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Construction of lower and upper solutions for firstorder periodic problem
Boundary Value Problems volume 2015, Article number: 190 (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.
Introduction
The method of lower and upper solutions is an elementary but powerful tool in the existence theory of solutions to initial value problems and for periodic boundary value problems, even in cases where no special structure is assumed on the nonlinearity. Starting with the pioneering work of Moretto [1] for locally Lipschitzian ordinary differential equations, the method of upper and lower solutions for the periodic boundary value problem
has been extended to the case of a continuous righthand side \(f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}\) (see, e.g., [2–6]).
In Franco et al. [6], the method of lower and upper solutions was applied to the periodic problem
and they also obtained a similar result to Lemma 1.1 below.
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 firstorder 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.
In order to explain the main results of the paper, let us introduce some notation: \(C[0,T]\) stands for the set of functions continuous on \([0,T]\). For \(x\in C[0,T]\), we denote
If \(x\in C[0,T]\), then we write \(x^{+}(t)=\max\{x(t),0\}\) and \(x^{}(t)=\max \{x(t),0\}\). For \(e\in C[0,T]\), we put
and note that \(E=E_{+}E_{}\).
Let us recall first some classical definitions and results. They are taken from [3].
Definition 1.1
A lower solution α (respectively, an upper solution β) of problem (1.1) is a function \(\alpha\in C^{1}[0,T]\) such that
(respectively, \(\beta\in C^{1}[0,T]\) and
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 firstorder periodic problem with strong singularity.
Construction of lower and upper solutions
In this section we consider
where \(f:[0,T]\times\mathbb{R}\rightarrow\mathbb{R}\) and \(e:[0, T]\rightarrow\mathbb{R}\) are continuous.
Let us consider an auxiliary boundary value problem
where \(\delta\in C[0,T]\) satisfies \(\bar{\delta}=0\) and \(\zeta\in \mathbb{R}\).
Clearly, for all \(\zeta\in\mathbb{R}\), problem (2.2) possesses a unique solution \(x_{\zeta}\) as follows:
Proposition 2.1
Assume that there are \(B_{1}\in\mathbb{R}\), \(c_{1}\in C[0,T]\) such that
where \(\eta_{1}(t)=\phi^{}(t)\phi^{+}(t)\frac{\Phi_{}}{\Phi_{+}}\) and \(\phi =c_{1}e\). If
then there exists an upper solution β of (2.1) such that
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).
Case 2. Assume that \(\Phi_{+}>0\). Using
it follows that there exists ω such that
Taking \(x_{0}=\frac{1}{\Phi_{+}}\) and
it is easy to see that \(\beta(0)=\beta(T)\). Since \(\int^{t}_{0} x_{0}[\phi^{+} (s)\Phi_{}\phi^{} (s)\Phi_{+}]\,ds=\int^{t}_{0} \eta _{1}(s)\,ds\leq \ \eta_{1}\_{1}\), choosing \(\zeta=B_{1}+\ \eta_{1}\_{1}\), we obtain
which means that (2.5) holds.
Now, using (2.4), it follows that \(\Phi_{+}\leq\Phi_{}\), implying that
From (2.3) and (2.5) we deduce that
and the proof is completed. □
Using similar arguments, one can prove the following proposition.
Proposition 2.2
Assume that there are \(B_{2}\in\mathbb {R}\), \(c_{2}\in C[0,T]\) such that
where \(\eta_{2}(t)=\phi^{}(t)\phi^{+}(t)\frac{\Phi_{}}{\Phi_{+}}\) and \(\phi =c_{2}e\). If
then there exists a lower solution α of (2.1) such that
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
Assume that there are \(B_{i}\in\mathbb{R}\), \(c_{i}\in C[0,T]\) such that
for all \(t\in[0,T]\), \(x\in[B_{i},B_{i}+2\ \eta_{i}\_{1}]\) and all \(i\in\{1,2\}\),
Then problem (2.1) possesses a solution x such that
(\(\eta_{1}\), \(\eta_{2}\) are given by Proposition 2.1 and Proposition 2.2, respectively).
Theorem 2.2
Assume that there are \(B_{1}, B_{2}\in\mathbb {R}\), \(c_{1}, c_{2}\in C[0,T]\) such that the assumptions of Theorem 2.1 are satisfied with
instead of (2.6). Then problem (2.1) possesses a solution x such that
Now, we consider problem (1.1), i.e., \(e\equiv0\).
Proposition 2.3
Assume that there are \(A_{1}\in\mathbb {R}\), \(b_{1}\in C[0,T]\) such that \(\bar{b}_{1}=0\) and
Then there exists a lower solution α of (1.1) such that
Proof
Problem (2.2) with \(\delta(t)=b_{1}(t)\) on \([0,T]\) has a unique solution
Since \(\ b_{1}\_{1}\leq\int^{t}_{0} b_{1}(s)\,ds\leq\ b_{1}\_{1}\), choosing \(\zeta=A_{1}+\ b_{1}\_{1} \), we obtain
Let us take \(\alpha(t)=x_{\zeta}(t)\), which means that (2.8) holds. According to (2.7) this yields
Furthermore, we have \(\alpha(0)=\alpha(T)\), by Definition 1.1, the function α is a lower solution of (1.1). □
The following assertion is due to Proposition 2.1 and its proof can be omitted.
