A Mixed Monotone Operator Method for the Existence and Uniqueness of Positive Solutions to Impulsive Caputo Fractional Differential Equations

Abstract

We establish some sufficient conditions for the existence and uniqueness of positive solutions to a class of initial value problem for impulsive fractional differential equations involving the Caputo fractional derivative. Our analysis relies on a fixed point theorem for mixed monotone operators. Our result can not only guarantee the existence of a unique positive solution but also be applied to construct an iterative scheme for approximating it. An example is given to illustrate our main result.

1. Introduction

The purpose of this paper is to investigate the existence and uniqueness of positive solutions to the following impulsive initial value problem (IVP for short) for Caputo fractional-order differential equations: ^𝑐𝐷^𝛼𝑦(𝑡)(𝑡)(𝑡(𝑡,[]=𝑓𝑡,𝑦,𝑦)+𝑔𝑡,𝑦)𝑡∈0,𝑇,𝑡≠𝑡_𝑘,Δ𝑦_{𝑡=𝑡𝑘}=𝐼_𝑘𝑦𝑡⁻_𝑘,𝑦(0)=𝑦₀, (1) where 𝑘=1,2,…,𝑚, 0<𝛼≤1, ^𝑐𝐷^𝛼 is the Caputo fractional derivative, 𝑓∶[0,𝑇]×𝐑×𝐑→𝐑 and 𝑔∶[0,𝑇]×𝐑→𝐑 are given functions, 𝐼_𝑘∶𝐑→𝐑, 𝑦₀∈𝐑, 0=𝑡₀<𝑡₁<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑇, Δ𝑦|_{𝑡=𝑡𝑘}=𝑦(𝑡⁺_𝑘)−𝑦(𝑡⁻_𝑘), and 𝑦(𝑡⁺_𝑘)=lim_ℎ→0⁺𝑦(𝑡_𝑘+ℎ) and 𝑦(𝑡⁻_𝑘)=lim_ℎ→0⁻𝑦(𝑡_𝑘+ℎ) represent the right and left limits of 𝑦(𝑡) at 𝑡=𝑡_𝑘.

Fractional differential equations arise in many fields, such as physics, mechanics, chemistry, economics, engineering, and biological sciences; see [1–6], for example. In the recent years, there has been a significant development in ordinary and partial differential equations involving fractional derivatives; see the monographs of Miller and Ross [3], Podlubny [5], Kilbas et al. [6], and the papers [7–21] and the references therein. In these papers, many authors have investigated the existence of positive solutions for nonlinear fractional differential equation boundary value problems. On the other hand, the uniqueness of positive solutions for nonlinear fractional differential equation boundary value problems has been studied by some authors; see [10, 15, 19, 22, 23], for example.

In [7], Ahmad and Sivasundaram considered the following impulsive hybrid boundary value problem for nonlinear fractional differential equations: ^𝑐𝐷^𝑞𝑥(𝑡)((𝑡[]⧵𝑡=𝑓𝑡,𝑥)),1<𝑞≤2,𝑡∈0,1₁,𝑡₂,…,𝑡_𝑝,Δ𝑥𝑡_𝑘=𝐼_𝑘𝑥𝑡⁻_𝑘,Δ𝑥^′𝑡_𝑘=𝐽_𝑘𝑥𝑡⁻_𝑘,𝑡_𝑘∈()𝑥(0)0,1,𝑘=1,2,…,𝑝,+𝑥^′(0)=0,𝑥(1)+𝑥^′(1)=0, (2) where ^𝑐𝐷^𝑞 is the Caputo fractional derivative, 𝑓∶[0,1]×𝐑→𝐑 is a continuous function, 𝐼_𝑘,𝐽_𝑘∶𝐑→𝐑, Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘) with 𝑥(𝑡⁺_𝑘)=lim_ℎ→0⁺𝑥(𝑡_𝑘+ℎ), and 𝑥(𝑡⁻_𝑘)=lim_ℎ→0⁻𝑥(𝑡_𝑘+ℎ), 𝑘=1,2,…,𝑝 for 0=𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑝<𝑡_𝑝+1=1. Based on contraction mapping principle and the Krasnoselskii fixed point theorem, they discussed some existence results for (2).

