Research Article | Open Access
Existence of Solutions of Nonlinear Mixed Two-Point Boundary Value Problems for Third-Order Nonlinear Differential Equation
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 equation. Some new existence results are obtained by developing the upper and lower solution method. Some applications are also presented.
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: together with the nonlinear mixed two-point boundary conditions where the functions , , and are continuous and monotonic, is a homeomorphic mapping.
We will develop the upper and lower solution method for the boundary value problem and establish some new existence results. Furthermore, some applications are also presented.
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 and satisfy Then and are, respectively, called the lower and upper solutions of the BVP (1.3).
Because of Definition 2.1, it is clear that . Let .
Definition 2.2. Let denote the class of continuous functions from into , and let and be lower and upper solutions of BVP (1.3). Suppose that there is a function such that for every , where with Then we say that satisfies Nagumo’s condition on the set relative to .
We assume throughout this paper the following.There are lower and upper solutions and of BVP (1.3) as Definition 2.1. Function satisfies Nagumo’s condition on the set relative to . Function is nonincreasing in . is a homeomorphism with Function is continuous on and nondecreasing in and satisfies Function is continuous on and nondecreasing in and nonincreasing in , and it satisfies
It is not difficult to obtain the following lemma.
Lemma 2.3. The boundary value problem has a Green function with
It is easy to prove the following lemma similarly to [12, page 25, Theorem ].
Lemma 2.4. Assume that hold. Then for any solution of with on , there exists a constant depending only on , such that and one calls is Nagumo’s constant.
Lemma 2.5. Assume that hold. Then for any constant , the boundary value problem at least has a solution , with on .
Proof. By Lemma 2.3, it is clear that BVP (2.12) is equivalent to integral equation
where is a polynomial satisfying .
Denote where is Nagumo’s constant.
Then is bounded and continuous on . Suppose ; , .
Now, define an operator on the set by If , the norm is defined by It is clear that
This shows that maps the closed, bounded, and convex set into itself. Also, is continuous and is bounded. All of these considerations imply that is completely continuous by Ascoli’s theorem. Schauder’ fixed point theorem then yields the fixed point of on . In other words, the following boundary value problem has a solution , and satisfying and , we have , .
In the following we prove that 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 , for some , then there is a such that
Now, let They imply that and We have which contradicts (2.24); hence, (2.22) is true.
Further, by the definition of is a solution of the boundary value problem
Because there is a , such that so has a maximal subinterval with interior point , for any , . Hence, for , is the solution of BVP (2.12). And by Lemma 2.4, we have ; this contracts that is the maximal subinterval, so we know . Consequently, is a solution of BVP (2.12).
3. Main Results
Theorem 3.1. Assume hold, then BVP has a solution , satisfying on .
Proof. By Lemma 2.5, we know that the boundary value problem
has a solution , with
on . For any .
For fixed , if , then . By , we know On the other hand, if , then . By , we have
Define the following sets: Obviously, is nonempty. If the theorem is not true, we know that and are all nonempty, and . we claim that is closed. To see this, let , with . Consider the following boundary value problem: By Lemma 2.5, it is known that, for every , BVP (3.8) has a solution , satisfying and, by Lemma 2.4, we know .
Clearly, sequences , , are uniformly bounded and equicontinuous on . Consequently, there exists a subsequence of which converges uniformly on , to a solution of the BVP: with By assumption, equality cannot occur, so that and thus . Consequently, is closed. Likewise, we may show 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 hold, then BVP has a solution , satisfying on .
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
It is easy to know that are lower and upper solutions of the linear boundary value problem, respectively, where and all assumptions of Theorem 3.2 hold. So the linear boundary value problem has a solution satisfying Obviously, the trivial solution of the linear boundary value problem is one.
Example 4.2. Consider nonlinear boundary value problem
It is easy to verify that are lower and upper solutions of the nonlinear boundary value problem, respectively, where and all assumptions of Theorem 3.2 hold, so the BVP has a solution satisfying
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.
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.
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Copyright © 2012 Yongxin Gao and Fengqin Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.