Monotone Iterative Solutions for Nonlinear Boundary Value Problems of Fractional Differential Equation

Abstract

By means of the method of quasi-lower and quasi-upper solutions and monotone iterative technique, we consider the nonlinear boundary value problems with Caputo fractional derivative and introduce two well-defined monotone sequences of quasi-lower and quasi-upper solutions which converge uniformly to the actual solution of the problem, and then the existence results of the solution for the problems are established. A numerical iterative scheme is introduced to obtain an accurate approximate solution and to give one example to demonstrate the accuracy and efficiency of the new approach.

1. Introduction

We are interested in the existence and uniqueness of solution of the following nonlinear boundary value problem of fractional order 𝐷^𝜇𝑢(𝑡)()(),𝑢(0)(1)+𝐹𝑡,𝑢=0,𝑡∈0,1=𝛼,𝑢=𝛽, (1) where 1<𝜇≤2, 𝐹(𝑡,𝑢)=𝑓₀(𝑡,𝑢)+𝑓₁(𝑡,𝑢)+𝑓₂(𝑡,𝑢), and 𝑓_𝑖∶[0,1]×𝑅→𝑅(𝑖=0,1,2) are a continuous function, 𝛼,𝛽∈𝑅, and 𝐷^𝜇 is the Caputo fractional derivative of order 𝜇.

It is well known that the differential equations with fractional order are generalization of ordinary differential equations to noninteger order, they occur more frequently in different research areas and engineering, such as physics, control of dynamical systems, chemistry and so forth. We also remark that several kinds of fractional derivatives were introduced to investigate the fractional differential equation, see, for example, [1, 2], and references therein.

Roughly speaking, it is a difficult task to give exact solutions for fractional differential equations. Recently, there are a number of numerical and analytical techniques to concerned with such problems, for instance, the homotopy analysis method, the Adomian decomposition method and the critical point theory have been discussed the fractional differential equations, such as [3–9]. On the other hand, very recently, the monotone iterative technique, combined with the method of lower and upper solutions were introduced to study the problems [10–13]. In [10, 11], the authors used the method of lower and upper solutions to investigate the existence of solutions for a class of fractional initial value problems. In [12] the authors considered fractional boundary value problem and proved the existence of solution. In [13], the author discussed the Monotone iterative technique for boundary value problems of a nonlinear fractional differential equation with deviating arguments. Specially, here it is worth mentioning, that Al-Refai and Ali Hajji [14] introduce two well-defined monotone sequences of lower and upper solutions which converge uniformly to the actual solution of the problem. However, the existence results in [14] mainly depend upon a restrictive condition; that is, 𝐹(𝑡,𝑢)isstrictlydecreasingwithrespectto𝑢. (2) It is a critical condition in order to discuss the monotone iterative sequences. It is, therefore, natural to ask how to discuss the problem (1), if 𝐹(𝑡,𝑢) is not necessary decreasing with respect to 𝑢. So, being directly inspired by [14], the purpose of this paper is to study the nonlinear boundary value problems of fractional order with Dirichlet boundary conditions. We introduce a method based on quasi-lower and quasi-upper solutions to prove the existence of a unique solution with no claim on the strictwe monotone. And also give an algorithm to construct two monotone sequences of quasi-lower and quasi-upper solutions. Moreover, the constructed sequences are proved to converge uniformly to the unique solution of the problem.

The paper is organized as follows. Preliminaries are in Section 2. Then in Section 3, we construct the monotone sequences of quasi-lower and quasi-upper solutions and prove their uniform convergence to the unique solution of the problem. Finally, in Section 4, we establish the numerical approach employed to obtain accurate numerical solution, and give one example to demonstrate the accuracy and efficiency of the new approach.

2. Preliminaries

In this sections, we present the definition of a pair of quasi-lower and quasi-upper solutions and some lemmas which will be needed in the next section.

For 𝜃>0, 𝑚−1<𝜃≤𝑚, 𝑚∈𝑁, the Caputo derivative (see [1, 2]) is defined by 𝐷^𝜃ℎ(𝑡)=1Γ(𝑚−𝜃)^𝑡₀(𝑡−𝑠)^{𝑚−1−𝜃}ℎ^(𝑚)(𝑠)𝑑𝑠. (3) The Caputo derivative defined in (3) is related to the Riemann-Liuville fractional integral 𝐼^𝜃 of 𝜃∈𝑅⁺ by 𝐷^𝜃ℎ(𝑡)=𝐼^𝑚−𝜃ℎ^(𝑚)(𝑡), (4) where for any 𝛿>0𝐼^𝛿ℎ(𝑡)=1Γ(𝛿)^𝑡₀(𝑡−𝑠)^𝛿−1ℎ(𝑠)𝑑𝑠. (5) Moreover, 𝐼^𝜃𝐷^𝜃ℎ(𝑡)=ℎ(𝑡)−^𝑚−1_𝑘=0𝑐_𝑘𝑡^𝑘,𝐷^𝜃𝐼^𝜃ℎ(𝑡)=ℎ(𝑡), (6) where 𝑐_𝑘=ℎ^(𝑘)(0⁺)/𝑘!, 1≤𝑘≤𝑚−1.

