Positive Solutions of a Singular Nonlocal Fractional Order Differential System via Schauder’s Fixed Point Theorem

Abstract

We establish the existence of positive solutions to a class of singular nonlocal fractional order differential system depending on two parameters. Our methods are based on Schauder’s fixed point theorem.

1. Introduction

Differential equations of fractional order have recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Particularly, fractional derivatives provide an excellent tool for the description of memory and hereditary properties of many materials and processes. With this advantage, fractional order models are more realistic and practical than the classical integer-order models in physics, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, fitting of experimental data, and so forth [1–16]. Recently, Rehman and Khan [17] studied the fractional order multipoint boundary value problem: 𝐷^𝛼_𝑡𝑦(𝑡)=𝑓𝑡,𝑦(𝑡),𝐷^𝛽_𝑡𝑦(𝑡)(),𝑦(0),𝑡∈0,1=0,𝐷^𝛽_𝑡𝑦(1)−^𝑚−2_𝑖=1𝜁_𝑖𝐷^𝛽_𝑡𝑦𝜉_𝑖=𝑦₀, (1) where 1<𝛼≤2, 0<𝛽<1, 𝜁_𝑖∈[0,+∞), 0<𝜉_𝑖<1, with ∑^𝑚−2_𝑖=1𝜁_𝑖𝜉_𝑖<1. By using the contraction mapping principle, the existence and uniqueness of positive solutions were established. In [18], Zhang et al. discussed the existence and uniqueness of positive solutions for the following fractional differential equation with derivatives: −𝒟_𝐭^𝛼𝑥(𝑡)=𝑓𝑡,𝑥(𝑡),−𝒟_𝐭^𝛽𝑥(𝑡)(),𝒟,𝑡∈0,1_𝐭𝛽𝑥(0)=0,𝒟_𝐭^𝛾𝑥(1)=^𝑝−2_𝑗=1𝑎_𝑗𝒟_𝐭^𝛾𝑥𝜉_𝑗, (2) where 1<𝛼≤2, 𝛼−𝛽>1, 0<𝛽≤𝛾<1, 0<𝜉₁<𝜉₂<⋯<𝜉_𝑝−2<1, 𝑎_𝑗∈[0,+∞) with 𝑐=∑^𝑝−2_𝑗=1𝑎_𝑗𝜉_𝑗^{𝛼−𝛾−1}<1, and 𝒟_𝐭 is the standard Riemann-Liouville derivative. 𝑓∶(0,1)×[0,+∞)×(−∞,+∞)→[0,+∞) is continuous, and 𝑓(𝑡,𝑢,𝑣) may be singular at 𝑡=0,1. By means of monotone iterative technique, the existence and uniqueness of the positive solution for a fractional differential equation with derivatives are established, and the iterative sequence of the solution, an error estimation, and the convergence rate of the positive solution are also given.

However, the research on the systems of fractional differential equations has not received much attention. So motivated by the results mentioned above, in this paper, we study the existence of positive solutions for the following singular nonlocal fractional order differential system depending on two parameters: −𝐷^𝛼_𝑡𝑢(𝑡)=𝜆𝑓𝑡,𝐷^𝛽1_𝑡𝑣(𝑡),−𝐷^𝛽_𝑡𝑣(𝑡)=𝜇𝑔𝑡,𝐷^𝛼1_𝑡𝑢(𝑡),𝐷0<𝑡<1,_𝛼1𝑡𝑢(0)=0,𝐷^𝛾1_𝑡𝑢(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^𝛾1_𝑡𝑢𝜁_𝑗,𝐷^𝛽1_𝑡𝑣(0)=0,𝐷^𝛿1_𝑡𝑣(1)=^𝑚−2_𝑗=1𝑎_𝑗𝐷^𝛿1_𝑡𝑣𝜉_𝑗, (3) where 1<𝛼, 𝛽≤2, 𝛼−𝛾₁≥1, 𝛽−𝛿₁≥1, 0≤𝛽₁≤𝛿₁, 0≤𝛼₁≤𝛾₁, 𝑎_𝑗, 𝑏_𝑗∈[0,+∞), 𝜉_𝑖,𝜁_𝑖∈(0,1) with 0<∑^𝑚−2_𝑗=1𝑎_𝑗𝜉_𝑗, ∑^𝑚−2_𝑗=1𝑏_𝑗𝜁_𝑗<1, 𝐷^𝛼_𝑡, 𝐷^𝛽_𝑡 are the standard Riemann-Liouville derivatives, and 𝜆 and 𝜇 are positive parameters. 𝑓∶(0,1)×(0,+∞)→[0,+∞) and 𝑔∶[0,1]×[0,+∞)→(0,+∞) are continuous and 𝑓(𝑡,𝑥) may be singular at 𝑡=0,1 and 𝑥=0. The system (3) is an abstract model arising from biological dynamic system, which was introduced by Perelson [5] to describe the primary infection with HIV in integer-order version, and was extended to a fractional order version of HIV-1 infection of CD4⁺ T-cells by Arafa et al. [11].

The present paper has several interesting features. Firstly, the system depends on two parameters and the nonlinear terms 𝑓 and 𝑔 are allowed to have different nonlinear character; that is, 𝑓 is decreasing on 𝑦 and 𝑔 is increasing on 𝑥; secondly, 𝑓 may be singular at 𝑦=0 and 𝑡=0,1; so far fewer work was done when 𝑓 can be singular at 𝑦=0; thirdly, the boundary conditions of the system are nonlocal and involve fractional derivatives of the unknown functions.

