Existence and Uniqueness of the Solutions for Fractional Differential Equations with Nonlinear Boundary Conditions

Abstract

We study the existence and uniqueness of the solutions for the boundary value problem of fractional differential equations with nonlinear boundary conditions. By using the upper and lower solutions method in reverse order and monotone iterative techniques, we obtain the sufficient conditions of both the existence of the maximal and minimal solutions between an upper solution and a lower solution and the uniqueness of the solutions for the boundary value problem and present the iterative sequence for calculating the approximate analytical solutions of the boundary value problem and the error estimate. An example is also given to illustrate the main results.

1. Introduction

In this paper, we study the following fractional differential equation with nonlinear boundary conditions: ^𝐶𝐷^𝛿₀₊𝑢(𝑡)(𝑡)((𝑡𝑢−𝑀𝑢=𝑓𝑡,𝑢)),𝑡∈𝐽,_′(0)=0,𝑢(0)=𝑔(𝑢(𝑇)), (1) where 𝐽=[0,𝑇], 0<𝑇<+∞. 𝑀≥0 is a constant. 𝑓∈𝐶(𝐽×ℝ,ℝ), 𝑔∈𝐶¹(ℝ,ℝ) are given functions. ^𝐶𝐷^𝛿₀₊ is the standard Caputo fractional derivative of order 𝛿 with 1<𝛿<2.

With the development of mathematics, fractional derivative occurs more and more frequently in different research areas, such as physics, mechanics, electricity, and economics (see [1, 2]). At the same time, significant progress has also been made on the studies of fractional differential equations (see [3–12]).

The upper and lower solutions method and monotone iterative techniques have been widely used in the studies for the boundary value problems of integer-order differential equations (see [13–15] and the references therein). In [13], the authors studied first order differential equation 𝑢^′(𝑡)((𝑡)(𝜃(𝑡,𝑔(𝑢(0(𝑇).=𝑓𝑡,𝑢,𝑢)))))=𝑢 (2) By using the upper and lower solutions method and the monotone iterative techniques, the authors concluded that (2) exists the maximal and minimal solutions in [𝛽(𝑡),𝛼(𝑡)], where 𝛼(𝑡), 𝛽(𝑡) are the lower and upper solutions of (2).

As for the fractional differential problems, upper and lower solutions and monotone iterative techniques have also been widely used in the studies of boundary value problem (see [16–18]). For example, in [16], by using the method, the authors discussed the existence and uniqueness of solutions for the fractional differential equation with linear boundary condition 𝐷^𝛿𝑢(𝑡)(𝑡)((𝑡𝑢(0)(𝑇),−𝑀𝑢=𝑓𝑡,𝑢)),𝑡∈𝐽,0<𝛿<1,=𝑟𝑢 (3) where 𝑀≥0, 0<𝑟<1/𝐸_𝛿,1(𝑀𝑇^𝛿).

In this paper, we study the existence and uniqueness of the solutions for the boundary value problem of the fractional differential equation (1), which has nonlinear boundary conditions. It is very difficult to get the iterative sequence which is used to obtain the solutions since the boundary value problem has nonlinear boundary conditions. By using the upper and lower solutions method in reverse order and monotone iterative techniques, we not only obtain the sufficient conditions that the boundary value problem (1) has the maximal and minimal solutions between an upper solution and a lower solution as well as having the unique solution, but also present the iterative sequence for calculating the approximate analytical solutions of the boundary value problem and the error estimate. An example is also given to illustrate the main results.

The organization of this paper is as follows. In Section 2, we provide the necessary background and comparison principles which are used to prove our main results. In Section 3, we consider a linear problem associated with (1). In Section 4, by using the method of upper and lower solutions and the monotone iterative techniques, we obtain the existence and uniqueness solutions of (1). In Section 5, the calculation method of the approximate analytical solutions for the boundary value problems (1) is obtained. In Section 6, an example is presented to illustrate our main results.

2. Preliminaries and Comparison Principles

In this section, we present some basic definitions, lemmas, and comparison principles which play important roles in our investigation.

Definition 1 (see [3, 4]). Let 𝛿>0 and function 𝑢∶(0,+∞)→ℝ. The Riemann-Liouville fractional integral operator of order 𝛿 of 𝑢 is defined by 𝐼^𝛿𝑢(𝑡)=1Γ(𝛿)^𝑡₀(𝑡−𝑠)^𝛿−1𝑢(𝑠)d𝑠, (4) provided the integral exists.
The Caputo derivative of order 𝛿 for function 𝑢∶(0,+∞)→ℝ is given by ^𝐶𝐷^𝛿𝑢(𝑡)=1Γ(𝑛−𝛼)^𝑡₀𝑢^(𝑛)(𝑠)(𝑡−𝑠)^{𝛿+1−𝑛}d𝑠, (5) provided the right side is pointwise defined on (0,+∞), where 𝑛∈ℕ⁺ with 𝑛−1<𝛿<𝑛.
If 𝛿=𝑛∈ℕ⁺, then ^𝐶𝐷^𝛿𝑢(𝑡)=𝑢^(𝑛)(𝑡).

Definition 2. Let 𝐴𝐶[0,1] be the space of the functions 𝑢 which are absolutely continuous on [0,1]. We denote 𝐴𝐶^𝑛[0,1] by the set of the functions 𝑢 which have continuous derivatives up to order 𝑛−1 on [0,1] such that 𝑢^(𝑛−1)∈𝐴𝐶[0,1]. In particular, 𝐴𝐶¹[0,1]=𝐴𝐶[0,1].

Lemma 3 (see [3]). If 𝑢∈𝐴𝐶^𝑛[0,1], then the Caputo fractional derivative ^𝐶𝐷^𝛿𝑢(𝑡) exists almost everywhere on [0,1], where 𝑛 is the smallest integer greater than or equal to 𝛿.

Lemma 4 (see [3]). Suppose 𝛿>0 and 𝑢∈𝐴𝐶^𝑛(𝐽). Then 𝐼^𝛿𝐶𝐷^𝛿𝑢(𝑡)=𝑢(𝑡)+𝑐₀+𝑐₁𝑡+𝑐₂𝑡²+⋯+𝑐_𝑛−1𝑡^𝑛−1,𝑐_𝑘=𝑢^(𝑘)(0)𝑘!,𝑘=0,1,2,…,𝑛−1, (6) where 𝑛 is the smallest integer greater than or equal to 𝛿.

Definition 5. One says 𝑢 is a solution of the boundary value problem (1) if 𝑢∈𝐴𝐶²(𝐽) and ^𝐶𝐷^𝛿𝑢∈𝐶(𝐽) and satisfies (1).

Definition 6 (see [4]). Let the real 𝑛>0. The function 𝐸_𝑛 is defined by 𝐸_𝑛(𝑧)=^∞_𝑗=0𝑧^𝑗Γ,𝑗𝑛+1 (7) whenever the series converges called the Mittag-Leffler function of order 𝑛.

Definition 7 (see [4]). Let 𝑛₁,𝑛₂>0. The function 𝐸_𝑛1,𝑛2 is defined by 𝐸_𝑛1,𝑛2(𝑧)=^∞_𝑗=0𝑧^𝑗Γ𝑗𝑛₁+𝑛₂, (8) whenever the series converges called the two-parameter Mittag-Leffler function with parameters 𝑛₁ and 𝑛₂.

Remark 8 (see [4]). It is evident that the one-parameter Mittag-Leffler functions may be defined in terms of their two-parameter counterparts via the relation 𝐸_𝑛(𝑧)=𝐸_𝑛,1(𝑧).

Lemma 9 (see [4], Theorem 4.1). The two-parameter Mittag-Leffler function 𝐸_𝑛1,𝑛2(𝑧) for some 𝑛₁,𝑛₂>0. The power series defining 𝐸_𝑛1,𝑛2(𝑧) is convergent for all 𝑧∈ℝ.

