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 fourthorder 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.
1. Introduction
In this paper, we consider the existence of positive solutions for the following nonlinear fourthorder singular Laplacian differential equations with integral boundary conditions:
where , , , , , , , , and is nonnegative. Let , . Throughout this paper, we always assume that , and nonlinear term satisfies the following hypothesis:
(H) is continuous, nondecreasing on and nonincreasing on for each fixed , and there exists a real number such that, for any ,
there exists a function , and is integrable on such that
Remark 1.1.
Condition is used to discuss the existence and uniqueness of smooth positive solutions in [1].
(i)Inequality (1.2) implies that
Conversely, (1.4) implies (1.2).
(ii)Inequality (1.3) implies that
Conversely, (1.5) implies (1.3).
Remark 1.2.
Typical functions that satisfy condition are those taking the form = , where , , ; .
Remark 1.3.
It follows from (1.2) and (1.3) that
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 [14] and Agarwal and O'Regan [15].
Recently, Zhang et al. [13] studied the existence and nonexistence of symmetric positive solutions for the following nonlinear fourthorder boundary value problems:
where , , , , is nonnegative, symmetric on the interval , is continuous, and are nonnegative, symmetric on .
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 [18], Graef and Kong [19], Zhang and Liu [20], 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
A function and is called a (positive) solution of BVP (1.1) if it satisfies (1.1) ( for ). A (positive) solution of (1.1) is called a psuedo (positive) solution if , for . Denote that
Definition 2.1.
A function is called a lower solution of BVP (1.1) if satisfies
Definition 2.2.
A function is called an upper solution of BVP (1.1) if satisfies
Suppose that , and
To prove the main results, we need the following maximum principle.
Lemma 2.3 (Maximum principle).
If , such that , , then , ,
Proof.
Set
then , , , and
Let
then
By integration of (2.12), we have
Integrating again, we get
Let in (2.15), we obtain that
Substituting (2.13) and (2.16) into (2.15), we obtain that
where
Notice that
therefore,
Substituting (2.20) into (2.17), we have
where
Obviously, , , . From (2.21), it is easily seen that for By (2.11), we know that that is, Thus, we have proved that , . Similarly, the solution of (2.5) and (2.7) can be expressed by
where
By (2.23), we can get that ,
Lemma 2.4.
Suppose that holds. Let be a positive solution of BVP (1.1). Then there exist two constants such that
Proof.
Assume that is a positive solution of BVP (1.1). Then can be stated as
where
It is easy to see that
By (2.26), for , we have that
From (2.26) and (2.27), we get that
Setting
then from (2.29) and (2.30), we have (2.25).
Lemma 2.5.
Suppose that holds. And assume that there exist lower and upper solutions of BVP (1.1), respectively, and , such that , for . Then BVP (1.1) has at least one positive solution such that , . If, in addition, there exists such that
then the solution of BVP (1.1) is a pseudo positive solution.
Proof.
For each , for all , , we defined an auxiliary function
By condition , we have that is continuous.
Let be sequences satisfying , and as and let , , be sequences satisfying
For each , consider the following nonsingular problem:
For convenience, we define linear operators as follows:
By the proof of Lemma 2.3, is a solution of problem (2.35) if and only if it is the fixed point of the following operator equation:
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
We claim that
From this it follows that
Indeed, suppose by contradiction that on . By the definition of , we have
Therefore,
On the other hand, since is an upper solution of (1.1), we also have
Then setting
By (2.41) and (2.42), we obtain that
By Lemma 2.3, we can conclude that
Hence,
Set
Then
By Lemma 2.3, we can conclude that
which contradicts the assumption that Therefore, is impossible.
Similarly, we can show that So, we have shown that (2.38) holds.
Using the method of [21] and Theorem .2 in [22], we can obtain that there is a positive solution of (1.1) such that and a subsequence of converging to on any compact subintervals of .
In addition, if (2.32) holds, then . Hence, is absolutely integrable on . This implies that is a pseudo positive solution of (1.1).
3. The Main Results
Theorem 3.1.
Suppose that holds, then a necessary and sufficient condition for BVP (1.1) to have a pseudo positive solution is that the following integral condition holds:
Proof.
The proof is divided into two parts, necessity and suffeciency.
Necessity.
Suppose that is a pseudo positive solution of (1.1). Then both and exist. By Lemma 2.4, there exist two constants such that
Without loss of generality, we may assume that . This together with condition implies that
On the other hand, since is a pseudo positive solution of (1.1), we have
Otherwise, let . By the proof of Lemma 2.3, we have that , , that is, which contradicts that is a pseudo positive solution. Therefore, there exists a positive such that . Obviously, . By (1.6) we have
Consequently, , which implies that
It follows from (3.3) and (3.6) that
which is the desired inequality.
Sufficiency.
First, we prove the existence of a pair of upper and lower solutions. Since is integrable on , we have
Otherwise, if , then there exists a real number such that when , which contradicts the condition that is integrable on . In view of condition and (3.8), we obtain that
where .
Suppose that (3.1) holds. Firstly, we define the linear operators and as follows:
where is given by (2.27). Let
