An application of a measure of noncompactness in the study of asymptotic stability
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.
scientific article; zbMATH DE number 1878922