Lemma 10 (see [3], Theorem 4.3). Let ℎ∈𝐶(𝐽). The Cauchy problem ^𝐶𝐷^𝛿₀₊𝑢(𝑡)(𝑡)(𝑡)𝑢(0)−𝑀𝑢=ℎ,𝑡∈𝐽,=𝑏,𝑢^′(0)=𝑑,𝑏,𝑑∈ℝ, (9) with 1<𝛿<2 and 𝑀∈ℝ has a unique solution 𝑢(𝑡)=𝑏𝐸_𝛿𝑀𝑡^𝛿+𝑑𝐸_𝛿,2𝑀𝑡^𝛿𝑡+^𝑡₀(𝑡−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑡−𝑠)^𝛿ℎ(𝑠)d𝑠. (10)

Lemma 11 (see [19]). Let 𝐸 be a partially ordered Banach space, 𝑥_𝑛≤𝑦_𝑛(𝑛=1,2,3,…), if 𝑥_𝑛→𝑥^∗, 𝑦_𝑛→𝑦^∗₀. Then 𝑥^∗≤𝑦^∗.

In this paper, we assume the following conditions are satisfied.(H0)A given function 𝑐∈𝐴𝐶²(𝐽) and ^𝐶𝐷^𝛿₀₊𝑐∈𝐶(𝐽). 𝑐 is monotone decreasing on 𝐽, and 𝑐(0)=𝑐^′(0)=0, 𝑐(𝑇)≤−1.(H1)𝑔∈𝐶¹(ℝ,ℝ), 𝑔(0)≥0, and 0<𝑔^′(𝑥)<1/𝐸_𝛿(𝑀𝑇^𝛿).

Definition 12. Let 𝛼∈𝐴𝐶²(𝐽) and ^𝐶𝐷^𝛿₀₊𝛼∈𝐶(𝐽); we say that 𝛼 is a lower solution of the boundary value problem (1), if ^𝐶𝐷^𝛿₀₊𝛼(𝑡)−𝑀𝛼(𝑡)≥𝑓(𝑡,𝛼(𝑡))−𝑎_𝛼(𝑡)𝛼,𝑡∈𝐽,_′(0)≥0, (11) where 𝑎_𝛼(𝑡)=0,𝑔(𝛼(𝑇))≤𝛼_𝐶(0),𝐷^𝛿₀₊𝑐(𝑡)(𝑡)×𝛼(𝑇)−𝑀𝑐−𝑔⁻¹(𝛼(0)),𝑔(𝛼(𝑇))>𝛼(0). (12) Let 𝛽∈𝐴𝐶²(𝐽) and ^𝐶𝐷^𝛿₀₊𝛽∈𝐶(𝐽); we say that 𝛽 is an upper solution of the boundary value problem (1), if ^𝐶𝐷^𝛿₀₊𝛽(𝑡)−𝑀𝛽(𝑡)≤𝑓𝑡,𝛽(𝑡)+𝑏_𝛽(𝑡)𝛽,𝑡∈𝐽,_′(0)≤0, (13) where 𝑏_𝛽(𝑡)=0,𝑔𝛽(𝑇)≥𝛽_𝐶(0),𝐷^𝛿₀₊𝑐(𝑡)(𝑡)×𝑔−𝑀𝑐₋₁𝛽(0)−𝛽(𝑇),𝑔𝛽(𝑇)<𝛽(0), (14) and 𝑐(𝑡) is defined in (H0).

Lemma 13. Let 𝑔₁∈𝐶¹(ℝ,ℝ) be any function with 𝑔₁(0)≥0 and 0<𝑔^′₁(𝑢)<1/𝐸_𝛿(𝑀𝑇^𝛿). If 𝑢∈𝐴𝐶²(𝐽) and ^𝐶𝐷^𝛿₀₊𝑢∈𝐶(𝐽) and satisfies ^𝐶𝐷^𝛿₀₊𝑢(𝑡)(𝑡)𝑢−𝑀𝑢≥0,𝑡∈𝐽,_′(0)≥0,𝑢(0)≥𝑔₁(𝑢(𝑇)), (15) then 𝑢(𝑡)≥0 for 𝑡∈𝐽.

Proof. If 𝑢(0)<0, by the boundary conditions, we have 0>𝑢(0)≥𝑔₁(𝑢(𝑇))=𝑔₁(0)+𝑔^′₁(𝜉)𝑢(𝑇)≥𝑔^′₁(𝜉)𝑢(𝑇), (16) where 𝜉 is a constant between 0 and 𝑢(𝑇).
Because 𝑔₁(0)≥0 and 0<𝑔^′₁(𝑢)<1/𝐸_𝛿(𝑀𝑇^𝛿), then 𝑢(𝑇)<0 and 𝑢(0)>(1/𝐸_𝛿(𝑀𝑇^𝛿))𝑢(𝑇); that is, 𝑢(𝑇)<𝑢(0)𝐸_𝛿𝑀𝑇^𝛿. (17)
Let ℎ(𝑡)=^𝐶𝐷^𝛿₀₊𝑢(𝑡)−𝑀𝑢(𝑡), 𝑢(0)=𝑏, and 𝑢^′(0)=𝑑, and then ℎ(𝑡)≥0 for 𝑡∈𝐽 and 𝑑≥0.
By Lemma 10, we can get that the Cauchy problem ^𝐶𝐷^𝛿₀₊𝑢(𝑡)(𝑡)(𝑡)𝑢(0)−𝑀𝑢=ℎ,𝑡∈𝐽,=𝑏,𝑢^′(0)=𝑑, (18) has a unique solution 𝑢(𝑡)=𝑏𝐸_𝛿𝑀𝑡^𝛿+𝑑𝐸_𝛿,2𝑀𝑡^𝛿𝑡+^𝑡₀(𝑡−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑡−𝑠)^𝛿ℎ(𝑠)d𝑠. (19) So we can obtain that 𝑢(𝑇)=𝑏𝐸_𝛿𝑀𝑇^𝛿+𝑑𝑇𝐸_𝛿,2𝑀𝑇^𝛿+^𝑇₀(𝑇−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑇−𝑠)^𝛿ℎ(𝑠)d𝑠≥𝑏𝐸_𝛿𝑀𝑇^𝛿=𝑢(0)𝐸_𝛿𝑀𝑇^𝛿, (20) which is contradictory to (17).
Hence, 𝑏=𝑢(0)≥0. By (19), we can obtain that 𝑢(𝑡)≥0 for 𝑡∈𝐽.

Let the function 𝑔₁(𝑥)=𝑟₁𝑥 in Lemma 13; we can obtain Corollary 14.

Corollary 14. If there exists a constant 𝑟₁ with 0<𝑟₁<1/𝐸_𝛿(𝑀𝑇^𝛿) such that 𝑢∈𝐴𝐶²(𝐽) and ^𝐶𝐷^𝛿₀₊𝑢∈𝐶(𝐽) and satisfies ^𝐶𝐷^𝛿₀₊𝑢(𝑡)(𝑡)𝑢≥𝑀𝑢,𝑡∈𝐽,_′(0)≥0,𝑢(0)≥𝑟₁𝑢(𝑇), (21) then 𝑢(𝑡)≥0 on 𝐽.

Corollary 15. Suppose (H0) holds, and there exists a constant 𝑟₂>𝐸_𝛿(𝑀𝑇^𝛿) such that 𝑢∈𝐴𝐶²(𝐽) and ^𝐶𝐷^𝛿₀₊𝑢∈𝐶(𝐽) and satisfies ^𝐶𝐷^𝛿₀₊𝑢(𝑡)≥𝑀𝑢(𝑡)−^𝐶𝐷^𝛿₀₊𝑐(𝑡)(𝑡)×𝑢(𝑇)−𝑀𝑐−𝑟₂𝑢(0)𝑢,𝑡∈𝐽,_′(0)≥0,𝑢(𝑇)≥𝑟₂𝑢(0); (22) then 𝑢(𝑡)≥0 for 𝑡∈𝐽.

