 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) of uniformly bounded operators in 𝐸. Assume that the hypotheses (H1), (H2), (H3) and (H4) are satisfied, then the NPFEE (1) have at least one mild solution in 𝐶(𝐽,𝐸) provided that 𝑀𝜌₂+𝑀𝜌₁𝑎^𝑞−𝑞1Γ𝑞1−𝑞₁𝑞−𝑞₁^1−𝑞1<1. (23)

Proof. Consider the operator 𝑄∶𝐶(𝐽,𝐸)→𝐶(𝐽,𝐸) defined by (𝑄𝑢)(𝑡)=𝒯_𝑞(𝑡)𝑔(𝑢)+^𝑡₀(𝑡−𝑠)^𝑞−1×𝒮_𝑞(𝑡−𝑠)𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠),𝑆𝑢(𝑠))𝑑𝑠,𝑡∈𝐽. (24) By direct calculation, we know that 𝑄 is well defined. From Definition 4, it is easy to see that the mild solution of the NPFEE (1) is equivalent to the fixed point of the operator 𝑄. In the following, we will prove 𝑄 has a fixed point by applying the fixed point theorem with respect to convex-power condensing operator.
Firstly, we prove that there exists a positive constant 𝑅, such that 𝑄(Ω_𝑅)⊂Ω_𝑅. If this is not true, then for each 𝑟>0, there would exist 𝑢_𝑟∈Ω_𝑟 and 𝑡_𝑟∈𝐽 such that ‖(𝑄𝑢_𝑟)(𝑡_𝑟)‖>𝑟. It follows from Lemma 3(1), the hypotheses (H2) and (H4) and Hölder inequality that 𝑟<𝑄𝑢_𝑟𝑡_𝑟≤𝑀𝑔𝑢_𝑟+𝑀Γ𝑞^𝑡𝑟₀𝑡_𝑟−𝑠^𝑞−1×𝑓𝑠,𝑢_𝑟(𝑠),𝐺𝑢_𝑟(𝑠),𝑆𝑢_𝑟(𝑠)(𝑟)+𝑀𝑑𝑠≤𝑀ΦΓ𝑞^𝑡𝑟₀𝑡_𝑟−𝑠^{(𝑞−1)/(1−𝑞1)}𝑑𝑠^1−𝑞1×𝜑_{𝑟𝐿11/𝑞[0,𝑡𝑟]}≤𝑀Φ(𝑟)+𝑀𝑎^𝑞−𝑞1Γ𝑞1−𝑞₁𝑞−𝑞₁^1−𝑞1𝜑_{𝑟𝐿11/𝑞[0,𝑎]}. (25) Dividing both sides of (25) by 𝑟 and taking the lower limit as 𝑟→+∞, we get 𝑀𝜌₂+𝑀𝜌₁𝑎^𝑞−𝑞1Γ𝑞1−𝑞₁𝑞−𝑞₁^1−𝑞1≥1, (26) which contradicts (23).
Secondly, we prove that 𝑄 is continuous in Ω_𝑅. To this end, let {𝑢_𝑛}^∞_𝑛=1⊂Ω_𝑅 be a sequence such that lim_𝑛→+∞𝑢_𝑛=𝑢 in Ω_𝑅. By the continuity of 𝑔 and the Carathéodory continuity of 𝑓, we have lim_𝑛→+∞𝑔𝑢_𝑛=𝑔(𝑢),lim_𝑛→+∞𝑓𝑠,𝑢_𝑛(𝑠),𝐺𝑢_𝑛(𝑠),𝑆𝑢_𝑛(𝑠)=𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠),𝑆𝑢(𝑠)),a.e.𝑠∈𝐽. (27) From the hypothesis (H2), we get for each 𝑡∈𝐽(𝑡−𝑠)^𝑞−1𝑓𝑠,𝑢_𝑛(𝑠),𝐺𝑢_𝑛(𝑠),𝑆𝑢_𝑛(𝑠((𝑠)(𝑠)(𝑠())−𝑓𝑠,𝑢,𝐺𝑢,𝑆𝑢))≤2𝑡−𝑠^𝑞−1𝜑_𝑅(𝑠). (28) Using the fact that the function 𝑠→2(𝑡−𝑠)^𝑞−1𝜑_𝑅(𝑠) is Lebesgue integrable for 𝑠∈[0,𝑡],𝑡∈𝐽, by (27) and (28) and the Lebesgue dominated convergence theorem, we get that 𝑄𝑢_𝑛(𝑡)−(𝑡𝑔𝑢𝑄𝑢)()≤𝑀_𝑛(𝑢+𝑀−𝑔)Γ𝑞×^𝑡₀(𝑡−𝑠)^𝑞−1×𝑓𝑠,𝑢_𝑛(𝑠),𝐺𝑢_𝑛(𝑠),𝑆𝑢_𝑛(𝑠)−𝑓(𝑠,𝑢(𝑠),𝐺𝑢(𝑠),𝑆𝑢(𝑠))𝑑𝑠⟶0as𝑛⟶∞. (29) Therefore, we know that (𝑄𝑢_𝑛)−(𝑄𝑢)_𝐶⟶0as𝑛⟶∞, (30) which means that 𝑄 is continuous.
