Liu, Zeqing; Kang, Shin Min Existence and asymptotic stability of solutions to a functional-integral equation. (English) Zbl 1145.45003 Taiwanese J. Math. 11, No. 1, 187-196 (2007). Using a fixed point theorem of Darbo type in the space of bounded continuous functions, the existence of a bounded solution of the equation \[ x(t)=f(t,x(t))+g(t,x(t))\int_{0}^{t}u(t,s,x(s))\,ds \]is obtained when \(| u(t,s,x)|\leq a(t)b(s)\) 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 same equation (with not too large norm) satisfies \(x(t)-y(t)\to0\) as \(t\to\infty\). Reviewer: Martin Väth (Gießen) Cited in 1 ReviewCited in 11 Documents MSC: 45G10 Other nonlinear integral equations 47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc. 45M05 Asymptotics of solutions to integral equations 45M10 Stability theory for integral equations Keywords:asymptotic stability; bounded solution; functional-integral equation; condensing operator PDF BibTeX XML Cite \textit{Z. Liu} and \textit{S. M. Kang}, Taiwanese J. Math. 11, No. 1, 187--196 (2007; Zbl 1145.45003) Full Text: DOI OpenURL