A Second-Order Boundary Value Problem with Nonlinear and Mixed Boundary Conditions: Existence, Uniqueness, and Approximation

Abstract

A second-order boundary value problem with nonlinear and mixed two-point boundary conditions is considered, 𝐿𝑥=𝑓(𝑡,𝑥,𝑥^′), 𝑡∈(𝑎,𝑏), 𝑔(𝑥(𝑎),𝑥(𝑏),𝑥^′(𝑎),𝑥^′(𝑏))=0, 𝑥(𝑏)=𝑥(𝑎) in which 𝐿 is a formally self-adjoint second-order differential operator. Under appropriate assumptions on 𝐿, 𝑓, and 𝑔, existence and uniqueness of solutions is established by the method of upper and lower solutions and Leray-Schauder degree theory. The general quasilinearization method is then applied to this problem. Two monotone sequences converging quadratically to the unique solution are constructed.

1. Introduction

The investigation of boundary value problems (denoted as BVPs for short) of ordinary differential equations is of great significance. On one hand, it makes a great impact on the studies of partial differential equations [1]. On the other hand, BVPs of ordinary differential equations can be used to describe a large number of mechanical, physical, biological, and chemical phenomena; see [2–5] for example. So far a lot of work has been carried out, including second-order, third-order, and higher-order BVPs with various boundary conditions.

As far as we know, for a long term most of works focused on existence and uniqueness of solutions. The works relating to approximation of solutions are relatively rare. In recent years, some approximate methods, such as the shooting method [6], monotone iterative technique [7], homotopy analysis method [8], and general quasilinearization method have been applied to BVPs for obtaining approximations of solutions. Among these methods, the general quasilinearization method becomes more and more popular.

The quasilinearization method was originally proposed by Bellman and Kalaba [9]. It is a very powerful approximation technique and unlike perturbation methods, is not dependent on the existence of a small or large parameter. The method, whose sequence of solutions of linear problems convergences to the solution of the original nonlinear problem, is quadratic and monotone, which is one of the reasons for the popularity of this technique. This method was generalized by Lakshmikantham and Vatsala [10] in which the convexity or concavity assumption on the nonlinear functions involved in the problems is relaxed.

So far, the general quasilinearization method, coupled with the method of upper and lower solutions, has been applied to obtain approximation of solutions for a large number of nonlinear problems, for example, BVPs of ordinary differential equations, such as first-order BVP with nonlinear boundary condition [11] and second-order BVPs with Dirichlet boundary condition [12], periodic boundary condition [13], three-point boundary condition [14], four-point boundary condition [15], and 𝑚-point boundary condition [16]; BVPs of partial differential equations, such as parabolic initial-boundary value problem [17], elliptic problems with nonlinear boundary condition [18] and 𝑝-Laplacian equations with nonlinear boundary condition [19], and so forth; BVPs of impulsive differential equations [20] and impulse functional differential equations with anti-periodic boundary condition [21]; BVPs of some practically nonlinear problems, such as Duffing equation involving both integral and nonintegral forcing terms with Robin boundary condition [22] and forced Duffing equation with discontinuous-type integral boundary condition [23]; as well as some abstract problems such as fixed point theorems in ordered Banach space [24].

Recently, El-Gebeily and O’Regan [25, 26] consider the singular and nonsingular second-order ordinary differential equations𝑙(𝑢(𝑡))=𝑓(𝑡,𝑢(𝑡)),𝑡∈(𝑎,𝑏) (1.1) with nonlinear boundary conditions as follows:𝑔(𝑢(𝑎),𝑢(𝑏),𝑝𝑢^′(𝑎))=0,ℎ(𝑢(𝑎),𝑢(𝑏),𝑝𝑢^′(𝑏))=0, (1.2)(1.3) in which 𝑙𝑢=−(𝑝𝑢^′)^′+𝑞𝑢 is a formally self-adjoint second-order differential operator. This type of BVPs arises in a variety of problems in applied mathematics and physics [27]. By defining the upper and lower solutions of BVP (1.1)–(1.3) suitably, Gebeily and Regan established the existence and uniqueness of solutions and constructed two monotonic iterative sequences converging to the unique solution quadratically. However, it can be noted that the 𝑢^′(𝑏) and 𝑢^′(𝑎) terms are not involved in (1.2) and (1.3), respectively. For second-order two-point BVPs, if all the terms 𝑢(𝑎),𝑢(𝑏),𝑢^′(𝑎) and 𝑢^′(𝑏) are involved in the boundary conditions, it seems quite difficult to deal with.

In this paper, we consider a second-order BVP with nonlinear and mixed two-point boundary conditions as follows: 𝐿𝑥=𝑓(𝑡,𝑥,𝑥^′),𝑡∈(𝑎,𝑏),𝑔(𝑥(𝑎),𝑥(𝑏),𝑥^′(𝑎),𝑥^′(𝑏))=0,𝑥(𝑏)=𝑥(𝑎), (1.4)(1.5)(1.6) where 𝐿𝑥=−(𝑝𝑥^′)^′+𝑞𝑥, in which 𝑝∈𝐶¹[𝑎,𝑏], 𝑝>0, and 𝑞∈𝐶[𝑎,𝑏], 𝑞≥0.

In BVP (1.4)–(1.6), it can be found that the boundary condition (1.5) is dependent on all the 𝑥(𝑎),𝑥(𝑏),𝑥^′(𝑎), and 𝑥^′(𝑏) terms. First, existence and uniqueness of solutions of BVP (1.4)–(1.6) is established by combining the method of upper and lower solutions with Leray-Schauder degree theory. Then, the general quasilinearization method is applied to construct the approximations of the unique solution. Two monotone sequences of iterations converging to the unique solution quadratically are obtained.

2. Preliminaries

In this section, several definitions and lemmas needed to the main results are given first.

Definition 2.1. 𝛽(𝑡),𝛼(𝑡)∈𝐶²[𝑎,𝑏] are called the upper and lower solutions of BVP (1.4)–(1.6), respectively, if 𝐿𝛽≥𝑓(𝑡,𝛽,𝛽^′),𝑡∈(𝑎,𝑏),𝑔(𝛽(𝑎),𝛽(𝑏),𝛽^′(𝑎),𝛽^′(𝑏))≤0,𝛽(𝑏)=𝛽(𝑎),𝐿𝛼≤𝑓(𝑡,𝛼,𝛼^′),𝑡∈(𝑎,𝑏),𝑔(𝛼(𝑎),𝛼(𝑏),𝛼^′(𝑎),𝛼^′(𝑏))≥0,𝛼(𝑏)=𝛼(𝑎). (2.1)

Definition 2.2. Let 𝐸 be a subset of [𝑎,𝑏]×𝑅²; it is said that the right-hand side function of (1.4) satisfies Nagumo condition on 𝐸 if |𝑓(𝑡,𝑥,𝑥^′)|≤ℎ(|𝑥^′|)=O(|𝑥^′|²) (2.2) holds for (𝑡,𝑥,𝑥^′)∈𝐸 and |𝑥^′|→+∞.