2. Preliminaries and Lemmas

In this section, we firstly define an appropriate invariant set and then make a change of variables for the system (3) so that Schauder’s fixed point theorem can be applied. Our work is based on fractional framework; for further background knowledge of fractional calculus, we refer readers to the monographs [1–4] or the papers [6, 8, 17, 18] and the references therein.

Throughout this paper, we mean by 𝐶[0,1] the Banach space of all continuous functions on [0,1] with the usual norm ‖𝑥‖=max_0≤𝑡≤1|𝑥(𝑡)|. Let 𝑃={𝑥∈𝐶[0,1]∶𝑥(𝑡)≥0,𝑡∈[0,1]}; (4) then 𝑃 is a normal cone in the Banach space 𝐸. Thus the space 𝐶[0,1] can be equipped with a partial order given by 𝑥,𝑦∈𝐶[0,1],𝑥−𝑦∈𝑃⟺𝑥(𝑡)≥𝑦(𝑡),for𝑡∈[0,1]. (5) Now define a subcone of 𝑃 as follows: 𝐵=𝑥(𝑡)∈𝑃∶thereexisttwopositivenumbers𝐿_𝑥>1>𝑙_𝑥suchthat𝑙_𝑥𝑡^{𝛼−𝛼1−1}≤𝑥(𝑡)≤𝐿_𝑥𝑡^{𝛼−𝛼1−1},𝑡∈[0,1]. (6) Obviously, 𝐵 is nonempty since 𝑡^{𝛼−𝛼1−1}∈𝐵.

Lemma 1. Let 𝑢(𝑡)=𝐼^𝛼1𝑥(𝑡), 𝑣(𝑡)=𝐼^𝛽1𝑦(𝑡), 𝑥(𝑡), and 𝑦(𝑡)∈𝐶[0,1]. Then system (3) is turned into the equivalent one: −𝐷^𝛼−𝛼1_𝑡𝑥(𝑡)=𝜆𝑓𝑡,𝑦(𝑡),−𝐷^𝛽−𝛽1_𝑡𝑦(𝑡)((𝑡,𝑥(0)=𝜇𝑔𝑡,𝑥))0<𝑡<1,=0,𝐷^{𝛾1−𝛼1}_𝑡𝑥(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝑥𝜁_𝑗,𝑦(0)=0,𝐷^{𝛿1−𝛽1}_𝑡𝑦(1)=^𝑚−2_𝑗=1𝑎_𝑗𝐷^{𝛿1−𝛽1}_𝑡𝑦𝜉_𝑗, (7) and if (𝑥,𝑦)∈𝐶[0,1]×𝐶[0,1] is a solution of the problem (7), then (𝐼^𝛼1𝑥(𝑡),𝐼^𝛽1𝑦(𝑡)) is a solution of the system (3).

Proof. By using semigroup property of the fractional integration operator (see [1] page 73, Lemma 2.3 or [4] Sections 2.3 and 2.5), one has 𝐷^𝛼_𝑡𝑢(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝐼^𝑛−𝛼𝐼^𝛼1𝑥(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝐼^{𝑛−𝛼+𝛼1}𝑥(𝑡)=𝐷^𝛼−𝛼1_𝑡𝑥(𝑡),𝐷^𝛾1_𝑡𝑢(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝐼^𝑛−𝛾1𝐼^𝛼1𝑥(𝑡)=𝑑^𝑛𝑑𝑡^𝑛𝐼^{𝑛−𝛾1+𝛼1}𝑥(𝑡)=𝐷^{𝛾1−𝛼1}_𝑡𝑥(𝑡),𝐷^𝛼1_𝑡𝑢(𝑡)=𝐷^𝛼1_𝑡𝐼^𝛼1𝑥(𝑡)=𝑥(𝑡). (8) And then, it follows from (8) that 𝐷^𝛼1_𝑡𝑢(0)(0)𝐷=𝑥=0,_{𝛾1−𝛼1𝑡}𝑥(1)=𝐷^𝛾1_𝑡𝑢(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^𝛾1_𝑡𝑢𝜁_𝑗=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝑥𝜁_𝑗. (9) In the same way, we also have 𝑦(0)=0,𝐷^{𝛿1−𝛽1}_𝑡𝑦(1)=^𝑚−2_𝑗=1𝑎_𝑗𝐷^{𝛿1−𝛽1}_𝑡𝑦𝜉_𝑗. (10) It follows from 𝑢(𝑡)=𝐼^𝛼1𝑥(𝑡) and 𝑣(𝑡)=𝐼^𝛽1𝑦(𝑡) that 𝑥(𝑡)=𝐷^𝛼1_𝑡𝑢(𝑡) and 𝑦(𝑡)=𝐷^𝛽1_𝑡𝑣(𝑡). So substituting the above formulas into (3), we obtain (7).
On the other hand, if (𝑥,𝑦) is a solution of (7), (8) yields −𝐷^𝛼_𝑡𝑢(𝑡)𝑑=−^𝑛𝑑𝑡^𝑛𝐼^𝑛−𝛼𝐼^𝛼1𝑥(𝑡)𝑑=−^𝑛𝑑𝑡^𝑛𝐼^{𝑛−𝛼−𝛼1}𝑥(𝑡)=−𝐷^𝛼−𝛼1_𝑡𝑥(𝑡)=𝜆𝑓𝑡,𝑦(𝑡)=𝜆𝑓𝑡,𝐷^𝛽1_𝑡𝑣(𝑡),−𝐷^𝛽_𝑡𝑣(𝑡)=−𝐷^𝛽−𝛽1_𝑡𝑦(𝑡)=𝜇𝑔(𝑡,𝑥(𝑡))=𝜇𝑔𝑡,𝐷^𝛼1_𝑡𝑢(𝑡),𝐷^𝛼1_𝑡𝑢(𝑡)=𝐷^𝛼1_𝑡𝐼^𝛼1𝑥(𝑡)=𝑥(𝑡),𝐷^𝛾1_𝑡𝑢(𝑡)=𝐷^{𝛾1−𝛼1}_𝑡𝑥(𝑡),𝐷^𝛽1_𝑡𝑣(𝑡)=𝐷^𝛽1_𝑡𝐼^𝛽1𝑦(𝑡)=𝑦(𝑡),𝐷^𝛿1_𝑡𝑣(𝑡)=𝐷^{𝛿1−𝛽1}_𝑡𝑦(𝑡). (11) So from (11), we have 𝐷^𝛼1_𝑡𝑢(0)=0,𝐷^𝛾1_𝑡𝑢(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^𝛾1_𝑡𝑢𝜁_𝑗,𝐷^𝛽1_𝑡𝑣(0)=0,𝐷^𝛿1_𝑡𝑣(1)^𝑚−2_𝑗=1𝑎_𝑗𝐷^{𝛿1−𝛽1}_𝑡𝑣𝜉_𝑗. (12) Moreover 𝐼^𝛼1𝑥(𝑡),𝐼^𝛽1𝑦(𝑡)∈𝐶[0,1]. (13) Consequently, (𝐼^𝛼1𝑥(𝑡),𝐼^𝛽1𝑦(𝑡)) is a positive solution of (3).

