Existence of Solutions of Nonlinear Mixed Two-Point Boundary Value Problems for Third-Order Nonlinear Differential Equation

Abstract

The authors use the upper and lower solution method to study the existence of solutions of nonlinear mixed two-point boundary value problems for third-order nonlinear differential equation𝑦′′′=𝑓(𝑥,𝑦,𝑦′,𝑦′′),𝑦′(𝑏)=ℎ(𝑦′(𝑎)),𝑝(𝑦(𝑎),𝑦(𝑏),𝑦′(𝑎),𝑦′(𝑏))=0,𝑔(𝑦(𝑎),𝑦(𝑏),𝑦′(𝑎),𝑦′(𝑏),𝑦′′(𝑎),𝑦′′(𝑏))=0. Some new existence results are obtained by developing the upper and lower solution method. Some applications are also presented.

1. Introduction

It is well known that the upper and lower solution method is a powerful tool for proving existence results for boundary value problems. The upper and lower solution method has been used to deal with the multipoint boundary value problems for second-order ordinary differential equations [1–4] and for higher-order ordinary differential equations [5–11]. There are fewer results on nonlinear mixed two-point boundary value problems for higher-order equations in the literature of ordinary differential equations. For this reason, we consider the third-order nonlinear ordinary differential equation: 𝑦^′′′=𝑓𝑥,𝑦,𝑦^′,𝑦^′′ (1.1) together with the nonlinear mixed two-point boundary conditions 𝑦′(𝑏)=ℎ𝑦^′(𝑎),𝑝𝑦(𝑎),𝑦(𝑏),𝑦^′(𝑎),𝑦^′(𝑏)𝑔𝑦(𝑎)(𝑏)=0,,𝑦,𝑦^′(𝑎),𝑦^′(𝑏),𝑦^′′(𝑎),𝑦^′′(𝑏)=0, (1.2) where the functions 𝑓, 𝑝, and 𝑔 are continuous and monotonic, ℎ is a homeomorphic mapping.

We will develop the upper and lower solution method for the boundary value problem 𝑦^′′′=𝑓𝑥,𝑦,𝑦^′,𝑦^′′,𝑦^′(𝑏)=ℎ𝑦^′(𝑎),𝑝𝑦(𝑎),𝑦(𝑏),𝑦^′(𝑎),𝑦^′(𝑏𝑔𝑦(𝑎)(𝑏))=0,,𝑦,𝑦^′(𝑎),𝑦^′(𝑏),𝑦^′′(𝑎),𝑦^′′(𝑏)=0 (1.3) and establish some new existence results. Furthermore, some applications are also presented.

2. Preliminaries

In this section, we will give some preliminary considerations and some lemmas which are essential to our main results.

Definition 2.1. Suppose the functions 𝛼(𝑥) and 𝛽(𝑥)∈𝐶⁽³⁾[𝑎,𝑏] satisfy 𝛼⁽³⁾(𝑥)≥𝑓𝑥,𝛼(𝑥),𝛼^′(𝑥),𝛼^′′(𝑥),𝛽⁽³⁾(𝑥)≤𝑓𝑥,𝛽(𝑥),𝛽^′(𝑥),𝛽^′′(𝑥),𝛼^′(𝑥)≤𝛽^′(𝑥)[],𝛼(𝑎)(𝑎),𝑥∈𝑎,𝑏<𝛽,𝛼^′(𝑎)<𝛽^′(𝑎). (2.1) Then 𝛼(𝑥) and 𝛽(𝑥) are, respectively, called the lower and upper solutions of the BVP (1.3).

Because of Definition 2.1, it is clear that 𝛼(𝑥)≤𝛽(𝑥),𝑥∈[𝑎,𝑏]. Let 𝐷=[𝑎,𝑏]×[𝛼(𝑥),𝛽(𝑥)]×[𝛼′(𝑥),𝛽′(𝑥)].

