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# Existence and uniqueness of solutions for the second order periodic-integrable boundary value problem

*Boundary Value Problems*
**volume 2017**, Article number: 109 (2017)

## Abstract

This paper is mainly devoted to studying one kind of the second order differential equation. Under periodic-integrable boundary value condition, the existence of the solutions of this equation is discussed by the method of the operator theory and the Schauder fixed point theorem.

## Introduction and the main results

Recently, the existence of solutions of ordinary differential equation with the periodic-integral boundary value conditions has been studied in some articles [1–6]. In [7] existence and uniqueness of solutions of second order periodic-integrable boundary value problems are discussed by using the lemma on bilinear forms and Schauder’s fixed point theorem. In [8] Cong *et al.* obtained existence and uniqueness of periodic solutions for \((2n+1)\)th order differential equations. In [9] the existence of solutions has been presented for the following second order differential equation:

Based on the above work, the purpose of this paper is to study the following periodic-integrable boundary value problem of the second order differential equations (denoted as PIBVP for short):

where \(f:[0,2\pi]\times R^{2}\longrightarrow R\) is continuous. We need the solution of PIBVP (1). To this aim, we introduce the following four assumptions.

### Assumption *A*
_{1}

There exist two continuous functions \(a(t)\) and \(b(t)\), and a nonnegative constant \(M_{1}\), such that

for any \((t,x,y)\) with \(\vert x \vert \geq M_{1}\) and \((t,y)\in[0,2\pi]\times R\).

### Assumption *A*
_{2}

There exist two nonnegative constants \(M_{2}\) and \(M_{3}\) such that

for any \((t,x)\in[0,2\pi]\times R\) whereas \(\vert y \vert \geq M_{3}\).

### Assumption *A*
_{3}

There exist two continuous functions \(\alpha(t)\) and \(\beta(t)\) such that

for any \((t,x,y)\in[0,2\pi]\times R^{2}\).

### Assumption *A*
_{4}

There exists a positive integer *M*, such that, for all \(t \in[0,2\pi]\) and \((x,y)\in R^{2} \),

We can now state our two main results by the following theorems.

### Theorem 1

*If Assumptions*
\(A_{1}\)
*and*
\(A_{2}\)
*hold*, *then the PIBVP* (1) *has at least one solution*.

### Theorem 2

*If Assumptions*
\(A_{3}\)
*and*
\(A_{4}\)
*hold*, *then the PIBVP* (1) *has a unique solution*.

In Section 2, we introduce two lemmas which will be used in later sections. In Section 3, the linear problem will be discussed by the theory of ordinary differential equation, thus the uniqueness of solutions of linear equations is proved. In Sections 4 and 5, we apply the conclusions in Sections 2 and 3 and Schauder’s fixed point theorem to proving Theorems 1 and 2. In Section 6, as applications of the main results, we introduce two examples.

## Preliminary

Let us first state some lemmas which will be used in the proof of the main results.

### Lemma 1

*Let*
\(x(t)\)
*be a continuous and differentiable function*, *and*

*Then*

### Proof

Expand \(x(t)\) as a Fourier series and substitute the expressions into the integrals. Thus, the proof is completed. □

Define

From Assumption
*A*
_{1}
, we have \(a\leq g_{0}\leq b\) for all \((t,x,y)\in[0,2\pi]\times R^{2}\). Let

Denote

It is easy to see

Likewise, we define

From Assumption
*A*
_{2}
and (9), we have \(\vert g_{1} \vert \leq 2M_{2}\) for all \((t,x,y)\in[0,2\pi]\times R^{2}\). Let

Denote

It is obvious that

From (11), we conclude

From the above steps, we can deduce the following lemma.

### Lemma 2

*The function*
*f*
*is denoted by*
\(f(t,x,y) = h_{1}(t,x,y)+xg_{0}(t,x,y)+yg_{1}(t,x,y)\), *whereas*
\(\vert h_{1}(t,x,y) \vert \leq4\sup\limits_{0\leq t\leq2\pi, \vert x \vert \leq M_{1}, \vert y \vert \leq M_{3}} \vert f(t,x,y) \vert \), \(a\leq g_{0}\leq b\)
*and*
\(\vert g_{1} \vert \leq2M_{2}\).

## Linear equation

Consider the following linear periodic-integrable boundary value problem:

where \(\hat{h_{1}}, \hat{g_{0}}\) and \(\hat{g_{1}}\) satisfy the inequalities in Lemma 2. Furthermore, we consider the corresponding homogeneous linear equation.

