- Research Article
- Open Access
© The Author(s) Xingqiu Zhang and Yujun Cui. 2010
Received: 7 April 2010
Accepted: 12 August 2010
Published: 19 August 2010
By employing upper and lower solutions method together with maximal principle, we establish a necessary and sufficient condition for the existence of pseudo- as well as positive solutions for fourth-order singular -Laplacian differential equations with integral boundary conditions. Our nonlinearity may be singular at , , and . The dual results for the other integral boundary condition are also given.
Condition is used to discuss the existence and uniqueness of smooth positive solutions in .
Conversely, (1.4) implies (1.2).
Conversely, (1.5) implies (1.3).
Boundary value problems with integral boundary conditions arise in variety of different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, and plasma physics can be reduced to nonlocal problems with integral boundary conditions. They include two point, three point, and nonlocal boundary value problems (see [2–5]) as special cases and have attracted much attention of many researchers, such as Gallardo, Karakostas, Tsamatos, Lomtatidze, Malaguti, Yang, Zhang, and Feng (see [6–13], e.g.). For more information about the general theory of integral equations and their relation to boundary value problems, the reader is referred to the book by Corduneanu  and Agarwal and O'Regan .
To seek necessary and sufficient conditions for the existence of solutions to the ordinary differential equations is important and interesting, but difficult. Professors Wei [16, 17], Du and Zhao , Graef and Kong , Zhang and Liu , and others have done much excellent work under some suitable conditions in this direction. To the author's knowledge, there are no necessary and sufficient conditions available in the literature for the existence of solutions for integral boundary value problem (1.1). Motivated by above papers, the purpose of this paper is to fill this gap. It is worth pointing out that the nonlinearity permits singularity not only at but also at . By singularity, we mean that the function is allowed to be unbounded at the points and .
2. Preliminaries and Several Lemmas
To prove the main results, we need the following maximum principle.
Lemma 2.3 (Maximum principle).
then from (2.29) and (2.30), we have (2.25).
By (2.33), it is easy to verify that is continuous and is a bounded set. Moreover, by the continuity of , we can show that is a compact operator and is a relatively compact set. So, is a completely continuous operator. In addition, is a solution of (2.35) if and only if is a fixed point of operator . Using the Shauder's fixed point theorem, we assert that has at least one fixed point , by , we can get
3. The Main Results
The proof is divided into two parts, necessity and suffeciency.
which is the desired inequality.
under the following condition:
The proof is similar to that of Theorem 3.1; we omit the details.
The proof is divided into two parts, necessity and suffeciency.
which is the desired inequality.
4. Dual Results
By analogous methods, we have the following results.
The project is supported financially by a Project of Shandong Province Higher Educational Science and Technology Program (no. J10LA53) and the National Natural Science Foundation of China (no. 10971179).
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