In [24], Benchohra and Slimani concerned the existence and uniqueness of solutions for the following initial value problem for Caputo fractional-order differential equations: ^𝑐𝐷^𝛼𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),𝑡∈𝐽=[0,𝑇],𝑡≠𝑡_𝑘,Δ𝑦_{𝑡=𝑡𝑘}=𝐼_𝑘𝑦𝑡⁻_𝑘,𝑦(0)=𝑦₀, (3) where 𝑘=1,…,𝑚, 0<𝛼≤1, 𝑓∶[0,𝑇]×𝐑→𝐑 is a given function, 𝐼_𝑘∶𝐑→𝐑, 𝑦₀∈𝐑, 0=𝑡₀<𝑡₁<⋯<𝑡_𝑚<𝑡_𝑚+1=𝑇, Δ𝑦|_{𝑡=𝑡𝑘}=𝑦(𝑡⁺_𝑘)−𝑦(𝑡⁻_𝑘), and 𝑦(𝑡⁺_𝑘)=lim_ℎ→0⁺𝑦(𝑡_𝑘+ℎ) and 𝑦(𝑡⁻_𝑘)=lim_ℎ→0⁻𝑦(𝑡_𝑘+ℎ) represent the right and left limits of 𝑦(𝑡) at 𝑡=𝑡_𝑘. They gave an existence and uniqueness result for the IVP (3) which was based on the Banach fixed point theorem and also obtained two existence and uniqueness results; the first one was based on the Schaefer fixed point theorem and the second one was based on the nonlinear alternative of the Leray-Schauder type.

Different from the above works [7, 24], in this paper, we will use a fixed point theorem for mixed monotone operators to study the existence and uniqueness of positive solutions for the IVP (1). Our result can not only guarantee the existence of unique positive solution but also be applied to construct iterative scheme for approximating it.

With this context in mind, the outline of this paper is as follows. In Section 2, we will recall certain results from the theory of fractional calculus and some definitions, notations, and results of mixed monotone operators. In Section 3, we will provide some conditions under which the IVP (1) will have a unique positive solution. Finally, in Section 4, we will provide one example, which explicates the applicability of our main result.

2. Preliminaries

For the convenience of the reader, we present here some definitions, lemmas, and basic results that will be used in the proof of our main theorem.

Definition 1 (see [9]). Let 𝜈>0 with 𝜈∈𝐑. Suppose that 𝑦∶[𝑎,∞)→𝐑. Then the 𝜈th Riemann-Liouville fractional integral is defined to be 𝐷_𝑎^−𝜈₊𝑦(𝑡)1∶=Γ(𝜈)^𝑡_𝑎𝑦(𝑠)(𝑡−𝑠)^𝜈−1𝑑𝑠, (4) whenever the right-hand side is defined. Similarly, with 𝜈>0 and 𝜈∈𝐑, one defines the 𝜈th Riemann-Liouville fractional derivative to be 𝐷^𝜈_𝑎+𝑦(𝑡)1∶=Γ()𝑑𝑛−𝜈𝑑𝑡^𝑛𝑛𝑡_𝑎𝑦(𝑠)(𝑡−𝑠)^{𝜈+1−𝑛}𝑑𝑠, (5) where 𝑛∈𝑁 is the unique positive integer satisfying 𝑛−1≤𝜈<𝑛 and 𝑡>𝑎.

Definition 2 (see [11]). For a function 𝑦 given on the interval [𝑎,𝑏], the Caputo fractional-order derivative of order 𝜈 of 𝑦 is defined by ^𝑐𝐷^𝜈_𝑎+𝑦(𝑡)1∶=Γ(𝑛−𝜈)^𝑡_𝑎𝑦^(𝑛)(𝑠)(𝑡−𝑠)^{𝜈+1−𝑛}𝑑𝑠, (6) where 𝑛=[𝜈]+1.
In the sequel, we present some basic concepts in the ordered Banach spaces for completeness and one fixed point theorem which will be used later. For convenience of readers, we suggest that one refers to [23, 25, 26] for details.
Suppose that (𝐸,‖⋅‖) is a real Banach space which is partially ordered by a cone 𝑃⊂𝐸. That is, 𝑥≤𝑦 if and only if 𝑦−𝑥∈𝑃. If 𝑥≤𝑦 and 𝑥≠𝑦, then we denote that 𝑥<𝑦 or 𝑦>𝑥. By 𝜃 we denote the zero element of 𝐸. Recall that a nonempty closed convex set 𝑃⊂𝐸 is a cone if it satisfies (𝑖)  𝑥∈𝑃, 𝜆≥0⇒𝜆𝑥∈𝑃; (𝑖𝑖)  𝑥∈𝑃, −𝑥∈𝑃⇒𝑥=𝜃.
𝑃 is called normal if there exists a constant 𝑁>0 such that, for all 𝑥, 𝑦∈𝐸, 𝜃≤𝑥≤𝑦 implies that ‖𝑥‖≤𝑁‖𝑦‖; in this case 𝑁 is called the normality constant of 𝑃. If 𝑥₁, 𝑥₂∈𝐸, the set [𝑥₁,𝑥₂]={𝑥∈𝐸∣𝑥₁≤𝑥≤𝑥₂} is called the order interval between 𝑥₁ and 𝑥₂.