Proof. Let 𝑔₁(𝑥)=(1/𝑟₂)𝑥 and 𝑦(𝑡)=𝑢(𝑡)+𝑐(𝑡)(𝑢(𝑇)−𝑟₂𝑢(0)), 𝑡∈𝐽.
Since 𝑐(𝑡) is monotone decreasing and 𝑐(0)=0, then 𝑐(𝑡)≤0 on 𝐽. So 𝑦(𝑡)≤𝑢(𝑡),𝑡∈𝐽.
By (22), we have ^𝐶𝐷^𝛿₀₊𝑦(𝑡)(𝑡)=−𝑀𝑦_𝐶𝐷^𝛿₀₊𝑢(𝑡)(𝑡)+−𝑀𝑢_𝐶𝐷^𝛿₀₊𝑐(𝑡)−𝑀𝑐(𝑡)𝑢(𝑇)−𝑟₂𝑢(0)≥0, (23) and 𝑦^′(0)=𝑢^′(0)≥0.
Because 𝑐(0)=0 and 𝑐(𝑇)≤−1, then 𝑦(𝑇)=𝑢(𝑇)+𝑐(𝑇)(𝑢(𝑇)−𝑟₂𝑢(0))≤𝑟₂𝑢(0)=𝑟₂𝑦(0).
So 𝑦(0)≥(1/𝑟₂)𝑦(𝑇)=𝑔₁(𝑢(𝑇)).
By Lemma 13, we can get that 𝑦(𝑡)≥0, which implies that 𝑢(𝑡)≥0 for 𝑡∈𝐽.

3. Boundary Value Problems for the Linear Equation

In this section, we consider the existence and uniqueness of solutions for the linear fractional differential equation with nonlinear boundary conditions ^𝐶𝐷^𝛿₀₊𝑢(𝑡)(𝑡)(𝑡)𝑢−𝑀𝑢=ℎ,𝑡∈𝐽,_′(0)=0,𝑢(0)=𝑔(𝑢(𝑇)), (24) where ℎ∈𝐶(𝐽).

Theorem 16. Assume that (H0) and (H1) hold; there exist a lower solution 𝛼 and an upper solution 𝛽 of the boundary value problem (24) with 𝛼(𝑡)≥𝛽(𝑡) on 𝐽. Then the boundary value problem (24) has a unique solution 𝑢. Moreover, 𝛼≥𝑢≥𝛽; that is, 𝛼(𝑡)≥𝑢(𝑡)≥𝛽(𝑡) for 𝑡∈𝐽.