Now we recall some useful lemmas by [18], which are important to the proof of our main results.

Lemma 2 (see [18]). Let ℎ∈𝐿¹[0,1], if 1<𝛼−𝛼₁≤2; then the unique solution of the linear problem −𝐷^𝛼−𝛼1_𝑡𝑥(𝑡)(𝑡)(),𝑥(0)=ℎ,𝑡∈0,1=0,𝐷^{𝛾1−𝛼1}_𝑡𝑥(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝑥𝜁_𝑗, (14) is given by 𝑥(𝑡)=¹₀𝐾_𝛼(𝑡,𝑠)ℎ(𝑠)𝑑𝑠, (15) where 𝐾_𝛼(𝑡,𝑠)=𝐾_𝛼1()+𝑡𝑡,𝑠^{𝛼−𝛼1−1}1−∑^𝑚−2_𝑗=1𝑏_𝑗𝜁^𝛼−𝛾1_𝑗^−1𝑚−2_𝑗=1𝑏_𝑗𝐾_𝛼2𝜁_𝑗,𝐾,𝑠_𝛼1()=1𝑡,𝑠Γ𝛼−𝛼₁𝑡^{𝛼−𝛼1−1}(1−𝑠)^{𝛼−𝛾1−1}−(𝑡−𝑠)^{𝛼−𝛼1−1},𝑡0≤𝑠≤𝑡≤1,_{𝛼−𝛼1−1}(1−𝑠)^{𝛼−𝛾1−1},𝐾0≤𝑡≤𝑠≤1,_𝛼2()=1𝑡,𝑠Γ𝛼−𝛼₁(𝑡(1−𝑠))^{𝛼−𝛾1−1}−(𝑡−𝑠)^{𝛼−𝛾1−1},(𝑡(0≤𝑠≤𝑡≤1,1−𝑠))^{𝛼−𝛾1−1},0≤𝑡≤𝑠≤1. (16) Moreover, for any 𝑡,𝑠∈[0,1], 𝐾_𝛼𝑖()≤()𝑡,𝑠1−𝑠^{𝛼−𝛾1−1}Γ𝛼−𝛼₁,𝑖=1,2. (17)

By Lemma 2, similar results are valid for the problem −𝐷^𝛽−𝛽1_𝑡𝑦(𝑡)(𝑡)[],𝑦(0)=ℎ,𝑡∈0,1=0,𝐷^{𝛿1−𝛽1}_𝑡𝑦(1)=^𝑚−2_𝑗=𝑖𝑎_𝑗𝐷^{𝛿1−𝛽1}_𝑡𝑦𝜉_𝑗. (18) For convenience, we adopt the following corresponding notations: 𝐾_𝛽(𝑡,𝑠)=𝐾_𝛽1()+𝑡𝑡,𝑠^{𝛽−𝛽1−1}1−∑^𝑚−2_𝑗=1𝑎_𝑗𝜉^𝛽−𝛿1_𝑗^−1𝑚−2_𝑗=1𝑎_𝑗𝐾_𝛽2𝜉_𝑗,𝐾,𝑠_𝛽1()=1𝑡,𝑠Γ𝛽−𝛽₁𝑡^{𝛽−𝛽1−1}(1−𝑠)^{𝛽−𝛿1−1}−(𝑡−𝑠)^{𝛽−𝛽1−1},𝑡0≤𝑠≤𝑡≤1,_{𝛽−𝛽1−1}(1−𝑠)^{𝛽−𝛿1−1},𝐾0≤𝑡≤𝑠≤1,_𝛽2()=1𝑡,𝑠Γ𝛽−𝛽₁(𝑡(1−𝑠))^{𝛽−𝛿1−1}−(𝑡−𝑠)^{𝛽−𝛿1−1},(𝑡(0≤𝑠≤𝑡≤1,1−𝑠))^{𝛽−𝛿1−1},0≤𝑡≤𝑠≤1. (19)

