Positive Solution of a Nonlinear Fractional Differential Equation Involving Caputo Derivative

Abstract

This paper is concerned with a nonlinear fractional differential equation involving Caputo derivative. By constructing the upper and lower control functions of the nonlinear term without any monotone requirement and applying the method of upper and lower solutions and the Schauder fixed point theorem, the existence and uniqueness of positive solution for the initial value problem are investigated. Moreover, the existence of maximal and minimal solutions is also obtained.

1. Introduction

Fractional differential equation can be extensively applied to various disciplines such as physics, mechanics, chemistry, and engineering, see [1–3]. Hence, in recent years, fractional differential equations have been of great interest and there have been many results on existence and uniqueness of the solution of FDE, see [4–8]. Especially, Diethelm and Ford [9] have gained existence, uniqueness, and structural stability of solution of the type of fractional differential equation 𝐷^𝑞𝑦−𝑇_𝑚−1𝑦(𝑥)=𝑓𝑥,𝑦(𝑥),𝑦^(𝑘)(0)=𝑦₀^(𝑘),𝑘=0,1,…,𝑚−1, (1.1)

where 𝑞>0 is a real number, 𝐷^𝑞 denotes the Riemann-Liouville differential operator of order 𝑞, and 𝑇_𝑚−1[𝑦] is the Taylor polynomial of order (𝑚−1) for the function 𝑦(𝑥) at 𝑥₀=0. Recently, Daftardar-Gejji and Jafari [10] have discussed the existence, uniqueness, and stability of solution of the system of nonlinear fractional differential equation 𝐷^𝛼_∗𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),0<𝑡<1,𝑦^(𝑘)(0)=𝑐_𝑘,0≤𝑘≤𝑚,0≤𝑐_𝑘, (1.2) where 𝑚<𝛼≤𝑚+1 and 𝐷^𝛼_∗ denotes Caputo fractional derivative (see Definition 2.3). Delbosco and Rodino [11] have proved existence and uniqueness theorems for the nonlinear fractional equation 𝐷^𝛿𝑢=𝑓(𝑡,𝑢),0<𝑡<1,𝑢(0)=0, (1.3) where 0<𝛿<1, 𝐷^𝛿 is the Riemann-Liouville fractional derivative. Zhang [12] used the method of the upper and lower solution and cone fixed point theorem to obtain the existence and uniqueness of positive solution to (1.3). Yao [13] considered the existence of positive solution to (1.3) controlled by the power function employing Krasnosel’skii fixed point theorem of cone expansion-compression type. The existence of the local and global solution for (1.3) was obtained by Lakshmikantham and Vatsala [14] utilizing classical differential equation theorem.

More recently, Zhang [15] shows the existence of positive solutions to the singular boundary value problem for fractional differential equation 𝐷^𝛼₀₊𝑢(𝑡)+𝑔(𝑡)𝑓𝑢,𝑢^′,…,𝑢^(𝑛−2)𝑢(0)=0,0<𝑡<1,=𝑢^′(0)=⋯=𝑢^(𝑛−2)(0)=𝑢^(𝑛−2)(1)=0, (1.4) where 𝐷^𝛼₀₊ is the Riemann-Liouville fractional derivative of order 𝑛−1<𝛼≤2, 𝑛≥2.

However, in the previous works, the nonlinear term has to satisfy the monotone or others control conditions. In fact, the fractional differential equations with nonmonotone function can respond better to impersonal law, so it is very important to weaken monotone condition. Considering this, in this paper, we mainly investigate the fractional differential Equation (1.2) without any monotone requirement on nonlinear term by constructing upper and lower control function and exploiting the method of upper and lower solutions and the Schauder fixed point theorem. The existence and uniqueness of positive solution for (1.2) are obtained. Some properties concerning the maximal and minimal solutions are also given. This work is motivated by the above references and my previous work [16, 17]. Other related results on the fractional differential equations can be found in [18–24].

This paper is organized as follow. In Section 2, we recall briefly some notions of the fractional calculus and the theory of the operators for integration and differentiation of fractional order. Section 3 is devoted to the study of the existence and uniqueness of positive solution for (1.2) utilizing the upper and lower solution method and the Schauder fixed point theorem. The existence of maximal and minimal solutions for (1.2) is given in Section 4.

2. Preliminaries and Notations

First, we give some basic definitions and theorems which are basically used throughout this paper. 𝐶[0,1] denotes the space of continuous functions defined on [0,1] and 𝐶^𝑛[0,1] denotes the class of all real valued functions defined on [0,1] which have continuous 𝑛th order derivative.