Now, we demonstrate that the operator 𝑄∶Ω_𝑅→Ω_𝑅 is equicontinuous. For any 𝑢∈Ω_𝑅 and 0≤𝑡₁<𝑡₂≤𝑎, we get that (𝑄𝑢)𝑡₂−(𝑄𝑢)𝑡₁=𝒯_𝑞𝑡₂𝑔(𝑢)−𝒯_𝑞𝑡₁𝑔(𝑢)+^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝒮_𝑞𝑡₂((𝑠)(𝑠)(𝑠−𝑠×𝑓𝑠,𝑢,𝐺𝑢,𝑆𝑢))𝑑𝑠+^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1×𝒮_𝑞𝑡₂𝑓((𝑠)(𝑠)(𝑠+−𝑠𝑠,𝑢,𝐺𝑢,𝑆𝑢))𝑑𝑠_𝑡10𝑡₁−𝑠^𝑞−1𝒮_𝑞𝑡₂−𝑠−𝒮_𝑞𝑡₁((𝑠)(𝑠)(𝑠−𝑠×𝑓𝑠,𝑢,𝐺𝑢,𝑆𝑢))𝑑𝑠=𝐼₁+𝐼₂+𝐼₃+𝐼₄, (31) where 𝐼₁=𝒯_𝑞𝑡₂𝑔(𝑢)−𝒯_𝑞𝑡₁𝑔(𝑢),𝐼₂=^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝒮_𝑞𝑡₂((𝑠)(𝑠)(𝑠𝐼−𝑠×𝑓𝑠,𝑢,𝐺𝑢,𝑆𝑢))𝑑𝑠,₃=^𝑡1₀𝑡₂−𝑠^𝑞−1−𝑡₁−𝑠^𝑞−1×𝒮_𝑞𝑡₂𝑓((𝑠)(𝑠)(𝑠𝐼−𝑠𝑠,𝑢,𝐺𝑢,𝑆𝑢))𝑑𝑠,₄=^𝑡1₀𝑡₁−𝑠^𝑞−1𝒮_𝑞𝑡₂−𝑠−𝒮_𝑞𝑡₁((𝑠)(𝑠)(𝑠−𝑠×𝑓𝑠,𝑢,𝐺𝑢,𝑆𝑢))𝑑𝑠. (32) It is obvious that (𝑄𝑢)𝑡₂−(𝑄𝑢)𝑡₁≤⁴_𝑖=1𝐼_𝑖. (33) Therefore, we only need to check ‖𝐼_𝑖‖ tend to 0 independently of 𝑢∈Ω_𝑅 when 𝑡₂−𝑡₁→0,𝑖=1,2,…,4.
For 𝐼₁, by Lemma 3(3), ‖𝐼₁‖→0 as 𝑡₂−𝑡₁→0.
For 𝐼₂, by the hypothesis (H2), Lemma 3(1), and Hölder inequality, we have 𝐼₂≤𝑀Γ𝑞^𝑡2_𝑡1𝑡₂−𝑠^𝑞−1𝜑_𝑅(𝑠)≤𝑀𝑑𝑠Γ𝑞^𝑡2_𝑡1𝑡₂−𝑠^{(𝑞−1)/(1−𝑞1)}𝑑𝑠^1−𝑞1×𝜑_{𝑅𝐿11/𝑞[𝑡1,𝑡2]}≤𝑀𝜑_{𝑅𝐿11/𝑞[0,𝑎]}Γ𝑞1−𝑞₁𝑞−𝑞₁^1−𝑞1𝑡₂−𝑡₁^𝑞−𝑞1⟶0as𝑡₂−𝑡₁⟶0. (34) For 𝐼₃, by the hypothesis (H2), Lemmas 3(1), and 9 and Hölder inequality, we get that 𝐼₃≤𝑀Γ𝑞^𝑡1₀𝑡₁−𝑠^𝑞−1−𝑡₂−𝑠^𝑞−1𝜑_𝑅(𝑠)≤𝑀𝑑𝑠Γ𝑞^𝑡1₀𝑡₁−𝑠^𝑞−1−𝑡₂−𝑠^{𝑞−11/(1−𝑞1)}𝑑𝑠^1−𝑞1×𝜑_{𝑅𝐿11/𝑞[0,𝑡1]}≤𝑀Γ𝑞^𝑡1₀𝑡₁−𝑠^{(𝑞−1)/(1−𝑞1)}−𝑡₂−𝑠^{(𝑞−1)/(1−𝑞1)}𝑑𝑠^1−𝑞1×𝜑_{𝑅𝐿11/𝑞[0,𝑎]}≤𝑀𝜑_{𝑅𝐿11/𝑞[0,𝑎]}Γ𝑞1−𝑞₁𝑞−𝑞₁^1−𝑞1×𝑡^{(𝑞−𝑞1)/(1−𝑞1)}₁−𝑡^{(𝑞−𝑞1)/(1−𝑞1)}₂+𝑡₂−𝑡₁^{(𝑞−𝑞1)/(1−𝑞1)1−𝑞1}≤𝑀2^1−𝑞1𝜑_{𝑅𝐿11/𝑞[0,𝑎]}Γ𝑞1−𝑞₁𝑞−𝑞₁^1−𝑞1𝑡₂−𝑡₁^𝑞−𝑞1⟶0as𝑡₂−𝑡₁⟶0. (35) For 𝑡₁=0, 0<𝑡₂≤𝑎, it is easy to see that ‖𝐼₄‖=0. For 𝑡₁>0 and 𝜖>0 small enough, by the hypothesis (H2), Lemma 3 and the equicontinuity of 𝑇(𝑡), we know that 𝐼₄≤^𝑡1₀^−𝜖𝑡₁−𝑠^𝑞−1×𝒮_𝑞𝑡₂−𝑠−𝒮_𝑞𝑡₁−𝑠𝜑_𝑅(𝑠)+𝑑𝑠_{𝑡1𝑡1−𝜖}𝑡₁−𝑠^𝑞−1×𝒮_𝑞𝑡₂−𝑠−𝒮_𝑞𝑡₁−𝑠𝜑_𝑅(𝑠)𝑑𝑠≤sup_{𝑠∈0,𝑡1−𝜖}𝒮_𝑞𝑡₂−𝑠−𝒮_𝑞𝑡₁×−𝑠_{𝑡10−𝜖}𝑡₁−𝑠^𝑞−1𝜑_𝑅(𝑠)+𝑑𝑠2𝑀Γ𝑞^𝑡1_𝑡1−𝜖𝑡₁−𝑠^𝑞−1𝜑_𝑅(𝑠)𝑑𝑠≤sup_{𝑠∈0,𝑡1−𝜖}𝒮_𝑞𝑡₂−𝑠−𝒮_𝑞𝑡₁⋅𝜑−𝑠_{𝑅𝐿11/𝑞[0,𝑎]}1−𝑞₁𝑞−𝑞₁^1−𝑞1⋅𝑡^{(𝑞−𝑞1)/(1−𝑞1)}₁−𝜖^{(𝑞−𝑞1)/(1−𝑞1)1−𝑞1}+2𝑀𝜑_{𝑅𝐿11/𝑞[0,𝑎]}Γ𝑞1−𝑞₁𝑞−𝑞₁^1−𝑞1𝜖^𝑞−𝑞1⟶0as𝑡₂−𝑡₁⟶0. (36) As a result, ‖(𝑄𝑢)(𝑡₂)−(𝑄𝑢)(𝑡₁)‖ tends to zero independently of 𝑢∈Ω_𝑅 as 𝑡₂−𝑡₁→0, which means that 𝑄∶Ω_𝑅→Ω_𝑅 is equicontinuous.