Definition 3 (see [23, 25, 26]). Consider that 𝐴∶𝑃×𝑃→𝑃 is said to be a mixed monotone operator if 𝐴(𝑥,𝑦) is increasing in 𝑥 and decreasing in 𝑦; that is, 𝑢_𝑖, 𝑣_𝑖(𝑖=1,2)∈𝑃, 𝑢₁≤𝑢₂, and 𝑣₁≥𝑣₂ imply that 𝐴(𝑢₁,𝑣₁)≤𝐴(𝑢₂,𝑣₂). Element 𝑥∈𝑃 is called a fixed point of 𝐴 if 𝐴(𝑥,𝑥)=𝑥.

Lemma 4 (see [25]). Let 𝑃 be normal and let 𝐴∶𝑃×𝑃→𝑃 be a mixed monotone operator. Suppose that(𝑎) there exist 𝑣>𝜃 and 𝑐>0, such that 𝜃<𝐴(𝑣,𝜃)≤𝑣 and 𝐴(𝜃,𝑣)≥𝑐𝐴(𝑣,𝜃),(𝑏) for any 0<𝑎<𝑏<1, there exists 𝜂=𝜂(𝑎,𝑏)>0, such that 𝐴𝑡𝑥,𝑡⁻¹𝑦≥𝑡1+𝜂𝐴𝑥,𝑦,∀𝑎≤𝑡≤𝑏,𝜃≤𝑦≤𝑥≤𝑣. (7)Then 𝐴 has a unique fixed point 𝑥^∗ in [𝜃,𝑣] with 𝑥^∗>𝜃. Moreover, for any initial values 𝑥₀, 𝑦₀∈[𝜃,𝑣], constructing successively the sequences 𝑥_𝑛=𝐴𝑥_𝑛−1,𝑦_𝑛−1,𝑦_𝑛=𝐴𝑦_𝑛−1,𝑥_𝑛−1,𝑛=1,2,…, (8) one has 𝑥_𝑛→𝑥^∗ and 𝑦_𝑛→𝑥^∗ as 𝑛→∞.

3. Main Result

In this section, we apply Lemma 4 to study the IVP (1) and then we obtain a new result on the existence and uniqueness of positive solutions. The existence and uniqueness result is relatively new to the fractional differential equations in this literature.

In our considerations we will work in the Banach space 𝐶[0,𝑇]={𝑥∶[0,𝑇]→𝐑iscontinuous} with the standard norm ‖𝑥‖=sup{|𝑥(𝑡)|∶𝑡∈[0,𝑇]}. Notice that this space can be equipped with a partial order given by 𝑥,𝑦∈𝐶[0,𝑇],𝑥≤𝑦⟺𝑥(𝑡)≤𝑦(𝑡)for𝑡∈[0,𝑇]. (9) Set 𝑃={𝑥∈𝐶[0,𝑇]∣𝑥(𝑡)≥0,𝑡∈[0,𝑇]}, the standard cone. It is clear that 𝑃 is a normal cone in 𝐶[0,𝑇] and the normality constant is 1. Consider the following set of functions: []=[]𝑡𝑃𝐶0,𝑇𝑦∶0,𝑇⟶𝐑∶𝑦∈𝐶_𝑘,𝑡_𝑘+1,𝑘=0,1,…,𝑚andthereexist𝑦𝑡⁻_𝑘and𝑦𝑡⁺_𝑘,𝑘=1,2,…,𝑚with𝑦𝑡⁻_𝑘=𝑦𝑡_𝑘. (10) Then 𝑃𝐶[0,𝑇] is a Banach space with the norm 𝑦_𝑃𝐶=sup_{[]𝑡∈0,𝑇}𝑦(𝑡). (11) Denote 𝑃_𝐶 by 𝑃_𝐶={𝑥∈𝑃𝐶[0,𝑇]∣𝑥(𝑡)≥0,𝑡∈[0,𝑇]}; then 𝑃_𝐶 is a standard and normal cone in 𝑃𝐶[0,𝑇].

