Existence Results of Periodic Solutions to First-Order Neutral Differential Equations on Time Scales

Abstract

The purpose of this paper is to study the existence of periodic solutions for the first-order nonlinear neutral differential equation on time scales. Burton–Krasnoselskii’s fixed point theorem will be sufficiently general for application to the considered equation. An example has been carried out to show our results. It should be pointed out that the problem of periodic solutions is one of the current hot topics in the study of dynamic equations, which contains rich symmetry ideas and methods.

1. Introduction

In this paper, we discuss the existence of periodic solutions for the first-order neutral differential equation on time scales as follows:

$$x^{\Delta }\left(t\right)=-a\left(t\right)x\left(\sigma \left(t\right)\right)-x\left(t\right)\left[c\left(t\right)x^{\tilde{\Delta }}(t-\tau \left(t\right)))+f(t,x\left(t\right),x(t-\tau \left(t\right)))\right],t\in \mathbb{T}$$

(1)

where $x^{\Delta }$ is the $\Delta$-derivative on $\mathbb{T}$, $x^{\tilde{\Delta }}$ is the $\Delta$-derivative on $(id-\tau )(\mathbb{T}$), $\tau :\mathbb{T}\to \mathbb{R}$, $id-\tau :\mathbb{T}\to \mathbb{T}$ is strictly increasing, $a\in {\mathcal{R}}^{+}$ with $a\left(t\right)\le 0$ for all $t\in \mathbb{T}$, c is continuously delta-differentiable, $\tau$ is twice continuously delta-differentiable and $id$ is the identity function on $\mathbb{T}$; $a,c$ and $id-\tau$ are periodic functions with period $\omega >0$; in other words,

$$a(t+\omega )=a\left(t\right),c(t+\omega )=c\left(t\right),(id-\tau )(t+\omega )=(id-\tau )\left(t\right)\mathrm{for}t\in \mathbb{T},$$

$f(t,x,y)$ is continuous and periodic in the first variable $t\in \mathbb{T}$, i.e., $f(t+\omega ,x,y)=f(t,x,y)$. The means of $\mathbb{T},$ $x^{\Delta },x^{\tilde{\Delta }},$ $\mathcal{R},{\mathcal{R}}^{+}$ and the periodic time scale can be founded in the Preliminaries section.

When $\mathbb{T}=\mathbb{R}$, Equation (1) changes into the following equation:

$$x^{\prime }\left(t\right)=-a\left(t\right)x\left(t\right)-x\left(t\right)\left[c\left(t\right)x^{\prime }(t-\tau \left(t\right)))+f(t,x\left(t\right),x(t-\tau \left(t\right)))\right].$$

(2)

Many mathematical models can be characterized and described by Equation (2). In particualr, when $f(t,x(t),x(t-\tau \left(t\right)\left)\right)=\alpha \left(t\right)x\left(t\right)+\beta \left(t\right)x(t-\tau (t\left)\right)$, Equation (2) is a classic single-species population growth model with delay. In [1], Kuang systematically studied the application of delay equations in population dynamics systems and laid the foundation for a general theory for population dynamics systems such as the global analysis for autonomous and nonautonomous systems with various delays, the possible delay influence on the dynamics of the system, the long time coexistence of populations, the oscillatory aspects of the dynamics and so on. Li [2] considered the periodic solution problem of the neutral differential equation by means of a topological degree theory. Fang and Li [3] obtained the existence of a periodic solution for the neutral single-species population model. By applying the continuation theorem of the coincidence degree theorem, Lu [4] investigated the existence of positive periodic solutions for neutral functional differential equations with multiple deviating arguments. Furthermore, a large number of Lotka–Volterra models can also be characterized and described by Equation (2); for more details, see, e.g., [5,6,7,8] and the references cited therein.

In 1988, Hilger [9] took the lead in proposing a time scale theory, which greatly promotes the development of differential equations and dynamic systems, especially in studying continuous systems and discrete systems under a unified framework. The time scale theory is widely used, especially in the study of systems with both discrete and continuous characteristics. In [10], Khuddush, etc., studied infinitely many positive solutions for an iterative system of singular multi-point boundary value problems on time scales by means of Krasnoselskii’s fixed point theorem. Almost periodically positive solution problems for a SIR epidemic model with delays and saturated treatment on time scales have been discussed in [11]. In very recent years, Negi and Torra [12] introduced and studied $\Delta$-Choquet integrals on time scales, which is a special case of the Choquet integral on abstract fuzzy (non-additive) measure spaces. For more results about dynamic equations, see [13,14,15,16,17] and the references cited therein.

The methods for studying the existence of the periodic solution include fixed-point theorems, variational methods, coincidence degree theories, Yoshizawa-type theorem, Massera-type theorem, etc. In this paper, we will use Burton–Krasnoselskii’s fixed point for studying the existence of a periodic solution to Equation (1). Burton–Krasnoselskii’s fixed point theorem is a reformulated version of the classic Krasnoselskii’s fixed point theorem, which can be easily used to study the existence of periodic solutions. Motivated by the papers mentioned above, in this paper, we are devoted to investigating the existence of periodic solutions for Equation (1). The following sections are organized as follows: Section 2 gives some preliminaries. The existence of the periodic solutions of Equation (1) is obtained in Section 3. In Section 4, an example is given to show the feasibility of our results. Section 5 concludes this article with a summary of our results.

2. Preliminaries

A time scale $\mathbb{T}$ is a closed subset of $\mathbb{R}.$ We give some notations, which their means can be founded in [18] as follows: the forward jump $\sigma \left(t\right)$, backward jump operator $\rho \left(t\right)$, the forward graininess $\mu \left(t\right)$, the set of all regressive rd-continuous functions $\mathcal{R}$ (or ${\mathcal{R}}^{+}$) and the exponential function $e_p(t,s)$. The following Lemmas and Definitions have important applications in this paper.