Definition 1. The functions 𝑣⁽⁰⁾,𝑤⁽⁰⁾∈𝐶²([0,1],𝑅) are called a pair of quasi-lower and quasi-upper solutions of the problem (1), if it satisfies 𝐷^𝜇𝑣⁽⁰⁾(𝑡)+𝑓₀𝑡,𝑣⁽⁰⁾+𝑓₁𝑡,𝑣⁽⁰⁾+𝑓₂𝑡,𝑤⁽⁰⁾𝑣≥0,₍₀₎(0)≤𝛼,𝑣⁽⁰⁾(1)𝐷≤𝛽,_𝜇𝑤⁽⁰⁾(𝑡)+𝑓⁽⁰⁾𝑡,𝑤⁽⁰⁾+𝑓₁𝑡,𝑤⁽⁰⁾+𝑓₂𝑡,𝑣⁽⁰⁾𝑤≤0,₍₀₎(0)≥𝛼,𝑤⁽⁰⁾(1)≥𝛽. (7) If the above inequalities are equalities, we call 𝑣⁽⁰⁾ and 𝑤⁽⁰⁾ a pair of quasi-solutions of the problem (1).

Lemma 2 (see [14]). Let 𝑝∈𝐶²([0,1],𝑅) and 𝐶>0 is a constant. If 𝑝(𝑡) satisfies the relations 𝐷^𝜇𝑝(𝑡)(𝑡)(),𝑝(0)(1)−𝐶𝑝≤0,𝑡∈0,1,𝑝≥0, (8) then 𝑝(𝑡)≥0 for 𝑡∈[0,1].

Lemma 3 (see [14]). A function 𝑢(𝑡) is a solution of the linear fractional BVP problems 𝐷^𝜇𝑢(𝑡)(𝑡)(𝑡)(),𝑢(0)(1)−𝐶𝑢=ℎ,𝑡∈0,1=𝛼,𝑢=𝛽, (9) if and only if it is a solution of integral equation 𝑢(𝑡)=𝛼+𝛽−𝛼𝑡−¹₀𝐺(𝑠,𝑡)(𝐶𝑢(𝑠)+ℎ(𝑠))𝑑𝑠, (10) where 𝐺()=1𝑠,𝑡Γ𝜇𝑡(1−𝑠)^𝜇−1−(𝑡−𝑠)^𝜇−1𝑡(),0≤𝑠≤𝑡≤1,1−𝑠^𝜇−1,0≤𝑡≤𝑠≤1. (11)

Lemma 4. For all 𝑡∈[0,1], we have 0≤¹₀𝐺()1𝑠,𝑡𝑑𝑠≤𝜇,ΛΓ (12) where Λ=𝜇^{𝜇/(𝜇−1)}⋅𝜇.𝜇−1 (13)

Proof. The proof of this lemma is easy, so we omit it.
In the rest of the paper, for convenience sake, we define the set 𝑥,𝑦=𝑧∈𝐶²([0,1],𝑅)∶𝑥(𝑡)≤𝑧(𝑡)≤𝑦(𝑡),𝑡∈[0,1], (14) let ‖𝑥‖=sup_𝑡∈[0,1]|𝑥(𝑡)|, (15) and list some conditions. (𝐶₁)  𝑣⁽⁰⁾, 𝑤⁽⁰⁾ is a pair of quasi-lower and quasi-upper solutions of the problem (1) and 𝑣⁽⁰⁾(𝑡)≤𝑤⁽⁰⁾(𝑡), 𝑡∈[0,1].(𝐶₂) The function 𝑓₁(𝑡,𝑢) is increasing with respect to 𝑢, and 𝑓₂ is decreasing with respect to 𝑢. And there exists a constant 𝑀∈(0,ΛΓ(𝜇)) such that 𝑓₀(𝑡,𝑢)−𝑓₀(𝑡,𝑢)≥−𝑀(𝑢−𝑢),∀𝑣⁽⁰⁾≤𝑢≤𝑢≤𝑤⁽⁰⁾. (16)

Lemma 5. Let (𝐶₁) and (𝐶₂) hold; then for any fixed 𝜂₁,𝜂₂∈[𝑣⁽⁰⁾,𝑤⁽⁰⁾], the linear problems 𝐷^𝜇𝑢(𝑡)=𝑀𝑢(𝑡)−𝜂₁(𝑡)−𝑓₀𝑡,𝜂₁(𝑡)−𝑓₁𝑡,𝜂₁(𝑡)−𝑓₂𝑡,𝜂₂(𝑡)()𝑢(0)(1),𝑡∈0,1=𝛼,𝑢=𝛽, (17) have a unique solution 𝑢_𝜂1,𝜂2∈[𝑣⁽⁰⁾,𝑤⁽⁰⁾], 𝑡∈[0,1].