Lemma 5 (see [24]). Let 0<𝛼≤1 and let ℎ∶𝐽→𝐑 be continuous. A function 𝑦∈𝑃𝐶[0,𝑇] is a solution of the fractional integral equation: 𝑦(𝑡)=𝑦₀+1Γ(𝛼)^𝑡₀(𝑡−𝑠)^𝛼−1ℎ(𝑠)𝑑𝑠,𝑖𝑓𝑡∈0,𝑡₁,𝑦₀+1Γ(𝛼)^𝑘_𝑖=1^𝑡𝑖_𝑡𝑖−1𝑡_𝑖−𝑠^𝛼−1ℎ(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1ℎ(𝑠)+𝑑𝑠_𝑘𝑖=1𝐼_𝑖𝑦𝑡⁻_𝑖𝑖𝑓𝑡∈𝑡_𝑘,𝑡_𝑘+1, (12) where 𝑘=1,…,𝑚, if and only if 𝑦 is a solution of the fractional IVP ^𝑐𝐷^𝛼𝑦(𝑡)=ℎ(𝑡),𝑡∈[0,𝑇],𝑡≠𝑡_𝑘,Δ𝑦_{𝑡=𝑡𝑘}=𝐼_𝑘𝑦𝑡⁻_𝑘𝑦(0),𝑘=1,…,𝑚,=𝑦₀. (13)

Remark 6. The concept of solutions for fractional differential equations has been argued extensively; see [7, 18, 21, 24] and the references therein. In [18], the authors gave a counterexample to show that the result in Lemma 5 is not reasonable. However, the counterexample is inappropriate. From the very recent paper [21], the approach for finding the solution of impulsive fractional differential equations in [18] is inappropriate and some arguments like Lemma 5 are plausible.
For the convenience, we list the following conditions:(𝐻₁)𝑓∶[0,𝑇] × [0,∞) × [0,∞)→[0,∞) is continuous and 𝑔∶[0,𝑇] × [0,∞)→[0,∞) is continuous;(𝐻₂)𝑓(𝑡,𝑢,𝑣) is increasing in 𝑢∈[0,∞) for fixed 𝑡∈[0,𝑇] and 𝑣∈[0,∞) and decreasing in 𝑣∈[0,∞) for fixed 𝑡∈[0,𝑇] and 𝑢∈[0,∞), and 𝑔(𝑡,𝑣) is decreasing in 𝑣∈[0,∞) for fixed 𝑡∈[0,𝑇];(𝐻₃) for any 0<𝑝<𝑞<1, there exist 𝛽₁, 𝛽₂, and 𝛽₃∈(0,∞) which depend on 𝑞, such that 𝑓𝑡,𝜆𝑢,𝜆⁻¹𝑣≥𝜆1−𝜆^𝛽1𝑓(),[][),𝑔𝑡,𝑢,𝑣∀𝑡∈0,𝑇,𝑝≤𝜆≤𝑞,𝑢,𝑣∈0,∞𝑡,𝜆⁻¹𝑣≥𝜆1−𝜆^𝛽2𝑔(),[][),𝐼𝑡,𝑣∀𝑡∈0,𝑇,𝑝≤𝜆≤𝑞,𝑣∈0,∞_𝑘()≥𝜆𝜆𝑢1−𝜆^𝛽3𝐼_𝑘(𝑢),∀𝑘=1,2,…𝑚,𝑝≤𝜆≤𝑞,𝑢∈[0,∞); (14)(𝐻₄) the functions 𝐼_𝑘∶[0,∞)→𝐑 are continuous and nondecreasing and there exists a constant 𝑀^∗>0 such that |𝐼_𝑘(𝑢)|≤𝑀^∗ for each 𝑢∈[0,∞), 𝑘=1,…,𝑚;(𝐻₅)𝑦₀>𝑚𝑀^∗>0.