Let 𝐹=co𝑄(Ω_𝑅), where co means the closure of convex hull. Then it is easy to verify that 𝑄 maps 𝐹 into itself and 𝐹⊂𝐶(𝐽,𝐸) is equicontinuous. Now, we are in the position to prove that 𝑄∶𝐹→𝐹 is a convex-power condensing operator. Take 𝑢₀∈𝐹; we will prove that there exists a positive integer 𝑛₀ such that for any bounded and nonprecompact subset 𝐷⊂𝐹𝛼𝑄^{(𝑛0,𝑢0)}(𝐷)<𝛼(𝐷). (37) For any 𝐷⊂𝐹 and 𝑢₀∈𝐹, by (17) and the equicontinuity of 𝐹, we get that 𝑄^(𝑛,𝑢0)(𝐷)⊂Ω_𝑅 is also equicontinuous. Therefore, we know from Lemma 5 that 𝛼𝑄^(𝑛,𝑢0)(𝐷)=max_𝑡∈𝐽𝛼𝑄^(𝑛,𝑢0)(𝐷)(𝑡),𝑛=1,2,…. (38) By Lemma 7, there exists a countable set 𝐷₁={𝑢¹_𝑛}⊂𝐷, such that 𝛼(𝑄(𝐷)(𝑡))≤2𝛼𝑄𝐷₁(𝑡). (39) Thanks to the fact that ^𝑎₀𝑢(𝑠)𝑑𝑠∈𝑎co{𝑢(𝑠)∣𝑠∈𝐽},𝑢∈𝐶(𝐽,𝐸), (40) we have 𝛼^𝑡₀𝐾(𝑡,𝑠)𝑢(𝑠)𝑑𝑠∣𝑢∈𝐷,𝑡∈𝐽≤𝑎𝐾₀𝛼𝑢(𝑡),𝛼({∣𝑢∈𝐷,𝑡∈𝐽})_𝑎0𝐻(𝑡,𝑠)𝑢(𝑠)𝑑𝑠∣𝑢∈𝐷,𝑡∈𝐽≤𝑎𝐻₀𝛼({𝑢(𝑡)∣𝑢∈𝐷,𝑡∈𝐽}), (41) where 𝐾₀=max_{(𝑡,𝑠)∈Δ}|𝐾(𝑡,𝑠)|, 𝐻₀=max_{(𝑡,𝑠)∈Δ0}|𝐻(𝑡,𝑠)|. Therefore, by (24), (39), and (41), Lemma 6 and the hypotheses (H3) and (H4), we get that 𝛼𝑄^(1,𝑢0)(𝐷𝑡)(𝑄(𝐷𝑡𝑄𝐷)(=𝛼)())≤2𝛼₁(𝑡)≤2𝛼𝒯_𝑞(𝑡)𝑔𝑢¹_𝑛+^𝑡₀(𝑡−𝑠)^𝑞−1𝒮_𝑞(𝑡−𝑠)×𝑓𝑠,𝑢¹_𝑛(𝑠),𝐺𝑢¹_𝑛(𝑠),𝑆𝑢¹_𝑛(𝑠)≤𝑑𝑠4𝑀Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1×𝛼𝑓𝑠,𝑢¹_𝑛(𝑠),𝐺𝑢¹_𝑛(𝑠),𝑆𝑢¹_𝑛(𝑠)≤𝑑𝑠4𝑀Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1×𝐿₁𝛼𝐷₁(𝑠)+𝐿₂𝛼𝐺𝐷₁(𝑠)+𝐿₃𝛼𝑆𝐷₁𝑠)≤(𝑑𝑠4𝑀Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1×𝐿₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝛼𝐷₁(𝑠)≤𝐿𝑑𝑠4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑡^𝑞Γ𝛼(𝐷).1+𝑞 (42)
Again by Lemma 7, there exists a countable set 𝐷₂={𝑢²_𝑛}⊂co{𝑄^(1,𝑢0)(𝐷),𝑢₀}, such that 𝛼𝑄co𝑄^(1,𝑢0)(𝐷),𝑢₀(𝑡)≤2𝛼𝑄𝐷₂(𝑡). (43) Therefore, by (24), (41), and (43), Lemma 6, and the hypotheses (H3) and (H4), we have that 𝛼𝑄^(2,𝑢0)(𝐷𝑡)𝑄)(=𝛼co𝑄^(1,𝑢0)(𝐷),𝑢₀(𝑡)≤2𝛼𝑄𝐷₂(𝑡)≤2𝛼^𝑡₀(𝑡−𝑠)^𝑞−1𝒮_𝑞(𝑡−𝑠)×𝑓𝑠,𝑢²_𝑛(𝑠),𝐺𝑢²_𝑛(𝑠),𝑆𝑢²_𝑛(𝑠)≤𝑑𝑠4𝑀Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1×𝛼𝑓𝑠,𝑢²_𝑛(𝑠),𝐺𝑢²_𝑛(𝑠),𝑆𝑢²_𝑛(𝑠)≤𝑑𝑠4𝑀Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1×𝐿₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝛼𝐷₂(𝑠)≤𝐿𝑑𝑠4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃Γ𝑞×^𝑡₀(𝑡−𝑠)^𝑞−1⋅𝛼co𝑄^(1,𝑢0)(𝐷),𝑢₀𝑠)≤𝐿(𝑑𝑠4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃²Γ𝑞Γ×1+𝑞_𝑡0(𝑡−𝑠)^𝑞−1𝑠^𝑞𝛼(𝐷)=𝐿𝑑𝑠4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑡^𝑞2Γ𝐁×1+2𝑞1+𝑞,𝑞₁₀(1−𝑠)^𝑞−1𝑠^𝑞(𝐷)=𝐿𝑑𝑠𝛼4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑡^𝑞2Γ𝛼(𝐷),1+2𝑞 (44) where 𝐁(𝑝,𝑞)=∫¹₀𝑠^𝑝−1(1−𝑠)^𝑞−1𝑑𝑠 is the Beta function.