Lemma 1

([18]). Suppose that $g:\mathbb{T}\to \mathbb{R}$ is a strict function and $\tilde{\mathbb{T}}=g\left(\mathbb{T}\right)$ is a time scale. Let $f:\tilde{\mathbb{T}}\to \mathbb{R}$. If $g^{\Delta }\left(t\right)$ and $f^{\tilde{\Delta }}\left(t\right)$ exist for $t\in {\mathbb{T}}^k$, then

$${(f\circ g)}^{\Delta }=(f^{\tilde{\Delta }}\circ g)g^{\Delta }.$$

Lemma 2

([18]). Suppose that $g:\mathbb{T}\to \mathbb{R}$ is a strict function and $\tilde{\mathbb{T}}=g\left(\mathbb{T}\right)$ is a time scale. If $f:\mathbb{T}\to \mathbb{R}$ is $rd$-continuous function and g is differentiable with $rd$-continuous derivative, then for $a,b\in \mathbb{T},$

$${\int }_a^{b}f\left(t\right)g^{\Delta }\left(t\right)\Delta t={\int }_{g\left(a\right)}^{g\left(b\right)}(f\circ g^{-1})\left(s\right)\tilde{\Delta }s.$$

Lemma 3

([18]). Let $p,q\in \mathcal{R}.$ Then, the following is the case.

[i] 

$e_0(t,s)\equiv 1$ and $e_p(t,t)\equiv 1;$

[ii] 

$e_p(\rho \left(t\right),s)=(1-\mu \left(t\right)p\left(t\right))e_p(t,s);$

[iii] 

$e_p(t,s)=\frac{1}{e_p(s,t)}=e_{\ominus p}(s,t);$

[iv] 

$e_p(t,s)e_p(s,r)=e_p(t,r);$

[v] 

$e_p(t,s)e_q(t,s)=e_{p\oplus q}(t,s);$

[vi] 

$e_p^{\Delta }(·,s)=p{e}_p(·,s)$.

Remark 1.

In view of [19], $e_p(t,s)$ has the following properties of inequality. If $p\in {\mathcal{R}}^{+}$, then

$$0<e_p(t,s)\le exp\left({\int }_s^{t}p\left(\tau \right)\Delta \tau \right)\mathrm{for}s,t\in \mathbb{T}.$$

For $s,t\in \mathbb{T}$ with $s\le t$, if $p\in {\mathcal{R}}^{+}$ and $p\le 0$, then

$$0<e_p(t,s)\le exp\left({\int }_s^{t}p\left(\tau \right)\Delta \tau \right)<1.$$

Definition 1

([20]). A time scale $\mathbb{T}$ is periodic if there exists $m>0$ such that if $t\in \mathbb{T}$ then $t\pm m\in \mathbb{T}$. For $\mathbb{T}\ne \mathbb{R}$, the smallest positive m is called the period of the time scale.

Definition 2

([20]). Let $\mathbb{T}\ne \mathbb{R}$ be a periodic time scale with the period m. The function $g:\mathbb{T}\to \mathbb{R}$ is periodic with period ω if there exists a natural number n such that $\omega =nm,g(t\pm \omega )=g\left(t\right)$ for all $t\in \mathbb{T}$. When $\mathbb{T}=\mathbb{R}$, g is a periodic function if ω is the smallest positive number such that $g(t\pm \omega )=g\left(t\right)$.

Remark 2

([20]). If $\mathbb{T}$ is a periodic scale with period m, then $\sigma (t\pm nm)=\sigma \left(t\right)\pm nm$ and $\mu (t\pm nm)=\mu \left(t\right)$ for $t\in \mathbb{T}$.

Definition 3.

(Large Contraction). Ref. [21]. Let $(\mathcal{M},d)$ be a metric space and $G:\mathcal{M}\to \mathcal{M}$ is said to be a large contraction if for $\varphi ,\psi \in \mathcal{M}$ with $\varphi \ne \psi$, then $d(\mathcal{M}\varphi ,\mathcal{M}\psi )\le d(\varphi ,\psi )$. and if for all $\epsilon >0$ and $\varphi ,\psi \in \mathcal{M}$, there exists a $0<\delta <1$ such that

$$d(\varphi ,\psi )\ge \epsilon \Rightarrow d(\mathcal{M}\varphi ,\mathcal{M}\psi )\le \delta d(\varphi ,\psi ).$$

Lemma 4.

(Burton–Krasnoselskii). Ref. [21]. Let $\mathcal{M}$ be a bounded convex nonempty subset of a Banach space B. Suppose that $A_1$ and $A_2$ map $\mathcal{M}$ into B such that the following is the case:

(i) 

$x,y\in \mathcal{M}$ implies $A_{1}x+A_{2}y\in \mathcal{M}$;

(ii) 

$A_1$ is continuous and $A_1\mathcal{M}$ is contained in a compact subset of $\mathcal{M}$;

(iii) 

$A_2$ is a large contraction mapping.

3. Existence of Periodic Solutions

We give the following notations: ${[a,b]}_{\mathbb{T}}=\{t\in \mathbb{T},a\le t\le b\}$; the intervals ${[a,b)}_{\mathbb{T}},{(a,b]}_{\mathbb{T}}$ and ${[a,b]}_{\mathbb{T}}$ are defined similarly, with $C_{\omega }=\{x\in C(\mathbb{T},\mathbb{R}):x(t+\omega )=x\left(t\right)\}$ with the norm $\left|\right|x\left|\right|={max}_{t\in {[0,\omega ]}_{\mathbb{T}}}\left|x\left(t\right)\right|.$ Obviously, let $x\in C_{\omega }$; then, $\left|\right|x\left|\right|=\left|\right|x^{\sigma }\left|\right|$.

Throughout this paper, we need the following assumptions.