Proof. Firstly, we prove that if  𝑢_𝜂1,𝜂2 is a solution of (17), then 𝑢_𝜂1,𝜂2∈[𝑣⁽⁰⁾,𝑤⁽⁰⁾]. From (𝐶₁), (𝐶₂) and if 𝑣⁽⁰⁾ is a quasi-lower solution of the problem (1), then we have 𝐷^𝜇𝑣⁽⁰⁾(𝑡)≥−𝑓₀𝑡,𝑣⁽⁰⁾−𝑓₁𝑡,𝑣⁽⁰⁾−𝑓₂𝑡,𝑤⁽⁰⁾≥𝑀𝑣⁽⁰⁾(𝑡)−𝜂₁(𝑡)−𝑓₀𝑡,𝜂₁(𝑡)−𝑓₁𝑡,𝜂₁(𝑡)−𝑓₂𝑡,𝜂₂(𝑡),𝑣⁽⁰⁾(0)≤𝛼,𝑣⁽⁰⁾(1)≤𝛽. (18) Thus from (18) and if 𝑢_𝜂1,𝜂2 is a solution of (17), set 𝑝(𝑡)=𝑢_𝜂1,𝜂2(𝑡)−𝑣⁽⁰⁾(𝑡); then 𝐷^𝜇𝑝(𝑡)(𝑡)()𝑝(0)(1)−𝑀𝑝≤0,𝑡∈0,1,𝑝≥0. (19) By Lemma 2, we have that 𝑝(𝑡)≥0,𝑡∈[0,1]. (20) That is, 𝑢_𝜂1,𝜂2≥𝑣⁽⁰⁾. Similarly, we show 𝑢_𝜂1,𝜂2≤𝑤⁽⁰⁾. Therefore, 𝑢_𝜂1,𝜂2∈[𝑣⁽⁰⁾,𝑤⁽⁰⁾].
Next, we show that (17) has a unique solution.
From Lemma 3, the problems (17) are equivalent to the following integral equation 𝑢(𝑡)=𝛼+𝛽−𝛼𝑡−¹₀𝐺(𝑠,𝑡)𝑀𝑢(𝑠)−ℎ_𝜂1,𝜂2(𝑠)𝑑𝑠, (21) where ℎ_𝜂1,𝜂2(𝑡)=𝑓₀(𝑡,𝜂₁(𝑡))+𝑓₁(𝑡,𝜂₁(𝑡))+𝑓₂(𝑡,𝜂₂(𝑡))+𝑀𝜂₁(𝑡).
Let 𝒜𝑢(𝑡)=𝛼+𝛽−𝛼𝑡−¹₀𝐺(𝑠,𝑡)𝑀𝑢(𝑠)−ℎ_𝜂1,𝜂2(𝑠)[].𝑑𝑠,𝑡∈0,1 (22) For any 𝑢,𝑢∈𝐶²([0,1],𝑅), by Lemma 4 and condition (𝐶₂), we have |𝒜𝑢(𝑡)−𝒜𝑢(𝑡)|≤𝑀¹₀𝐺(𝑠,𝑡)|𝑢(𝑠)−𝑢(𝑠)|𝑑𝑠≤𝑀¹₀𝐺(𝑠,𝑡)𝑑𝑠‖𝑢−𝑢‖≤𝑀𝜇‖ΛΓ𝑢−𝑢‖<‖𝑢−𝑢‖. (23) So ‖𝒜𝑢−𝒜𝑢‖<‖𝑢−𝑢‖. (24) By the Banach fixed point theorem, the operator 𝒜 has a unique point. This shows that (17) has a unique solution 𝑢∈[𝑣⁽⁰⁾,𝑤⁽⁰⁾], 𝑡∈[0,1].

3. Monotone Sequences of Quasi-Lower and Quasi-Upper Solutions

In this sections we construct the monotone sequences of quasi-lower and quasi-upper solutions and prove their uniform convergence to the unique solution of the problem.