Definition 2.2. Let 𝐶[𝐷×ℝ,ℝ] denote the class of continuous functions from 𝐷×ℝ into ℝ, and let 𝑓(𝑥,𝑦,𝑦′,𝑦^′′)∈𝐶[𝐷×ℝ,ℝ] and 𝛼(𝑥),𝛽(𝑥)∈𝐶⁽³⁾[𝑎,𝑏] be lower and upper solutions of BVP (1.3). Suppose that there is a function 𝑊(𝑠)∈𝐶[ℝ₊,(0,+∞)] such that 𝑓𝑥,𝑦,𝑦^′,𝑦^′′≤𝑊𝑦^′′, (2.2) for every (𝑥,𝑦,𝑦′,𝑦^′′)∈𝐷×ℝ, where ^∞_𝜆𝑠𝑊(𝑠)𝑑𝑠>max_{𝑥∈[𝑎,𝑏]}𝛽^′(𝑥)−min_{[]𝑥∈𝑎,𝑏}𝛼^′(𝑥), (2.3) with 𝜆(𝑏−𝑎)=max𝛼^′(𝑎)−𝛽^′(𝑏),𝛼^′(𝑏)−𝛽^′(𝑎). (2.4) Then we say that 𝑓 satisfies Nagumo’s condition on the set 𝐷 relative to 𝛼(𝑥),𝛽(𝑥).

We assume throughout this paper the following.(𝐻₁)There are lower and upper solutions 𝛼(𝑥) and 𝛽(𝑥) of BVP (1.3) as Definition 2.1. (𝐻₂)Function 𝑓(𝑥,𝑦,𝑦′,𝑦^′′) satisfies Nagumo’s condition on the set 𝐷 relative to 𝛼(𝑥),𝛽(𝑥).(𝐻₃) Function 𝑓(𝑥,𝑦,𝑦′,𝑦^′′)∈𝐶[[𝑎,𝑏]×ℝ³,ℝ] is nonincreasing in 𝑦.(𝐻₄)ℎ∶[𝛼′(𝑎),𝛽′(𝑎)]→[𝛼′(𝑏),𝛽′(𝑏)] is a homeomorphism with ℎ𝛼^′(𝑎)=𝛼^′(𝑏),ℎ𝛽^′(𝑎)=𝛽^′(𝑏). (2.5)(𝐻₅) Function 𝑝(𝑠,𝑡,𝑢,𝑣) is continuous on ℝ⁴ and nondecreasing in 𝑡,𝑢,𝑣 and satisfies 𝑝𝛽(𝑎),𝛽(𝑏),𝛽^′(𝑎),𝛽^′(𝑏)𝑝𝛼(𝑎)(𝑏)≤0,,𝛼,𝛼^′(𝑎),𝛼^′(𝑏)≥0. (2.6)(𝐻₆) Function 𝑔(𝑥,𝑦,𝑧,𝑝,𝑞,𝑟) is continuous on ℝ⁶ and nondecreasing in 𝑥,𝑦,𝑞 and nonincreasing in 𝑟, and it satisfies 𝑔𝛽(𝑎),𝛽(𝑏),𝛽^′(𝑎),𝛽^′(𝑏),𝛽^′′(𝑎),𝛽^′′(𝑏)𝑔𝛼(𝑎)(𝑏)≤0,,𝛼,𝛼^′(𝑎),𝛼^′(𝑏),𝛼^′′(𝑎),𝛼^′′(𝑏)≥0. (2.7)

It is not difficult to obtain the following lemma.

Lemma 2.3. The boundary value problem 𝑦^′′′=𝑓𝑥,𝑦,𝑦^′,𝑦^′′,𝑦(𝑎)=0,𝑦^′(𝑎)=0,𝑦^′(𝑏)=0 (2.8) has a Green function 𝐺()=()𝑥,𝑠𝑥−𝑠²2−(𝑏−𝑠)(𝑎−𝑥)²2()−()𝑏−𝑎,𝑎≤𝑠≤𝑥≤𝑏,𝑏−𝑠)(𝑎−𝑥²2(𝑏−𝑎),𝑎≤𝑥≤𝑠≤𝑏 (2.9) with ^𝑏_𝑎|𝐺(()𝑥,𝑠)|𝑑𝑡≤𝑏−𝑎³,12_𝑏𝑎𝐺_𝑥(()𝑥,𝑠)𝑑𝑡≤𝑏−𝑎²4_𝑏𝑎,𝐺_𝑥𝑥(𝑥,𝑠)𝑑𝑡≤𝑏−𝑎. (2.10)

It is easy to prove the following lemma similarly to [12, page 25, Theorem 1.4.1].