### Lemma 3

*If*
\(\hat{g_{0}}(t)\geq0\)
*and*
\(\hat{g_{0}}(t)\not\equiv0\)
*on*
\([0,{2\pi}]\), *then the following problem*:

*has only a trivial solution*.

### Proof

Assume that there exists a nontrivial solution \(x(t)\), that is, \(x(t)\neq0\). From the assumption of \(\hat{g_{0}}(t)\), we known that \(x(t)\) is not constant. So we assert that there exist \(t_{0}\) and \(t_{1}\), such that \(t_{0}< t_{1}\), and we have

Now we prove it. There are two cases:

Case 1. \(x(0)=\eta<0\). Let \(t_{0} =\operatorname{inf}\{{t| t\in[0,{2\pi}]}\mbox{ and } x(t)=0\}\), which implies that \(x(t_{0})=0 \mbox{ and } x'(t_{0})>0\). Define \(t_{\star} =\operatorname{inf}\{{t | t\in(t_{0},{2\pi}]}\mbox{ and }x(t)=0\}\). If \(t_{\star}= t_{0}\), then there will exist the sequences \(\{t^{i}\}\), \(x(t^{i})= 0\) as \(t^{i}\rightarrow t_{0}\) \((i\rightarrow\infty)\). By Rolle’s theorem, there is a number \(\xi^{i}\) in \([t^{i-1},t^{i}]\), such that \(x'(\xi^{i})= 0\), meanwhile \(\xi^{i}\rightarrow t_{0} \), so \(x'(t_{0})= 0\), a contradiction. Therefore \(t_{0}\) is the first zero point and \(t_{\star}\) is the next zero point. By the periodic-integral boundary conditions, there exists \(t_{1}\in[t_{0},t_{\star}]\), such that \(x(t)>0\), for \(t\in(t_{0},t_{1})\), \(x'(t_{1})= 0\).

Case 2. \(x(0)=\eta>0\). By the linear property of the problem, \(-x(t)\) is also a solution. Thus, the case is translated into Case 1.

Multiplying both sides of (14) by \(\operatorname{exp}\{- {\int_{t_{0}}^{t}}{\widehat{g}_{1}(s)\,ds} \}\) and integrating from \(t_{0} \) and \(t_{1}\), we derive

which leads to a contradiction. This proof of Lemma 3 is completed. □

### Lemma 4

*Problem* (13) *has a unique solution*.

### Proof

Let \(x_{1}(t)\) and \(x_{2}(t)\) be two linear independent solutions of the linear homogeneous equation \(x''=\hat{g_{1}}(t)x'+\hat{g_{0}}(t)x\), and \(x=c_{1}x_{1}(t)+c_{2}x_{2}(t)\) is its general solution. Then, by the PIBVP condition, we have

By Lemma 3, Problem (14) has only a trivial solution, which implies

Assume that \(x_{*}(t)\) is a special solution of equation \(x''=\hat{g_{1}}(t)x'+\hat{g_{0}}(t)x+\hat{h_{1}}(t)\), and \(x=c_{3}x_{1}(t)+c_{4}x_{2}(t)+x_{*}(t)\) is its general solution. By the PIBVP condition, we have

Constants \(c_{3}\) and \(c_{4}\) are unique because of (15) and (16), and therefore Problem (13) has only one solution. The proof is completed. □

## The proof of Theorem 1

In this section, we will investigate the existence of the solution of Theorem 1 by the Schauder fixed point theorem. Define

with the norm \(\Vert \bullet \Vert \) defined as follows:

It is clear that \(\mathcal{\mathcal{C}}\) is a Banach space.

Applying Lemma 2, for any \({x}\in\mathcal{\mathcal{C}}\), consider

Define the linear operator \(\overline{P}:\mathcal{\mathcal{C}}\rightarrow\mathcal{\mathcal{C}}\). For each \({x}\in \mathcal{\mathcal{C}}\), \(\overline{P}[x](t)=y(t)\) is a solution of (17). Thus the existence of the solution of (1) is equivalent to the existence of the fixed point of *P̅* in Banach space \(\mathcal{C}\). We will prove that *P̅* is continuous and compact, and \(\overline{P}(\mathcal{\mathcal{C}})\) is a bounded subset of \(\mathcal{\mathcal{C}}\). The proof is divided into three steps.