Theorem 6. Let (𝐶₁) and (𝐶₂) hold, and assume that (𝐶₃)𝑣^(𝑘),𝑤^(𝑘),𝑘≥1 is a pair of solutions of 𝐷^𝜇𝑣^(𝑘)=𝑀𝑣^(𝑘)−𝑀𝑣^(𝑘−1)−𝑓₀𝑡,𝑣^(𝑘−1)−𝑓₁𝑡,𝑣^(𝑘−1)−𝑓₂𝑡,𝑤^(𝑘−1)(),𝑣,𝑡∈0,1_(𝑘)(0)=𝛼,𝑣^(𝑘)(1)𝐷=𝛽,_𝜇𝑤^(𝑘)=𝑀𝑤^(𝑘)−𝑀𝑤^(𝑘−1)−𝑓₀𝑡,𝑤^(𝑘−1)−𝑓₁𝑡,𝑤^(𝑘−1)−𝑓₂𝑡,𝑣^(𝑘−1)(),𝑤,𝑡∈0,1_(𝑘)(0)=𝛼,𝑤^(𝑘)(1)=𝛽. (25) Then one has the following. (i) The sequence 𝑣^(𝑘), 𝑤^(𝑘) is a pair of quasi-lower and quasi-upper solutions of (1). Moreover, 𝑣⁽⁰⁾≤𝑣⁽¹⁾≤⋯≤𝑣^(𝑘)≤⋯≤𝑤^(𝑘)≤⋯≤𝑤⁽¹⁾≤𝑤⁽⁰⁾. (26)(ii) The sequence 𝑣^(𝑘), 𝑤^(𝑘) converges uniformly to 𝑣_∗ and 𝑤^∗, respectively, with 𝑣⁽⁰⁾≤𝑣_∗≤𝑤^∗≤𝑤⁽⁰⁾. Moreover, 𝑣_∗ and 𝑤^∗ are a pair of minimal-maximal quasi-solutions of (1) in [𝑣⁽⁰⁾,𝑤⁽⁰⁾]. (iii) Suppose further that there exist constants 𝐿_𝑖>0(𝑖=0,1,2) such that 𝐿₁+𝐿₂<𝐿₀≤𝑀, and 𝑓₀(𝑡,𝑢)−𝑓₀(𝑡,𝑢)≤−𝐿₀(𝑢−𝑢),𝑓₁(𝑡,𝑢)−𝑓₁(𝑡,𝑢)≤𝐿₁(𝑢−𝑢),∀𝑣⁽⁰⁾≤𝑢≤𝑢≤𝑤⁽⁰⁾,𝑓₂(𝑡,𝑢)−𝑓₂(𝑡,𝑢)≥−𝐿₂(𝑢−𝑢). (27) Then 𝑣_∗=𝑤^∗ is the actual solution of (1) in [𝑣⁽⁰⁾,𝑤⁽⁰⁾].