Lemma 2.4. Assume that (𝐻₁),(𝐻₂) hold. Then for any solution 𝑦 of 𝑦^′′′=𝑓(𝑥,𝑦,𝑦′,𝑦^′′) with 𝛼(𝑥)≤𝑦(𝑥)≤𝛽(𝑥),𝛼′(𝑥)≤𝑦′(𝑥)≤𝛽′(𝑥) on [𝑎,𝑏], there exists a constant 𝑁>0 depending only on 𝛼,𝛽,𝑊, such that 𝑦^′′(𝑥)≤𝑁,𝑥∈[𝑎,𝑏] (2.11) and one calls 𝑁 is Nagumo’s constant.

Lemma 2.5. Assume that (𝐻₁)-(𝐻₃) hold. Then for any constant 𝐴∈[𝛼(𝑎),𝛽(𝑎)],𝐵∈[𝛼′(𝑎),𝛽′(𝑎)],𝐶∈[𝛼′(𝑏),𝛽′(𝑏)], the boundary value problem 𝑦^′′′=𝑓𝑥,𝑦,𝑦^′,𝑦^′′,𝑦(𝑎)𝑦=𝐴,_′(𝑎)𝑦=𝐵,_′(𝑏)=𝐶 (2.12) at least has a solution 𝑦∈𝐶⁽³⁾[𝑎,𝑏], with 𝛼(𝑥)(𝑥)(𝑥),𝛼≤𝑦≤𝛽_′(𝑥)≤𝑦^′(𝑥)≤𝛽^′(𝑥) (2.13) on [𝑎,𝑏].