Step 1: *P̅* is continuous. For given any convergent sequence \(\{{x}_{k}\}\subset\mathcal{\mathcal{C}}\), we have \({x}_{k}\rightarrow{x}_{0}\) as \(k\rightarrow\infty\). Let \({y}_{k}=\overline{P}{x}_{k}\), then

We assert that \(\{{y}_{k}\}\) is the bounded sequence in \(\mathcal{\mathcal{C}}\). Otherwise, there exists a subsequence of \(\{{y}_{k_{j}}\}\), such that \(\Vert {y}_{k_{j}} \Vert \rightarrow \infty\) as \(j\rightarrow\infty\). Let \(\omega_{k_{j}}= \frac{{y}_{k_{j}}}{ \Vert {y}_{k_{j}} \Vert }\). For \(\{\omega_{k_{j}}\}\subset \mathcal{\mathcal{C}}\), then \(\Vert \omega_{k_{j}} \Vert =1\). By Lemma 2 we have

So \(\Vert \omega''_{k_{j}} \Vert \leq 2M_{2}+\max\limits_{t\in[0,{2\pi}]}{b}(t)+C<\infty\), where *C* is a constant. Thus \(\{\omega''_{k_{j}}\}\) is bounded. Obviously,

Hence, \(\{\omega'_{k_{j}}\}\) and \(\{\omega_{k_{j}}\}\) are both uniformly family bounded degree of equicontinuous functions. By the Ascoli-Arzela theorem, \(\{\omega_{k_{j}}\}\) and \(\{ \omega'_{k_{j}}\}\) contain a uniformly convergent subsequence, respectively. For convenience, we use the same notation and we have

Let \(j\rightarrow\infty\). From (21) and (22), we obtain

Hence,

By Lemma 3, we can conclude that \(\omega_{0}\equiv0\), and conflicts with \(\Vert \omega_{0} \Vert =1\).

Hence, from (18), we derive \(\{{y''_{k}}\}\) is bounded. So \(\{ y_{k}\}\) and \(\{y'_{k}\}\) are both uniformly family bounded degree of equicontinuous functions. By the Ascoli-Arzela theorem, \(\{y_{k}\}\) and \(\{y'_{k}\}\) contain a uniformly convergent subsequence, respectively. For the sake of convenience, we use the same notation, such that

Thus,

Let \(k\rightarrow\infty\). From (23) and (24), we obtain

Hence, by the uniqueness we know \({y}_{0}=\overline{P}{x}_{0}\). Thus the operator *T* is continuous.

Step 2: *P̅* is compact. For any bounded set \(S\subset{\mathcal{C}}\), we assert that \(\overline{P}(S)\) is the bounded set in \({\mathcal{C}}\). If not, similar to the proof of step 1, we will be led to a contradiction. For any \({x}\in{S}\), \(y=\overline{P}x\) is defined by (17). Because \(\vert {y'} \vert \), \(\vert {y} \vert \), \(\vert {f}_{x} \vert \) and \(\vert {f}_{x'} \vert \) are all bounded, proceeding as the proof of step 1, we show that \(\{y_{k}\}\) and \(\{y'_{k}\}\) are both uniformly family bounded degree of equicontinuous. By the Ascoli-Arzela theorem, *P̅* is a compact operator.

Step 3: \(\overline{P}(\mathcal{C})\) is a bounded set. If not, there exists a subsequence \(\{x_{k}\},k=1,2,\ldots \) , such that \(\Vert \overline{P}(x_{k}) \Vert \rightarrow\infty\) as \(k\rightarrow\infty\). Let \(y_{k}=\overline{P}x_{k}\), and Problem (4.2) holds. Let \(\omega_{k}= \frac{{y}_{k}}{ \Vert {y}_{k} \Vert }\), then \(\Vert \omega_{k} \Vert =1\) for \(\{\omega_{k}\}\subset{\mathcal{C}}\), and (19), (20), (21) and (22) hold. From step 1, we know \(\{\omega_{k}\}\) and \(\{\omega'_{k}\}\) are both uniformly family bounded degree of equicontinuous functions and contain a uniformly convergent subsequence, respectively. For the sake of convenience, we use the same notation, such that

The sequences \(g_{0}(t,x_{k},x'_{k})\) and \(g_{1}(t,x_{k},x'_{k})\) are both bounded set in \(L^{2}[0,2\pi] \) and contain a weakly convergent subsequence, respectively, such that

Obviously, as \(k\rightarrow\infty\),

From (23) and (24), for a.e. \(t\in{[0,2\pi]}\), we have

Hence,

We obtain \(\omega_{0}\equiv0\), this contradicts \(\Vert \omega_{0} \Vert =1\). Then there exists a constant \(K>0\), such that \(\Vert \overline{P}x \Vert \leq{K}\), where \({x}\in{\mathcal{C}}\).

