- Research
- Open Access
- Published:
Existence of positive periodic solutions for third-order differential equation with strong singularity
Advances in Difference Equations volume 2014, Article number: 162 (2014)
Abstract
Sufficient conditions are presented for the existence of positive periodic solutions for a third-order nonlinear differential equation with singularity. Besides, an example is given to illustrate the results.
MSC:34K13, 34B16, 34B18.
1 Introduction
In this paper, we consider the following third-order differential equation with singularity:
where f, h are continuous function and T-periodic about t, , is an -Carathéodory function, i.e., it is measurable in the first variable and continuous in the second variable, and for every there exists such that for all and a.e. , f is ω-periodic function about t. Equation (1.1) is singular at 0, which means that becomes unbounded when . We say that (1.1) is of repulsive type (resp. attractive type) if (resp. ) when .
The study of singular differential equations began with the paper of Taliaferro. In 1979, Taliaferro [1] discussed the model equation with singularity
subject to
and obtained the existence of a solution for the problem. Here , with on and . We call the equation a strong force condition if and we call it a weak force condition if .
Taliaferro’s work has attracted the attention of many specialists in differential equations and they have contributed to the research of singular differential equations (see, e.g., [2–10]). Among these results, some are obtained for a second-order equation with strong force condition; see, e.g., [5, 9]. With a strong singularity, the energy near the origin becomes infinite and this fact is helpful for obtaining either a priori bounds, which are needed for a classical application of the degree theory, or the fast rotation, which is needed in recent versions of the Poincaré-Birkhoff theorem. Afterwards, in 2007 Torres [10] considered the periodic problem for a singular second-order equation with the weak force condition and showed that weak singularities may help periodic solutions to exist, which has driven the study of weak singularities (see [7]).
At the beginning, most of work concentrated on second-order singular differential equation, as in the references we mentioned above. Recently there have been published some results on third-order singular differential equation (see [11–17]). For example, in [13], Sun and Liu considered the singular nonlinear third-order periodic boundary value problem
with , , where and f is singular at , , and . Under suitable growth conditions, it is proved by constructing a special cone in and employing fixed point index theory that the problem has at least one solution or at least two positive solutions. Afterwards, Li [15] investigated the third-order ordinary differential equation
where is ω-periodic in t, and may be singular at . By applying of a fixed point theorem in cones, the author obtained that existence results of positive ω-periodic solutions for (1.4). Recently, Ren et al. [17] studied the third-order nonlinear singular differential equation
Using Green’s function for a third-order differential equation and some fixed point theorems, i.e., the Leray-Schauder alternative principle and Schauder’s fixed point theorem, they established three new existence results of periodic solutions for (1.5).
Based on the above work, in this paper we will study (1.1) and obtain the existence of periodic solutions by using topological degree theorem. The rest of this paper is organized as follows. In Section 2, we give some lemmas. In Section 3, by using topological degree theorem by Mawhin [18], some sufficient conditions are obtained for the existence of positive periodic solutions of (1.1). We, respectively, consider repulsive type and attractive type. In Section 4, an example is given to show the feasibility of the main result of this paper.
2 Some lemmas
Lemma 2.1 [[18], Theorem 2.4]
Let X, Y be real normed spaces and a linear Fredholm map of index zero. Assume that is an open bounded set and is an L-compact mapping. Assume that the following conditions are satisfied:
-
(i)
, for each ;
-
(ii)
, for each ;
-
(iii)
, where is a continuous projector such that and is the Brouwer degree,
then the equation has at least one solution in .
For the sake of convenience, throughout this paper we will adopt the following notation:
Lemma 2.2 [19]
If and , then
where , .
Remark 2.1 When , .
Lemma 2.3 [20]If with , then
Lemma 2.4 If with , and such that , then
Proof Let , and then . By Lemma 2.2 and Minkowski’s inequality, we have
This completes the proof of Lemma 2.4. □
Lemma 2.5 If with , and such that , then
Proof From Lemma 2.3 and Lemma 2.4, we know that, when , we have
This completes the proof of Lemma 2.5. □
3 Main results
First we consider (1.1) when is of attractive type. Assume that
exists uniformly a.e. , i.e., for any there is such that
for all and a.e. . Moreover, and .
