# Positive periodic solutions for third-order ordinary differential equations with delay

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## Abstract

*ω*-periodic solutions for third-order ordinary differential equation with delay

*ω*-periodic in

*t*, and \(\tau>0\) is a constant denoting the time delay. We show the existence of positive

*ω*-periodic solutions when \(0< M<(\frac{2\pi}{\sqrt {3}\omega})^{3}\) and

*f*satisfies some order conditions. The discussion is based on the theory of fixed point index.

## Keywords

Third-order differential equation Positive*ω*-periodic solution Positive cone Delay Fixed point index theory in cones

## MSC

34B18 34C25## 1 Introduction

*ω*-periodic solutions for the third-order ordinary differential equation with delay

*ω*-periodic in

*t*, \(\tau>0\) is a constant which denotes the time delay.

*p*-Laplacian boundary value problems. In recent years, the existence of positive periodic solutions for third-order ODEs has been studied by using fixed point theorems of cone mapping. By utilizing Krasnoselskii’s fixed point theorem in cones Feng [3] proved the existence and multiplicity of positive periodic solutions of the third-order equation

*ω*-periodic solutions for the fully third-order ODE

*ω*-periodic solutions of Eq. (1.3). However, all these works contain no delay terms, and few researchers consider the existence of positive periodic solutions for the third-order delayed Eq. (1.1). In this paper, we show the existence of positive

*ω*-periodic solutions for the third-order delayed Eq. (1.1) when \(0< M<(\frac{2\pi }{\sqrt{3}\omega})^{3}\) and

*f*satisfies some order conditions. The discussion is based on the theory of fixed point index.

The rest of this paper is organized as follows. In Sect. 2, we introduce some preliminary facts and establish the existence of *ω*-periodic solution for third-order linear differential equation with delay. In Sect. 3, we prove two existence theorems of positive *ω*-periodic solutions for the third-order delayed Eq. (1.1).

## 2 Preliminaries

Let \(C_{\omega}(\mathbb{R})\) denote the Banach space of all continuous *ω*-periodic function \(u(t)\) with norm \(\|u\|_{C}=\max_{0\leq t\leq\omega}|u(t)|\). We denote by \(C^{+}_{\omega}(\mathbb {R})\) the cone of positive functions in \(C_{\omega}(\mathbb{R})\).

### Lemma 1

([11], Lemma 2.2)

*Let*\(0< M<(\frac{2\pi}{\sqrt{3}\omega})^{3}\). *Then the solution* Ψ *of the BVP* (2.1) *is positive on*\([0,\omega]\).

*ω*-periodic solution of the linear third-order ordinary differential equation

### Lemma 2

([11], Lemma 2.1)

*Let*\(M >0\).

*Then for any*\(h\in C_{\omega}(\mathbb{R})\),

*the linear Eq*. (2.2)

*has a unique*

*ω*-

*periodic solution*\(u(t)\)

*expressed by*

*Moreover*, \(P: C_{\omega}(\mathbb{R})\rightarrow C_{\omega}(\mathbb {R})\)

*is a completely continuous linear operator*.

### Remark 1

By Lemma 1, \(\Psi(t)>0\) for every \(t\in[0,\omega]\) when \(0< M<(\frac{2\pi}{\sqrt{3}\omega})^{3}\). Combining this fact with Lemma 2 we have that \(P: C_{\omega}(\mathbb{R})\rightarrow C_{\omega}(\mathbb{R})\) is a positive operator when \(0< M<(\frac{2\pi }{\sqrt{3}\omega})^{3}\).

*ω*-periodic solution of the linear third-order ordinary differential equation

### Lemma 3

*Let*\(0< M<(\frac{2\pi}{\sqrt{3}\omega})^{3}\)

*and*\(0< M_{1}<M\).

*Then Eq*. (2.4)

*has a unique*

*ω*-

*periodic solution*\(u\in C_{\omega }(\mathbb{R})\)

*given by*

*where*\(B_{1}: C_{\omega}(\mathbb{R})\rightarrow C_{\omega}(\mathbb {R})\)

*is defined by*

### Proof

*ω*-periodic solution given by

*ω*-periodic solution of Eq. (2.4). This completes the proof of Lemma 3. □

*Q*is a positive operator.