Proof. By Lemma 2.3, it is clear that BVP (2.12) is equivalent to integral equation 𝑦(𝑥)=^𝑏_𝑎𝐺(𝑥,𝑠)𝑓𝑠,𝑦(𝑠),𝑦^′(𝑠),𝑦^′′(𝑠)𝑑𝑠+𝑊(𝑥), (2.14) where 𝑊(𝑥) is a polynomial satisfying 𝑦^′′′=0,𝑦(𝑎)=𝐴,𝑦^′(𝑎)=𝐵,𝑦′(𝑏)=𝐶.
Denote 𝑚=maxmax_{[]𝑥∈𝑎,𝑏}𝛼^′′(𝑥),max_{[]𝑥∈𝑎,𝑏}𝛽^′′(𝑥),𝑁+1, (2.15) where 𝑁 is Nagumo’s constant.
Assume 𝑓_𝑚𝑥,𝑦,𝑦′,𝑦^′′=𝑓𝑥,𝑦,𝑦′,𝑚,𝑦^′′𝑓>𝑚,𝑥,𝑦,𝑦′,𝑦^′′,−𝑚≤𝑦^′′𝑓≤𝑚,𝑥,𝑦,𝑦′,−𝑚,𝑦^′′𝐺<−𝑚,𝑥,𝑦,𝑦^′,𝑦^′′=𝑓_𝑚𝑥,𝛽(𝑥),𝑦^′,𝑦^′′(𝑥),𝑓,𝑦>𝛽_𝑚𝑥,𝑦,𝑦^′,𝑦^′′(𝑥)(𝑥),𝑓,𝛼≤𝑦≤𝛽_𝑚𝑥,𝛼(𝑥),𝑦^′,𝑦^′′(𝑥),𝐹,𝑦<𝛼𝑥,𝑦,𝑦^′,𝑦^′′=𝐺𝑥,𝑦,𝛽^′(𝑥),𝑦^′′+𝑦′−𝛽^′(𝑥)1+𝑦^′2,𝑦′>𝛽^′(𝑥),𝐺𝑥,𝑦,𝑦^′,𝑦^′′,𝛼^′(𝑥)≤𝑦′≤𝛽^′(𝑥),𝐺𝑥,𝑦,𝛼^′(𝑥),𝑦^′′−𝛼^′(𝑥)−𝑦′1+𝑦^′2,𝑦′<𝛼^′(𝑥). (2.16)
Then 𝐹(𝑥,𝑦,𝑦′,𝑦^′′) is bounded and continuous on [𝑎,𝑏]×ℝ³. Suppose |𝐹(𝑥,𝑦,𝑦′,𝑦^′′)|≤𝑀; |𝑊^(𝑖)(𝑥)|≤𝐾(𝑖=0,1,2), 𝑥∈[𝑎,𝑏].
Now, define an operator 𝑇 on the set 𝐸=𝐶⁽²⁾[[𝑎,𝑏],ℝ] by 𝑇𝑦(𝑥)=^𝑏_𝑎𝐺(𝑥,𝑠)𝐹𝑠,𝑦(𝑠),𝑦^′(𝑠),𝑦^′′(𝑠)𝑑𝑠+𝑊(𝑥). (2.17) If 𝑦∈𝐸, the norm is defined by 𝑦=max_{𝑥∈[𝑎,𝑏]}𝑦(𝑥)+𝑦^′(𝑥)+𝑦^′′(𝑥). (2.18) It is clear that (𝑥≤()𝑇𝑦)𝑏−𝑎³12𝑀+𝐾,𝑇𝑦^′(𝑥≤())𝑏−𝑎²4𝑀+𝐾,𝑇𝑦^′′(𝑥)≤(𝑏−𝑎)𝑀+𝐾. (2.19)
This shows that 𝑇 maps the closed, bounded, and convex set 𝑦≤()𝐵=𝑦∣𝑦∈𝐸,𝑏−𝑎³𝑀+()12𝑏−𝑎²𝑀4+(𝑏−𝑎)𝑀+3𝐾 (2.20) into itself. Also, 𝑇 is continuous and (𝑇𝑦)′ is bounded. All of these considerations imply that 𝑇 is completely continuous by Ascoli’s theorem. Schauder’ fixed point theorem then yields the fixed point 𝑦 of 𝑇 on 𝐵. In other words, the following boundary value problem 𝑦^′′′=𝐹𝑥,𝑦,𝑦^′,𝑦^′′,𝑦(𝑎)𝑦=𝐴,_′(𝑎)𝑦=𝐵,_′(𝑏)=𝐶 (2.21) has a solution 𝑦∈𝐶⁽³⁾[𝑎,𝑏], and satisfying 𝛼′(𝑥)≤𝑦′(𝑥)≤𝛽′(𝑥) and 𝛼(𝑎)≤𝑦(𝑎)≤𝛽(𝑎), we have 𝛼(𝑥)≤𝑦(𝑥)≤𝛽(𝑥), 𝑥∈[𝑎,𝑏].