Theorem 7. Assume that (𝐻₁)--(𝐻₅) are satisfied and there exists 𝑅>0 such that 𝑦₀+(𝑚+1)𝑀₁+𝑀₂Γ()𝑇𝛼+1^𝛼+𝑚𝑀^∗≤𝑅, (15) where 𝑀₁=max_{𝑡∈[0,𝑇]}𝑓(𝑡,𝑅,0), 𝑀₂=max_{𝑡∈[0,𝑇]}𝑔(𝑡,0).
Then the IVP (1) has a unique positive solution 𝑢^∗ in 𝑃_[𝜃,𝑅]={𝑦∈𝑃_𝐶∣0≤𝑦(𝑡)≤𝑅,𝑡∈[0,𝑇]} with 𝑢^∗(𝑡)≢0, 𝑡∈[0,𝑇]. Moreover, for any initial values 𝑥₀, 𝑦₀∈𝑃_𝐶, constructing successively the sequences 𝑥_𝑛+1(𝑡)=𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓𝑠,𝑥_𝑛(𝑠),𝑦_𝑛(𝑠)+𝑔𝑠,𝑦_𝑛(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑥_𝑛(𝑠),𝑦_𝑛(𝑠)+𝑔𝑠,𝑦_𝑛(𝑠)+𝑑𝑠_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑥_𝑛𝑡⁻_𝑘,𝑦_𝑛+1(𝑡)=𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓𝑠,𝑦_𝑛(𝑠),𝑥_𝑛(𝑠)+𝑔𝑠,𝑥_𝑛(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑦_𝑛(𝑠),𝑥_𝑛(𝑠)+𝑔𝑠,𝑥_𝑛(𝑠)𝑑𝑠+_0<𝑡𝑘<t𝐼_𝑘𝑦_𝑛𝑡⁻_𝑘, (16) where 𝑛=0,1,2,…, then one has ‖𝑥_𝑛−𝑢^∗‖→0 and ‖𝑦_𝑛−𝑢^∗‖→0 as 𝑛→∞.