Proof. (1) We show that the solution of (24) is unique if it exists.
Suppose that 𝑢₁, 𝑢₂ are two solutions of (24) and let 𝑣=𝑢₁−𝑢₂. Then, ^𝐶𝐷^𝛿₀₊𝑣(𝑡)(𝑡)𝑣−𝑀𝑣=0,𝑡∈𝐽,_′(0)𝑣(0)𝑢=0,=𝑔₁(𝑇)−𝑔𝑢₂(𝑇)=𝑔^′𝜏₁𝑣(𝑇), (25) where 𝜏₁ is a constant between 𝑢₁(𝑇) and 𝑢₂(𝑇) and 0<𝑔^′(𝜏₁)<1/𝐸_𝛿(𝑀𝑇^𝛿) is a constant.
By Corollary 14, we have that 𝑣(𝑡)≥0; that is, 𝑢₁(𝑡)≥𝑢₂(𝑡) for 𝑡∈𝐽.
Similarly, we can also obtain that 𝑢₂(𝑡)≥𝑢₁(𝑡), for 𝑡∈𝐽.
Hence, 𝑢₁=𝑢₂.
(2) We prove that 𝛽≤𝑢≤𝛼 if 𝑢 is a solution of the boundary value problem (24).
Let 𝑚=𝛼−𝑢.
If 𝑔(𝛼(𝑇))≤𝛼(0), then 𝑎_𝛼(𝑡)=0 on 𝐽. We have ^𝐶𝐷^𝛿₀₊𝑚(𝑡)(𝑡)𝑚−𝑀𝑚≥0,𝑡∈𝐽,_′(0)𝑚(0)(𝛼(𝑇(𝑢(𝑇≥0,≥𝑔))−𝑔))=𝑔^′𝜏₂𝑚(𝑇), (26) where 𝜏₂ is a constant between 𝑢(𝑇) and 𝛼(𝑇).
We denote 𝑟₁=𝑔^′(𝜏₂). Hence, 0<𝑟₁<1/𝐸_𝛿(𝑀𝑇^𝛿).
By Corollary 14, we can obtain that 𝑚(𝑡)=𝛼(𝑡)−𝑢(𝑡)≥0 on 𝐽.
If 𝑔(𝛼(𝑇))>𝛼(0), then 𝑎_𝛼(𝑡)=(^𝐶𝐷^𝛿₀₊𝑐(𝑡)−𝑀𝑐(𝑡))(𝛼(𝑇)−𝑔⁻¹(𝛼(0))). We have that ^𝐶𝐷^𝛿₀₊𝑚(𝑡)(𝑡)=−𝑀𝑚_𝐶𝐷^𝛿₀₊𝛼(𝑡)−𝑀𝛼(𝑡)−^𝐶𝐷^𝛿₀₊𝑢(𝑡)+𝑀𝑢(𝑡)≥−^𝐶𝐷^𝛿₀₊𝑐(𝑡)−𝑀𝑐(𝑡)𝛼(𝑇)−𝑔⁻¹(𝛼(0))=−^𝐶𝐷^𝛿₀₊𝑐(𝑡)(𝑡)×𝛼(𝑇)(𝑇)−𝑀𝑐−𝑢+𝑔⁻¹(𝑢(0))−𝑔⁻¹(𝛼(0))=−^𝐶𝐷^𝛿₀₊𝑐(𝑡)−𝑀𝑐(𝑡)𝑚(𝑇)−𝑔^−1′𝜏₃𝑚(0), (27) where 𝜏₃ is a constant between 𝛼(0) and 𝑢(0).
We denote 𝑟₂=(𝑔⁻¹)^′(𝜏₃). Hence, 𝑟₂>𝐸_𝛿(𝑀𝑇^𝛿).
We can show 𝑚^′(0)≥0, 𝑚(𝑇)≥𝑔⁻¹(𝛼(0))−𝑔⁻¹(𝑢(0))=𝑟₂𝑚(0).
By Corollary 15, we have 𝑚(𝑡)≥0; that is, 𝛼(𝑡)≥𝑢(𝑡) on 𝐽. So 𝛼(𝑡)≥𝑢(𝑡) on 𝐽.
Similarly, we can obtain that 𝑢(𝑡)≥𝛽(𝑡) for 𝐽. Therefore, 𝛽≤𝑢≤𝛼.
(3) We prove that the problem (24) has a unique solution.
Let 𝑝(𝑡)=𝛼(𝑡)(𝛼(𝑇(0),𝛼(𝑡)(𝑡)×𝛼(𝑇),𝑔))≤𝛼+𝑐−𝑔⁻¹(𝛼(0)),𝑔(𝛼(𝑇))>𝛼(0),𝑞(𝑡)=𝛽(𝑡)𝛽(𝑇)(0),𝛽(𝑡)(𝑡)×𝑔,𝑔≥𝛽−𝑐₋₁𝛽(0)−𝛽(𝑇),𝑔𝛽(𝑇)<𝛽(0). (28)(29)
It is obvious that 𝑝,𝑞∈𝐴𝐶²(𝐽), ^𝐶𝐷^𝛿₀₊𝑝, ^𝐶𝐷^𝛿₀₊𝑞∈𝐶(𝐽), and 𝑝(0)=𝛼(0), 𝑞(0)=𝛽(0). If 𝑔(𝛼(𝑇))≤𝛼(0), we have 𝑝(𝑇)=𝛼(𝑇). If 𝑔(𝛼(𝑇))>𝛼(0), we have 𝑝(𝑇)≤𝑔⁻¹(𝛼(0)); that is, 𝑔𝑝(𝑇)≤𝑝(0). (30) Similarly, we can get 𝑔𝑞(𝑇)≥𝑞(0). (31)
If 𝑔(𝛼(𝑇))≤𝛼(0), we have ^𝐶𝐷^𝛿₀₊𝑝(𝑡)−𝑀𝑝(𝑡)≥ℎ(𝑡),𝑡∈𝐽. (32) If 𝑔(𝛼(𝑇))>𝛼(0), since 𝛼 is a lower solution of boundary value problem (24), then ^𝐶𝐷^𝛿₀₊𝛼(𝑡)−𝑀𝛼(𝑡)≥ℎ(𝑡)−𝑎_𝛼(𝑡). And by (12), we can obtain that ^𝐶𝐷^𝛿₀₊𝑝(𝑡)(𝑡)=−𝑀𝑝_𝐶𝐷^𝛿₀₊𝛼(𝑡)(𝑡)+−𝑀𝛼_𝐶𝐷^𝛿₀₊𝑐(𝑡)−𝑀𝑐(𝑡)𝛼(𝑇)−𝑔⁻¹(𝛼(0(𝑡)))≥ℎ−𝑎_𝛼(𝑡)+^𝐶𝐷^𝛿₀₊𝑐(𝑡)−𝑀𝑐(𝑡)𝛼(𝑇)−𝑔⁻¹(𝛼(0(𝑡)))=ℎ,𝑡∈𝐽. (33) Hence, we obtain ^𝐶𝐷^𝛿₀₊𝑝(𝑡)−𝑀𝑝(𝑡)≥ℎ(𝑡),𝑡∈𝐽. (34) Using the same way as mentioned above, we can get that ^𝐶𝐷^𝛿₀₊𝑞(𝑡)−𝑀𝑞(𝑡)≤ℎ(𝑡),𝑡∈𝐽. (35) Since (H0) holds, it is easy to see that 𝑐(𝑡)≤0 for 𝑡∈𝐽. We can easily get that 𝛽(𝑡)≤𝑞(𝑡) and 𝑝(𝑡)≤𝛼(𝑡) for 𝑡∈𝐽.
Let 𝑛(𝑡)=𝑝(𝑡)−𝑞(𝑡). It follows from (34) and (35) that ^𝐶𝐷^𝛿₀₊𝑛(𝑡)(𝑡)=−𝑀𝑛_𝐶𝐷^𝛿₀₊𝑝(𝑡)−𝑀𝑝(𝑡)−^𝐶𝐷^𝛿₀₊𝑞(𝑡)(𝑡)𝑛+𝑀𝑞≥0,_′(0)=𝑝^′(0)−𝑞^′(0)=𝛼^′(0)−𝛽^′(0)𝑛(0)(0)(0)𝑝(𝑇)𝑞(𝑇)≥0,=𝑝−𝑞≥𝑔−𝑔=𝑔^′𝜏₄𝑛(𝑇), (36) where 𝜏₄ is a constant between 𝑝(𝑇) and 𝑞(𝑇) and 0<𝑔^′(𝜏₄)<1/𝐸_𝛿(𝑀𝑇^𝛿) is a constant.
By Corollary 14, we have 𝑛(𝑡)≥0 for 𝑡∈𝐽; that is, 𝑝(𝑡)≥𝑞(𝑡)for𝑡∈𝐽. (37)
According to Lemma 10, for each 𝜆∈ℝ, the Cauchy problem ^𝐶𝐷^𝛿₀₊𝑢(𝑡)(𝑡)(𝑡),𝑢−𝑀𝑢=ℎ_′(0)=0,𝑢(0)=𝜆, (38) has a unique solution 𝑢(𝑡,𝜆)=𝜆𝐸_𝛿𝑀𝑡^𝛿+^𝑡₀(𝑡−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑡−𝑠)^𝛿ℎ(𝑠)d𝑠,𝑡∈𝐽. (39)
In the following, we show that 𝑞(𝑇)≤𝑢(𝑇,𝜆)≤𝑝(𝑇) for each 𝜆∈[𝑔(𝑞(𝑇)),𝑔(𝑝(𝑇))], where 𝑢(𝑡,𝜆) is defined by (39).
If 𝑢(𝑇,𝜆)<𝑞(𝑇), we denote 𝑙(𝑡)=𝑢(𝑡,𝜆)−𝑞(𝑡), and then 𝑙(𝑇)<0,𝑙(0)=𝑢(0,𝜆)−𝑞(0)≥𝜆−𝑔𝑞(𝑇)≥0. (40)(41) So 𝑙(0)≥𝑟₁𝑙(𝑇), for any 𝑟₁∈(0,1/𝐸_𝛿(𝑀𝑇^𝛿)).
It follows that 𝑐^′(0)=0 from (H0), and by (29), we have that 𝑞^′(0)=𝛽^′(0). Then 𝑙^′(0)=𝑢^′(0,𝜆)−𝑞^′(0)=−𝛽^′(0)≥0. (42) And from (35), we have ^𝐶𝐷^𝛿₀₊𝑙(𝑡)(𝑡)=−𝑀𝑙_𝐶𝐷^𝛿₀₊𝑢(𝑡)−𝑀𝑢(𝑡)−^𝐶𝐷^𝛿₀₊𝑞(𝑡)(𝑡)(𝑡)−−𝑀𝑞=ℎ^𝐶𝐷^𝛿₀₊𝑞(𝑡)−𝑀𝑞(𝑡)≥0. (43) By Corollary 14, we can obtain that 𝑙(𝑡)≥0, for all 𝑡∈𝐽, which is a contradiction to (40). So 𝑢(𝑇,𝜆)≥𝑞(𝑇). (44) Similarly, we can get 𝑢(𝑇,𝜆)≤𝑝(𝑇). (45)
Let 𝐻(𝜆)=𝑔(𝑢(𝑇,𝜆))−𝜆. Since 𝐻(𝑔(𝑞(𝑇)))𝐻(𝑔(𝑝(𝑇)))≤0 and 𝐻^′(𝜆)=𝑔^′(𝑢(𝑇,𝜆))𝑢^′_𝜆()<1𝑇,𝜆−1𝐸_𝛿𝑀𝑇^𝛿⋅𝐸_𝛿𝑀𝑇^𝛿−1=0, (46) then  𝐻(𝜆) is strictly monotone decreasing and continuous for 𝜆∈[𝑔(𝑞(𝑇)),𝑔(𝑝(𝑇))].
Hence, we can get that 𝐻(𝜆)=0 has a unique solution 𝜆₀∈[𝑔(𝑞(𝑇)),𝑔(𝑝(𝑇))] with 𝑔(𝑢(𝑇,𝜆₀))=𝜆₀=𝑢(0).
It is easy to see that 𝑢(𝑡,𝜆₀) is the unique solution of the boundary value problem (24).

4. Existence and Uniqueness of the Solutions for Boundary Value Problem

In this section, we study the existence and uniqueness of the solutions for fractional differential equation with nonlinear boundary conditions (1).

Let 𝐸=𝐶(𝐽) endowed with norm ‖𝑢‖=max_𝑡∈𝐽|𝑢(𝑡)| for 𝑢∈𝐸. Then 𝐸 is a Banach space. Denote 𝛽,𝛼=𝑢∈𝐶(𝐽)∶𝛽(𝑡)≤𝑢(𝑡)≤𝛼(𝑡),𝑡∈𝐽, (47) and 𝛽≤𝛼 if and only if 𝛽(𝑡)≤𝛼(𝑡) on 𝐽.

