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An application of a measure of noncompactness in the study of asymptotic stability. (English) Zbl 1015.47034
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 $\lim_{t\to \infty}a(t) =0$. 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. As an application, the functional-integral equation $x(t)=f(t,x(t))+ \int^t_0 u(t,s,x(s))ds$ is studied.

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
Full Text: DOI
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[6] Banaś, J.: Measures of noncompactness in the space of continuous tempered functions. Demonstratio math. 14, 127-133 (1981)