Suppose that 𝛼𝑄^(𝑘,𝑢0)(𝐷𝑡)≤𝐿)(4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑡^𝑞𝑘Γ𝛼(𝐷)1+𝑘𝑞,∀𝑡∈𝐽. (45) Then by Lemma 7, there exists a countable set 𝐷_𝑘+1={𝑢_𝑛^𝑘+1}⊂co{𝑄^(𝑘,𝑢0)(𝐷),𝑢₀}, such that 𝛼𝑄co𝑄^(𝑘,𝑢0)(𝐷),𝑢₀(𝑡)≤2𝛼𝑄𝐷_𝑘+1(𝑡). (46) From (24), (41), and (46), using Lemma 6, and the hypotheses (H3) and (H4), we get that 𝛼𝑄^{(𝑘+1,𝑢0)}(𝐷𝑡)𝑄)(=𝛼co𝑄^(𝑘,𝑢0)(𝐷),𝑢₀(𝑡)≤2𝛼𝑄𝐷_𝑘+1(𝑡)≤2𝛼^𝑡₀(𝑡−𝑠)^𝑞−1𝒮_𝑞(𝑡−𝑠)×𝑓𝑠,𝑢_𝑛^𝑘+1(𝑠),𝐺𝑢_𝑛^𝑘+1(𝑠),𝑆𝑢_𝑛^𝑘+1(𝑠)≤𝑑𝑠4𝑀Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1×𝐿₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝛼𝐷_𝑘+1(𝑠)≤𝐿𝑑𝑠4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃Γ𝑞×^𝑡₀(𝑡−𝑠)^𝑞−1𝛼co𝑄^(𝑘,𝑢0)(𝐷),𝑢₀𝑠)≤𝐿(𝑑𝑠4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃^𝑘+1Γ𝑞Γ×1+𝑘𝑞_𝑡0(𝑡−𝑠)^𝑞−1𝑠^𝑘𝑞𝛼(𝐷)=𝐿𝑑𝑠4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑡^𝑞𝑘+1Γ()𝑞𝐁×1+𝑘+11+𝑘𝑞,𝑞₁₀(1−𝑠)^𝑞−1𝑠^𝑘𝑞(𝐷)=𝐿𝑑𝑠𝛼4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑡^𝑞𝑘+1Γ()𝑞𝛼(𝐷).1+𝑘+1 (47) Hence, by the method of mathematical induction, for any positive integer 𝑛 and 𝑡∈𝐽, we have 𝛼𝑄^(𝑛,𝑢0)(𝐷𝑡≤𝐿)()4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑡^𝑞𝑛Γ𝛼(𝐷).1+𝑛𝑞 (48) Consequently, from (38) and (48), we have 𝛼𝑄^(𝑛,𝑢0)(𝐷)=max_𝑡∈𝐽𝛼𝑄^(𝑛,𝑢0)(𝐷𝑡)≤𝐿)(4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑎^𝑞𝑛Γ𝛼(𝐷).1+𝑛𝑞 (49) By the well-known Stirling's formula, we know that Γ1+𝑛𝑞=2𝜋𝑛𝑞𝑛𝑞𝑒^𝑛𝑞𝑒^{𝜈/12𝑛𝑞},0<𝜈<1. (50) Thus, 4𝑀𝐿₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑎^𝑞𝑛Γ=𝐿1+𝑛𝑞4𝑀₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑎^𝑞𝑛2𝜋𝑛𝑞𝑛𝑞/𝑒^𝑛𝑞𝑒^{𝜈/12𝑛𝑞}⟶0as𝑛⟶∞. (51) Therefore, there exists a large enough positive integer 𝑛₀ such that 4𝑀𝐿₁+𝑎𝐾₀𝐿₂+𝑎𝐻₀𝐿₃𝑎^𝑞𝑛0Γ1+𝑛₀𝑞<1. (52) Hence, we get that 𝛼𝑄^{(𝑛0,𝑢0)}(𝐷)<𝛼(𝐷). (53) Thus, 𝑄∶𝐹→𝐹 is a convex-power condensing operator. It follows from Lemma 8 that 𝑄 has at least one fixed point 𝑢∈𝐹, which is just a mild solution of the NPFEE (1). This completes the proof of Theorem 10.