(H${}_1$) There are positive constants $k_1,k_2$ and $k_3$ such that

$$|f(t,x_1,y_1)-f(t,x_2,y_2)|\le k_1|x_1-x_2|+k_2|y_1-y_2|\mathrm{for}\mathrm{all}{x}_1,x_2,y_1,y_2\in \mathbb{R},t\in \mathbb{T},$$

$$\left|f(t,x,y)\right|\le k_3\mathrm{for}\mathrm{all}x,y\in \mathbb{R},t\in \mathbb{T}.$$

(H${}_2$) ${\tau }^{\Delta }\left(t\right)\ne 1$ for $t\in {[0,\omega ]}_{\mathbb{T}}$.

Lemma 5.

Suppose (H${}_2$) holds. If $x\in C_{\omega }$, then x is a solution of Equation (1) if and only if

$$\begin{array}{cc}\hfill x\left(t\right)& =\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^{t}x\left(s\right)f(s,x\left(s\right),x(s-\tau \left(s\right)))e_{\ominus a}(t,s)\Delta s\hfill \\ & -x(t-\tau \left(t\right)))\frac{x\left(t\right)c\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}-\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t{h}_1\left(s\right)x\left(s\right)x^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t,s)\Delta s\hfill \\ & -\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t{h}_2\left(s\right)x^{\Delta }\left(s\right)x^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t,s)\Delta s\hfill \\ & -\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t{h}_3\left(s\right)x\left(\sigma \left(s\right)\right)x^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t,s)\Delta s,\hfill \end{array}$$

(3)

where

$$h_1\left(s\right)=\frac{[c^{\Delta }\left(s\right)(1-{\tau }^{\Delta }\left(s\right))+c\left(s\right){\tau }^{\Delta \Delta }\left(s\right))]}{[1-{\tau }^{\Delta }\left(s\right)][1-{\tau }^{\Delta }\left(\sigma \left(s\right)\right)]},$$

$$h_2\left(s\right)=\frac{c\left(\sigma \right(s\left)\right))}{1-{\tau }^{\Delta }\left(\sigma \left(s\right)\right)},$$

$$h_3\left(s\right)=\frac{a\left(s\right)c\left(\sigma \right(s\left)\right))}{1-{\tau }^{\Delta }\left(\sigma \left(s\right)\right)}.$$

Proof. 

Let $x\in C_{\omega }$ be a solution of (1). Rewrite (1) as

$$x^{\Delta }\left(t\right)+a\left(t\right)x\left(\sigma \left(t\right)\right)=-x\left(t\right)\left[c\left(t\right)x^{\tilde{\Delta }}(t-\tau \left(t\right)))+f(t,x\left(t\right),x(t-\tau \left(t\right)))\right].$$

(4)

Multiply both sides of (4) by $e_a(t,0)$ and then integrate from $t-\omega$ to t to obtain

$$\begin{array}{cc}\hfill {\int }_{t-\omega }^t{\left(e_a(s,0)x\left(s\right)\right)}^{\Delta }\Delta s& =-{\int }_{t-\omega }^{t}x\left(s\right)c\left(s\right)x^{\tilde{\Delta }}(s-\tau \left(s\right)))e_a(s,0)\Delta s\hfill \\ & -{\int }_{t-\omega }^{t}x\left(s\right)f(s,x\left(s\right),x(s-\tau \left(s\right)))e_a(s,0)\Delta s\hfill \end{array}$$

and then

$$\begin{array}{cc}\hfill e_a(t,0)x\left(t\right)-e_a(t-\omega ,0)x(t-\omega )& =-{\int }_{t-\omega }^{t}x\left(s\right)c\left(s\right)x^{\tilde{\Delta }}(s-\tau \left(s\right)))e_a(s,0)\Delta s\hfill \\ & -{\int }_{t-\omega }^{t}x\left(s\right)f(s,x\left(s\right),x(s-\tau \left(s\right)))e_a(s,0)\Delta s.\hfill \end{array}$$

Dividing both sides of the above equation by $e_a(t,0)$, we have

$$\begin{array}{cc}\hfill \left(e_{\ominus a}(t,t-\omega )-1\right)x\left(t\right)& ={\int }_{t-\omega }^{t}x\left(s\right)c\left(s\right)x^{\tilde{\Delta }}(s-\tau \left(s\right)))e_{\ominus a}(t,s)\Delta s\hfill \\ & +{\int }_{t-\omega }^{t}x\left(s\right)f(s,x\left(s\right),x(s-\tau \left(s\right)))e_{\ominus a}(t,s)\Delta s.\hfill \end{array}$$

(5)

Consider the integral ${\int }_{t-\omega }^{t}x\left(s\right)c\left(s\right)x^{\tilde{\Delta }}(s-\tau \left(s\right)))e_{\ominus a}(t,s)\Delta s$ in (5). By (H${}_2$), we have

$$\begin{array}{cc}& {\int }_{t-\omega }^{t}x\left(s\right)c\left(s\right)x^{\tilde{\Delta }}(s-\tau \left(s\right)))e_{\ominus a}(t,s)\Delta s\hfill \\ & ={\int }_{t-\omega }^t{x}^{\tilde{\Delta }}(s-\tau \left(s\right)))(1-{\tau }^{\Delta }\left(s\right))\frac{x\left(s\right)c\left(s\right)e_{\ominus a}(t,s)}{1-{\tau }^{\Delta }\left(s\right)}\Delta s.\hfill \end{array}$$

Using the integration by the parts formula,

$${\int }_{t-\omega }^t{f}^{\Delta }\left(s\right)g\left(s\right)\Delta s=\left(fg\right)\left(t\right)-\left(fg\right)(t-\omega )-{\int }_{t-\omega }^t{f}^{\sigma }\left(s\right)g^{\Delta }\left(s\right)\Delta s,$$

By Lemmas 1 and 2, we have

$$\begin{array}{cc}& {\int }_{t-\omega }^{t}x\left(s\right)c\left(s\right)x^{\tilde{\Delta }}(s-\tau \left(s\right)))e_{\ominus a}(t,s)\Delta s\hfill \\ & =x(t-\tau \left(t\right)))\frac{x\left(t\right)c\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}(1-e_{\ominus a}(t,t-\omega ))\hfill \\ & -{\int }_{t-\omega }^t{h}_1\left(s\right)x\left(s\right)x^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t,s)\Delta s-{\int }_{t-\omega }^t{h}_2\left(s\right)x^{\Delta }\left(s\right)x^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t,s)\Delta s\hfill \\ & -{\int }_{t-\omega }^t{h}_3\left(s\right)x\left(\sigma \left(s\right)\right)x^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t,s)\Delta s,\hfill \end{array}$$

(6)

where $h_1\left(s\right),h_2\left(s\right)$ and $h_3\left(s\right)$ are defined by (3). Substituting the right hand side of (6) into (5), we complete the proof. □

Remark 3.