Lemma 3 (see [18]). The Green functions 𝐾_𝛼(𝑡,𝑠)and 𝐾_𝛽(𝑡,𝑠) have the following properties:(1)𝐾_𝛼(𝑡,𝑠)>0 and 𝐾_𝛽(𝑡,𝑠)>0 for 𝑡,𝑠∈(0,1),(2)there exist functions 𝑚_𝛼(𝑠), 𝑚_𝛽(𝑠), 𝑀_𝛼(𝑠), and 𝑀_𝛽(𝑠) such that 𝑚_𝛼(𝑠)𝑡^{𝛼−𝛼1−1}≤𝐾_𝛼(𝑡,𝑠)≤𝑀_𝛼(𝑠)𝑡^{𝛼−𝛼1−1}(),𝑚,𝑡,𝑠∈0,1_𝛽(𝑠)𝑡^{𝛽−𝛽1−1}≤𝐾_𝛽(𝑡,𝑠)≤𝑀_𝛽(𝑠)𝑡^{𝛽−𝛽1−1},𝑡,𝑠∈(0,1), (20)where 𝑚_𝛼(𝑠)=∑^𝑚−2_𝑗=1𝑏_𝑗𝐾_𝛼2𝜁_𝑗,𝑠1−∑^𝑚−2_𝑗=1𝑏_𝑗𝜁^𝛼−𝛾1_𝑗⁻¹,𝑀_𝛼(𝑠)=1Γ𝛼−𝛼₁+∑^𝑚−2_𝑗=1𝑏_𝑗𝐾_𝛼2𝜁_𝑗,𝑠1−∑^𝑚−2_𝑗=1𝑏_𝑗𝜁^𝛼−𝛾1_𝑗⁻¹,𝑚_𝛽(𝑠)=∑^𝑚−2_𝑗=1𝑎_𝑗𝐾_𝛽2𝜉_𝑗,𝑠1−∑^𝑚−2_𝑗=1𝑎_𝑗𝜉^𝛽−𝛿1_𝑗⁻¹,𝑀_𝛽(𝑠)=1Γ𝛽−𝛽₁+∑^𝑚−2_𝑗=1𝑎_𝑗𝐾_𝛽2𝜉_𝑗,𝑠1−∑^𝑚−2_𝑗=1𝑎_𝑗𝜉^𝛽−𝛿1_𝑗⁻¹. (21)

Clearly, the following maximum principle is direct conclusion of Lemma 2.

Lemma 4. If 1<𝛼−𝛼₁≤2 and 𝑥∈𝐶([0,1],𝑅) satisfies 𝑥(0)=0,𝐷^{𝛾1−𝛼1}_𝑡𝑥(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝑥𝜁_𝑗, (22) and −𝐷^𝛼−𝛼1_𝑡𝑥(𝑡)≥0 for any 𝑡∈(0,1), then 𝑥(𝑡)≥0 for 𝑡∈(0,1).

It is well known that (𝑥,𝑦)∈𝐶(0,1)×𝐶[0,1] is a solution of the system (7) if and only if (𝑥,𝑦) is a solution of the nonlinear integral system of equations 𝑥(𝑡)=𝜆¹₀𝐾_𝛼()𝑓(𝑠)𝑦(𝑡)𝑡,𝑠𝑠,𝑦𝑑𝑠,=𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔(𝑠,𝑥(𝑠))𝑑𝑠, (23) and the system (23) is equivalent to the following nonlinear integral equation 𝑥(𝑡)=𝜆¹₀𝐾_𝛼(𝑡,𝑠)𝑓𝑠,𝜇¹₀𝐾_𝛽()𝑔((𝜏[].𝑠,𝜏𝜏,𝑥))𝑑𝜏𝑑𝑠,𝑡∈0,1 (24)

Let us define a nonlinear operator (𝐹𝑥)(𝑡) by (𝐹𝑥)(𝑡)=𝜆¹₀𝐾_𝛼(𝑡,𝑠)𝑓𝑠,𝜇¹₀𝐾_𝛽()𝑔((𝜏[].𝑠,𝜏𝜏,𝑥))𝑑𝜏𝑑𝑠,𝑡∈0,1 (25) Then the existence of solutions to the system of (7) is equivalent to the existence of fixed point of the nonlinear operator 𝐹; that is, if 𝑥^∗(𝑡) is a fixed point of 𝐹 in 𝐶[0,1], then system (7) has at least one solution (𝑥^∗(𝑡),𝑦^∗(𝑡)) which can be written by 𝑥(𝑡)=𝑥^∗(𝑡),𝑦^∗(𝑡)=𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝑥^∗(𝑠)𝑑𝑠, (26) and then system (3) has at least one solution: 𝐼^𝛼1𝑥^∗(𝑡),𝐼^𝛽1𝑦^∗(𝑡). (27)

In order to find the fixed point of 𝐹, we need the definitions of the upper solution and lower solution for the following integrodifferential equation: −𝐷^𝛼−𝛼1_𝑡𝑥(𝑡)=𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽()𝑔((𝑠,𝑥(0)𝑡,𝑠𝑠,𝑥))𝑑𝑠=0,𝐷^{𝛾1−𝛼1}_𝑡𝑥(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝑥𝜁_𝑗. (28)

Definition 5. A continuous function 𝜙(𝑡) is called a lower solution of the problem (28), if it satisfies −𝐷^𝛼−𝛼1_𝑡𝜙(𝑡)≤𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽()𝑔(𝑠),𝜙(0)𝑡,𝑠𝑠,𝜙𝑑𝑠≥0,𝐷^{𝛾1−𝛼1}_𝑡𝜙(1)≥^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝜙𝜁_𝑗. (29)