Theorem 17. Suppose that (H0) and (H1) hold; there exist 𝛼₀, 𝛽₀ which are lower and upper solutions of the boundary value problem (1) with 𝛽₀(𝑡)≤𝛼₀(𝑡) for 𝑡∈𝐽. And 𝑓 satisfies(H2)𝑓(𝑡,𝑥₂)−𝑓(𝑡,𝑥₁)≥0 for any 𝛽₀≤𝑥₁≤𝑥₂≤𝛼₀, 𝑡∈𝐽.Then there exist monotone sequences {𝛼_𝑛}, {𝛽_𝑛} such that lim_𝑛→∞𝛼_𝑛(𝑡)=𝛼^∗(𝑡),lim_𝑛→∞𝛽_𝑛(𝑡)=𝛽^∗(𝑡), (48)converging uniformly on 𝐽, and 𝛽^∗, 𝛼^∗ are the minimal and the maximal solutions of (1) on [𝛽₀,𝛼₀].

Proof. We denote 𝐷=[𝛽₀,𝛼₀]. For any 𝛾∈𝐷, we consider the following boundary value problem: ^𝐶𝐷^𝛿₀₊𝑢(𝑡)(𝑡)(𝑡)𝑢−𝑀𝑢=𝑓𝑡,𝛾,𝑡∈𝐽,_′(0)=0,𝑢(0)=𝑔(𝑢(𝑇)). (49) Since 𝛼₀(𝑡), 𝛽₀(𝑡) are lower and upper solutions of the boundary value problem (1), by (H2), we have ^𝐶𝐷^𝛿₀₊𝛼₀(𝑡)−𝑀𝛼₀(𝑡)≥𝑓𝑡,𝛼₀(𝑡)−𝑎_𝛼(𝑡)≥𝑓𝑡,𝛾(𝑡)−𝑎_𝛼(𝑡),_𝐶𝐷^𝛿₀₊𝛽₀(𝑡)−𝑀𝛽₀(𝑡)≤𝑓𝑡,𝛽₀(𝑡)−𝑏_𝛽(𝑡)≤𝑓𝑡,𝛾(𝑡)−𝑏_𝛽(𝑡), (50) for 𝑡∈𝐽.
Therefore, 𝛼₀(𝑡), 𝛽₀(𝑡) are also the lower and upper solutions of the boundary value problem (49), respectively.
In view of Theorem 16, the boundary value problem (49) has a unique solution 𝑢 and 𝑢∈𝐷.
Define 𝑃∶[𝛽₀,𝛼₀]→[𝛽₀,𝛼₀] by 𝑃𝛾=𝑢. Hence, 𝛼₀≥𝑃𝛼₀, 𝛽₀≤𝑃𝛽₀.
We will show that 𝑃𝑥₁≤𝑃𝑥₂ if 𝛽₀≤𝑥₁≤𝑥₂≤𝛼₀.
Let 𝑚=𝑃𝑥₂−𝑃𝑥₁. By (H2) and (49), we have ^𝐶𝐷^𝛿₀₊𝑚(𝑡)−𝑀𝑚(𝑡)=𝑓𝑡,𝑥₂(𝑡)−𝑓𝑡,𝑥₁(𝑡)𝑚≥0,_′(0)=0,𝑚(0)=𝑔^′𝜏₅𝑚(𝑇), (51) where 𝜏₅ is a constant between 𝑃𝑥₁(𝑇) and 𝑃𝑥₂(𝑇), and denote 𝑟₁=𝑔^′(𝜏₅)<1/𝐸_𝛿,1(𝑀𝑇^𝛿). By Corollary 14, 𝑚(𝑡)≥0, which implies 𝑃𝑥₁≤𝑃𝑥₂. Hence, 𝑃 is monotone increasing in [𝛽₀,𝛼₀].
Let 𝛼_𝑛=𝑃𝛼_𝑛−1 and 𝛽_𝑛=𝑃𝛽_𝑛−1 for 𝑛=1,2,…. We can get monotone iterative sequences 𝛽₁≤𝛽₂≤⋯≤𝛽_𝑛≤⋯≤𝛼_𝑛≤⋯≤𝛼₂≤𝛼₁. (52) Therefore, there exist 𝛼^∗,𝛽^∗∈𝐸, such that lim_𝑛→∞𝛼_𝑛(𝑡)=𝛼^∗(𝑡),lim_𝑛→∞𝛽_𝑛(𝑡)=𝛽^∗(𝑡). (53)
It is easy to see that 𝛼_𝑛(𝑡) satisfies ^𝐶𝐷^𝛿₀₊𝛼_𝑛(𝑡)−𝑀𝛼_𝑛(𝑡)=𝑓𝑡,𝛼_𝑛−1(𝑡),𝛼^′_𝑛(0)=0,𝛼_𝑛(0)=𝑔𝛼_𝑛(𝑇). (54) By Lemma 10, we have 𝛼_𝑛(𝑡)=𝛼_𝑛(0)𝐸_𝛿𝑀𝑡^𝛿+^𝑡₀(𝑡−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑡−𝑠)^𝛿𝑓𝑠,𝛼_𝑛−1(𝑠)d𝑠. (55) Because 𝑓 is continuous and 𝛼_𝑛−1∈𝐷, there exists a constant 𝐿>0 such that 𝑓𝑡,𝛼_𝑛−1(𝑡)𝑓≤𝐿,𝑡∈𝐽,𝑛=1,2,…,𝑡,𝛼_𝑛−1(𝑡)⟶𝑓𝑡,𝛼^∗(𝑡),𝑡∈𝐽,𝑛⟶∞. (56) By Lebesgue dominated convergence theorem, we can get that lim_𝑛→∞𝛼_𝑛(𝑡)=lim_𝑛→∞𝛼_𝑛(0)𝐸_𝛿𝑀𝑡^𝛿+lim_𝑛→∞^𝑡₀(𝑡−𝑠)^𝛿−1𝑓𝑠,𝛼_𝑛−1(𝑠)d𝑠=𝛼^∗(0)𝐸_𝛿𝑀𝑡^𝛿+^𝑡₀(𝑡−𝑠)^𝛿−1𝑓𝑠,𝛼^∗(𝑠)d𝑠. (57) That is, 𝛼^∗(𝑡)=𝛼^∗(0)𝐸_𝛿𝑀𝑡^𝛿+^𝑡₀(𝑡−𝑠)^𝛿−1𝑓𝑠,𝛼^∗(𝑠)d𝑠. (58)
Therefore, 𝛼^∗(𝑡)=𝑃𝛼^∗(𝑡)for𝑡∈𝐽. (59) It is similar to show that 𝛽^∗(𝑡)=𝑃𝛽^∗(𝑡)for𝑡∈𝐽. (60)
It is clear that 𝛼^∗, 𝛽^∗ are solutions of the boundary value problem (1).
Assume 𝑢∈[𝛽₀,𝛼₀] is a solution of the boundary value problem (1). We can easily obtain that 𝑃𝛽₀(𝑡)≤𝑃𝑢(𝑡)≤𝑃𝛼₀(𝑡) by the fact that 𝑃 is increasing in [𝛽₀,𝛼₀]. That is, 𝛽₁(𝑡)≤𝑢(𝑡)≤𝛼₁(𝑡). Doing this repeatedly, we have 𝛽_𝑛(𝑡)≤𝑢(𝑡)≤𝛼_𝑛(𝑡), for 𝑛=1,2,…. From Lemma 11, we obtain that 𝛽^∗(𝑡)≤𝑢(𝑡)≤𝛼^∗(𝑡), as 𝑛→∞.
Hence, 𝛼^∗, 𝛽^∗ are the maximal and the minimal solutions of the boundary value problem (1), respectively.