From the proof of Lemma 4 in [22], we know that $e_{\ominus a}(t,t-\omega )-1$ does not depend on $t\in \mathbb{T}$.

To use Lemma 4, let $B=C_{\omega }$ and $\mathcal{M}=\{\varphi \in C_{\omega }:|\left|\varphi \right||\le L_1,|\left|{\varphi }^{\Delta }\right||\le L_2\}$, where $L_1$ and $L_2$ are positive constants with $L_2\le L_1$. Rewrite (3) as the following operator equation:

$$\varphi \left(t\right)=\left(A_1\varphi \right)\left(t\right)+\left(A_2\varphi \right)\left(t\right),t\in \mathbb{T},$$

where $A_1,A_2:\mathcal{M}\to C_{\omega }$ are defined by the following.

$$\left(A_1\varphi \right)\left(t\right)=\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t\varphi \left(s\right)f(s,\varphi \left(s\right),\varphi (s-\tau \left(s\right)))e_{\ominus a}(t,s)\Delta s-\varphi (t-\tau \left(t\right)))\frac{\varphi \left(t\right)c\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)},$$

(7)

$$\begin{array}{cc}\hfill \left(A_2\varphi \right)\left(t\right)& =-\frac{1}{e_{\ominus a}(t,t-\omega )-1}[{\int }_{t-\omega }^t{h}_1\left(s\right)\varphi \left(s\right){\varphi }^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t,s)\Delta s\hfill \\ & +{\int }_{t-\omega }^t{h}_2\left(s\right){\varphi }^{\Delta }\left(s\right){\varphi }^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t,s)\Delta s+{\int }_{t-\omega }^t{h}_3\left(s\right)\varphi \left(\sigma \left(s\right)\right){\varphi }^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t,s)\Delta s].\hfill \end{array}$$

(8)

Lemma 6.

Let $\left({\mathcal{B}}_1\varphi \right)\left(t\right)=\varphi \left(t\right){\varphi }^{\sigma }(t-\tau \left(t\right))$ for $\varphi \in \mathcal{M}$ and $t\in \mathbb{T}$. Then, ${\mathcal{B}}_1$ is a large contraction of the set $\mathcal{M}$ if

$$L_1<\frac{1}{2}.$$

(9)

Proof. 

For all $t\in \mathbb{T}$ and $\varphi ,\psi \in \mathcal{M}$, we have the following.

$$\begin{array}{cc}\hfill |\left({\mathcal{B}}_1\varphi \right)\left(t\right)-\left({\mathcal{B}}_1\psi \right)\left(t\right)|& =|\varphi \left(t\right){\varphi }^{\sigma }(t-\tau \left(t\right))-\psi \left(t\right){\psi }^{\sigma }(t-\tau \left(t\right))|\hfill \\ & \le \left|\varphi \left(t\right)\right|{\varphi }^{\sigma }(t-\tau \left(t\right))-{\psi }^{\sigma }(t-\tau \left(t\right))|+|{\psi }^{\sigma }(t-\tau \left(t\right))\left|\right|\varphi \left(t\right)-\psi \left(t\right)|\hfill \\ & \le 2L_1\left|\right|\varphi -\psi \left|\right|.\hfill \end{array}$$

Thus,

$$\left|\right|{\mathcal{B}}_1\varphi -{\mathcal{B}}_1\psi \left|\right|\le 2L_1\left|\right|\varphi -\psi \left|\right|$$

which together with (9) results in ${\mathcal{B}}_1$ being a large contraction of set $\mathcal{M}$. □

Lemma 7.

Let $\left({\mathcal{B}}_2\varphi \right)\left(t\right)={\varphi }^{\Delta }\left(t\right){\varphi }^{\sigma }(t-\tau \left(t\right))$ for $\varphi \in \mathcal{M}$ and $t\in \mathbb{T}$. Then, ${\mathcal{B}}_2$ is a large contraction of set $\mathcal{M}$ if (9) holds.

Proof. 

For each $t\in \mathbb{T}$ and $\varphi ,\psi \in \mathcal{M}$, we have the following.

$$\begin{array}{cc}\hfill |\left({\mathcal{B}}_2\varphi \right)\left(t\right)-\left({\mathcal{B}}_2\psi \right)\left(t\right)|& =|{\varphi }^{\Delta }\left(t\right){\varphi }^{\sigma }(t-\tau \left(t\right))-{\psi }^{\Delta }\left(t\right){\psi }^{\sigma }(t-\tau \left(t\right))|\hfill \\ & \le |{\varphi }^{\Delta }\left(t\right)|{\varphi }^{\sigma }(t-\tau \left(t\right))-{\psi }^{\sigma }(t-\tau \left(t\right))|+|{\psi }^{\sigma }(t-\tau \left(t\right))\left|\right|{\varphi }^{\Delta }\left(t\right)-{\psi }^{\Delta }\left(t\right)|\hfill \\ & \le 2L_1\left|\right|\varphi -\psi \left|\right|.\hfill \end{array}$$

Thus,

$$\left|\right|{\mathcal{B}}_2\varphi -{\mathcal{B}}_2\psi \left|\right|\le 2L_1\left|\right|\varphi -\psi \left|\right|$$

which together with (9) results in the fact that ${\mathcal{B}}_2$ is a large contraction of the set $\mathcal{M}$. □

Lemma 8.

