Geodesic
Existence and uniform asymptotic stability for an abstract differential equation with infinite delay
Electronic journal of differential equations, Tome 2012 (2012)
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.
Classification : 35B35, 37L15
Keywords: infinite delay, sectorial operator, mild solution, uniform asymptotic stability, fixed point method 
@article{EJDE_2012__2012__a63,
     author = {Cung The Anh and Le Van Hieu},
     title = {Existence and uniform asymptotic stability for an abstract differential equation with infinite delay},
     journal = {Electronic journal of differential equations},
     year = {2012},
     volume = {2012},
     zbl = {1248.34114},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a63/}
}
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Cung The Anh; Le Van Hieu. Existence and uniform asymptotic stability for an abstract differential equation with infinite delay. Electronic journal of differential equations, Tome 2012 (2012). http://geodesic.mathdoc.fr/item/EJDE_2012__2012__a63/