Abstract
We study the existence of positive periodic solutions of second-order singular differential equations. The proof relies on Schauder’s fixed point theorem. Our results generalized and extended those results contained in the studies by Chu and Torres (2007) and Torres (2007) . In some suitable weak singularities, the existence of periodic solutions may help.
1. Introduction
The theory of second-order periodic differential equation has been widely studied [1–20] because of great practical importance. In this paper, we apply Schauder’s fixed point theorem to study the existence of positive solutions of the second-order periodic differential equation where , are continuous and 1-periodic functions and the nonlinearity is continuous in , 1-periodic in , and singular at . In addition, may take some negative values.
Beginning with the paper of Lazer and Solimini [13], the semilinear singular differential equation where , , and , has received the attention of many researchers during the last two decades [5, 7, 10, 21]. In [2], the author investigated the existence of positive solutions of (2) for three cases when , , and through a basic application of Schauder’s fixed point theorem. Here, let and denote the essential supremum and infimum of a function , if they exist. For any continuous function in , we set
In [1], the authors generalized the results of [2] and considered the second-order periodic semilinear singular equation with , , , and , . By Schauder’s fixed point theorem, they established the existence of positive periodic solutions to (4), if and .
Although [1, Theorem 3.1] can deal with the cases of the existence of positive periodic solutions to (4), the authors there did not consider these two cases as applications of the existence of positive periodic solutions of (4) if or . We also generalize some of the results of [1, 2] in certain ways.
The remaining part of the paper is organized as follows. In Section 2, some preliminary results are given. In Sections 3 and 4, we will state and prove the main results of the paper, as well as some applications to (2) and (4). In Section 5, we apply the results established in this paper to two specific equations.
2. Preliminaries
Throughout this paper, we assume that the equation, known as Hill’s equation, with periodic boundary conditions satisfies the following standing hypothesis:(A)associated Green’s function is nonnegative for all .
In other words, the antimaximum principle holds. Under this assumption, the function is just the unique 1-periodic solution of the linear equation
Now we make condition (A) clear. When , condition (A) is equivalent to . Note that is the first eigenvalue of the linear problem with Dirichlet conditions . For a nonconstant function , there is a -criterion proved in [16]. Let denote the best Sobolev constant in the following inequality: The explicit formula for is where is the gamma function.
We write if for a.e. and is positive in a set of positive measures.
Lemma 1 (see [16]). Assume that and for some . If then the standing hypothesis (A) holds.
Remark 2. In [3, 12, 16], the existence results are based on the positiveness of , which plays a very important role in employing some fixed point theorems in cones for completely continuous operators. Our assumption (A) only needs that is nonnegative, and therefore our results cover the critical case, which was not covered in the above three papers.
3. Main Results
In this section, we establish the main result about the existence of positive periodic solutions for (1).
Theorem 3. Suppose that the following conditions hold.There exist nonnegative continuous functions , , on and , on such that , is strictly decreasing, and is increasing. Moreover, there holds for any and There exists a positive constant such that Then (1) has at least one positive periodic solution.
Proof. Let denote the set of all continuous 1-periodic functions. We define a completely continuous map by
It is clear that a periodic solution of (1) is just a fixed point of . It suffices to prove the existence of the fixed point of map .
Let be the positive constant in () and set
Combining (12) with the fact that is positive and strictly decreasing, we have . We introduce a set by
Obviously, is a closed convex set. For any and , by assumption (11), we get
Since and is strictly decreasing, there holds
Together with assumption (13), the above inequality leads to
Similar discussion shows that there holds, for any and ,
Consequently, we get the conclusion that there holds . By Schauder’s fixed point theorem, we obtain that there exists at least one fixed point of , which completes the proof.
Applying Theorem 3 to a special case, we have the following result.
Corollary 4. Suppose that and there hold condition () and the following one.()There exists a positive constant such that Then (1) has at least one positive periodic solution.
4. Applications
4.1. The Case
In this subsection, we consider (4) in the cases when . To meet the condition in Corollary 4, we let We note that to verify the existence of the positive periodic solution of (4) by Corollary 4 an important step is to find such that where , , are the functions in condition (). Let . Then finding with (23) is equivalent to finding satisfying To this end, we consider the following function: By direct calculation, we have Suppose that and . Set It is clear that is continuous and increasing on . Since there hold and , there exists a unique such that . It follows that . In addition, we note that for and for . Hence, we get the conclusion that possesses a minimum at . Based upon the above discussion, we have that is the best choice for satisfying inequality (24).
