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An application of a measure of noncompactness in the study of asymptotic stability. (English) Zbl 1015.47034

Let BC( + ) be the Banach space of all real functions which are defined, bounded and continuous on + with the sup|x| norm. Let F be an operator transforming the space B( + ) into itself and such that

( F x ) ( t ) - ( F y ) ( t )kx ( t ) - y ( t )+a(t)

for all functions x,yBC( + ) and for any t + , k(0,1) and a: + + is a continuous function such that lim t a(t)=0.

Further, assume that x=x(t) (xBC( + )) is a solution of the operator equation

x=Fx·(*)

In the paper under review, the following result is proved: The function x is an asymptotically stable solution of equation (*) if for any ε>0 there exists T>0 such that for every tT and for every other solution y of equation (*) the inequality |x(t)-y(t)|ε holds.

As an application, the functional-integral equation x(t)=f(t,x(t))+ 0 t u(t,s,x(s))ds is studied.


MSC:
47H09Mappings defined by “shrinking” properties
45G10Nonsingular nonlinear integral equations
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47N20Applications of operator theory to differential and integral equations