Let $\left({\mathcal{B}}_3\varphi \right)\left(t\right)={\varphi }^{\sigma }\left(t\right){\varphi }^{\sigma }(t-\tau \left(t\right))$ for $\varphi \in \mathcal{M}$ and $t\in \mathbb{T}$. Then ${\mathcal{B}}_3$ is a large contraction of set $\mathcal{M}$ if (9) holds.

Proof. 

The proof of Lemma 8 is similar to the proof of Lemma 6; thus, we omit it. □

Lemma 9.

Assume that (H${}_1$) and (H${}_2$) hold. Then $A_1:\mathcal{M}\to \mathcal{M}$ is continuous and maps $\mathcal{M}$ into a compact subset of $\mathcal{M}$ provided that

$$k_1+k_2+\left|f(t,0,0)\right|\le L_1\ominus a\left(t\right)\mathrm{for}t\in \mathbb{T},$$

(10)

$$\alpha L_1+L_1\le 1,$$

(11)

where $\alpha ={max}_{t\in {[0,\omega ]}_{\mathbb{T}}}\left|\frac{c\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}\right|$.

Proof. 

First, we show $A_1:\mathcal{M}\to \mathcal{M}$. By (7), we have

$$\begin{array}{cc}\hfill \left(A_1\varphi \right)(t+\omega )& =\frac{1}{e_{\ominus a}(t+\omega ,t)-1}{\int }_t^{t+\omega }\varphi \left(s\right)f(s,\varphi \left(s\right),\varphi (s-\tau \left(s\right)))e_{\ominus a}(t+\omega ,s)\Delta s\hfill \\ & -\varphi (t+\omega -\tau (t+\omega )))\frac{\varphi (t+\omega )c(t+\omega )}{1-{\tau }^{\Delta }(t+\omega )}\hfill \\ & =\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_t^{t+\omega }\varphi \left(s\right)f(s,\varphi \left(s\right),\varphi (s-\tau \left(s\right)))e_{\ominus a}(t+\omega ,s)\Delta s\hfill \\ & -\varphi (t-\tau \left(t\right)))\frac{\varphi \left(t\right)c\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}.\hfill \end{array}$$

(12)

Let $u=s-\omega$; by (12), we have

$$\begin{array}{cc}\hfill \left(A_1\varphi \right)(t+\omega )& =\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t\varphi (u+\omega )f(u+\omega ,\varphi (u+\omega ),\varphi (u+\omega -\tau (u+\omega )))e_{\ominus a}(t+\omega ,u+\omega )\Delta u\hfill \\ & -\varphi (t-\tau \left(t\right)))\frac{\varphi \left(t\right)c\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}\hfill \\ & =\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t\varphi \left(s\right)f(s,\varphi \left(s\right),\varphi (s-\tau \left(s\right)))e_{\ominus a}(t,s)\Delta s\hfill \\ & -\varphi (t-\tau \left(t\right)))\frac{\varphi \left(t\right)c\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}\hfill \\ & =\left(A_1\varphi \right)\left(t\right).\hfill \end{array}$$

Hence, $A_1:C_{\omega }\to C_{\omega }$. By $a<0$, we have $\frac{1}{e_{\ominus a}(t,t-\omega )-1}>0.$ Note that (H${}_1$), we obtain the following.

$$\begin{array}{cc}\hfill \left|f\right(t,x,y\left)\right|& =\left|f\right(t,x,y)-f(t,0,0)+f(t,0,0\left)\right|\hfill \\ & \le k_1\left|\right|x\left|\right|+k_2\left|\right|y\left|\right|+\left|f(t,0,0)\right|.\hfill \end{array}$$

So, for each $\varphi \in \mathcal{M}$, by (10)–(12), we have

$$\begin{array}{cc}\hfill \left(A_1\varphi \right)\left(t\right)& \le \frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t\left|\varphi \left(s\right)f(s,\varphi \left(s\right),\varphi (s-\tau \left(s\right)))\right|e_{\ominus a}(t,s)\Delta s\hfill \\ & +\left|\varphi (t-\tau \left(t\right))\right)|\left|\frac{\varphi \left(t\right)c\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}\right|\hfill \\ & \le \frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t{L}_1(k_1{L}_1+k_2{L}_1+f(t,0,0))e_{\ominus a}(t,s)\Delta s+\alpha L_1^2\hfill \\ & \le L_1^2+\alpha L_1^2\hfill \\ & \le L_1.\hfill \end{array}$$

Hence, $A_1:\mathcal{M}\to \mathcal{M}$. Next, we show that $A_1$ is continuous. For each $\varphi ,\psi \in \mathcal{M}$, we have

$$\begin{array}{cc}& |\left(A_1\varphi \right)\left(t\right)-\left(A_1\psi \right)\left(t\right)|\hfill \\ & \le \frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t|\varphi \left(s\right)f(s,\varphi \left(s\right),\varphi (s-\tau \left(s\right)))-\psi \left(s\right)f(s,\psi \left(s\right),\psi (s-\tau \left(s\right)))|e_{\ominus a}(t,s)\Delta s\hfill \\ & +\left|\frac{c\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}\right||\varphi \left(t\right)\varphi (t-\tau \left(t\right))-\psi \left(t\right)\psi (t-\tau \left(t\right))|\hfill \\ & \le {\gamma }_1{\int }_{t-\omega }^t\left|\varphi \left(s\right)\right|f(s,\varphi \left(s\right),\varphi (s-\tau \left(s\right)))-f(s,\psi \left(s\right),\psi (s-\tau \left(s\right)))|\Delta s\hfill \\ & +{\gamma }_1{\int }_{t-\omega }^t\left|f(s,\psi \left(s\right),\psi (s-\tau \left(s\right)))\right||\varphi \left(s\right)-\psi (s-\tau \left(s\right))|\Delta s\hfill \\ & +\alpha \left|\varphi \right(t\left)\right|\left|\varphi \right(t-\tau \left(t\right))-\psi (t-\tau \left(t\right)\left)\right|+\alpha \left|\psi \right(t-\tau \left(t\right)\left)\right|\left|\varphi \right(t)-\psi (t\left)\right|\hfill \\ & \le {\gamma }_1{L}_1(k_1+k_2)\omega \left|\right|\varphi -\psi \left|\right|+{\gamma }_1{k}_3\omega \left|\right|\varphi -\psi \left|\right|\hfill \\ & =\left[{\gamma }_1{L}_1(k_1+k_2)\omega +{\gamma }_1{k}_3\omega \right]\left|\right|\varphi -\psi \left|\right|.\hfill \end{array}$$