### Lemma 4

*Let*\(0< M<(\frac{2\pi}{\sqrt{3}\omega})^{3}\)*and*\(0< M_{1}<\sigma ^{2}M\). *Then*\(Q: C_{\omega}(\mathbb{R})\rightarrow C_{\omega}(\mathbb {R})\)*is a positive operator*, *where**Q**is defined by* (2.8).

### Proof

*K*in \(C^{+}_{\omega}(\mathbb{R})\) by

### Lemma 5

*Let*\(0< M<(\frac{2\pi}{\sqrt{3}\omega})^{3}\)*and*\(0< M_{1}<\sigma ^{2}M\). *Then*\(Q: K\rightarrow K\)*is completely continuous*, *where**Q**is defined by* (2.8).

### Proof

Applying the fixed point index theory in cones to prove the existence of *ω*-periodic solutions of Eq. (1.1), we recall some concepts and conclusions on the fixed point index in [2, 4]. Let *E* be a Banach space, and let \(K\subset E\) be a closed convex cone in *E*. Assume that Ω is a bounded open subset of *E* with boundary *∂*Ω and \(K\cap\partial\Omega=\theta\), where *θ* denotes the zero element in *E*. Let \(A: K\cap\overline{\Omega }\rightarrow K\) be a completely continuous mapping. If \(Au\neq u\) for any \(u\in K\cap\partial\Omega\), then the fixed point index \(i(A, K\cap\Omega, K)\) is defined. If \(i(A, K\cap\Omega, K)\neq0\), then *A* has a fixed point in \(K\cap\Omega\). The following lemmas can be found in [4].

### Lemma 6

*Let*Ω

*be a bounded open subset of*

*E*

*with*\(\theta\in\Omega\),

*and let*\(A: K\cap\overline{\Omega}\rightarrow K\)

*be a completely continuous mapping*.

*If*

*then*\(i(A, K\cap\Omega, K)=1\).

### Lemma 7

*Let*Ω

*be a bounded open subset of*

*E*,

*and let*\(A: K\cap\overline{\Omega}\rightarrow K\)

*be a completely continuous mapping*.

*If there exists*\(e\in K\setminus \{\theta\}\)

*such that*

*then*\(i(A, K\cap\Omega, K)=0\).

## 3 Existence of positive periodic solutions

### Theorem 1

*Let*\(f(t,x,y): {\mathbb{R}}\times {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\rightarrow {\mathbb{R}}^{+}\)

*be continuous and*

*ω*-

*periodic in*

*t*.

*Suppose that*\(0< M<(\frac{2\pi}{\sqrt{3}\omega})^{3}\)

*and*

*f*

*satisfies the following conditions*:

- (H
_{1}) -
*there exist*\(a_{1}>0\)*and*\(a_{2}>0\)*with*\(a_{1}+a_{2}< M\)*and*\(\delta>0\)*such that*$$f(t,x,y)\leq a_{1}x+a_{2}y $$*for any*\(t\in\mathbb{R}\)*and*\(x,y\in[0,\delta]\); - (H
_{2}) -
*there exist*\(b_{1}>0\)*and*\(b_{2}>0\)*with*\(b_{1}+b_{2}>M\)*and*\(h_{0}\in C^{+}_{\omega}(\mathbb{R})\)*such that*$$f(t,x,y)\geq b_{1}x+b_{2}y-h_{0}(t) $$*for any*\(t\in\mathbb{R}\)*and*\(x,y\in\mathbb{R}^{+}\).

*Then Eq*. (1.1)

*has at least one positive*

*ω*-

*periodic solution*.

### Proof

_{1}) \(F:C^{+}_{\omega}(\mathbb{R})\rightarrow C^{+}_{\omega }(\mathbb{R})\) is bounded. If \(0< M_{1}<\sigma^{2}M\), then by Lemma 3 and 5 we get that \(A=Q\circ F: C^{+}_{\omega}(\mathbb{R})\rightarrow C^{+}_{\omega}(\mathbb{R})\) is completely continuous, where

*Q*is defined by (2.8). For any \(0< r< R<+\infty\), let

*A*has a fixed point in \(K\cap(\Omega_{R}\setminus \overline{\Omega}_{r})\) when

*r*is small enough and

*R*is large enough.