Lemma 2.3 (see [28]). Let 𝑓∶[𝑎,𝑏]×𝑅²→𝑅 be a continuous function satisfying Nagumo condition on 𝐸={(𝑡,𝑥,𝑥^′)∈[𝑎,𝑏]×𝑅²∶𝛼(𝑡)≤𝑥(𝑡)≤𝛽(𝑡)}, (2.3) where 𝛼,𝛽∶[𝑎,𝑏]→𝑅 are continuous functions such that 𝛼(𝑡)≤𝛽(𝑡) for all 𝑡∈[𝑎,𝑏]. Then there exists a constant 𝑁>0 such that every solution 𝑥(𝑡) of second-order equations 𝑥^′′=𝑓(𝑡,𝑥,𝑥^′) with 𝛼(𝑡)≤𝑥(𝑡)≤𝛽(𝑡),𝑡∈[𝑎,𝑏] (2.4) satisfies ‖𝑥^′‖_∞≤𝑁, in which 𝑁 is called the Nagumo constant.

Lemma 2.4. Boundary value problem as follows: 𝐿𝑥=−x,𝑡∈(𝑎,𝑏),𝑥(𝑎)=0,𝑥(𝑏)=0 (2.5)(2.6) has only the trivial solution.

Proof. Assume that 𝑥₀(𝑡) is an arbitrarily nontrivial solution of BVP (2.5)-(2.6). From the boundary conditions (2.6), it can be concluded that 𝑥₀(𝑡) can achieve its positive maximum or negative minimum in the interior of [𝑎,𝑏], suppose at 𝑡₀, 𝑡₀∈(𝑎,𝑏).
If 𝑥₀(𝑡) achieves its positive maximum, then 𝑥₀(𝑡₀)>0,𝑥^′₀(𝑡₀)=0,𝑥₀^′′(𝑡₀)≤0 (2.7) which means that 𝐿𝑥₀(𝑡₀)=−𝑝(𝑡₀)𝑥₀^′′(𝑡₀)−𝑝^′(𝑡₀)𝑥^′₀(𝑡₀)+𝑞(𝑡₀)𝑥(𝑡₀)≥0. (2.8) On the other hand, it can be derived from (2.5) that 𝐿𝑥₀(𝑡₀)=−𝑥₀(𝑡₀)<0. (2.9) It is a contradiction.
If 𝑥₀(𝑡) achieves its negative minimum, similar arguments lead to a contradiction too. Hence, BVP (2.5)-(2.6) has only the trivial solution.

Lemma 2.5 (see [26]). Define a linear operator 𝑙∶𝐶¹[𝑎,𝑏]→𝐶[𝑎,𝑏]×𝑅×𝑅 (2.10) by 𝑙𝑥(𝑡)=(𝑝𝑥^′(𝑎)−𝑝𝑥^′(𝑡)+^𝑡_𝑎𝑞(𝑠)𝑥(𝑠)𝑑𝑠,𝑥(𝑎),𝑥(𝑏)). (2.11) Then 𝑙⁻¹ exists and is continuous.

Lemma 2.6. Assume that(1)𝛼(𝑡),𝛽(𝑡) are the lower and upper solutions of BVP (1.4)–(1.6), respectively;(2)𝑓(𝑡,𝑥,𝑥^′) is continuous on [𝑎,𝑏]×𝑅² and is strictly decreasing in 𝑥 on 𝐷={(𝑡,𝑥,𝑥^′)∈[𝑎,𝑏]×𝑅²∶min{𝛼(𝑡),𝛽(𝑡)}≤𝑥≤max{𝛼(𝑡),𝛽(𝑡)}}; (2.12)(3)𝑔 is continuously differentiable on 𝑅⁴, strictly decreasing in the first and second variables, and nondecreasing and nonincreasing, respectively, in the third and forth variables.
Then, 𝛼(𝑡)≤𝛽(𝑡),𝑡∈[𝑎,𝑏]. (2.13)

Proof. Suppose that 𝛼(𝑡)>𝛽(𝑡) for some 𝑡∈[𝑎,𝑏]. Then there exist some 𝑡₀∈[𝑎,𝑏] such that max_{𝑡∈[𝑎,𝑏]}[𝛼(𝑡)−𝛽(𝑡)]=𝛼(𝑡₀)−𝛽(𝑡₀)>0. (2.14)
If 𝑡₀∈(𝑎,𝑏), then 𝛼(𝑡₀)>𝛽(𝑡₀), 𝛼^′(𝑡₀)=𝛽^′(𝑡₀), and 𝛼^′′(𝑡₀)≤𝛽^′′(𝑡₀) and consequently, 𝐿[𝛼(𝑡₀)−𝛽(𝑡₀)]=𝐿𝛼(𝑡₀)−𝐿𝛽(𝑡₀)≥0. (2.15) However, it follows from Definition 2.1 and mean value theorem that 𝐿𝛼(𝑡₀)−𝐿𝛽(𝑡₀)≤𝑓(𝑡₀,𝛼(𝑡₀),𝛼^′(𝑡₀))−𝑓(𝑡₀,𝛽(𝑡₀),𝛽^′(𝑡₀))=𝑓^′_𝑥(𝑡₀,𝜉,𝛽^′(𝑡₀))(𝛼(𝑡₀)−𝛽(𝑡₀))<0, (2.16) in which 𝜉∈(𝛽(𝑡₀),𝛼(𝑡₀)), and the last inequality follows from the strictly decreasing property of 𝑓 in 𝑥. It is a contradiction.
If 𝑡₀=𝑎, then 𝛼(𝑎)>𝛽(𝑎) and 𝛼^′(𝑎)−𝛽^′(𝑎)=lim_𝑡→𝑎+[𝛼(𝑡)−𝛽(𝑡)]−[𝛼(𝑎)−𝛽(𝑎)]𝑡−𝑎≤0. (2.17) By the definitions of the lower and upper solutions, we have 𝛼(𝑏)=𝛼(𝑎)>𝛽(𝑎)=𝛽(𝑏). (2.18) Moreover, 𝛼^′(𝑏)−𝛽^′(𝑏)=lim_{𝑡→𝑏−}(𝛼(𝑡)−𝛽(𝑡))−(𝛼(𝑏)−𝛽(𝑏))𝑡−𝑏=lim_{𝑡→𝑏−}(𝛼(𝑡)−𝛽(𝑡))−(𝛼(𝑎)−𝛽(𝑎))𝑡−𝑏≥0. (2.19) Consequently, in view of the monotonicity of 𝑔 in its variables, it follows that 0≥𝑔(𝛽(𝑎),𝛽(𝑏),𝛽^′(𝑎),𝛽^′(𝑏))>𝑔(𝛼(𝑎),𝛼(𝑏),𝛼^′(𝑎),𝛼^′(𝑏))≥0, (2.20) which is a contradiction.
If 𝑡₀=𝑏, similar deductions lead to a contradiction too. Hence, Lemma 2.6 is proved.

3. Existence and Uniqueness of Solutions

Theorem 3.1. Assume that(1)𝛼(𝑡),𝛽(𝑡) are the lower and upper solutions of BVP (1.4)–(1.6), respectively;(2)𝑓(𝑡,𝑥,𝑥^′) is continuous on [𝑎,𝑏]×𝑅² and is strictly decreasing in 𝑥 on 𝐷={(𝑡,𝑥,𝑥^′)∈[𝑎,𝑏]×𝑅²∶𝛼(𝑡)≤𝑥(𝑡)≤𝛽(𝑡)} (3.1) and satisfies Nagumo condition on 𝐷;(3)𝑔 is continuously differentiable, strictly decreasing in the first and second variables, nondecreasing and nonincreasing, respectively, in the third and forth variables on [𝛼(𝑎),𝛽(𝑎)]×[𝛼(𝑏),𝛽(𝑏)]×𝑅².