Definition 2.1. Let 𝑓(𝑥)∈𝐶[0,1] and 𝛼>0, then the expression 𝐼^𝛼₀𝑓(𝑥)=1Γ(𝛼)^𝑥₀(𝑥−𝑡)^𝛼−1𝑓(𝑡)𝑑𝑡,0<𝑥<1 (2.1) is called the (left-sided) Riemann-Liouville integral of order 𝛼.

Definition 2.2. Let 𝑛−1<𝛼≤𝑛,𝑛∈𝑁, then the expression 𝐷^𝛼₀𝑓(𝑥)=𝑑^𝑛𝑑𝑥^𝑛𝐼₀^𝑛−𝛼𝑓(𝑥),0<𝑥<1 (2.2) is called the (left-sided) Riemann-Liouville derivative of 𝑓(𝑥) of order 𝛼 whenever the expression on the right-hand side is defined.

Definition 2.3. Let 𝑓(𝑥)∈𝐶^𝑛[0,1] and 𝑛−1<𝛼≤𝑛,𝑛∈𝑁, then the expression 𝐷^𝛼_∗0𝑓(𝑥)=𝐼₀^𝑛−𝛼𝑓^(𝑛)(𝑥) (2.3) is called the (left-sided) Caputo derivative of 𝑓(𝑥) of order 𝛼.

In further discussion we will denote 𝐷^𝛼₀, 𝐼^𝛼₀, and 𝐷^𝛼_∗0 as 𝐷^𝛼, 𝐼^𝛼, and 𝐷^𝛼_∗, respectively.

Lemma 2.4 (see [25, 26]). Let 𝑓(𝑥)∈𝐶^𝑛[0,1] and 𝑛−1<𝛼≤𝑛,𝑛∈𝑁, then we one has 𝐼^𝜇𝐼^𝜈𝑓(𝑥)=𝐼^𝜇+𝜈𝑓(𝑥)𝐷,𝜇,𝜈≥0,_𝛼𝐼^𝛼𝑓(𝑥)(𝑥),𝐼=𝑓_𝛼𝐷^𝛼_∗𝑓(𝑥)=𝑓(𝑥)−^𝑛−1_𝑘=0𝑥^𝑘𝑓𝑘!^(𝑘)(0+). (2.4)(2.5)(2.6)

Lemma 2.5 (see, [10]). If the function 𝑓(𝑡,𝑦(𝑡)) is 𝐶¹[0,1], then the initial value problem (1.2) is equivalent to the Volterra integral equations 𝑦(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼𝑓𝑡,𝑦(𝑡),0<𝑡<1,𝑚<𝛼≤𝑚+1. (2.7)

Proof. Suppose 𝑦(𝑡) satisfies the initial value problem (1.2), then applying 𝐼^𝛼 to both sides of (1.2) and using Lemma 2.4 (2.7) follows. Conversely, suppose 𝑦(𝑡) satisfies (2.7). Then observe that 𝐷^(𝑚+1)𝑦(𝑡) exists and is integrable, because 𝑦^(𝑚+1)(𝑡)=𝐷^𝑚+1𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼𝑓𝑡,𝑦(𝑡)=𝐷^𝑚+1𝐼^𝛼𝑓𝑡,𝑦(𝑡)=𝐷𝐷^𝑚𝐼^𝑚𝐼^𝛼−𝑚𝑓𝑡,𝑦(𝑡)=𝐷𝐼^𝛼−𝑚𝑓𝑡,𝑦(𝑡)=𝐷^{𝑚+1−𝛼}𝑓𝑡,𝑦(𝑡), (2.8) which exists and is integrable as 𝑓(𝑡,𝑦(𝑡)) is 𝐶¹[0,1]. Thus 𝐼^{𝑚+1−𝛼}𝑦^(𝑚+1)(𝑡)=𝐷^𝛼_∗𝑦(𝑡) exists.
Applying 𝐷^𝛼_∗ on both sides of (2.7), one has 𝐷^𝛼_∗𝑦(𝑡)=𝐼^{𝑚+1−𝛼}𝐷^𝑚+1𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼𝑓𝑡,𝑦(𝑡)=𝐼^{𝑚+1−𝛼}𝐷^{𝑚+1−𝛼}𝑓𝑡,𝑦(𝑡)=𝑓𝑡,𝑦(𝑡), (2.9) as 𝑓(𝑡,𝑦(𝑡)) is continuous and 0<𝑚+1−𝛼<1. Hence 𝑦(𝑡) satisfies (1.2). Moreover, from (2.4), 𝑦^(𝑘)(0)=𝑐_𝑘,0≤𝑘≤𝑚 hold.

