Browsing by Author "Yao, Jen-Chih"

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Editorial for the Special Issue of "fractional Differential and Fractional Integro-Differential Equations: Qualitative Theory, Numerical Simulations, and Symmetry Analysis
(Mdpi, 2024) Tunc, Cemil; Yao, Jen-Chih; Benchohra, Mouffak; El-Sayed, Ahmed M. A.
Global Existence and Uniqueness of Solutions of Integral Equations With Multiple Variable Delays and Integro Differential Equations: Progressive Contractions
(Mdpi, 2024) Tunc, Osman; Tunc, Cemil; Yao, Jen-Chih
In this work, we delve into a nonlinear integral equation (IEq) with multiple variable time delays and a nonlinear integro-differential equation (IDEq) without delay. Global existence and uniqueness (GEU) of solutions of that IEq with multiple variable time delays and IDEq are investigated by the fixed point method using progressive contractions, which are due to T.A. Burton. We prove four new theorems including sufficient conditions with regard to GEU of solutions of the equations. The results generalize and improve some related published results of the relevant literature.
New and Improved Criteria on Fundamental Properties of Solutions of Integro-Delay Differential Equations With Constant Delay
(Mdpi, 2021) Tunc, Cemil; Wang, Yuanheng; Tunc, Osman; Yao, Jen-Chih
This paper is concerned with certain non-linear unperturbed and perturbed systems of integro-delay differential equations (IDDEs). We investigate fundamental properties of solutions such as uniformly stability (US), uniformly asymptotically stability (UAS), integrability and instability of the un-perturbed system of the IDDEs as well as the boundedness of the perturbed system of IDDEs. In this paper, five new and improved fundamental qualitative results, which have less conservative conditions, are obtained on the mentioned fundamental properties of solutions. The technique used in the proofs depends on Lyapunov-Krasovski functionals (LKFs). In particular cases, three examples and their numerical simulations are provided as numerical applications of this paper. This paper provides new, extensive and improved contributions to the theory of IDDEs.
New Fundamental Results on the Continuous and Discrete Integro-Differential Equations
(Mdpi, 2022) Tunc, Osman; Tunc, Cemil; Yao, Jen-Chih; Wen, Ching-Feng
This work studies certain perturbed and un-perturbed nonlinear systems of continuous and discrete integro-delay differential equations (IDDEs). Using the Lyapunov-Krasovskii functional (LKF) method and the Lyapunov-Razumikhin method (LRM), uniform asymptotic stability (UAS), uniform stability (US), integrability and boundedness of solutions as well as exponential stability (ES) and instability of solutions are discussed. In this paper, five new theorems and a corollary are given and three numerical applications are provided with their simulations. With this work, we aim to make new contributions to the theory of the continuous and discrete integro-differential equations.
New Results on Ulam Stabilities of Nonlinear Integral Equations
(Mdpi, 2024) Tunc, Osman; Tunc, Cemil; Yao, Jen-Chih
This article deals with the study of Hyers-Ulam stability (HU stability) and Hyers-Ulam-Rassias stability (HUR stability) for two classes of nonlinear Volterra integral equations (VIEqs), which are Hammerstein-type integral and Hammerstein-type functional integral equations, respectively. In this article, both the HU stability and HUR stability are obtained for the first integral equation and the HUR stability is obtained for the second integral equation. Among the used techniques, we present fixed point arguments and the Gronwall lemma as a basic tool. Two supporting examples are also provided to demonstrate the applications and effectiveness of the results.
