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

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.


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
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