Research Article | Open Access
Dongming Yan, Qiang Zhang, Zhigang Pan, "Existence of Positive Solutions for Neumann Boundary Value Problem with a Variable Coefficient", International Journal of Differential Equations, vol. 2011, Article ID 376753, 11 pages, 2011. https://doi.org/10.1155/2011/376753
Existence of Positive Solutions for Neumann Boundary Value Problem with a Variable Coefficient
We consider the existence of positive solutions for the Neumann boundary value problem , where and is continuous. The theorem obtained is very general and complements previous known results.
The existence of solutions of Neumann boundary value problem of second-order ordinary differential equations has been studied by many authors; see Sun et al. , Cabada and Pouso , Cabada et al. , Canada et al. , Chu et al. , Jiang, and Liu , Yazidi , Sun and Li  and the references therein.
Recently, Chu et al.  have studied the existence of positive solution to the Neumann boundary value problem where is a constant, and nonlinearity may be singular at . Their approach was based upon the nonlinear alternative principle of Leray-Schauder and Green's function, , of the associated linear problem Notice that Green's function can be explicitly expressed by
In this paper, we will consider the more general problem where , and is continuous.
Of course, the natural question is what would happen when the constant in (1.1) is replaced with a function ? Obviously, Green's function of the associated linear problem cannot be explicitly expressed by elementary functions! The primary contribution of this paper is to construct Green's function associated with the Neumann boundary value problem with a variable coefficient (1.5) and study the properties of the Green's function. We apply the Krasnoselskii and Guo fixed point theorem as an application. This application was first made by Erbe and Wang  to ordinary differential equations. Since that time, there has been a tremendous amount of work to study the existence of positive solutions to BVPs for ordinary differential equations. Once we obtain Theorem 2.2, many of those applications would work here as well.
The rest of the paper is organized as follows: Section 2 is devoted to constructing Green's function and proves some preliminary results. In Section 3, we state and prove our main results. In Section 4, an example illustrates the applicability of the main existence result.
2. Preliminaries and Lemmas
Let us fix some notation to be used. Given , we write if for a.e. , and it is positive in a set of positive measure. Let us denote by and the essential supremum and infimum of a given function if they exist. To study the boundary value problem (1.4), we need restriction on .
To rewrite (1.4) to an equivalent integral equation, we need to construct Green's function of the corresponding linear problem. To do this, we need the following.
Lemma 2.1. Let hold. Suppose and be the solution of the linear problems respectively. Then(i) on , and on ;(ii)(ii) on , and on .
Proof. We will give a proof for (i) only. The proof of (ii) follows in a similar manner.
It is easy to see that the problem has the unique solution and . From , we know that On the other hand, for all , we have By using comparison theorem (see ), we obtain Therefore, we have from (2.3) and (2.5) that Thus From the fact and (2.7), we obtain on .
Theorem 2.2. Let hold. Then for any , the problem is equivalent to the integral equation
Proof. First we show that the unique solution of (2.9) can be represented by (2.10).
In fact, we know that the equation has known two linear independent solutions and since .
Now by the method of variation of constants, we can obtain that the unique solution of the problem (2.9) can be represented by where is as (2.8).
Next we check that the function defined by (2.10) is a solution of (2.9).
From (2.10), we know that So that Finally, it is easy to see that
From Lemma 2.1, we know that Let . Then and .
In order to prove the main result of this paper, we need the following fixed-point theorem of cone expansion-compression type due to Krasnoselskii's (see ).
Theorem 2.3. Let be a Banach space, and is a cone in . Assume that and are open subsets of with and . Let be a completely continuous operator. In addition, suppose that either (i) and or(ii) and holds.Then has a fixed point in .
3. Main Results
In this section, we state and prove the main results of this paper.
Let us define the function which is just the unique solution of the linear problem (2.9) with . For our constructions, let , with norm . Define a cone , by
Theorem 3.1. Let hold. Suppose that there exist a constant such that ()there exist continuous, nonnegative functions , and , such that is nonincreasing, and is nondecreasing in ;(), here ;()there exists a continuous function such that (). Then problem (1.4) has at least one positive solution with .
