Research Article A Novel Implementation of Krasnoselskii’s Fixed-Point Theorem to a Class of Nonlinear Neutral Differential Equations Ali Rezaiguia, 1,2 Abdelkader Moumen , 1 Abdelaziz Mennouni , 3 Mohammad Alshammari, 1 and Taher S. Hassan 1,4 1 Department of Mathematics, Faculty of Science, University of Ha’il, Ha’il 55425, Saudi Arabia 2 Laboratory of Computer Science and Mathematics, Mouhamed Cherif Messadia University, Souk Ahras, Algeria 3 Department of Mathematics, LTM, University of Batna 2, Mostefa Ben Boulaïd, Fesdis, Batna 05078, Algeria 4 Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Correspondence should be addressed to Abdelaziz Mennouni; a.mennouni@univ-batna2.dz Received 20 May 2022; Accepted 2 August 2022; Published 2 September 2022 Academic Editor: Serena Matucci Copyright © 2022 Ali Rezaiguia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 consider- able 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 non- linear periodic differential equation are investigated in [11]. The authors of [12] discussed nearly periodic solutions to non- linear Duffing equations. Among them, the fixed-point princi- ple 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: d 4 dt 4 xt ðÞ − gt , xt − τ t ðÞ ð Þ ð Þ ð Þ = −at ðÞxt ðÞ + ft , xt − τ t ðÞ ð Þ ð Þ: ð1Þ Under the assumptions: (i) a, τ ∈ Cðℝ, ð0,∞ÞÞ (ii) g ∈ Cðℝ × ½0,∞Þ, ℝÞ and f ∈ Cðℝ × ½0,∞Þ, ½0,∞ÞÞ (iii) a, τ, gðt , xÞ, f ðt , xÞ are ω-periodic in t , ω is a positive constant Krasnoselskii’s fixed-point theorem offers sufficient con- ditions 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 theo- rem was proposed in 2012 in [4] to show the existence of positive periodic solutions to the nonlinear neutral differen- tial equation with variable delay of the form d dt xt ðÞ − gt , xt − τ t ðÞ ð Þ ð Þ ð Þ = rt ðÞxt ðÞ − ft , xt − τ t ðÞ ð Þ ð Þ: ð2Þ The same researchers evaluated the existence of positive periodic solutions for two types of second-order nonlinear Hindawi Journal of Function Spaces Volume 2022, Article ID 9242541, 7 pages https://doi.org/10.1155/2022/9242541