Then there exists a unique solution 𝑥(𝑡)∈𝐶²[𝑎.𝑏] of BVP (1.4)–(1.6) such that 𝛼(𝑡)≤𝑥(𝑡)≤𝛽(𝑡),𝑡∈[𝑎,𝑏]. (3.2)

Proof. Define 𝑤(𝑡,𝑥)=𝛼(𝑡),𝑥(𝑡)<𝛼(𝑡),𝑥(𝑡),𝛼(𝑡)≤𝑥(𝑡)≤𝛽(𝑡),𝛽(𝑡),𝑥(𝑡)>𝛽(𝑡). (3.3) Introduce the following auxiliary BVPs with homotopy character: 𝐿𝑥=𝜆𝑓(𝑡,𝑤(𝑡,𝑥),𝑥^′)−𝑥+𝜆𝑤(𝑡,𝑥),𝑥(𝑎)=𝜆[𝑔(𝑤(𝑎,𝑥(𝑎)),𝑤(𝑏,𝑥(𝑏)),𝑥^′(𝑎),𝑥^′(𝑏))+𝑤(𝑎,𝑥(𝑎))],𝑥(𝑏)=𝑥(𝑎) (3.4) in which 𝜆∈[0,1] is called the embedded parameter.
By the continuity of 𝑓 and 𝑔, and by the boundedness of 𝛼(𝑡) and 𝛽(𝑡), we can select a sufficiently large constant 𝑀₀>0 such that −𝑀₀<𝛼(𝑡)≤𝛽(𝑡)<𝑀₀,𝑓(𝑡,𝛽(𝑡),0)+𝛽(𝑡)<𝑀₀,𝑔(𝛽(𝑎),𝛽(𝑏),0,0)+𝛽(𝑎)<𝑀₀,−𝑀₀<𝑓(𝑡,𝛼(𝑡),0)+𝛼(𝑡),−𝑀₀<𝑔(𝛼(𝑎),𝛼(𝑏),0,0)+𝛼(𝑎). (3.5)(3.6)(3.7)(3.8)(3.9)
In what follows, the proof of the existence of solutions is divided into four steps.
Step 1. Show that, for 𝜆∈[0,1], every solution 𝑥(𝑡) of BVPs (3.4) satisfies |𝑥(𝑡)|<𝑀₀,𝑡∈[𝑎,𝑏]. (3.10)
Suppose that the estimate |𝑥(𝑡)|<𝑀₀ is not true. Then, there must be some points in [𝑎,𝑏] such that either 𝑥(𝑡)≥𝑀₀ or 𝑥(𝑡)≤−𝑀₀. For the former case, 𝑥(𝑡) has a positive maximum, suppose at 𝑡₁, that is, max_{𝑡∈[𝑎,𝑏]}𝑥(𝑡)=𝑥(𝑡₁)(≥𝑀₀>0) (3.11) can be assumed. We have three cases to consider.
Case 1 (𝑡₁∈(𝑎,𝑏)). In this case, 𝑥(𝑡₁)≥𝑀₀>𝛽(𝑡₁), 𝑥^′(𝑡₁)=0 and 𝑥^′′(𝑡₁)≤0 which leads to 𝐿𝑥(𝑡₁)≥0. On the other hand, since the definition of 𝑤(𝑡,𝑥) yields 𝑤(𝑡₁,𝑥(𝑡₁))=𝛽(𝑡₁); hence, for 𝜆∈(0,1], we have 𝐿𝑥(𝑡₁)=𝜆𝑓(𝑡₁,𝛽(𝑡₁),0)−𝑥(𝑡₁)+𝜆𝛽(𝑡₁)=𝜆[𝑓(𝑡₁,𝛽(𝑡₁),0)+𝛽(𝑡₁)]−𝑥(𝑡₁)≤𝜆[𝑓(𝑡₁,𝛽(𝑡₁),0)+𝛽(𝑡₁)−𝑀₀]<0, (3.12) in which the last inequality is obtained by the inequality (3.6). It is a contradiction.
For 𝜆=0, it can be derived that 0≤𝐿𝑥(𝑡₁)=−𝑥(𝑡₁)≤−𝑀₀<0, (3.13) which is also a contradiction.Case 2 (𝑡₁=𝑎). In this case, 𝑥(𝑎)≥𝑀₀>𝛽(𝑎), 𝑥^′(𝑎)≤0. Furthermore, we have 𝑥(𝑏)=𝑥(𝑎)≥𝑀₀𝑥>𝛽(𝑏),_′(𝑏)=lim_{𝑡→𝑏−}𝑥(𝑡)−𝑥(𝑏)𝑡−𝑏=lim_{𝑡→𝑏−}𝑥(𝑡)−𝑥(𝑎)𝑡−𝑏≥0. (3.14) The definition of 𝑤(𝑡,𝑥) means that 𝑤(𝑎,𝑥(𝑎))=𝛽(𝑎) and 𝑤(𝑏,𝑥(𝑏))=𝛽(𝑏). Hence, for 𝜆∈(0,1], in view of 𝑥^′(𝑎)≤0 and 𝑥^′(𝑏)≥0, it follows from the monotonicity of 𝑔 in its third and forth variables and the inequality (3.7) that 𝑀₀≤𝑥(𝑎)=𝜆[𝑔(𝑤(𝑎,𝑥(𝑎)),𝑤(𝑏,𝑥(𝑏)),𝑥^′(𝑎),𝑥^′(𝑏))+𝑤(𝑎,𝑥(𝑎))]=𝜆[𝛽(𝑎)+𝑔(𝛽(𝑎),𝛽(𝑏),𝑥^′(𝑎),𝑥^′(𝑏))]≤𝜆[𝛽(𝑎)+𝑔(𝛽(𝑎),𝛽(𝑏),0,0)]<𝜆𝑀₀≤𝑀₀ (3.15) which is a contradiction.
For 𝜆=0, we can obtain from the second equation in (3.4) that 0<𝑀₀≤𝑥(𝑎)=0, (3.16) which is a contradiction too.Case 3 (𝑡₁=𝑏). In this case, 𝑥(𝑏)≥𝑀₀>𝛽(𝑏), 𝑥^′(𝑏)≥0, and furthermore, 𝑥(𝑎)=𝑥(𝑏)≥𝑀₀𝑥>𝛽(𝑎),_′(𝑎)=lim_𝑡→𝑎+𝑥(𝑡)−𝑥(𝑎)𝑡−𝑎=lim_𝑡→𝑎+𝑥(𝑡)−𝑥(𝑏)𝑡−𝑎≤0. (3.17) The same deductions with those in Case 2 yield, for 𝜆∈(0,1], that 𝑀₀≤𝑥(𝑏)=𝑥(𝑎)≤𝜆[𝛽(𝑎)+𝑔(𝛽(𝑎),𝛽(𝑏),0,0)]<𝜆𝑀₀≤𝑀₀. (3.18) It is also a contradiction. For 𝜆=0, it can be deduced from the boundary conditions of (3.4) that 0<𝑀₀≤𝑥(𝑏)=𝑥(𝑎)=0, (3.19) which is a contradiction too.Step 2. Show that there exists a positive constant 𝑀₁ such that, for 𝜆∈[0,1], every solution 𝑥(𝑡) of BVPs (3.4) satisfies |𝑥^′(𝑡)|<𝑀₁,𝑡∈[𝑎,𝑏]. (3.20)
Let 𝐸={(𝑡,𝑥,𝑥^′)∈[𝑎,𝑏]×𝑅²∶|𝑥(𝑡)|<𝑀₀}. (3.21) Define 𝐹_𝜆(𝑡,𝑥,𝑥^′1)=−𝑝[𝑝^′𝑥^′−𝑞𝑥+𝜆𝑓(𝑡,𝑤(𝑡,𝑥),𝑥^′)−𝑥+𝜆𝑤(𝑡,𝑥)],(𝑡,𝑥,𝑥^′)∈𝐸. (3.22) Consequently, (1.4) can be rewritten as 𝑥^′′=𝐹_𝜆(𝑡,𝑥,𝑥^′). (3.23) Since the function 𝑓 satisfies Nagumo condition on 𝐸, |𝐹_𝜆(𝑡,𝑥,𝑥^′1)|=𝑝|𝑝^′𝑥^′−𝑞𝑥+𝜆𝑓(𝑡,𝑤(𝑡,𝑥),𝑥^′)−𝑥+𝜆𝑤(𝑡,𝑥)|=O(|𝑥^′|²),for(𝑡,𝑥,𝑥^′)∈𝐸,|𝑥^′|→+∞ (3.24) is obvious if the boundedness of 𝑥(𝑡) and 𝑤(𝑡,𝑥), 𝑝∈𝐶¹[𝑎,𝑏], 𝑝>0, and 𝑞∈𝐶[𝑎,𝑏] is kept in mind. That is, 𝐹_𝜆(𝑡,𝑥,𝑥^′) satisfies Nagumo condition on 𝐸. Hence, by Lemma 2.3, the estimate |𝑥^′(𝑡)|<𝑀₁, 𝑡∈[𝑎,𝑏], can be obtained.Step 3. Show that for 𝜆=1, BVP (3.4) has at least one solution 𝑥₁(𝑡).
Define a nonlinear operator 𝑁_𝜆∶𝐶¹[𝑎,𝑏]→𝐶[𝑎,𝑏]×𝑅×𝑅 (3.25) by 𝑁_𝜆𝑥(𝑡)=(^𝑡_𝑎[𝜆𝑓(𝑠,𝑤(𝑠,𝑥),𝑥^′)−𝑥+𝜆𝑤(𝑠,𝑥)]𝑑𝑠,𝐴_𝜆,𝐵_𝜆) (3.26) with 𝐴_𝜆=𝜆[𝑔(𝑤(𝑎,𝑥(𝑎)),𝑤(𝑏,𝑥(𝑏)),𝑥^′(𝑎),𝑥^′𝐵(𝑏))+𝑤(𝑎,𝑥(𝑎))],_𝜆=𝑥(𝑎). (3.27) Consequently, BVPs (3.4) are equivalent to the following operator equations: (𝐼−𝑙⁻¹𝑁_𝜆)𝑥=0, (3.28) in which 𝑙 is the linear operator defined in Lemma 2.5, and 𝐼 is the unit operator.
Define the norm in 𝐶[𝑎,𝑏]×𝑅×𝑅 as ‖𝑥(𝑡)‖_{𝐶[𝑎,𝑏]×𝑅×𝑅}=‖𝑥(𝑡)‖_{𝐶[𝑎,𝑏]}+|𝑅₁|+|𝑅₂| (3.29) for 𝑥(𝑡)∈𝐶[𝑎,𝑏]×𝑅×𝑅, in which 𝑅₁,𝑅₂ are two real numbers.
