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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)) 0 1 u(t,s,x(s))ds,tt[0,1]·(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,). 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.


MSC:
45G10Nonsingular nonlinear integral equations
45M05Asymptotic theory of integral equations
47H09Mappings defined by “shrinking” properties
45M10Stability theory of integral equations