Exponential stability of integro-differential equations and applications

Ngoc, Pham Huu Anh; Hieu, Le Trung

doi:10.1016/j.aml.2021.107127

Cited by 3 publications

(4 citation statements)

References 6 publications

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“…and (8), transform Equation (7) into the form of a neutral-type equation Remark 2. The representation (6) in particular means that the level of nonlinearity of Equation ( 5) is more than one. In this case, it is known that sufficient conditions for the asymptotic mean square stability of the zero solution of the linear Equation (7) are also sufficient conditions for stability in probability of the zero solution of the nonlinear Equation (5) and therefore are sufficient conditions for stability in probability of the equilibrium x * of Equation (4) [4].…”

Section: Statement Of the Problemmentioning

confidence: 99%

“…It is known that after the works of Krasovskii N.N. [1][2][3], the method of Lyapunov functionals or the so-called method of Lyapunov-Krasovskii functionals is one of the most powerful methods of stability investigation for functional differential equations (see, for instance [4][5][6][7][8] and the references therein). The special procedure of Lyapunov functionals construction allows for the construction of different Lyapunov functionals for one differential equation with delay and, as a result, obtains different stability conditions for the considered equation [4].…”

Section: Introductionmentioning

confidence: 99%

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Some Generalization of the Method of Stability Investigation for Nonlinear Stochastic Delay Differential Equations

Shaikhet

2022

Symmetry

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It is known that the method of Lyapunov functionals is a powerful method of stability investigation for functional differential equations. Here, it is shown how the previously proposed method of stability investigation for nonlinear stochastic differential equations with delay and a high order of nonlinearity can be extended to nonlinear mathematical models of a much more general form. An important feature is the combination of the method of Lyapunov functionals with the method of Linear Matrix Inequalities (LMIs). Some examples of applications of the proposed method of stability research to known mathematical models are given.

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Section: Statement Of the Problemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Some Generalization of the Method of Stability Investigation for Nonlinear Stochastic Delay Differential Equations

Shaikhet

2022

Symmetry

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“…In particular, some problems of standard closed electric RLC circuits, heat transfer, fluid mechanics, radiation, population dynamics, etc. are described by IDEs and IDDEs ( [2], [4], [5], [6], [9], [10], [12]). In particular, we would like to mention that the subject of qualitative theory of IDDEs and IDEs both with and without time retardation(s) is a hot and attractive topic, and it deserves investigation.…”

Section: Introductionmentioning

confidence: 99%

On the stability, integrability and boundedness analysis of systems of integro-differential equations with time-delay

Cemil,

Osman,

Jen-Chih

2023

FPT-RO

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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.

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“…The stability of differential equations has grown to be one of the considerable areas in the field of mathematical analysis, and we find many different types of stability, such as exponential [7,17,23], Mittag-Leffler [20,27], Hyers-Ulam (HU) stability and other types of stability [13,18,19,21,22]. Among these kinds of stability, Hyers-Ulam stability and its various types provide a bridge between the exact and numerical solutions, so researchers devoted their work to the study of different kinds of HU stability for nonlinear fractional differential equation; see [5,10,12,29].…”

Section: Introductionmentioning

confidence: 99%

Existence and stability analysis of solutions for a new kind of boundary value problems of nonlinear fractional differential equations

Liu

Liu²

2022

NAMC

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This research work is dedicated to an investigation for a new kind of boundary value problem of nonlinear fractional differential equation supplemented with general boundary condition. A full analysis of existence and uniqueness of positive solutions is respectively proved by Leray–Schauder nonlinear alternative theorem and Boyd–Wong’s contraction principles. Furthermore, we prove the Hyers–Ulam (HU) stability and Hyers–Ulam–Rassias (HUR) stability of solutions. An example illustrating the validity of the existence result is also discussed.

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Exponential stability of integro-differential equations and applications

Cited by 3 publications

References 6 publications

Some Generalization of the Method of Stability Investigation for Nonlinear Stochastic Delay Differential Equations

Some Generalization of the Method of Stability Investigation for Nonlinear Stochastic Delay Differential Equations

On the stability, integrability and boundedness analysis of systems of integro-differential equations with time-delay

Existence and stability analysis of solutions for a new kind of boundary value problems of nonlinear fractional differential equations

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