Theorem 18. Suppose that the conditions (H0) and (H1) hold; there exist 𝛼₀,𝛽₀ lower and upper solutions of the boundary value problem (1) with 𝛽₀(𝑡)≤𝛼₀(𝑡) for 𝑡∈𝐽, respectively. If there exists a constant 𝑘 with 0<𝑘<𝛿(1−𝑎𝐸_𝛿(𝑀𝑇^𝛿))/𝑇^𝛿𝐸_𝛿,𝛿(𝑀𝑇^𝛿), where 𝑎=max_{𝑢∈[𝛽0,𝛼0]}𝑔^′(𝑢), and 𝑓 satisfies(H3)𝑓(𝑡,𝑥₂)−𝑓(𝑡,𝑥₁)≤𝑘(𝑥₂−𝑥₁) for any 𝛽₀≤𝑥₁≤𝑥₂≤𝛼₀, 𝑡∈𝐽,then the boundary value problem (1) has a unique solution 𝑢^∗ on [𝛽₀,𝛼₀] and for any 𝑢₀∈[𝛽₀,𝛼₀], the iterative sequence 𝑢_𝑛=𝑃𝑢_𝑛−1, 𝑛=0,1,2,…, converging uniformly to 𝑢^∗ on 𝐽 and its error estimate is 𝑢_𝑛−𝑢^∗≤𝑘𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑎𝐸_𝛿𝑀𝑇^𝛿𝑛𝛼₀−𝛽₀. (61)

Proof. It is easy to check that the conditions of Theorem 17 are satisfied. Then the boundary value problem (1) has the maximal solution and the minimal solution which are denoted by 𝛼^∗, 𝛽^∗, respectively.
For 𝛽₀≤𝜔₁≤𝜔₂≤𝛼₀, 𝑡∈𝐽, we have 𝑃𝜔₂(𝑡)−𝑃𝜔₁(𝑡)=𝑃𝜔₂(0)−𝑃𝜔₁(0)𝐸_𝛿𝑀𝑡^𝛿+^𝑡₀(𝑡−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑡−𝑠)^𝛿×𝑓𝑠,𝜔₂(𝑠)−𝑓𝑠,𝜔₁(𝑠)d𝑠, (62) and from (H3), 𝑃𝜔₂(0)−𝑃𝜔₁(0)=𝑔𝑃𝜔₂(𝑇)−𝑔𝑃𝜔₁(𝑇)=𝑔^′(𝜏)𝑃𝜔₂(𝑇)−𝑃𝜔₁(𝑇)=𝑔^′(𝜏)𝑃𝜔₂(0)−𝑃𝜔₁(0)𝐸_𝛿𝑀𝑇^𝛿+^𝑇₀(𝑇−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑇−𝑠)^𝛿×𝑓𝑠,𝜔₂(𝑠)−𝑓𝑠,𝜔₁(𝑠)d𝑠≤𝑎𝑃𝜔₂(0)−𝑃𝜔₁(0)𝐸_𝛿𝑀𝑇^𝛿+^𝑇₀(𝑇−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑇−𝑠)^𝛿×𝑓𝑠,𝜔₂(𝑠)−𝑓𝑠,𝜔₁(𝑠)d𝑠≤𝑎𝑃𝜔₂(0)−𝑃𝜔₁(0)𝐸_𝛿𝑀𝑇^𝛿+𝑘𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿𝜔₂−𝜔₁, (63) where 𝜏∈[𝜔₁(𝑇),𝜔₂(𝑇)]⊂[𝛽₀(𝑇),𝛼₀(𝑇)] and 𝑃 is defined in Theorem 17.
It follows that 1−𝑎𝐸_𝛿(𝑀𝑇^𝛿)>0 from 0<𝑔^′(𝑥)<1/𝐸_𝛿(𝑀𝑇^𝛿) for 𝑥∈ℝ. Hence, 0≤𝑃𝜔₂(0)−𝑃𝜔₁(0)≤𝑘𝑎𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑎𝐸_𝛿𝑀𝑇^𝛿𝜔₂−𝜔₁,0≤𝑃𝜔₂(𝑡)−𝑃𝜔₁(𝑡)≤𝐸_𝛿𝑀𝑇^𝛿𝑘𝑎𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑎𝐸_𝛿𝑀𝑇^𝛿𝜔₂−𝜔₁+𝑘𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿𝜔₂−𝜔₁. (64)
We can easily get 𝑃𝜔₂−𝑃𝜔₁≤𝑘𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑎𝐸_𝛿𝑀𝑇^𝛿𝜔₂−𝜔₁. (65) Because 0<𝑘<𝛿(1−𝑎𝐸_𝛿(𝑀𝑇^𝛿))/𝑇^𝛿𝐸_𝛿,𝛿(𝑀𝑇^𝛿), we have 0<𝑘𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑎𝐸_𝛿𝑀𝑇^𝛿<1. (66) It implies that 𝑃 is contraction mapping. By using the contraction mapping principle, the 𝑃 has a unique fixed point.
Therefore, the boundary value problem (1) has a unique solution; that is, 𝛼^∗=𝛽^∗, and we denote it by 𝑢^∗.
For any 𝑢₀∈[𝛽₀,𝛼₀], let the iterative sequence 𝑢_𝑛=𝑃𝑢_𝑛−1, 𝑛=0,1,2,…. Similar to (65), we can get 𝑢_𝑛−𝑢^∗≤𝑘𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑎𝐸_𝛿𝑀𝑇^𝛿𝑃𝑢_𝑛−1−𝑢^∗≤≤⋯𝑘𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑎𝐸_𝛿𝑀𝑇^𝛿𝑛𝑢₀−𝑢^∗. (67) It follows that {𝑢_𝑛} converging uniformly to 𝑢^∗ on 𝐽 from (66).
Since 𝑢₀,𝑢^∗∈[𝛽₀,𝛼₀], we can obtain that the error estimate of {𝑢_𝑛} is 𝑢_𝑛−𝑢^∗≤𝑘𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑎𝐸_𝛿𝑀𝑇^𝛿𝑛𝛼₀−𝛽₀. (68)

5. The Calculation Method of the Approximate Analytical Solutions

In this section, we give the calculation method of the approximate analytical solutions for the boundary value problem (1).

Theorem 19. Suppose (H0) and (H1) hold; there exist a lower solution 𝛼₀ and an upper solution 𝛽₀ of the boundary value problem (1) with 𝛽₀(𝑡)≤𝛼₀(𝑡) for 𝑡∈𝐽. And there exists a constant 𝜅 with 0<𝜅<𝛿(1−𝑏𝐸_𝛿(𝑀𝑇^𝛿))/𝑇^𝛿𝐸_𝛿,𝛿(𝑀𝑇^𝛿), where 𝑏=sup_𝑢∈ℝ𝑔^′(𝑢)<1/𝐸_𝛿(𝑀𝑇^𝛿), such that 𝑓 satisfies(H4)0≤𝑓(𝑡,𝑥₂)−𝑓(𝑡,𝑥₁)≤𝜅(𝑥₂−𝑥₁) for any 𝑥₁≤𝑥₂, 𝑡∈𝐽.Then for any initial value 𝑦₀∈[𝛽₀,𝛼₀], the sequence {𝑦_𝑛} is defined by 𝑦_𝑛(𝑡)=𝑔𝑦_𝑛−1(𝑇)𝐸_𝛿𝑀𝑡^𝛿+^𝑡₀(𝑡−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑡−𝑠)^𝛿𝑓𝑠,𝑦_𝑛−1(𝑠)d𝑠, (69) converging uniformly to the unique solution 𝑢^∗ of the boundary value problem (1) on the [𝛽₀,𝛼₀]. Furthermore, the error estimate is 𝑦_𝑛−𝑢^∗≤𝜚^𝑛₀−𝜚₁^𝑛+11−𝜚₁𝛼₀−𝛽₀, (70) where 𝜚₀=𝑏𝐸_𝛿𝑀𝑇^𝛿+𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿,𝜚₁=𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑏𝐸_𝛿𝑀𝑇^𝛿. (71)

