About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 782363, 7 pages
http://dx.doi.org/10.1155/2013/782363
Research Article

Solvability of a Fourth-Order Boundary Value Problem with Integral Boundary Conditions

Department of Mathematics, Beihua University, Jilin City 132013, China

Received 21 December 2012; Accepted 13 February 2013

Academic Editor: Meng Fan

Copyright © 2013 Hui Li 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.

Abstract

We investigate the existence of solutions and positive solutions for a nonlinear fourth-order differential equation with integral boundary conditions of the form , , , , , where , . By using a fixed point theorem due to D. O'Regan, the existence of solutions and positive solutions for the previous boundary value problems is obtained. Meanwhile, as applications, some examples are given to illustrate our results.

1. Introduction

It is well known that fourth-order boundary value problems (BVPs) arise in a variety of different areas of the flexibility mechanics and engineering physics and thus have been extensively studied; for instance, see [129] and references therein. Boundary value problems with integral boundary conditions appear in heat conduction, thermoelasticity, chemical engineering underground water flow, and plasma physics; see [12, 14, 21, 24, 26, 29] and references therein.

Motivated by the previous works and [30], in this paper, we consider fully nonlinear fourth-order differential equation subject to the integral boundary conditions as well as its simplified form with the integral boundary conditions where and are continuous functions.

We notice that if in problems (1), (2) and (3), (4), then the models are known as the one endpoint simply supported and the other one sliding clamped beam. The study of this class of problems was considered by some authors via various methods; we refer the reader to the papers [2, 5, 8, 11, 22].

The aim of this paper is to establish the existence results of solutions and positive solutions for problems (1), (2) and (3), (4), respectively. By positive solution, we mean a solution such that for . Our main tool is the fixed point theorem due to D. O'Regan [31].

2. Preliminary

In this section, we present some lemmas which are needed for our main results.

Let denote the Banach space of real-valued continuous functions on with the norm . is the Banach space of times continuously differentiable functions defined on , with the norm .

Throughout this paper, we always assume that (or ) and (or ) are continuous.

We consider a priori bound of solutions of the following one-parameter family of boundary value problem: where . Simple computations lead to the following lemma.

Lemma 1. BVP (5), (6) with has only the trivial solution, and the corresponding Green function exists and is given by

Now, define a linear operator by

Then, we can easily show that is a Fredholm operator with index zero, and its inverse is given by where .

Define Nemytskii operators induced by as and induced by as

Also, define an operator as

Simple computations yield the following lemma.

Lemma 2. BVP (5), (6) is equivalent to the abstract equation in ; that is, is a solution of BVP (5), (6) if and only if is a solution of the integral equation where .

Let us denote the operators , as

Then, can be written as

Now, we can easily give some properties of the Green function and by direct computation.

Lemma 3. Let be as in Lemma 1 and . Then,(1), ; (2), , , , ; (3), , , , .

Lemma 4. Suppose that(i)for each fixed , is nondecreasing in and ;(ii)there exists a constant such that for , , (iii)there exist and nondecreasing continuous function such that for , and for all .
Then, any solution of BVP (5), (6) satisfies where .

Proof. Let us first show that
Note that if in (5) and (6), then BVP (5), (6) has only the trivial solution, and thus (23) holds. Hence, we may assume that . Suppose now that (23) is not true. Then, there exists such that . Let
Then, , and from , we have
It is easy to see that . In fact, if , then . From (6) and (iii), it follows that for some , which is a contradiction, and thus . Furthermore, by definition of and (6), we have . Hence, from assumptions (i) and (ii) and (25), we have
We may assume that ; then, . Thus, by the continuity of on , there exists such that for . Since , it follows that for ; namely, is decreasing on , which contradicts the fact that attains its positive maximum value at . In summary, inequality (23) is true, which implies from that
This completes the proof of the lemma.

Remark 5. In Lemma 4, if condition (i) is replaced by
(i′), there exists a constant such that whenever and all , then, the conclusion of Lemma 4 remains true.

The following fixed point result due to D. O'Regan plays a crucial role.

Lemma 6 (see [31]). Let be an open set in a closed, convex set of a Banach space . Assume that , is bounded, and is given by , where is continuous and completely continuous and is a nonlinear contraction. Then, either(A1) has a fixed point in , or (A2) there is a point and with .

3. Main Results

Firstly in this section, we state and prove our existence results of solutions for BVP (1), (2).

Theorem 7. Suppose that(i)for each fixed , is nondecreasing in and ;(ii)there exists a constant such that for , , (iii)there exist and nondecreasing continuous function such that for , and for all ;(iv) satisfies the Nagumo condition; that is, there exists a positive-valued continuous function on with such that for all , where
Then, BVP (1), (2) has at least one solution.