Proof. To begin with, from Lemma 5, the IVP (1) has an integral formulation given by 𝑢(𝑡)=𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓((𝑠)(𝑠((𝑠+1𝑠,𝑢,𝑢))+𝑔𝑠,𝑢))𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓((𝑠)(𝑠((𝑠𝑠,𝑢,𝑢))+𝑔𝑠,𝑢))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢𝑡⁻_𝑘. (17) Define an operator 𝐴∶𝑃_𝐶×𝑃_𝐶→𝑃𝐶[0,𝑇] by 𝐴(𝑢,𝑣)(𝑡)=𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓((𝑠)(𝑠((𝑠+1𝑠,𝑢,𝑣))+𝑔𝑠,𝑣))𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓((𝑠)(𝑠((𝑠𝑠,𝑢,𝑣))+𝑔𝑠,𝑣))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢𝑡⁻_𝑘. (18) It is easy to prove that 𝑢 is the solution of the IVP (1) if and only if 𝑢 is the fixed point of 𝐴. From (𝐻₁), (𝐻₄), and (𝐻₅), we know that 𝐴(𝑢,𝑣)(𝑡)=𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓((𝑠)(𝑠((𝑠+1𝑠,𝑢,𝑣))+𝑔𝑠,𝑣))𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓((𝑠)(𝑠((𝑠𝑠,𝑢,𝑣))+𝑔𝑠,𝑣))𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢𝑡⁻_𝑘,≥𝑦₀−𝑚𝑀^∗>0, (19) for any 𝑢,𝑣∈𝑃_𝐶, 𝑡∈[0,𝑇]. So, 𝐴∶𝑃_𝐶×𝑃_𝐶→𝑃_𝐶. In the sequel we check that 𝐴 satisfies all assumptions of Lemma 4.
Firstly, we prove that 𝐴 is a mixed monotone operator. In fact, for 𝑢_𝑖,𝑣_𝑖∈𝑃_𝐶, 𝑖=1,2 with 𝑢₁≥𝑢₂, 𝑣₁≤𝑣₂, we know that 𝑢₁(𝑡)≥𝑢₂(𝑡), 𝑣₁(𝑡)≤𝑣₂(𝑡), and 𝑡∈[0,𝑇], and by (𝐻₂) and Definition 3, we have that 𝐴𝑢₁,𝑣₁(𝑡)=𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓𝑠,𝑢₁(𝑠),𝑣₁(𝑠)+𝑔𝑠,𝑣₁(𝑠+1)𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑢₁(𝑠),𝑣₁(𝑠)+𝑔𝑠,𝑣₁(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢₁𝑡⁻_𝑘≥𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓𝑠,𝑢₂(𝑠),𝑣₂(𝑠)+𝑔𝑠,𝑣₂(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓𝑠,𝑢₂(𝑠),𝑣₂(𝑠)+𝑔𝑠,𝑣₂(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘𝑢₂𝑡⁻_𝑘=𝐴𝑢₂,𝑣₂(𝑡). (20) That is, 𝐴(𝑢₁,𝑣₁)≥𝐴(𝑢₂,𝑣₂).
Next, we show that 𝐴 satisfies the condition (𝑎) of Lemma 4. Now, we set 𝑣(𝑡)≡𝑅(𝑡∈[0,𝑇]). Then, 𝑣>𝜃.
On the one hand, from (𝐻₁) and (𝐻₅), we have 𝐴(𝑣,𝜃)(𝑡)=𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓()(+1𝑠,𝑅,0+𝑔𝑠,0)𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓(𝑠,𝑅,0)+𝑔(𝑠,0)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑅),≥𝑦₀−𝑚𝑀^∗>0. (21) That is to say, 𝐴(𝑣,𝜃)>𝜃. From (15), for any 𝑡∈[0,𝑇], we obtain 𝐴(𝑣,𝜃)(𝑡)≤𝑦₀+𝑚𝑀₁+𝑀₂Γ()𝑇𝛼+1^𝛼+𝑀₁+𝑀₂Γ()𝑇𝛼+1^𝛼+𝑚𝑀^∗≤𝑅=𝑣(𝑡), (22) That is, 𝐴(𝑣,𝜃)≤𝑣.
