Existence of positive periodic solutions for third-order differential equation with strong singularity
© Cheng; licensee Springer. 2014
Received: 14 January 2014
Accepted: 27 May 2014
Published: 3 June 2014
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.
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 .
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  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 ).
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 , 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 [, Theorem 2.4]
, for each ;
, for each ;
, where is a continuous projector such that and is the Brouwer degree,
then the equation has at least one solution in .
Lemma 2.2 
where , .
Remark 2.1 When , .
This completes the proof of Lemma 2.4. □
This completes the proof of Lemma 2.5. □
3 Main results
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:
(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:
Then (1.1) has at least one positive T-periodic solution.
Proof Let , endowed with the -norm. Let with the -norm.
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.
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 .
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 .
where , .
where is as in (H2).
The case can be treated similarly.
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:
() (Strong force condition at ) .
Then (1.1) has at least one positive T-periodic solution.
The rest of the proof is the same as that of Theorem 3.1. □
Finally, we present some examples to illustrate our result.
So by Theorem 3.1, we know (4.1) has at least one positive π-periodic solution.
ZBC worked together in the derivation of the mathematical results. The author read and approved the final manuscript.
ZBC would like to thank the referee for invaluable comments and insightful suggestions. This work was supported by NSFC project (No. 11326124, 11271339).
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