Existence and asymptotic stability of solutions to a functional-integral equation. (English) Zbl 1145.45003

Liu, Zeqing; Kang, Shin Min

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)

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