For the sake of convenience, we list the following assumptions which will be used repeatedly in the sequel:
(H1) There exist two positive constants such that
(H2) (Decomposition condition) , where and is an -Carathéodory function, i.e., it is measurable in the first variable and continuous in the second variable, and for any there is such that
(H3) (Strong force condition at ) .
(H4) There exists a positive constant A such that , for all .
Theorem 3.1 Assume that (3.2), , and (H1)-(H4) hold. We have the following condition:
(H5) .
Then (1.1) has at least one positive T-periodic solution.
Proof Let , endowed with the -norm. Let with the -norm.
Let and let be the operator defined by
Define a nonlinear mapping by
Then (1.1) can be converted to the abstract equation . Define the projectors and by
The real number Px and Qy are seen as elements of X and Y inasmuch constant function. It is easy to see that , , , , and then L is a Fredholm linear mapping with zero index.
Let K denote the inverse of . Then we have
From (3.3), (3.4), and (3.5), it follows that QN and are continuous, and is bounded and then is compact for any open bounded , which means N is L-compact on .
Now we consider the following (homotopy) family of (1.1):
i.e., the abstract equation . We need to show that the set of all possible solutions of the family of (3.6) is, a priori, bounded in by a constant independent of .
Suppose that x is a solution to (3.6) for some . Let , be, respectively, the global maximum point and global minimum point of on ; Firstly, we consider . Since , we know that there exist two points , such that , . So, we get . Because is the maximum value of , . Furthermore, we conclude
So, we have
since and , and we get
From (H1) we obtain
Similarly, we get
From (H1) we obtain
From (3.8) and (3.9), we know that there exists a point such that
Therefore, we have
Multiplying by on both sides of (3.6) and integrating from 0 to T, we have
Since , from (H4) and , we have
For any , let be as in (3.2). Thus we have
Therefore,
Since , we can get . So, we have
From , we know that there exists a point such that . So, we have
From (3.12) and the Hölder inequality, we have
From (3.10) and Lemma 2.5, we have
From (3.13) and Lemma 2.3, we have
Since ε sufficiently small, from (H5) we know that . Thus, it is easy to see that there exists a positive constant such that
So, by the Hölder inequality, we have
From (3.11), we have
On the other hand, from , we know that there exists a point such that . From (3.2), (3.14), and (3.15), and by (3.6), we have
where , .
Next, multiplying (3.6) by we get
Let , for any , we integrate (3.17) on and get
By (3.14), (3.15), and (3.16) we have
where is as in (H2).
From these inequalities we can derive form (3.18) that
for some constant which is independent on λ, x and t. In view of the strong force condition (H3), we know that there exists a constant such that
The case can be treated similarly.
From (3.8), (3.14), (3.15), (3.16), and (3.20), we let
where , , and . Then the conditions (i) and (ii) of Lemma 2.1 are satisfied. For a constant , , we have
The degree condition (H1) shows that
Thus (iii) of Lemma 2.1 is also verified. Therefore has at least one solution in , which means (1.1) has at least one positive T-periodic solution. □
Next we consider (1.1) when is of repulsive type.
Theorem 3.2 Assume that (3.2), , (H2), and (H4), (H5) are satisfied. We have the following condition:
() there exist two positive constants such that
() (Strong force condition at ) .
Then (1.1) has at least one positive T-periodic solution.
Proof Let and consider the following:
Let , be, respectively, the global maximum point and global minimum point of on . First, we consider . Since , we know that there exist two points , such that , . So, we get . Because is the minimum value of , . Furthermore, we can conclude
So, we have
since and , we get
Hence, from () we know that there exists a positive constant such that
Similarly, we get
Hence, from () we know that there exists a positive constant such that
From (3.24) and (3.25), we know that there exists a point such that
Therefore, we have
The rest of the proof is the same as that of Theorem 3.1. □
4 Examples
Finally, we present some examples to illustrate our result.
Example 4.1 Consider the three-order differential equation with singularity:
where .
It is clear that , , , , . It is obvious that (H1)-(H4) hold. Now we consider the assumption (H5). Since , , we have
So by Theorem 3.1, we know (4.1) has at least one positive π-periodic solution.
Author’s contributions
ZBC worked together in the derivation of the mathematical results. The author read and approved the final manuscript.