Proposition 2.4
Assume that there are \(A_{2}\in\mathbb{R}\), \(b_{2}\in C[0,T]\) such that \(\bar{b}_{2}=0\) and
Then there exists an upper solution β of (1.1) such that
It is easy to see that the following results can be shown in a similar manner as in Theorem 2.1.
Theorem 2.3
Assume that there are \(A_{i}\in\mathbb{R}\), \(b_{i}\in C[0,T]\) such that \(\bar{b}_{i}=0\) and
for all \(t\in[0,T]\), \(x\in[A_{i},A_{i}+2\ b_{i}\_{1}]\) and all \(i\in\{1,2\}\),
Then problem (1.1) possesses a solution x such that
Theorem 2.4
Assume that there are \(A_{1}, A_{2}\in\mathbb {R}\), \(b_{1}, b_{2}\in C[0,T]\) such that the assumptions of Theorem 2.3 are satisfied with
instead of (2.10). Then problem (1.1) possesses a solution x such that
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
Assume that there exists a positive number M such that \(Mxf(t,x)\geq0\) for \(t\in[0,T]\) and \(x\geq0\). If

(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 \(f^{0}=\limsup_{x\rightarrow\infty} \max_{t\in[0,T]}\frac{f(t,x)}{x}<0\).
Then problem (1.1) has at least one positive solution.
In this paper, we do not require f to satisfy (i) or (ii). Now we give an example.
Example 2.1
Consider the periodic boundary value problem
It is easy to see that
So we can not obtain existence result by Theorem A. But if we choose \(b_{1}(t)=\frac{\pi}{3} t\frac{\pi}{6}\), compute that \(\bar{b}_{1}= \int^{1}_{0} \frac{\pi}{3} s\frac{\pi}{6}\,ds=0\), and (2.7) holds for \(A_{1}=\frac{\sqrt {3}}{3}+1\) and
By Proposition 2.3, problem (2.11) has a lower solution. In fact,
is a lower solution.
According to Proposition 2.4, we choose \(b_{2}(t)=\frac{\pi}{4}t\frac {\pi}{8}\), \(A_{2}=1\frac{\sqrt{3}}{3}\frac{\pi}{8}\), compute that \(A_{2}+2\ b_{2} \_{1}=1\frac{\sqrt{3}}{3}\) and (2.9) hold. Then problem (2.11) has an upper solution
Obviously, \(A_{2}+2\ b_{2} \_{1} < A_{1}\). Thus, from Theorem 2.2, problem (2.11) has a solution \(x(t)\) satisfying
i.e., x is a positive solution of (2.11).
Periodic problem with strong singularity
We will consider the following singular equation with periodic conditions:
where \(g\in C(0,\infty)\), \(e \in C[0,T] \) and g has strong singularity at 0, i.e.,
Theorem 3.1
Assume that (3.2) is fulfilled, there exists \(\tilde{A}_{1}\in(1,\infty)\) such that
where
Then problem (3.1) has a positive solution.
Proof
If \(g\in C(0,\infty)\) satisfies (3.2), then \(\limsup_{x\rightarrow0_{+}} g(x)=\infty\), which implies the existence of a sequence \(\{\varepsilon_{n}\}^{\infty}_{n=1}\subset(0,1)\) such that
Let \(e_{M}=\max_{t\in[0,T]}e(t)\). By (3.3), there exists N such that
We choose \(\varepsilon=\varepsilon_{N+1}\) such that \(g(\varepsilon)>e_{M} \). Let \(R\geq\tilde{B}_{1}\) be a constant. For the proof of Theorem 3.1, we deal with the auxiliary family of problems
where
Then \(\tilde{g}(x)\) fulfils the assumptions of Proposition 2.1 with \(c_{1}=\bar{e}\). Thus, by Proposition 2.1, problem (3.4) has an upper solution β such that
Since \(\varepsilon<\tilde{A}_{1}\leq\tilde{B}_{1}\leq R\), then \(\varepsilon <\beta\leq R\) is a lower solution of (3.1).
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
Consider the periodic boundary value problem with strong singularity
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 \(\widehat{\phi}=\bar {e}e=\frac{1}{2}t\).
We have
We can choose \(A_{1}=2\), then \(B_{1}=A_{1}+2\\eta\_{1}=\frac{5}{2}\) and g satisfies
By Theorem 3.1, problem (3.6) has a positive solution.
In fact, \(\alpha=\frac{1}{2}\) is a lower solution of (3.6).
is an upper solution of (3.6).
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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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RM completed the main study, carried out the results of this article. LZ drafted the manuscript, checked the proofs and verified the calculation. All the authors read and approved the final manuscript.
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Ma, R., Zhang, L. Construction of lower and upper solutions for firstorder periodic problem. Bound Value Probl 2015, 190 (2015). https://doi.org/10.1186/s1366101504577
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DOI: https://doi.org/10.1186/s1366101504577
MSC
 34B10
 34B18
Keywords
 periodic boundary value problem
 singular problem
 lower and upper solutions