where ${\gamma }_1={max}_{t\in {[0,\omega ]}_{\mathbb{T}}}\frac{1}{\ominus a\left(t\right)}.$ Hence, we have

$$\left|\right|A_1\varphi -A_1\psi \left|\right|\le \left[{\gamma }_1{L}_1(k_1+k_2)\omega +{\gamma }_1{k}_3\omega \right]\left|\right|\varphi -\psi \left|\right|.$$

Let $\rho ={\gamma }_1{L}_1(k_1+k_2)\omega +{\gamma }_1{k}_3\omega$. For all $\epsilon >0$, define $\vartheta =\frac{\epsilon }{\rho }$. Then, for $\left|\right|\varphi -\psi \left|\right|<\vartheta$, we obtain $\left|\right|A_1\varphi -A_1\psi \left|\right|\le \epsilon$. This proves that $A_1$ is continuous.

Finally, we show that $A_1$ is compact on $\mathcal{M}$. Let the sequence ${\varphi }_n\in \mathcal{M}$. Due to the above proof, $\left|\right|A_1{\varphi }_n\left|\right|\le L_1$. By (7), we have

$$\begin{array}{cc}\hfill {\left(A_1{\varphi }_n\right)}^{\Delta }\left(t\right)& =\frac{{\varphi }^{\sigma }(t-\tau \left(t\right)){\varphi }^{\sigma }\left(t\right)c^{\Delta }\left(t\right)+{\varphi }^{\Delta }(t-\tau \left(t\right))\varphi \left(t\right)c\left(t\right)+{\varphi }^{\sigma }(t-\tau \left(t\right))c^{\Delta }\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}\hfill \\ & +\frac{\varphi (t-\tau \left(t\right))\varphi \left(t\right)c\left(t\right){\tau }^{\Delta \Delta }\left(t\right)}{(1-{\tau }^{\Delta }\left(t\right))(1-{\tau }^{\Delta }\left(\sigma \left(t\right)\right))}\hfill \\ & +\frac{1}{e_{\ominus a}(t,t-\omega )-1}\varphi \left(t\right)f(t,\varphi \left(t\right),\varphi (t-\tau \left(t\right)))[e_{\ominus a}(\sigma \left(t\right),t)-e_{\ominus a}(\sigma \left(t\right)+\omega ,t)]\hfill \\ & +\frac{1}{e_{\ominus a}(t,t-\omega )-1}\ominus a\left(t\right){\int }_{\tilde{a}}^t\varphi \left(s\right)f(s,\varphi \left(s\right),\varphi (s-\tau \left(s\right)))e_{\ominus a}(t,s)\Delta s\hfill \\ & -\frac{1}{e_{\ominus a}(\tilde{a},t-\omega )-1}\ominus a\left(t\right){\int }_{\tilde{a}}^{t-\omega }\varphi \left(s\right)f(s,\varphi \left(s\right),\varphi (s-\tau \left(s\right)))e_{\ominus a}(t,s)\Delta s\hfill \\ & =\frac{{\varphi }^{\sigma }(t-\tau \left(t\right)){\varphi }^{\sigma }\left(t\right)c^{\Delta }\left(t\right)+{\varphi }^{\Delta }(t-\tau \left(t\right))\varphi \left(t\right)c\left(t\right)+{\varphi }^{\sigma }(t-\tau \left(t\right))c^{\Delta }\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}\hfill \\ & +\frac{\varphi (t-\tau \left(t\right))\varphi \left(t\right)c\left(t\right){\tau }^{\Delta \Delta }\left(t\right)}{(1-{\tau }^{\Delta }\left(t\right))(1-{\tau }^{\Delta }\left(\sigma \left(t\right)\right))}\hfill \\ & +\frac{1}{e_{\ominus a}(t,t-\omega )-1}\varphi \left(t\right)f(t,\varphi \left(t\right),\varphi (t-\tau \left(t\right)))[1-e_{\ominus a}(t,t-\omega )]\hfill \\ & +\ominus a\left(t\right)\left[\left(A_1{\varphi }_n\right)\left(t\right)+\varphi (t-\tau \left(t\right)))\frac{\varphi \left(t\right)c\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}\right].\hfill \end{array}$$

(13)

Consequently, by the use of (H${}_1$) and (13) we obtain

$$|{\left(A_1{\varphi }_n\right)}^{\Delta }\left(t\right)|\le L_1^2{\gamma }_2+L_1{L}_2\alpha +L_1{\gamma }_2+L_1^2{\gamma }_3+k_3{L}_1+{\gamma }_4(L_1+L_1^2\alpha ),$$

where

$${\gamma }_2=\underset{t\in {[0,\omega ]}_{\mathbb{T}}}{max}\left|\frac{c^{\Delta }\left(t\right)}{1-{\tau }^{\Delta }\left(t\right)}\right|,{\gamma }_3=\underset{t\in {[0,\omega ]}_{\mathbb{T}}}{max}\left|\frac{c\left(t\right){\tau }^{\Delta \Delta }\left(t\right)}{(1-{\tau }^{\Delta }\left(t\right))(1-{\tau }^{\Delta }\left(\sigma \left(t\right)\right))}\right|,{\gamma }_4=\underset{t\in {[0,\omega ]}_{\mathbb{T}}}{max}\ominus a\left(t\right).$$

Hence, sequence $A_1{\varphi }_n$ is uniformly bounded and equicontinuous. It follows by the Ascoli–Arzel$\stackrel{`}{a}$ theorem that a subsequence $A_1{\varphi }_{n_k}$ of $A_1{\varphi }_n$ converges uniformly to a continuous $\omega$-periodic function. Thus, $A_1$ is continuous, and $A_1\mathcal{M}$ is a compact set. □

Lemma 10.

