1. Introduction
In this paper, we discuss the existence of periodic solutions for the first-order neutral differential equation on time scales as follows:
where
is the
-derivative on
,
is the
-derivative on
),
,
is strictly increasing,
with
for all
,
c is continuously delta-differentiable,
is twice continuously delta-differentiable and
is the identity function on
;
and
are periodic functions with period
; in other words,
is continuous and periodic in the first variable
, i.e.,
. The means of
and the periodic time scale can be founded in the Preliminaries section.
When
, Equation (
1) changes into the following equation:
Many mathematical models can be characterized and described by Equation (
2). In particualr, when
, 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
-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.
3. Existence of Periodic Solutions
We give the following notations: ; the intervals and are defined similarly, with with the norm Obviously, let ; then, .
Throughout this paper, we need the following assumptions.
(H
) There are positive constants
and
such that
(H) for .
Lemma 5. Suppose (H) holds. If , then x is a solution of Equation (1) if and only ifwhere Proof. Let
be a solution of (1). Rewrite (1) as
Multiply both sides of (4) by
and then integrate from
to
t to obtain
and then
Dividing both sides of the above equation by
, we have
Consider the integral
in (5). By (H
), we have
Using the integration by the parts formula,
By Lemmas 1 and 2, we have
where
and
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 does not depend on . To use Lemma 4, let
and
, where
and
are positive constants with
. Rewrite (3) as the following operator equation:
where
are defined by the following.
Lemma 6. Let for and . Then, is a large contraction of the set if Proof. For all
and
, we have the following.
Thus,
which together with (9) results in
being a large contraction of set
. □
Lemma 7. Let for and . Then, is a large contraction of set if (9) holds.
Proof. For each
and
, we have the following.
Thus,
which together with (9) results in the fact that
is a large contraction of the set
. □
Lemma 8. Let for and . Then is a large contraction of set 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) and (H) hold. Then is continuous and maps into a compact subset of provided thatwhere . Proof. First, we show
. By (7), we have
Let
; by (12), we have
Hence,
. By
, we have
Note that (H
), we obtain the following.
So, for each
, by (10)–(12), we have
Hence,
. Next, we show that
is continuous. For each
, we have
where
Hence, we have
Let . For all , define . Then, for , we obtain . This proves that is continuous.
Finally, we show that
is compact on
. Let the sequence
. Due to the above proof,
. By (7), we have
Consequently, by the use of (H
) and (13) we obtain
where
Hence, sequence is uniformly bounded and equicontinuous. It follows by the Ascoli–Arzel theorem that a subsequence of converges uniformly to a continuous -periodic function. Thus, is continuous, and is a compact set. □
Lemma 10. Assume that (H) and (9) hold. Then, is a large contraction provided thatwhere . Proof. First, we show that
. For each
, it is clear that
is continuous by the continuity of
. In view of (3),
and (8), we have
Let
; note that
and (15). Then, the following is the case.
Hence,
and
. Furthermore, for each
, from (8), (9) and (14), we have the following.
Thus,
and
. Finally, we show that
is a large contraction with a unique fixed point in
. Using Lemmas 6–8, we have the following.
Due to (14), is a large contraction on . □
Theorem 1. Suppose (H), (H), (9)–(11) and (14) hold. Then, Equation (1) has a ω-periodic solution u in the subset provided that Proof. In view of Lemmas 9 and 10, we obtain that
is continuous and maps
into a compact subset of
, and
is a large contraction. From the proof of Lemmas 9 and 10, for each
, we have
In view of (16), we obtain the following:
and
. Then, all conditions of Lemma 4 hold. Hence, there exists a fixed point
such that
. Hence, Equation (
1) has a
-periodic solution in
. □
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. 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.