Definition 6. A continuous function 𝜑(𝑡) is called an upper solution of the problem (28), if it satisfies −𝐷^𝛼−𝛼1_𝑡𝜑(𝑡)≥𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽()𝑔(𝑠),𝜑(0)𝑡,𝑠𝑠,𝜑𝑑𝑠≤0,𝐷^{𝛾1−𝛼1}_𝑡𝜑(1)≤^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝜑𝜁_𝑗. (30)

3. Main Result

For the convenience in presentation, we now present some assumptions to be used in the rest of the paper.(A1)𝑓∶(0,1)×(0,+∞)→[0,+∞) is continuous and decreasing on 𝑦 in (0,+∞) and 𝑔∶[0,1]×[0,+∞)→(0,+∞) is continuous and increasing on 𝑥 in [0,+∞).(A2)For any real numbers 𝑙, 𝜇>0, ¹₀𝑀_𝛼(𝑠)𝑓𝑠,𝜇¹₀𝐾_𝛽(𝑠,𝜏)𝑔𝜏,𝑙𝜏^{𝛼−𝛼1−1}𝑑𝜏𝑑𝑠<+∞. (31)

Theorem 7. Suppose (A1) and (A2) hold; then for any (𝜆,𝜇)∈(0,+∞)×(0,+∞), the system (3) has at least one positive solution (𝑢^∗,𝑣^∗), and there exist positive constants 𝑟^∗₁, 𝑟^∗₂, 𝑑^∗₁, and 𝑑^∗₂ such that 𝑟^∗₁𝑡^𝛼−1≤𝑢^∗(𝑡)≤𝑟^∗₂𝑡^𝛼−1,𝑑^∗₁𝑡^𝛽−1≤𝑣^∗(𝑡)≤𝑑^∗₂𝑡^𝛽−1,𝑡∈[0,1]. (32)

