Journal of Applied Mathematics

Journal of Applied Mathematics / 2014 / Article

Research Article | Open Access

Volume 2014 |Article ID 895862 | https://doi.org/10.1155/2014/895862

Haitong Li, Minghe Pei, Libo Wang, "Nontrivial Solutions of a Fully Fourth-Order Periodic Boundary Value Problem", Journal of Applied Mathematics, vol. 2014, Article ID 895862, 7 pages, 2014. https://doi.org/10.1155/2014/895862

Nontrivial Solutions of a Fully Fourth-Order Periodic Boundary Value Problem

Academic Editor: Daqing Jiang
Received15 May 2014
Accepted23 Jul 2014
Published06 Aug 2014

Abstract

We investigate the solvability of a fully fourth-order periodic boundary value problem of the form where satisfies Carathéodory conditions. By using the coincidence degree theory, the existence of nontrivial solutions is obtained. Meanwhile, as applications, some examples are given to illustrate our results.

1. Introduction

In this paper, we consider a fully nonlinear fourth-order periodic boundary value problem of the form subject to the boundary conditions where satisfies Carathéodory conditions; that is, (i)for a.e. , the function is continuous;(ii)for every , the function is measurable;(iii)for each , there is a real valued function such that for a.e. and .

It is well known that fourth-order periodic boundary value problems are important research topics which arise in a variety of different areas, such as nonlinear oscillations, fluid mechanical, and nonlinear elastic mechanical phenomena, and thus have been extensively studied; for instance, see [130] and references therein. However, most of the works in the above-mentioned references allow only having , or , , in the right-hand side nonlinear function ; see [211, 13, 1518, 2030]. The works on the fully nonlinear cases of which contains explicitly and every derivative of up to order three have been quite rarely seen; see [1, 12, 14, 19].

The aim of this paper is to establish the existence of solutions and nontrivial solutions for the fully nonlinear fourth-order PBVP (1), (2). Our main tool is the coincidence degree theory. The paper [31] motivated our study.

2. Preliminary

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

At first, we will briefly recall some notations that are needed for our discussion.

Let , be real Banach spaces. A linear mapping will be called a Fredholm mapping of index zero if the following two conditions hold:(i) is a closed subspace of ;(ii).

Let be a Fredholm mapping of index zero; then there exist continuous projectors and such that so that It follows that is invertible. We denote the inverse of that map by . Let be an open bounded subset of such that ; the map will be called -compact on , if and are compact.

Lemma 1 (see [32]). Let be a linear Fredholm mapping of index zero and let be an open bounded set. Let be -compact on and let be -completely continuous such that(i);(ii)for every , and assume that . Then equation has at least one solution in .

Lemma 2 (see [33]). Let be a linear Fredholm mapping of index zero and let be an open bounded set. Let be -compact on and the coincidence degree is well defined. If there exists with such that then .

In the following, we take Banach space with the norm , and . Define a linear map by where and is the usual Sobolev space. It is easy to see that is a Fredholm mapping of index zero. Also define a nonlinear map by

Define two projects and as follows:

Let be Green function for the homogeneous BVP Then can be given by Hence the map is continuous. We note that if satisfying Carathéodory conditions, then is bounded and continuous by Lebesgue's dominated convergence theorem. Furthermore, is -compact on every bounded set .

3. Main Results

For and we put

In order to introduce our main theorem, we need some lemmas.

Lemma 3. Let be a nonnegative function and . Let , , , , and fulfil (14). Then for any , the inequalities imply

Proof. Since , there exists such that We will show that By contradiction, assume that there exists such that Then there exists such that There are two cases to consider.
Case 1. Consider on . In this case, integrating (16) from to and using (15) and (21) we infer that Thus , which contradicts (20).
Case 2. Consider on . Similar to Case 1, we have Thus, , which contradicts (20). Therefore (19) is true. Furthermore, from the fact it follows that .
Finally, we show that Suppose on the contrary that there exists satisfying Then there exists such that There are two cases to consider.
Case  1 Consider on . Similar to Case 1, we have , which contradicts (25).
Case  2 Consider on . Similar to Case 2, one has , which also contradicts (25). Hence (24) is true.
In summary, from (19) and (24) it follows that estimate (17) holds. This completes the proof of the lemma.

Lemma 4. Let and let be a nonnegative function. Let , , , , and fulfil (14). Then for any function the inequalities imply

Proof. For every , from (28), there exist such that Integrating (27) by (14) and (30) we get This completes the proof of the lemma.

Now, we apply Lemma 1 to establish the existence results of solutions for the fourth-order PBVP (1), (2).