If we replace the hypotheses (H2) and (H4) by the following hypotheses:(H2)^′ there exist a function 𝜑∈𝐿^1/𝑞1(𝐽,ℝ⁺), 𝑞₁∈[0,𝑞) and a nondecreasing continuous function Ψ∶ℝ⁺→ℝ⁺ such that 𝑓(𝑡,𝑢,𝐺𝑢,𝑆𝑢)≤𝜑(𝑡)Ψ(‖𝑢‖), (54) for all 𝑢∈𝐸 and a.e. 𝑡∈𝐽;(H4)^′ the nonlocal function 𝑔∶𝐶(𝐽,𝐸)→𝐸 is continuous and compact; there exist constants 0<𝑏<1/𝑀 and 𝑐>0 such that for all 𝑢∈𝐶(𝐽,𝐸), ‖𝑔(𝑢)‖≤𝑏‖𝑢‖_𝐶+𝑐;

we have the following existence result.

Theorem 11. Let 𝐸 be a real Banach space; let 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 be a closed linear operator and −𝐴 generate an equicontinuous 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) of uniformly bounded operators in 𝐸. Assume that the hypotheses (H1), (H2)′, (H3), and (H4)′ are satisfied, then the NPFEE (1) has at least one mild solution in 𝐶(𝐽,𝐸) provided that there exists a constant 𝑅 with Ψ(𝑅)𝑀𝑐+𝑀𝑎^𝑞−𝑞1Γ𝑞1−𝑞₁𝑞−𝑞₁^1−𝑞1𝜑_{𝐿11/𝑞[0,𝑎]}×(1−𝑀𝑏)⁻¹≤𝑅. (55)

Proof. From the proof of Theorem 10, we know that the mild solution of the NPFEE (1) is equivalent to the fixed point of the operator 𝑄 defined by (24). For any 𝑢∈Ω_𝑅, by (24), (55), and the hypotheses (H2)^′ and (H4)^′, we have 𝑡𝑔(𝑢+𝑀‖(𝑄𝑢)()‖≤𝑀)Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1×𝑓((𝑠)(𝑠)(𝑠𝑏‖𝑢‖𝑠,𝑢,𝐺𝑢,𝑆𝑢))𝑑𝑠≤𝑀_𝐶+Ψ(𝑅)𝑀+𝑐Γ𝑞×^𝑡₀(𝑡−𝑠)^{(𝑞−1)/(1−𝑞1)}𝑑𝑠^1−𝑞1𝜑_{𝐿11/𝑞[0,𝑡]}()+Ψ(𝑅)≤𝑀𝑏𝑅+𝑐𝑀𝑎^𝑞−𝑞1Γ𝑞×1−𝑞₁𝑞−𝑞₁^1−𝑞1𝜑_{𝐿11/𝑞[0,𝑎]}≤𝑅, (56) which implies 𝑄(Ω_𝑅)⊂Ω_𝑅. By adopting completely similar method with the proof of Theorem 10, we can prove that the NPFEE (1) have at least one mild solution in 𝐶(𝐽,𝐸). This completes the proof of Theorem 11.

