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Zbl 1145.45003
Liu, Zeqing; Kang, Shin Min
Existence and asymptotic stability of solutions to a functional-integral equation.
(English)
[J] Taiwanese J. Math. 11, No. 1, 187-196 (2007). ISSN 1027-5487

Using a fixed point theorem of Darbo type in the space of bounded continuous functions, the existence of a bounded solution of the equation $$x(t)=f(t,x(t))+g(t,x(t))\int\sb{0}\sp{t}u(t,s,x(s))\,ds$$ is obtained when $\vert u(t,s,x)\vert\le a(t)b(s)$ with small $a,b$ and $f(t,\cdot)$ and $g(t,\cdot)$ are contractions with small constants $k$ and $m(t)$ where $m(t)a(t)\to0$ sufficiently fast as $t\to\infty$. It is also claimed that the solution $x$ is asymptotically stable; however, the authors mean by this apparently only that the solution is asymptotically unique, i.e.\ any solution $y$ of the {\it same\/} equation (with not too large norm) satisfies $x(t)-y(t)\to0$ as $t\to\infty$.
[Martin Väth (Giessen)]
MSC 2000:
*45G10 Nonsingular nonlinear integral equations
47H09 Mappings defined by "shrinking" properties
45M05 Asymptotic theory of integral equations
45M10 Stability theory of integral equations

Keywords: asymptotic stability; bounded solution; functional-integral equation; condensing operator

Cited in: Zbl 1190.47090

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