Existence of Positive Solutions for an Elastic Beam Equation with Nonlinear Boundary Conditions
We study the existence and nonexistence of positive solutions for the following fourth-order two-point boundary value problem subject to nonlinear boundary conditions , where are parameters, and , and are continuous. By using the fixed-point index theory, we prove that the problem has at least one positive solution for sufficiently small and has no positive solution for large enough.
In this paper, we study the existence and nonexistence of positive solutions for the following fourth-order boundary value problem with nonlinear boundary conditions where , are parameters and and are real functions. If , , in mechanics, problem (1) is called cantilever beam equation [1, 2]. The equation describes the deflection of an elastic beam fixed at the left and freed at the right. There are some papers discussing the existence of solutions of the equation by using various methods, such as the lower and upper solution method, the Leray-Schauder continuation method, fixed-point theory, and the monotone iterative method; see [3–11]. If , problem (1) has also been studied; see [12–17].
In the case , , Yang et al.  obtained sufficient conditions of the existence of two solutions of problem (1) by using variational technique and a three-critical-point theorem. Recently, Li and Zhang  are concerned with the existence and uniqueness of monotone positive solution of problem (1) with , by using a new fixed-point theorem of generalized concave operators, but some monotone assumptions on and are needed.
A natural question is what would happen if and ?
Motivated by the above papers, we will prove the existence and nonexistence of positive solution for problem (1) by using the fixed-point index theory with the nonlinearity satisfying superlinear growth condition at infinity.
We make the following assumptions. (A1) is continuous. (A2) is continuous and is continuous. (A3) For any , .
The main results of the present paper are summarized as follows.
Theorem 1. Assume that (A1)–(A3) hold. Then problem (1) has at least one positive solution for , sufficiently small.
Theorem 2. Assume that (A1)–(A3) hold. Then problem (1) has no positive solution for large enough.
Remark 3. The results obtained in this paper are not a consequence of the previous theorem in the previous literature. Clearly, the boundary condition of (1) is more general than the above pieces of literature and problem (1) considers two different parameters which is more extensive.
Remark 4. It is pointed out that we do not need any monotone assumption on , , and , which is weaker than the corresponding assumptions on , in . References [12, 14] are only considered with the existence of solution for problem (1) with nonlinear boundary condition; however, we study the existence and nonexistence of positive solution of (1).
The remainder of this paper is arranged as follows. Section 2 presents some preliminaries. The proofs of Theorems 1 and 2 are given in Section 3. Finally, we give a simple example to illustrate our main results.
In this section we collect some preliminary results that will be used in subsequent sections. Let ; then is a Banach space under the norm . Define ; then is a nonnegative cone.
Lemma 5 (see [11, Lemma 2.2]). Let ; then the solution of the problem is where
Note that for any the function satisfies the boundary conditions in (1) by the definition of Green’s function . In view of Lemma 5, it is easy to see that is a fixed point of the operator if and only if is a solution of problem (1).
Lemma 7. has the following properties:(i), ;(ii), .
Proof. First, we will prove (i). For any , it follows that
From (A2), . So is nondecreasing for . Moreover,
so (i) holds.
Now, we prove (ii). For any , obviously, .
From (i), is nondecreasing for ; then .
For any fixed , define the cone Then is a positive cone; letting , with is a constant.
Lemma 8. Assume that (A1) and (A2) hold; then is completely continuous.
Proof. Obviously, for any , , it follows that .
For any , from (6) and Lemma 7(i), we have Hence
For any fixed , (11) together with Lemma 7(ii), we get So, .
According to (A1), (A2), and Arzela-Ascoli theorem, it is not difficult to verify that is completely continuous.
The following well-known fixed-point index theorem in cones is crucial to our arguments.
Lemma 9 (see ). Let be a Banach space and let be a cone. For , define . Assume that is compact map such that for .(i)If for , then .(ii)If for , then .
3. Proof of Main Results
In this section, we will prove our main results.
Proof of Theorem 1. Let
Let ; set
For any number , let . Choose so that
Then, for , and , combining (6) with Lemma 7, we have
which implies that
Thus, Lemma 9 implies that
According to (A3), for any ], there exists a such that where is chosen so that For any , choose , and set . If , then It follows from (A2) and Lemma 6 that which implies that Hence, from Lemma 9, we get From (18) and (24), it follows that Therefore, has a fixed point in ; that is, problem (1) has at least one positive solution.
Proof of Theorem 2. From (A1) and (A3), for any , there is a constant such that Let be a positive solution of (1); then satisfies (6). Choose large enough such that For any , we have which is a contradiction. Therefore, problem (1) has no positive solution for large enough.
Finally, we give an example to illustrate our main result.
Example 10. Consider the following fourth-order two-point problem with nonlinear boundary conditions: Clearly, the nonlinearity
It is easy to check that (A1)–(A3) are satisfied. By simple computation, we have . Set ; then .
If (where can be any real number greater than 0), , then , , and take ; then obviously holds for .
Choosing , it is easy to verify that
For any , there is a constant , so that If , then problem (29) has no positive solution for .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by NSFC (nos. 11361054 and 11201378), SRFDP (no. 2012 6203110004), and Gansu Provincial National Science Foundation of China (no. 1208RJZA258).