Assume that (H${}_2$) and (9) hold. Then, $A_2:\mathcal{M}\to \mathcal{M}$ is a large contraction provided that

$$2L_1(h_1^M+h_2^M+h_3^M)<1,$$

(14)

where $h_i^M={max}_{t\in {[0,\omega ]}_{\mathbb{T}}}\left|h_i\left(t\right)\right|,i=1,2,3$.

Proof. 

First, we show that $A_2:\mathcal{M}\to \mathcal{M}$. For each $\varphi \in \mathcal{M}$, it is clear that $A_2\varphi$ is continuous by the continuity of $\varphi$. In view of (3), $h_1\left(t\right),h_2\left(t\right),h_3\left(t\right)\in C_{\omega }$ and (8), we have

$$\begin{array}{cc}\hfill \left(A_2\varphi \right)(t+\omega )& =-\frac{1}{e_{\ominus a}(t+\omega ,t)-1}[{\int }_t^{t+\omega }{h}_1\left(s\right)\varphi \left(s\right){\varphi }^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t+\omega ,s)\Delta s\hfill \\ & +{\int }_t^{t+\omega }{h}_2\left(s\right){\varphi }^{\Delta }\left(s\right){\varphi }^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t+\omega ,s)\Delta s\hfill \\ & +{\int }_t^{t+\omega }{h}_3\left(s\right)\varphi \left(\sigma \left(s\right)\right){\varphi }^{\sigma }(s-\tau \left(s\right))e_{\ominus a}(t+\omega ,s)\Delta s].\hfill \end{array}$$

(15)

Let $u=s-\omega$; note that $e_{\ominus a}(t+\omega ,u+\omega )=e_{\ominus a}(t,u),e_{\ominus a}(t+\omega ,t)=e_{\ominus a}(t,t-\omega ,),\sigma (u+\omega )=\sigma \left(u\right)+\omega$ and (15). Then, the following is the case.

$$\begin{array}{cc}\hfill \left(A_2\varphi \right)(t+\omega )& =-\frac{1}{e_{\ominus a}(t+\omega ,t)-1}[{\int }_t^{t+\omega }{h}_1(u+\omega )\varphi (u+\omega ){\varphi }^{\sigma }(u+\omega -\tau (u+\omega ))e_{\ominus a}(t+\omega ,u+\omega )\Delta u\hfill \\ & +{\int }_t^{t+\omega }{h}_2(u+\omega ){\varphi }^{\Delta }(u+\omega ){\varphi }^{\sigma }(u+\omega -\tau (u+\omega ))e_{\ominus a}(t+\omega ,u+\omega )\Delta u\hfill \\ & +{\int }_t^{t+\omega }{h}_3(u+\omega )\varphi \left(\sigma (u+\omega )\right){\varphi }^{\sigma }(u+\omega -\tau (u+\omega ))e_{\ominus a}(t+\omega ,u+\omega )\Delta u]\hfill \\ & =-\frac{1}{e_{\ominus a}(t,t-\omega )-1}[{\int }_{t-\omega }^t{h}_1\left(u\right)\varphi \left(u\right){\varphi }^{\sigma }(u-\tau \left(u\right))e_{\ominus a}(t,u)\Delta u\hfill \\ & +{\int }_{t-\omega }^t{h}_2\left(u\right){\varphi }^{\Delta }\left(u\right){\varphi }^{\sigma }(u-\tau \left(u\right))e_{\ominus a}(t,u)\Delta u\hfill \\ & +{\int }_{t-\omega }^t{h}_3\left(u\right)\varphi \left(\sigma \left(u\right)\right){\varphi }^{\sigma }(u-\tau \left(u\right))e_{\ominus a}(t,u)\Delta s].\hfill \end{array}$$

Hence, $\left(A_2\varphi \right)(t+\omega )=\left(A_2\varphi \right)\left(t\right)$ and $A_2:C_{\omega }\to C_{\omega }$. Furthermore, for each $\varphi \in \mathcal{M}$, from (8), (9) and (14), we have the following.

$$\begin{array}{cc}\hfill |\left(A_2\varphi \right)\left(t\right)|& \le h_1^M{L}_1^2\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t{e}_{\ominus a}(t,s)\Delta s\hfill \\ & +h_2^M{L}_1{L}_2\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t{e}_{\ominus a}(t,s)\Delta s+h_3^M{L}_1^2\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t{e}_{\ominus a}(t,s)\Delta s\hfill \\ & \le h_1^M{L}_1^2+h_2^M{L}_1^2+h_3^M{L}_1^2\hfill \\ & <2L_1<1.\hfill \end{array}$$

Thus, $A_2\varphi \in \mathcal{M}$ and $A_2:\mathcal{M}\to \mathcal{M}$. Finally, we show that $A_2$ is a large contraction with a unique fixed point in $\mathcal{M}$. Using Lemmas 6–8, we have the following.

$$\begin{array}{cc}\hfill |\left(A_2\varphi \right)\left(t\right)-\left(A_2\psi \right)\left(t\right)|& \le 2L_1{h}_1^M\left|\right|\varphi -\psi \left|\right|\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t{e}_{\ominus a}(t,s)\Delta s\hfill \\ & +2L_1{h}_2^M\left|\right|\varphi -\psi \left|\right|\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t{e}_{\ominus a}(t,s)\Delta s\hfill \\ & +2L_1{h}_3^M\left|\right|\varphi -\psi \left|\right|\frac{1}{e_{\ominus a}(t,t-\omega )-1}{\int }_{t-\omega }^t{e}_{\ominus a}(t,s)\Delta s\hfill \\ & =2L_1(h_1^M+h_2^M+h_3^M)\left|\right|\varphi -\psi \left|\right|.\hfill \end{array}$$

Due to (14), $A_2$ is a large contraction on $\mathcal{M}$. □

Theorem 1.