Let 𝑋=𝐶[0,1] be the Banach space endowed with the infinity norm and 𝐾 a nonempty closed subset of 𝑋 defined as 𝐾={𝑦(𝑡)∈𝑋∣0<𝑦(𝑡)≤𝑙,0<𝑡≤1,𝑦^(𝑘)(0)=𝑐_𝑘,1≤𝑘≤𝑚,0≤𝑐_𝑘}. The positive solution which we consider in this paper is a function such that 𝑦(𝑡)∈𝐾.

According to Lemma 2.5, (1.2) is equivalent to the fractional integral Equation (2.7). The integral equation (2.7) is also equivalent to fixed point equation 𝑇𝑦(𝑡)=𝑦(𝑡),𝑦(𝑡)∈𝐶[0,1], where operator 𝑇∶𝐾→𝐾 is defined as 𝑇𝑦(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼𝑓𝑡,𝑦(𝑡), (2.10) then we have the following lemma.

Lemma 2.6. Let 𝑓∶[0,1]×[0,𝑙]→𝑅⁺ a given continuous function. Then the operator 𝑇∶𝐾→𝐾 is completely continuous.

Proof. Let 𝑀⊂𝐾 be bounded, that is, there exists a positive constant 𝑙^∗ such that ‖𝑦‖_∞≤𝑙^∗ for any 𝑦(𝑡)∈𝑀. Since 𝑓(𝑡,𝑦(𝑡)) is a given continuous function, we have max_0≤𝑡≤1𝑓𝑡,𝑦(𝑡)≤max_{(𝑡,𝑦)∈𝐷}𝑓𝑡,𝑦,forany𝑦(𝑡)∈𝑀, (2.11) where 𝐷={(𝑡,𝑦)∣0≤𝑡≤1,0≤𝑦≤𝑙^∗}.
Let 𝐿=max_{(𝑡,𝑦)∈𝐷}𝑓(𝑡,𝑦), then for any 𝑦(𝑡)∈𝑀, we have 𝑇𝑦(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼𝑓(𝑡)=𝑡,𝑦_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠)≤𝑠,𝑦𝑑𝑠_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑠≤𝑠,𝑦)𝑑𝑠_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐿Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑑𝑠≤^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐿Γ()𝑡1+𝛼^𝛼. (2.12) Thus, 𝑇𝑦≤^𝑚_𝑘=0𝑐_𝑘+𝐿𝑘!Γ().1+𝛼 (2.13) Hence 𝑇∶𝐾→𝐾 is uniformly bounded.
Now, we prove that 𝑇∶𝐾→𝐾 is continuous. Since 𝑓(𝑡,𝑦(𝑡)) is continuous function in a compact set [0,1]×[0,𝑙], then it is uniformly continuous there. Thus given 𝜀>0, we can find 𝜇>0 such that ‖𝑓(𝑡,𝑦)−𝑓(𝑡,𝑧)‖<𝜀^∗ whenever ‖𝑦−𝑧‖<𝜇, where 𝜀^∗=𝜀Γ(𝛼+1). Then (𝑡)(𝑡≤1𝑇𝑦−𝑇𝑧)Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑦(𝑠)−𝑓(𝑠,𝑧(𝑠))𝑑𝑠≤𝜀^∗1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑑𝑠=𝜀^∗𝑡^𝛼Γ(𝛼+1)<𝜀, (2.14) proving the continuity of the operators 𝑇∶𝐾→𝐾.
Now, we will prove that the operator 𝑇∶𝐾→𝐾 is equicontinuous. For each 𝑦(𝑡)∈𝑀, any 𝜀>0,𝑡₁,𝑡₂∈[0,1] and 𝑡₁<𝑡₂. Let 𝛿=min{(𝜀/2)(∑^𝑚_𝑘=1(𝑐_𝑘/(𝑘−1)!)⁻¹,(𝜀Γ(1+𝛼)/4𝐿)^1/𝛼}, then when |𝑡₂−𝑡₁|<𝛿, we have 𝑇𝑦𝑡₁−𝑇𝑦𝑡₂=^𝑚_𝑘=0𝑡^𝑘₁𝑐𝑘!^𝑘+𝐼^𝛼𝑓𝑡₁,𝑦𝑡₁−^𝑚_𝑘=0𝑡^𝑘₂𝑐𝑘!^𝑘−𝐼^𝛼𝑓𝑡₂,𝑦𝑡₂≤^𝑚_𝑘=0𝑡^𝑘₁𝑐𝑘!^𝑘−^𝑚_𝑘=0𝑡^𝑘₂𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡1₀𝑡₁−𝑠^𝛼−1𝑓((𝑠1𝑠,𝑢))𝑑𝑠−Γ(𝛼)^𝑡2₀𝑡₂−𝑠^𝛼−1𝑓((𝑠≤𝑡𝑠,𝑢))𝑑𝑠₂−𝑡₁^𝑚_𝑘=1𝑐_𝑘()!+1𝑘−1Γ(𝛼)^𝑡1₀𝑡₁−𝑠^𝛼−1−𝑡₂−𝑠^𝛼−1𝑓((𝑠+1𝑠,𝑢))𝑑𝑠Γ(𝛼)^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1𝑓((𝑠≤𝑡𝑠,𝑢))𝑑𝑠₂−𝑡₁^𝑚_𝑘=1𝑐_𝑘()!+𝐿𝑘−1Γ(𝛼)^𝑡1₀𝑡₁−𝑠^𝛼−1−𝑡₂−𝑠^𝛼−1+𝐿𝑑𝑠Γ(𝛼)^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1=𝑡𝑑𝑠₂−𝑡₁^𝑚_𝑘=1𝑐_𝑘()!+𝐿𝑘−1Γ(𝛼)^𝑡1₀𝑡₁−𝑠^𝛼−1𝑑𝑠−^𝑡1₀𝑡₂−𝑠^𝛼−1𝑑𝑠+^𝑡2_𝑡1𝑡₂−𝑠^𝛼−1=𝑡𝑑𝑠₂−𝑡₁^𝑚_𝑘=1𝑐_𝑘()!+𝐿𝑘−1Γ()𝑡1+𝛼^𝛼1+𝑡₂−𝑡₁^𝛼−𝑡^𝛼₂+𝑡₂−𝑡₁^𝛼≤𝑡₂−𝑡₁^𝑚_𝑘=1𝑐_𝑘()!+𝑘−12𝐿Γ()𝑡1+𝛼²−𝑡₁^𝛼<𝛿^𝑚_𝑘=1𝑐_𝑘()!+𝑘−12𝐿Γ()𝛿1+𝛼^𝛼=𝜀. (2.15) The Arzela-Ascoli Theorem implies that 𝑇 is completely continuous. The proof is therefore completed.

