Abstract

In this work, we examine a class of nonlinear neutral differential equations. Krasnoselskii’s fixed-point theorem is used to provide sufficient conditions for the existence of positive periodic solutions to this type of problem.

1. Introduction

In recent years, differential equations have garnered considerable interest (cf. [1, 2] and references therein). Important types of these problems include differential equations with delay. For instance, in [1, 3–10], the authors employed a variety of techniques to determine the existence of positive periodic solutions. The uniqueness and positivity of a first-order nonlinear periodic differential equation are investigated in [11]. The authors of [12] discussed nearly periodic solutions to nonlinear Duffing equations. Among them, the fixed-point principle has established itself as a critical tool for studying the existence and periodicity of positive solutions. Numerous studies, including [4, 6, 11], examined this method.

In this work, we investigate the following fourth-order nonlinear neutral differential equation:

Under the assumptions: (i)(ii) and (iii) are -periodic in , is a positive constant

Krasnoselskii’s fixed-point theorem offers sufficient conditions for the existence of positive periodic solutions to the aforesaid problem.

Neutral differential equations are employed in various technological and natural science applications. For example, they are widely employed to investigate distributed networks with lossless transmission lines (see [7]). Therefore, their qualitative qualities are significant.

It is worth noting that Krasnoselskii’s fixed-point theorem was proposed in 2012 in [4] to show the existence of positive periodic solutions to the nonlinear neutral differential equation with variable delay of the form

The same researchers evaluated the existence of positive periodic solutions for two types of second-order nonlinear neutral differential equations with variable delay the following year in [5]. where Krasnoselskii’s fixed-point theorem is also used as a tool. The authors of [10] investigated the following third-order nonlinear neutral differential equations with variable delay.

The existence of positive periodic solutions is demonstrated using Krasnoselskii’s fixed-point theorem. In [3], the authors investigated the fourth-order nonlinear neutral differential equations with variable delay of the form

Krasnoselskii’s fixed-point theorem is used to derive some sufficient conditions for the existence of positive periodic solutions to the aforementioned problem.

The remainder of this paper is organized as follows: in the next Section, we deliver the definitions and lemmas required to prove our main results. In particular, we state some Green’s function properties related to the problem (1). Section 3 establishes some necessary conditions for the existence of positive solutions to our problem (1).

2. Preliminaries

For a fixed , we consider a set of continuous scalar functions which are periodic in , with period . We recall that and are in and is a Banach space with the supremum norm [13, 14].

Define

Lemma 1. The equation has a unique -periodic solution where

Proof. First, it is evident that the homogeneous equation associated with (8) has a solution Using the parameter variation method, we obtain Keeping in mind that , , , and are periodic functions, we obtain Hence, where is identified by (10).

Lemma 2. Assume that Then,

Proof. The definition of gives On the other hand, it is simple to demonstrate that only if .
Hence, Since we get So, Consequently,

Lemma 3. If Then, solves equation (1) if and only if

Proof. Let be a solution of (1). Equation (1) reads as According to Lemma 1, we obtain which implies that This completes the proof. Let us define the two operators as follows: We formulate equation (26) in Lemma 3 as follows:

Remark 4. Any solution to equation (31) is a solution to problem (1).
Let us introduce the following hypotheses, which are assumed hereafter:
The function is Lipschitz continuous in . That is to say, there exists a positive constant such that

Lemma 5. Assume that (32) holds and Then, is a contraction.

Proof. It is evident that is continuous for all . Moreover, So, for all , we have Thus, Consequently, it follows from (33) that is a contraction.

Lemma 6. Assume that and . Then, is completely continuous.

Proof. Firstly, we show that is continuous. To this end, let be a sequence such that in . We have It follows from the continuity of and that Thus, is continuous.
Secondly, we prove that maps bounded sets into bounded sets in To this end, let be a bounded ball in we have From Lemma 2 and since we get The estimation of implies This shows that is uniformly bounded.

Finally, we prove that sends bounded sets into equicontinuous sets. Let , , , and be a bounded set of For all , we have

Denote . So, we obtain

As , the right-hand side of the above inequality tends to zero. By the Arzela-Ascoli theorem, we conclude that is a completely continuous operator. This completes the proof. This section will be concluded by referring to Krasnoselskii’s fixed-point theorem (see [9]).

Theorem 7. (Krasnoselskii). Let be a closed convex nonempty subset of a Banach space . Suppose that and map into such that (i), implies (ii) is a contraction mapping(iii) is completely continuousThen, there exists with

3. Existence of Positive Periodic Solutions

We will examine the existence of positive periodic solutions to problem (1) using Krasnoselskii’s fixed-point theorem. For this purpose, we consider and for some positive constant and . Moreover, define the set which is a closed convex and bounded subset of the Banach space

By looking at the three cases , and for all , we can prove the existence of a positive periodic solution of (1).

3.1. The Case

We assume that there exist nonpositive constants and such that

Theorem 8. Assume that and the function satisfies Then, problem (1) has a positive -periodic solution in the subset .

Proof. Let us start by proving that In fact, On the other hand, which leads to We conclude from Lemma 5 that is a contraction. Also, Lemma 6 implies that the operator is completely continuous.
We deduce from Krasnoselskii’s fixed-point theorem (see [15], p.~31) that has a fixed point which is a solution to (31). As a result of Remark 4, is a solution to problem (1). This completes the proof.

3.2. The Case

Theorem 9. Assume that , and Then, equation (1) has a positive -periodic solution in the subset .

Proof. According to [16], we have . Similarly to the proof of Theorem 8, we show that (1) has a nonnegative -periodic solution . Since , it is easy to see that ; i.e., (1) has a positive -periodic solution .

3.3. The Case

We assume that there exist nonnegative constants and such that

Theorem 10. Assume that and the function satisfies Then, problem (1) has a positive -periodic solution in the subset .

Proof. According to Lemma 5, it follows that the operator is a contraction, and from Lemma 6, the operator is completely continuous.
Now, we prove that We have Also, Thus, By Krasnoselskii’s theorem (see [15], p. 31), we deduce that has a fixed point which is a solution to (31), so problem (1) has a positive -periodic solution in the subset .

4. Conclusion

In this work, we established sufficient conditions for the existence of positive periodic solutions to the fourth-order nonlinear neutral differential equations with variable delay. Our proof relies on Krasnoselskii’s fixed-point theorem, which is an excellent tool when the conditions of the Banach or Schauder fixed-point theorems are not fulfilled.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare no conflict of interest.

Authors’ Contributions

All authors contributed to the design and implementation of the research, to the analysis of the results, and to the writing of the manuscript.

Acknowledgments

This research has been funded by the Scientific Research Deanship at University of Hail, Saudi Arabia, through project number RG-21 008.