1. Introduction
Impulsive systems turned out to be the most effective tools for describing many evolutionary progresses that experience instantaneous changes of state at certain moments. Pioneering studies on impulsive differential equations and their dynamics are detailed in [
1], establishing a fundamental theoretical framework for impulsive systems. Additionally, a comprehensive analysis of the system properties within impulsive systems is provided in the monographs [
2]. Beyond the theory [
3,
4,
5,
6,
7,
8,
9], impulsive systems are widely used in biological systems, control systems, ecological systems, and neural network systems [
10,
11,
12]. As a result of these successful applications in various fields, impulsive differential equations have attracted considerable attention.
The theory of differential equations with causal operators is experiencing an important development because it is a framework richer than the corresponding theory of ordinary differential equations. A causal operator is a non-anticipative operator and has been adopted from the engineering literature; some results have been introduced in the monograph [
13]. Recently, various types of causal differential equations have been studied widely, such as ordinal differential equations [
14,
15], functional differential equations [
16], differential equations in Banach spaces [
17], difference equations [
18], and integral differential equations [
19]. In addition, Jabeen et al. [
20] investigated impulsive differential equations with causal operators and gave the existence of an optimal solution for the control problem. Inspired by the above results, the aim of this paper is to study the impulsive differential equations further enriched by the causal operators, while subject to nonlinear periodic boundary conditions. Therefore, we discuss the following impulsive differential equations involving a causal operator with nonlinear periodic boundary conditions:
where
,
E is a real separable Banach space of continuous functions from
to the set of real numbers
,
is a causal operator,
,
,
, and
, where
and
denote the right and left limits of
at
.
One point of interest of our study lies in the fact that the periodic boundary conditions are nonlinear and encompass the usual linear boundary conditions (such as initial, periodic, and anti-periodic) and other general conditions, such as
and
= C (C is a constant). In addition, note that the impulsive differential equations (
1) reduce to ordinary differential equations for
, which has been studied in [
14], initial boundary problems for
, which has been studied in [
20], and other differential equations, such as
, which has been studied in [
21]. Thus, our work includes more types of differential equations.
In this paper, we extend the notion of causal operators to the nonlinear periodic boundary value problems for impulsive differential equations. The rest of this paper is organized as follows.
Section 2 establishes a new differential inequality.
Section 3 presents the existence of extremal solutions, following the definition of upper and lower solutions.
Section 4 gives the existence of extremal quasi-solutions after the definition of coupled lower and upper solutions. In
Section 5, a weakly coupled extremal quasi-solution is studied, which is dependent on the introduction of weakly coupled lower and upper solutions. In addition, examples are added in each section to verify the theoretical results.
2. Preliminaries
Let
denote the set of real valued continuous functions and
. Let us introduce the space:
Clearly, , are Banach spaces with the following respective norms:
, .
Let
. A function
is called a solution of (
1) if it satisfies (
1).
Definition 1. An operator is a causal operator if for all , with , such that for every , we have .
Lemma 1. Suppose that satisfieswhere , , , and is a positive linear operator, i.e., wherever , satisfiesThen for . Proof. If the assumption does not hold, two cases arise:
Case 1: There exists such that and for all .
Then, from (
2), we have
for
and
, hence
is nonincreasing on
J. If
,
shows that
(
c is a constant), then
. Notice that
, we have
, which is a contradiction. For
, it follows that
, presenting another contradiction.
Case 2: There exist and , such that and .
Given , , and for a specific , there is a , such that or . We only focus on the case where , as the proof for the case is analogous.
Subcase (1): If
, from (
2), we have
and hence
which yields
which contradicts (
3).
Subcase (2): If
, from (
2), we get
and
Using
,
, we get
which is a contradiction. Then, we get
,
. The proof is complete. □
Next, we give the following linear problems and lemmas, which help to validate our main results.
where
,
,
,
,
.
Lemma 2. A function is a solution to (4) if and only if satisfies the following integral equation:where , , are constants with and Proof. Assume
is a solution to (
4); we have
It is easy to obtain the following formula:
Since
, we obtain
We see that if
is a solution to (
4), then
is also a solution to (
5). The proof is complete. □
Apparently, . In the remainder of the paper, we denote .
Lemma 3. Let be a positive linear operator; , and Then, problem (4) has a unique solution. Proof. For any
, let us define
and
, as defined in Lemma
4. For any
, we have
Hence, by the Banach contraction principle, there exists a function
such that
. Apparently,
x is also the unique solution to (
4), thus concluding the proof. □
Remark 1. If , , is a positive linear operator, and Then, problem (4) has a unique solution. 4. Extremal Quasi-Solutions of Problem (1)
This part aims to verify that the monotone iterative technique remains applicable. It will be used for establishing the existence of extremal quasi-solutions.
Definition 2. Functions are called coupled lower and upper solutions of (1) ifand Definition 3. Functions are called quasi-solutions of (1) ifand Definition 4. A paired quasi-solution of problems (1) is said to be the minimal and maximal quasi-solution of (1) if . If both the minimal and maximal solutions are present, they are referred to as the extremal quasi-solutions of problem (1). Theorem 2. Assume that , , (3) and (6) hold. Moreover, is a causal operator, and are coupled lower and upper solutions of (1) with ; there exist , such that , is nonincreasing in the first variable, satisfying Then, problem (1) has extremal quasi-solutions within the internal . Proof. Let
and
for
, where
,
.
It follows from Lemma 3 that both (
9) and (
10) have a unique solution, respectively. Now, we conclude the proof through three steps.
Step 1: One proves that and , .
Set
. Employing
, we have
and
From Lemma 1 and , we get , so .
Utilizing the induction, we deduce that the sequence is monotonically nondecreasing. Analogously, one finds that is monotonically nonincreasing.
Step 2: We demonstrate that if .
Let
. Using
and
, we get
and
Subsequently, by Lemma 1, one has , which indicates .
Next, we will prove that
are coupled lower and upper solutions for (
1). Using the assumptions
,
, and
,
, we obtain
It is evident that
is a coupled lower solution to (
1). Analogously,
is a coupled upper solution to (
1). Using the induction, one has
,
.
Step 3: Based on the above two steps, it can be seen that
and each
satisfy (
9) and (
10). Apparently,
are uniformly bounded and equicontinuous, using the Ascoli–Arzela theorem and passing to the limit when
; we find that
satisfy the following equations
and
It proves that
are the extremal quasi-solutions of problem (
1). This ends the proof. □
Example 2. Setting Clearly, is a coupled lower solution, and is a coupled upper solution with . It is easy to see that (3), (6), , , and hold with , , , , , , , . From Theorem (2), the problem (11) has extremal quasi-solutions within the interval . 6. Conclusions
This study extends the notion of casual operators to impulsive differential equations with nonlinear periodic boundary conditions. Under the assumption of the existence of (coupled or weakly coupled) upper and lower solutions, we applied the monotone iterative technique to prove the existence of extremal solutions, extremal quasi-solutions, and weakly coupled extremal quasi-solutions for impulsive differential equations.
To validate and illustrate the theoretical results obtained, we present three different examples that summarise the essence of our findings. These examples are carefully designed to demonstrate the applicability of our theoretical framework and the effectiveness of the monotone iterative technique in the context of nonlinear impulsive differential equations with causal operators. Each example is analyzed in detail, and the results are shown to be consistent with our theoretical investigation. Through these examples, we highlight the significance of our work in advancing the study of impulsive differential equations and providing new insights into their solution behaviour.