*K*is defined by (2.9). In fact, if there exist \(u_{0}\in K\cap\partial \Omega_{r}\) and \(0<\lambda_{0}\leq1\) such that

_{1}) we have

*ω*and using the periodicity of \(u_{0}\), we have

*A*satisfies the conditions of Lemma 6. By Lemma 6 we have

*R*is large enough. In fact, if there exist \(u_{1}\in K\cap\partial\Omega_{R}\) and \(\mu_{1}\geq0\) such that

_{2}) we have

*ω*and using the periodicity of \(u_{1}\), we have

*A*satisfies the conditions of Lemma 7 in \(\Omega_{R}\). By Lemma 7 we have

*A*has at least one fixed point in \(K\cap(\Omega_{R}\setminus \overline{\Omega}_{r})\), which is a positive

*ω*-periodic solution of Eq. (1.1). This completes the proof of Theorem 1. □

### Theorem 2

*Let*\(f(t,x,y): {\mathbb{R}}\times {\mathbb{R}}^{+}\times {\mathbb{R}}^{+}\rightarrow {\mathbb{R}}^{+}\)

*be continuous and*

*ω*-

*periodic in*

*t*.

*Suppose that*\(0< M<(\frac{2\pi}{\sqrt{3}\omega})^{3}\)

*and*

*f*

*satisfies the following conditions*:

- (H
_{3}) -
*there exist*\(b_{1}>0\)*and*\(b_{2}>0\)*with*\(b_{1}+b_{2}>M\)*and*\(\delta>0\)*such that*$$f(t,x,y)\geq b_{1}x+b_{2}y $$*for any*\(t\in\mathbb{R}\)*and*\(x,y\in[0,\delta]\); - (H
_{4}) -
*there exist*\(a_{1}>0\)*and*\(a_{2}>0\)*with*\(a_{1}+a_{2}< M\)*and*\(h_{1}\in C^{+}_{\omega}(\mathbb{R})\)*such that*$$f(t,x,y)\leq a_{1}x+a_{2}y+h_{1}(t) $$*for any*\(t\in\mathbb{R}\)*and*\(x,y\in\mathbb{R}^{+}\).

*Then Eq*. (1.1)

*has at least one positive*

*ω*-

*periodic solution*.

### Proof

Let \(F(u)(t)=f(t,u(t),u(t-\tau))+M_{1}u(t-\tau)\), and \(A=Q\circ F\). Then \(A: C^{+}_{\omega}(\mathbb{R})\rightarrow C^{+}_{\omega }(\mathbb{R})\) is completely continuous when \(0\leq M_{1}<\sigma^{2}M\). For any \(0< r< R<+\infty\), we prove that *A* has a fixed point in \(K\cap (\Omega_{R}\setminus \overline{\Omega}_{r})\) when *r* is small enough and *R* is large enough.

*A*and Lemma 3\(u_{0}\) satisfies

_{3}) we have

*ω*and using the periodicity of \(u_{0}\), we have

*A*satisfies the conditions of Lemma 7. By Lemma 7 we have

*A*satisfies the condition of Lemma 6 in \(K\cap\Omega_{R}\) when

*R*is large enough. In fact, if there exist \(u_{1}\in K\cap\partial\Omega_{R}\) and \(0<\lambda_{1}\leq1\) such that

_{4}) we have

*ω*and using the periodicity of \(u_{1}\), we have

*A*satisfies the conditions of Lemma 6. By Lemma 6 we have

*A*has at least one fixed point in \(K\cap(\Omega_{R}\setminus \overline{\Omega}_{r})\), which is a positive

*ω*-periodic solution of Eq. (1.1). This completes the proof of Theorem 2. □

## 4 Conclusion

In this paper, by utilizing the fixed point index in cones, we prove the existence of positive periodic solutions for the general third-order Eq. (1.1). The results are obtained in the case that *f* satisfies some order conditions. A similar method can be used to prove the existence of positive periodic solutions for other differential equations.

## Notes

### Authors’ contributions

Both authors contributed equally in writing this paper. Both authors read and approved the final manuscript.

### Funding

The research is supported by the National Natural Science Function of China (No. 11701457) and Gansu Technology Plan (No. 17JR5RA071).

### Competing interests

None of the authors has any competing interests in the manuscript.

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