Existence of positive periodic solutions of second-order differential equations with weak singularities
© Ma; licensee Springer 2014
Received: 25 June 2014
Accepted: 23 July 2014
Published: 25 September 2014
We establish the existence of positive periodic solutions of the second-order differential equation via Schauder’s fixed point theorem, where , , f is a Caratheodory function and it is singular at . Our main results generalize some recent results by Torres (J. Differ. Equ. 232:277-284, 2007).
MSC: 34B10, 34B18.
with , , is a Caratheodory function, and f is singular at .
Moreover, if (weak force condition) they found examples of functions c with negative mean values and such that periodic solutions do not exist.
If compared with the literature available for strong singularities, see – and the references therein, the study of the existence of periodic solutions in the presence of a weak singularity is much more recent and the number of references is considerably smaller. The likely reason may be that with a weak singularity, the energy near the origin becomes finite, and this fact is not helpful for obtaining the a priori bound needed for a classical application of the degree theory, and also is not helpful for the fast rotation needed in recent versions of the Poincaré-Birkhoff theorem. The first existence result with a weak force condition appears in Rachunková et al.. Since then, Eq. (1.1) with f having weak singularities has been studied by several authors; see Torres , , Franco and Webb , Chu and Li  and Li and Zhang .
Recently, Torres  showed how a weak singularity can play an important role if Schauder’s fixed point theorem is chosen in the proof of the existence of positive periodic solutions for (1.1). From now on, for a given function , we denote the essential supremum and infimum of ξ by and , respectively. We write if for a.e. and it is positive in a set of positive measure. Under the following assumption:
Torres showed the following.
, Theorem 1]
If, then there exists a positive T-periodic solution of (1.1).
, Theorem 2]
Let (H1) hold. Assume that
If, then (1.1) has a positive T-periodic solution.
, Theorem 4]
then (1.1) has a positive T-periodic solution.
Obviously, (H2) and (H3) are too restrictive so that the above mentioned results are only applicable to (1.1) with nonlinearity which is bounded at origin and infinity by a function of the form . Very recently, Ma et al. generalized Theorems A-C under some conditions which allow the nonlinearity f to be bounded by two different functions and .
Notice that in (H2), and in (H3). Of course the natural question is: what would happen if we allow that the functions b and may change sign?
is still open; see Bravo and Torres  and Hakl and Torres . Notice that (1.4) plays an important role in the study of stabilization of matter-wave breathers in Bose-Einstein condensates , the propagation of guided waves in optical fibers , and in the electromagnetic trapping of a neutral atom near a charged wire .
It is the purpose of this paper to general Theorems A-C under some assumptions which allow the nonlinearity b and to change sign. The main tool is Schauder’s fixed point theorem.
2 Existence of periodic solutions
Let. Assume (H1) and
then there exists a positive T-periodic solution of (1.1).
Thus, . Clearly, , so the proof is finished. □
If in , then , and accordingly . In this case, (2.6) is satisfied for all and γ with . So, Theorem 2.1 generalizes Theorem A.
Let. Assume (H1) and
Then there exists a positive T-periodic solution of (1.1).
Obviously, conditions (2.15) and (2.16) guarantee that (2.20) and (2.21) hold for . □
Let. Assume (H1) and (A2) and
Then there exists a positive T-periodic solution of (1.1).
If we fix and , then and (2.22), (2.23) ensure that (2.24) and (2.25) hold. □
As in the proof of , Theorem 4], we may take , then (2.26) can be reduced by the condition (2.22).
Since for all , Theorem 2.1 guarantees that the set (2.27), (2.28) has at least one positive 1-periodic solution.
YM completed the study, carried out the results of this article and drafted the manuscript, and checked the proofs and verified the calculation. The author read and approved the final manuscript.
This work was supported by the NSFC (No. 11361054).
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