Theorem 5. Assume that there exist , , and a nonnegative function . Suppose further that (H0) satisfies the Carathéodory conditions;(H1)if , , , , then (H2)if , ,  ,  , then where , , , fulfil (14). Then PBVP (1), (2) has at least one solution such that

Proof. Let Then iff there exist some such that
Now, we show that where , . To do this, we assume that is the solution of the following periodic boundary value problem: Integrating the equation as above on , we obtain Thus, by Wirtinger inequality, Hence from (38) it follows that If , then, from , the following contradiction holds: Therefore, .
Finally, we show that, for every , To do this, let and let be a solution of the following PBVP: Then . In fact, let Then, by (33), for a.e. . Applying Lemma 3, we obtain Thus according to (32), we have for a.e. . It follows from Lemma 4 that Thus . This implies that condition (ii) of Lemma 1 is valid.
In summary, all conditions of Lemma 1 are satisfied. Therefore the conclusion of Theorem 5 holds. This completes the proof of the theorem.

Next, we establish the existence result of nontrivial solutions for the fourth-order PBVP (1), (2) by means of Lemma 2.

Theorem 6. Assume that all conditions in Theorem 5 hold with the exception of (H1), which is replaced by the following:there exists a constant such that if , , , , then and if , , , , then Then PBVP (1), (2) has at least one nontrivial solution satisfying (34).

Proof. From the proof of Theorem 5 and Lemma 1, it follows that has a solution in and . Without loss of generality, we assume that and . We also assume that for all .
Now we assert that In fact, suppose that there exist and such that Applying to both sides of above equality, it follows that that is, Notice that and ; we have Consequently, from assumption one has This together with (56) it follows that which is a contradiction. This implies that Thus from Lemma 2 it follows that Hence Therefore has a solution in ; that is, PBVP (1), (2) has at least one nontrivial solution. This completes the proof of the theorem.

Finally, we give some examples to illustrate our results.

Example 7. Consider the fourth-order periodic boundary value problem where is a parameter, is nonnegative, and is a constant.
Let Then satisfies Carathéodory conditions. Taking any , then , , , and are well defined by (14).
Now, we assert that all conditions of Theorem 5 are satisfied when In fact, without loss of generality, we can assume . In this case, we choose . It is easy to see that, for every , and, for every , Hence condition of Theorem 5 is satisfied. In addition, for , , , , we have Therefore, condition of Theorem 5 is also satisfied. Hence, from Theorem 5, the fourth-order PBVP (63) has at least a solution , provided

Example 8. Consider the fourth-order periodic boundary value problem where is a parameter, , is odd, is even, and is a nonnegative function.
Let Then satisfies Carathéodory conditions. We choose ; then is well defined by (14).
Now, we assert that satisfies all conditions of Theorem 6 when In fact, without loss of generality, we can assume that . Choose and . Then it is easy to see that, for every , and, for every , On the other hand, for every , In summary, all conditions of Theorem 6 are satisfied. Therefore, from Theorem 6, the fourth-order PBVP (71) has at least one nontrivial solution , provided .

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This work was supported by the National Natural Science Foundation of China (11201008).