In the following we prove that 𝛼^′(𝑥)≤𝑦^′(𝑥)≤𝛽^′(𝑥),𝑥∈[𝑎,𝑏]. (2.22) In fact, if it is invalid, there is no harm in setting the right inequality to be not true (the case that the left inequality is not true can be proved in the same way). By the assumption, if 𝑦′(𝑥)>𝛽′(𝑥), for some 𝑥∈[𝑎,𝑏], then there is a 𝑥₀∈(𝑎,𝑏) such that 𝑦^′𝑥₀−𝛽^′𝑥₀=max_{()𝑥∈𝑎,𝑏}𝑦^′(𝑥)−𝛽^′(𝑥𝑦)>0,_′′𝑥₀=𝛽^′′𝑥₀,𝑦^′′′𝑥₀≤𝛽^′′′𝑥₀. (2.23)(2.24)
Now, let 𝜔𝑥₀=𝛽𝑥₀,𝑦𝑥₀>𝛽𝑥₀,𝑦𝑥₀,𝛼𝑥₀≤𝑦𝑥₀≤𝛽𝑥₀,𝛼𝑥₀,𝑦𝑥₀<𝛼𝑥₀. (2.25) They imply that 𝛼(𝑥₀)≤𝜔(𝑥₀)≤𝛽(𝑥₀) and 𝐺𝑥₀,𝑦𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀=𝑓_𝑚𝑥₀,𝜔𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀. (2.26) We have 𝑦^′′′𝑥₀−𝛽^′′′𝑥₀≥𝐹𝑥₀,𝑦𝑥₀,𝑦^′𝑥₀,𝑦^′′𝑥₀−𝑓𝑥₀,𝛽𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀=𝐹𝑥₀,𝑦𝑥₀,𝑦^′𝑥₀,𝛽^′′𝑥₀−𝑓𝑥₀,𝛽𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀=𝐺𝑥₀,𝑦𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀−𝑓𝑥₀,𝛽𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀+𝑦^′𝑥₀−𝛽^′𝑥₀1+𝑦^′𝑥₀²>𝐺𝑥₀,𝑦𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀−𝑓𝑥₀,𝛽𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀=𝑓_𝑚𝑥₀,𝜔𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀−𝑓𝑥₀,𝛽𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀=𝑓𝑥₀,𝜔𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀−𝑓𝑥₀,𝛽𝑥₀,𝛽^′𝑥₀,𝛽^′′𝑥₀≥0 (2.27) which contradicts (2.24); hence, (2.22) is true.
Further, by the definition of 𝐹,𝑦 is a solution of the boundary value problem 𝑦^′′′=𝑓_𝑚𝑥,𝑦,𝑦^′,𝑦^′′,𝑦(𝑎)𝑦=𝐴,_′(𝑎)𝑦=𝐵,_′(𝑏)=𝐶. (2.28)
Because there is a 𝜉∈[𝑎,𝑏], such that 𝑦^′′(𝜉=𝑦)^′(𝑏)−𝑦^′(𝑎)𝑏−𝑎≤𝜆<𝑚, (2.29) so [𝑎,𝑏] has a maximal subinterval [𝑐,𝑑] with interior point 𝜉, for any 𝑥∈[𝑐,𝑑], |𝑦^′′(𝑥)|≤𝑚. Hence, for 𝑥∈[𝑐,𝑑], 𝑦(𝑥) is the solution of BVP (2.12). And by Lemma 2.4, we have |𝑦^′′(𝑥)|≤𝑁<𝑚; this contracts that [𝑐,𝑑] is the maximal subinterval, so we know [𝑐,𝑑]=[𝑎,𝑏]. Consequently, 𝑦 is a solution of BVP (2.12).

