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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:$$x(t)= f(t, x(t)) \int^1_0 u(t, s,x(s)) ds,\quad t\in t\in [0,1].\tag 1$$ 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.

45G10Nonsingular nonlinear integral equations
45M05Asymptotic theory of integral equations
47H09Mappings defined by “shrinking” properties
45M10Stability theory of integral equations
Full Text: DOI
[1] Argyros, I. K.: Quadratic equations and applications to Chandrasekhar’s and related equations. Bull. austral. Math. soc. 32, 275-292 (1985) · Zbl 0607.47063
[2] Banaś, J.: Measures of noncompactness in the space of continuous tempered functions. Demonstratio math. 14, 127-133 (1981)
[3] Banaś, J.; Goebel, K.: Measures of noncompactness in Banach spaces. Lecture notes in pure and applied mathematics 60 (1980)
[4] Deimling, K.: Nonlinear functional analysis. (1985) · Zbl 0559.47040