Proof. (i) For any fixed 𝜂₁,𝜂₂∈[𝑣⁽⁰⁾,𝑤⁽⁰⁾], consider the linear problems 𝐷^𝜇𝑢(𝑡)=𝑀𝑢(𝑡)−𝜂₁(𝑡)−𝑓₀𝑡,𝜂₁(𝑡)−𝑓₁𝑡,𝜂₁(𝑡)−𝑓₂𝑡,𝜂₂(𝑡)()𝑢(0)(1),𝑡∈0,1=𝛼,𝑢=𝛽. (28) By Lemma 5, the linear problems (28) have a unique solution 𝑢_𝜂1,𝜂2∈[𝑣⁽⁰⁾,𝑤⁽⁰⁾]. Define operator ℬ∶[𝑣⁽⁰⁾,𝑤⁽⁰⁾]×[𝑣⁽⁰⁾,𝑤⁽⁰⁾]→[𝑣⁽⁰⁾,𝑤⁽⁰⁾] as follows: ℬ𝜂₁,𝜂₂=𝑢_𝜂1,𝜂2. (29) Again, it is easy that to prove that 𝜂₁, 𝜂₂ are a pair of quasi-solutions of the problem (1) if and only if 𝜂₁=ℬ𝜂₁,𝜂₂,𝜂₂=ℬ𝜂₂,𝜂₁. (30) Furthermore, from (28) and condition (𝐶₂), we can prove ℬ[𝜂₁,𝜂₂] is increasing with respect to 𝜂₁, and is decreasing with respect to 𝜂₂; that is, ∀𝜂,𝜂₁,𝜂₂∈𝑣⁽⁰⁾,𝑤⁽⁰⁾,𝜂₁≤𝜂₂⟹ℬ𝜂₁,𝜂≤ℬ𝜂₂,ℬ,𝜂𝜂,𝜂₁≥ℬ𝜂,𝜂₂. (31) In fact, setting 𝑢₁=ℬ[𝜂₁,𝜂], 𝑢₂=ℬ[𝜂₂,𝜂], by (28) and condition (𝐶₂), we have 𝐷^𝜇𝑢₁(𝑡)−𝑀𝑢₁(𝑡)=−𝑀𝜂₁(𝑡)−𝑓₀𝑡,𝜂₁(𝑡)−𝑓₁𝑡,𝜂₁(𝑡)−𝑓₂𝑡,𝜂(𝑡)≥−𝑀𝜂₂(𝑡)−𝑓₀𝑡,𝜂₂(𝑡)−𝑓₁𝑡,𝜂₂(𝑡)−𝑓₂𝑡,𝜂(𝑡)=𝐷^𝜇𝑢₂(𝑡)−𝑀𝑢₂(𝑡),𝑢₁(0)=𝛼=𝑢₂(0),𝑢₁(1)=𝛽=𝑢₂(1). (32) Thus, by setting 𝑝(𝑡)=𝑢₂(𝑡)−𝑢₁(𝑡), then 𝐷^𝜇𝑝(𝑡)(𝑡)𝑝(0)(1)−𝑀𝑝≤0,,𝑝=0. (33) By Lemma 2, we have that 𝑝(𝑡)≥0,𝑡∈[0,1]. (34) That is, 𝑢₂≥𝑣₁, or the first inequality holds in (31). Similarly, the second inequality holds in (31) too.
Again, similar to the previous argument, we can show that 𝑣⁽⁰⁾≤ℬ𝑣⁽⁰⁾,𝑤⁽⁰⁾,ℬ𝑤⁽⁰⁾,𝑣⁽⁰⁾≤𝑤⁽⁰⁾. (35)
Now, let 𝑣^(𝑘)=ℬ𝑣^(𝑘−1),𝑤^(𝑘−1),𝑤^(𝑘)=ℬ𝑤^(𝑘−1),𝑣^(𝑘−1),𝑘=1,2,…. (36) Thus, from (31), (35), and (36), we have 𝑣⁽⁰⁾≤𝑣⁽¹⁾≤⋯≤𝑣^(𝑘)≤⋯≤𝑤^(𝑘)≤⋯≤𝑤⁽¹⁾≤𝑤⁽⁰⁾. (37) Moreover, it is clear that 𝑣^(𝑘), 𝑤^(𝑘) satisfy (36) if and only if 𝑣^(𝑘), 𝑤^(𝑘) are a pair of solutions of (25).
To prove (𝑖), we only need to show that 𝑣^(𝑘), 𝑤^(𝑘) are a pair of quasi-lower and quasi-upper solutions of (1). In fact, from (25) and condition (𝐶₂), we can obtain 𝐷^𝜇𝑣^(𝑘)+𝑓₀𝑡,𝑣^(𝑘)+𝑓₁𝑡,𝑣^(𝑘)+𝑓₂𝑡,𝑤^(𝑘)=𝑀𝑣^(𝑘)−𝑣^(𝑘−1)+𝑓₀𝑡,𝑣^(𝑘)−𝑓₀𝑡,𝑣^(𝑘−1)+𝑓₁𝑡,𝑣^(𝑘)−𝑓₁𝑡,𝑣^(𝑘−1)+𝑓₂𝑡,𝑤^(𝑘)−𝑓₂𝑡,𝑤^(𝑘−1)𝑣≥0,_(𝑘)(0)=𝛼,𝑣^(𝑘)(1)𝐷=𝛽,_𝜇𝑤^(𝑘)+𝑓₀𝑡,𝑤^(𝑘)+𝑓₁𝑡,𝑤^(𝑘)+𝑓₂𝑡,𝑣^(𝑘)=𝑀𝑤^(𝑘)−𝑤^(𝑘−1)+𝑓₀𝑡,𝑤^(𝑘)−𝑓₀𝑡,𝑤^(𝑘−1)+𝑓₁𝑡,𝑤^(𝑘)−𝑓₁𝑡,𝑤^(𝑘−1)+𝑓₂𝑡,𝑣^(𝑘)−𝑓₂𝑡,𝑣^(𝑘−1)𝑣≤0,_(𝑘)(0)=𝛼,𝑣^(𝑘)(1)=𝛽. (38) Hence, 𝑣^(𝑘), 𝑤^(𝑘) is a pair of quasi-lower and quasi-upper solutions of (1). This proves (i).