Similar with Theorem 11, we have the following result.

Corollary 12. Let 𝐸 be a real Banach space; let 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 be a closed linear operator and −𝐴 generate an equicontinuous 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) of uniformly bounded operators in 𝐸. Assume that the hypotheses (H1), (H2)′, (H3) and (H4)′ are satisfied, then the NPFEE (1) have at least one mild solution in 𝐶(𝐽,𝐸) provided that ^𝑎₀𝜑^1/𝑞1(𝑠)𝑑𝑠^𝑞1<liminf_𝑟→+∞(𝑟(1−𝑀𝑏)+𝑀𝑐)Γ𝑞Ψ(𝑟)𝑀𝑎^𝑞−𝑞1×𝑞−𝑞₁1−𝑞₁^1−𝑞1. (57)

4. Existence of Positive Mild Solutions

In this section, we discuss the existence of positive mild solutions for the NPFEE (1). we first introduce some notations and definitions which will be used in this section.

Let 𝐸 be an ordered Banach space with the norm ‖⋅‖ and let 𝑃={𝑢∈𝐸∣𝑢≥𝜃} be a positive cone in 𝐸 which defines a partial ordering in 𝐸 by 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝑃, where 𝜃 denotes the zero element of 𝐸. 𝑃 is said to be normal if there exists a positive constant 𝑁 such that 𝜃≤𝑥≤𝑦 implies ‖𝑥‖≤𝑁‖𝑦‖; the infimum of all 𝑁 with the property above is called the normal constant of 𝑃. For more definitions and details of the cone 𝑃, we refer to the monographs [40, 43].

Definition 13. A 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) in 𝐸 is called to be positive, if order inequality 𝑇(𝑡)𝑢≥𝜃 holds for each 𝑢≥𝜃, 𝑢∈𝐸 and 𝑡≥0.

For more details of the properties of positive 𝐶₀-semigroup, we refer to the monograph [44] and the paper [45].

Lemma 14. If 𝑇(𝑡)(𝑡≥0) is a positive 𝐶₀-semigroup in 𝐸, then 𝒯_𝑞(𝑡)(𝑡≥0) and 𝒮_𝑞(𝑡)(𝑡≥0) are also positive operators.

Proof. Since the semigroup 𝑇(𝑡)(𝑡≥0) and the function ℎ_𝑞(𝑠) defined by (10) are positive, by (9) we can easily conclude that the operators 𝒯_𝑞(𝑡)(𝑡≥0) and 𝒮_𝑞(𝑡)(𝑡≥0) are also positive. This completes the proof.

Here, we will obtain positive mild solutions under the following assumptions.(A1) The function 𝑓∶𝐽×𝑃×𝑃×𝑃→𝑃 satisfies the Carathéodory type conditions.(A2) There exist a constant 𝑞₁∈[0,𝑞) and a function 𝜑∈𝐿^1/𝑞1(𝐽,ℝ⁺) such that, for all 𝑢∈𝑃 and a.e. 𝑡∈𝐽, 𝑓(𝑡,𝑢,𝐺𝑢,𝑆𝑢)≤𝜑(𝑡).(A3) There exist positive constants 𝐿_𝑖(𝑖=1,2,3) such that for any bounded and countable sets 𝐷_𝑖⊂𝐶(𝐽,𝑃)(𝑖=1,2,3) and a.e. 𝑡∈𝐽, 𝛼𝑓𝑡,𝐷₁(𝑡),𝐷₂(𝑡),𝐷₃(𝑡)≤³_𝑖=1𝐿_𝑖𝛼𝐷_𝑖(𝑡). (58)(A4) The nonlocal function 𝑔∶𝐶(𝐽,𝑃)→𝑃 is continuous and compact, and there exist constants 0<b<1/(NM) and c>0 such that for all 𝑢∈𝐶(𝐽,𝑃), 𝑔(𝑢)≤𝑏𝑢+𝑐, where 𝑁 is the normal constant of the positive cone 𝑃.

Theorem 15. Let 𝐸 be an ordered Banach space, 𝑃 be a normal positive cone with normal constant N, A∶D(A)⊂E→E be a closed linear operator, and −A generate a positive and equicontinuous 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) of uniformly bounded operators in 𝐸. Assume that the assumptions (A1), (A2), (A3), and (A4) are satisfied, then the NPFEE (1) have at least one positive mild solution in 𝐶(𝐽,𝑃).