Lemma 2.7. If the operator 𝐴∶𝑋→𝑋 is the contraction mapping, where 𝑋 is the Banach space, then 𝐴 has a unique fixed point in 𝑋.
Let 𝑓∶[0,1]×[0,𝑙)→𝑅⁺ be a given function. Take 𝑎,𝑏∈𝑅⁺, and 𝑎<𝑏<𝑙. For any 𝑦∈[𝑎,𝑏] one defines the upper-control function 𝐻(𝑡,𝑦)=sup_{𝑎≤𝜂≤𝑦}𝑓(𝑡,𝜂), and lower-control function ℎ(𝑡,𝑦)=inf_{𝑦≤𝜂≤𝑏}𝑓(𝑡,𝜂), obviously 𝐻(𝑡,𝑦),ℎ(𝑡,𝑦) is monotonous nondecreasing on 𝑦 and ℎ(𝑡,𝑦)≤𝑓(𝑡,𝑦)≤𝐻(𝑡,𝑦).

Definition 2.8. Let 𝑦(𝑡),𝑦(𝑡)∈𝐾,𝑏≥𝑦(𝑡)≥𝑦(𝑡)≥𝑎, and satisfy 𝐷^𝛼_∗𝑦(𝑡)≥𝑓𝑡,𝑦(𝑡),𝑦^(𝑘)(0)≥𝑐_𝑘,𝑚<𝛼≤𝑚+1,1≤𝑘≤𝑚,𝑐_𝑘𝑦(𝑡)≥≥0,or^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼𝑓𝑦(𝑡),𝐷𝑡,,𝑚<𝛼≤𝑚+1_𝛼∗𝑦(𝑡)≤𝑓𝑡,𝑦(𝑡),𝑦^(𝑘)(0)≤𝑐_𝑘,𝑚<𝛼≤𝑚+1,1≤𝑘≤𝑚,𝑐_𝑘𝑦(𝑡)≤≥0,or^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼𝑓𝑡,𝑦(𝑡),𝑚<𝛼≤𝑚+1, (2.16) then the functions 𝑦(𝑡),𝑦(𝑡) are called a pair of order upper and lower solutions for (1.2).

3. Existence and Uniqueness of Positive Solution

Now, we give and prove the main results of this paper.