Let {𝑥_𝑛(𝑡)} be the bounded sequence on 𝐶¹[𝑎,𝑏]. It then follows from Steps 1 and 2 that {𝑁_𝜆𝑥_𝑛} and {𝑁^′_𝜆𝑥_𝑛} are both uniformly bounded in the sense of the norm defined above. Thus, {𝑁_𝜆𝑥_𝑛} is equicontinuous on 𝐶[𝑎,𝑏]×𝑅×𝑅. Consequently, Arzela-Ascoli theorem yields that {𝑁_𝜆𝑥_𝑛} is compact on 𝐶[𝑎,𝑏]×𝑅×𝑅. Therefore, 𝑁_𝜆𝑥 is a completely continuous operator.
Furthermore, the operator 𝑇_𝜆∶𝐶¹[𝑎,𝑏]→𝐶¹[𝑎,𝑏] (3.30) defined by 𝑇_𝜆=𝑙⁻¹𝑁_𝜆 (3.31) is also a completely continuous operator.
Define a bounded and open domain as follows: Ω={𝑥(𝑡)∈𝐶¹[𝑎,𝑏]∶|𝑥(𝑡)|<𝑀₀,|𝑥^′(𝑡)|<𝑀₁}. (3.32) It follows from Steps 1 and 2 that (𝐼−𝑙⁻¹𝑁_𝜆)𝜕Ω≠0. (3.33) Therefore, the degree Deg(𝐼−𝑙⁻¹𝑁_𝜆,Ω,0) is well defined. Then the invariance of degree under homotopy yields Deg(𝐼−𝑙⁻¹𝑁₁,Ω,0)=Deg(𝐼−𝑙⁻¹𝑁₀,Ω,0). (3.34)
Since the operator equation (𝐼−𝑙⁻¹𝑁₀)𝑥=0 is equivalent to BVP (2.5)-(2.6) which has only the trivial solution, therefore Deg(𝐼−𝑙⁻¹𝑁₁,Ω,0)=Deg(𝐼−𝑙⁻¹𝑁₀,Ω,0)=±1. (3.35) Consequently, the operator equation (𝐼−𝑙⁻¹𝑁₁)𝑥=0 gives 𝐿𝑥=𝑓(𝑠,𝑤(𝑠,𝑥),𝑥^′)−𝑥+𝑤(𝑠,𝑥),𝑥(𝑎)=𝑔(𝑤(𝑎,𝑥(𝑎)),𝑤(𝑏,𝑥(𝑏)),𝑥^′(𝑎),𝑥^′(𝑏))+𝑤(𝑎,𝑥(𝑎)),𝑥(𝑏)=𝑥(𝑎), (3.36)which has at least one solution 𝑥₁(𝑡)∈𝐶²[𝑎,𝑏].Step 4. Show that every solution 𝑥(𝑡)∈𝐶²[𝑎,𝑏] of BVP (3.36) satisfies 𝛼(𝑡)≤𝑥(𝑡)≤𝛽(𝑡),𝑡∈[𝑎,𝑏]. (3.37)
The right-hand side of this inequality is first proved. For the sake of contradiction, suppose that 𝑥(𝑡)>𝛽(𝑡) for some 𝑡∈[𝑎,𝑏]. Define ℎ(𝑡)=𝑥(𝑡)−𝛽(𝑡), then ℎ(𝑡) has a positive maximum at some 𝑡₁∈[𝑎,𝑏].
Case 1 (𝑡₁∈(𝑎,𝑏)). In this case, ℎ(𝑡₁)=𝑥(𝑡₁)−𝛽(𝑡₁)>0, ℎ^′(𝑡₁)=𝑥^′(𝑡₁)−𝛽^′(𝑡₁)=0, and ℎ^′′(𝑡₁)≤0 and consequently, 𝐿ℎ(𝑡₁)≥0.
On the other hand, 𝐿ℎ(𝑡₁)=𝐿𝑥(𝑡₁)−𝐿𝛽(𝑡₁)≤𝑓(𝑡₁,𝑤(𝑡₁,𝑥(𝑡₁)),𝑥^′(𝑡₁))−𝑥(𝑡₁)+𝑤(𝑡_,𝑥(𝑡₁))−𝑓(𝑡₁,𝛽(𝑡₁),𝛽^′(𝑡₁))=𝑓(𝑡₁,𝛽(𝑡₁),𝛽^′(𝑡₁))−𝑥(𝑡₁)+𝛽(𝑡₁)−𝑓(𝑡₁,𝛽(𝑡₁),𝛽^′(𝑡₁))=−𝑥(𝑡₁)+𝛽(𝑡₁)<0, (3.38) which is a contradiction.Case 2 (𝑡₁=𝑎). In this case, ℎ(𝑎)=𝑥(𝑎)−𝛽(𝑎)>0,ℎ^′(𝑎)=𝑥^′(𝑎)−𝛽^′(𝑎)≤0. (3.39) Furthermore, we have 𝑥𝛽(𝑏)=𝛽(𝑎)<𝑥(𝑎)=𝑥(𝑏),_′(𝑏)−𝛽^′(𝑏)=ℎ^′(𝑏)=lim_{𝑡→𝑏−}ℎ(𝑡)−ℎ(𝑏)𝑡−𝑏=lim_{𝑡→𝑏−}ℎ(𝑡)−ℎ(𝑎)𝑡−𝑏≥0. (3.40) The definition of 𝑤(𝑡,𝑥) yields 𝑤(𝑎,𝑥(𝑎))=𝛽(𝑎) and 𝑤(𝑏,𝑥(𝑏))=𝛽(𝑏). In view of 𝑥^′(𝑎)≤𝛽^′(𝑎) and 𝑥^′(𝑏)≥𝛽^′(𝑏), it then follows from the monotonicity of 𝑔 in its third and forth variables that 𝛽(𝑎)<𝑥(𝑎)=𝑔(𝑤(𝑎,𝑥(𝑎)),𝑤(𝑏,𝑥(𝑏)),𝑥^′(𝑎),𝑥^′(𝑏))+𝑤(𝑎,𝑥(𝑎))≤𝑔(𝛽(𝑎),𝛽(𝑏),𝛽^′(𝑎),𝛽^′(𝑏))+𝛽(𝑎)≤𝛽(𝑎). (3.41) It is also a contradiction.Case 3 (𝑡₁=𝑏). In this case, ℎ(𝑏)=𝑥(𝑏)−𝛽(𝑏)>0,ℎ^′(𝑏)=𝑥^′(𝑏)−𝛽^′(𝑏)≥0. (3.42) Furthermore, 𝑥𝛽(𝑎)=𝛽(𝑏)<𝑥(𝑏)=𝑥(𝑎),_′(𝑎)−𝛽^′(𝑎)=ℎ^′(𝑎)=lim_𝑡→𝑎+ℎ(𝑡)−ℎ(𝑎)𝑡−𝑎=lim_𝑡→𝑎+ℎ(𝑡)−ℎ(𝑏)𝑡−𝑎≤0. (3.43) Similar deductions lead to a contradiction too.
Therefore, the inequality 𝑥(𝑡)≤𝛽(𝑡) holds. In a similar way, 𝛼(𝑡)≤𝑥(𝑡) can be proved.
Till now, by Steps 1–4, the proof of the existence of solutions is completed. In what follows, we turn to the proof of the uniqueness of solutions.
We may assume that 𝑥₁(𝑡) and 𝑥₂(𝑡) are two arbitrary solutions of BVP (1.4)–(1.6). Define 𝑦(𝑡)=𝑥₁(𝑡)−𝑥₂(𝑡),𝑡∈[𝑎,𝑏]. (3.44) If 𝑦(𝑡)≡0 for 𝑡∈[𝑎,𝑏], the uniqueness of solutions is obvious. Otherwise, there must be some points in [𝑎,𝑏] such that either 𝑦(𝑡)>0 or 𝑦(𝑡)<0. For the former case, we define max_{𝑡∈[𝑎,𝑏]}𝑦(𝑡)∶=𝑦(𝑡₁). (3.45) Similarly, we only need to consider the following three cases.
Case 1 (𝑡₁∈(𝑎,𝑏)). In this case, 𝑦(𝑡₁)=𝑥₁(𝑡₁)−𝑥₂(𝑡₁)>0,𝑦^′(𝑡₁)=𝑥^′₁(𝑡₁)−𝑥^′₂(𝑡₁)=0,𝑦^′′(𝑡₁)≤0. (3.46) Thus, 0≤𝐿𝑦(𝑡₁). On the other hand, the mean value theorem and the strictly decreasing property of 𝑓(𝑡,𝑥,𝑥^′) in 𝑥 results that 𝐿𝑦(𝑡₁)=𝐿𝑥₁(𝑡₁)−𝐿𝑥₂(𝑡₁)=𝑓(𝑡₁,𝑥₁(𝑡₁),𝑥^′₁(𝑡₁))−𝑓(𝑡₁,𝑥₂(𝑡₁),𝑥^′₂(𝑡₁))=𝑓^′_𝑥(𝑡₁,𝜃,𝑥^′₁(𝑡₁))(𝑥₁(𝑡₁)−𝑥₂(𝑡₁))<0, (3.47) which is a contradiction, where 𝜃∈(𝑥₂(𝑡₁),𝑥₁(𝑡₁)).Case 2 (𝑡₁=𝑎). In this case, 𝑦(𝑎)=𝑥₁(𝑎)−𝑥₂(𝑎)>0,𝑦^′(𝑎)=𝑥^′₁(𝑎)−𝑥^′₂(𝑎)≤0. (3.48) Moreover, 𝑦(𝑏)=𝑥₁(𝑏)−𝑥₂(𝑏)=𝑥₁(𝑎)−𝑥₂𝑥(𝑎)=𝑦(𝑎)>0,_′1(𝑏)−𝑥^′₂(𝑏)=𝑦^′(𝑏)=lim_{𝑡→𝑏−}𝑦(𝑡)−𝑦(𝑏)𝑡−𝑏=lim_{𝑡→𝑏−}𝑦(𝑡)−𝑦(𝑎)𝑡−𝑏≥0. (3.49) Nevertheless, in view of the monotonicity of 𝑔 in its variables, it follows from the mean value theorem that 0=𝑔(𝑥₁(𝑎),𝑥₁(𝑏),𝑥^′₁(𝑎),𝑥^′₁(𝑏))−𝑔(𝑥₂(𝑎),𝑥₂(𝑏),𝑥^′₂(𝑎),𝑥^′₂(𝑏))=𝑔^′₁[𝑥₁(𝑎)−𝑥₂(𝑎)]+𝑔^′₂[𝑥₁(𝑏)−𝑥₂(𝑏)]+𝑔^′₃[𝑥^′₁(𝑎)−𝑥^′₂(𝑎)]+𝑔^′₄[𝑥^′₁(𝑏)−𝑥^′₂(𝑏)]<0, (3.50) which is a contradiction, in which 𝑔^′_𝑖=𝑔^′_𝑖(𝜉₁,𝜉₂,𝜉₃,𝜉₄), 𝑖=1,2,3,4, 𝜉₁∈(𝑥₂(𝑎),𝑥₁(𝑎)) and 𝜉₂,𝜉₃,𝜉₄ can be located by analogy.Case 3 (𝑡₁=𝑏). For this case, in the same way, it can be proved that this case is also impossible.Consequently, the conclusion in Theorem 3.1 is proved.