Proof. We start by showing that (3) has at least one positive solution (𝑢^∗,𝑣^∗). For this purpose, we firstly prove that the operator 𝐹 is well defined and 𝐹(𝐵)⊂𝐵.
For any 𝑥∈𝐵, there exist two positive numbers 𝑙_𝑥<1<𝐿_𝑥 such that 𝑙_𝑥𝑡^{𝛼−𝛼1−1}≤𝑥(𝑡)≤𝐿_𝑥𝑡^{𝛼−𝛼1−1}, so it follows from (25), (20), and (31) that (𝐹𝑥)(𝑡)=𝜆¹₀𝐾_𝛼(𝑡,𝑠)×𝑓𝑠,𝜇¹₀𝐾_𝛽(𝑠,𝜏)𝑔(𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠≤𝜆𝑡^{𝛼−𝛼1−1}×¹₀𝑀_𝛼(𝑠)×𝑓𝑠,𝜇¹₀𝐾_𝛽(𝑠,𝜏)𝑔𝜏,𝑙_𝑥𝜏^{𝛼−𝛼1−1}(𝑡)𝑑𝜏𝑑𝑠<+∞,𝐹𝑥)(≥𝜆𝑡^{𝛼−𝛼1−11}₀𝑚_𝛼(𝑠)𝑓𝑠,𝜇¹₀𝐾_𝛽(𝑠,𝜏)𝑔(𝜏,𝑥(𝜏))𝑑𝜏𝑑𝑠≥𝜆𝑡^{𝛼−𝛼1−1}×¹₀𝑚_𝛼(𝑠)𝑓𝑠,𝜇¹₀𝐾_𝛽(𝑠,𝜏)𝑔𝜏,𝐿_𝑥𝜏^{𝛼−𝛼1−1}𝑑𝜏𝑑𝑠. (33)
Take 𝑙^′_𝑥×=min1,𝜆₁₀𝑚_𝛼(𝑠)×𝑓𝑠,𝜇¹₀𝐾_𝛽(𝑠,𝜏)𝑔𝜏,𝐿_𝑥𝜏^{𝛼−𝛼1−1},𝐿𝑑𝜏𝑑𝑠_′𝑥×=max1,𝜆₁₀𝑀_𝛼(𝑠)×𝑓𝑠,𝜇¹₀𝐾_𝛽(𝑠,𝜏)𝑔𝜏,𝑙_𝑥𝜏^{𝛼−𝛼1−1}𝑑𝜏𝑑𝑠; (34) then 𝑙^′_𝑥𝑡^{𝛼−𝛼1−1}≤(𝐹𝑥)(𝑡)≤𝐿^′_𝑥𝑡^{𝛼−𝛼1−1}. (35) So (𝐹𝑥)(𝑡) is well defined and 𝐹(𝐵)⊆𝐵. It follows from (A1) that the operator (𝐹𝑥)(𝑡) is decreasing on 𝑥. Moreover, by Lemma 2, we have −𝐷^𝛼−𝛼1_𝑡(𝐹𝑥)(𝑡)=𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽()𝑔((𝑠,(0)𝑡,𝑠𝑠,𝐹𝑥))𝑑𝑠𝐹𝑥)(=0,𝐷^{𝛾1−𝛼1}_𝑡(𝐹𝑥)(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝐹𝑥𝜁_𝑗. (36)
Let 𝑒(𝑡)=min𝑡^{𝛼−𝛼1−1},𝐹𝑡^{𝛼−𝛼1−1},𝑒(𝑡)=max𝑡^{𝛼−𝛼1−1},𝐹𝑡^{𝛼−𝛼1−1}; (37) if 𝑡^{𝛼−𝛼1−1}=𝐹(𝑡^{𝛼−𝛼1−1}), then 𝑥^∗(𝑡)=𝑡^{𝛼−𝛼1−1},𝑦^∗(𝑡)=𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝑠^{𝛼−𝛼1−1}𝑑𝑠, (38) is positive solution of (7). Thus the system (3) has at least one solution 𝐼^𝛼1𝑥^∗(𝑡),𝐼^𝛽1𝑦^∗(𝑡) (39) which satisfies Theorem 7. If 𝑡^{𝛼−𝛼1−1}≠𝐹(𝑡^{𝛼−𝛼1−1}), we have 𝑒(𝑡),𝑒(𝑡)∈𝐵,𝑒(𝑡)≤𝑡^{𝛼−𝛼1−1}≤𝑒(𝑡). (40)
Let 𝜑(𝑡)=𝐹𝑒(𝑡),𝜙(𝑡)=(𝐹𝑒)(𝑡). (41) From (A1), we know that 𝐹 is nonincreasing on 𝑥; thus by (37) and (40), one gets 𝜙(𝑡)=(𝐹𝑒)(𝑡)≤𝐹𝑒(𝑡)(𝑡),(𝐹=𝜑𝑒)(𝑡)≤𝐹𝑡^{𝛼−𝛼1−1}≤𝑒(𝑡),𝐹𝑒(𝑡)≥𝐹𝑡^{𝛼−𝛼1−1}≥𝑒(𝑡), (42) and 𝜙(𝑡) and 𝜑(𝑡)∈𝐵. Thus from (36)–(42), we have 𝐷^𝛼−𝛼1_𝑡𝜑(𝑡)+𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝜑(𝑠)𝑑𝑠=𝐷^𝛼−𝛼1_𝑡𝐹𝑒(𝑡)+𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝐹𝑒(𝑠)𝑑𝑠≤𝐷^𝛼−𝛼1_𝑡𝐹𝑒(𝑡)+𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝑒(𝑠)𝜑(0)𝑑𝑠=0,=0,𝐷^{𝛾1−𝛼1}_𝑡𝜑(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝜑𝜁_𝑗,𝐷^𝛼−𝛼1_𝑡𝜙(𝑡)+𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝜙(𝑠)𝑑𝑠=𝐷^𝛼−𝛼1_𝑡(𝐹𝑒)(𝑡)+𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔(𝑠,(𝐹𝑒)(𝑠))𝑑𝑠≥𝐷^𝛼−𝛼1_𝑡(𝐹𝑒)(𝑡)+𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔(𝑠,𝑒(𝑠𝜙(0)))𝑑𝑠=0,=0,𝐷^{𝛾1−𝛼1}_𝑡𝜙(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝜙𝜁_𝑗. (43) So it follows from (42)-(43) that 𝜑(𝑡) and 𝜙(𝑡) are, respectively, upper solution and lower solution of the problem (28) and 𝜑(𝑡), 𝜙(𝑡)∈𝐵.
Define a function 𝑊∶(0,1)×𝐸→𝐸 as follows: 𝑊((𝑡=𝑓𝑡,𝑥))𝑡,𝜇¹₀𝐾_𝛽()𝑔(𝑠(𝑡)(𝑡),𝑓𝑡,𝑠𝑠,𝜙)𝑑𝑠,𝑥<𝜙𝑡,𝜇¹₀𝐾_𝛽()𝑔((𝑠(𝑡)(𝑡)(𝑡),𝑓𝑡,𝑠𝑠,𝑥))𝑑𝑠,𝜙≤𝑥≤𝜑𝑡,𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝜑(𝑠)𝑑𝑠,𝑥(𝑡)>𝜑(𝑡). (44) Obviously 𝑊∈𝐶((0,1)×[0,+∞)).