References

  1. P. Amster and M. C. Mariani, “Oscillating solutions of a nonlinear fourth order ordinary differential equation,” Journal of Mathematical Analysis and Applications, vol. 325, no. 2, pp. 1133–1141, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  2. Z. Bai, “Iterative solutions for some fourth-order periodic boundary value problems,” Taiwanese Journal of Mathematics, vol. 12, no. 7, pp. 1681–1690, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. C. Bereanu, “Periodic solutions of some fourth-order nonlinear differential equations,” Nonlinear Analysis: Theory, Methods and Applications, vol. 71, no. 1-2, pp. 53–57, 2009. View at: Publisher Site | Google Scholar
  4. A. Cabada, “The method of lower and upper solutions for second, third, fourth, and higher order boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 302–320, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. A. Cabada and S. Lois, “Maximum principles for fourth and sixth order periodic boundary value problems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 29, no. 10, pp. 1161–1171, 1997. View at: Publisher Site | Google Scholar
  6. A. Cabada and J. J. Nieto, “Quasilinearization and rate of convergence for higher-order nonlinear periodic boundary-value problems,” Journal of Optimization Theory and Applications, vol. 108, no. 1, pp. 97–107, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. P. C. Carrião, L. F. O. Faria, and O. H. Miyagaki, “Periodic solutions for extended Fisher-Kolmogorov and Swift-Hohenberg equations by truncature techniques,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 11, pp. 3076–3083, 2007. View at: Publisher Site | Google Scholar | MathSciNet
  8. J. Chen and D. O'Regan, “On periodic solutions for even order differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1138–1144, 2008. View at: Publisher Site | Google Scholar | MathSciNet
  9. F. Cong, “Periodic solutions for 2kth order ordinary differential equations with nonresonance,” Nonlinear Analysis: Theory, Methods and Applications, vol. 32, no. 6, pp. 787–793, 1998. View at: Publisher Site | Google Scholar | MathSciNet
  10. M. Conti, S. Terracini, and G. Verzini, “Infinitely many solutions to fourth order superlinear periodic problems,” Transactions of the American Mathematical Society, vol. 356, no. 8, pp. 3283–3300, 2004. View at: Publisher Site | Google Scholar | MathSciNet
  11. Q. Fan, W. Wang, and J. Zhou, “Periodic solutions of some fourth-order nonlinear differential equations,” Journal of Computational and Applied Mathematics, vol. 233, no. 2, pp. 121–126, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  12. J. Fialho and F. Minhós, “On higher order fully periodic boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 395, no. 2, pp. 616–625, 2012. View at: Publisher Site | Google Scholar | MathSciNet
  13. M. R. Grossinho, L. Sanchez, and S. A. Tersian, “On the solvability of a boundary value problem for a fourth-order ordinary differential equation,” Applied Mathematics Letters, vol. 18, no. 4, pp. 439–444, 2005. View at: Publisher Site | Google Scholar | MathSciNet
  14. C. P. Gupta, “Solvability of a fourth-order boundary value problem with periodic boundary conditions II,” International Journal of Mathematics and Mathematical Sciences, vol. 14, pp. 127–138, 1991. View at: Publisher Site | Google Scholar
  15. 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 Site | Google Scholar | Zentralblatt MATH | MathSciNet
  16. Y. Li, “Positive solutions of fourth-order periodic boundary value problems,” Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods, vol. 54, no. 6, pp. 1069–1078, 2003. View at: Publisher Site | Google Scholar | MathSciNet
  17. Y. Li, “On the existence and uniqueness for higher order periodic boundary value problems,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 2, pp. 711–718, 2009. View at: Publisher Site | Google Scholar | MathSciNet
  18. Y. Liu, “Solvability of periodic boundary value problems for nth-order ordinary differential equations,” Computers & Mathematics with Applications, vol. 52, no. 6-7, pp. 1165–1182, 2006. View at: Publisher Site | Google Scholar | MathSciNet
  19. Z. Liu, “Periodic solutions for nonlinear nth order ordinary differential equations,” Journal of Mathematical Analysis and Applications, vol. 204, no. 1, pp. 46–64, 1996. View at: Publisher Site | Google Scholar | MathSciNet
  20. 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 Site | Google Scholar | MathSciNet
  21. R. Y. Ma, “Solvability of a class of fourth-order periodic boundary value problems,” Acta Mathematica Scientia. Series A, vol. 15, no. 3, pp. 315–318, 1995. View at: Google Scholar | MathSciNet
  22. J. Mawhin and F. Zanolin, “A continuation approach to fourth order superlinear periodic boundary value problems,” Topological Methods in Nonlinear Analysis, vol. 2, no. 1, pp. 55–74, 1993. View at: Google Scholar | MathSciNet
  23. F. I. Njoku and P. Omari, “Singularly perturbed higher order periodic boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 289, no. 2, pp. 639–649, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  24. L. A. Peletier and W. C. Troy, “Spatial patterns described by the extended Fisher-Kolmogorov equation: periodic solutions,” SIAM Journal on Mathematical Analysis, vol. 28, no. 6, pp. 1317–1353, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  25. V. Šeda, J. J. Nieto, and M. Gera, “Periodic boundary value problems for nonlinear higher order ordinary differential equations,” Applied Mathematics and Computation, vol. 48, no. 1, pp. 71–82, 1992. View at: Google Scholar
  26. S. Tersian and J. Chaparova, “Periodic and homoclinic solutions of extended Fisher-Kolmogorov equations,” Journal of Mathematical Analysis and Applications, vol. 260, no. 2, pp. 490–506, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  27. Z. Wang, L. Qian, and S. Lu, “On the existence of periodic solutions to a fourth-order p-Laplacian differential equation with a deviating argument,” Nonlinear Analysis: Real World Applications, vol. 11, no. 3, pp. 1660–1669, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  28. J. R. Ward Jr., “Asymptotic conditions for periodic solutions of ordinary differential equations,” Proceedings of the American Mathematical Society, vol. 81, no. 3, pp. 415–420, 1981. View at: Publisher Site | Google Scholar | MathSciNet
  29. Q. Yao, “Existence, multiplicity and infinite solvability of positive solutions to a nonlinear fourth-order periodic boundary value problem,” Nonlinear Analysis: Theory, Methods and Applications, vol. 63, no. 2, pp. 237–246, 2005. View at: Publisher Site | Google Scholar
  30. C. Zhao, W. Chen, and J. Zhou, “Periodic solutions for a class of fourth-order nonlinear differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 72, no. 3-4, pp. 1221–1226, 2010. View at: Publisher Site | Google Scholar | MathSciNet
  31. I. Rachůnková, “Periodic boundary value problems for third-order differential equations,” Mathematica Slovaca, vol. 41, no. 3, pp. 241–248, 1991. View at: Google Scholar | MathSciNet
  32. J. Mawhin, Topological Degree Methods in Nonlinear Boundary Value Problems, vol. 40 of NSF-CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1979.
  33. F. B. Zhang, “Lack of direction property of the coincidence degree and applications,” Journal of Mathematical Research and Exposition, vol. 19, no. 4, pp. 693–698, 1999 (Chinese). View at: Google Scholar | MathSciNet

Copyright © 2014 Haitong 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.


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