Theorem 3.1. Assume 𝑓∶[0,1]×[0,𝑙)→[0,+∞) is continuous, and 𝑦(𝑡),𝑦(𝑡) are a pair of order upper and lower solutions of (1.2), then the boundary value problem (1.2) exists one solution 𝑦(𝑡)∈𝐶[0,1]; moreover, 𝑦(𝑡)≥𝑦(𝑡)≥𝑦(𝑡),𝑡∈[0,1]. (3.1)

Proof. Let 𝑆=𝑧(𝑡)∣𝑧(𝑡)∈𝐾,𝑦(𝑡)≤𝑧(𝑡)≤𝑦(𝑡),𝑡∈[0,1], (3.2) endowed with the norm ‖𝑧‖=max_𝑡∈[0,1]𝑧(𝑡), then we have ‖𝑧‖≤𝑏. Hence 𝑆 is a convex, bounded, and closed subset of the Banach space 𝑋. According to Lemma 2.6, the operator 𝑇∶𝐾→𝐾 is completely continuous. Then we need only to prove 𝑇∶𝑆→𝑆.
For any 𝑧(𝑡)∈𝑆, we have 𝑦(𝑡)≥𝑧(𝑡)≥𝑦(𝑡), then 𝑇𝑧(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼𝑓((𝑡=𝑡,𝑧))_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓((𝑠≤𝑠,𝑧))𝑑𝑠_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐻((𝑠≤𝑠,𝑧))𝑑𝑠_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐻𝑦(𝑠)≤𝑦(𝑡),(𝑡)=𝑠,𝑑𝑠𝑇𝑧^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼𝑓((𝑡=𝑡,𝑧))_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓((𝑠≥𝑠,𝑧))𝑑𝑠_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1ℎ((𝑠≥𝑠,𝑧))𝑑𝑠_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1ℎ𝑦(𝑠)≥𝑦(𝑡).𝑠,𝑑𝑠 (3.3) Hence 𝑦(𝑡)≥𝑇𝑧(𝑡)≥𝑦(𝑡),1>𝑡>0, that is, 𝑇∶𝑆→𝑆. According to Schauder fixed point theorem, the operator 𝑇 exists at least one fixed point 𝑦(𝑡)∈𝑆,0<𝑡<1. Therefore the boundary value problem (1.2) exists at least one solution 𝑦(𝑡)∈𝐶[0,1], and 𝑦(𝑡)≥𝑦(𝑡)≥𝑦(𝑡),𝑡∈[0,1].

Corollary 3.2. Assume 𝑓∶[0,1]×[0,𝑙)→[0,+∞) is continuous, and there exist 𝑝₂>𝑝₁≥0, such that 𝑝₁≤𝑓(𝑡,𝑠)≤𝑝₂,(𝑡,𝑠)∈[0,1]×[0,𝑙), (3.4) then the boundary value problem (1.2) exists at least one positive solution 𝑦(𝑡)∈𝐶[0,1], moreover ^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑝₁𝑡^𝛼Γ()(𝑡)≤𝛼+1≤𝑦^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑝₂𝑡^𝛼Γ().𝛼+1 (3.5)

Proof. By assumption (3.4) and the definition of control function, we have 𝑝₁≤ℎ(𝑡,𝑠)≤𝐻(𝑡,𝑠)≤𝑝₂,(𝑡,𝑠)∈[0,1]×[𝑎,𝑏]. (3.6) Now, we consider the equation 𝐷^𝛼_∗𝑤(𝑡)=𝑝₂,𝑤^(𝑘)(0)=𝑐_𝑘,𝑚<𝛼≤𝑚+1,1≤𝑘≤𝑚,𝑐_𝑘≥0. (3.7) Obviously, (3.7) has a positive solution 𝑤(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑝₂𝑑𝑠=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑝₂𝑡^𝛼Γ()[],𝑤(𝑡)=𝛼+1,𝑡∈0,1_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑝₂𝑡^𝛼Γ()=𝛼+1_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑝₂≥𝑑𝑠_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐻(𝑡,𝑤(𝑡))𝑑𝑠, (3.8) namely, 𝑤(𝑡) is a upper solution of (1.2). In the similar way, we obtain 𝑣(𝑡)=∑^𝑚_𝑘=0((𝑡^𝑘/𝑘!)𝑐_𝑘)+𝑝₁𝑡^𝛼/Γ(𝛼+1) is the lower solution of (1.2). An application of Theorem 3.1 now yields that the boundary value problem (1.2) exists at least one positive solution 𝑦(𝑡)∈𝐶[0,1], moreover ^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑝₁𝑡^𝛼Γ()(𝑡)≤𝛼+1≤𝑦^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑝₂𝑡^𝛼Γ().𝛼+1 (3.9)

Corollary 3.3. Assume 𝑓∶[0,1]×[0,+∞)→[𝑐,+∞) is continuous, where 𝑐>0, moreover 𝑐<lim_𝑦→+∞𝑓𝑡,𝑦<+∞,𝑡∈[0,1], (3.10) then the boundary value problem (1.2) has at least one positive solution 𝑢(𝑡)∈𝐶[0,1].