(ii) Since the sequence 𝑣^(𝑘) is monotone nondecreasing and is bounded from above mention, the sequence 𝑤^(𝑘) is monotone nonincreasing and is bounded from below, therefore the pointwise limits exist and these limits are denoted by 𝑣_∗ and 𝑤^∗. The sequence 𝑣^(𝑘), 𝑤^(𝑘) are sequences of continuous functions defined on the compact set [0,1], hence By Dini’s theorem [15], the convergence is uniform. This is 𝑣⁽⁰⁾≤𝑣_∗=lim_𝑘→∞𝑣^(𝑘)≤lim_𝑘→∞𝑤^(𝑘)=𝑤^∗≤𝑤⁽⁰⁾ (39) uniformly on 𝑡∈[0,1].
Moveover, from (25), we have 𝐷^𝜇𝑣_∗=−𝑓₀𝑡,𝑣_∗−𝑓₁𝑡,𝑣_∗−𝑓₂𝑡,𝑤^∗(),𝑣,𝑡∈0,1_∗(0)=𝛼,𝑣_∗(1)𝐷=𝛽,_𝜇𝑤^∗=−𝑓₀𝑡,𝑤^∗−𝑓₁𝑡,𝑤^∗−𝑓₂𝑡,𝑣_∗(),𝑤,𝑡∈0,1_∗(0)=𝛼,𝑤^∗(1)=𝛽. (40) This show that 𝑣_∗, 𝑤^∗ are a pair of quasi-solutions of the problem (1).
Now, we prove that 𝑣_∗ and 𝑤^∗ are a pair of minimal-maximal quasi-solutions of (1).
Let 𝑣,𝑤∈[𝑣⁽⁰⁾,𝑤⁽⁰⁾] be a pair of quasi-solutions of the problem (1), in the following, we show this using induction arguments. In fact, we have 𝑣⁽⁰⁾≤𝑣,𝑤≤𝑤⁽⁰⁾. (41) Assume that 𝑣^(𝑘−1)≤𝑣,𝑤≤𝑤^(𝑘−1) (42) is true. Thus from (31) and (36), we can obtain 𝑣^(𝑘)=ℬ𝑣^(𝑘−1),𝑤^(𝑘−1)[]𝑣≤ℬ𝑣,𝑤=𝑣,_(𝑘)=ℬ𝑣^(𝑘−1),𝑤^(𝑘−1)[]𝑤≤ℬ𝑤,𝑣=𝑤,_(𝑘)=ℬ𝑤^(𝑘−1),𝑣^(𝑘−1)[]𝑤≥ℬ𝑣,𝑤=𝑣,_(𝑘)=ℬ𝑤^(𝑘−1),𝑣^(𝑘−1)≥ℬ[𝑤,𝑣]=𝑤. (43) Therefore, 𝑣^(𝑘)≤𝑣,𝑤≤𝑤^(𝑘),𝑘=0,1,2,…. (44) Thus, taking limit in (44) as 𝑘→∞, we have 𝑣_∗≤𝑣,𝑤≤𝑤^∗. (45) That is, 𝑣_∗ and 𝑤^∗ are a pair of minimal-maximal quasi-solutions of (1) in [𝑣⁽⁰⁾,𝑤⁽⁰⁾].
(iii) Since 𝑣_∗(𝑡)≤𝑤^∗(𝑡), 𝑡∈[0,1], it is sufficient to prove that 𝑣_∗(𝑡)≥𝑤^∗(𝑡), 𝑡∈(0,1). In fact, from (40) and supposition (iii), we have (notice that 𝑣_∗≤𝑤^∗) 𝐷^𝜇𝑣_∗−𝑤^∗=𝑓₀𝑡,𝑤^∗−𝑓₀𝑡,𝑣_∗+𝑓₁𝑡,𝑤^∗−𝑓₁𝑡,𝑣_∗+𝑓₂𝑡,𝑣_∗−𝑓₂𝑡,𝑤^∗≤−𝐿₀𝑤^∗−𝑣_∗+𝐿₁𝑤^∗−𝑣_∗+𝐿₂𝑤^∗−𝑣_∗=−𝐿₀−𝐿₁−𝐿₂𝑤^∗−𝑣_∗,𝑣_∗(0)−𝑤^∗(0)=0,𝑣_∗(1)−𝑤^∗(1)=0. (46) Thus, by setting 𝑝(𝑡)=𝑣_∗−𝑤^∗, from (46), we obtain 𝐷^𝜇𝑝(𝑡)−𝐿₀−𝐿₁−𝐿₂𝑝(𝑡)𝑝(0)(1)≤0,,𝑝=0. (47) By Lemma 2 (notice that 𝐿₀−𝐿₁−𝐿₂>0), we have that 𝑝(𝑡)≥0,𝑡∈[0,1]. (48) Therefore, 𝑣_∗≥𝑤^∗, 𝑡∈[0,1]. Hence 𝑣_∗=𝑤^∗ is the actual solution of (1) in [𝑣⁽⁰⁾,𝑤⁽⁰⁾]. Thus, we complete this proof.

