Recent Existence Results for Second-Order Singular Periodic Differential Equations
© J. Chu and J. J. Nieto. 2009
Received: 12 February 2009
Accepted: 29 April 2009
Published: 8 June 2009
We present some recent existence results for second-order singular periodic differential equations. A nonlinear alternative principle of Leray-Schauder type, a well-known fixed point theorem in cones, and Schauder's fixed point theorem are used in the proof. The results shed some light on the differences between a strong singularity and a weak singularity.
is that the mean value of is negative, , here , which is a strong force condition in a terminology first introduced by Gordon . Moreover, if , which corresponds to a weak force condition, they found examples of functions with negative mean values and such that periodic solutions do not exist. Since then, the strong force condition became standard in the related works; see, for instance, [2, 8–10, 13, 19–21], and the recent review . With a strong singularity, the energy near the origin becomes infinity and this fact is helpful for obtaining the a priori bounds needed for a classical application of the degree theory. Compared with the case of a strong singularity, the study of the existence of periodic solutions under the presence of a weak singularity by topological methods is more recent but has also attracted many researchers [4, 6, 23–28]. In , for the first time in this topic, Torres proved an existence result which is valid for a weak singularity whereas the validity of such results under a strong force assumption remains as an open problem. Among topological methods, the method of upper and lower solutions [6, 29, 30], degree theory [8, 20, 31], some fixed point theorems in cones for completely continuous operators [25, 32–34], and Schauder's fixed point theorem [27, 35, 36] are the most relevant tools.
In this paper, we select several recent existence results for singular equation (1.1) via different topological tools. The remaining part of the paper is organized as follows. In Section 2, some preliminary results are given. In Section 3, we present the first existence result for (1.1) via a nonlinear alternative principle of Leray-Schauder. In Section 4, the second existence result is established by using a well-known fixed point theorem in cones. The condition imposed on in Sections 3 and 4 is that the Green function associated with the linear periodic equations is positive, and therefore the results cannot cover the critical case, for example, when is a constant, , , and is the first eigenvalue of the linear problem with Dirichlet conditions . Different from Sections 3 and 4, the results obtained in Section 5, which are established by Schauder's fixed point theorem, can cover the critical case because we only need that the Green function is nonnegative. All results in Sections 3–5 shed some lights on the differences between a strong singularity and a weak singularity.
In this paper, we will use the following notation. Given , we write if for a.e. and it is positive in a set of positive measure. For a given function essentially bounded, we denote the essential supremum and infimum of by and , respectively.
In Sections 3 and 4, we assume that
In Section 5, we assume that
3. Existence Result (I)
In this section, we state and prove the first existence result for (1.1). The proof is based on the following nonlinear alternative of Leray-Schauder, which can be found in . This part can be regarded as the scalar version of the results in .
From this claim, the Leray-Schauder alternative principle guarantees that
The fact and (3.19) show that is a bounded and equicontinuous family on . Now the Arzela-Ascoli Theorem guarantees that has a subsequence, , converging uniformly on to a function . Moreover, satisfies the integral equation
From the proof of Theorem 3.2 and Lemma 3.3, we see that the strong force condition ( ) is only used when we prove (3.18). From the next theorem, we will show that, for the case , we can remove the strong force condition ( ), and replace it by one weak force condition.
4. Existence Result (II)
In this section, we establish the second existence result for (1.1) using a well-known fixed point theorem in cones. We are mainly interested in the superlinear case. This part is essentially extracted from .
First we recall this fixed point theorem in cones, which can be found in . Let be a cone in and is a subset of , we write and
Theorem 4.1 (see ).
be a completely continuous operator such that
Combined Theorem 4.2 with Theorems 3.2 or 3.4, we have the following two multiplicity results.
Since , it is easy to see that the right-hand side goes to 0 as . Thus, for any given , it is always possible to find such that (4.9) is satisfied. Thus, (1.5) has an additional positive periodic solution .
5. Existence Result (III)
In this section, we prove the third existence result for (1.1) by Schauder's fixed point theorem. We can cover the critical case because we assume that the condition (B) is satisfied. This part comes essentially from , and the results for the vector version can be found in .
The validity of (ii) in Corollary 5.3 under strong force conditions remains still open to us. Such an open problem has been partially solved by Corollary 3.5. However, we do not solve it completely because we need the positivity of in Corollary 3.5, and therefore it is not applicable to the critical case. The validity for the critical case remains open to the authors.
The authors express their thanks to the referees for their valuable comments and suggestions. The research of J. Chu is supported by the National Natural Science Foundation of China (Grant no. 10801044) and Jiangsu Natural Science Foundation (Grant no. BK2008356). The research of J. J. Nieto is partially supported by Ministerio de Education y Ciencia and FEDER, Project MTM2007-61724, and by Xunta de Galicia and FEDER, project PGIDIT06PXIB207023PR.
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