A Generalization of Bihari's Lemma for Discontinuous Functions and Its Application to the Stability Problem of Differential Equations with Impulse Disturbance

Abstract This paper presents a generalization of nonlinear integral inequalities of the Gronwall–Bellman–Bihari type for discontinuous functions and its application to the investigation of the practical stability of solutions of systems of integro-differential equations with impulse perturbations at fixed moments of time.

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Implicit nonlinear fractional differential equations of variable order

Boundary Value Problems ◽

10.1186/s13661-021-01540-7 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Amar Benkerrouche ◽

Mohammed Said Souid ◽

Kanokwan Sitthithakerngkiet ◽

Ali Hakem

Keyword(s):

Differential Equations ◽

Boundary Value Problem ◽

Fractional Differential Equations ◽

Boundary Value ◽

Variable Order ◽

Stability Of Solutions ◽

Nonlinear Fractional Differential Equations ◽

Caputo Fractional Differential Equations ◽

Fractional Differential ◽

The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

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On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva ◽

10.15507/2079-6900.20.201802.260-272 ◽

2018 ◽

Vol 20 (3) ◽

pp. 260-272

Author(s):

A. S. Andreev ◽

O. A. Peregudova

Keyword(s):

Differential Equations ◽

Stability Problem ◽

Infinite Delay ◽

Zero Solution ◽

Model Systems ◽

Lyapunov Functionals ◽

Structure Equations ◽

Positive Limit Set ◽

The Stability ◽

The Right

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

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Stability of delay differential equations in the sense of Ulam on unbounded intervals

An International Journal of Optimization and Control Theories & Applications (IJOCTA) ◽

10.11121/ijocta.01.2019.00628 ◽

2019 ◽

Vol 9 (2) ◽

pp. 125-131 ◽

Cited By ~ 1

Author(s):

Süleyman Öğrekçi

Keyword(s):

Differential Equations ◽

Delay Differential Equations ◽

Stability Problem ◽

The Stability ◽

Delay Differential ◽

Unbounded Intervals

In this paper, we consider the stability problem of delay differential equations in the sense of Hyers-Ulam-Rassias. Recently this problem has been solved for bounded intervals, our result extends and improve the literature by obtaining stability in unbounded intervals. An illustrative example is also given to compare these results and visualize the improvement.

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Stability of Solutions of Ordinary Differential Equations with Respect to a Closed Set

Canadian Journal of Mathematics ◽

10.4153/cjm-1968-070-3 ◽

1968 ◽

Vol 20 ◽

pp. 720-726

Author(s):

T. G. Hallam ◽

V. Komkov

Keyword(s):

Differential Equation ◽

Ordinary Differential Equation ◽

Differential Equations ◽

Ordinary Differential Equations ◽

Compact Set ◽

Stability Of Solutions ◽

Closed Set ◽

The Stability ◽

Stability Results

The stability of the solutions of an ordinary differential equation will be discussed here. The purpose of this note is to compare the stability results which are valid with respect to a compact set and the stability results valid with respect to an unbounded set. The stability of sets is a generalization of stability in the sense of Liapunov and has been discussed by LaSalle (5; 6), LaSalle and Lefschetz (7, p. 58), and Yoshizawa (8; 9; 10).

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