Research Article | Open Access
Symmetric Positive Solutions for Nonlinear Singular Fourth-Order Eigenvalue Problems with Nonlocal Boundary Condition
We investigate nonlinear singular fourth-order eigenvalue problems with nonlocal boundary condition , , , , where , , may be singular at and/or . Moreover may also have singularity at and/or . By using fixed point theory in cones, an explicit interval for is derived such that for any in this interval, the existence of at least one symmetric positive solution to the boundary value problem is guaranteed. Our results extend and improve many known results including singular and nonsingular cases. The associated Green's function for the above problem is also given.
Boundary value problems for ordinary differential equations arise in different areas of applied mathematics and physics and so on, and the existence of positive solutions for such problems has become an important area of investigation in recent years. To identify a few, we refer the reader to [1–7] and references therein.
At the same time, a class of boundary value problems with nonlocal boundary conditions appeared in heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics. Such problems include two-point, three-point, multipoint boundary value problems as special cases and have attracted the attention of Gallardo , Karakostas and Tsamatos , and Lomtatidze and Malaguti  (and see the references therein). For more information about the general theory of integral equations and their relation to boundary value problems we refer the reader to the book of Corduneanu  and Agarwal and O'Regan .
Motivated by the works mentioned above, in this paper, we study the existence of symmetric positive solutions of the following fourth-order nonlocal boundary value problem (BVP):
where , , may be singular at and/or . Moreover may also have singularity at and/or .
The main features of this paper are as follows. Firstly, comparing with [4–7], we discuss the boundary value problem with nonlocal boundary conditions, that is, BVP (1.1) including fourth-order two-point, three-point, multipoint boundary value problems as special cases. Secondly, comparing with [4–7], we discuss the boundary value problem when nonlinearity contains second-derivatives . Thirdly, here we not only allow have singularity at and/or but also allow have singularity at and/or . Finally, in [4–7], authors only studied the existence of positive solutions. However, they did not further provide characters of positive solutions, such as symmetry. It is now natural to consider the existence of symmetric positive solutions. To our knowledge, no paper has considered the existence of symmetric positive solutions and nonlinearity with singularity at and (or) for fourth-order equation with nonlocal boundary condition. Hence we improve and generalize the results of [4–7] to some degree, and so it is interesting and important to study the existence of symmetric positive solutions for problem (1.1). The arguments are based upon a specially constructed cone and the fixed point theory for cones.
Let be a cone in a Banach space and let and , where .
Our main tool of this paper is the following fixed point theorem.
Lemma 1.1 (see ). Let be a positive cone in real Banach space , , and let be a completely continuous operator and such that (i) for ;(ii)there exists such that for any and Then has a fixed point in .
Remark 1.2. If (i) and (ii) are satisfied for and , respectively, then Lemma 1.1 is still true.
The following concept will also be utilized.
Definition 1.3. If is a continuous function and for , then one says that is symmetric on .
2. Preliminaries and Lemmas
In this section, we present some lemmas that are important to prove our main results.
Lemma 2.1. Suppose that , , , then BVP has a unique solution where
Proof. Integrating both sides of (2.1) on , we have Again integrating (2.5) from 0 to , we get In particular, By (2.2) we get By and (2.6), we can get So, By (2.6), (2.8), and (2.10), we have This completes the proof of Lemma 2.1.
It is easy to verify the following properties of and .
Lemma 2.2. If , and , then (1), , , ;(2), , ;(3), where = , = .
So we may denote Green's function of boundary value problem
by and , respectively. By Lemma 2.1, we know that and can be written by
Obviously, and have the same properties with in Lemma 2.2.
Remark 2.3. For notational convenience, we introduce the following constants: Obviously, , .
Now we define an integral operator by
Then, we have
Lemma 2.4. The fourth-order nonlocal boundary value problem (1.1) has a positive solution if and only if the following integral-differential boundary value problem has a positive solution.