Theorem 5. Suppose that , , , and there exist nonnegative continuous functions , , on such that and If there holds where is the unique positive solution of the equation then (4) has at least one positive periodic solution.
Proof. Let , , and . It is obvious that is strictly decreasing and is nondecreasing. According to the assumption about , , , we have that condition () of Theorem 3 holds.
Let . It follows from assumption (29) that there holds
It suffices to verify that satisfies
which is equivalent to the inequality
Due to (30), we get
which implies
We also obtain by (30)
which leads to
Combining (35) with (37), we have
which completes the proof.
4.2. The Case
In this subsection, we consider (2) and (4) in the case when . In order to study (2) by Theorem 3, we have to set , , and . Let be the functions in condition (). For the purpose of looking for satisfying we introduce a function on by It suffices to find such that . To this end, we compute the derivative of . There holds Assume . By letting we note that is increasing, with , and . Then the equation has a unique solution , which is also the unique solution of . Since for and for , attains the minimum at , which shows that we get the best choice of positive constant satisfying .
Theorem 6. Suppose that and there exist continuous and nonnegative functions , such that . If and where is the unique positive solution of the equation then (2) has at least one positive periodic solution.
Proof. Let , , . Since , we have, for any and any and , . Hence, we conclude that condition () in Theorem 3 is satisfied. Taking , we obtain by inequality (43) It suffices to verify which is equivalent to It follows from (44) that , which implies Together with the assumption that , the above inequality shows The proof is finished.
In the following, we consider (2). We also let , , and . We will deal with this case in a similar manner. Let , , be the functions in condition (). For this purpose, we note that looking for satisfying is equivalent to finding such that Based upon this observation, we have to define The derivative of can be calculated as follows: Since the function is increasing on and , and , the equation has a unique solution . Thus the function attains a minimum at , which provides a suitable satisfying (51).
Theorem 7. Suppose that , , and there exist continuous and nonnegative functions , , on such that and If and where is the unique positive solution of the equation then (4) has at least one positive periodic solution.
Proof. By similar discussion in Theorem 5, we have that condition () of Theorem 3 is satisfied under the assumption in this theorem. Set . Assumption (56) shows To verify that satisfies we note that, by (57), there holds which leads to Furthermore, we also have that is, Combining (61) with (63), we get Substituting into the above inequality, we get (59), which completes the proof.
5. Example
In this section, we apply the results established in this paper to two second-order singular equations.
Example 1. Consider the periodic problem for a second-order singular equation We will make use of Theorem 5 to show that this equation has at least one positive solution. We first note that (65) is nonresonant and the associated Green function is nonnegative. In fact, we have Furthermore, there also holds . In order to verify the conditions in Theorem 5, we set for any and any It is easy to see that there hold and, for any and , To verify (29), we point out that which lead to , , and . By numerical calculation, we get the unique positive solution of equation as . Hence, we have the desired inequalities Since all the conditions are satisfied, Theorem 5 guarantees that (65) has a positive solution (see Figure 1, numerical simulation for ).
Example 2. Consider the periodic problem for a second-order singular equation where In this example, we note that there hold To meet the conditions in Theorem 7, we also set for any and any It is clear that there hold and, for any and , By direct calculation, we have , , and which leads us to consider the equation By numerical calculation, we get the unique positive solution of the above equation as . Thus, we get Together with the fact that , the above inequality shows that all the conditions of Theorem 7 are satisfied. Consequently, Theorem 7 guarantees that (72) has a positive solution (see Figure 2, numerical simulation).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The work was supported by Program for Changjiang Scholars and Innovative Research Team in University, NSFC of China (no. 10971021) and Key Subject of Chinese Ministry of Education (no. 109051), Scientific Research Fund of Heilongjiang Provincial Education Department (no. 11544032), a Grant from the Ph.D. Programs Foundation of Ministry of Education of China (no. 200918), and Youth Education Jun Projects of Jilin University of Finance and Economics.