Proof. Because 0<𝜅<𝛿(1−𝑏𝐸_𝛿(𝑀𝑇^𝛿))/𝑇^𝛿𝐸_𝛿,𝛿(𝑀𝑇^𝛿), where 𝑏=sup_𝑢∈ℝ𝑔^′(𝑢)<1/𝐸_𝛿(𝑀𝑇^𝛿), then 𝑏≥𝑎 and 𝜅<𝛿(1−𝑎𝐸_𝛿(𝑀𝑇^𝛿))/𝑇^𝛿𝐸_𝛿,𝛿(𝑀𝑇^𝛿), where 𝑎 is defined by Theorem 18.
Hence, the conditions of Theorem 18 hold, and we can get that the boundary value problem (1) has a unique solution 𝑢^∗ on [𝛽₀,𝛼₀] and for any 𝑢₀∈[𝛽₀,𝛼₀], the iterative sequence 𝑢_𝑛=𝑃𝑢_𝑛−1, 𝑛=1,2,…, converging uniformly to 𝑢^∗ on 𝐽. That is, 𝑢_𝑛−𝑢^∗⟶0,as𝑛⟶∞. (72) By Lemma 10 and the definition of operator 𝑃, we can easily obtain that 𝑢_𝑛=𝑃𝑢_𝑛−1 is equivalent to 𝑢_𝑛(𝑡)=𝑢_𝑛(0)𝐸_𝛿𝑀𝑡^𝛿+^𝑡₀(𝑡−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑡−𝑠)^𝛿𝑓𝑠,𝑢_𝑛−1(𝑠)d𝑠, (73) and 𝑢_𝑛(0)=𝑔(𝑢_𝑛(𝑇)), 𝑛=0,1,2,….
Let 𝑦₀=𝑢₀, 𝑛=1,2,3,…. By Lemma 10, the Cauchy problem ^𝐶𝐷^𝛿₀₊𝑦_𝑛(𝑡)−𝑀𝑦_𝑛(𝑡)=𝑓𝑡,𝑦_𝑛−1(𝑡),𝑦^′_𝑛(0)=0,𝑦_𝑛(0)=𝑔𝑦_𝑛−1(𝑇), (74) has a unique solution 𝑦_𝑛(𝑡)=𝑔𝑦_𝑛−1(𝑇)𝐸_𝛿𝑀𝑡^𝛿+^𝑡₀(𝑡−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑡−𝑠)^𝛿𝑓𝑠,𝑦_𝑛−1(𝑠)d𝑠, (75) which is (69).
In the following, we prove 𝑦_𝑛 converging uniformly to 𝑢^∗ on 𝐽.
For 𝑡∈𝐽, we have 𝑦_𝑛(𝑡)−𝑢_𝑛(𝑡)≤𝐸_𝛿𝑀𝑇^𝛿𝑔𝑦_𝑛−1(𝑇)−𝑔𝑢_𝑛(𝑇)+^𝑇₀(𝑇−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀𝑇^𝛿×𝑓𝑠,𝑦_𝑛−1(𝑠)−𝑓𝑠,𝑢_𝑛−1(𝑠)d𝑠≤𝐸_𝛿𝑀𝑇^𝛿𝑔^′𝜉_𝑛𝑦_𝑛−1(𝑇)−𝑢_𝑛(𝑇+)𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿𝑦_𝑛−1−𝑢_𝑛−1≤𝑏𝐸_𝛿𝑀𝑇^𝛿𝑢_𝑛−𝑢_𝑛−1+𝑦_𝑛−1−𝑢_𝑛−1+𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿𝑦_𝑛−1−𝑢_𝑛−1≤𝑏𝐸_𝛿𝑀𝑇^𝛿𝑢_𝑛−𝑢_𝑛−1+𝑏𝐸_𝛿𝑀𝑇^𝛿+𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿𝑦_𝑛−1−𝑢_𝑛−1, (76) where 𝜉_𝑛 is between 𝑢_𝑛(𝑇) and 𝑦_𝑛−1(𝑇).
Therefore, 𝑦_𝑛−𝑢_𝑛≤𝑏𝐸_𝛿𝑀𝑇^𝛿𝑢_𝑛−𝑢_𝑛−1+𝜚₀𝑦_𝑛−1−𝑢_𝑛−1. (77)
Because 0<𝜅<𝛿(1−𝑏𝐸_𝛿(𝑀𝑇^𝛿))/𝑇^𝛿𝐸_𝛿,𝛿(𝑀𝑇^𝛿), we can show 0<𝜚₀=𝑏𝐸_𝛿𝑀𝑇^𝛿+𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿<1. (78)
On the other hand, by (73), we have 𝑢_𝑛−𝑢_𝑛−1=max_𝑡∈𝐽𝑢_𝑛(0)−𝑢_𝑛−1(0)𝐸_𝛿𝑀𝑡^𝛿+^𝑡₀(𝑡−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀(𝑡−𝑠)^𝛿×𝑓𝑠,𝑢_𝑛−1(𝑠)−𝑓𝑠,𝑢_𝑛−2(𝑠)d𝑠≤𝐸_𝛿𝑀𝑇^𝛿𝑔^′𝜂_𝑛𝑢_𝑛−𝑢_𝑛−1+^𝑇₀(𝑇−𝑠)^𝛿−1𝐸_𝛿,𝛿𝑀𝑇^𝛿×𝑓𝑠,𝑢_𝑛−1(𝑠)−𝑓𝑠,𝑢_𝑛−2(𝑠)d𝑠≤𝑏𝐸_𝛿𝑀𝑇^𝛿𝑢_𝑛−𝑢_𝑛−1+𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿𝑢_𝑛−1−𝑢_𝑛−2, (79) where 𝜂_𝑛 is between 𝑢_𝑛(0) and 𝑢_𝑛−1(0).
In view of 𝑏<1/𝐸_𝛿(𝑀𝑇^𝛿), we can get𝑢_𝑛−𝑢_𝑛−1≤𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑏𝐸_𝛿𝑀𝑇^𝛿𝑢_𝑛−1−𝑢_𝑛−2=𝜚₁𝑢_𝑛−1−𝑢_𝑛−2. (80)It follows that 0<𝜅<𝛿(1−𝑏𝐸_𝛿(𝑀𝑇^𝛿))/𝑇^𝛿𝐸_𝛿,𝛿(𝑀𝑇^𝛿) from 0<𝜚₁<1.
By (80), we can easily get 𝑢_𝑛−𝑢_𝑛−1≤𝜚₁𝑢_𝑛−1−𝑢_𝑛−2≤𝜚²₁𝑢_𝑛−2−𝑢_𝑛−2≤⋯≤𝜚₁^𝑛−1𝑢₁−𝑢₀. (81)
Substituting (81) into (77), we have 𝑦_𝑛−𝑢_𝑛≤𝑏𝐸_𝛿𝑀𝑇^𝛿𝜚₁^𝑛−1𝑢₁−𝑢₀+𝜚₀𝑦_𝑛−1−𝑢_𝑛−1. (82) It is easy to show that 𝑦_𝑛−𝑢_𝑛≤𝑏𝐸_𝛿𝑀𝑇^𝛿𝜚₁^𝑛−1+𝜚₀𝜚₁^𝑛−2𝑢₁−𝑢₀+𝜚²₀𝑦_𝑛−2−𝑢_𝑛−2≤𝑏𝐸_𝛿𝑀𝑇^𝛿𝜚₁^𝑛−1+𝜚₀𝜚₁^𝑛−2+𝜚²₀𝜚₁^𝑛−3𝑢₁−𝑢₀+𝜚³₀𝑦_𝑛−3−𝑢_𝑛−3≤⋯≤𝑏𝐸_𝛿𝑀𝑇^𝛿𝜚₁^𝑛−1+𝜚₀𝜚₁^𝑛−2+𝜚²₀𝜚₁^𝑛−3+⋯+𝜚₀^𝑛−1×𝑢₁−𝑢₀+𝜚^𝑛₀𝑦₀−𝑢₀=𝑏𝐸_𝛿𝑀𝑇^𝛿𝜚₁^𝑛−1+𝜚₀𝜚₁^𝑛−2+𝜚²₀𝜚₁^𝑛−3+⋯+𝜚₀^𝑛−1×𝑢₁−𝑢₀. (83)
We claim 𝜚₀>𝜚₁ from (71).
If 𝜚₀≤𝜚₁, we have 𝑏𝐸_𝛿𝑀𝑇^𝛿+𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿≤𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑏𝐸_𝛿𝑀𝑇^𝛿. (84) So 𝛿𝜅≥1−𝑏𝐸_𝛿𝑀𝑇^𝛿𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿, (85) which is a contradiction to the hypothesis of the theorem.
Therefore, it follows from (78) that 𝑦_𝑛−𝑢_𝑛≤𝑏𝐸_𝛿𝑀𝑇^𝛿𝜚^𝑛₀−𝜚^𝑛₁𝜚₀−𝜚₁𝑢₁−𝑢₀⟶0,as𝑛⟶∞. (86) By (72) and (86), we can show that 𝑦_𝑛−𝑢^∗≤𝑦_𝑛−𝑢_𝑛+𝑢_𝑛−𝑢^∗⟶0,as𝑛⟶∞. (87) Then for any initial value 𝑢₀∈[𝛽₀,𝛼₀], the sequence {𝑦_𝑛} is defined by (69) converging uniformly to the unique solution 𝑢^∗ of the boundary value problem (1) on the [𝛽₀,𝛼₀].
In view of (61), we have 𝑢_𝑛−𝑢^∗≤𝜅𝑇^𝛿𝐸_𝛿,𝛿𝑀𝑇^𝛿𝛿1−𝑎𝐸_𝛿𝑀𝑇^𝛿𝑛𝛼₀−𝛽₀=𝜚^𝑛₁𝛼₀−𝛽₀. (88) According to (86), we can show 𝑦_𝑛−𝑢_𝑛≤𝑏𝐸_𝛿𝑀𝑇^𝛿𝜚^𝑛₀−𝜚^𝑛₁𝜚₀−𝜚₁𝛼₀−𝛽₀. (89)
It is easy to see that 𝜚₀−𝜚₁=𝑏𝐸_𝛿𝑀𝑇^𝛿1−𝜚₁. (90) By (88) and (89), we can obtain 𝑦_𝑛−𝑢^∗≤𝜚^𝑛₁𝛼₀−𝛽₀+𝑏𝐸_𝛿𝑀𝑇^𝛿𝜚^𝑛₀−𝜚^𝑛₁𝜚₀−𝜚₁𝛼₀−𝛽₀=𝜚^𝑛₁+𝜚^𝑛₀−𝜚^𝑛₁1−𝜚₁𝛼₀−𝛽₀=𝜚^𝑛₀−𝜚₁^𝑛+11−𝜚₁𝛼₀−𝛽₀. (91) Hence, the error estimate of {𝑦_𝑛} is 𝑦_𝑛−𝑢^∗≤𝜚^𝑛₀−𝜚₁^𝑛+11−𝜚₁𝛼₀−𝛽₀. (92)

