An Existence Result of Positive Solutions for Fully Second-Order Boundary Value Problems

Abstract

An existence result of positive solutions is obtained for the fully second-order boundary value problem       where is continuous. The nonlinearity may be sign-changing and superlinear growth on and . Our discussion is based on the method of lower and upper solution.

1. Introduction and Main Results

In this paper we discuss the existence of positive solution for second-order boundary value problem (BVP) with fully nonlinear term:where is continuous, . For the special case of BVP (1) that is nonnegative and does not contain derivative term , namely, the simply second-order boundary value problem the existence of positive solutions has been discussed by many authors; see [1–7]. In these works, the positivity of the corresponding Green function plays an important role. The positivity guarantees that BVP (2) can be converted to a fixed point problem of a cone mapping in , where . Hence, these authors can apply the fixed point theorems of cone mapping to obtain the existence of positive solutions for BVP (2). But their argument methods are not applicable to BVP (1), since these methods cannot deal with the derivative term .

For the more general BVP (1), the existence of solutions and multiple solutions has been discussed by some authors; see [8–11]. In [8–11], the authors have obtained some of the existence results of one solution or multiple solutions by using lower and upper solutions method. But, there are only a few results [12, 13] on the existence of positive solutions to the general Dirichlet BVP (1). In [12], Zhang discussed the existence of positive solutions by using Leray-Schauder degree theory and proved that BVP (1) has at least one positive solution if is nonnegative and sublinear growth on and ; see [12, Theorem 2.1]. Usually the superlinear problems are more difficult to treat than the sublinear problems. In [13], Agarwal et al. obtained existence results of positive solutions of BVP (1) by using the fixed point index in cones when is nonnegative. They allow that may be superlinear growth on rather than .

The purpose of this paper is to obtain existence result of positive solution for BVP (1) under the more general case that may be sign-changing and superlinear growth on and . Our main result is as follows.

Theorem 1. Let be continuous. If satisfies the conditions, (F1)there exists a positive constant such that (F2)there exist nonnegative constants satisfying and a positive constant such that (F3)given any , there is a positive continuous function on satisfying  such that then BVP (1) has at least one positive solution.

In Theorem 1, besides that nonlinearity may be sign-changing, condition (F3), a Nagumo-type condition, allows that may be superlinear growth on and but restricts on to quadric growth. See Example 4. This case has not been discussed in [12, 13].

The proofs of Theorem 1 are based on the method of lower and upper solution. If a function satisfies we call it a lower solution of BVP (1). If all of the inequality in (7) is inverse, we call it an upper solution of BVP (1). We will use the following well-known lower and upper solution theorem to prove Theorem 1 in the next section.

Theorem A. Let be continuous, and BVP (1) has a lower solution and an upper solution with . If the nonlinearity satisfies the Nagumo-type condition (F3), then BVP (1) has at least one solution between and .

For Theorem A, see [11, Theorem ].

2. Proof of the Main Results

Let . Let denote the Banach space of all continuous function on with norm , and generally for , denote the space of all th-order continuously differentiable function on with the norm . Let be the cone of nonnegative functions in . Let be the usual Hilbert space with the interior product and the norm . Let be the usual Sobolev space. means that , is absolutely continuous on , and . Let . Then is a Hilbert space with the norm . It is well-known that every satisfies the Poincaré inequality: .

Given , we consider the linear boundary value problem (LBVP):

Lemma 2. For every , LBVP (8) has a unique solution . Moreover, the solution operator is completely continuous and its norm satisfies where denotes the norm of the bounded linear operator space . When , the solution , and the solution operator is completely continuous.

Proof. Let . By the Green function expression of solution of linear boundary value problem, LBVP (8) has a unique solution expressed by where is the corresponding Green function given by By (10), we have From this and (10), we easily see that the solution operator is completely continuous. When , the solution , and the solution operator is completely continuous.
Since sine system is a complete orthogonal system of , every can be expressed by the Fourier series expansion where , , and the Parseval equalityholds. Let , then is the unique solution of LBVP (8), and and can be expressed by the Fourier series expansion of the sine system. Since , by the integral formula of Fourier coefficient, we obtain that On the other hand, since cosine system is another complete orthogonal system of , every can be expressed by the cosine series expansion where , . For the above , by the integral formula of the coefficient of cosine series, we obtain the cosine series expansions of : Now from (17), (13), and Parseval equality, it follows thatThis means that , namely (9) holds.

Lemma 3. Let be nonnegative constants and satisfy ; then for every nonzero , the boundary value problem has a unique positive solution .

Proof. We define a mapping by By Lemma 2, the composite mapping is well defined and the solution of BVP (19) is equivalent to the fixed point of . For every , by definition (20) and the Poincáre inequality, we have Hence by (9), Since , this means that is a contraction mapping. Hence, has a unique fixed point . By Lemma 2, . By (20), , and hence is unique solution of BVP (19).
Set . Then is nonzero and is a solution of the linear boundary problemBy the Green function expression of solution of linear boundary value problem, is expressed by where is the corresponding Green function. By the assumption of Lemma 3, . If , is given by (11), and if , is given by By (25) or (11), for every . Hence from (24) it follows that for every . This means that is a unique positive solution of BVP (19).

Proof of Theorem 1. We use Theorem A to prove Theorem 1. Choose a positive constant and let , where , , and are the constants in Assumptions (F1) and (F2). SinceBy Assumption (F1), we haveHence is a positive lower solution of BVP (1).
On the other hand, by Lemma 3, the following boundary value problem has a unique positive solution . By Assumption (F2), we have and hence is a positive upper solution of BVP (1).
Next we show that . Set ; then by the definitions of and , we have Let . Then by (30), and is a solution of linear boundary value problem (23). Hence can be expressed byBy the positivity of the Green function , for every . This means that . Hence, by Theorem A BVP (1) has a solution between and , which is a positive solution.

Example 4. Consider the following superlinear second-order boundary value problem: where , and are positive constants. Clearly, is a trivial solution of BVP (32). We show that BVP (32) has an explicit positive solution when . Corresponding to BVP (1), We easily see that satisfies Condition (F3). Choosing for every , by (33) we have Hence satisfies Condition (F1). Letting , we show that satisfies Condition (F2). Choose a constant such that . Then for every , by (33) we have This means that satisfies Condition (F2) for , , and . Hence by Theorem 1, BVP (32) has a positive solution when besides the trivial zero solution.

Conflict of Interests

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

Acknowledgments

This research was supported by NNSFs of China (11261053, 11361055) and the NFS of Gansu Province (1208RJZA129).