Let \(E=\{{x}\in{\mathcal{C}}\vert \Vert x \Vert \leq{K}\}\). By the fixed point theorem, \(\overline{P}:E\rightarrow E\) has at least one fixed point and thus the PIBVP (1) has at least one solution. The proof of Theorem 1 is completed.

## The proof of Theorem 2

Firstly, we consider the uniqueness of the solutions of Theorem 2. Let \(x_{1}(t)\) and \(x_{2}(t)\) be any two solutions of the PIBVP (1), then \(u(t)=x_{2}(t)-x_{1}(t)\) is a solution of the PIBVP.

Here \({0}\leq\theta_{1}\leq {1}\), \({0}\leq\theta_{2}\leq{1} \). According to Assumption
*A*
_{3}
, we know

Hence, by Lemma 3, \(u(t)\equiv0\) on \([0,2\pi]\), that is, \(x_{1}(t)=x_{2}(t)\).

Next, we will prove the existence of Theorem 2 by the Schauder fixed point theorem. According to the integral mean value theorem, we rewrite the equation of PIBVP (1) in the equivalent form

By Lemma 4, the following problem (25) has a unique solution for any \({x}\in {\mathcal{C}}\):

Define the linear operator \({T}:{\mathcal{C}}\rightarrow{\mathcal{C}}\). For each \({x}\in {\mathcal{C}}\), \({T}[x](t)=y(t)\) is the unique solution of (25). Thus, the existence of the solution of Problem (1) is equivalent to the existence of the fixed point of *T* in Banach space \(\mathcal{C}\). We will prove that *T* is continuous and compact, and \({T({\mathcal{C}})}\) is a bounded subset in \({\mathcal{C}}\).

Step 1: *T* is continuous. Given any convergent sequence \(\{{x}_{j}\}\subset{\mathcal{C}}\), such that \({x}_{j}\rightarrow{x}_{0}\) as \(j\rightarrow\infty\). Let \({y}_{j}={T}{x}_{j}\), then

We will prove the existence of \({y}_{0}\), such that \({y}_{j}\rightarrow{y}_{0}\) as \(j\rightarrow\infty\), and

We assert that \(\{{y}_{j}\}\) is the bounded sequence in \({\mathcal{C}}\). If not, there exists a subsequence of \(\{{y}_{j}\}\). For the sake of convenience, this subsequence is still expressed as \(\{{y}_{j}\}\), such that \(\Vert {y}_{j} \Vert \rightarrow\infty\), as \(j\rightarrow\infty\). Take \(\omega_{j}= \frac{{y}_{j}}{ \Vert {y}_{j} \Vert }\). Then \(\Vert \omega_{j} \Vert =1\) for \(\{\omega_{j}\}\subset{\mathcal {C}}\). We have

So \(\Vert \omega''_{j} \Vert \leq M+\max\limits_{t\in[0,{2\pi}]}{\beta}(t)+1<\infty\). Thus \(\{\omega''_{j}\}\) is bounded. It is easy to see that \(\{\omega'_{j}\}\) and \(\{\omega_{j}\}\) are both uniformly family bounded degree of equicontinuous functions, and

By the Ascoli-Arzela theorem, \(\{\omega'_{j}\}\) and \(\{\omega_{j}\}\) contain a uniformly convergent subsequence, respectively, and satisfy

Obviously \(\omega_{0}\) and \(\nu_{0}\in{\mathcal{C}}\). Let \(j\rightarrow \infty\). From (27) and (28), we obtain

By Lemma 4, \(\omega_{0}\equiv0\), this contradicts \(\Vert \omega_{0} \Vert =1\).

By (26), we derive that \(\{y''_{j}\}\) is bounded. So \(\{y_{j}\}\) and \(\{y'_{j}\}\) are both uniformly family bounded degree of equicontinuous functions. By the Ascoli-Arzela theorem, \(\{y_{j}\}\) and \(\{y'_{j}\}\) contain a uniformly convergent subsequence, respectively. For the sake of convenience, we use the same notation, thus

We know

Let \(j\rightarrow\infty\). From (30) and (31), we obtain

Hence, by the uniqueness we know \({y}_{0}={T}{x}_{0}\). Thus, operator *T* is continuous.

