- Research
- Open Access
Construction of lower and upper solutions for first-order periodic problem
- Ruyun Ma^{1}Email author and
- Lu Zhang^{1}
- Received: 18 November 2014
- Accepted: 9 October 2015
- Published: 20 October 2015
Abstract
In this paper, we construct nonconstant lower and upper solutions for the periodic boundary value problem \(x'+f(t,x)=e(t)\), \(x(0)=x(T)\) and find their estimates. We prove the existence of positive solutions for the singular problem \(x'+g(x)=e(t)\), \(x(0)=x(T)\) by using these results.
Keywords
- periodic boundary value problem
- singular problem
- lower and upper solutions
MSC
- 34B10
- 34B18
1 Introduction
In order to apply these results, finding upper and lower solutions is very important. But the problem of construction of lower and upper solutions has been solved very rarely (see, e.g., [7, 8]). In this paper we fill this gap and present conditions ensuring the existence of nonconstant lower and upper solutions to the first-order periodic boundary value problem (1.1) and find their estimates. This enables us to prove the existence result for the periodic problem with strong singularity.
Let us recall first some classical definitions and results. They are taken from [3].
Definition 1.1
The basic existence theorem of the method of upper and lower solutions for (1.1) can be stated as follows.
Lemma 1.1
If problem (1.1) has a lower solution α and upper solution β such that \(\alpha(t)\leq\beta(t)\) for all \(t\in[0,T]\) (resp. \(\beta(t)\leq\alpha(t)\) for all \(t\in[0,T]\)), then problem (1.1) has at least one solution x such that \(\alpha(t)\leq x(t)\leq\beta (t)\) for all \(t\in[0,T]\) (resp. \(\beta(t)\leq x(t)\leq\alpha(t)\) for all \(t\in[0,T]\)).
The paper is organized as follows. In Section 2 we develop a method to construct lower and upper solutions for (1.1). As the application, in Section 3, we establish an existence result for a nonlinear first-order periodic problem with strong singularity.
2 Construction of lower and upper solutions
Proposition 2.1
Proof
Consider the function \(\phi=c_{1}-e\). We have two cases.
Case 1. Assume that \(\Phi_{+}=\int^{T}_{0}\phi^{+}(t)\,dt=0\). Taking \(\beta=\xi\in[B_{1},B_{1}+2\| \eta_{1} \|_{1}]\) and using \(c_{1}-e\leq0\), it follows from (2.3) that β is an upper solution of (2.1).
Using similar arguments, one can prove the following proposition.
Proposition 2.2
Theorem 2.1 and Theorem 2.2 are simple examples of existence results which follow immediately from Lemma 1.1, Propositions 2.1 and 2.2.
Theorem 2.1
Theorem 2.2
Now, we consider problem (1.1), i.e., \(e\equiv0\).
Proposition 2.3
Proof
The following assertion is due to Proposition 2.1 and its proof can be omitted.
Proposition 2.4
It is easy to see that the following results can be shown in a similar manner as in Theorem 2.1.
Theorem 2.3
Theorem 2.4
Problem (1.1) has been considered by several authors (see, e.g., [6, 9]). In [9], Peng gave the existence result as follows.
Theorem A
- (i)
\(f_{0}=\liminf_{x\rightarrow0^{+}} \min_{t\in[0,T]}\frac {f(t,x)}{x}>0\) and \(f^{\infty}=\limsup_{x\rightarrow\infty} \max_{t\in[0,T]}\frac{f(t,x)}{x}<0\); or
- (ii)
\(f_{\infty}=\liminf_{x\rightarrow\infty} \min_{t\in [0,T]}\frac{f(t,x)}{x}>0\) and \(f^{0}=\limsup_{x\rightarrow\infty} \max_{t\in[0,T]}\frac{f(t,x)}{x}<0\).
In this paper, we do not require f to satisfy (i) or (ii). Now we give an example.
Example 2.1
3 Periodic problem with strong singularity
Theorem 3.1
Proof
It is easy to see that \(\alpha=\varepsilon\) is a lower solution of (3.1). Thus, by Lemma 1.1, problem (3.1) has a positive solution x such that \(0<\varepsilon<x\leq B_{1}\). □
Example 3.1
Let \(g(x)=\frac{1}{x}\), \(e(t)=t\). We can prove that \(g(x)\) satisfies condition (3.2). Since \(\bar{e}=\int^{1}_{0} t \,dt=\frac{1}{2}\), then \(\widehat{\phi}=\bar {e}-e=\frac{1}{2}-t\).
Declarations
Acknowledgements
The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by NSFC (No. 11361054 and No. 11201378), SRFDP (No. 2012 6203110004), and Gansu provincial National Science Foundation of China (No. 1208RJZA258).
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Authors’ Affiliations
References
- Moretto, S: Sull’esistenza di soluzioni periodiche per l’equazione \(y' =f(t, y)\). Ann. Univ. Ferrara 8, 61-67 (1958) MathSciNetGoogle Scholar
- Bereanu, C, Mawhin, J: Upper and lower solutions for periodic problems: first order difference vs first order differential equations. In: Mathematical Analysis and Applications. AIP Conf. Proc., vol. 835, pp. 30-36. Am. Inst. Phys., Melville (2006) (Reviewer: Johnny Henderson) 34C25 (39A11) Google Scholar
- Mawhin, J: First order ordinary differential equations with several periodic solutions. Z. Angew. Math. Phys. 38, 257-265 (1987) MATHMathSciNetView ArticleGoogle Scholar
- Obersnel, F, Omari, P: Old and new results for first order periodic ODEs without uniqueness: a comprehensive analysis via lower and upper solutions. Adv. Nonlinear Stud. 4, 323-376 (2004) MATHMathSciNetGoogle Scholar
- Lakshmikantham, V, Leela, S: Existence and monotone method for periodic solutions of first order differential equations. J. Math. Anal. Appl. 91, 237-243 (1983) MATHMathSciNetView ArticleGoogle Scholar
- Franco, D, Nieto, J, O’Regan, D: Upper and lower solutions for first order problems with nonlinear boundary conditions. Extr. Math. 18, 153-160 (2003) MATHMathSciNetGoogle Scholar
- Bereanu, C, Gheorghe, D, Zamora, M: Periodic solutions for singular perturbations of the singular ϕ-Laplacian operator. Commun. Contemp. Math. 15, 1-22 (2013) MathSciNetView ArticleGoogle Scholar
- Rachu̇nková, I, Tvrdý, M: Construction of lower and upper functions and their application to regular and singular periodic boundary value problems. Nonlinear Anal. 47, 3937-3948 (2001) MathSciNetView ArticleGoogle Scholar
- Peng, S: Positive solutions for first order periodic boundary value problem. Appl. Math. Comput. 158, 345-351 (2004) MATHMathSciNetView ArticleGoogle Scholar