1. Introduction
In actuarial science, Volterra integral equations are used in ruin theory, which analyzes the risk of insolvency, and several researchers have directed their interest to equations containing Volterra terms, see for instance [
1,
2,
3].
In the applied sciences, neutral equations play an important role. A positive periodic solution for two neutral functional differential equations was investigated by Luo et al. [
4] using Krasnoselskii’s fixed-point theorem. Various mathematical ecological and population models are covered by these functional differential equations, including hematopoietic models [
5,
6], the models of Nicholson’s blowflies [
7,
8] and the models of blood-cell production [
9].
The properties of exponential dichotomies and trichotomies have been studied by many researchers over the last decades due to their importance in the theory of differential and integro-differential equations. Some fundamental works on the theory of periodic solutions related to this subject are here [
10,
11,
12,
13,
14,
15,
16,
17,
18,
19].
The differential system, including many delay terms (Sa Ngiamsunthorn [
20]), has been studied for the periodicity of solutions under an integrable dichotomy. Similar systems have been studied in [
21,
22] under an exponential-type condition.
Motivated by the abovementioned references, a periodic solution to the following nonlinear neutral differential equations interested us:
in which
,
, and
are
-periodic continuous functions on
,
.
and
are
-periodic continuous matrix functions with respect to
defined on
such that
is nonsingular.
This paper is arranged as follows: In the second part of our paper, we provide definitions and fixed-point theorems for integrable dichotomies and previous results. In
Section 3, we establish some criteria for determining whether periodic solutions of the system (1) exist and whether they are unique.
Section 4 illustrates the main results with an example. Finally, we end the paper with a conclusion.
2. Preliminaries
Several results and definitions of integrable dichotomies are presented in this section and will be crucial for proving our results, see [
13,
14].
First, we recall some basic facts about integrable dichotomies. Consider the system
where
is
continuous matrix function defined on
. Let
be the fundamental matrix of (2) that satisfies
.
Let
be the set of continuous and bounded functions. Let
P be the projection matrix and
, the Green matrix associated with
P given by
Definition 1 ([
14] Definition 1, p. 3).
We say that the linear system (2) has an integrable dichotomy if there exists a constant and a projection matrix P such that the associated Green matrix satisfies Hence, under an integrable dichotomy condition, we consider the nonhomogeneous linear system
and we need the following results. See for instance [
14].
Proposition 1 ([
14] Proposition 1, p. 4).
Assume that there is an integrable dichotomy of (2). Then the trivial solution is the only bounded solution to (2). Proposition 2 ([
14] Proposition 2, p. 4).
Suppose that there is an integrable dichotomy for the homogeneous system (2). If , then system (3) has a bounded unique solution . Furthermore, Proposition 3 ([
14] Propositions 4 and 5, p. 5).
Assume that there is an integrable dichotomy for the system (2) such that is bounded. If , then is also Υ-periodic. Furthermore, if is Υ-periodic, then (3) has a unique periodic solution satisfying (4). Our objective is to demonstrate the existence and uniqueness of periodic solutions for system (1) by using the following fixed point theorems; see [
2,
23].
Theorem 1 (
Banach)
. For any complete metric space and . If there is a constant such that for ,then there a unique point with . Theorem 2 (Krasnoselskii). Let Λ be a nonempty bounded, convex and closed subset of a Banach space Y. Suppose that and map Λ into Y such that
is a contraction mapping on Λ,
is completely continuous on Λ, and
, gives .
Then there is , which satisfies .
Let a constant
and denote
it is easily seen that
is a nonempty-bounded, convex and closed subset of
.
Assume that for all
, there exists
such that the function
Q satisfies
3. Main Results
Under the conditions stated previously, we will show in this section the existence and the uniqueness of the solution for (1), which can then be written as
where
By Proposition 2, system (1) holds the integral equation
which is
We define the operators
and
by
and
for
. Clearly, if the operator
has a fixed point, then it is a periodic solution of (1).
Lemma 1. The operators and defined by (8) and (9) are, respectively, from Λ into ; that is, .
Proof. Secondly, for
, we get
Since all quantities in and are periodic, then . □
Lemma 2. The operator given by (8) is a contraction if .
Proof. Therefore, is a contraction. □
Lemma 3. If we assume (5) holds, then the operator is completely continuous.
Proof. We need to prove
is continuous, so let
such that
as
with
n as a positive integer. Hence, by condition (5) we have
Therefore, we have
and by the Dominated Convergence Theorem, we conclude the continuity of
.
Next we will prove that the image of the operator
is relatively compact. For any
, by (10) we have
A simple calculation of
gives
Hence, is equicontinuous and uniformly bounded. According to the Ascoli–Arzela theorem is relatively compact. □
In the following Lemma, we prove for any that .
Lemma 4. For any , we have since (5) and (6) hold.
Proof. Let
. Then
. By conditions (5) and (6), we have
it follows that
for all
. Hence
. □
Theorem 3. Assume that there is an integrable dichotomy for the system , and suppose conditions (5) and (6) hold. Then (1) has a Υ-periodic solution.
Proof. The assumptions in the Krasnoselskii theorem are satisfied by Lemmas 1–4, so there exists a fixed point , which is a solution of (1) such that . Hence, (1) has a -periodic solution. □
Theorem 4. Assume that system has an integrable dichotomy. Ifthen system (1) has a unique Υ-periodic solution. Proof. Consider the operator
as the following
For
, we obtain
Since (11) holds, then is contraction. Therefore, system (1) has a unique -periodic solution. □
4. An Example
Consider the system
with
,
and
Then, we have
,
,
,
,
, and we use
to get
Then and .
For , Theorem 3 give us -periodic solution not necessarily unique for the system (12).
Now, if , then condition (11) holds, and Theorem 3 gives us the uniqueness of a periodic solution for the system (12).
5. Conclusions
This manuscript dealt with the study of neutral systems containing Volterra terms. The notable thing is that there is no explicit term that we use for an integrable dichotomy, so the kernel has been assumed to be nonsingular for this purpose. The fixed point theorems of Banach and Krasnoselskii played a pivotal role in proving the existence and uniqueness of periodic solutions.
By the analysis in this paper, our work generalized some previous papers such as [
24].
Author Contributions
Methodology, M.B.M.; formal analysis, M.B.M.; investigation, M.B.M.; writing—original draft preparation, M.B.M.; writing—review and editing, M.B.M.; visualization, M.B.M.; funding acquisition, M.B.M.; funding acquisition, M.A.; writing—review and editing, M.A.; supervision, W.W.M.; project administration, W.W.M.; writing—review and editing, W.W.M. All authors have read and agreed to the published version of the manuscript.
Funding
This research received no external funding.
Data Availability Statement
Not applicable.
Acknowledgments
The authors are thankful to the Deanship of Scientific Research at University of Bisha for supporting this work through the Fast-Track Research Support Program.
Conflicts of Interest
The authors declare no conflict of interest.
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