On the other hand, we take 𝑐∈(0,1) such that 𝑚𝑐≤min¹+𝑚₂𝑀₁+𝑀₂,𝑚₃𝑀^∗, (23) where 𝑚₁=min_{𝑡∈[0,𝑇]}𝑓(𝑡,0,𝑅), 𝑚₂=min_{𝑡∈[0,𝑇]}𝑔(𝑡,𝑅), and 𝑚₃=min_{𝑘∈{1,…,𝑚}}𝐼_𝑘(0). For any 𝑡∈[0,𝑇], 𝐴(𝜃,𝑣)(𝑡)=𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓()(+1𝑠,0,𝑅+𝑔𝑠,𝑅)𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓(𝑠,0,𝑅)+𝑔(𝑠,𝑅)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝐼_𝑘(0)≥𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑚₁+𝑚₂+1𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑚₁+𝑚₂𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑚₃≥𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑐𝑀₁+𝑀₂+1𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑐𝑀₁+𝑀₂𝑑𝑠+_{0<𝑡𝑘<𝑡}𝑐𝑀^∗≥𝑐𝑦₀+𝑐Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓()(+𝑐𝑠,𝑅,0+𝑔𝑠,0)𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓(𝑠,𝑅,0)+𝑔(𝑠,0)𝑑𝑠+𝑐_{0<𝑡𝑘<𝑡}𝐼_𝑘(𝑅)=𝑐𝐴(𝑣,𝜃)(𝑡). (24) That is to say, 𝐴(𝜃,𝑣)≥𝑐𝐴(𝑣,𝜃). Therefore, condition (𝑎) of Lemma 4 holds.
Finally, we show that 𝐴 satisfies condition (𝑏) of Lemma 4. Let 0<𝑎<𝑏<1 and 1𝜂=min𝑏1−1,1−𝑎^𝛽11−1,1−𝑎^𝛽21−1,1−𝑎^𝛽3−1>0. (25) For any 𝑎≤𝜆≤𝑏, 𝑥,𝑦∈𝑃_𝐶, and 𝑡∈[0,T], by (𝐻₃) we have 𝐴𝜆𝑥,𝜆⁻¹𝑦(𝑡)=𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝑓𝑠,𝜆𝑥(𝑠),𝜆⁻¹𝑦(𝑠)+𝑔𝑠,𝜆⁻¹𝑦(𝑠)+1𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝑓𝑠,𝜆𝑥(𝑠),𝜆⁻¹𝑦(𝑠)+𝑔𝑠,𝜆⁻¹𝑦(𝑠)+𝑑𝑠_{0<𝑡𝑘<𝑡}𝐼_𝑘𝜆𝑥𝑡⁻_𝑘≥𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝜆1−𝜆^𝛽1𝑓(𝑠)(𝑠)+𝜆𝑠,𝑥,𝑦1−𝜆^𝛽2𝑔(𝑠+1𝑠,𝑦)𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝜆1−𝜆^𝛽1𝑓(𝑠)(𝑠)+𝜆𝑠,𝑥,𝑦1−𝜆^𝛽2𝑔𝑠,𝑦(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝜆1−𝜆^𝛽3𝐼_𝑘𝑥𝑡⁻_𝑘≥𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝜆1−𝑎^𝛽1𝑓(𝑠)(𝑠)+𝜆𝑠,𝑥,𝑦1−𝑎^𝛽2𝑔(𝑠)+1𝑠,𝑦𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝜆1−𝑎^𝛽1𝑓(𝑠)(𝑠+𝜆𝑠,𝑥,𝑦)1−𝑎^𝛽2𝑔𝑠,𝑦(𝑠)𝑑𝑠+_{0<𝑡𝑘<𝑡}𝜆1−𝑎^𝛽3𝐼_𝑘𝑥𝑡⁻_𝑘≥𝑏1+𝜂𝑦₀+1Γ(𝛼)×_{0<𝑡𝑘<𝑡}^𝑡𝑘_𝑡𝑘−1𝑡_𝑘−𝑠^𝛼−1𝜆𝑓(𝑠)(𝑠𝑔(𝑠)+11+𝜂𝑠,𝑥,𝑦)+𝜆1+𝜂𝑠,𝑦𝑑𝑠Γ(𝛼)^𝑡_𝑡𝑘(𝑡−𝑠)^𝛼−1𝜆𝑓(𝑠)(𝑠𝑔(𝑠)1+𝜂𝑠,𝑥,𝑦)+𝜆1+𝜂𝑠,𝑦𝑑𝑠+_{0<𝑡𝑘<𝑡}𝜆1+𝜂𝐼_𝑘𝑥𝑡⁻_𝑘≥𝜆1+𝜂𝐴𝑥,𝑦(𝑡). (26) That is to say, 𝐴(𝜆𝑥,𝜆⁻¹𝑦)≥𝜆(1+𝜂)𝐴(𝑥,𝑦), for all 𝑎≤𝜆≤𝑏, 𝑥,𝑦∈𝑃_𝐶. Hence, condition (𝑏) of Lemma 4 is satisfied. So, the conclusion of Theorem 7 follows from Lemma 4.