4. Approximations of the Unique Solution

In this section, a series of boundary value problems whose sequence of solutions converging to the unique solution of BVP (1.4)–(1.6) is constructed.

Theorem 4.1. Assume that (1)there exist the upper and lower solutions 𝛽₀(𝑡),𝛼₀(𝑡) of BVP (1.4)–(1.6);(2)𝑓(𝑡,𝑥,𝑥^′) is continuous on [𝑎,𝑏]×𝑅² and satisfies Nagumo condition on 𝐷={(𝑡,𝑥,𝑥^′)∈[𝑎,𝑏]×𝑅²∶𝛼₀(𝑡)≤𝑥(𝑡)≤𝛽₀(𝑡)}; (4.1)(3)𝑓_𝑥𝑥(𝑡,𝑥,𝑥^′) exists and is continuous on [𝑎,𝑏]×𝑅² and 𝑓_𝑥(𝑡,𝑥,𝑥^′)<0, 𝑓_𝑥𝑥(𝑡,𝑥,𝑥^′)≤0 on 𝐷×[−𝐶,𝐶], where 𝐶≥max{𝑁,max_{𝑡∈[𝑎,𝑏]}𝛼^′₀(𝑡),max_{𝑡∈[𝑎,𝑏]}𝛽^′₀(𝑡)} (4.2) is chosen sufficiently large, and 𝑁 is the Nagumo constant;(4)𝑔(𝑢,𝑣,𝑧,𝑤)∈𝐶¹([𝛼₀(𝑎),𝛽₀(𝑎)]×[𝛼₀(𝑏),𝛽₀(𝑏)]×𝑅²,𝑅), and it is strictly decreasing in 𝑢 and 𝑣, no-decreasing and nonincreasing, respectively, in 𝑧 and 𝑤.Then there exist a monotone nonincreasing sequence {𝛽_𝑛(𝑡)} and a monotone nondecreasing sequence {𝛼_𝑛(𝑡)} converging uniformly to the unique solution of BVP (1.4)–(1.6). Moreover, the rate of the convergence is quadratic.