6. Example

We consider the boundary value problem ^𝐶𝐷^3/2₀₊𝑢(𝑡)−14𝑢(𝑡)=𝑡(𝑡)+𝑡10arctan𝑢²,𝑢10_′(0)(0)=1=0,𝑢5arctan𝑢(1). (93)

Let 𝑓(𝑡,𝑢)=(𝑡/10)arctan𝑢+𝑡²/10 and 𝑔(𝑢)=(1/5)arctan𝑢.

It is obvious that 𝑀=1/4, 𝑇=1, 𝛿=3/2, and 𝑏=sup_𝑢∈ℝ𝑔^′(𝑢)=1/5<1/𝐸_𝛿(𝑀𝑇^𝛿)(≈0.834179). Then (H1) holds.

We take 𝛽(𝑡)=0 for 𝑡∈𝐽.

Hence, 𝛽(0)=0, 𝑔(𝛽(1))=0, and 𝑏_𝛽(𝑡)=0. ^𝐶𝐷^𝛿₀₊𝛽(𝑡)=0, 𝑓(𝑡,𝛽(𝑡))=0, and then 𝛽(𝑡) is an upper solution of the boundary value problem (93).

Let 𝛼(𝑡)=5𝑡^3/2/6√𝜋+5𝑡/8√𝜋+35/48√𝜋≤35/16√𝜋 for 𝑡∈𝐽.

We have 𝛼^′(0)=5/8√𝜋, 𝛼(0)=35/48√𝜋≈0.411388, 𝑔(𝛼(1))=(1/5)arctan(35/16√𝜋)≈0.177966, and 𝑎_𝛼(𝑡)=0. Consider ^𝐶𝐷^𝛿₀₊𝛼(𝑡)−14𝛼(𝑡)≥58−354×16√𝜋≈0.316459>0.188983≈𝑓1,3516√𝜋≥𝑓(𝑡,𝛼(𝑡)), (94) for 𝑡∈𝐽. Then 𝛼(𝑡) is a lower solution of the boundary value problem (93). And 𝛽(𝑡)≤𝛼(𝑡).

It is easy to verify that the (H2) holds. By using Theorem 17, we know the problem (93) has minimal and maximal solutions in [𝛽,𝛼].

Let 𝜅=1/10; we know 𝜅<𝛿(1−𝑏𝐸_𝛿(𝑀𝑇^𝛿))/𝑇^𝛿𝐸_𝛿,𝛿(𝑀𝑇^𝛿)≈0.905853.

By calculating, we have 𝜚₀≈0.323682,𝜚₁≈0.110393. (95)

It is obvious that 0≤𝑓𝑡,𝑥₂−𝑓𝑡,𝑥₁≤1𝑥10²−𝑥₁=𝜅𝑥₂−𝑥₁,forany𝑥₁≤𝑥₂,𝑡∈𝐽. (96) By Theorem 19, we can get the problem (93) has a unique solution on the [𝛽₀,𝛼₀].

For any initial value 𝑢₀∈[𝛽₀,𝛼₀], the sequence {𝑦_𝑛} defined by 𝑦_𝑛(𝑡)=15arctan𝑦_𝑛−1(1)𝐸_3/214𝑡^3/2+^𝑡₀(𝑡−𝑠)^1/2𝐸_3/2,3/214(𝑡−𝑠)^3/2×𝑠10arctan𝑦_𝑛−1(𝑠)+𝑠²10d𝑠, (97) converging uniformly to the unique solution 𝑢^∗ of the boundary value problem (93) on the [𝛽₀,𝛼₀], and its error estimate is 𝑦_𝑛−𝑢^∗≤𝜚^𝑛₀−𝜚₁^𝑛+11−𝜚₁𝛼₀−𝛽₀=35𝜚^𝑛₀−𝜚₁^𝑛+116√𝜋1−𝜚₁, (98) where 𝑦₀=𝑢₀.

We take 𝑢₀=0. For 𝑛=3, its error is not more than 0.046841; for 𝑛=5, its error is not more than 0.00492662; for 𝑛=8, its error is not more than 0.000167155; for 𝑛=10, its error is not more than 0.0000175132.

Conflict of Interests

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

Acknowledgments

This work is supported by Natural Science Foundation of China (no. 11171220) and Support Projects of University of Shanghai for Science and Technology (no. 14XPM01).