scholarly journals Asymptotic equivalence of impulsive differential equations in a Banach space

1990 ◽  
Vol 34 ◽  
pp. 249-257 ◽  
Author(s):  
D. D. Bainov ◽  
S. I. Kostadinov ◽  
A. D. Myshkis
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Impulsive Differential Equations ◽  
Asymptotic Equivalence

scholarly journals The sufficient conditions for polystability of solutions of~nonlinear systems of ordinary differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 304-317
Author(s):  
Pavel A. Shamanaev ◽  
Olga S. Yazovtseva
Keyword(s):  
Banach Space ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Approximation ◽  
Critical Case ◽  
Sufficient Conditions ◽  
Asymptotic Equivalence ◽  
Theorem Proof

The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.

scholarly journals Lp-equivalence of a linear and a nonlinear impulsive differential equation in a Banach space

1993 ◽  
Vol 36 (1) ◽  
pp. 17-33 ◽  
Author(s):  
D. D. Bainov ◽  
S. I. Kostadinov ◽  
P. P. Zabreiko
Keyword(s):  
Differential Equation ◽  
Banach Space ◽  
Differential Equations ◽  
Impulsive Differential Equation ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations

In the present paper by means of the Schauder-Tychonoff principle sufficient conditions are obtained for Lp-equivalence of a linear and a nonlinear impulsive differential equations.

scholarly journals The sufficient conditions for polystability of solutions of nonlinear systems of ordinary differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 304-317
Author(s):  
P. A. Shamanaev ◽  
O. S. Yazovtseva
Keyword(s):  
Banach Space ◽  
Nonlinear System ◽  
Nonlinear Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Approximation ◽  
Critical Case ◽  
Sufficient Conditions ◽  
Asymptotic Equivalence ◽  
Theorem Proof

The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.

scholarly journals Nonlinear Impulsive Differential Equations with Weighted Exponential or Ordinary Dichotomous Linear Part in a Banach Space

2015 ◽  
Vol 2015 ◽  
pp. 1-7 ◽  
Author(s):  
Hristo Kiskinov ◽  
Andrey Zahariev
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Differential Equations ◽  
Linear Part ◽  
Sufficient Conditions ◽  
Impulsive Differential Equations ◽  
Bounded Solutions ◽  
Fixed Point Principle

We consider nonlinear impulsive differential equations withψ-exponential andψ-ordinary dichotomous linear part in a Banach space. By the help of Banach’s fixed-point principle sufficient conditions are found for the existence ofψ-bounded solutions of these equations onRandR+.

scholarly journals An Alternative Method for the Study of Impulsive Differential Equations of Fractional Orders in a Banach Space

2013 ◽  
Vol 2013 ◽  
pp. 1-12 ◽  
Author(s):  
Asma Bouzaroura ◽  
Saïd Mazouzi
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Fixed Point Theorem ◽  
Differential Equations ◽  
Nonlocal Condition ◽  
Impulsive Differential Equations ◽  
Fractional Equations ◽  
Contraction Theorem ◽  
Fractional Orders ◽  
Stability Of The Solution

This paper is concerned with the existence, uniqueness, and stability of the solution of some impulsive fractional problem in a Banach space subjected to a nonlocal condition. Meanwhile, we give a new concept of a solution to impulsive fractional equations of multiorders. The derived results are based on Banach's contraction theorem as well as Schaefer's fixed point theorem.

Existence of integral manifolds for impulsive differential equations in a Banach space

1989 ◽  
Vol 28 (7) ◽  
pp. 815-833 ◽  
Author(s):  
D. D. Bainov ◽  
S. I. Kostadinov ◽  
Nguy�� H�ng Th�i ◽  
P. P. Zabreiko
Keyword(s):  
Banach Space ◽  
Differential Equations ◽  
Impulsive Differential Equations ◽  
Integral Manifolds
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