Qualitative Properties of Nonlinear Neutral Transmission Line Models and Their Applications

Abstract

Neutral transmission line models are essential for analyzing stability and periodicity in systems influenced by nonlinear and delayed dynamics, particularly in modern smart grids. This study utilizes Krasnoselskii’s fixed-point theorem to establish sufficient conditions for the existence and asymptotic stability of periodic solutions, eliminating the need for differentiability in delay terms and coefficients. The results extend existing findings and are validated through a single test example, demonstrating the theoretical applicability of the proposed approach. These findings provide a mathematical framework for understanding the behavior of power distribution systems under nonlinear and delayed influences.

1. Introduction

Delay differential equations (DDEs) have emerged as indispensable tools for modeling complex dynamic systems influenced by delays, nonlinearities, and periodic behaviors. Their applications span a wide range of disciplines, including population dynamics, mechanical systems, and electrical circuits. Foundational studies, such as those by Burton [1] and Kuang [2], have laid the theoretical groundwork, exploring key properties like stability, positivity, and periodicity of solutions. These studies underscore the importance of DDEs in addressing intricate dynamical phenomena that arise in real-world systems. In addition, comprehensive discussions and significant advancements in these topics are documented across numerous studies [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24].

The phenomenon of delay in dynamic systems is often associated with memory effects and hereditary properties, which have been mathematically described using integro-differential equations (IDEs). The foundational work of Vito Volterra [25,26,27,28,29] established the principles of hereditary systems, demonstrating how past states influence future evolution through integral formulations. These principles have significantly influenced the development of delay differential equations (DDEs), which serve as a more explicit representation of delayed interactions in various applications. By incorporating memory effects, IDEs and DDEs provide a rigorous mathematical framework for modeling real-world systems where delays play a critical role, such as energy transmission networks and biological systems.

Neutral transmission line models, a specific application of DDEs, are critical for analyzing the stability and periodicity of systems characterized by nonlinear and delayed dynamics. Their relevance has grown significantly in the context of modern smart grids, where the integration of renewable energy sources, variable loads, and communication delays presents unique challenges to stability and efficiency. Recent studies have established criteria for periodicity and stability in simplified and generalized neutral differential equation frameworks [30,31,32]. Notably, Ding and Li [31] leveraged Krasnoselskii’s fixed-point theorem to demonstrate periodic solutions in simplified models, while Mansouri et al. [32] extended these results to accommodate more complex dynamics involving variable delays and nonlinearities.

Despite these advancements, existing approaches are often constrained by assumptions such as the differentiability of delay terms and coefficients, which limit their applicability to real-world systems. In modern smart grids, such assumptions fail to capture the nondifferentiable dynamics introduced by renewable energy integration, time-varying power demands, and communication delays [33,34,35,36,37]. Addressing these gaps requires a more robust and flexible mathematical framework capable of capturing the interplay between nonlinear, delayed, and periodic behaviors.

This study aims to bridge this gap by exploring a more comprehensive class of nonlinear neutral transmission line models, removing the need for differentiability in delay terms and coefficients. By leveraging Krasnoselskii’s fixed-point theorem, this work introduces novel sufficient conditions for the existence and asymptotic stability of periodic solutions in the space $C^0$. The derived results not only extend theoretical insights but also demonstrate practical relevance in optimizing power distribution within smart grids. The considered problem is given by

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{d\varsigma }}\left\{\varkappa \left(\varsigma \right)-q\left(\varsigma \right)\Phi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right)\right\}\hfill \\ \hfill & =p\left(\varsigma \right)-a\left(\varsigma \right)\varkappa \left(\varsigma \right)-a\left(\varsigma \right)q\left(\varsigma \right)\Phi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right)-b\left(\varsigma \right)\Psi \left(\varkappa \left(\varsigma \right)\right)+b\left(\varsigma \right)q\left(\varsigma \right)\Psi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right),\hfill \end{array}$$

(1)

where a, b, and $\delta$ are positive $\varpi$-periodic continuous functions with $a\left(\varsigma \right)>\alpha >0$; q and p are $\varpi$-periodic continuous functions; $\Psi$ is a continuous function; and $\Phi$ is a locally Lipschitz continuous function, i.e., for $\lambda >0$, there is ${\varrho }_{\Phi }>0$ such that

$$\left|\Phi \left(\varkappa \right)-\Phi \left(y\right)\right|\le {\varrho }_{\Phi }\left|\varkappa -y\right|\mathrm{for}\mathrm{all}\left|\varkappa \right|,\left|y\right|\le \lambda .$$

A notable application of this research lies in addressing critical challenges in smart grid optimization. Renewable energy sources, such as solar and wind power, introduce periodic and nonlinear power variations that complicate grid stability. Additionally, power transmission delays, communication inefficiencies, and variable load conditions demand advanced analytical tools to maintain grid reliability and efficiency. The proposed model captures these complexities by incorporating nonlinear, delayed, and periodic dynamics, enabling precise analysis and control of voltage behavior. Furthermore, this approach aligns with contemporary advancements in machine learning and decentralized control, which further enhance grid performance and resilience [38,39].

Through theoretical analysis and practical examples, this work advances the mathematical modeling of nonlinear systems while providing actionable solutions to the pressing challenges of modern power systems. These findings pave the way for improved stability, efficiency, and resilience in smart grid operations, addressing the dynamic needs of contemporary energy networks.

To provide a clear roadmap for the reader, the remainder of this paper is structured as follows: In Section 2, we establish the existence of periodic solutions using fixed-point methods. Section 3 presents conditions for asymptotic stability, ensuring long-term system behavior remains predictable. In Section 4, we apply our theoretical findings to smart grid optimization, demonstrating their relevance in real-world power distribution challenges. Finally, Section 5 concludes this paper with a summary of our contributions and potential future research directions.

2. Existence of Periodic Solutions

Let $C\left(\mathbb{R}\right)$ be the set of all continuous functions $\varkappa :\mathbb{R}\to \mathbb{R}$. $C_{\varpi }=\{\varkappa \in C\left(\mathbb{R}\right):\varkappa (\varsigma +\varpi )=\varkappa \left(\varsigma \right)\}$ is equipped with the supremum norm $∥.∥$ in a period interval.

The following two lemmas will be applied in this work.

Lemma 1.

For $\mathsf{Ϝ}\in C_{\varpi }$, the differential equation ${\varkappa }^{\prime }\left(\varsigma \right)+a\left(\varsigma \right)\varkappa \left(\varsigma \right)=\mathsf{Ϝ}\left(\varsigma \right)$ admits a unique ϖ-periodic solution

$$\varkappa \left(\varsigma \right)=\eta {\int }_{\varsigma }^{\varsigma +\varpi }{e}^{{\int }_{\varsigma +\varpi }^{s}a\left(\iota \right)d\iota }\mathsf{Ϝ}\left(s\right)ds,$$

where

$$\eta ={\left(1-e^{-{\int }_0^{\varpi }a\left(\iota \right)d\iota }\right)}^{-1}.$$

Proof. 