Proof. By assumption (3.9), there are positive constants 𝑁,𝑅, such that 𝑓(𝑡,𝑦)≤𝑁 whenever 𝑢>𝑅. Let 𝑀=max_{0≤𝑡≤1,0≤𝑦≤𝑅}𝑓(𝑡,𝑦), then 𝑓(𝑡,𝑦)≤𝑁+𝑀, 0≤𝑦<+∞. By the definition of control function, one has 𝐻(𝑡,𝑦)≤𝑁+𝑀,0≤𝑡≤1,0≤𝑦<+∞.
Now, we consider the equation 𝐷^𝛼_∗𝑤(𝑡)=𝑁+𝑀,𝑤^(𝑘)(0)=𝑐_𝑘,𝑚<𝛼≤𝑚+1,1≤𝑘≤𝑚,𝑐_𝑘≥0. (3.11) Obviously, (3.11) has a positive solution 𝑤(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1(𝑁+𝑀)𝑑𝑠=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+(𝑁+𝑀)𝑡^𝛼Γ()[],𝑤(𝑡)=𝛼+1,𝑡∈0,1_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+(𝑁+𝑀)𝑡^𝛼Γ()=𝛼+1_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1()≥𝑁+𝑀𝑑𝑠_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐻(𝑡,𝑤(𝑡))𝑑𝑠, (3.12) namely, 𝑤(𝑡) is the upper solution of (1.2). In the similar way, we obtain 𝑣(𝑡)=∑^𝑚_𝑘=0(𝑡^𝑘/𝑘!)𝑐_𝑘+𝑐𝑡^𝛼/Γ(𝛼+1) is the lower solution of (1.2). Therefore, the boundary value problem of (1.2) has at least one positive solution 𝑦(𝑡)∈𝐶[0,1], what is more, we have ^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑐𝑡^𝛼Γ()(𝑡)≤𝛼+1≤𝑦^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+(𝑁+𝑀)𝑡^𝛼Γ().𝛼+1 (3.13)

Corollary 3.4. Assume 𝑓∶[0,1]×[0,+∞)→[𝑐,+∞) is continuous, where 𝑐>0, moreover 𝑐<lim_𝑢→+∞max_0≤𝑡≤1𝑓𝑡,𝑦𝑦=𝑀<Γ(𝛼+1), (3.14) then the boundary value problem (1.2) exists at least at one positive solution 𝑢(𝑡)∈𝐶[0,1].

Proof. According to 𝑐<lim_𝑦→+∞max_0≤𝑡≤1(𝑓(𝑡,𝑦)/𝑦)=𝑀<+∞, there exists 𝐷>0, such that for any 𝑦(𝑡)∈𝑋, we have 𝑓𝑡,𝑦(𝑡)≤𝑀𝑦(𝑡)+𝐷. (3.15) By the definition of control function, we have 𝐻𝑡,𝑦(𝑡)≤𝑀𝑦(𝑡)+𝐷. (3.16) We now consider the equation 𝐷^𝛼_∗𝑤(𝑡)=𝑀𝑦(𝑡)+𝐷,𝑤^(𝑘)(0)=𝑐_𝑘,𝑚<𝛼≤𝑚+1,1≤𝑘≤𝑚,𝑐_𝑘≥0. (3.17) According to Lemma 2.5, (3.17) is equivalent to the integral equation 𝑦(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼(𝑡)=𝑀𝑦+𝐷_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀𝑦(𝑠)+𝐷𝑑𝑠,𝑚<𝛼≤𝑚+1. (3.18) Let 𝐴∶𝐾→𝐾 be an operator as follows: 𝐴𝑦(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼(𝑡)=𝑀𝑦+𝐷_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀𝑦(𝑠)+𝐷𝑑𝑠,𝑚<𝛼≤𝑚+1. (3.19) by Lemma 2.6, the operator 𝐴 is completely continuous.
Let 𝐵_𝑅=𝑢(𝑡)∈𝐾∣𝑢−𝐷𝑡^𝛼Γ(),𝛼+1≤𝑅<+∞ (3.20) where 𝑅>0 and satisfies that ∑^𝑚_𝑘=0(𝑐_𝑘/𝑘!)+(𝑀/Γ(𝛼+1))[(𝐷/Γ(𝛼+1))+𝑅]−𝑅≤0, then 𝐵_𝑅 is convex, bounded, and closed subset of the Banach space 𝐶[0,1]. For any 𝑦(𝑡)∈𝐵_𝑅, we have ‖𝑢‖≤𝐷Γ(𝛼+1)+𝑅, (3.21) then 𝐴𝑦(𝑡)−𝐷𝑡^𝛼Γ()=𝛼+1^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀𝑦(𝑠)+𝐷𝑑𝑠−𝐷𝑡^𝛼Γ()=𝛼+1_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑀Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑦(𝑠)≤𝑑𝑠_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘𝑦(𝑡𝑡+𝑀)^𝛼Γ()≤𝛼+1_𝑚𝑘=0𝑐_𝑘+𝑀𝑘!Γ()𝐷𝛼+1𝑅+Γ(𝛼+1)≤𝑅, (3.22) thus 𝐴𝑦(𝑡)−𝐷𝑡^𝛼Γ(𝛼+1)≤𝑅. (3.23) Hence, the Schauder fixed theorem assures that the operator 𝐴 has at least one fixed point and then (3.17) has at least one positive solution 𝑦^∗(𝑡), therefore we have 𝑦^∗(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑀𝑦^∗(𝑠)+𝐷𝑑𝑠,𝑚<𝛼≤𝑚+1. (3.24) Combining condition (3.16), we have 𝑦^∗(𝑡)≥^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝐻𝑠,𝑦^∗(𝑠)𝑑𝑠,𝑚<𝛼≤𝑚+1. (3.25) Obviously, 𝑦^∗(𝑡) is the upper solution of initial value problem (1.2), and 𝑣(𝑡)=∑^𝑚_𝑘=0(𝑐_𝑘/𝑘!)+𝑐𝑡^𝛼/Γ(𝛼+1) is the lower solution. By Theorem 3.1, system (1.2) has at least one positive solution 𝑢(𝑡)∈𝐶[0,1].