3. Main Results

Theorem 3.1. Assume (𝐻₁)-(𝐻₄),(𝐻₆) hold, then BVP 𝑦^′′′=𝑓𝑥,𝑦,𝑦^′,𝑦^′′,𝑦(𝑎)𝑦=𝐴,_′(𝑏)=ℎ𝑦^′(𝑎),𝑔𝑦(𝑎),𝑦(𝑏),𝑦^′(𝑎),𝑦^′(𝑏),𝑦^′′(𝑎),𝑦^′′(𝑏)=0 (3.1) has a solution 𝑦∈𝐶⁽³⁾[𝑎,𝑏], satisfying 𝛼(𝑥)(𝑥)(𝑥),𝛼≤𝑦≤𝛽_′(𝑥)≤𝑦^′(𝑥)≤𝛽^′(𝑥) (3.2) on [𝑎,𝑏].

Proof. By Lemma 2.5, we know that the boundary value problem 𝑦^′′′=𝑓𝑥,𝑦,𝑦^′,𝑦^′′,𝑦(𝑎)𝑦=𝐴,_′(𝑎)𝑦=𝐵,_′(𝑏)=ℎ(𝐵) (3.3) has a solution 𝑦∈𝐶⁽³⁾[𝑎,𝑏], with 𝛼(𝑥)(𝑥)(𝑥),𝛼≤𝑦≤𝛽_′(𝑥)≤𝑦^′(𝑥)≤𝛽′(𝑥) (3.4) on [𝑎,𝑏]. For any 𝐴∈[𝛼(𝑎),𝛽(𝑎)],𝐵∈[𝛼′(𝑎),𝛽′(𝑎)].
For fixed 𝐴, if 𝐵=𝛼′(𝑎), then 𝑦^′′(𝑎)≥𝛼^′′(𝑎),𝑦^′′(𝑏)≤𝛼^′′(𝑏). By (𝐻₆), we know 𝑔𝑦(𝑎),𝑦(𝑏),𝑦^′(𝑎),𝑦^′(𝑏),𝑦^′′(𝑎),𝑦^′′(𝑏)≥𝑔𝛼(𝑎),𝛼(𝑏),𝛼^′(𝑎),𝛼^′(𝑏),𝛼^′′(𝑎),𝛼^′′(𝑏)≥0. (3.5) On the other hand, if 𝐵=𝛽′(𝑎), then 𝑦^′′(𝑎)≤𝛽^′′(𝑎),𝑦^′′(𝑏)≥𝛽^′′(𝑏). By (𝐻₆), we have 𝑔𝑦(𝑎),𝑦(𝑏),𝑦^′(𝑎),𝑦^′(𝑏),𝑦^′′(𝑎),𝑦^′′(𝑏)≤𝑔𝛽(𝑎),𝛽(𝑏),𝛽^′(𝑎),𝛽^′(𝑏),𝛽^′′(𝑎),𝛽^′′(𝑏)≤0. (3.6)
Define the following sets: Π𝑦=𝑦∣𝑦^′′′=𝑓𝑥,𝑦,𝑦^′,𝑦^′′,𝑦(𝑎)=𝐴,𝑦^′(𝑎)=𝐵,𝑦^′(𝑏)=ℎ(𝐵);𝐴∈𝛼(𝑎),𝛽(𝑎),𝐵∈𝛼^′(𝑎),𝛽^′(𝑎,𝑀₁)=𝐵∣𝐵∈𝛼^′(𝑎),𝛽^′(𝑎(𝑥)𝑦,𝑔𝑦(𝑎)(𝑏)),𝑦∈Π,𝑦,𝑦^′(𝑎),𝑦^′(𝑏),𝑦^′′(𝑎),𝑦^′′(𝑏),𝑀>0₂=(𝑎)(𝑎(𝑥)𝑦,𝑔𝑦(𝑎)(𝑏)𝐵∣𝐵∈𝛼′,𝛽′),𝑦∈Π,𝑦,𝑦^′(𝑎),𝑦^′(𝑏),𝑦^′′(𝑎),𝑦^′′(𝑏)<0. (3.7) Obviously, Π(𝑦) is nonempty. If the theorem is not true, we know that 𝑀₁ and 𝑀₁ are all nonempty, and 𝑀₁⋃𝑀₂=[𝛼′(𝑎),𝛽′(𝑎)]. we claim that 𝑀₁ is closed. To see this, let 𝐵_𝑛∈𝑀₁, with 𝐵_𝑛→𝐵₀(𝑛→∞). Consider the following boundary value problem: 𝑦^′′′=𝑓𝑥,𝑦,𝑦^′,𝑦^′′,𝑦(𝑎)𝑦=𝐴,_′(𝑎)=𝐵_𝑛,𝑦^′(𝑏)=ℎ𝐵_𝑛. (3.8) By Lemma 2.5, it is known that, for every 𝑛∈𝑁, BVP (3.8) has a solution 𝑦_𝑛(𝑥)∈𝐶⁽³⁾[𝑎,𝑏], satisfying 𝛼(𝑥)≤𝑦_𝑛(𝑥)≤𝛽(𝑥),𝛼^′(𝑥)≤𝑦^′_𝑛(𝑥)≤𝛽^′(𝑥),𝑥∈[𝑎,𝑏] (3.9) and, by Lemma 2.4, we know |𝑦_𝑛^′′(𝑥)|≤𝑁.
Clearly, sequences {𝑦_𝑛(𝑥)}, {𝑦^′_𝑛(𝑥)}, {𝑦_𝑛^′′(𝑥)} are uniformly bounded and equicontinuous on [𝑎,𝑏]. Consequently, there exists a subsequence of 𝑦_𝑛(𝑥) which converges uniformly on [𝑎,𝑏], to a solution 𝑦₀(𝑥) of the BVP: 𝑦^′′′=𝑓𝑥,𝑦,𝑦^′,𝑦^′′,𝑦^′′′,𝑦(𝑎)𝑦=𝐴,_′(𝑎)=𝐵₀,𝑦′(𝑏)=ℎ𝐵₀ (3.10) with 𝑔𝑦₀(𝑎),𝑦₀(b),𝑦^′₀(𝑎),𝑦^′₀(𝑏),𝑦₀^′′(𝑎),𝑦₀^′′(𝑏)≥0. (3.11) By assumption, equality cannot occur, so that 𝑔𝑦₀(𝑎),𝑦₀(𝑏),𝑦^′₀(𝑎),𝑦^′₀(𝑏),𝑦₀^′′(𝑎),𝑦₀^′′(𝑏)>0, (3.12) and thus 𝐵₀∈𝑀₁. Consequently, 𝑀₁ is closed. Likewise, we may show 𝑀₂ is closed. This is a contradiction and proves the theorem.

Similar to the proof of Theorem 3.1, we can obtain the following theorem.