The proof can be found in ODE books (see for example, [4]). □

Lemma 2

(Krasnoselskii’s fixed point [40]). Let Ω be a closed bounded convex nonempty subset of a Banach space $(\mathbb{E},\Vert ·\Vert )$. Suppose that ${\Lambda }_1$ and ${\Lambda }_2$ map Ω into $\mathbb{E}$ such that

(i) 

$\varkappa ,y\in \Omega$ implies ${\Lambda }_1\varkappa +{\Lambda }_{2}y\in \Omega$;

(ii) 

${\Lambda }_1$ is compact and continuous;

(iii) 

${\Lambda }_2$ is a contraction mapping.

Then, there is $\varphi \in \Omega$ with $\varphi ={\Lambda }_1\varphi +{\Lambda }_2\varphi$.

In this section, by using Lemmas 1 and 2, we demonstrate the existence of $\varpi$-periodic solutions of (1).

Theorem 1.

Suppose that there exist $\rho \in \left(0,1\right)$ and $\lambda >0$ such that

$$\rho ={\varrho }_{\Phi }\parallel q\parallel ,\underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|\le {\displaystyle \frac{\alpha }{∥b∥}}\lambda ,$$

$$∥q∥<{\displaystyle \frac{\alpha \lambda -\parallel b\parallel \underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|}{3\alpha \left({\varrho }_{\Phi }\lambda +\left|\Phi \left(0\right)\right|\right)+\parallel b\parallel \underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|}},$$

(2)

and

$$∥p∥<\alpha \lambda -3\alpha \parallel q\parallel ({\varrho }_{\Phi }\lambda +\left|\Phi \left(0\right)\right|)-\parallel b\parallel \left(1+\parallel q\parallel \right)\underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|.$$

(3)

Then, (1) admits a ϖ-periodic solution.

Proof. 

From (3), we have

$${\displaystyle \frac{\parallel p\parallel }{\alpha }}+3\parallel q\parallel ({\varrho }_{\Phi }\lambda +\left|\Phi \left(0\right)\right|)+{\displaystyle \frac{\parallel b\parallel }{\alpha }}\left(1+\parallel q\parallel \right)\underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|\le \lambda .$$

(4)

Let

$$\Omega =\{\varkappa \in C_{\varpi }:∥\varkappa ∥\le \lambda \},$$

which is a closed bounded convex set of $C_{\varpi }$.

For all $\varkappa \in \Omega$, we rewrite (1) as

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{d\varsigma }}\left\{\varkappa \left(\varsigma \right)-q\left(\varsigma \right)\Phi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right)\right\}+a\left(\varsigma \right)\left\{\varkappa \left(\varsigma \right)-q\left(\varsigma \right)\Phi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right)\right\}\hfill \\ \hfill & =p\left(\varsigma \right)-2a\left(\varsigma \right)q\left(\varsigma \right)\Phi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right)-b\left(\varsigma \right)\Psi \left(\varkappa \left(\varsigma \right)\right)+b\left(\varsigma \right)q\left(\varsigma \right)\Psi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right).\hfill \end{array}$$

(5)

By applying Lemma 1, we obtain

$$\begin{array}{cc}\hfill & \varkappa \left(\varsigma \right)-q\left(\varsigma \right)\Phi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right)\hfill \\ \hfill & =\eta {\int }_{\varsigma }^{\varsigma +\varpi }{e}^{{\int }_{\varsigma +\varpi }^{s}a\left(\iota \right)d\iota }\left[p\left(s\right)-2a\left(s\right)q\left(s\right)\Phi \left(\varkappa \right(s-\delta \left(s\right)\left)\right)\right.\hfill \\ \hfill & \left.-b\left(s\right)\Psi \left(\varkappa \right(s\left)\right)+b\left(s\right)q\left(s\right)\Psi \left(\varkappa \right(s-\delta \left(s\right)\left)\right)\right]ds.\hfill \end{array}$$

Then,

$$\begin{array}{cc}\hfill \varkappa \left(\varsigma \right)& =\eta {\int }_{\varsigma }^{\varsigma +\varpi }{e}^{{\int }_{\varsigma +\varpi }^{s}a\left(\iota \right)d\iota }\left[p\left(s\right)-2a\left(s\right)q\left(s\right)\Phi \left(\varkappa \right(s-\delta \left(s\right)\left)\right)\right.\hfill \\ \hfill & \left.-b\left(s\right)\Psi \left(\varkappa \right(s\left)\right)+b\left(s\right)q\left(s\right)\Psi \left(\varkappa \right(s-\delta \left(s\right)\left)\right)\right]ds+q\left(\varsigma \right)\Phi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right).\hfill \end{array}$$

Define Mappings ${\Lambda }_1$ and ${\Lambda }_2$ by

$$\begin{array}{cc}\hfill \left({\Lambda }_1\varkappa \right)\left(\varsigma \right)& =\eta {\int }_{\varsigma }^{\varsigma +\varpi }{e}^{{\int }_{\varsigma +\varpi }^{s}a\left(\iota \right)d\iota }\left[p\left(s\right)-2a\left(s\right)q\left(s\right)\Phi \left(\varkappa \right(s-\delta \left(s\right)\left)\right)\right.\hfill \\ \hfill & \left.-b\left(s\right)\Psi \left(\varkappa \right(s\left)\right)+b\left(s\right)q\left(s\right)\Psi \left(\varkappa \right(s-\delta \left(s\right)\left)\right)\right]ds,\hfill \end{array}$$

and

$$\left({\Lambda }_2\varkappa \right)\left(\varsigma \right)=q\left(\varsigma \right)\Phi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right).$$

To demonstrate that (1) admits a $\varpi$-periodic solution, we shall make sure that ${\Lambda }_1$ and ${\Lambda }_2$ satisfy the assumptions of Lemma 2. For all $\varkappa ,y\in \Omega$, we have $\varkappa (\varsigma +\varpi )=\varkappa \left(\varsigma \right)$, $y(\varsigma +\varpi )=y\left(\varsigma \right)$ and $∥\varkappa ∥\le \lambda$, $∥y∥\le \lambda$. Now, let us discuss ${\Lambda }_1\varkappa +{\Lambda }_{2}y$. We have

$$\begin{array}{cc}\hfill \left({\Lambda }_1\varkappa \right)(\varsigma +\varpi )& =\eta {\int }_{\varsigma +\varpi }^{\varsigma +2\varpi }{e}^{{\int }_{\varsigma +2\varpi }^{s}a\left(\iota \right)d\iota }\left[p\left(s\right)-2a\left(s\right)q\left(s\right)\Phi \left(\varkappa \right(s-\delta \left(s\right)\left)\right)\right.\hfill \\ \hfill & \left.-b\left(s\right)\Psi \left(\varkappa \right(s\left)\right)+b\left(s\right)q\left(s\right)\Psi \left(\varkappa \right(s-\delta \left(s\right)\left)\right)\right]ds\hfill \\ \hfill & =\eta {\int }_{\varsigma }^{\varsigma +\varpi }{e}^{{\int }_{\varsigma +\varpi }^{s}a\left(\iota \right)d\iota }\left[p\left(s\right)-2a\left(s\right)q\left(s\right)\Phi \left(\varkappa \right(s-\delta \left(s\right)\left)\right)\right.\hfill \\ \hfill & \left.-b\left(s\right)\Psi \left(\varkappa \right(s)+b(s\left)q\right(s)\Psi (\varkappa (s-\delta (s\left)\right))\right]ds\hfill \\ \hfill & =\left({\Lambda }_1\varkappa \right)\left(\varsigma \right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left({\Lambda }_{2}y\right)(\varsigma +\varpi )& =q(\varsigma +\varpi )\Phi \left(y\right(\varsigma +\varpi -\delta (\varsigma +\varpi )\left)\right)\hfill \\ \hfill & =q\left(\varsigma \right)\Phi (y(\varsigma -\delta \left(\varsigma \right))=\left({\Lambda }_{2}y\right)\left(\varsigma \right).\hfill \end{array}$$

Therefore, $({\Lambda }_1\varkappa +{\Lambda }_{2}y)(\varsigma +\varpi )=({\Lambda }_1\varkappa +{\Lambda }_{2}y)\left(\varsigma \right)$. Meanwhile, we get

$$\begin{array}{cc}\hfill \left|\left({\Lambda }_1\varkappa \right)\left(\varsigma \right)\right|& =\left|\eta {\int }_{\varsigma }^{\varsigma +\varpi }{e}^{{\int }_{\varsigma +\varpi }^{s}a\left(\iota \right)d\iota }a\left(s\right)\left[{\displaystyle \frac{p\left(s\right)}{a\left(s\right)}}-2q\left(s\right)\Phi \left(\varkappa (s-\delta \left(s\right))\right)\right.\right.\hfill \\ \hfill & \left.\left.-{\displaystyle \frac{b\left(s\right)}{a\left(s\right)}}\Psi \left(\varkappa \left(s\right)\right)+{\displaystyle \frac{b\left(s\right)}{a\left(s\right)}}q\left(s\right)\Psi \left(\varkappa (s-\delta \left(s\right))\right)\right]ds\right|\hfill \\ \hfill & \le {\displaystyle \frac{\parallel p\parallel }{\alpha }}+2\parallel q\parallel ({\varrho }_{\Phi }\lambda +\left|\Phi \left(0\right)\right|)+{\displaystyle \frac{\parallel b\parallel }{\alpha }}\left(1+\parallel q\parallel \right)\underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \left|\left({\Lambda }_2\varkappa \right)\left(\varsigma \right)\right|& =\left|q\left(\varsigma \right)\Phi \left(\varkappa \right(\varsigma -\delta \left(\varsigma \right)\left)\right)\right|\hfill \\ \hfill & \le \parallel q\parallel ({\varrho }_{\Phi }\lambda +|\Phi \left(0\right)\left|\right).\hfill \end{array}$$

So,

$$∥{\Lambda }_1\varkappa ∥\le {\displaystyle \frac{\parallel p\parallel }{\alpha }}+2\parallel q\parallel ({\varrho }_{\Phi }\lambda +\left|\Phi \left(0\right)\right|)+{\displaystyle \frac{\parallel b\parallel }{\alpha }}\left(1+\parallel q\parallel \right)\underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|,$$

and

$$∥{\Lambda }_{2}y∥\le \parallel q\parallel ({\varrho }_{\Phi }\lambda +|\Phi \left(0\right)|).$$

Therefore,

$$\begin{array}{cc}\hfill & \parallel {\Lambda }_1\varkappa +{\Lambda }_{2}y\parallel \hfill \\ \hfill & \le ∥{\Lambda }_1\varkappa ∥+\parallel {\Lambda }_{2}y\parallel \hfill \\ \hfill & \le {\displaystyle \frac{\parallel p\parallel }{\alpha }}+3\parallel q\parallel ({\varrho }_{\Phi }\lambda +\left|\Phi \left(0\right)\right|)+{\displaystyle \frac{\parallel b\parallel }{\alpha }}\left(1+\parallel q\parallel \right)\underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|\hfill \\ \hfill & \le \lambda ,\hfill \end{array}$$

by (4). Consequently, ${\Lambda }_1\varkappa +{\Lambda }_{2}y\in \Omega$.

For all $\varkappa \in \Omega$, $∥{\Lambda }_1\varkappa ∥\le \lambda$. On the other hand, we have

$$\begin{array}{cc}\hfill {\left({\Lambda }_1\varkappa \right)}^{\prime }\left(\varsigma \right)& =-a\left(\varsigma \right)\eta {\int }_{\varsigma }^{\varsigma +\varpi }{e}^{{\int }_{\varsigma +\varpi }^{s}a\left(\iota \right)d\iota }[p\left(s\right)-2a\left(s\right)q\left(s\right)\Phi \left(\varkappa (s-\delta \left(s\right))\right)\hfill \\ \hfill & -b\left(s\right)\Psi \left(\varkappa \right(s\left)\right)+b\left(s\right)q\left(s\right)\Psi \left(\varkappa \right(s-\delta \left(s\right)\left)\right)]ds+p(\varsigma )\hfill \\ \hfill & -2a\left(\varsigma \right)q\left(\varsigma \right)\Phi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right)-b\left(\varsigma \right)\Psi \left(\varkappa \left(\varsigma \right)\right)+b\left(\varsigma \right)q\left(\varsigma \right)\Psi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right)\hfill \\ \hfill & =-a\left(\varsigma \right)\left({\Lambda }_1\varkappa \right)\left(\varsigma \right)+p\left(\varsigma \right)-2a\left(\varsigma \right)q\left(\varsigma \right)\Phi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right)\hfill \\ \hfill & -b\left(\varsigma \right)\Psi \left(\varkappa \left(\varsigma \right)\right)+b\left(\varsigma \right)q\left(\varsigma \right)\Psi \left(\varkappa (\varsigma -\delta \left(\varsigma \right))\right).\hfill \end{array}$$

Thus,

$$\left|{\left({\Lambda }_1\varkappa \right)}^{\prime }\left(\varsigma \right)\right|\le \parallel a\parallel \lambda +∥p∥+2\parallel a\parallel ∥q∥({\varrho }_{\Phi }\lambda +\left|\Phi \left(0\right)\right|)+\parallel b\parallel \left(1+\parallel q\parallel \right)\underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|.$$

Therefore, there is a constant ${\lambda }_1>0$ such that ${sup}_{\varsigma \in \mathbb{R}}\left|{\left({\Lambda }_1\varkappa \right)}^{\prime }\left(\varsigma \right)\right|\le {\lambda }_1$. From the Ascoli–Arzela theorem, for all of the sequence $\left\{{\varkappa }_n\right\}$ of $\Omega$, there is a subsequence $\left\{{\varkappa }_{n_k}\right\}$ of $\left\{{\varkappa }_n\right\}$, such that ${\Lambda }_1{\varkappa }_{n_k}\to {\varkappa }_0\in C_{\varpi }$ as $k\to +\infty$. Hence, $\left\{{\Lambda }_1{\varkappa }_n\right\}$ is contained in a compact set. So, ${\Lambda }_1$ is a compact mapping.

Assume that $\left\{{\varkappa }_n\right\}\subset \Omega$, $\varkappa \in \Omega$, ${\varkappa }_n\to \varkappa$. Then, $\parallel {\varkappa }_n-\varkappa \parallel \to 0$ as $n\to \infty$, and we obtain

$$\begin{array}{cc}\hfill & \left|\left({\Lambda }_1{\varkappa }_n\right)\left(\varsigma \right)-\left({\Lambda }_1\varkappa \right)\left(\varsigma \right)\right|\hfill \\ \hfill & \le \eta {\int }_{\varsigma }^{\varsigma +\varpi }{e}^{{\int }_{\varsigma +\varpi }^{s}a\left(\iota \right)d\iota }a\left(s\right)[2∥q∥\left|\Phi \left({\varkappa }_n(s-\delta \left(s\right))\right)-\Phi \left(\varkappa (s-\delta \left(s\right))\right)\right|\hfill \\ \hfill & +{\displaystyle \frac{∥b∥}{\alpha }}\left|\Psi \left({\varkappa }_n\left(s\right)\right)-\Psi \left(\varkappa \left(s\right)\right)\right|+{\displaystyle \frac{∥b∥}{\alpha }}∥q∥\left|\Psi \left({\varkappa }_n(s-\delta \left(s\right))\right)-\Psi \left(\varkappa (s-\delta \left(s\right))\right)\right|]ds\hfill \\ \hfill & \le 2{\varrho }_{\Phi }∥q∥∥{\varkappa }_n-\varkappa ∥+{\displaystyle \frac{∥b∥}{\alpha }}\left(1+∥q∥\right)\underset{\varsigma \in \left[0,\varpi \right]}{sup}\left|\Psi \left({\varkappa }_n\left(s\right)\right)-\Psi \left(\varkappa \left(s\right)\right)\right|.\hfill \end{array}$$

Thus,

$$∥{\Lambda }_1{\varkappa }_n-{\Lambda }_1\varkappa ∥\le 2{\varrho }_{\Phi }∥q∥∥{\varkappa }_n-\varkappa ∥+{\displaystyle \frac{∥b∥}{\alpha }}\left(1+∥q∥\right)\underset{\varsigma \in \left[0,\varpi \right]}{sup}\left|\Psi \left({\varkappa }_n\left(s\right)\right)-\Psi \left(\varkappa \left(s\right)\right)\right|,$$

when $\parallel {\varkappa }_n-\varkappa \parallel \to 0$ as $n\to \infty$ and $\left|{\varkappa }_n\left(\varsigma \right)-\varkappa \left(\varsigma \right)\right|\to 0$ for $\varsigma \in [0,\varpi ]$ uniformly. And since $\Psi$ is continuous, $\parallel {\Lambda }_1{\varkappa }_n-{\Lambda }_1\varkappa \parallel \to 0$. Consequently, ${\Lambda }_1$ is continuous.

Now, for all $\varkappa ,y\in \Omega$, we have

$$\begin{array}{cc}\hfill & \left|\left({\Lambda }_2\varkappa \right)\left(\varsigma \right)-\left({\Lambda }_{2}y\right)\left(\varsigma \right)\right|\hfill \\ \hfill & =\left|q\left(\varsigma \right)\right|\left|\Phi \left(\varkappa \right(\varsigma -\delta \left(\varsigma \right)\left)\right)-\Phi \left(y\right(\varsigma -\delta \left(\varsigma \right)\left)\right)\right|\hfill \\ \hfill & \le {\varrho }_{\Phi }\parallel q\parallel \parallel \varkappa -y\parallel .\hfill \end{array}$$

Then,

$$∥{\Lambda }_2\varkappa -{\Lambda }_{2}y∥\le \rho ∥\varkappa -y∥.$$

Therefore, ${\Lambda }_2$ is a contraction mapping.

Thus, the assumptions of Lemma 2 are satisfied. Hence, there is a $\varkappa \in \Omega$, such that $\varkappa ={\Lambda }_1\varkappa +{\Lambda }_2\varkappa$. It is a $\varpi$-periodic solution for (1). □

Example 1.

Consider the nonlinear neutral transmission line model (1) where $\delta \left(\varsigma \right)=1$, $q\left(\varsigma \right)={\displaystyle \frac{1}{2}}sin\left(\varsigma \right)$, $\Phi \left(\varkappa \right)={\displaystyle \frac{1}{3}}sin\left(\varkappa \right)$, $p\left(\varsigma \right)={\displaystyle \frac{1}{10}}+{\displaystyle \frac{1}{10}}cos\left(\varsigma \right)$, $a\left(\varsigma \right)=2+sin\left(\varsigma \right)$, $b\left(\varsigma \right)=cos\left(\varsigma \right)$, $\Psi \left(\varkappa \right)={\displaystyle \frac{1}{4}}arctan\left(\varkappa \right)$. For $\lambda =1$, one may easily verify that

$$\begin{array}{cc}\hfill ∥b∥& =1,∥q∥={\displaystyle \frac{1}{2}},∥p∥={\displaystyle \frac{1}{5}},\alpha =1,{\varrho }_{\Phi }=1,\left|\Phi \left(0\right)\right|=0,\hfill \\ \hfill \underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|& ={\displaystyle \frac{\pi }{16}}\le 1={\displaystyle \frac{\alpha }{∥b∥}}\lambda ,\hfill \\ \hfill ∥q∥& ={\displaystyle \frac{1}{2}}<0.67175\approx {\displaystyle \frac{\alpha \lambda -\parallel b\parallel \underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|}{3\alpha \left({\varrho }_{\Phi }\lambda +\left|\Phi \left(0\right)\right|\right)+\parallel b\parallel \underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|}},\hfill \\ \hfill ∥p∥& ={\displaystyle \frac{1}{5}}<0.20548\approx \alpha \lambda -3\alpha \parallel q\parallel ({\varrho }_{\Phi }\lambda +\left|\Phi \left(0\right)\right|)-\parallel b\parallel \left(1+\parallel q\parallel \right)\underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|.\hfill \end{array}$$

Then, all assumptions of Theorem 1 are satisfied. Hence, (1) admits a $2\pi$-periodic solution.

3. Asymptotic Stability

We study the asymptotic stability of periodic solutions in this section. Let ${\varkappa }^{\ast }$ be a periodic solution of (1). For $\upsilon =\varkappa -{\varkappa }^{\ast }$, (1) is transformed to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{d\varsigma }}\left\{\upsilon \left(\varsigma \right)-q\left(\varsigma \right)\left[\Phi \left(\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right))\right]\right\}\hfill \\ \hfill & =-a\left(\varsigma \right)\upsilon \left(\varsigma \right)-a\left(\varsigma \right)q\left(\varsigma \right)\left[\Phi \left(\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right]\hfill \\ \hfill & -b\left(\varsigma \right)\left[\Psi \left(\upsilon \left(\varsigma \right)+{\varkappa }^{\ast }\left(\varsigma \right)\right)-\Psi \left({\varkappa }^{\ast }\left(\varsigma \right)\right)\right]\hfill \\ \hfill & +b\left(\varsigma \right)q\left(\varsigma \right)\left[\Psi \left(\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)-\Psi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right].\hfill \end{array}$$

(6)

Clearly, (6) admits the zero solution. To demonstrate that the zero solution of (6) is asymptotically stable, we apply Krasnoselskii’s fixed-point theorem. Let $\mathbb{E}$ be the Banach space of bounded continuous functions $\upsilon :[m_0,\infty )\to \mathbb{R}$ with the supremum norm $\parallel .\parallel$ and $m_0=inf\{\varsigma -\delta \left(\varsigma \right)$, $\varsigma \ge 0\}$. Moreover, for a given initial function $\psi$, denote the norm of $\psi$ by $∥\psi ∥=\underset{\varsigma \in [m_0,0]}{sup}\left|\psi \left(\varsigma \right)\right|$.

Definition 1

([1]). The zero solution of (6) is asymptotically stable if it is Lyapunov-stable and if there exists a $\sigma >0$ for any $\varsigma \ge m_0$ such that $∥\psi ∥\le \sigma$ implies $\left|\upsilon \left(\varsigma \right)\right|\to 0$ as $\varsigma \to \infty$.

Theorem 2.

Suppose that all assumptions of Theorem 1 are satisfied and that Ψ satisfies the locally Lipschitz condition. Also, assume that

$$\varsigma -\delta \left(\varsigma \right)\to \infty \mathrm{as}\varsigma \to \infty ,$$

and that there is $Q>\lambda$ such that

$${\displaystyle \frac{∥b∥}{\alpha }}\underset{\mid \varkappa \mid \le \lambda +Q}{sup}\left|\Psi \left(\varkappa \right)\right|<Q-\lambda ,$$

(7)

$$∥q∥<{\displaystyle \frac{Q-\lambda -{\displaystyle \frac{∥b∥}{\alpha }}\underset{\mid \varkappa \mid \le \lambda +Q}{sup}\left|\Psi \left(\varkappa \right)\right|}{3{\varrho }_{\Phi }Q+\lambda +{\displaystyle \frac{∥b∥}{\alpha }}\underset{\mid \varkappa \mid \le \lambda +Q}{sup}\left|\Psi \left(\varkappa \right)\right|}},$$

(8)

and

$$∥\psi ∥\le {\displaystyle \frac{Q-3{\varrho }_{\Phi }∥q∥Q-\left(1+∥q∥\right)\lambda -{\displaystyle \frac{∥b∥}{\alpha }}\left(1+∥q∥\right)\underset{\mid \varkappa \mid \le \lambda +Q}{sup}\left|\Psi \left(\varkappa \right)\right|}{1+{\varrho }_{\Phi }∥q∥}}.$$

(9)

Then, the solution of (6) $\upsilon \left(\varsigma \right)\to 0$ is $\varsigma \to \infty$.

Proof. 

From (9), we obtain

$$3{\varrho }_{\Phi }∥q∥Q+\left(1+∥q∥\right)\lambda +{\displaystyle \frac{∥b∥}{\alpha }}\left(1+∥q∥\right)\underset{\left|\varkappa \right|\le \lambda +Q}{sup}\left|\Psi \left(\varkappa \right)\right|+(1+{\varrho }_{\Phi }\parallel q\parallel )∥\psi ∥\le Q.$$

(10)

There is a unique solution $\upsilon$ of (6) for the given initial function $\psi$. Let

$${\Omega }_{\psi }=\{\upsilon \in \mathbb{E}:∥\upsilon ∥\le Q,\upsilon \left(\varsigma \right)=\psi \left(\varsigma \right)\mathrm{for}\varsigma \in [m_0,0],\upsilon \left(\varsigma \right)\to 0\mathrm{as}\varsigma \to \infty \},$$

which is a closed convex bounded subset of $\mathbb{E}$. Now, we rewrite (6) as

$$\begin{array}{cc}\hfill & {\displaystyle \frac{d}{d\varsigma }}\left\{\upsilon \left(\varsigma \right)-q\left(\varsigma \right)\left[\Phi \left(\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right))\right]\right\}\hfill \\ \hfill & +a\left(\varsigma \right)\left\{\upsilon \left(\varsigma \right)-q\left(\varsigma \right)\left[\Phi \left(\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right]\right\}\hfill \\ \hfill & =-2a\left(\varsigma \right)q\left(\varsigma \right)\left[\Phi (\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right)))-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right]\hfill \\ \hfill & -b\left(\varsigma \right)\left[\Psi \left(\upsilon \left(\varsigma \right)+{\varkappa }^{\ast }\left(\varsigma \right)\right)-\Psi \left({\varkappa }^{\ast }\left(\varsigma \right)\right)\right]\hfill \\ \hfill & +b\left(\varsigma \right)q\left(\varsigma \right)\left[\Psi \left(\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)-\Psi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right].\hfill \end{array}$$

(11)

Then,

$$\begin{array}{cc}\hfill \upsilon \left(\varsigma \right)& =\left\{\upsilon \left(0\right)-q\left(0\right)\left[\Phi \left(\psi (-\delta \left(0\right))+{\varkappa }^{\ast }(-\delta \left(0\right))\right)-\Phi ({\varkappa }^{\ast }(-\delta \left(0\right))\right]\right\}{e}^{-{\int }_0^{\varsigma }a\left(\iota \right)d\iota }\hfill \\ \hfill & +q\left(\varsigma \right)\left[\Phi \left(\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right]\hfill \\ \hfill & +{\int }_0^{\varsigma }{e}^{-{\int }_s^{\varsigma }a\left(\iota \right)d\iota }\left\{-2a\left(s\right)q\left(s\right)\left[\Phi \left(\upsilon (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right))\right)-\Phi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right.\hfill \\ \hfill & -b\left(s\right)\left[\Psi \left(\upsilon \left(s\right)+{\varkappa }^{\ast }\left(s\right)\right)-\Psi \left({\varkappa }^{\ast }\left(s\right)\right)\right]\hfill \\ \hfill & \left.+b\left(s\right)q\left(s\right)\left[\Psi \left(\upsilon (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right))\right)-\Psi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right\}ds.\hfill \end{array}$$

Let $\left(L\upsilon \right)\left(\varsigma \right)=\psi \left(\varsigma \right)$, $\varsigma \in [m_0,0]$, and

$$\begin{array}{cc}\hfill \left(L\upsilon \right)\left(\varsigma \right)& =\left\{\psi \left(0\right)-q\left(0\right)\left[\Phi (\psi (-\delta \left(0\right))+{\varkappa }^{\ast }(-\delta \left(0\right)))-\Phi ({\varkappa }^{\ast }(-\delta \left(0\right))\right]\right\}{e}^{-{\int }_0^{\varsigma }a\left(\iota \right)d\iota }\hfill \\ \hfill & +q\left(\varsigma \right)\left[\Phi (\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right)))-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right]\hfill \\ \hfill & +{\int }_0^{\varsigma }{e}^{-{\int }_s^{\varsigma }a\left(\iota \right)d\iota }\left\{-2a\left(s\right)q\left(s\right)\left[\Phi (\upsilon (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right)))-\Phi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right.\hfill \\ \hfill & -b\left(s\right)\left[\Psi (\upsilon \left(s\right)+{\varkappa }^{\ast }\left(s\right))-\Psi \left({\varkappa }^{\ast }\left(s\right)\right)\right]\hfill \\ \hfill & \left.+b\left(s\right)q\left(s\right)\left[\Psi (\upsilon (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right)))-\Psi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right\}ds.\hfill \end{array}$$

For all $\upsilon \in {\Omega }_{\psi }$, define the Mappings ${\Lambda }_1$ and ${\Lambda }_2$ by

$$\left({\Lambda }_1\upsilon \right)\left(\varsigma \right)=\left\{\begin{array}{c}0,\varsigma \in [m_0,0],\hfill \\ {\int }_0^{\varsigma }{e}^{-{\int }_s^{\varsigma }a\left(\iota \right)d\iota }\left\{-2a\left(s\right)q\left(s\right)\left[\Phi (\upsilon (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right)))-\Phi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right.\hfill \\ -b\left(s\right)\left[\Psi (\upsilon \left(s\right)+{\varkappa }^{\ast }\left(s\right))-\Psi \left({\varkappa }^{\ast }\left(s\right)\right)\right]\hfill \\ \left.+b\left(s\right)q\left(s\right)\left[\Psi (\upsilon (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right)))-\Psi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right\}ds,\varsigma \ge 0,\hfill \end{array}\right.$$

$$\left({\Lambda }_2\upsilon \right)\left(\varsigma \right)=\left\{\begin{array}{c}\psi \left(\varsigma \right),\varsigma \in [m_0,0],\hfill \\ \left\{\psi \left(0\right)-q\left(0\right)\left[\Phi (\psi (-\delta \left(0\right))+{\varkappa }^{\ast }(-\delta \left(0\right)))-\Phi ({\varkappa }^{\ast }(-\delta \left(0\right))\right]\right\}{e}^{-{\int }_0^{\varsigma }a\left(\iota \right)d\iota }\hfill \\ +q\left(\varsigma \right)\left[\Phi (\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right)))-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right],\varsigma \ge 0.\hfill \end{array}\right.$$

(i) For all $\varkappa ,y\in {\Omega }_{\psi }$, $\varkappa \left(\varsigma \right)\to 0$ and $y\left(\varsigma \right)\to 0$, as $\varsigma \to \infty$. Then,

$$\begin{array}{cc}\hfill \underset{\varsigma \to \infty }{lim}\left({\Lambda }_{2}y\right)\left(\varsigma \right)& =\underset{\varsigma \to \infty }{lim}\left\{\psi \left(0\right)-q\left(0\right)\left[\Phi (\psi (-\delta \left(0\right))+{\varkappa }^{\ast }(-\delta \left(0\right)))-\Phi ({\varkappa }^{\ast }(-\delta \left(0\right))\right]\right\}{e}^{-{\int }_0^{\varsigma }a\left(\iota \right)d\iota }\hfill \\ \hfill & +q\left(\varsigma \right)\left[\Phi (y(\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right)))-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right]\hfill \\ \hfill & =0,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill & \underset{\varsigma \to \infty }{lim}\left({\Lambda }_1\varkappa \right)\left(\varsigma \right)\hfill \\ \hfill & =\underset{\varsigma \to \infty }{lim}{e}^{-{\int }_0^{\varsigma }a\left(\iota \right)d\iota }{\int }_0^{\varsigma }{e}^{{\int }_0^{s}a\left(\iota \right)d\iota }\left\{-2a\left(s\right)q\left(s\right)\left[\Phi (\varkappa (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right)))-\Phi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right.\hfill \\ \hfill & -b\left(s\right)\left[\Psi \left(\varkappa \left(s\right)+{\varkappa }^{\ast }\left(s\right)\right)-\Psi \left({\varkappa }^{\ast }\left(s\right)\right)\right]\hfill \\ \hfill & \left.+b\left(s\right)q\left(s\right)\left[\Psi \left(\varkappa (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right))\right)-\Psi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right\}ds\hfill \\ \hfill & =0.\hfill \end{array}$$

Therefore, ${lim}_{\varsigma \to \infty }({\Lambda }_1\varkappa +{\Lambda }_{2}y)\left(\varsigma \right)=0$. We have

$$\begin{array}{cc}\hfill \left|\left({\Lambda }_1\varkappa \right)\left(\varsigma \right)\right|& =\left|{\int }_0^{\varsigma }{e}^{-{\int }_s^{\varsigma }a\left(\iota \right)d\iota }a\left(s\right)\left\{-2q\left(s\right)\left[\Phi (\varkappa (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right)))-\Phi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right.\right.\hfill \\ \hfill & -{\displaystyle \frac{b\left(s\right)}{a\left(s\right)}}\left[\Psi \left(\varkappa \left(s\right)+{\varkappa }^{\ast }\left(s\right)\right)-\Psi \left({\varkappa }^{\ast }\left(s\right)\right)\right]\hfill \\ \hfill & \left.\left.+{\displaystyle \frac{b\left(s\right)}{a\left(s\right)}}q\left(s\right)\left[\Psi \left(\varkappa (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right))\right)-\Psi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right\}ds\right|\hfill \\ \hfill & \le 2{\varrho }_{\Phi }∥q∥Q+{\displaystyle \frac{∥b∥}{\alpha }}\left(1+∥q∥\right)\left(\underset{\left|\varkappa \right|\le \lambda +Q}{sup}\left|\Psi \left(\varkappa \right)\right|+\underset{\left|\varkappa \right|\le \lambda }{sup}\left|\Psi \left(\varkappa \right)\right|\right)\hfill \\ \hfill & \le 2{\varrho }_{\Phi }∥q∥Q+\left(1+∥q∥\right)\lambda +{\displaystyle \frac{∥b∥}{\alpha }}\left(1+∥q∥\right)\underset{\left|\varkappa \right|\le \lambda +Q}{sup}\left|\Psi \left(\varkappa \right)\right|,\hfill \end{array}$$

and

$$\left|\left({\Lambda }_2\varkappa \right)\left(\varsigma \right)\right|\le (1+{\varrho }_{\Phi }\parallel q\parallel )\parallel \psi \parallel +{\varrho }_{\Phi }\parallel q\parallel Q.$$

Then,

$$∥{\Lambda }_1\varkappa ∥\le 2{\varrho }_{\Phi }∥q∥Q+\left(1+∥q∥\right)\lambda +{\displaystyle \frac{∥b∥}{\alpha }}\left(1+∥q∥\right)\underset{\left|\varkappa \right|\le \lambda +Q}{sup}\left|\Psi \left(\varkappa \right)\right|,$$

and

$$∥{\Lambda }_{2}y∥\le (1+{\varrho }_{\Phi }\parallel q\parallel )\parallel \psi \parallel +{\varrho }_{\Phi }\parallel q\parallel Q.$$

Thus,

$$\begin{array}{cc}\hfill & \parallel {\Lambda }_1\varkappa +{\Lambda }_{2}y\parallel \hfill \\ \hfill & \le \parallel {\Lambda }_1\varkappa \parallel +\parallel {\Lambda }_{2}y\parallel \hfill \\ \hfill & \le 3{\varrho }_{\Phi }∥q∥Q+\left(1+∥q∥\right)\lambda +{\displaystyle \frac{∥b∥}{\alpha }}\left(1+∥q∥\right)\underset{\left|\varkappa \right|\le \lambda +Q}{sup}\left|\Psi \left(\varkappa \right)\right|+(1+{\varrho }_{\Phi }\parallel q\parallel )\parallel \psi \parallel .\hfill \end{array}$$

By using Condition (10), we obtain $∥{\Lambda }_1\varkappa +{\Lambda }_{2}y∥\le Q$. Hence, we continue with ${\Lambda }_1\varkappa +{\Lambda }_{2}y\in {\Omega }_{\psi }$.

(ii) For all $\varkappa \in {\Omega }_{\psi }$, $∥\varkappa ∥\le Q$, we have $\left|{\left({\Lambda }_1\varkappa \right)}^{\prime }\left(\varsigma \right)\right|=0$, $\varsigma \in [m_0,0]$ and

$$\begin{array}{cc}\hfill \left|{\left({\Lambda }_1\varkappa \right)}^{\prime }\left(\varsigma \right)\right|& =\left|-a\left(\varsigma \right)\left({\Lambda }_1\varkappa \right)\left(\varsigma \right)-2a\left(\varsigma \right)q\left(\varsigma \right)\left[\Phi (\varkappa (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right)))-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right]\right.\hfill \\ \hfill & -b\left(\varsigma \right)\left[\Psi (\varkappa \left(\varsigma \right)+{\varkappa }^{\ast }\left(\varsigma \right))-\Psi \left({\varkappa }^{\ast }\left(\varsigma \right)\right)\right]\hfill \\ \hfill & \left.+b\left(\varsigma \right)q\left(\varsigma \right)\left[\Psi (\varkappa (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right)))-\Psi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right]\right|\hfill \\ \hfill & \le \parallel a\parallel Q+2{\varrho }_{\Phi }∥a∥∥q∥Q+\alpha \lambda +\parallel b\parallel (1+\parallel q\parallel )\underset{\left|\varkappa \right|\le \lambda +Q}{sup}|\Psi \left(\varkappa \right)|.\hfill \end{array}$$

We can see that $\left|{\left({\Lambda }_1\varkappa \right)}^{\prime }\left(\varsigma \right)\right|$ is bounded for all $\psi \in {\Omega }_{\psi }$ and that ${\Lambda }_1{\Omega }_{\psi }$ is a relatively compact subset of $\mathbb{E}$. Hence, ${\Lambda }_1$ is compact.

Let $\left({\varkappa }_n\right)\subset {\Omega }_{\psi }$, $\varkappa \in \mathbb{E}$, ${\varkappa }_n\to \varkappa$ as $n\to \infty$. Then, $\left|{\varkappa }_n\left(\varsigma \right)-\varkappa \left(\varsigma \right)\right|\to 0$ uniformly for $\varsigma \ge m_0$ as $n\to \infty$. Since

$$\begin{array}{cc}\hfill & \parallel {\Lambda }_1{\varkappa }_n-{\Lambda }_1\varkappa \parallel \hfill \\ \hfill & \le 2{\varrho }_{\Phi }∥q∥∥{\varkappa }_n-\varkappa ∥+{\displaystyle \frac{∥b∥}{\alpha }}\left(1+∥q∥\right)\underset{\varsigma \in \left[m_0,\infty \right)}{sup}\left|\Psi \left({\varkappa }_n\left(\varsigma \right)+{\varkappa }^{\ast }\left(\varsigma \right)\right)-\Psi \left(\varkappa \left(\varsigma \right)+{\varkappa }^{\ast }\left(\varsigma \right)\right)\right|,\hfill \end{array}$$

and $\Psi$ is continuous, $∥{\Lambda }_1{\varkappa }_n-{\Lambda }_1\varkappa ∥\to 0$ as $n\to \infty$, and ${\Lambda }_1$ is continuous.

(iii) For all $\varkappa ,y\in {\Omega }_{\psi }$, we have

$$\begin{array}{cc}\hfill & \left|\left({\Lambda }_2\varkappa \right)\left(\varsigma \right)-\left({\Lambda }_{2}y\right)\left(\varsigma \right)\right|\hfill \\ \hfill & =\left|q\left(\varsigma \right)\right|\left|\Phi (\varkappa (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right)))-\Phi (y(\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right)))\right|\hfill \\ \hfill & \le {\varrho }_{\Phi }\parallel q\parallel \parallel \varkappa -y\parallel .\hfill \end{array}$$

Then,

$$∥{\Lambda }_2\varkappa -{\Lambda }_{2}y∥\le \rho \parallel \varkappa -y\parallel .$$

Therefore, ${\Lambda }_2$ is a contraction mapping.

Consequently, there is a $\upsilon \in {\Omega }_{\psi }$ such that $({\Lambda }_1+{\Lambda }_2)\upsilon =\upsilon$ by applying Krasnoselskii’s fixed-point theorem. Hence, $\upsilon$ is a solution for (6). Since the solution $\upsilon$ through $\psi$ for the equation is unique, the solution $\upsilon \left(\varsigma \right)\to 0$ as $\varsigma \to \infty$. □

When $\lambda$ in Theorem 1 and Q in Theorem 2 exist, $\Psi$ and $\Phi$ satisfy the locally Lipschitz condition. Thus, there are constants ${\varrho }_{\Psi },{\varrho }_{\Phi }>0$ such that

$$\left|\Psi (\upsilon \left(\varsigma \right)+{\varkappa }^{\ast }\left(\varsigma \right))-\Psi \left({\varkappa }^{\ast }\left(\varsigma \right)\right)\right|<{\varrho }_{\Psi }\left|\upsilon \left(\varsigma \right)\right|,$$

and

$$\left|\Phi (\upsilon \left(\varsigma \right)+{\varkappa }^{\ast }\left(\varsigma \right))-\Phi \left({\varkappa }^{\ast }\left(\varsigma \right)\right)\right|<{\varrho }_{\Phi }\left|\upsilon \left(\varsigma \right)\right|.$$

Since $\upsilon$ satisfies

$$\begin{array}{cc}\hfill \upsilon \left(\varsigma \right)& =\left\{\psi \left(0\right)-q\left(0\right)\left[\Phi (\psi (-\delta \left(0\right))+{\varkappa }^{\ast }(-\delta \left(0\right)))-\Phi ({\varkappa }^{\ast }(-\delta \left(0\right))\right]\right\}{e}^{-{\int }_0^{\varsigma }a\left(\iota \right)d\iota }\hfill \\ \hfill & +q\left(\varsigma \right)\left[\Phi (\upsilon (\varsigma -\delta \left(\varsigma \right))+{\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right)))-\Phi \left({\varkappa }^{\ast }(\varsigma -\delta \left(\varsigma \right))\right)\right]\hfill \\ \hfill & +{\int }_0^{\varsigma }{e}^{-{\int }_s^{\varsigma }a\left(\iota \right)d\iota }\left\{-2a\left(s\right)q\left(s\right)\left[\Phi (\upsilon (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right)))-\Phi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right.\hfill \\ \hfill & -b\left(s\right)\left[\Psi (\upsilon \left(s\right)+{\varkappa }^{\ast }\left(s\right))-\Psi \left({\varkappa }^{\ast }\left(s\right)\right)\right]\hfill \\ \hfill & \left.+b\left(s\right)q\left(s\right)\left[\Psi (\upsilon (s-\delta \left(s\right))+{\varkappa }^{\ast }(s-\delta \left(s\right)))-\Psi \left({\varkappa }^{\ast }(s-\delta \left(s\right))\right)\right]\right\}ds,\hfill \end{array}$$

then

$$∥\upsilon ∥\le (1+{\varrho }_{\Phi }\parallel q\parallel )\parallel \psi \parallel +3{\varrho }_{\Phi }\parallel q\parallel \parallel \upsilon \parallel +{\displaystyle \frac{\parallel b\parallel }{\alpha }}(1+\parallel q\parallel ){\varrho }_{\Psi }\parallel \upsilon \parallel ,$$

that is,

$$\left[1-3{\varrho }_{\Phi }\parallel q\parallel -{\displaystyle \frac{\parallel b\parallel }{\alpha }}(1+\parallel q\parallel ){\varrho }_{\Psi }\right]\parallel \upsilon \parallel \le (1+{\varrho }_{\Phi }\parallel q\parallel )\parallel \psi \parallel .$$

Then, there clearly exists a $\sigma >0$ for each $ϵ>0$ such that $\left|\upsilon \left(\varsigma \right)\right|<ϵ$ for all $\varsigma \ge m_0$ if $∥\psi ∥<\sigma$. Thus, we obtain the next theorem.

Theorem 3.

Suppose that ${\varrho }_{\Psi }$ and ${\varrho }_{\Phi }$ satisfy

$$1-3{\varrho }_{\Phi }\parallel q\parallel -{\displaystyle \frac{\parallel b\parallel }{\alpha }}(1+\parallel q\parallel ){\varrho }_{\Psi }>0.$$

Then, the zero solution for (6) is stable.

4. Application in Smart Grid Optimization

In this section, we explore the application of the modified model (1) to optimize the power distribution in smart grids. A smart grid is an advanced electrical grid system that integrates digital communication, automation, and modern control technologies to improve the efficiency, reliability, and sustainability of electricity production, distribution, and consumption. It enables two-way communication between power providers and consumers, allowing for real-time monitoring, demand response, and better integration of renewable energy sources. Modern smart grids face significant challenges due to the following:

• Variability in Renewable Energy Sources: Solar and wind power introduce periodic and nonlinear fluctuations in energy supply [34,37].
• Delays in Power Transmission and Communication: Signal transmission and control delays impact the stability of power distribution [33,35].
• Dynamic Load Demand: Fluctuations in electricity consumption require advanced models to maintain stability and efficiency [36].

4.1. Problem Statement

The findings in Section 2 and Section 3 provide a mathematical foundation for addressing these challenges. Specifically, the Existence of Periodic Solutions: Theorem 1 establishes sufficient conditions for periodic solutions in nonlinear neutral transmission line models. Traditional power system models often assume differentiability and linearity, which do not accurately capture the real-world behavior of smart grids. This directly applies to smart grids where voltage variations follow periodic patterns due to renewable energy sources [38,39]. Thus, there is a need for a robust mathematical framework that can address the following:

• Periodic and nonlinear variations in voltage and power flow.
• The impact of transmission delays on grid stability.
• The long-term asymptotic behavior of voltage fluctuations in the presence of nonlinearities.

4.2. Application of Theoretical Results

Equation (1) can address these challenges by capturing the essential dynamics of power distribution networks, where

• $\varkappa \left(\varsigma \right)$ represents the voltage at a node in the network.
• $p\left(\varsigma \right)$ models the periodic power input from renewable sources.
• $\delta \left(\varsigma \right)$ accounts for delays due to transmission distances.
• $\Phi (·)$ and $\Psi (·)$ describe nonlinear load characteristics.

Using Equation (1), the voltage dynamics at each node can be modeled, considering the following:

• The nonlinear behavior of loads and power sources.
• Delayed feedback due to communication or signal propagation.
• Periodic variations in power generation and consumption.

The periodicity conditions in Theorem 1 ensure that under appropriate parameter constraints, the voltage will exhibit stable periodic behavior, avoiding instabilities in the grid.

4.3. Simulation of Voltage Dynamics

For the simulation of Model (1), the following parameters and functions were defined:

• T: Period of the system, $T=2\pi$.
• $\delta \left(\varsigma \right)$: Delay term, defined as $\delta \left(\varsigma \right)=0.5+0.2sin\left(\varsigma \right)$.
• $a\left(\varsigma \right)$: Periodic coefficient, defined as $a\left(\varsigma \right)=2+sin\left(\varsigma \right)$.
• $b\left(\varsigma \right)$: Periodic coefficient, defined as $b\left(\varsigma \right)=cos\left(\varsigma \right)+1.5$.
• $p\left(\varsigma \right)$: Periodic power input, defined as $p\left(\varsigma \right)=1+0.5sin\left(2\varsigma \right)$.
• $q\left(\varsigma \right)$: Periodic coefficient, defined as $q\left(\varsigma \right)=0.5+0.1sin\left(\varsigma \right)$.
• $\Phi \left(\varkappa \right)$: Nonlinear function, defined as $\Phi \left(\varkappa \right)=sin\left(\varkappa \right)$.
• $\Psi \left(\varkappa \right)$: Nonlinear function, defined as $\Psi \left(\varkappa \right)=tanh\left(\varkappa \right)$.

These parameters and functions capture the nonlinear, delayed, and periodic behaviors relevant to the dynamics of voltage in a smart grid environment. The simulation of Model (1) was performed to analyze the voltage dynamics ($\varkappa \left(\varsigma \right)$) over time. Figure 1 illustrates the result of the simulation, showing the periodic behavior.

5. Conclusions

In this paper, we developed new sufficient conditions for the existence and asymptotic stability of periodic solutions in nonlinear neutral transmission line models using Krasnoselskii’s fixed-point theorem. By removing differentiability assumptions on delay terms and coefficients, our results extend existing theoretical studies in the field. A test example was presented to illustrate the applicability of the proposed conditions.

Our findings contribute to the theoretical understanding of nonlinear neutral systems, particularly in the context of smart grid stability. However, additional research is needed to explore broader applications and validate the theoretical results with real-world data. Future work could consider numerical simulations, experimental validation, and extensions to more complex models incorporating stochastic effects or distributed parameter systems.