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.
here , and is a given parameter. The corresponding results are also valid for the general case
with . Some open problems for (1.5) or (1.6) are posed.
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
(A)the Green function associated with (2.1)–(2.2), is positive for all .
In Section 5, we assume that
(B)the Green function associated with (2.1)–(2.2), is nonnegative for all
When condition (A) is equivalent to and condition (B) is equivalent to . In this case, we have
The explicit formula for is
Obviously, and .
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 .
Assume is a relatively compact subset of a convex set in a normed space . Let be a compact map with . Then one of the following two conclusions holds:
(a) has at least one fixed point in
(b)thereexist and such that
Suppose that satisfies (A) and satisfies the following.
(H1)There exist constants and such that
(H2)There exist continuous, nonnegative functions and such that
is nonincreasing and is nondecreasing in .
(H3)There exists a positive number such that and
Then for each , (1.1) has at least one positive periodic solution with for all and .
Since ( ) holds, we can choose such that and
We claim that any fixed point of (3.9) for any must satisfy . Otherwise, assume that is a fixed point of (3.9) for some such that . Note that
This is a contradiction to the choice of and the claim is proved.
From this claim, the Leray-Schauder alternative principle guarantees that
has a periodic solution with . Since for all and is actually a positive periodic solution of (3.17).
In the next lemma, we will show that there exists a constant such that
for large enough.
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
where the uniform continuity of on is used. Therefore, is a positive periodic solution of (3.4).
There exist a constant and an integer such that any solution of (3.17) satisfies (3.18) for all .
Take such that and let . For , let
We claim first that . Otherwise, suppose that for some . Then from (3.24), it is easy to verify
This is a contradiction. Thus .
Now we consider the minimum values . Let . Without loss of generality, we assume that , otherwise we have (3.18). In this case,
Thus for , we have As , for all and the function is strictly increasing on . We use to denote the inverse function of restricted to .
In order to prove (3.18) in this case, we first show that, for ,
Otherwise, suppose that for some . Then there would exist such that and
On the other hand, by the strong force condition ( ), we can choose large enough such that
for all . So (3.30) holds for
Finally, multiplying (3.17) by and integrating from to , we obtain
if Thus we know that for some constant .
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.
Assume that ( ) and ( )–( ) are satisfied. Suppose further that
(H4)for each constant , there exists a continuous function such that for all .
Then for each with (1.1) has at least one positive periodic solution with for all and .
Assume that satisfies ( ) and . Then
(i)if then for each (1.5) has at least one positive periodic solution for all ;
(ii)if , then for each (1.5) has at least one positive periodic solution for each here is some positive constant.
(iii)if , then for each with (1.5) has at least one positive periodic solution for all ;
(iv)if , then for each with (1.5) has at least one positive periodic solution for each .
Note that if and if . Thus we have (i)–(iv).
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
(b)There exists such that and all
Then has a fixed point in
In applications below, we take with the supremum norm and define
Suppose that satisfies ( ) and satisfies ( )–( ). Furthermore, assume that
is nonincreasing and is nondecreasing in
Then (1.1) has one positive periodic solution with .
As in the proof of Theorem 3.2, we only need to show that (3.4) has a positive periodic solution with and
Let be a cone in defined by (4.2). Define the open sets
For each , we have . Thus for all Since is continuous, then the operator is well defined and is continuous and completely continuous. Next we claim that:
(i) for and
(ii)there exists such that and all
We start with (i). In fact, if then and for all Thus we have
Next we consider (ii). Let then Next, suppose that there exists and such that Since then for all As a result, it follows from ( ) and ( ) that, for all
Hence this is a contradiction and we prove the claim.
Now Theorem 4.1 guarantees that has at least one fixed point with Note by (4.7).
Combined Theorem 4.2 with Theorems 3.2 or 3.4, we have the following two multiplicity results.
Suppose that satisfies ( ) and satisfies ( )–( ) and ( )–( ). Then (1.1) has two different positive periodic solutions and with .
Suppose that satisfies ( ) and satisfies ( )–( ). Then (1.1) has two different positive periodic solutions and with .
Assume that satisfies ( ) and . Then
(i)if , then for each (1.5) has at least two positive periodic solutions for each ;
(ii)if , then for each with (1.5) has at least two positive periodic solutions for each .
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 .
Assume that conditions ( ) and ( ), ( ) are satisfied. Furthermore, suppose that
(H7)there exists a positive constant such that and here
Then (1.1) has at least one positive -periodic solution.
A -periodic solution of (1.1) is just a fixed point of the map defined by (4.6). Note that is a completely continuous map.
Let be the positive constant satisfying ( ) and Then we have . Now we define the set
Obviously, is a closed convex set. Next we prove
In fact, for each , using that and condition ( ),
In conclusion, . By a direct application of Schauder's fixed point theorem, the proof is finished.
As an application of Theorem 5.1, we consider the case . The following corollary is a direct result of Theorem 5.1.
Assume that conditions ( ) and ( ), ( ) are satisfied. Furthermore, assume that
If then (1.1) has at least one positive -periodic solution.
Suppose that satisfies ( ) and , , then for each with one hasthe following:
(i)if then (1.5) has at least one positive periodic solution for each .
(ii)if then (1.5) has at least one positive -periodic solution for each where is some positive constant.
for some .
So (1.5) has at least one positive -periodic solution for
Note that if and if . We have the desired results (i) and (ii).
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 next results explore the case when .
Suppose that satisfies ( ) and satisfies condition ( ). Furthermore, assume that
If then (1.1) has at least one positive -periodic solution.
We follow the same strategy and notation as in the proof of Theorem 5.1. Let be the positive constant satisfying ( ) and then since . Next we prove
For each , by the nonnegative sign of and , we have
In conclusion, and the proof is finished by Schauder's fixed point theorem.
Suppose that satisfies ( ) and , then for each with , one has the following:
(i)if then (1.5) has at least one positive -periodic solution for each
(ii)if , then (1.5) has at least one positive -periodic solution for each where is some positive constant.
Note that if and if . We have the desired results (i) and (ii).
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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