Corollary 3.5. Assume 𝑓∶[0,1]×[0,+∞)→[𝑐,+∞) is continuous and there exists 𝑑>0,𝑒>0, such that max𝑓(𝑡,𝑙)∶(𝑡,𝑙)∈[0,1]×[0,𝑑]≤𝑒, (3.26) then the boundary value problem (1.2) has at least one positive solution 𝑦(𝑡)∈𝐶[0,1], moreover ^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑐𝑡^𝛼Γ()(𝑡)≤𝛼+1≤𝑦^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑒𝑡^𝛼Γ().𝛼+1 (3.27)

Proof. By the definition of control function, we have 𝑐≤ℎ(𝑡,𝑙)≤𝐻(𝑡,𝑙)≤𝑒,(𝑡,𝑙)∈[0,1]×[0,𝑑]. (3.28) By Corollary 3.2, the boundary value problem (1.2) has at least one positive solution 𝑦(𝑡)∈𝐶[0,1], moreover ^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑐𝑡^𝛼Γ()(𝑡)≤𝛼+1≤𝑦^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝑒𝑡^𝛼Γ().𝛼+1 (3.29)

Theorem 3.6. Let the conditions in Theorem 3.1 hold. Moreover for any 𝑦₁(𝑡),𝑦₂(𝑡)∈𝑋,0<𝑡<1, there exists 𝑙>0, such that 𝑓𝑡,𝑦₁−𝑓𝑡,𝑦₂≤𝑙𝑦₁−𝑦₂, (3.30) then when 𝑙/Γ(𝛼+1)<1, the boundary value problem (1.2) has a unique positive solution 𝑦(𝑡)∈𝑆.

Proof. According to Theorem 3.1, if the conditions in Theorem 3.1 hold, then the boundary value problem (1.2) has at least one positive solution in 𝑆. Hence we need only to prove that the operator 𝑇 defined in (2.10) is the contraction mapping in 𝑋. In fact, for any 𝑦₁(𝑡),𝑦₂(𝑡)∈𝑋, by assumption (3.30), we have 𝑇𝑦₁(𝑡)−𝑇𝑦₂(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝐼^𝛼𝑓𝑡,𝑦₁(𝑡)−^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘−𝐼^𝛼𝑓𝑡,𝑦₂(𝑡)=1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑦₁(𝑠)1𝑑𝑠−Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑦₂(𝑠)≤1𝑑𝑠Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑦₁(𝑠)−𝑓𝑠,𝑦₂(𝑠)≤𝑑𝑠𝑙𝑡^𝛼Γ()𝑦𝛼+1¹−𝑦₂<𝑙Γ()𝑦𝛼+1¹−𝑦₂. (3.31) Thus, when 𝑙/Γ(𝛼+1)<1, the operator 𝑇 is the contraction mapping. Then by Lemma 2.7, the boundary value problem (1.2) has a unique positive solution 𝑦(𝑡)∈𝑆.

4. Maximal and Minimal Solutions Theorem

In this section, we consider the existence of maximal and minimal solutions for (1.2).

Definition 4.1. Let 𝑚(𝑡) be a solution of (1.2) in [0,1], then 𝑚(𝑡) is said to be a maximal solution of (1.2), if for every solution 𝑦(𝑡) of (1.2) existing on [0,1] the inequality 𝑦(𝑡)≤𝑚(𝑡),𝑡∈[0,1] holds. A minimal solution may be defined similarly by reversing the last inequality.