4. An Example

We present one example to illustrate Theorem 7.

Example 1. Consider the following fractional IVP: ^𝑐𝐷^1/5𝑦(𝑡)(𝑡)(𝑡)+𝑦(𝑡)=𝜏+𝛿(𝑡)+21+𝑦(𝑡)[]11+𝑦+2,𝑡∈0,1,𝑡≠3,Δ𝑦_𝑡=(1/3)=𝑦(1/3⁻)4+𝑦(1/3⁻)+1,𝑦(0)=2, (27) where 𝜏, 𝛿∈𝐶([0,1,0,∞)).
In this example, 1𝛼=5,𝑇=1,𝑡₁=13,𝑚=1,𝑦₀𝑓(𝑡)𝑥=2,𝑡,𝑥,𝑦=𝜏+1++11+𝑥,𝑔(𝑡)11+𝑦𝑡,𝑦=𝛿+1+,𝐼1+𝑦₁(𝑥)𝑥=1+.4+𝑥 (28) Then, 𝑓(𝑡,𝑥,𝑦)∈𝐶([0,1]×[0,∞)×[0,∞), [0,∞)) is continuous and increasing in 𝑥∈[0,∞) for fixed 𝑡∈[0,1] and 𝑦∈[0,∞) and decreasing in 𝑦∈[0,∞) for fixed 𝑡∈[0,1] and 𝑥∈[0,∞). And 𝑔(𝑡,𝑦)∈𝐶([0,1]×[0,∞), [0,∞)) is continuous and decreasing in 𝑦∈[0,∞) for fixed 𝑡∈[0,1]. Moreover, for any 0<𝑝<𝑞<1, we take 𝛽₁≥log_𝑞1−3𝑞1+2𝑞,𝛽₂≥log_𝑞1−2𝑞,𝛽1+𝑞₃≥log_𝑞1−2𝑞.1+𝑞 (29) For any 𝑝≤𝜆≤𝑞, 𝑡∈[0,1], and 𝑥,𝑦∈[0,∞), we have 𝑓𝑡,𝜆𝑥,𝜆⁻¹𝑦=𝜏(𝑡)+1+𝜆𝑥+𝜆1+𝜆𝑥≥𝜆𝜆+𝑦𝑞(𝜏(𝑡))𝑥+1+𝜆+11+𝑥.1+𝑦 (30) We show that 𝑓𝑡,𝜆𝑥,𝜆⁻¹𝑦≥𝜆1−𝜆^𝛽1𝑓𝑡,𝑥,𝑦, (31) if and only if we can prove that 1𝑞(𝜏(𝑡))+𝑥+1+11+𝑥≥11+𝑦1−𝜆^𝛽1𝜏(𝑡)𝑥+1++11+𝑥.1+𝑦 (32) That is to say, (𝜏(𝑡))1+1𝑞−11−𝜆^𝛽1≥11−𝜆^𝛽1𝑥−1+11+𝑥.1+𝑦 (33) From (29), we get 𝜆^𝛽1≤𝑞^𝛽1≤1−3𝑞,1+2𝑞 (34) and then 1𝑞−11−𝜆^𝛽11≥21−𝜆^𝛽1−1. (35) Hence, (33) is satisfied. So, (31) holds.
Similarly, we obtain that 𝑔𝑡,𝜆⁻¹𝑦(𝑡)𝜆=𝛿+1+≥𝜆𝜆+𝑦𝑞(𝛿(𝑡))+𝜆+1≥𝜆1+𝑦1−𝜆^𝛽2𝛿(𝑡)1+1+=𝜆1+𝑦1−𝜆^𝛽2𝑔,𝐼𝑡,𝑦₁(𝜆𝑥)=1+𝜆𝑥≥𝜆4+𝜆𝑥𝑞+𝜆𝑥≥𝜆4+𝑥1−𝜆^𝛽3𝑥1+=𝜆4+𝑥1−𝜆^𝛽3𝐼₁(𝑥). (36) Further, 𝐼₁(𝑥) is continuous, increasing in 𝑥∈[0,∞), and 𝐼₁(𝑥=𝑥)4+𝑥≤1≡𝑀^∗,∀𝑥∈[0,∞). (37) And, we also have 𝑦₀=2>1=𝑚𝑀^∗>0.
Finally, owing to 𝜏,𝛿∈𝐶([0,1],[0,∞)), we have that 𝜏^∗=max_[]𝑡∈0,1𝜏(𝑡),𝛿^∗=max_[]𝑡∈0,1𝛿(𝑡). (38) We take 𝑅>0 such that 23+5+𝜏^∗+𝛿^∗Γ((1/5)+1)≤𝑅. (39) And thus, 𝑀₁=max_[]𝑡∈0,1𝑓(𝑡,𝑅,0)=max_[]𝑡∈0,1𝜏(𝑡)𝑅+1+1+𝑅+1≤3+𝜏^∗,𝑀₂=max_[]𝑡∈0,1𝑔(𝑡,0)=max_[]𝑡∈0,1(𝛿(𝑡)+1+1)=𝛿^∗+2. (40) Then, 𝑦₀+(𝑚+1)𝑀₁+𝑀₂Γ()𝑇𝛼+1^𝛼+𝑚𝑀^∗2≤2+Γ((1/5)+1)3+𝜏^∗+2+𝛿^∗2+1=3+5+𝜏^∗+𝛿^∗Γ((1/5)+1)≤𝑅. (41) Hence, (15) holds. Therefore, all the conditions of Theorem 7 are satisfied. An application of Theorem 7 implies that problem (27) has a unique positive solution.

Acknowledgments

This research was supported by the Youth Science Foundation of China (11201272), the Science Foundation of Shanxi Province (2013011003-3), and the Science Foundation of Business College of Shanxi University (2012050).