Electronic Journal of Differential Equations,
Vol. 2012 (2012), No. 51, pp. 1-14.

Title: Existence and uniform asymptotic stability for an abstract differential equation
       with infinite delay

Author: Cung The Anh (Hanoi National Univ., Vietnam)
        Le Van Hieu  (Academy of Journalism and Comm., Hanoi, Vietnam)

Abstract:
 Using the Contraction Mapping Principle, we study the existence, uniqueness,
 and uniform asymptotic stability of solutions to an abstract differential
 equation with infinite delay of the form $du(t)/dt+Au(t)=B(t,u_t)$,
 where A is  a positive sectorial operator and the nonlinear part B
 is Lipschitz continuous with respect to a fractional power of A in the
 second variable and the Lipschitz coefficient may depend on time t.
 Some special cases and examples are provided to illustrate the results 
 obtained.
 
Submitted June 3, 2011. Published March 29, 2012.
Math Subject Classifications: 35B35, 37L15.
Key Words: Infinite delay; sectorial operator; mild solution; 
           uniform asymptotic stability; fixed point method.