Proof. From the proof of Theorem 10, we know that the positive mild solution of the NPFEE (1) is equivalent to the fixed point of operator 𝑄 defined by (24) in 𝐶(𝐽,𝑃). We choose 𝑅>0 big enough such that 𝑅≥𝑁𝑀𝑎1−𝑁𝑀𝑏𝑐+^𝑞−𝑞1Γ𝑞1−𝑞₁𝑞−𝑞₁^1−𝑞1𝜑_{𝐿11/𝑞[0,𝑎]}. (59) Then for any 𝑢∈Ω_𝑅(𝑃)={𝑢∈𝐶(𝐽,𝑃)∶‖𝑢‖_𝐶≤𝑅}, by (24) and the assumptions (A2) and (A4), we have that 𝜃≤(𝑄𝑢)(𝑡)≤𝒯_𝑞(𝑡)+)(𝑏𝑢+𝑐_𝑡0(𝑡−𝑠)^𝑞−1𝒮_𝑞(𝑡−𝑠)𝜑(𝑠)𝑑𝑠. (60) By the normality of the cone 𝑃, (59), (60), the assumption (A4), and Hölder inequality, we get ‖(𝑄𝑢)(𝑡)‖≤𝑁𝑀𝑏‖𝑢‖_𝐶++𝑐𝑁𝑀Γ𝑞^𝑡₀(𝑡−𝑠)^{(𝑞−1)/(1−𝑞1)}𝑑𝑠^1−𝑞1×𝜑_{𝐿11/𝑞[0,𝑡]}≤𝑁𝑀(𝑏𝑅+𝑐)+𝑁𝑀𝑎^𝑞−𝑞1Γ𝑞1−𝑞₁𝑞−𝑞₁^1−𝑞1×𝜑_{𝐿11/𝑞[0,𝑎]}≤𝑅. (61) Thus, 𝑄∶Ω_𝑅(𝑃)→Ω_𝑅(𝑃). Let 𝐹=co𝑄(Ω_𝑅(𝑃)). Similar to the proof of Theorem 10, we can prove that 𝑄∶𝐹→𝐹 is a convex-power condensing operator. It follows from Lemma 8 that 𝑄 has at least one fixed point 𝑢∈𝐹, which is just a positive mild solution of the NPFEE (1). This completes the proof of Theorem 15.

Theorem 16. Let 𝐸 be an ordered Banach space, 𝑃 be a normal positive cone with normal constant 𝑁, 𝐴∶𝐷(𝐴)⊂𝐸→𝐸 be a closed linear operator and −𝐴 generate a positive and equicontinuous 𝐶₀-semigroup 𝑇(𝑡)(𝑡≥0) of uniformly bounded operators in 𝐸. Assume that the assumptions (A1), (A3) and the following assumptions:(A2)^∗ There exist nonnegative continuous functions 𝑔₁(𝑡), 𝑔₂(𝑡) and abstract continuous function ℎ∶𝐽→𝑃 such that 𝑓(𝑡,𝑢,𝑣,𝑤)≤𝑔₁(𝑡)𝑢+𝑔₂(𝑡)𝑣+ℎ(𝑡),a.e.𝑡∈𝐽,∀𝑢,𝑣,𝑤∈𝑃, (62)(A4)^∗ The nonlocal function 𝑔∶𝐶(𝐽,𝑃)→𝑃 is continuous and compact, and there exists a constant 𝑐>0 such that, for any 𝑢∈𝐶(𝐽,𝑃), 𝑔(𝑢)≤𝑐,are satisfied, then the NPFEE (1) have at least one positive mild solution in 𝐶(𝐽,𝑃).

