LINEAR DIFFERENTIAL EQUATIONS IN SCALES OF BANACH SPACES

Analysis ◽  
1992 ◽  
Vol 12 (1-2) ◽  
pp. 31-46 ◽  
Author(s):  
J. Appell ◽  
P.P. Zabrejko
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Linear Differential Equations ◽  
Linear Differential ◽  
Scales Of Banach Spaces

Differential equations in Banach spaces and the extension of Lyapunov's method

1963 ◽  
Vol 59 (2) ◽  
pp. 373-381 ◽  
Author(s):  
V. Lakshmikantham
Keyword(s):  
Differential Equations ◽  
Banach Spaces ◽  
Existence Theorems ◽  
Linear Operators ◽  
Linear Differential Equations ◽  
Bounded Operators ◽  
Lyapunov's Method ◽  
Linear Differential ◽  
Lyapunov’S Method ◽  
Image Position

The concept of Lyapunov's function is an important tool in studying various problems of ordinary differential equations. In the present paper we shall extend the Lyapunov's method to study some problems of differential equations in Banach spaces. Continuing the theory of one parameter semi-groups of linear and bounded operators founded by Hille and Yoshida, Kato(4) presented some uniqueness and existence theorems for the solutions of linear differential equations of the typewhere A(t) is a given function whose values are linear operators in Banach space. Krasnoselskii, Krein and Soboleveskii (5,6) also considered such equations including non-linear differential equations of the typeMlak (9) obtained some results concerning the limitations of solutions of the latter equation.

Stabilization of Functions and its Application

1994 ◽  
Vol 1 (2) ◽  
pp. 183-195
Author(s):  
L. D. Kudryavtsev
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Banach Spaces ◽  
Existence And Uniqueness ◽  
Fundamental System ◽  
Linear Differential Equations ◽  
Uniqueness Of Solutions ◽  
Strong Stabilization ◽  
Linear Differential ◽  
Polynomial Stabilization

Abstract The concepts of polynomial stabilization, strong polynomial stabilization, and strong stabilization are introduced for a fundamental system of solutions of linear differential equations. Some criteria of such kinds of stabilizations and applications to the theory of existence and uniqueness of solutions of ordinary differential equations are given. An abstract scheme of the obtained results is presented for Banach spaces.

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