Theorem 4.2. Let 𝑓∶[0,1]×[0,+∞)→[0,+∞) be a given continuous and monotone nondecreasing with respect to the second variable. Assume that there exist two positive constants 𝜆,𝜇(𝜇>𝜆) such that 𝜆≤𝑓𝑡,𝑦≤𝜇. (4.1) Then there exist maximal solution 𝜑(𝑡) and minimal solution 𝜂(𝑡) of (1.2) on [0,1], moreover ^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝜆𝑡^𝛼Γ()(𝑡)(𝑡)≤𝛼+1≤𝜂≤𝜑^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝜇𝑡^𝛼Γ(𝛼+1),0≤𝑡≤1. (4.2)

Proof. It is easy to know that ∑^𝑚_𝑘=0((𝑡^𝑘/𝑘!)𝑐_𝑘)+((𝜇𝑡^𝛼)/Γ(𝛼+1)) and ∑^𝑚_𝑘=0((𝑡^𝑘/𝑘!)𝑐_𝑘)+((𝜆𝑡^𝛼)/Γ(𝛼+1)) are the upper and lower solutions of (1.2), respectively. Then by using 𝑦⁽⁰⁾=∑^𝑚_𝑘=0((𝑡^𝑘/𝑘!)𝑐_𝑘)+𝜇𝑡^𝛼/Γ(𝛼+1),𝑦⁽⁰⁾=∑^𝑚_𝑘=0((𝑡^𝑘/𝑘!)𝑐_𝑘)+𝜆𝑡^𝛼/Γ(𝛼+1) as a pair of coupled initial iterations we construct two sequences {𝑦^(𝑚)},{𝑦^(𝑚)}from the following linear iteration process: 𝑦^(𝑚)(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑦^(𝑚−1)(𝑡𝑦)𝑑𝑠,_(𝑚)(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑦^(𝑚−1)(𝑡)𝑑𝑠. (4.3) It is easy to show from the monotone property of 𝑓(𝑡,𝑦) and condition (4.1) that the sequences {𝑦^(𝑚)},{𝑦^(𝑚)} possess the following monotone property: ^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝜆𝑡^𝛼Γ(𝛼+1)=𝑦⁽⁰⁾≤𝑦^(𝑚)≤𝑦^(𝑚+1)≤𝑦^(𝑚+1)≤𝑦^(𝑚)≤𝑦⁽⁰⁾=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝜇𝑡^𝛼Γ()().𝛼+1𝑚=1,2,… (4.4) The above property implies that lim_𝑚→∞𝑦(𝑡)^(𝑚)=𝜑(𝑡),lim_𝑚→∞𝑦(𝑡)^(𝑚)=𝜂(𝑡) (4.5) exist and satisfy the relation ^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝜆𝑡^𝛼Γ()(𝑡)(𝑡)≤𝛼+1≤𝜂≤𝜑^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+𝜇𝑡^𝛼Γ(𝛼+1),0≤𝑡≤1. (4.6) Letting 𝑚→∞ in (4.3) shows that 𝜑(𝑡) and 𝜂(𝑡) satisfy the equations 𝜑(𝑡)=^𝑚_𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓(𝑡𝜂(𝑡)=𝑠,𝜑)𝑑𝑠,_𝑚𝑘=0𝑡^𝑘𝑐𝑘!^𝑘+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1𝑓𝑠,𝜂(𝑡)𝑑𝑠. (4.7) It is easy to verify that the limits 𝜑(𝑡) and 𝜂(𝑡) are maximal and minimal solutions of (1.2) in 𝑆^∗={𝜓(𝑡)∣𝜓(𝑡)∈𝐾,∑^𝑚_𝑘=0((𝑡^𝑘/𝑘!)𝑐_𝑘)+𝜆𝑡^𝛼/Γ(𝛼+1)≤𝜓(𝑡)≤∑^𝑚_𝑘=0((𝑡^𝑘/𝑘!)𝑐_𝑘)+𝜇𝑡^𝛼/Γ(𝛼+1),s𝑡∈[0,1],‖𝜓(𝑡)‖=max_0≤𝑡≤1𝜓(𝑡)}, respectively, furthermore, if 𝜑(𝑡)=𝜂(𝑡)(≡𝜁(𝑡)) then 𝜁(𝑡) is the unique solution in 𝑆^∗, and hence the proof is completed.

Acknowledgments

The authors are grateful to the referee for the comments. This work is supported by the Science and Technology Project of Chongqing Municipal Education Committee (Grants nos. KJ110501 and KJ120520) of China, Natural Science Foundation Project of CQ CSTC (Grant no. cstc2012jjA20016) of China, and the NSFC (Grant no. 11101298) of China.