It is easy to know from (3.11) and (3.12) that By Lemma 2.4, we know that there exists a positive number such that
Take sufficiently small, then by (3.10), we get that , that is,
Let
Thus, from (3.14) and (3.16), we have
Considering , it follows from (3.15), (3.17), and condition that
From (3.13) and (3.16), it follows that
Thus, we have shown that and are lower and upper solutions of BVP (1.1), respectively.
Additionally, when , , by (3.17) and condition , we have
From (3.1), we have So, it follows from Lemma 2.5 that BVP (1.1) admits a pseudo positive solution such that
Remark 3.2.
Lin et al. [23, 24] considered the existence and uniqueness of solutions for some fourthorder and conjugate boundary value problems when , where
under the following condition:
for and , there exists such that
Lei et al. [25] and Liu and Yu [26] investigated the existence and uniqueness of positive solutions to singular boundary value problems under the following condition:
for all , where and is nondecreasing on and nonincreasing on .
Obviously, (3.21)(3.22) imply condition and condition implies condition . So, condition is weaker than conditions and . Thus, functions considered in this paper are wider than those in [23–26].
In the following, when admits the form , that is, nonlinear term is not mixed monotone on , but monotone with respect , BVP (1.1) becomes
If satisfies one of the following:
is continuous, nondecreasing on , for each fixed , there exists a function , and is integrable on such that
Theorem 3.3.
Suppose that holds, then a necessary and sufficient condition for BVP (3.23) to have a pseudo positive solution is that the following integral condition holds
Proof.
The proof is similar to that of Theorem 3.1; we omit the details.
Theorem 3.4.
Suppose that holds, then a necessary and sufficient condition for problem (3.23) to have a positive solution is that the following integral condition holds
Proof.
The proof is divided into two parts, necessity and suffeciency.
Necessity.
Assume that is a positive solution of BVP (3.23). By Lemma 2.4, there exist two constants and , , such that
Let be a constant such that . By condition , we have
By virtue of (3.28), we obtain that
By boundary value condition, we know that there exists a such that
For by integration of (3.29), we get
Integrating (3.31), we have
Exchanging the order of integration, we obtain that
Similarly, by integration of (3.29), we get
Equations (3.33) and (3.34) imply that
Since is a positive solution of BVP (1.1), there exists a positive such that . Obviously, . On the other hand, choose , then . By condition , we have
Consequently, , which implies that
It follows from (3.35) and (3.37) that
which is the desired inequality.
Sufficiency.
Suppose that (3.26) holds. Let
It is easy to know, from (3.11) and (3.26), that
Thus, (3.12), (3.39), and (3.40) imply that By Lemma 2.4, we know that there exists a positive number such that
Take sufficiently small, then by (3.10), we get that , that is,
Let
Thus, from (3.41) and (3.43), we have
Notice that , it follows from (3.42)–(3.44) and condition that
From (3.39) and (3.43), it follows that
Thus, we have shown that and are lower and upper solutions of BVP (1.1), respectively.
From the first conclusion of Lemma 2.5, we conclude that problem (1.1) has at least one positive solution .
4. Dual Results
Consider the fourthorder singular Laplacian differential equations with integral conditions:
Firstly, we define the linear operator as follows:
where is given by (2.27).
By analogous methods, we have the following results.
Assume that is a positive solution of problem (4.1). Then can be expressed by
Theorem 4.1.
Suppose that holds, then a necessary and sufficient condition for (4.1) to have a pseudo positive solution is that the following integral condition holds:
Theorem 4.2.
Suppose that holds, then a necessary and sufficient condition for problem (4.2) to have a pseudo positive solution is that the following integral condition holds:
Theorem 4.3.
Suppose that holds, then a necessary and sufficient condition for problem (4.2) to have a positive solution is that the following integral condition holds:
Consider the fourthorder singular Laplacian differential equations with integral conditions:
Define the linear operator as follows:
If is a positive solution of problem (4.8). Then can be expressed by
Theorem 4.4.
Suppose that holds, then a necessary and sufficient condition for problem (4.8) to have a pseudo positive solution is that the following integral condition holds:
Theorem 4.5.
Suppose that holds, then a necessary and sufficient condition for problem (4.9) to have a pseudo positive solution is that the following integral condition holds:
Theorem 4.6.
Suppose that holds, then a necessary and sufficient condition for problem (4.9) to have a positive solution is that the following integral condition holds:
Consider the fourthorder singular Laplacian differential equations with integral conditions:
Define the linear operator as follows:
If is a positive solution of problem (4.15). Then can be expressed by
Theorem 4.7.
Suppose that holds, then a necessary and sufficient condition for problem (4.15) to have a pseudo positive solution is that the following integral condition holds:
Theorem 4.8.
Suppose that holds, then a necessary and sufficient condition for problem (4.16) to have a pseudo positive solution is that the following integral condition holds:
Theorem 4.9.
Suppose that holds, then a necessary and sufficient condition for problem (4.16) to have a positive solution is that the following integral condition holds:
Acknowledgments
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).
References