4. Numerical Implementation

In this section, the numerical procedures are introduced to obtain an approximate solution of (1).

For the problem (25), from Lemma 3, we have 𝑣^(𝑘)(𝑡)𝑡−=𝛼+𝛽−𝛼₁₀𝐺(𝑠,𝑡)𝑀𝑣^(𝑘)(𝑠)−𝑀𝑣^(𝑘−1)(𝑠)−𝑓₀𝑠,𝑣^(𝑘−1)(𝑠)−𝑓₁𝑠,𝑣^{(𝑘−1)(𝑠)}−𝑓₂𝑠,𝑤^(𝑘−1)(𝑠)𝑤𝑑𝑠,_(𝑘)(𝑡)𝑡−=𝛼+𝛽−𝛼₁₀𝐺(𝑠,𝑡)𝑀𝑤^(𝑘)(𝑠)−𝑀𝑤^(𝑘−1)(𝑠)−𝑓₀𝑠,𝑤^(𝑘−1)(𝑠)−𝑓₁𝑠,𝑤^(𝑘−1)(𝑠)−𝑓₂𝑠,𝑣^(𝑘−1)(𝑠)𝑑𝑠. (49) Thus by Theorem 6, the unique solution to problem (1) is given by lim_𝑘→∞𝑣^(𝑘)(𝑡)=𝑣_∗(𝑡)=𝑤^∗(𝑡)=lim_𝑘→∞𝑤^(𝑘)(𝑡). (50) Setting Φ(𝑠,𝑡)=𝐺(𝑠,𝑡)𝑀𝑣^(𝑘)(𝑠)−𝑀𝑣^(𝑘−1)(𝑠)−𝑓₀𝑠,𝑣^(𝑘−1)(𝑠)−𝑓₁𝑠,𝑣^{(𝑘−1)(𝑠)}−𝑓₂𝑠,𝑤^(𝑘−1)(𝑠),Ψ(𝑠,𝑡)=𝐺(𝑠,𝑡)𝑀𝑤^(𝑘)(𝑠)−𝑀𝑤^(𝑘−1)(𝑠)−𝑓₀𝑠,𝑤^(𝑘−1)(𝑠)−𝑓₁𝑠,𝑤^(𝑘−1)(𝑠)−𝑓₂𝑠,𝑣^(𝑘−1)(𝑠). (51) Discretize the interval [0,1] with the nodes ℎ=1/𝑁, 𝑡_𝑖=𝑖ℎ, 𝑠_𝑗=𝑗ℎ. Using Simpson’s quadrature rule, we can obtain the function 𝑆_Φ(𝑡), and Ψ(𝑡), to approximate the integral ∫¹₀Φ(𝑠,𝑡)𝑑𝑠 and ∫¹₀Ψ(𝑠,𝑡)𝑑𝑠 on the right hand sid of (49), where 𝑆_Φ(𝑡)=ℎ3^𝑁_𝑗=0𝑑_𝑗Φ𝑠_𝑗,𝑆,𝑡_Ψ(𝑡)=ℎ3^𝑁_𝑗=0𝑑_𝑗Ψ𝑠_𝑗,𝑡, (52) where {𝑑_𝑗} are the coefficients in composite Simpson’s quadrature rule: 𝑑₀=𝑑_𝑁=1, 𝑑_2𝑗−1=4, 𝑑_2𝑗=2, 1≤𝑗≤𝑁/2.

Now, let 𝑣_𝑖^(𝑘)≈𝑣^(𝑘)(𝑡_𝑖), 𝑤_𝑖^(𝑘)≈𝑤^(𝑘)(𝑡_𝑖), from (49) and the above; we obtain the following linear system of equations mentioned 𝑣_𝑖^(𝑘)=𝑣^(𝑘)𝑡_𝑖=𝛼+𝛽−𝛼𝑡_𝑖−𝑆_Φ𝑡_𝑖=𝛼+𝛽−𝛼𝑡_𝑖−ℎ3^𝑁_𝑗=0𝑑_𝑗Φ𝑠_𝑗,𝑡_𝑖=𝛼+𝛽−𝛼𝑡_𝑖−ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖𝑀𝑣_𝑗^(𝑘)+ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖×𝑀𝑣_𝑗^(𝑘−1)+𝑓₀𝑠_𝑗,𝑣_𝑗^(𝑘−1)+𝑓₁𝑠_𝑗,𝑣_𝑗^(𝑘−1)+ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖𝑓₂𝑠_𝑗,𝑤_𝑗^(𝑘−1),𝑤_𝑖^(𝑘)=𝑤^(𝑘)𝑡_𝑖=𝛼+𝛽−𝛼𝑡_𝑖−𝑆_Ψ𝑡_𝑖=𝛼+𝛽−𝛼𝑡_𝑖−ℎ3^𝑁_𝑗=0𝑑_𝑗Ψ𝑠_𝑗,𝑡_𝑖=𝛼+𝛽−𝛼𝑡_𝑖−ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖𝑀𝑤_𝑗^(𝑘)+ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖×𝑀𝑤_𝑗^(𝑘−1)+𝑓₀𝑠_𝑗,𝑤_𝑗^(𝑘−1)+𝑓₁𝑠_𝑗,𝑤_𝑗^(𝑘−1)+ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖𝑓₂𝑠_𝑗,𝑣_𝑗^(𝑘−1). (53) or 𝑣_𝑖^(𝑘)+ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖𝑀𝑣_𝑗^(𝑘)=𝛼+𝛽−𝛼𝑡_𝑖+ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖×𝑀𝑣_𝑗^(𝑘−1)+𝑓₀𝑠_𝑗,𝑣_𝑗^(𝑘−1)+𝑓₁𝑠_𝑗,𝑣_𝑗^(𝑘−1)+ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖𝑓₂𝑠_𝑗,𝑤_𝑗^(𝑘−1),𝑤_𝑖^(𝑘)+ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖𝑀𝑤_𝑗^(𝑘)=𝛼+𝛽−𝛼𝑡_𝑖+ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖×𝑀𝑤_𝑗^(𝑘−1)+𝑓₀𝑠_𝑗,𝑤_𝑗^(𝑘−1)+𝑓₁𝑠_𝑗,𝑤_𝑗^(𝑘−1)+ℎ3^𝑁_𝑗=0𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖𝑓₂𝑠_𝑗,𝑣_𝑗^(𝑘−1). (54) Setting 𝐺_𝑖𝑗=ℎ3𝑑_𝑗𝐺𝑠_𝑗,𝑡_𝑖,𝒢=𝐺_{𝑖𝑗(𝑁+1)×(𝑁+1)}𝒜_𝑖𝑣^(𝑘−1)=𝛼+𝛽−𝛼𝑡_𝑖+^𝑁_𝑗=0𝐺_𝑖𝑗𝑀𝑣_𝑗^(𝑘−1)+𝑓₀𝑠_𝑗,𝑣_𝑗^(𝑘−1)+𝑓₁𝑠_𝑗,𝑣_𝑗^(𝑘−1),ℬ_𝑖𝑤^(𝑘−1)=𝛼+𝛽−𝛼𝑡_𝑖+^𝑁_𝑗=0𝐺_𝑖𝑗𝑀𝑤_𝑗^(𝑘−1)+𝑓₀𝑠_𝑗,𝑤_𝑗^(𝑘−1)+𝑓₁𝑠_𝑗,𝑤_𝑗^(𝑘−1),𝒱^(𝑘)=𝑣₀^(𝑘),𝑣₁^(𝑘),…,𝑣_𝑁^(𝑘)𝑇,𝒲^(𝑘)=𝑤₀^(𝑘),𝑤₁^(𝑘),…,𝑤_𝑁^(𝑘)𝑇,𝒜𝑣^(𝑘−1)=𝒜₀𝑣^(𝑘−1),𝒜₁𝑣^(𝑘−1),…,𝒜_𝑁𝑣^{(𝑘−1)𝑇},ℬ𝑤^(𝑘−1)=ℬ₀𝑤^(𝑘−1),ℬ₁𝑤^(𝑘−1),…,ℬ_𝑁𝑤^{(𝑘−1)𝑇},𝒫𝑤^(𝑘−1)=𝑓₂𝑠₀,𝑤₀^(𝑘−1),𝑓₂𝑠₁,𝑤₁^(𝑘−1),…,𝑓₂𝑠_𝑁,𝑤_𝑁^{(𝑘−1)𝑇},𝒬𝑣^(𝑘−1)=𝑓₂𝑠₀,𝑣₀^(𝑘−1),𝑓₂𝑠₁,𝑣₁^(𝑘−1),…,𝑓₂𝑠_𝑁,𝑣_𝑁^{(𝑘−1)𝑇}. (55) Thus system (54) can be written as a matrix-vector equation (ℐ+𝑀𝒢)𝒱^(𝑘)=𝒜𝑣^(𝑘−1)+𝒢𝒫𝑤^(𝑘−1),(ℐ+𝑀𝒢)𝒲^(𝑘)=ℬ𝑤^(𝑘−1)+𝒢𝒬𝑣^(𝑘−1), (56) where ℐ is the identity matrix.