Proof. Let be a possible solution of BVP (5), (6). We now show that where and .
Suppose that (34) is not true. Then, there exists such that . Since , then there exists , () such that
Therefore, is positive or negative on by the continuity of the . Hence, from assumption (iv), the definition of , and Lemma 4, we can get the following contradiction:
Therefore, inequality (34) holds.
Let . It follows easily from the properties of the Green function and the continuity of that the operator is completely continuous.
We now show that is a nonlinear contraction. In fact, from assumption (iii), we have
Consequently, from Lemma 3, we have
Similarly, for all ,
Hence,
Since all possible solutions of BVP (5), (6) satisfy , it follows that there is no and such that . We conclude that (A2) of Lemma 6 does not hold. Consequently, has a fixed point, which is a solution of BVP (1), (2). This completes the proof of the theorem.

Remark 8. In Theorem 7, if condition (i) is replaced by(i′) there exists a constant such that whenever and all , then the conclusion of Theorem 7 remains true.

Remark 9. In Theorem 7, if and , then all the solutions of BVP (1), (2) are monotone and positive. This is clear because by Lemma 3 we have , and .
Next, we consider the existence of solutions and positive solutions for BVP (3), (4).

Theorem 10. Suppose that(i)there exists being continuous and nondecreasing such that (ii)there exists such that (iii)there exists such that where .
Then, BVP (3), (4) has at least one solution.

Proof. It is easy to see that is a solution of BVP (3), (4) if and only if is a solution of the integral equation (14) with . Moreover, is completely continuous, and is a nonlinear contraction.
It follows from (15), (i), and Lemma 3 that ,
Also, (16), (ii), and Lemma 3 yield
From (14), (45), and (46), we have that all possible solutions of satisfy
Let . Then, is open in , , and is bounded. Suppose that and satisfy . Then, , and (i) and (47) lead to that is, which contradicts (iii). Hence, (A2) of Lemma 6 does not hold, and consequently has a fixed point which is a solution of BVP (3), (4). This completes the proof of the theorem.

Remark 11. In Theorem 10, if and , then all the solutions of BVP (3), (4) are monotone and positive.

Now, we consider BVP (3), (4) with linear boundary conditions as where . Define

Then,

Theorem 12. Suppose that(i);(ii)there exists being continuous and nondecreasing such that
Then, the nonlinear fourth-order differential equation (3) with boundary conditions has at least one solution.

Proof. Notice that the existence of solutions of BVP (3), (54) is equivalent to the existence of fixed points of operator equation
As a linear operator on , from (52) and (i), we get , which implies that is invertible and its inverse is given by
Hence, we see from (55) that is a solution of BVP (3), (54) if and only if is a fixed point of the completely continuous operator .
Let us show that there exists such that any solution of operator equation () satisfies . In fact, any solution of () satisfies and, hence, by (ii) and Lemma 3, we have
The condition implies that there exists such that for all ; that is,
Comparing (58) and (59), we see that .
Now, let . By the Leray-Schauder continuation theorem has a fixed point in , which is a solution of BVP (3), (54). This completes the proof of the theorem.

Remark 13. In Theorem 12, if and , then all the solutions of BVP (3), (54) are monotone and positive since all the solutions of BVP (3), (54) satisfy and thus

Finally, we give some examples to illustrate our results.

Example 14. Consider the fourth-order boundary value problem
Let
It is easy to check that all the assumptions in Theorem 7 are satisfied. Hence, BVP (62), (63) has at least one solution.

Example 15. Consider the fourth-order boundary value problem where .
Let
It is easy to check that all the assumptions in Theorem 10 and Remark 11 are satisfied. Hence, BVP (65), (66) has at least one monotone positive solution.

Example 16. Consider the fourth-order boundary value problem where .
Let
It is easy to check that all the assumptions in Theorem 12 and Remark 13 are satisfied. Hence, BVP (68), (69) has at least one monotone positive solution.

Acknowledgment

This work was supported by NSFC (11126339).