References
Taliaferro S: A nonlinear singular boundary value problem. Nonlinear Anal. TMA 1979, 3: 897–904. 10.1016/0362-546X(79)90057-9
Agarwal RP, O’Regan D: Existence theory for single and multiple solutions to singular positone boundary value problems. J. Differ. Equ. 2001, 175: 393–414. 10.1006/jdeq.2001.3975
Cheng ZB, Ren JL: Periodic and subharmonic solutions for Duffing equation with a singularity. Discrete Contin. Dyn. Syst. 2012, 32: 1557–1574.
Chu JF, Torres P, Zhang MR: Periodic solution of second order non-autonomous singular dynamical systems. J. Differ. Equ. 2007, 239: 196–212. 10.1016/j.jde.2007.05.007
Fonda A, Manásevich R: Subharmonics solutions for some second order differential equations with singularities. SIAM J. Math. Anal. 1993, 24: 1294–1311. 10.1137/0524074
Martins RF: Existence of periodic solutions for second-order differential equations with singularities and the strong force condition. J. Math. Anal. Appl. 2006, 317: 1–13. 10.1016/j.jmaa.2004.07.016
Li X, Zhang ZH: Periodic solutions for damped differential equations with a weak repulsive singularity. Nonlinear Anal. TMA 2009, 70: 2395–2399. 10.1016/j.na.2008.03.023
del Poin M, Manásevich R: Infinitely many T -periodic solutions for a problem arising in nonlinear elasticity. J. Differ. Equ. 1993, 103: 260–277. 10.1006/jdeq.1993.1050
del Poin M, Manásevich R, Murua A: On the number of 2 π -periodic solutions for using the Poincaré-Birkhoff theorem. J. Differ. Equ. 1992, 95: 240–258. 10.1016/0022-0396(92)90031-H
Torres P: Weak singularities may help periodic solutions to exist. J. Differ. Equ. 2007, 232: 277–284. 10.1016/j.jde.2006.08.006
Sun YP: Positive solution of singular third-order three-point boundary value problem. J. Math. Anal. Appl. 2005, 306: 586–603.
Chu JF, Zhou ZC: Positive solutions for singular non-linear third-order periodic boundary value problems. Nonlinear Anal. TMA 2006, 64: 1528–1542. 10.1016/j.na.2005.07.005
Sun JX, Liu YS: Multiple positive solutions of singular third-order periodic boundary value problem. Acta Math. Sci. 2005, 25: 81–88.
Palamides AP, Smyrlis G: Positive solutions to a singular third-order three-point boundary value problem with an indefinitely signed Green’s function. Nonlinear Anal. TMA 2008, 68: 2014–2118.
Li YX: Positive periodic solutions for fully third-order ordinary differential equations. Comput. Math. Appl. 2010, 59: 3464–3471. 10.1016/j.camwa.2010.03.035
Liu ZQ, Ume JS, Kang SM: Positive solutions of a singular nonlinear third order two-point boundary value problem. J. Math. Anal. Appl. 2007, 326: 589–601. 10.1016/j.jmaa.2006.03.030
Ren JL, Cheng ZB, Chen YL: Existence results of periodic for third-order nonlinear singular differential equation. Math. Nachr. 2013, 286: 1022–1042. 10.1002/mana.200910173
Mawhin J: Topological degree and boundary value problems for nonlinear differental equations. Lecture Notes in Math. 1537. In Topological Methods for Ordinary Differential Equations. Springer, Berlin; 1993:74–142.
Zhang MR: Nonuniform nonresonance at the first eigenvalue of the p -Laplacian. Nonlinear Anal. TMA 1997, 29: 41–51. 10.1016/S0362-546X(96)00037-5
Ren JL, Cheng ZB: On high-order delay differential equation. Comput. Math. Appl. 2009, 57: 324–331. 10.1016/j.camwa.2008.10.071
Acknowledgements
ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (No. 11326124, 11271339).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that they have no competing interests.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Cheng, Z. Existence of positive periodic solutions for third-order differential equation with strong singularity. Adv Differ Equ 2014, 162 (2014). https://doi.org/10.1186/1687-1847-2014-162
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1847-2014-162
Keywords
- third order
- singularity
- positive periodic solution
- topological degree theorem