Remark 7. Applying the above numerical scheme, we can solve iteratively for 𝑣^(𝑘) and 𝑤^(𝑘) up to some large enough 𝑘=𝐾. For a given accuracy 𝜀, we take 𝑣^(𝑘) and 𝑤^(𝑘) as 𝜀-accurate approximations of 𝑣_∗ and 𝑤^∗, respectively. And we use 𝐸(𝑘)=𝑤^(𝑘)−𝑣^(𝑘)₂=¹₀𝑤^(𝑘)(𝑡)−𝑣^(𝑘)(𝑡)²𝑑𝑡^1/2 (57) to describe the error.

Example 8. Consider the fractional BVP 𝐷^3/2𝑢−4𝑢+𝑢²+𝑒^−𝑢(),𝑢(0)(1)=0,𝑡∈0,1=0,𝑢=1. (58) Let 3𝜇=2,𝛼=0,𝛽=1,𝑓₀()𝑓𝑡,𝑢=−4𝑢,₁(𝑡,𝑢)=𝑢²,𝑓₂(𝑡,𝑢)=𝑒^−𝑢. (59) It can be easily verified that 𝑣⁽⁰⁾(𝑡)=0,𝑤⁽⁰⁾(𝑡)=1 (60) are a pair of ordered quasi-lower and quasi-upper solutions of (58). Moreover, 𝑀=𝐿₀=4,𝐿₁=2,𝐿₂=1,Λ=818. (61) Thus 𝑀<Γ𝜇Λ,𝐿₁+Ł₂<𝐿₀≤𝑀. (62) Hence for this example, we found that, for 𝜀=10⁻¹⁰, it has 𝐾=10 iterations for 𝐸(𝑘)<𝜀. the graph of 𝑣^(𝑘) and 𝑤^(𝑘), for some values of 𝑘, is in Figure 1. Table 1 displays 𝐸(𝑘) versus 𝑘 for selected values of 𝑘.

Table 1 

𝐸(𝑘)∶𝑘=2𝑖, 𝑖=0,1,2,3,4,5.

(a)

(b)

(c)

(d)

Figure 1 

Graph of 𝑣(𝑘) and 𝑤(𝑘).

Acknowledgment

This work was partially supported by NNSF of China (Grant no. 11171122).