Next we define an operator 𝑇 in 𝐶[0,1] by (𝑇𝑥)(𝑡)=𝜆¹₀𝐾_𝛼(𝑡,𝑠)𝑊(𝑠,𝑥(𝑠))𝑑𝑠 (45) and consider the following boundary value problem: −𝐷^𝛼−𝛼1_𝑡𝑥(𝑡)((𝑡,𝑥(0)=𝜆𝑊𝑡,𝑥))=0,𝐷^{𝛾1−𝛼1}_𝑡𝑥(1)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝑥𝜁_𝑗. (46) It is easy to see that the fixed point of 𝑇 is the solution of (46).
Note that (𝑇𝑥)(𝑡)≤𝜆𝑡^{𝛼−𝛼1−11}₀𝑀_𝛼(𝑠)𝑊(𝑠,𝑥(𝑠))𝑑𝑠≤𝜆𝑡^{𝛼−𝛼1−11}₀𝑀_𝛼(𝑠)×𝑓𝑠,𝜇¹₀𝐾_𝛽(𝑠,𝜏)𝑔𝜏,𝜙(𝜏)𝑑𝜏𝑑𝑠≤𝜆𝑡^{𝛼−𝛼1−11}₀𝑀_𝛼(𝑠)×𝑓𝑠,𝜇¹₀𝐾_𝛽(𝑠,𝜏)𝑔𝜏,𝑙_𝜙𝜏^{𝛼−𝛼1−1}𝑑𝜏𝑑𝑠<+∞; (47) then 𝑇 is uniformly bounded. From the uniform continuity of 𝐾_𝛼(𝑡,𝑠) and Lebesgue dominated convergence theorem, we get 𝑇 is equicontinuous. So 𝑇 is completely continuous. It follows from Schauder’s fixed point theorem that 𝑇 has at least one fixed point 𝑥^∗(𝑡) such that 𝑥^∗(𝑡)=(𝑇𝑥^∗)(𝑡).
In what follows, we prove 𝜙(𝑡)≤𝑥^∗(𝑡)≤𝜑(𝑡),𝑡∈[0,1]. (48) In fact, as 𝑥^∗(𝑡) is a fixed point of 𝑇, we have 𝑥^∗(0)=0,𝐷^{𝛾1−𝛼1}_𝑡𝑥^∗(𝑡)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝑥^∗𝜁_𝑗. (49) We firstly prove 𝑥^∗(𝑡)≤𝜑(𝑡). Otherwise, suppose 𝑥^∗(𝑡)>𝜑(𝑡). According to the definition of 𝑊, we have −𝐷^𝛼−𝛼1_𝑡𝑥^∗(𝑡)=𝜆𝑊𝑡,𝑥^∗(𝑡)=𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝜑(𝑠)𝑑𝑠. (50) On the other hand, as 𝜑 is an upper solution of (28), we have −𝐷^𝛼−𝛼1_𝑡𝜑(𝑡)≥𝜆𝑓𝑡,𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝜑(𝑠)𝑑𝑠. (51) Letting 𝑧(𝑡)=𝜑(𝑡)−𝑥^∗(𝑡), (50)-(51) imply that 𝐷^𝛼−𝛼1_𝑡𝑧(𝑡)=𝐷^𝛼−𝛼1_𝑡𝜑(𝑡)−𝐷^𝛼−𝛼1_𝑡𝑥^∗(𝑡)𝑧(0)≤0,=0,𝐷^{𝛾1−𝛼1}_𝑡𝑧(𝑡)=^𝑚−2_𝑗=1𝑏_𝑗𝐷^{𝛾1−𝛼1}_𝑡𝑧𝜁_𝑗. (52) It follows from Lemma 4 that 𝑧(𝑡)≥0; (53) that is, 𝑥^∗(𝑡)≤𝜑(𝑡) on [0,1], which contradicts 𝑥^∗(𝑡)>𝜑(𝑡). Thus 𝜑(𝑡)≥𝑥^∗(𝑡), 𝑡∈(0,1). In the same way, 𝑥^∗(𝑡)≥𝜙(𝑡). Consequently, 𝜙(𝑡)≤𝑥^∗(𝑡)≤𝜑(𝑡), (54) and then 𝑥^∗(𝑡) is a positive solution of the problem (28).
It follows from 𝜙(𝑡), 𝜑(𝑡)∈𝐵 that there exist four numbers 0<𝑙_𝜙, 𝑙_𝜑<1, 𝐿_𝜙, 𝐿_𝜑>1 such that 𝑙_𝜙𝑡^{𝛼−𝛼1−1}≤𝜙(𝑡)≤𝐿_𝜙𝑡^{𝛼−𝛼1−1},𝑙_𝜑𝑡^{𝛼−𝛼1−1}≤𝜑(𝑡)≤𝐿_𝜑𝑡^{𝛼−𝛼1−1}, (55) and (54)-(55) yield 𝑟₁𝑡^{𝛼−𝛼1−1}=𝑙_𝜙𝑡^{𝛼−𝛼1−1}≤𝑥^∗(𝑡)≤𝐿_𝜑𝑡^{𝛼−𝛼1−1}=𝑟₂𝑡^{𝛼−𝛼1−1}. (56) Furthermore, 𝑑₁𝑡^{𝛽−𝛽1−1}≤𝜇𝑡^{𝛽−𝛽1−11}₀𝑚_𝛽(𝑠)𝑔𝑠,𝑟₁𝑠^{𝛼−𝛼1−1}(𝑠)𝑑𝑠≤𝑦^∗(𝑡)≤𝜇𝑡^{𝛽−𝛽1−11}₀𝑀_𝛽(𝑠)𝑔𝑠,𝑟₂𝑠^{𝛼−𝛼1−1}(𝑠)𝑑𝑠≤𝑑₂𝑡^{𝛽−𝛽1−1}, (57) where 𝑑₁1=min2,𝜇¹₀𝑚_𝛽(𝑠)𝑔𝑠,𝑟₁𝑠^{𝛼−𝛼1−1}(𝑠),𝑑𝑑𝑠₂=max2,𝜇¹₀𝑀_𝛽(𝑠)𝑔𝑠,𝑟₂𝑠^{𝛼−𝛼1−1}(𝑠)𝑑𝑠. (58) By Lemma 1, (56), and (57), we obtain that the problem (3) has a positive solution (𝑢^∗(𝑡),𝑣^∗(𝑡))=(𝐼^𝛼1𝑥^∗(𝑡),𝐼^𝛽1𝑦^∗(𝑡)), which satisfies 𝑟^∗₁𝑡^𝛼−1=Γ𝛼−𝛼₁𝑟₁Γ(𝛼)𝑡^𝛼−1≤𝑢^∗(𝑡)=1Γ𝛼₁^𝑡₀(𝑡−𝑠)^𝛼1−1𝑥^∗(𝑠)≤Γ𝑑𝑠𝛼−𝛼₁𝑟₂Γ(𝛼)𝑡^𝛼−1=𝑟^∗₂𝑡^𝛼−1,𝑑^∗₁𝑡^𝛽−1=Γ𝛽−𝛽₁𝑑₁Γ𝛽𝑡^𝛽−1≤𝑣^∗(𝑡)=1Γ𝛽₁^𝑡₀(𝑡−𝑠)^𝛽1−1𝑦^∗(𝑠)≤Γ𝑑𝑠𝛽−𝛽₁𝑑₂Γ𝛽𝑡^𝛽−1=𝑑^∗₂𝑡^𝛽−1. (59) The proof is completed.

