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Zbl 1015.47034
Banaś, J.; Rzepka, B.
An application of a measure of noncompactness in the study of asymptotic stability.
(English)
[J] Appl. Math. Lett. 16, No.1, 1-6 (2003). ISSN 0893-9659

Let $BC(\bbfR_+)$ be the Banach space of all real functions which are defined, bounded and continuous on $\bbfR_+$ with the $\sup|x |$ norm. Let $F$ be an operator transforming the space $B\subset (\bbfR_+)$ into itself and such that $$\bigl|(Fx)(t)-(Fy)(t) \bigr|\le k\bigl |x(t)-y(t) \bigr|+ a(t)$$ for all functions $x,y\in BC(\bbfR_+)$ and for any $t\in\bbfR_+$, $k\in(0,1)$ and $a:\bbfR_+ \to\bbfR_+$ is a continuous function such that $\lim_{t\to \infty}a(t) =0$.\par Further, assume that $x=x (t)$ $(x\in BC(\bbfR_+))$ is a solution of the operator equation $$x=Fx.\tag *$$ In the paper under review, the following result is proved: The function $x$ is an asymptotically stable solution of equation (*) if for any $\varepsilon>0$ there exists $T>0$ such that for every $t\ge T$ and for every other solution $y$ of equation (*) the inequality $|x(t)-y(t)|\le\varepsilon$ holds.\par As an application, the functional-integral equation $x(t)=f(t,x(t))+ \int^t_0 u(t,s,x(s))ds$ is studied.
[Rudolf Kodnár (Bratislava)]
MSC 2000:
*47H09 Mappings defined by "shrinking" properties
45G10 Nonsingular nonlinear integral equations
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47N20 Appl. of operator theory to differential and integral equations

Keywords: measure of noncompactness; modulus of continuity; functional-integral equation; asymptotic stability

Cited in: Zbl 1197.45005 Zbl 1066.45002

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