Proof of Theorem 3.1. Let Choose such that , where is a constant. Let . Fix . Consider the boundary value problem
We note that is a solution of () if and only if
Define an integral operator by
Then, () is equivalent to the fixed point equation . We seek a fixed point of in the cone .
Set . If , then Notice from (2.16), , and that, for on . Also, for , we have so that And next, if , we have by (3.10), As a consequence, . In addition, standard arguments show that is completely continuous.
If with , then and we have by , and Thus, . Hence,
If with , then and we have by and Thus, . Hence,
Applying (ii) of Theorem 2.3 to (3.14) and (3.17) yields that has a fixed point , and . As such, is a solution of (), and
Next we prove the fact for some constant and for all . To this end, integrating the first equation of () from 0 to 1, we obtain Then
The fact and (3.19) show that is a bounded and equicontinuous family on . Now the Arzela-Ascoli Theorem guarantees that has a subsequence, , converging uniformly on to a function . From the fact and (3.18), satisfies for all . Moreover, satisfies the integral equation Let , and we arrive at where the uniform continuity of on is used. Therefore, is a positive solution of boundary value problem (1.4). Finally, it is not difficult to show that, .
By Theorem 3.1, we have the following Corollary.
Corollary 3.3. Let hold. Assume that there exist continuous functions and such that , for all and .Then problem (1.4) has at least one positive solution if one of the following two conditions holds:(i);(ii), where .
Consider second-order Neumann boundary value problem Here . Obviously, is satisfied. Let , then we can check that , and are satisfied. In addition, for , we have On the other hand, by Lemma 2.1, we have By (4.3), we have Hence, . So that is satisfied. According to Theorem 3.1, the boundary value problem (4.1) has at least one positive solution with .
For boundary value problem (4.1), however, we cannot obtain the above conclusion by Theorem 3.1 of paper  since is not a constant. These imply that Theorem 3.1 in this paper complement and improve those obtained in .
- Y. Sun, Y. J. Cho, and D. O'Regan, “Positive solutions for singular second order Neumann boundary value problems via a cone fixed point theorem,” Applied Mathematics and Computation, vol. 210, no. 1, pp. 80–86, 2009.
- A. Cabada and R. R. L. Pouso, “Existence result for the problem with periodic and Neumann boundary conditions,” vol. 30, no. 3, pp. 1733–1742.
- A. Cabada, P. Habets, and S. Lois, “Monotone method for the Neumann problem with lower and upper solutions in the reverse order,” Applied Mathematics and Computation, vol. 117, no. 1, pp. 1–14, 2001.
- A. Canada, J. A. Montero, and S. Villegas, “Liapunov-type inequalities and Neumann boundary value problems at resonance,” Mathematical Inequalities and Applications, vol. 8, no. 3, pp. 459–475, 2005.
- J. F. Chu, Y. G. Sun, and H. Chen, “Positive solutions of Neumann problems with singularities,” Journal of Mathematical Analysis and Applications, vol. 337, no. 2, pp. 1267–1272, 2008.
- D. Jiang and H. Liu, “Existence of positive solutions to second order Neumann boundary value problems,” Journal of Mathematical Research and Exposition, vol. 20, no. 3, pp. 360–364, 2000.
- N. Yazidi, “Monotone method for singular Neumann problem,” Nonlinear Analysis, vol. 49, no. 5, pp. 589–602, 2002.
- J.-P. Sun and W.-T. Li, “Multiple positive solutions to second-order Neumann boundary value problems,” Applied Mathematics and Computation, vol. 146, no. 1, pp. 187–194, 2003.
- L. H. Erbe and H. Wang, “On the existence of positive solutions of ordinary differential equations,” Proceedings of the American Mathematical Society, vol. 120, no. 3, pp. 743–748, 1994.
- P. B. Bailey, L. F. Shampine, and P. E. Waltman, Nonlinear Two Point Boundary Value Problems, Academic Press, New York, NY, USA, 1968.
- D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, Mass, USA, 1988.
Copyright © 2011 Dongming Yan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.