On existence and asymptotic stability of solutions of a nonlinear integral equation. (English) Zbl 1029.45003

Banaś, Józef; Rzepka, Beata

J. Math. Anal. Appl. 284, No. 1, 165-173 (2003).

The authors prove an existence theorem for a nonlinear Volterra integral equation of a special type arising in traffic theory: \[ x(t)= f(t, x(t)) \int^1_0 u(t, s,x(s)) ds,\quad t\in t\in [0,1].\tag{1} \] It is an example of a quadratic integral equation. Using measures of noncompactness, the authors show that (1) has continuous and bounded solutions on \([0,\infty)\). Fixed points results are used. Furthermore, for suitable measure of noncompactness the authors prove that those solutions are asymptotically stable in some sense defined in the paper.

Reviewer: Yves Cherruault (Paris)

MSC:

45G10	Other nonlinear integral equations
45M05	Asymptotics of solutions to integral equations
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
45M10	Stability theory for integral equations

Keywords:

asymptotic stability; fixed points; nonlinear Volterra integral equation; traffic theory; quadratic integral equation; measures of noncompactness; continuous and bounded solutions

References:

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