Du, X, Zhao, Z: Existence and uniqueness of smooth positive solutions to a class of singular point boundary value problems. Boundary Value Problems. 2009, (2009)

Graef, JR, Webb, JRL: Third order boundary value problems with nonlocal boundary conditions. Nonlinear Analysis: Theory, Methods & Applications. 71(56), 1542–1551 (2009). PubMed Abstract  Publisher Full Text

Graef, JR, Yang, B: Positive solutions of a third order nonlocal boundary value problem. Discrete and Continuous Dynamical Systems. Series S. 1(1), 89–97 (2008)

Infante, G, Webb, JRL: Nonlinear nonlocal boundaryvalue problems and perturbed Hammerstein integral equations. Proceedings of the Edinburgh Mathematical Society. Series II. 49(3), 637–656 (2006). Publisher Full Text

Webb, JRL, Infante, G: Positive solutions of nonlocal boundary value problems: a unified approach. Journal of the London Mathematical Society. Second Series. 74(3), 673–693 (2006). Publisher Full Text

Gallardo, JM: Secondorder differential operators with integral boundary conditions and generation of analytic semigroups. The Rocky Mountain Journal of Mathematics. 30(4), 1265–1292 (2000). Publisher Full Text

Karakostas, GL, Tsamatos, PCh: Multiple positive solutions of some Fredholm integral equations arisen from nonlocal boundaryvalue problems. Electronic Journal of Differential Equations.(30), 1–17 (2002)

Lomtatidze, A, Malaguti, L: On a nonlocal boundary value problem for second order nonlinear singular differential equations. Georgian Mathematical Journal. 7(1), 133–154 (2000)

Yang, Z: Positive solutions to a system of secondorder nonlocal boundary value problems. Nonlinear Analysis: Theory, Methods & Applications. 62(7), 1251–1265 (2005). PubMed Abstract  Publisher Full Text

Yang, Z: Existence and nonexistence results for positive solutions of an integral boundary value problem. Nonlinear Analysis: Theory, Methods & Applications. 65(8), 1489–1511 (2006). PubMed Abstract  Publisher Full Text

Yang, Z: Positive solutions of a secondorder integral boundary value problem. Journal of Mathematical Analysis and Applications. 321(2), 751–765 (2006). Publisher Full Text

Feng, M, Ji, D, Ge, W: Positive solutions for a class of boundaryvalue problem with integral boundary conditions in Banach spaces. Journal of Computational and Applied Mathematics. 222(2), 351–363 (2008). Publisher Full Text

Zhang, X, Feng, M, Ge, W: Symmetric positive solutions for Laplacian fourthorder differential equations with integral boundary conditions. Journal of Computational and Applied Mathematics. 222(2), 561–573 (2008). Publisher Full Text

Corduneanu, C: Integral Equations and Applications,p. x+366. Cambridge University Press, Cambridge, UK (1991)

Agarwal, RP, O'Regan, D: Infinite Interval Problems for Differential, Difference and Integral Equations, Kluwer Academic Publishers, Dordrecht, The Netherlands (2001)

Wei, Z: Positive solutions for thorder singular sublinear point boundary value problems. Applied Mathematics and Computation. 182(2), 1280–1295 (2006). Publisher Full Text

Wei, Z: A necessary and sufficient condition for thorder singular superlinear point boundary value problems. Journal of Mathematical Analysis and Applications. 327(2), 930–947 (2007). Publisher Full Text

Du, X, Zhao, Z: A necessary and sufficient condition of the existence of positive solutions to singular sublinear threepoint boundary value problems. Applied Mathematics and Computation. 186(1), 404–413 (2007). Publisher Full Text

Graef, JR, Kong, L: Necessary and sufficient conditions for the existence of symmetric positive solutions of singular boundary value problems. Journal of Mathematical Analysis and Applications. 331(2), 1467–1484 (2007). Publisher Full Text

Zhang, X, Liu, L: A necessary and sufficient condition for positive solutions for fourthorder multipoint boundary value problems with Laplacian. Nonlinear Analysis: Theory, Methods & Applications. 68(10), 3127–3137 (2008). PubMed Abstract  Publisher Full Text

Zhang, Y: Positive solutions of singular sublinear EmdenFowler boundary value problems. Journal of Mathematical Analysis and Applications. 185(1), 215–222 (1994). Publisher Full Text

Hartman, P: Ordinary Differential Equations,p. xv+612. Birkhäuser, Boston, Mass, USA (1982)

Lin, X, Jiang, D, Li, X: Existence and uniqueness of solutions for singular fourthorder boundary value problems. Journal of Computational and Applied Mathematics. 196(1), 155–161 (2006). Publisher Full Text

Lin, X, Jiang, D, Li, X: Existence and uniqueness of solutions for singular conjugate boundary value problems. Computers & Mathematics with Applications. 52(34), 375–382 (2006). PubMed Abstract  Publisher Full Text

Lei, P, Lin, X, Jiang, D: Existence and uniqueness of positive solutions for singular nonlinear elliptic boundary value problems. Nonlinear Analysis: Theory, Methods & Applications. 69(9), 2773–2779 (2008). PubMed Abstract  Publisher Full Text

Liu, Y, Yu, H: Existence and uniqueness of positive solution for singular boundary value problem. Computers & Mathematics with Applications. 50(12), 133–143 (2005). PubMed Abstract  Publisher Full Text