On the Enhanced New Qualitative Results of Nonlinear Integro-Differential Equations
(Mdpi, 2023) Tunc, Cemil; Tunc, Osman; Yao, Jen-Chih
In this article, a class of scalar nonlinear integro-differential equations of first order with fading memory is investigated. For the considered fading memory problem, we discuss the effects of the memory over all the values of the parameter in the kernel of the equations. Using the Lyapunov-Krasovski functional method, we give various sufficient conditions of stability, asymptotic stability, uniform stability of zero solution, convergence and boundedness, and square integrability of nonzero solutions in relation to the considered scalar nonlinear integro-differential equations for various cases. As the novel contributions of this article, the new scalar nonlinear integro-differential equation with the fading memory is firstly investigated in the literature, and seven theorems, which have novel sufficient qualitative conditions, are provided on the qualitative behaviors of solutions called boundedness, convergence, stability, integrability, asymptotic stability and uniform stability of solutions. The novel outcomes and originality of this article are that the considered integro-differential equations are new mathematical models, they include former mathematical models in relation to the mathematical models of this paper as well as the given main seven qualitative results are also new. The outcomes of this paper enhance some present results and provide new contributions to the relevant literature. The results of the article have complementary properties for the symmetry of integro-differential equations.
On the Existence of Results for Multiple Retarded Differential and Integro-Differential Equations of Second Order
(Yokohama Publ, 2024) Tunc, Osman; Tunc, Cemil; Yao, Jen-Chih
. In this research paper, we bear in mind a class of nonlinear impulsive delay differential equations (IDDEs) and impulsive delay integro-differential equations (IDIDEs) of second order with multiple constant delays. We depict sufficient hypotheses for the existence of solutions (EOS) of these IDDEs and IDIDEs of second order. Based upon the Schaefer fixed point theorem (Schaefer FPT) and control by impulses, we gain the proofs in relation to the EOS of the IDDEs and the IDIDEs of second order. The effects of impulses are emphasized in the content of the proofs. The outcomes of this research paper have more general forms and make new improvements in relation to the some recent ones of the literature.
On the Stability, Integrability and Boundedness Analysis of Systems of Integro-Differential Equations With Time-Delay
(House Book Science-casa Cartii Stiinta, 2023) Tunc, Cemil; Tunc, Osman; Yao, Jen-Chih
In the paper "AIMS Math. 5 (2020), no. 6, 6448-6456", Tian et al. [15, Theorem 1] considered a linear system of integro-time delay differential equations (IDDEs) with constant time retardation. In [15], firstly, a generalized double integral inequality was obtained and then less conservative asymptotically stability criteria were proposed by using that double integral inequality and choosing a new Lyapunov-Krasovskii functional (LKF). To the best of the information, we would like to note that the asymptotically stability criteria of Tian et al. [15, Theorem 1] consist of very interesting but strong conditions. However, in this paper, we define a more suitable LKF, then we obtain the result of Tian et al. [15, Theorem 1] for uniformly asymptotically stability under very weaker conditions using the LKF and also investigate the integrability of the norm and boundedness of solutions. To show the effectiveness of our results, two numerical examples are proposed for the uniformly asymptotically stability as well as integrability and boundedness of solutions. By this work, we do contributions to the work of Tian et al. [15, Theorem 1] under very weaker conditions and those in the previous relevant literature, and we obtain two more new results on the integrability of the norm of solutions and boundedness of solutions. The results of this paper are new and they may be useful for researchers working on the topics of this paper.
Qualitative Analyses of Differential Systems With Time-Varying Delays Via Lyapunov-Krasovskii Approach
(Mdpi, 2021) Tunc, Cemil; Tunc, Osman; Wang, Yuanheng; Yao, Jen-Chih
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-Krasovskii functionals (LKFs) as the main tools in proofs. The technique of the proofs depends upon the Lyapunov-Krasovskii 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.
Qualitative Analyses of Integro-Fractional Differential Equations With Caputo Derivatives and Retardations Via the Lyapunov-Razumikhin Method
(Mdpi, 2021) Tunc, Osman; Atan, Ozkan; Tunc, Cemil; Yao, Jen-Chih
The purpose of this paper is to investigate some qualitative properties of solutions of nonlinear fractional retarded Volterra integro-differential equations (FrRIDEs) with Caputo fractional derivatives. These properties include uniform stability, asymptotic stability, Mittag-Leffer stability and boundedness. The presented results are proved by defining an appropriate Lyapunov function and applying the Lyapunov-Razumikhin method (LRM). Hence, some results that are available in the literature are improved for the FrRIDEs and obtained under weaker conditions via the advantage of the LRM. In order to illustrate the results, two examples are provided.