Proof. Define a piecewise function 𝑞(𝑥^′)=max{−𝐶,min{𝑥^′,𝐶}} for 𝑥^′∈𝑅, and introduce 𝑓(𝑡,𝑥,𝑞(𝑥^′)) on 𝐷×[−𝐶,𝐶] and 𝐹(𝑡,𝑥,𝑥^′;𝑧)=𝑓(𝑡,𝑧,𝑞(𝑥^′))+𝑓_𝑥(𝑡,𝑧,𝑞(𝑥^′))(𝑥−𝑧) (4.3) for (𝑡,𝑥,𝑥^′;𝑧)∈𝐷×[−𝐶,𝐶]×[𝛼₀(𝑡),𝛽₀(𝑡)].
Obviously, 𝐹(𝑡,𝑥,𝑥^′;𝑧) is linear in 𝑥 if 𝑧 is viewed as a known quantity. Moreover, 𝑓(𝑡,𝑥,𝑞(𝑥^′)) and 𝐹(𝑡,𝑥,𝑥^′;𝑧) both satisfy the Nagumo condition.
We begin by considering the following BVP: 𝐿𝑥=𝐹(𝑡,𝑥,𝑥^′;𝛽₀),𝑔(𝑥(𝑎),𝑥(𝑏),𝑥^′(𝑎),𝑥^′(𝑏))=0,𝑥(𝑏)=𝑥(𝑎). (4.4) In view of the definition of the upper solution 𝛽₀(𝑡) of BVP (1.4)–(1.6), it can be verified directly that 𝐿𝛽₀≥𝑓(𝑡,𝛽₀,𝛽^′₀)=𝐹(𝑡,𝛽₀,𝛽^′₀;𝛽₀),𝑔(𝛽₀(𝑎),𝛽₀(𝑏),𝛽^′₀(𝑎),𝛽^′₀𝛽(𝑏))≤0,₀(𝑏)=𝛽₀(𝑎). (4.5) Hence, 𝛽₀(𝑡) is the upper solution of BVP (4.4).
According to condition (3), it can be deduced that 𝐹(𝑡,𝛼₀,𝛼^′₀;𝛽₀)−𝑓(𝑡,𝛼₀,𝛼^′₀)=𝑓(𝑡,𝛽₀,𝛼^′₀)+𝑓_𝑥(𝑡,𝛽₀,𝛼^′₀)(𝛼₀−𝛽₀)−𝑓(𝑡,𝛼₀,𝛼^′₀)=𝑓(𝑡,𝛽₀,𝛼^′₀)+𝑓_𝑥(𝑡,𝛽₀,𝛼^′₀)(𝛼₀−𝛽₀)−[𝑓(𝑡,𝛽₀,𝛼^′₀)+𝑓_𝑥(𝑡,𝛽₀,𝛼^′₀)(𝛼₀−𝛽₀1)+2𝑓_𝑥𝑥(𝑡,𝜉,𝛼^′₀)(𝛼₀−𝛽₀)²]1=−2𝑓_𝑥𝑥(𝑡,𝜉₁,𝛼^′₀)(𝛼₀−𝛽₀)²≥0, (4.6) in which |𝛼^′₀|<𝐶 has been taken into account, and 𝜉₁∈(𝛼₀,𝛽₀). Consequently, by taking the definition of the lower solution 𝛼₀(𝑡) of BVP (1.4)–(1.6) into account, we have 𝐿𝛼₀≤𝑓(𝑡,𝛼₀,𝛼^′₀)≤𝐹(𝑡,𝛼₀,𝛼^′₀;𝛽₀),𝑔(𝛼₀(𝑎),𝛼₀(𝑏),𝛼^′₀(𝑎),𝛼^′₀𝛼(𝑏))≥0,₀(𝑏)=𝛼₀(𝑎), (4.7) which indicates that 𝛼₀(𝑡) is the lower solution of BVP (4.4). Since BVP (4.4) satisfies all the requirements in Theorem 3.1, hence BVP (4.4) has a unique solution 𝛽₁(𝑡)∈𝐶²[𝑎,𝑏] with 𝛼₀(𝑡)≤𝛽₁(𝑡)≤𝛽₀(𝑡) and |𝛽^′₁(𝑡)|<𝐶₁, 𝑡∈[𝑎,𝑏], where 𝐶₁≤𝐶 is a constant.
Next, we consider 𝐿𝑥=𝐹(𝑡,𝑥,𝑥^′;𝛽₁),𝑔(𝑥(𝑎),𝑥(𝑏),𝑥^′(𝑎),𝑥^′(𝑏))=0,𝑥(𝑏)=𝑥(𝑎). (4.8) It can be shown that 𝐿𝛽₁=𝐹(𝑡,𝛽₁,𝛽^′₁;𝛽₀)=𝑓(𝑡,𝛽₀,𝛽^′₁)+𝑓_𝑥(𝑡,𝛽₀,𝛽^′₁)(𝛽₁−𝛽₀)≥𝑓(𝑡,𝛽₀,𝛽^′₁)+𝑓_𝑥(𝑡,𝛽₀,𝛽^′₁)(𝛽₁−𝛽₀1)+2𝑓_𝑥𝑥(𝑡,𝜉₂,𝛽^′₁)(𝛽₁−𝛽₀)²=𝑓(𝑡,𝛽₁,𝛽^′₁)=𝐹(𝑡,𝛽₁,𝛽^′₁;𝛽₁) (4.9) in which 𝜉₂∈(𝛽₁,𝛽₀). Thus, 𝛽₁(𝑡) is the upper solution of BVP (4.8) since 𝑔(𝛽₁(𝑎),𝛽₁(𝑏),𝛽^′₁(𝑎),𝛽^′₁𝛽(𝑏))≤0,₁(𝑏)=𝛽₁(𝑎) (4.10) are obvious if we keep that 𝛽₁(𝑡) is the solution of BVP (4.4) in mind.
Similarly, we can show that 𝛼₀(𝑡) is the lower solution of BVP (4.8). Hence, according to Theorem 3.1, there exists a unique solution 𝛽₂(𝑡)∈𝐶²[𝑎,𝑏] of BVP (4.8) such that 𝛼₀(𝑡)≤𝛽₂(𝑡)≤𝛽₁(𝑡) and |𝛽^′₂(𝑡)|<𝐶₁, 𝑡∈[𝑎,𝑏], where 𝐶₁ is a constant.
By repeating the above procedure iteratively, we obtain a monotone nonincreasing sequence {𝛽_𝑛(𝑡)} with 𝛼₀(𝑡)≤⋯≤𝛽_𝑛(𝑡)≤⋯≤𝛽₁(𝑡)≤𝛽₀(𝑡),|𝛽^′_𝑖(𝑡)|≤𝐶₁,𝑡∈[𝑎,𝑏], (4.11) where {𝛽_𝑛(𝑡)} are, respectively, the unique solution of the BVPs as follows: 𝐿𝑥=𝐹(𝑡,𝑥,𝑥^′;𝛽_𝑛−1),𝑔(𝑥(𝑎),𝑥(𝑏),𝑥^′(𝑎),𝑥^′(𝑏))=0,𝑥(𝑏)=𝑥(𝑎). (4.12)
Formula (4.11), the properties of the operator 𝐿, and the continuity of 𝐹 yield the uniform boundedness of {𝛽_𝑛^′′(𝑡)} for 𝑡∈[𝑎,𝑏]. Therefore, the sequences {𝛽^𝑖_𝑛(𝑡)}(𝑖=0,1) are uniformly bounded and equicontinuous on [𝑎,𝑏]. Hence, Ascoli-Arzela theorem guarantees the existence of subsequences {𝛽^𝑖_𝑛𝑘(𝑡)}⊂{𝛽^𝑖_𝑛(𝑡)} and a function 𝑥(𝑡)∈𝐶¹[𝑎,𝑏] with {𝛽^𝑖_𝑛𝑘(𝑡)} converging uniformly to 𝑥^𝑖(𝑡)(𝑖=0,1) on [𝑎,𝑏] as 𝑘→+∞.
Now let 𝑓_𝑘=𝐹(𝑡,𝛽_𝑛𝑘,𝛽^′_𝑛𝑘;𝛽_𝑛𝑘−1𝑔),_𝑘=𝑔(𝛽_𝑛𝑘(𝑎),𝛽_𝑛𝑘(𝑏),𝛽^′_𝑛𝑘(𝑎),𝛽^′_𝑛𝑘(𝑏)). (4.13) The continuity of the functions 𝐹 and 𝑔 imply that, as 𝑘→+∞, 𝑓_𝑘=𝐹(𝑡,𝛽_𝑛𝑘,𝛽^′_𝑛𝑘;𝛽_𝑛𝑘−1)→𝑓(𝑡,𝑥,𝑥^′𝑔),_𝑘=𝑔(𝛽_𝑛𝑘(𝑎),𝛽_𝑛𝑘(𝑏),𝛽^′_𝑛𝑘(𝑎),𝛽^′_𝑛𝑘(𝑏))→𝑔(𝑥(𝑎),𝑥(𝑏),𝑥^′(𝑎),𝑥^′(𝑏)). (4.14) The Lebeague dominated convergence theorem then yields ^𝑡_𝑎𝑓_𝑘(𝑠)𝑑𝑠→^𝑡_𝑎𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠 (4.15) for each 𝑡∈[𝑎,𝑏] as 𝑘→+∞.
Integrating the equation 𝐿𝛽_𝑛𝑘=𝐹(𝑡,𝛽_𝑛𝑘,𝛽^′_𝑛𝑘;𝛽_𝑛𝑘−1) (4.16) from 𝑎 to 𝑡 results in 𝑝𝛽^′_𝑛𝑘(𝑎)−𝑝𝛽^′_𝑛𝑘(𝑡)+^𝑡_𝑎𝑞(𝑠)𝛽_𝑛𝑘(𝑠)𝑑𝑠=^𝑡_𝑎𝑓_𝑘(𝑠)𝑑𝑠. (4.17) Passing to the limit in (4.17), we obtain 𝑝𝑥^′(𝑎)−𝑝𝑥^′(𝑡)+^𝑡_𝑎𝑞(𝑠)𝑥(𝑠)𝑑𝑠=^𝑡_𝑎𝑓(𝑠,𝑥(𝑠),𝑥^′(𝑠))𝑑𝑠. (4.18) Hence, 𝑥(𝑡) is the unique solution of BVP (1.4)–(1.6).
To show the quadratic rate of convergence, we define the error function as follows: 𝑒_𝑛(𝑡)=𝛽_𝑛(𝑡)−𝑥(𝑡). (4.19) Hence, 𝑒_𝑛(𝑡)≥0 for 𝑡∈[𝑎,𝑏].𝐿𝑒_𝑛=𝐿(𝛽_𝑛−𝑥)=𝐿𝛽_𝑛−𝐿𝑥=𝐹(𝑡,𝛽_𝑛,𝛽^′_𝑛;𝛽_𝑛−1)−𝑓(𝑡,𝑥,𝑥^′)=𝑓(𝑡,𝛽_𝑛−1,𝛽^′_𝑛)+𝑓_𝑥(𝑡,𝛽_𝑛−1,𝛽^′_𝑛)(𝛽_𝑛−𝛽_𝑛−1)−𝑓(𝑡,𝑥,𝑥^′)=𝑓(𝑡,𝑥,𝛽^′_𝑛)−𝑓(𝑡,𝑥,𝑥^′)+𝑓(𝑡,𝛽_𝑛−1,𝛽^′_𝑛)−𝑓(𝑡,𝑥,𝛽^′_𝑛)+𝑓_𝑥(𝑡,𝛽_𝑛−1,𝛽^′_𝑛)(𝛽_𝑛−𝑥)+𝑓_𝑥(𝑡,𝛽_𝑛−1,𝛽^′_𝑛)(𝑥−𝛽_𝑛−1)=𝑓_𝑥′(𝑡,𝛽_𝑛−1,𝜂₁)𝑒^′_𝑛+𝑓_𝑥(𝑡,𝛽_𝑛−1,𝛽^′_𝑛)𝑒_𝑛−12𝑓_𝑥𝑥(𝑡,𝜂₂,𝛽^′_𝑛)(𝑥−𝛽_𝑛−1)²≤𝑓_𝑥′(𝑡,𝛽_𝑛−1,𝜂₁)𝑒^′_𝑛+𝐾₁𝑒_𝑛+𝐾₂‖𝑒_𝑛−1‖², (4.20) in which 𝜂₁ lies between 𝛽^′_𝑛 and 𝑥^′, 𝜂₂∈(𝑥,𝛽_𝑛−1), 𝐾₁=max_{𝑡∈[𝑎,𝑏]}𝑓_𝑥(𝑡,𝛽_𝑛−1,𝛽^′_𝑛) is a negative constant, 𝐾₂ is a constant with 0≤−(1/2)𝑓_𝑥𝑥(𝑡,𝜂₂,𝛽^′_𝑛)≤𝐾₂ for (𝑡,𝜂₂,𝛽^′_𝑛)∈𝐷×[−𝐶,𝐶], and ‖𝑒_𝑛−1‖=max_{𝑡∈[𝑎,𝑏]}|𝑒_𝑛−1(𝑡)|.
Moreover, since 𝛽_𝑛(𝑡),𝑥(𝑡) are the solutions of BVPs (4.12) and BVP (1.4)–(1.6), respectively, therefore, mean value theorem yields 𝑔^′₁𝑒_𝑛(𝑎)+𝑔^′₂𝑒_𝑛(𝑏)+𝑔^′₃𝑒^′_𝑛(𝑎)+𝑔^′₂𝑒^′_𝑛𝑒(𝑏)=0,_𝑛(𝑎)=𝑒_𝑛(𝑏), (4.21)(4.22) in which 𝑔^′_𝑖=𝑔^′_𝑖(𝜂₃,𝜂₄,𝜂₅,𝜂₆), 𝑖=1,2,3,4, where 𝜂₃∈(𝑥(𝑎),𝛽_𝑛(𝑎)), and so forth.
Define a constant function by 𝐾𝑊(𝑡)=−²𝐾₁‖𝑒_𝑛−1‖²≥0,𝑡∈[𝑎,𝑏]. (4.23) Then, 𝑊(𝑡) satisfies 𝐿𝑊≥𝑓_𝑥′(𝑡,𝛽_𝑛−1,𝜂₁)𝑊^′+𝐾₁𝑊+𝐾₂‖𝑒_𝑛−1‖² (4.24) and moreover, since 𝑊(𝑡) is a nonnegative constant function, therefore, we have 𝑊(𝑎)≥0,𝑊(𝑏)≥0 and 𝑊^′(𝑎)=0, 𝑊^′(𝑏)=0. Consequently, the condition (4) guarantees that 𝑔^′₁𝑊(𝑎)+𝑔^′₂𝑊(𝑏)+𝑔^′₃𝑊^′(𝑎)+𝑔^′₂𝑊^′(𝑏)≤0,𝑊(𝑎)=𝑊(𝑏) (4.25)(4.26) in which 𝑔^′_𝑖, 𝑖=1,2,3,4, are defined in (4.21).
If 𝑒_𝑛(𝑡)≤𝑊(𝑡) holds uniformly for 𝑡∈[𝑎,𝑏], the quadratic convergence of the iteration has been proved. Otherwise, there must be some points 𝑡∈[𝑎,𝑏] such that 𝑒_𝑛(𝑡)>𝑊(𝑡). We then introduce a function 𝑞_𝑛(𝑡)=𝑒_𝑛(𝑡)−𝑊(𝑡),𝑡∈[𝑎,𝑏] (4.27) and define max_{𝑡∈[𝑎,𝑏]}𝑞_𝑛(𝑡)∶=𝑞_𝑛(𝑡₁). (4.28)Case 1 (𝑡₁∈(𝑎,𝑏)). In this case, 𝑞_𝑛(𝑡₁)=𝑒_𝑛(𝑡₁)−𝑊(𝑡₁)>0, 𝑞^′_𝑛(𝑡₁)=𝑒^′_𝑛(𝑡₁)−𝑊^′(𝑡₁)=0, and 𝑞_𝑛^′′(𝑡₁)=𝑒_𝑛^′′(𝑡₁)−𝑊^′′(𝑡₁)≤0. Hence, 𝐿𝑞_𝑛(𝑡₁)≥0. On the other hand, it can be deduced that 𝐿𝑞_𝑛(𝑡₁)=𝐿𝑒_𝑛(𝑡₁)−𝐿𝑊(𝑡₁)≤𝑓_𝑥′(𝑡₁,𝛽_𝑛−1,𝜂₁)(𝑒^′_𝑛(𝑡₁)−𝑊^′(𝑡₁))+𝐾₁(𝑒_𝑛(𝑡₁)−𝑊(𝑡₁))=𝐾₁𝑞_𝑛(𝑡₁)<0 (4.29) which is a contradiction.Case 2 (𝑡₁=𝑎). In this case, 𝑞_𝑛(𝑎)>0, 𝑞^′_𝑛(𝑎)≤0. Furthermore, we can obtain 𝑞_𝑛(𝑏)=𝑒_𝑛(𝑏)−𝑊(𝑏)=𝑒_𝑛(𝑎)−𝑊(𝑎)=𝑞_𝑛𝑞(𝑎)>0,_′𝑛(𝑏)=lim_{𝑡→𝑏−}𝑞_𝑛(𝑡)−𝑞_𝑛(𝑏)𝑡−𝑏=lim_{𝑡→𝑏−}𝑞_𝑛(𝑡)−𝑞_𝑛(𝑎)𝑡−𝑏≥0. (4.30) Consequently, the monotonicity of 𝑔 given in the condition (4) yields 𝑔^′₁𝑞_𝑛(𝑎)+𝑔^′₂𝑞_𝑛(𝑏)+𝑔^′₃𝑞^′_𝑛(𝑎)+𝑔^′₂𝑞^′_𝑛(𝑏)<0. (4.31) However, equation (4.21) minus inequality (4.25) gives 𝑔^′₁𝑞_𝑛(𝑎)+𝑔^′₂𝑞_𝑛(𝑏)+𝑔^′₃𝑞^′_𝑛(𝑎)+𝑔^′₂𝑞^′_𝑛(𝑏)≥0 (4.32) which contradicts to (4.31).
Similarly, we can prove that Case 3, that is, 𝑡₁=𝑏, is also impossible.
Hence, the inequality 𝑒_𝑛𝐾(𝑡)≤𝑊(𝑡)=−²𝐾₁‖𝑒_𝑛−1‖² (4.33) holds uniformly for 𝑡∈[𝑎,𝑏], that is, ‖𝑒_𝑛(𝑡)‖≤𝐾‖𝑒_𝑛−1‖²,𝑡∈[𝑎,𝑏] (4.34) holds uniformly, in which 𝐾=−𝐾₂/𝐾₁≥0 and ‖𝑒_𝑛(𝑡)‖=max_{𝑡∈[𝑎,𝑏]}|𝑒_𝑛(𝑡)|.