Proof. In fact, if is a positive solution of (BVP) (1.1), let , then . This implies that is a solution of (2.17). Conversely, if is a positive solution of (2.17). Let , by (2.16), . Thus is a positive solution of (BVP) (1.1). This completes the proof of Lemma 2.4.
So we will concentrate our study on (2.17). Let and
is the supremum norm on . It is easy to see that is a cone in and . Now we define an operator by
Clearly is a solution of the BVP (2.17) if and only if is a fixed point of the operator .
In the rest of the paper, we make the following assumptions:(), , , , ;(), , ;(), , and for any , where .
Remark 2.5. If () holds, then for all , we have
Lemma 2.6. Assume that conditions , , and hold. Then is a completely continuous operator.
Proof. Firstly, for any , we will show
At the same time, this implies that is well defined.
In fact, by , for any , there exists a natural number such that
For any , let = . It follows from the concavity of on that So we obtain Consequently, from (2.25) for any , we have and where and is defined in Remark 2.3. Let By , we have that is, (2.22) holds. This also implies that is uniformly bounded for any bounded set from (2.28).
Next we prove that is equicontinuous on . In fact, by for any , there exists a natural number such that
Let where . Since is uniformly continuous on , for the above and fixed , there exists such that for and . Consequently, when and , we have This implies that is equicontinuous. Then by the Arzela-Ascoli theorem is compact.
Finally, we show that is continuous. Assume and . Then and . For any , by , there exists a natural number such that
On the other hand, by (2.25), for any , we have
where and is defined by Remark 2.3.
Since is uniformly continuous in , we have that
holds uniformly on . Then the Lebesgue dominated convergence theorem yields that Thus for above , there exists a natural number; , for , we have It follows from (2.33) and (2.37) that when , This implies that is continuous. Thus is completely continuous. This completes the proof of Lemma 2.6.
Lemma 2.7. One has .
Proof. For , we have
On the other hand, by Lemma 2.2 we have
which implies .
In addition, for , by Remark 2.5, , and we have
that is, , . Therefore, is symmetric on . Obviously, is concave on . Consequently, . This completes the proof of Lemma 2.7.
3. The Main Results
Our main results of this paper are as follows. For notational convenience, we let
Theorem 3.1. Suppose that conditions , , and are satisfied. Further assume that the following condition holds:()Then the BVP (1.1) has at least one symmetric positive solution for any where and are defined by Remark 2.3.
Proof. Let satisfy (3.2) and at be chosen such that
Next, by there exists such that
Take . Notice that
It follows from (3.4) and (3.5) that, for any ,
Thus, , for all .
On the other hand, for the above , by , there exists such that
Let and , . Then and
In the following we show . Otherwise, there exists and such that Let and notice that for any ,
Consequently for any , we have This implies that , which is a contradiction. It follows from Lemma 1.1 that has a fixed point with Thus is a symmetric positive solution of the BVP (2.17). Consequently, by Lemma 2.4, one can obtain that BVP (1.1) has a symmetric positive solution. This completes the proof of Theorem 3.1.
Remark 3.2. Since , we easily obtain , . Thus ; so when , Theorem 3.1 always holds.
Remark 3.3. From Theorem 3.1, we can see that need not be superlinear or sublinear. In fact, Theorem 3.1 still holds, if one of the following conditions is satisfied.(i)If , , then for each .(ii)If , , then for each .(iii)If , , then for each .
Theorem 3.4. Suppose that conditions , , and are satisfied. Further assume that the following condition holds.Then the BVP (1.1) has at least one symmetric positive solution for any where and are defined by Remark 2.3.
Proof. Let satisfy (3.10) and let be chosen such that and . By , there exists such that
Then by (2.22). Take , then
Notice that implies that
So for any , let and then for any , clearly
In addition, for any , let then . Thus, for any , we have
Therefore for any .
Next, let satisfy (3.10). Choose such that Then from , there exists such that
Let and , , then and
Now we prove Otherwise, there exists and such that Let and apply that ; then for any , we have