scholarly journals Uniform Stability In Nonlinear Infinite Delay Volterra Integro-differential Equations Using Lyapunov Functionals

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Youssef Raffoul ◽  
Habib Rai
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Exponential Stability ◽  
Lyapunov Functional ◽  
Infinite Delay ◽  
Zero Solution ◽  
Uniform Stability ◽  
Lyapunov Functionals ◽  
Integro Differential Equation ◽  
The Stability

AbstractIn [10] the first author used Lyapunov functionals and studied the exponential stability of the zero solution of finite delay Volterra Integro-differential equation. In this paper, we use modified version of the Lyapunov functional that were used in [10] to obtain criterion for the stability of the zero solution of the infinite delay nonlinear Volterra integro-differential equation

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

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.

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

2018 ◽  
Vol 20 (3) ◽  
pp. 260-272
Author(s):  
Aleksandr S. Andreev ◽  
Olga 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.

scholarly journals Functional differential equations with infinite delay in Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 497-508 ◽  
Author(s):  
Jin Liang ◽  
Tijun Xiao
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Exponential Stability ◽  
Functional Differential Equation ◽  
Infinite Delay ◽  
Functional Differential Equations ◽  
Zero Solution ◽  
Fading Memory ◽  
Fundamental Operator ◽  
Functional Differential

In this paper, a definition of the fundamental operator for the linear autonomous functional differential equation with infinite delay in a Banach space is given, and some sufficient and necessary conditions of the fundamental operator being exponentially stable in abstract phase spaces which satisfy some suitable hypotheses are obtained. Moreover, we discuss the relation between the exponential asymptotic stability of the zero solution of nonlinear functional differential equation with infinite delay in a Banach space and the exponential stability of the solution semigroup of the corresponding linear equation, and find that the exponential stability problem of the zero solution for the nonlinear equation can be discussed only in the exponentially fading memory phase space.

scholarly journals Some new results on the stability and boundedness of solutions of certain class of second order vector differential equations

2021 ◽  
Vol 7 (1) ◽  
pp. 108-115
Author(s):  
A.A. ADEYANJU ◽  
◽  
D.O. ADAMS ◽  
Keyword(s):  
Differential Equation ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Order Differential Equation ◽  
Zero Solution ◽  
Second Order ◽  
Boundedness Of Solutions ◽  
Second Order Differential Equation ◽  
All Solutions ◽  
The Stability

n this paper, we provide certain conditions that guarantee the stability of the zero solution when P(t, X, Y) = 0 and boundedness of all solutions when P(t, X, Y)# 0 of a certain system of second order differential equation using a suitable Lyapunov function. The results in this paper are quite new and complement those in the literature. Examples are given to demonstrate the correctness of the established results.

scholarly journals A Stability Result for the Solutions of a Certain System of Fourth-Order Delay Differential Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
A. M. A. Abou-El-Ela ◽  
A. I. Sadek ◽  
A. M. Mahmoud ◽  
R. O. A. Taie
Keyword(s):  
Differential Equation ◽  
Fourth Order ◽  
Delay Differential Equation ◽  
Lyapunov Functional ◽  
Sufficient Conditions ◽  
Stability Result ◽  
Zero Solution ◽  
Uniform Stability ◽  
Delay Differential

The main purpose of this work is to give sufficient conditions for the uniform stability of the zero solution of a certain fourth-order vector delay differential equation of the following form:X(4)+F(X˙,X¨)X⃛+Φ(X¨)+G(X˙(t-r))+H(X(t-r))=0.By constructing a Lyapunov functional, we obtained the result of stability.

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

Sign in / Sign up

Export Citation Format

Share Document