Step 2: *T* is compact. For any bounded set \(S\subset{\mathcal{C}}\), we assert that \({T}(S)\) is the bounded set in \({\mathcal{C}}\). If not, similar to the proof of step 1, we are led to a contradiction. For any \({x}\in{S}\), \(y={T}x\) is defined by (25). Because \(\vert {y'} \vert \), \(\vert {y} \vert \), \(\vert {f}_{x} \vert \), and \(\vert {f}_{x'} \vert \) are all bounded, and then \(\Vert {y'} \Vert <\infty\). Proceeding as in the proof of step 1, we show that \(\{y_{j}\}\) and \(\{y'_{j}\}\) are both uniformly family bounded degree of equicontinuous. By the Ascoli-Arzela theorem, *T* is a compact operator.

Step 3: \({T}(\mathcal{C})\) is a bounded set. If not, there exists a subsequence \(\{x_{j}\},j=1,2,\ldots \) , such that \(\Vert {T}(x_{j}) \Vert \rightarrow\infty\) as \(j\rightarrow\infty\). Let \(y_{j}={T}x_{j}\), and Problem (26) holds. Take \(\omega_{j}= \frac{{y}_{j}}{ \Vert {y}_{j} \Vert }\), then \(\Vert \omega_{j} \Vert =1\) for \(\{\omega_{j}\}\subset{\mathcal{C}}\), and (27), (28), (29) and (30) hold. From step 1, we know \(\{\omega_{j}\}\) and \(\{\omega'_{j}\}\) are both uniformly family bounded degree of equicontinuous functions and they contain a uniformly convergent subsequence, respectively. For the sake of convenience, we use the same notation, such that

The sequences \(\{\int_{0}^{1}f_{x'}(t,x_{j},\theta_{1}x'_{j})\,d\theta _{1}\}_{k=1}^{\infty}\) and \(\{\int_{0}^{1}f_{x}(t,\theta _{2}x_{j},0)\,d\theta_{2}\}_{k=1}^{\infty}\) are both bounded in \(L^{2}[0,2\pi] \) and contain a weakly convergence subsequence, respectively, such that

Obviously,

Moreover,

Let \(j\rightarrow\infty\). From (32) and (33), for a.e. \(t\in{[0,2\pi]}\), we have

Hence

We obtain \(\omega_{0}\equiv0\); this contradicts \(\Vert \omega_{0} \Vert =1\). Then there exists a constant \(K>0\), such that \(\Vert {T}x \Vert \leq{K}\) as \({x}\in{\mathcal{C}}\).

Let \(\overline{E}=\{{x}\in{\mathcal{C}} \vert \Vert x \Vert \leq{K}\}\). By the fixed point theorem, \({T}:\overline{E}\rightarrow\overline{E}\) has one fixed point. The proof of Theorem 2 is completed.

## Examples

In this section, to illustrate significance and effectiveness of the results, we introduce two examples.

### Example 1

Consider the PIBVP as follows:

where

It is clear that *f* is continuous on \([0,2\pi]\times R^{2}\). For \(\vert x \vert \geq1\), we have

Notice that \(x\sin^{2} \frac{1}{x}\) is a bounded function, that is, \(\vert x\sin^{2} \frac{1}{x} \vert \leq M\), where *M* is a positive constant, for any \(x\in R \). When \(\vert y \vert \geq1\), for all \(t\in[0,2\pi]\), we get

According to (35) and (36), we derive that *f* satisfies Assumptions
*A*
_{1}
and
*A*
_{2}
. By Theorem 1, the PIBVP (34) has at least one solution.

### Example 2

Consider the following PIBVP:

Let \(f(t,x,y)=\operatorname{sin}y\operatorname{sin}x+x(t^{2}+1)\). Because

and

we prove that *f* suits Assumptions
*A*
_{3}
and
*A*
_{4}
. By Theorem 2, the PIBVP (37) has a unique solution.

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## Acknowledgements

The authors express their thanks to the referee for their useful comments. The research of F. Cong was partially supported by NFSC Grant (11171350).

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Feng, X., Cong, F. Existence and uniqueness of solutions for the second order periodic-integrable boundary value problem.
*Bound Value Probl* **2017, **109 (2017). https://doi.org/10.1186/s13661-017-0840-7

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### Keywords

- periodic-integrable boundary value
- existence and uniqueness
- Schauder’s fixed point theorem