Subspaces of Stable and Unstable Solutions of a Functional Differential Equation in a Fading Memory Space: The Critical Case

1987 ◽  
Vol 18 (5) ◽  
pp. 1323-1340 ◽  
Author(s):  
G. S. Jordan ◽  
Olof J. Staffans ◽  
Robert L. Wheeler
Keyword(s):  
Differential Equation ◽  
Functional Differential Equation ◽  
Critical Case ◽  
Fading Memory ◽  
Memory Space ◽  
Unstable Solutions ◽  
Functional Differential

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.

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