5. An Example

Consider a nonlinear BVP as follows: −((𝑡²+1)𝑥^′)^′+𝑘𝑥=−(𝑡²+3)𝑒^𝑥−2(𝑥^′)²+𝑡²1+1,𝑡∈(−1,1),−𝑥(−1)−𝑥(1)+3(𝑥^′(−1))³−13(𝑥^′(1))³=0,𝑥(−1)=𝑥(1) (5.1) in which 𝑘 is a constant to be determined, that is, 𝐿𝑥=−((𝑡²+1)𝑥^′)^′+𝑘𝑥,𝑓(𝑡,𝑥,𝑥^′)=−(𝑡²+3)𝑒^𝑥−2(𝑥^′)²+𝑡²1+1,𝑔(𝑥,𝑦,𝑧,𝑤)=−𝑥−𝑦+3𝑧³−13𝑤³ (5.2) in this BVP.

It can be verified that 𝛼₀(𝑡)=−2 and 𝛽₀(𝑡)=𝑡² are, respectively, the lower and upper solutions of BVP (5.1) if 𝑘≥(3−𝑒²)/2𝑒² is chosen. Moreover, it is easy to show that all the assumptions in Theorem 4.1 are satisfied. Therefore, by Theorem 4.1, BVP (5.1) has a unique solution and it can be approximated quadratically by two monotone sequences.

Acknowledgment

This work is supported by the National Natural Science Foundation of Fujian Province of China under Grant (no. S0650010).