Theorem 3.2. Assume (𝐻₁)-(𝐻₆) hold, then BVP 𝑦^′′′=𝑓𝑥,𝑦,𝑦^′,𝑦^′′,𝑦^′(𝑏)=ℎ𝑦^′(𝑎),𝑝𝑦(𝑎),𝑦(𝑏),𝑦^′(𝑎),𝑦^′(𝑏𝑔𝑦(𝑎)(𝑏))=0,,𝑦,𝑦^′(𝑎),𝑦^′(𝑏),𝑦^′′(𝑎),𝑦^′′(𝑏)=0 (3.13) has a solution 𝑦∈𝐶⁽³⁾[𝑎,𝑏], satisfying 𝛼(𝑥)(𝑥)(𝑥),𝛼≤𝑦≤𝛽_′(𝑥)≤𝑦^′(𝑥)≤𝛽^′(𝑥) (3.14) on [𝑎,𝑏].

4. Applications

We all know it is difficult to find a solution of some nonlinear ordinary differential equation. But according to Theorem 3.2, we can know whether a boundary value problem, especially a nonlinear boundary value problem, has a solution and we also can know the existence regions of the solution and its derivative.

Example 4.1. Consider the following linear boundary value problem 𝑦^′′′(𝑥)=−𝑦(𝑥)+2𝑦^′′(𝑥)[],𝑦,𝑥∈0,1_′(1)−2𝑒𝑦^′(0)()𝑦(0)(1)=0,−3𝑒−2+𝑦+𝑦^′(0)+𝑦^′(1)𝑦(0)(1)=0,+𝑦−4𝑦^′(0)−5𝑦^′(1)+𝑦^′′(0)−𝑦^′′(1)=0. (4.1)
It is easy to know that 𝛼(𝑥)=−𝑥𝑒^𝑥−1,𝛽(𝑥)=𝑥𝑒^𝑥+1 are lower and upper solutions of the linear boundary value problem, respectively, where ℎ(𝑠)()=()𝑔=2𝑒𝑠,𝑝𝑠,𝑡,𝑢,𝑣−3𝑒−2𝑠+𝑡+𝑢+𝑣,𝑥,𝑦,𝑧,𝑝,𝑞,𝑟=𝑥+𝑦−4𝑧−5𝑝+𝑞−𝑟, (4.2) and all assumptions of Theorem 3.2 hold. So the linear boundary value problem has a solution 𝑦(𝑥) satisfying −𝑥𝑒^𝑥−1≤𝑦(𝑥)≤𝑥𝑒^𝑥+1,−𝑒^𝑥−𝑥𝑒^𝑥≤𝑦^′(𝑥)≤𝑒^𝑥+𝑥𝑒^𝑥. (4.3) Obviously, the trivial solution of the linear boundary value problem is one.

Example 4.2. Consider nonlinear boundary value problem 𝑦^′′′(𝑥)=−𝑦(𝑥)+𝑦^′′(𝑥)²−𝑦^′′(𝑥)𝜋,𝑥∈4,𝜋2,𝑦^′𝜋2−sin√2𝜋𝑦^′𝜋4𝑦=0,₂𝜋4+14𝑦𝜋2+18𝑦^′𝜋4+𝑦^′𝜋2𝑦𝜋=0,4+18𝑦𝜋2+𝑦^′𝜋4²+𝑦^′𝜋2²+𝑦^′′𝜋4−18𝑦^′′𝜋2=0. (4.4)
It is easy to verify that 𝛼(𝑥)=−sin(𝑥),𝛽(𝑥)=0 are lower and upper solutions of the nonlinear boundary value problem, respectively, where ℎ(𝑠)=sin√2𝜋𝑠,𝑝(𝑠,𝑡,𝑢,𝑣)=𝑠²+141𝑡+8𝑔1𝑢+𝑣,𝑥,𝑦,𝑧,𝑝,𝑞,𝑟=𝑥+8𝑦+𝑧²+𝑝²1+𝑞−8𝑟, (4.5) and all assumptions of Theorem 3.2 hold, so the BVP has a solution 𝑦(𝑥) satisfying −sin(𝑥)≤𝑦(𝑥)≤0,−cos(𝑥)≤𝑦^′(𝑥)≤0. (4.6)

5. Conclusion

In this paper, we study a nonlinear mixed two-point boundary value problem for a third-order nonlinear ordinary differential equation. Some new existence results are obtained by developing the upper and lower solution method. Furthermore, some applications are also presented.

Acknowledgments

The work is supported by the Fundamental Research Funds for the Central Universities (no. ZXH2012 K004) and Civil Aviation University of China Research Funds (no. 2012KYM05). The authors would like to thank the referees for their valuable comments.