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

Let $$BC(\mathbb{R}_+)$$ be the Banach space of all real functions which are defined, bounded and continuous on $$\mathbb{R}_+$$ with the $$\sup|x |$$ norm. Let $$F$$ be an operator transforming the space $$B\subset (\mathbb{R}_+)$$ into itself and such that $\bigl|(Fx)(t)-(Fy)(t) \bigr|\leq k\bigl |x(t)-y(t) \bigr|+ a(t)$ for all functions $$x,y\in BC(\mathbb{R}_+)$$ and for any $$t\in\mathbb{R}_+$$, $$k\in(0,1)$$ and $$a:\mathbb{R}_+ \to\mathbb{R}_+$$ is a continuous function such that $$\lim_{t\to \infty}a(t) =0$$.
Further, assume that $$x=x (t)$$ $$(x\in BC(\mathbb{R}_+))$$ 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 $$t\geq T$$ and for every other solution $$y$$ of equation (*) the inequality $$|x(t)-y(t)|\leq\varepsilon$$ 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.

MSC:

 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 45G10 Other nonlinear integral equations 47H10 Fixed-point theorems 47N20 Applications of operator theory to differential and integral equations
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References:

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