References

  1. R. P. Agarwal and Y. M. Chow, “Iterative methods for a fourth order boundary value problem,” Journal of Computational and Applied Mathematics, vol. 10, no. 2, pp. 203–217, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. Z. Bai, “The upper and lower solution method for some fourth-order boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 67, no. 6, pp. 1704–1709, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. A. Cabada and F. M. Minhós, “Fully nonlinear fourth-order equations with functional boundary conditions,” Journal of Mathematical Analysis and Applications, vol. 340, no. 1, pp. 239–251, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. M. A. Del Pino and R. F. Manásevich, “Existence for a fourth-order boundary value problem under a two-parameter nonresonance condition,” Proceedings of the American Mathematical Society, vol. 112, no. 1, pp. 81–86, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. J. Du and M. Cui, “Constructive proof of existence for a class of fourth-order nonlinear BVPs,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 903–911, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. J. Ehme, P. W. Eloe, and J. Henderson, “Upper and lower solution methods for fully nonlinear boundary value problems,” Journal of Differential Equations, vol. 180, no. 1, pp. 51–64, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. D. Franco, D. O'Regan, and J. Perán, “Fourth-order problems with nonlinear boundary conditions,” Journal of Computational and Applied Mathematics, vol. 174, no. 2, pp. 315–327, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. H. Feng, D. Ji, and W. Ge, “Existence and uniqueness of solutions for a fourth-order boundary value problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 10, pp. 3561–3566, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. R. Graef and L. Kong, “A necessary and sufficient condition for existence of positive solutions of nonlinear boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications, vol. 66, no. 11, pp. 2389–2412, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. R. Grossinho and St. A. Tersian, “The dual variational principle and equilibria for a beam resting on a discontinuous nonlinear elastic foundation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 41, no. 3-4, pp. 417–431, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. C. P. Gupta, “Existence and uniqueness theorems for the bending of an elastic beam equation,” Applicable Analysis, vol. 26, no. 4, pp. 289–304, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. T. Jankowski, “Positive solutions for fourth-order differential equations with deviating arguments and integral boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 73, no. 5, pp. 1289–1299, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. D. Jiang, W. Gao, and A. Wan, “A monotone method for constructing extremal solutions to fourth-order periodic boundary value problems,” Applied Mathematics and Computation, vol. 132, no. 2-3, pp. 411–421, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. P. Kang, Z. Wei, and J. Xu, “Positive solutions to fourth-order singular boundary value problems with integral boundary conditions in abstract spaces,” Applied Mathematics and Computation, vol. 206, no. 1, pp. 245–256, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Y. Li, “Positive solutions of fourth-order boundary value problems with two parameters,” Journal of Mathematical Analysis and Applications, vol. 281, no. 2, pp. 477–484, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. X.-L. Liu and W.-T. Li, “Existence and multiplicity of solutions for fourth-order boundary value problems with three parameters,” Mathematical and Computer Modelling, vol. 46, no. 3-4, pp. 525–534, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. Liu, “Multiple positive solutions of nonlinear singular boundary value problem for fourth-order equations,” Applied Mathematics Letters, vol. 17, no. 7, pp. 747–757, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Lu and S. Jin, “Existence of periodic solutions for a fourth-order P-Laplacian equation with a deviating argument,” Journal of Computational and Applied Mathematics, vol. 230, no. 2, pp. 513–520, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  19. T. F. Ma, “Positive solutions for a beam equation on a nonlinear elastic foundation,” Mathematical and Computer Modelling, vol. 39, no. 11-12, pp. 1195–1201, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. R. Ma and J. Xu, “Bifurcation from interval and positive solutions of a nonlinear fourth-order boundary value problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 72, no. 1, pp. 113–122, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. R. Ma and T. Chen, “Existence of positive solutions of fourth-order problems with integral boundary conditions,” Boundary Value Problems, vol. 2011, Article ID 297578, 17 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. F. Minhós, T. Gyulov, and A. I. Santos, “Lower and upper solutions for a fully nonlinear beam equation,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 1-2, pp. 281–292, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. P. K. Palamides, “Uniqueness of monotone positive solutions for singular boundary value problems,” Communications on Applied Nonlinear Analysis, vol. 9, no. 3, pp. 79–89, 2002. View at Zentralblatt MATH · View at MathSciNet
  24. P. Pietramala, “A note on a beam equation with nonlinear boundary conditions,” Boundary Value Problems, vol. 2011, Article ID 376782, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. M. Pei and S. K. Chang, “Monotone iterative technique and symmetric positive solutions for a fourth-order boundary value problem,” Mathematical and Computer Modelling, vol. 51, no. 9-10, pp. 1260–1267, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. J. R. L. Webb, G. Infante, and D. Franco, “Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions,” Proceedings of the Royal Society of Edinburgh A, vol. 138, no. 2, pp. 427–446, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. Z. Wei, “A class of fourth order singular boundary value problems,” Applied Mathematics and Computation, vol. 153, no. 3, pp. 865–884, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. Q. Yao, “Positive solutions for eigenvalue problems of fourth-order elastic beam equations,” Applied Mathematics Letters, vol. 17, no. 2, pp. 237–243, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. X. Zhang and W. Ge, “Positive solutions for a class of boundary-value problems with integral boundary conditions,” Computers & Mathematics with Applications, vol. 58, no. 2, pp. 203–215, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. A. Boucherif, S. M. Bouguima, N. Al-Malki, and Z. Benbouziane, “Third order differential equations with integral boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 12, pp. e1736–e1743, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. D. O'Regan, “Fixed-point theory for the sum of two operators,” Applied Mathematics Letters, vol. 9, no. 1, pp. 1–8, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet