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Keywords = Lyapunov–Krasovskiĭ functional

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18 pages, 296 KiB  
Article
On the Fundamental Analyses of Solutions to Nonlinear Integro-Differential Equations of the Second Order
by Cemil Tunç and Osman Tunç
Mathematics 2022, 10(22), 4235; https://doi.org/10.3390/math10224235 - 13 Nov 2022
Cited by 16 | Viewed by 1813
Abstract
In this article, a scalar nonlinear integro-differential equation of second order and a non-linear system of integro-differential equations with infinite delays are considered. Qualitative properties of solutions called the global asymptotic stability, integrability and boundedness of solutions of the second-order scalar nonlinear integro-differential [...] Read more.
In this article, a scalar nonlinear integro-differential equation of second order and a non-linear system of integro-differential equations with infinite delays are considered. Qualitative properties of solutions called the global asymptotic stability, integrability and boundedness of solutions of the second-order scalar nonlinear integro-differential equation and the nonlinear system of nonlinear integro-differential equations with infinite delays are discussed. In the article, new explicit qualitative conditions are presented for solutions of both the second-order scalar nonlinear integro-differential equations with infinite delay and the nonlinear system of integro-differential equations with infinite delay. The proofs of the main results of the article are based on two new Lyapunov–Krasovskiĭ functionals. In particular cases, the results of the article are illustrated with three numerical examples, and connections to known tests are discussed. The main novelty and originality of this article are that the considered integro-differential equation and system of integro-differential equations with infinite delays are new mathematical models, the main six qualitative results given are also new. Full article
(This article belongs to the Special Issue Mathematical Modeling with Differential Equations in Biology, Chemistry, Economics, Finance and Physics, Volume 2)
20 pages, 845 KiB  
Article
Delay-Dependent Stability, Integrability and Boundedeness Criteria for Delay Differential Systems
by Osman Tunç, Cemil Tunç and Yuanheng Wang
Axioms 2021, 10(3), 138; https://doi.org/10.3390/axioms10030138 - 29 Jun 2021
Cited by 17 | Viewed by 2438
Abstract
This paper deals with non-perturbed and perturbed systems of nonlinear differential systems of first order with multiple time-varying delays. Here, for the considered systems, easily verifiable and applicable uniformly asymptotic stability, integrability, and boundedness criteria are obtained via defining an appropriate Lyapunov–Krasovskiĭ functional [...] Read more.
This paper deals with non-perturbed and perturbed systems of nonlinear differential systems of first order with multiple time-varying delays. Here, for the considered systems, easily verifiable and applicable uniformly asymptotic stability, integrability, and boundedness criteria are obtained via defining an appropriate Lyapunov–Krasovskiĭ functional (LKF) and using the Lyapunov–Krasovskiĭ method (LKM). Comparisons with a former result that can be found in the literature illustrate the novelty of the stability theorem and show new contributions to the qualitative theory of solutions. A discussion of two illustrative examples and the obtained results are presented. Full article
(This article belongs to the Special Issue Special Issue in Honor of the 60th Birthday of Professor Hong-Kun Xu)
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20 pages, 294 KiB  
Article
Qualitative Analyses of Differential Systems with Time-Varying Delays via Lyapunov–Krasovskiĭ Approach
by Cemil Tunç, Osman Tunç, Yuanheng Wang and Jen-Chih Yao
Mathematics 2021, 9(11), 1196; https://doi.org/10.3390/math9111196 - 25 May 2021
Cited by 25 | Viewed by 2575
Abstract
In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and [...] Read more.
In this paper, a class of systems of linear and non-linear delay differential equations (DDEs) of first order with time-varying delay is considered. We obtain new sufficient conditions for uniform asymptotic stability of zero solution, integrability of solutions of an unperturbed system and boundedness of solutions of a perturbed system. We construct two appropriate Lyapunov–Krasovskiĭ functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov–Krasovskiĭ method. For illustration, two examples are provided in particular cases. An advantage of the new LKFs used here is that they allow to eliminate using Gronwall’s inequality. When we compare our results with recent results in the literature, the established conditions are more general, less restrictive and optimal for applications. Full article
(This article belongs to the Special Issue Mathematical Analysis and Analytic Number Theory 2020)
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