Suppose (H${}_1$), (H${}_2$), (9)–(11) and (14) hold. Then, Equation (1) has a ω-periodic solution u in the subset $\mathcal{M}$ provided that

$$L_1(1+\alpha +h_1^M+h_2^M+h_3^M)<1.$$

(16)

Proof. 

In view of Lemmas 9 and 10, we obtain that $A_1:\mathcal{M}\to \mathcal{M}$ is continuous and maps $\mathcal{M}$ into a compact subset of $\mathcal{M}$, and $A_2:\mathcal{M}\to \mathcal{M}$ is a large contraction. From the proof of Lemmas 9 and 10, for each $\varphi ,\psi \in \mathcal{M}$, we have

$$\begin{array}{cc}\hfill \left|\right|A_1\varphi +A_2\psi \left|\right|& \le \left|\right|A_1\varphi \left|\right|+\left|\right|A_2\psi \left|\right|\hfill \\ & \le L_1^2+\alpha L_1^2+(\alpha +h_1^M+h_2^M+h_3^M)L_1^2\hfill \\ & =(1+\alpha +h_1^M+h_2^M+h_3^M)L_1^2.\hfill \end{array}$$

In view of (16), we obtain the following:

$$\left|\right|A_1\varphi +A_2\psi \left|\right|\le L_1$$

and $A_1\varphi +A_2\psi \in \mathcal{M}$. Then, all conditions of Lemma 4 hold. Hence, there exists a fixed point $u\in \mathcal{M}$ such that $u=A_{1}u+A_{2}u$. Hence, Equation (1) has a $\omega$-periodic solution in $\mathcal{M}$. □

Remark 4.

In this paper, we are devoted to studying the existence of periodic solutions for Equation (1) by using Burton–Krasnoselskii’s fixed point theorem. Using the above theorem, we can only obtain the existence of the solution but not the unique existence of the solution. In fact, there are few results for the unique existence of the solution to dynamic equations. The existence and uniqueness of solutions for a initial value problem on time scales are studied; see Theorems 8.18 and 8.20 in [18]. We will study the uniqueness of the periodic solutions to Equation (1) in future work.

4. Example

When $\mathbb{T}=\mathbb{R}$, consider the the following equation:

$$x^{\prime }\left(t\right)=-(2-sint)x\left(t\right)-x\left(t\right)\left[0.5x^{\prime }(t-cos\frac{1}{3}(t+0.5\pi ))+0.01cosx\left(t\right)+0.01sinx(t-cos\frac{1}{3}(t+0.5\pi ))\right],$$

(17)

where

$$a\left(t\right)=-(2-sint)<0,c\left(t\right)=0.5,\tau \left(t\right)=cos\frac{1}{3}(t+0.5\pi ),$$

$$f(t,x,x(t-\tau \left(t\right)))=0.01cosx\left(t\right)+0.01sinx(t-cos\frac{1}{3}(t+0.5\pi )).$$

When $t\in [0,2\pi ]$, it follows that ${\tau }^{\prime }\left(t\right)\ne 1,k_1=k_2=0.01$ and $k_3=0.02$. Thus, assumptions (H${}_1$) and (H${}_2$) hold. Choose $L_1=0.1$, by a simple calculation, the following is obtained.

$$\alpha =\underset{t\in {[0,\omega ]}_{\mathbb{T}}}{max}\left|\frac{c\left(t\right)}{1-{\tau }^{\prime }\left(t\right)}\right|=6,h_1^M=0,h_2^M=0.4,h_3^M=1.8,$$

$$k_1+k_2+\left|f(t,0,0)\right|=0.03\le L_1\ominus a\left(t\right)=0.2-0.1sint,$$

$$\alpha L_1+L_1=0.7<1,$$

$$2L_1(h_1^M+h_2^M+h_3^M)=0.44<1,$$

$$L_1(1+\alpha +h_1^M+h_2^M+h_3^M)=0.92<1.$$

Hence, all conditions of Theorem 1 hold, and Equation (17) has a periodic solution.

5. Conclusions and Discussions

In the past few years, the study of dynamic equations on time scales attracted the persistent research interest of scholars because it unifies discrete analysis and continuous analysis. In this paper, the existence of a periodic solution for first-order nonlinear neutral differential equations on time scales is studied by applying Burton–Krasnoselskii’s theorem. It is important to point out that the existence conditions of the periodic solution in the present paper are easy to verify. The main contributions of our study are as follows: (1) We study a more general dynamic equation, including some classic equations, such as a single-specie population model and the Lotka–Volterra model, etc. (2) Due to the wide-ranging nature of dynamic equations on time scales, our results are novel for ordinary differential equations, difference equations, and discrete and continuous equations.

Burton–Krasnoselskii’s fixed point theorem (see [21,23]) constitutes a basis for our main results, which is a reformulated version of Krasnoselskii’s fixed point theorem. Obviously, if we want to apply the above theorem, we need to construct two mappings: One is large contraction, and the other is compact. Many nonlinear real systems from biology and physics, etc., can be easily studied by using Burton–Krasnoselskii’s fixed point theorem. Furthermore, we can only obtain the existence of the solution but not the uniqueness of the solution. We will study the uniqueness of periodic solutions relative to Equation (1) in future work. The methods of this paper can be extended to investigate other types of dynamic equations on time scales such as stochastic differential equations, impulsive differential equations, partial differential equations, fractional differential equations, etc. The above dynamic equations on time scales are the focus of our future research work.