Example 8. Consider the following singular fractional order differential system −𝐷_𝑡^7/4𝑢(𝑡)=𝜆sin𝑡+𝐷_𝑡^1/3𝑣(𝑡)^−1/3,−𝐷_𝑡^5/3𝑣(𝑡)=𝜇𝑡^−1/2+𝐷_𝑡^1/2𝑢(𝑡)²,𝐷_𝑡^1/2𝑢(0)=0,𝐷_𝑡^2/3𝑢(1)=¹⁰_𝑗=1𝑏_𝑗𝐷_𝑡^2/3𝑢𝜁_𝑗,𝐷_𝑡^1/3𝑣(0)=0,𝐷_𝑡^3/5𝑣(1)=¹⁰_𝑗=1𝑎_𝑗𝐷_𝑡^3/5𝑣𝜉_𝑗, (60) where 𝜆>0 and 𝜇>0 are parameters, 𝜉_𝑗=𝑗/12, 𝑎_𝑗=1/15𝑗^1/15, 𝜁_𝑗=1/(𝑗+1), and 𝑏_𝑗=(1/2)^𝑗, 𝑗=1,2,…,10. Obviously, 0<¹⁰_𝑗=1𝑎_𝑗𝜉_𝑗^1/15=¹⁰_𝑗=1115𝑗^1/15𝑗12^1/15=115¹⁰_𝑗=1112^1/15<1,0<¹⁰_𝑗=1𝑏_𝑗𝜁_𝑗^1/12=¹⁰_𝑗=112^𝑗1𝑗+1^1/12≤12^1/13<1. (61) Letting 𝑢(𝑡)=𝐼^1/2𝑥(𝑡), 𝑣(𝑡)=𝐼^1/3𝑦(𝑡), then (60) is turned into −𝐷_𝑡^5/4𝑥(𝑡)=𝜆sin𝑡+𝑦^−1/3(𝑡),−𝐷_𝑡^4/3𝑦(𝑡)=𝜇𝑡^−1/2+𝑥²(𝑡,𝑥(0))=0,𝐷_𝑡^1/3𝑥(1)=¹⁰_𝑗=1𝑏_𝑗𝐷_𝑡^1/3𝑥𝜁_𝑗,𝑦(0)=0,𝐷_𝑡^1/10𝑦(1)=¹⁰_𝑗=1𝑎_𝑗𝐷_𝑡^1/10𝑦𝜉_𝑗, (62) where 𝑓(𝑡,𝑦)=sin𝑡+𝑦^−1/3 and 𝑔(𝑡,𝑥)=𝑡^−1/2+𝑥²; it is easy to see that 𝑓(𝑡,𝑦) is decreasing with respect to 𝑦 and 𝑔(𝑡,𝑥) is increasing with respect to 𝑥, and then (A1) is satisfied.
It follows from (20) and (21) that there exists a positive number 𝑚₀<+∞ such that 𝐾_𝛽(𝑡,𝑠)≥𝑚_𝛽(𝑠)𝑡^1/3≥𝑚₀𝑡^1/3, (63) and for any constants 𝜇, 𝑙>0, we have 𝑓𝑡,𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝑙𝑠^{𝛼−𝛼1−1}𝑑𝑠=sin𝑡+𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑔𝑠,𝑙𝑠^{𝛼−𝛼1−1}𝑑𝑠^−1/3=sin𝑡+𝜇¹₀𝐾_𝛽(𝑡,𝑠)𝑠^−1/2+𝑙²𝑠²𝑑𝑠^−1/3≤sin𝑡+𝜇𝑚₀^−1/31₀𝑠^−1/2+𝑙²𝑠²𝑑𝑠^−1/3≤sin1+𝜇𝑚₀^−1/312+13𝑙^2−1/31<sin1+2𝜇𝑚₀^−1/3,𝑀_𝛼(𝑠)=1Γ()+∑5/4^10𝑗=1𝑏_𝑗𝐾_𝛼2𝜁_𝑗,𝑠1−𝑐₂≤1Γ()+()5/41−𝑠^1/12/Γ(5/4)∑¹⁰_𝑗=1𝑗+1(1/2)^𝑗1−𝑐₂≤1Γ()25/41+¹⁰−92¹¹1−𝑐₂, (64) where 𝑐₂=∑¹⁰_𝑗=1𝑏_𝑗𝜁_𝑗^1/12. So ¹₀𝑀_𝛼(𝑠)𝑓𝑠,𝜇¹₀𝐾_𝛽(𝑠,𝜏)𝑔𝜏,𝑙𝜏^{𝛼−𝛼1−1}≤1𝑑𝜏𝑑𝑠Γ(5/4)¹₀1sin1+2𝜇𝑚₀^−1/321+¹⁰−92¹¹1−𝑐₂𝑑𝑠<+∞, (65) and thus (A2) is satisfied. So from Theorem 7, for any (𝜆,𝜇)∈(0,+∞)×(0,+∞), the system (60) has at least one positive solution (𝑢^∗,𝑣^∗), and there exist positive constants 𝑟^∗₁, 𝑟^∗₂, 𝑑^∗₁, and 𝑑^∗₂ such that 𝑟^∗₁𝑡^3/4≤𝑢^∗(𝑡)≤𝑟^∗₂𝑡^3/4,𝑑^∗₁𝑡^2/3≤𝑣^∗(𝑡)≤𝑑^∗₂𝑡^2/3,𝑡∈[0,1]. (66)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors were supported financially by the National Natural Science Foundation of China (no. 11371221) and the Project of Shandong Province Higher Educational Science and Technology Program (no. J13LI12).