Proof. From the proof of Theorem 10, we know that the positive mild solution of the NPFEE (1) is equivalent to the fixed point of operator 𝑄 defined by (24) in 𝐶(𝐽,𝑃). Let (𝐵𝑢)(𝑡)=^𝑡₀(𝑡−𝑠)^𝑞−1𝒮_𝑞()×𝑔𝑡−𝑠₁(𝑠)𝑢(𝑠)+𝑔₂(𝑠)𝐺𝑢(𝑠)𝑑𝑠,𝑡∈𝐽. (63) We can prove that 𝑟(𝐵)=0, where 𝑟(⋅) denotes the spectral radius of bounded linear operator. In fact, for any 𝑡∈𝐽, by (63), we get that 𝑡≤𝑀‖(𝐵𝑢)()‖Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1×𝑔₁(𝑠)𝑢(𝑠)+𝑔₂(𝑠)^𝑠₀𝐾()𝑢(𝜏)≤𝑠,𝜏𝑑𝜏𝑑𝑠𝑀𝐺^∗1+𝑎𝐾₀‖𝑢‖_𝐶Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1=𝑑𝑠𝛽𝑡^𝑞‖𝑢‖_𝐶Γ,1+𝑞 (64) where 𝐺^∗=max{max_𝑡∈𝐽𝑔₁(𝑡),max_𝑡∈𝐽𝑔₂(𝑡)}, 𝛽=𝑀𝐺^∗(1+𝑎𝐾₀). Further, 𝐵²𝑢(𝑡≤)𝑀𝐺^∗Γ𝑞^𝑡₀(𝑡−𝑠)^𝑞−1×(𝐵𝑢)(𝑠)+^𝑠₀𝐾(𝜏)≤𝑠,𝜏)(𝐵𝑢)(𝑑𝜏𝑑𝑠𝑀𝐺^∗𝛽‖𝑢‖_𝐶Γ𝑞Γ1+𝑞×^𝑡₀(𝑡−𝑠)^𝑞−1𝑠^𝑞+^𝑠₀𝐾(𝑠,𝜏)𝜏^𝑞≤𝛽𝑑𝜏𝑑𝑠₂‖𝑢‖_𝐶Γ1+2𝑞𝐁1+𝑞,𝑞×^𝑡₀(𝑡−𝑠)^𝑞−1𝑠^𝑞=𝛽𝑑𝑠₂𝑡^2𝑞‖𝑢‖_𝐶Γ.1+2𝑞 (65) By the method of mathematical induction, for any positive integer 𝑛 and 𝑡∈𝐽, we obtain that 𝐵^𝑛𝑢(𝑡≤𝛽)^𝑛𝑡^𝑛𝑞‖𝑢‖_𝐶Γ.1+𝑛𝑞 (66) Therefore, we have 𝐵^𝑛_𝐶≤𝛽^𝑛𝑎^𝑛𝑞Γ.1+𝑛𝑞 (67) Thus, combining (50) we get that 𝑟(𝐵)=lim_𝑛→∞𝐵^𝑛_𝐶^1/𝑛≤lim_𝑛→∞𝛽𝑎^𝑞_2𝑛2𝜋𝑛𝑞𝑛𝑞/𝑒^𝑞𝑒^{𝜈/(12𝑛2𝑞)}=0. (68) Let 0<𝛾<1/𝑁. From [46] we know that there exists an equivalent norm ‖⋅‖^∗ in 𝐸 such that ‖𝐵‖^∗≤𝑟(𝐵)+𝛾=𝛾, (69) where ‖𝐵‖^∗ denotes the operator norm of 𝐵 with respect to the norm ‖⋅‖^∗.
Let 𝑀^∗=sup_{𝑡∈[0,+∞)}‖𝑇(𝑡)‖^∗_ℒ(𝐸) and ‖𝑢‖^∗_𝐶=sup_𝑡∈𝐽‖𝑢(𝑡)‖^∗. Choose 𝑅^∗=𝑁𝑀^∗Γ1+𝑞𝑐+𝑎^𝑞‖ℎ‖^∗_𝐶Γ.1+𝑞1−𝑁𝛾 (70) For any 𝑢∈Ω^∗_𝑅^∗(𝑃)={𝑢∈𝐶(𝐽,𝑃)∶‖𝑢‖^∗_𝐶≤𝑅^∗}, by (24) and the assumptions (A2)^∗ and (A4)^∗, we have that 𝜃≤(𝑄𝑢)(𝑡)≤𝒯_𝑞(𝑡)𝑔(𝑢)+^𝑡₀(𝑡−𝑠)^𝑞−1𝒮_𝑞()×𝑔𝑡−𝑠₁(𝑠)𝑢(𝑠)+𝑔₂(𝑠)𝐺𝑢(𝑠)+ℎ(𝑠)𝑑𝑠≤𝒯_𝑞(𝑡)(𝑢𝑡)+𝑐+𝐵)(_𝑡0(𝑡−𝑠)^𝑞−1𝒮_𝑞(𝑡−𝑠)ℎ(𝑠)𝑑𝑠. (71) By the normality of the cone 𝑃, (69), (70), and (71), we have ‖(𝑄𝑢)(𝑡)‖^∗≤𝑁𝒯_𝑞(𝑡)𝑐^∗+‖(𝐵𝑢)(𝑡)‖^∗+^𝑡₀(𝑡−𝑠)^𝑞−1𝒮_𝑞(𝑡−𝑠)ℎ(𝑠)𝑑𝑠^∗≤𝑁𝑀^∗𝑐+‖𝐵‖^∗‖𝑢‖^∗_𝐶+𝑀^∗𝑎^𝑞‖ℎ‖^∗_𝐶Γ1+𝑞≤𝑁𝑀^∗𝑐+𝑁𝛾𝑅^∗+𝑁𝑀^∗𝑎^𝑞‖ℎ‖^∗_𝐶Γ1+𝑞=𝑅^∗. (72) Thus, we have proved that 𝑄∶Ω^∗_𝑅^∗(𝑃)→Ω^∗_𝑅^∗(𝑃). Let 𝐹=co𝑄(Ω^∗_𝑅^∗(𝑃)). Similar to the proof of Theorem 10, we can prove that 𝑄∶𝐹→𝐹 is a convex-power condensing operator. It follows from Lemma 8 that 𝑄 has at least one fixed point 𝑢∈𝐹, which is just a positive mild solution of the NPFEE (1). This completes the proof of Theorem 16.

Remark 17. Positive operator semigroups are widely appearing in heat conduction equations, neutron transport equations, reaction diffusion equations, and so on [47]. Therefore, using Theorems 15 and 16 to these partial differential equations are very convenient.

Remark 18. Analytic semigroup and differentiable semigroup are equicontinuous semigroup [48]. In the application of partial differential equations, such as parabolic equations and strongly damped wave equations, the corresponding solution semigroup is analytic semigroup. Therefore, the results obtained in this paper have broad applicability.

Acknowledgments

This research was supported by NNSF of China (11261053 and 11061031), NSF of Gansu Province (1208RJZA129), and Project of NWNU-LKQN-11-3.